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    Microscopes, telescopes and spectroscopy

1

The light microscope was key to the development of the cell theory. The electron microscope has vastly expanded our knowledge of the cell but it has now become a key tool for nanotechnology. Flame atomic absorption spectroscopy can be used to detect metals and metalloids in environmental samples. Spectroscopy is also essential to space research. 7 The Hubble Space Telescope7 is being used to identify compounds such as carbon dioxide and methane in the atmosphere of extrasolar planets. It uses a combination of a near infra-red camera8 and spectroscopy to identify compounds.

                                            The Telescope and the Microscope 

Introduction: If there were two scientific instruments which propelled experimental science forward in the late Renaissance they are the microscope and the telescope. They were some of the first sensor devices capable of amplifying human senses.

Microscopes
"Magnification by simple lenses has been known from ancient times,
but the development of the modern microscope dates from the construction
of compound-lens systems, which occurred some time in the period between
1590 and 1610. The credit should probably go to the Dutch lensmakers
Hans and Zacharias Jannsen (father and son), who in about 1600 constructed
a simple instrument made of a pair of lenses mounted in a sliding
tube. A compound-lens system using a convex lens in the eyepiece was
described by Johannes Kepler in 1611, probably deriving from the Jannsens'
work.
Telescopes
"Tradition attributes the invention of the telescope to the accidental
alignment of two lenses of opposite curvature and diverse focal length
by Hans Lippershey in Holland in 1608. The principle, however, may
have been known to Roger Bacon in the 13th century and to the early
spectacle makers of Italy.
"Galileo Galilei constructed (1609) the first lens, or refracting,
telescope for astronomical purposes. Using several versions, he discovered
the four brightest Jovian satellites, lunar mountains, sunspots, the
starry nature of the Milky Way, and the apparent elongation of Saturn,
now known to be its rings. Galileo's simple lenses suffered from a
variety of aberrations, or defects of image formation: chromatic aberration,
or the variation of focal length with color; spherical aberration,
the variation of focal length with distance of parallel rays from
the lens axis; coma, the increasing blur of a point image with angle
of the rays to the axis; and distortion, the imaging of straight lines
in the object as curves. These aberrations were minimized in a variety
of ways. Christiaan Huygens constructed extremely long aerial telescopes
in which the objective lens was mounted on a pole and connected to
the eyepiece by only a taut wire. Despite extraordinary difficulties,
such instruments achieved useful results, especially in lunar mapping.''

These instruments have progressed far beyond the simple devices described above, but our study will deal only with the basic versions, using only convex (converging) lenses.

Theory: There are two principles essential to these instruments. First; light bends when moving from one medium to another. Second; the image formed by one lens can be the object for another.
1) The whys of the bending of light were described in a previous lab. However, that experiment dealt with planar surfaces. A lens has at least one spherical section as a surface. With planar or rectangular lenses the exit light ray is parallel to the incident ray; it is merely offset. See figure 1:

fig. 1

On the other hand, when a ray of light is incident on a spherical lens it is redirected. According to Snell's law the ray must be bent toward the normal when traveling from a less to a more optically dense medium. The normal is drawn tangent to the surface of the lens where the ray enters. See figure 2a. When the ray exits to a less optically dense medium it is bent away from the normal drawn to the tangent. See figure 2b.

fig. 2

 
The effect is to focus parallel rays of light to a point. See figure 3:

fig. 3

Moreover, due to symmetry, if the rays originate from a point the lens will cause them to be parallel; merely reverse the rays direction in figure 3. (There is a great deal of symmetry between telescopes and microscopes: one is the conjugate of the other.)

How much of a change of direction the light rays experience is dependent upon the curvature of the lens surface. Greater curvature causes greater bending. Lenses are normally characterized by the amount of bending, not by their curvature. Where two incident parallel rays which are subsequently bent cross is called the focal point, and the distance from the lens to this point is called the focal length. We rate lenses by their focal length, as in a long or a short focal length lens. Microscopes and telescopes employ both.
One can determine a focal length (f) by examining the distance (s) to an object seen through the lens and the distance (s') to the image formed. A simple equation, called the lensmakers' equation, generates the focal length:

eq. 1

It will be necessary to ascertain the focal lengths of the various lenses you will use. N.B.: Some texts replace s and s' with p and q; be flexible!
2) Unlike most human beings, a lens doesn't know if what it `sees' is a physical reality or a mere projection. Therefore if one lens produces an image a second lens can treat this as an object. The difference between a microscope and a telescope is where the object for that first lens, called the objective, is in relation to the lens. This relation determines where the image formed by the first lens will be formed. For instance, if the object for the first lens is a long way away, the image will be formed on the opposite side of the lens. When, for a converging lens, the image is formed on the opposite side of the lens from the object s' is positive. You can verify this with equation 1: make s > f. On the other hand, if the object is close to the lens, the image will be formed on the same side of the lens as the object. Here s' < 0.
A small image (or object) one meter away emits (or reflects) light in parallel rays to your eyes. Your eye is most comfortable viewing objects perhaps a meter away; there is little strain to see detail and the image is clear. Optical instruments should therefore produce an image to your eye that is about one meter distant (a ballpark figure).
Now the obvious difference between the function of a microscope and the function of a telescope is that the former views small things close up and the latter views large things far away. In both cases we want a comfortable image for the viewer's eye. For these reasons a microscope uses a short focal length objective and a telescope uses a long focal length objective.
With a microscope we place a sample close to a short focal length lens, just beyond its focal point. This produces an image on the other side of the objective. We then use a second, longer focal length lens, called the eyepiece, to view this image. The image formed by the objective is just inside the focal length of the eyepiece. The image the eyepiece forms is therefore in front of the lens. This serves the purpose of moving the "virtual'' object the eye sees to a comfortable distance while magnifying it. (Magnification for a single lens is determined by the negative ratio of s' to s, or alternatively, the height of the image to the height of the object). The spacing (L) between the lenses is critical; L must be slightly less than s' (for the objective) plus f for the eyepiece. Since s is approximately equal to f for the objective, with the lensmakers' equation and f for the eyepiece L can be determined. Small adjustments to focusing are made by small variations in L.
A telescope works in the opposite manner. The long focal length objective forms its image such that s' = f for the objective. This is due to the fact that s is infinite (or close enough) for a telescope. The eyepiece, a shorter focal length lens, focuses on this image. The distance between the lenses is exactly the sum of the two lenses' focal lengths, so s = f for the eyepiece. The eye sees an image at infinity (greater than a meter), which is a comfortable viewing position. The magnification in a telescope is the ratio of the focal lengths (objective over eyepiece obviously).
There are other kinds of microscopes and telescopes--electron microscope, reflecting telescope--but we will restrict ourselves to 'scopes explained above.

Task:

Build a microscope
Build a telescope

Equipment:

• lenses
• optic bench w/lens holders

Procedure:

1. Determine the focal lengths of all lenses. Use the optic bench and the lensmakers' equation.

-If you let the object distance go to infinity, finding the focal length should be easy.

2. Place both lenses on the optic bench some distance apart. Which lens is the eyepiece and which is the objective depends on the experiment performed.
3. The scale drawings can be done either in a graphics program or just on paper with the scale shown.

Microscope:

1. To find the magnifying power of the microscope you need a reference object of known size.
2. We suggest that this object have gradations like a meterstick, although an actual meterstick won't do.
   1. Make a card with regularly ruled lines about 20cm long. The separation should be around 0.5cm.
3. Aim your microscope at this card and adjust the focus by moving the lenses relative to each other.
   1. Describe this image in your report.
4. Re-aim the `scope off to one side just a bit so that both the microscope's image and the card are side-by-side when viewed through the 'scope.
5. Compare the ruled markings on the card with the view through the 'scope. There should be an obvious difference in the apparent spacing between lines.
   1. Suppose there are 6 marked lines for every 3 image lines. The magnification is therefore 6/3 or 2. Something like this:

2. Determine the magnification of your microscope.
3. Compare this with the theoretical value.
6. Draw a scale ray diagram for your instrument.

Telescope:

1. To find the magnifying power of the telescope you also need a reference object of known size. Try making a paper card with ruled markings which are barely discernible from around 3-5 or more meters away.
2. Aim your telescope at this target and adjust the focus by moving the lenses relative to each other. Describe this image in your report.
3. Re-aim the 'scope off to one side just a bit so that both the telescope's image and the target are side-by-side when viewed through the `scope.
4. Compare the ruled markings on the target with the view through the 'scope. There should be an obvious difference in the apparent spacing between marks.
   1. Suppose there are 7 target marks for every 2 image marks. The magnification is therefore 7/2 or 3.5. (see the discussion above).
   2. Determine the magnification of your telescope.
   3. Compare this with your theoretical value.
5. Draw a scale ray diagram for your instrument.

Questions:

1. Why is it obvious that the magnification of a telescope is the ratio of the objective's focal length over the eyepiece's focal length, and not the other way around?
2. Although prism binoculars contain converging lenses only, objects viewed with them are seen erect. Explain how this might be accomplished.
3. Show the following is true if the two lenses are directly next to each other (touching):

 

      

 

   XXX  .  V  Optical and digital microscopic imaging techniques and applications in pathology 

 

The conventional optical microscope has been the primary tool in assisting pathological examinations. The modern digital pathology combines the power of microscopy, electronic detection, and computerized analysis. It enables cellular-, molecular-, and genetic-imaging at high efficiency and accuracy to facilitate clinical screening and diagnosis. This paper first reviews the fundamental concepts of microscopic imaging and introduces the technical features and associated clinical applications of optical microscopes, electron microscopes, scanning tunnel microscopes, and fluorescence microscopes. The interface of microscopy with digital image acquisition methods is discussed. The recent developments and future perspectives of contemporary microscopic imaging techniques such as three-dimensional and in vivo imaging are analyzed for their clinical potentials.

 

1. Flash back of microscope development

More than 2000 years ago, around one century B.C., people already had discovered that one could view an enlarged image of a small object by using a transparent “lens” with a spherical shape. Although a single convex lens can magnify an object by more than ten times, it is quite insufficient for people clearly to observe the details of many small objectives [1]. Near the end of 16th century, Dutch eyeglass dealer, Janssen, and his son inserted several lenses into a cylinder and found that an object was enlarged significantly if viewed through this assembled cylinder. This was the first prototype of a modern microscope and telescope. Based on this accidental discovery, Janssen designed and assembled the first compound microscope that included two convex lenses. The basic concept of a compound microscope is that an object can be enlarged sequentially by two convex lenses.

While observing the soft-wood specimen using a microscope in 1665, the British scientist, Robert Hooke, surprisingly discovered that the specimen depicted a series of unique “elements.” Hooke named them “cells.” Years later, a new microscope designed and assembled by a Dutch scientist, Anthony van Leeuwenhoek, provided much higher magnification power, which enabled people to observe substantially more details of cells. Although it exceeded all precedent microscopes, it was not another compound microscope and actually used only one convex lens. However, due to Leeuwenhoek's excellent manufacturing skill, this specially manufactured single convex lens achieved more than 300 times of magnification power.

In the following two centuries, the magnification power, as well as the image quality of the compound microscope, was improved substantially. This occurred, in particular, after the discovery and use of the new lens assemblies were able to eliminate or minimize the chromatic and other optical aberrations. Compared to the microscopes developed in 19th century, the conventional optical microscopes used to date have no substantial difference or improvement, because the optical microscopes have reached their maximum limitation in the spatial resolution.

Due to the large range or fluctuation of optical wavelength, it is impossible for any optical instruments to create a perfect image of a natural object, even if all shape defects in manufacturing optical lens were eliminated. Due to the unavoidable diffraction of optical wave passing through the microscopic lens, any point depicted on an object is no longer the point in the image plane but a diffraction spot. If two diffraction spots are too close together, the two points are not distinguishable. Under such circumstance, increasing magnification power of the microscopic objective lenses cannot further increase the spatial resolution of the microscopes. For the microscopes using the light sources within the range of visible optical wavelength, their spatial resolution is limited to 0.2 μm. Thus, any structures smaller than 0.2 μm cannot be distinguished using this type of microscope.

One approach to increasing spatial resolution of microscopes is to reduce optical wavelength or use electron beam to replace visible light source. According to the de Broglie theory of matter wave, moving electrons is similar in nature to optical wave fluctuation. The faster that electrons move, the shorter the “wavelength” of the released energy. If the electrons can be accelerated sufficiently and the released energy can be converged, the moving electron beam also can enlarge the objects. Based on this concept, German engineers, Max Knoll and Ernst Ruska, assembled the first transmission electron microscope (TEM) in the world in 1938 [2]. British engineer, Charles Oatley, manufactured the first scanning electron microscope (SEM) in 1952 [3]. Electron microscopy was one of the most important inventions in 20th century. Because electrons can be accelerated to very high speed, the spatial resolution of electron microscopes reaches up to 0.3 nm. As a result, many invisible materials (i.e., virus) under the visible light become “visible” under the electron microscope.

In 1983, two scientists in IBM laboratory, Gerd Binning and Heinrich Rohrer, invented the scanning tunnel electron microscope (STM) [4]. This advanced microscope was built using totally different concepts than those used by the conventional microscopes. STM works based on the so called “tunnel effect”. STM has no lens; rather, it uses a probe. Once the voltage is added between the probe and the observed object, the tunnel effect occurs if the distance between the probe and the surface of the observed objects is sufficiently small (i.e., measured in nanometers). When the electrons pass through the tiny space between the probe and the object, weak electronic current is generated. If the distance between the probe and the object varies, the strength of the current varies as well. Hence, the three-dimensional shape of the object can be detected as one measures the electronic current change. The spatial resolution of the STM can reach the level of a single atom.

 

2. The basic concept of microscopic imaging

Although many different types of microscopes have been developed, the basic imaging concept and structures can be simply illustrated in Fig. 1. The optical system of a microscope mainly includes an objective lens and eyepieces. The purpose of an objective lens is to magnify an object so that it can be clearly observed by the user. During the observation, the specimen is placed near the focal plane of the objective lens in the object space, and a magnified real image of specimen is first created on the intermediate plane. The intermediate plane is located on the focal plane of the eyepiece, thus the eyepiece is working as a magnifier to further magnify the image projected on the intermediate image plane. Finally, a magnified, virtual, inverted image is provided for the observer.

Fig. 1

The optical principle of microscope imaging.

For a well designed microscope, the spatial resolution is mainly determined by the objective lens. Although an eyepiece can also magnify the image, it cannot improve the resolving power of the microscopes. The spatial resolution of an optical microscope is given by the Rayleigh equation as follows [5]:

Δr₀ = 0.62λ ∕ n sin α

(1)

where Δr₀ is the minimum resolvable distance, λ is the wavelength of the light source; n is the refractive index between the lens and the object, α is the half-angle aperture – the half inclination of the lens to the objective points, and n × sin(α) is the numerical aperture (NA) of the objective lens.

In the case of digital microscopic systems, the image magnified by the objective lens is directly acquired (or through a relay lens) by electronic detectors such as CCD or CMOS detectors [6, 7]. The electronic detectors are selected so that their pixel pitch is smaller than the magnified minimum resolvable distance determined by the objective lens. So that the resolving power of the microscope is not degraded by the digital detectors.

Based on the above equation, and considering the following practical limitations, (1) the use of visible light with the wavelength between 390 nm and 760 nm, (2) the maximally reachable aperture with the half-angle of 70–75 degree, and (3) the requirement of using immersion methods with water or oil to increase of refractive index, the resolution of a conventional optical microscope cannot exceed 200 nm.

Since the wavelength of electronic wave (beam) is much shorter than the visible light waves in several orders of magnitude, the resolution of a microscope using electronic beam in theory can reach approximately 0.3 nm. For this reason, electron microscopy [8, 9] is developed based on the principles of electronic optics, which can image the fine structure of objects at very high magnification power by using the electron beam and electron lens instead of optical lens. The transmission electron microscope (TEM) is the primary representation of the electron microscopes. The TEM is so named because the electron beam first penetrates the sample and then is magnified by the electronic imaging lens to produce the images. Its optic path is similar to the conventional optical microscope (as shown in Fig. 2). After passing through convergent lens, the parallel electron beam reaches the sample. Once the electron beam passes through the sample, it carries the information related to the sample characteristics. An electronic image is first generated once the electron beam passes though the objective lens and contrast aperture. After the electronic image is magnified again by the intermediate lens and projection lens, the final electron image is displayed in the electronic monitor screen. The contrast of the electron image is determined by the scattering level of electron beam once it interacts with the atoms in the sample. The electron beam also scatters less in the thinner samples or some parts with lower density. Thus, more electrons pass through lens aperture and result in the brighter image. On the other hand, thicker samples or denser parts appear darker in the image. If the sample is too thick or too dense, the image contrast is decreased substantially.

Fig. 2

Comparison of the optical microscopic and electron microscopic systems.

 

3. The microscopes used in biomedical fields

Although a large number of different types of microscopes have been developed and applied in different applications, the microscopes actually can be divided into three categories: (1) optical microscopes, (2) electron microscopes, and (3) scanning tunnel electron microscopes based on the microscope development history.

3.1. Optical microscopes

The simplified optical wave path of a conventional optical microscope is illustrated in Fig. 3. The modern optical microscope is able to magnify an object by 1500 times with the 0.2 μm limit in spatial resolution. The optical microscopes can be divided into many different types using a variety of criteria. For example, based on a lighting method, there are the transmission and reflection types of microscopes. In a transmission microscope, the light passes through transparent objects. In a refection microscope, the light source installed on the top of the microscopic lens illuminates the non-transparent objects, and the reflected light is collected by the lens. The microscopes can also be differentiated based on the observation methods, including bright field microscopes, dark field microscopes, phase difference microscopes, polarized light microscopes, interference microscopes, and fluorescent microscopes [5, 10–13]. Each microscope can use either the transmission or reflection approach. The bright field microscopes are the most popular and widely used of all microscopes. Using this type of microscope, the transmission (or absorption) ratio and reflection ratio of some observed objects vary according to the change of working environments. The amplitude of these objects varies with the change in lighting intensity. The colorless transparent objects are visible only when the phase of illuminated light changes. Because the bright field microscopes cannot change light phase, the colorless transparent specimens are invisible when using this type of microscope.

Fig. 3

Illustration of optical path of the microscope.

3.2. Electron microscopes

The resolution of the electron microscopes typically is represented by the small distance of two adjacent and distinguishable points. In the 1970s, transmission electron microscopes were able to reach the resolution of 0.3 nm, while the resolution limit of the human eye is around 0.1 mm. Compared to the optical microscopes that have the maximum magnification power of a few thousand times, the modern electron microscopes can increase the maximum magnification power to more than 300 million times. Using electron microscopes, one can directly observe some biological structures and/or the atomic structures of cells. Despite having superior spatial resolution over the optical microscopes, electron microscopes must work under the vacuum environment. Thus, they cannot be used to observe living biology samples. In addition, the electron beam can damage the illuminated specimens. Other issues (i.e., the optimal control of brightness of electron bean gun and the improvement of electron lens quality) still need investigations [14].

3.3. Scanning tunnel microscopes

The scanning tunnel microscope (STM) is also named “tunnel scanning microscope”. This instrument detects the surface structure of the objects based on the tunnel effect of the quantum mechanics. STM applies a very thinning tip (in an atom unit of its head) to detect the sample surface. As the tip is very close to the sample surface (<1 nm), the atoms in the tip head overlap with the electronic cloud of electrons in the sample surface. If a biasing voltage is applied between the tip and the samples, electrons will pass through the barrier between the tip and sample to generate the tunnel current in the order of nano-amperes (10⁻⁹ A). By controlling the distance between the tip and the sample surface and accurately moving the tip along the sample surface in the three-dimension, the detector can record the data related to the surface morphology, electronic surface states, and other relevant sample surface information [15]. The STM has the amazing ability to achieve the spatial resolution of less than 0.1 nm in the horizontal direction and less than 0.001 nm in vertical direction. Generally, when objects are in the solid state, the distance between atoms typically is from 0.001 nm to 0.1 nm. Because of this, the STM enables scientists to locate the single atom and observe the status of the atoms and molecules in the conductive material and surface structure. Hence, the STM generates substantially higher spatial resolution than other similar atomic microscopes. On the other hand, STMs can use the tip of the probe needle to manipulate accurately the atoms under low temperature environment. This enables the STM to be used as both a measurement and a process tool in the nanometer technology [16–18]. One unique characteristic of STM is that, for the first time, one can observe the permutation status of the atoms in the object surface and the related physical chemistry characteristics of the electrons in the surface. Due to the width and scope of its perspectives in the research and application fields related to the surface, material, and life sciences, the invention of the STM was considered by the international scientific community as one of the ten most important achievements of the science and technology in 1980s.

In summary, increasing microscope resolution has been an actively pursued goal in the microscopic imaging field. Using these three types of microscopes, one may observe the biological cells and microorganisms using an optical microscope, observe virus using an electron microscope, and detect or visualize the atoms using a scanning tunneling microscope.

3.4. Fluorescence microscopes

Although the fluorescence microscope can be characterized as a branch of the optical microscopes, it has many unique image generation and application characteristics in the biomedical imaging research fields. This section discusses this in more detail. The fluorescence microscope [19] uses short-wave light to illuminate the examined specimen and make certain objects inside the specimen emit the fluorescent light. Based on the received fluorescent light, one is able to observe the shape and location of the target objects. Comparing the difference between the fluorescence and conventional optical microscopes, one of the major differences is that the fluorescence microscope uses the high-voltage mercury lamp to provide all band light illumination, using a projection method in which the light is projected to the specimen. Due to the typically weakly reflected fluorescent light, the microscope must have a large numerical aperture to allow the objects to be observed. As a result, the numerical aperture of the fluorescent microscopes is larger than conventional optical microscopes. The fluorescent microscopes also have a special filtering system, including the assembly of the illumination filters and cut-off filters. Specifically, the fluorescent microscope has the following unique characteristics or requirements:

• 1)
  It requires a light source that has sufficient power to emit fluorescent light;
• 2)
  It is equipped with a set of optical filters that fit the requirement of stimulating different objects to emit different fluorescent light. By selecting the suitable wavelength of the emitting light, the wavelength of the emitted light can coincide with that of the absorption light, resulting in the maximum fluorescent light output;
• 3)
  To acquire the weak fluorescent images, it is equipped with a set of cut-off light filters that allow only the selected fluorescent light to pass through the imaging system and block the other types of scattering light to increase the signal-to-noise ratio of the system; and
• 4)
  The system of optical magnification needs to fit the characteristic of the fluorescent light in order eventually to obtain the fluorescent images with the highest spatial resolution.

Due to these requirements, a modern fluorescent microscope typically is assembled based on the structure of a duplex optical microscope. The equipped fluorescent device includes fluorescent light source, emission light path, stimulation/emission light filters, and the other components.

• 1)
  The fluorescent light source: It provides the light source that emits the light within the specific range of the wavelength or energy, which guarantees that the examined specimen acquires sufficient stimulation and generates strong fluorescent light. The fluorescent microscope typically uses the mercury lamp as the light source that is able to provide the exciting light with continuous wavelength. The mercury lamp has stronger energy in several commonly used light wavelengths (including 365 nm, 405 nm, 550 nm, and 600 nm).
• 2)
  An exciting light path: It serves as a fluorescent lighting device that contains a group of condenser lens and light focusing adjustment device. In the light path, the adjustable light and field apertures, (Neutral Density) ND filters, light splitter, and processing lens are installed.
• 3)
  Light filters: These are a set of the optical filtering components. Each has a certain wavelength width that selectively allows stimulated/emitted light pass through the optical system of the microscope.

Similar to the other conventional optical microscopes, the fluorescence microscopes can be divided into two categories based on their optical path [20]: transmission fluorescence microscopy (TFM) and epifluorescence microscopy. In TFM, the light passes through the condenser to excite the specimens to emit the fluorescence light. The TFM often uses the dark-field condenser that is able to adjust a reflector so that the light can illuminate the specimen from different directions. Compared to the old style fluorescent microscopes, the TFM has a number of advantages, which include producing strong fluorescence light when the specimen is observed in the low magnification. Disadvantages include the fact that as magnification levels increase, the fluorescence light weakens. Therefore, TFM is better used for observing larger specimens. As shown in Fig. 4, the light is emitted from one side of the specimen and the fluores-cent light is collected by the lens in the other side of the specimen.

Fig. 4

Illustration of optical path of a transmission fluorescence microscopy.

The recently developed epifluorescence microscopy is a new type of fluorescent microscopy [21]. The difference between the epifluorescence and the transmission fluorescence microscopy is that the objective lens in epifluorescence microscope serves first as a well-corrected condenser and then as the imageforming light gatherer. The illuminator directs light onto the specimen by first passing the light through the microscope objective lens on the way toward the specimen and then using the same objective lens to capture the emitted light. The dichroic beam splitter is placed between the optical path titled at 45°, and the reflected excitation light passes through the objective lens and eventually reaches the specimen. The fluorescent light emitted by the specimen, as well as the reflected exciting or scattering lights, passes through the objective lens simultaneously and arrives at the dichroic separator (mirror) that separates the fluorescent light and the original exciting light. The remaining scattering light is absorbed further by the cut-off filter. The advantages of the epifluorescence microscope include the uniform illumination, sharp image, and the strong fluorescent light as the increase of the magnification level.

The optical path of an epifluorescence microscope is illustrated in Fig. 5. After the reflection from the dichroic mirror, the excitation light converges at the specimen. The emitted fluorescent light from the specimen first passes through the objective lens and the dichroic mirror, and then is collected by detection sensor (probes). The key component of an epifluorescence microscope is the dichroic beam splitter. It includes a multilayer optical interference filter that is installed with a titled angle of 45° along the optical path. It is able selectively to reflect or transmit the light only in certain wavelength range.

Fig. 5

Illustration of the optical path of a typical epifluorescence microscopy.

Although the transmission fluorescence microscopes can yield images with sufficiently high contrast in low magnification, the image is often relatively dark compared to the use of the epifluorescence microscope. Thus, the epifluorescence microscope is more popular in clinical practice to date. In addition, compared to the transmission fluorescence microscope, the epifluorescence microscope has following characteristics.

• 1)
  The microscope objective lens also serves as a light condenser. Thus, the alignment issue of the condenser is avoided. In particular, the fluorescent light intensity of the image increases as the increase of the numerical aperture (NA) of the object.
• 2)
  Since excitation light does not pass through slides in epifluorescence microscopy, it reduces the light loss. Meanwhile, the excitation and fluorescent light travel in the opposite direction to the objective lens. Two light beams separate with each other without interference. Some scattered excitation light may reach the object, and they are mostly reflected by the dichroic mirror to the light source. Thus, a very thin filter typically is installed between the objective lens and the detector (probe) to absorb a small fraction of remaining scattering light passing through the objective lens.
• 3)
  Unlike the use of transmission-type of excitation, in which most of fluorescence light creates in the bottom section of the specimen and must transmit through the specimen slice producing unavoidable scatter light, in epifluorescence microscope, the light source and the observation plane locate in the same plane. Therefore, there is no loss of the fluorescence intensity and no effect on image quality. In particular, it has unique advantages when it is used to examine the thick specimen slices with bacteria and tissue culture.
• 4)
  Since the ultraviolet (UV) light needs to pass through the objective lens, the transmission coefficient of the objective lens must be considered. As a result, a series of special objective lenses that can transmit UV light have been designed and manufactured. However, such a special objective lens is not required in the transmission fluorescence microscope.

 

4. The application of microscopes in pathological imaging

The invention of the microscope has significantly promoted the development of biology and medicine. It demonstrates the mystery of the microscopic world and reveals the causes of disease from the microscopic view. By acquiring the microscopic and pathologic images of the pathogenic microorganisms, medical researchers have established a series of new academic subjects, including bacteriology, immunology, virology, cell pathology, and others. In particular, the invention of electron microscopes has raised the observation of the traditional pathology from the cellular level to the sub-cellular level. As a result, one cannot practice modern medicine without microscopes. New microscopes such as the confocal laser scanning microscope (CLSM) [22], atomic force microscope (AFM), and scanning tunneling microscopy (STM) further improve the resolution of pathology imaging [23]. In this section, we briefly discuss the application of microscopes in assisting the clinicians in biomedical engineering, particularly in diagnosing pathological images.

4.1. Optical microscopes

Optical microscopes have been used widely in the pathology related clinical laboratories to diagnose a variety of diseases based on the examination of the body fluid change, invaded virus, and the variation of atomic structures [24]. Such diagnostic information and reports provide the clinicians with valuable references to select and implement the optimal treatment plans for the patients and monitor their efficacy. In genetic engineering and microscopic surgery, microscopes are essential tools for clinicians. Although optical microscopes are still considered to be very important tools in the clinical facilities, the development of the originally optical microscopes is quite mature to date; in particular, the optical characteristics of the microscopes have reached perfect degree without much space left to make further improvement. Therefore, the current development of the optical microscopes focuses on finding new application fields, making the systems simple and easy to use, as well as making one system for the multiple applications.

Since the biological specimen is typically one kind of muddy medium, it can generate the strong light scattering and absorption. As a result, the light (or optical wave) cannot penetrate deeply into the interior of the biological structures; thus, it is unable to extract a clear image of the biological structures. With the rapid advance of photonics, new imaging methods based on photonics have been developed, including the laser confocal scanning microscope and the near-field optical scanning microscop [25, 26]. First, the laser confocal scanning microscope is the combination of laser source and the confocal microscope. Specifically, it is a system that uses both the confocal concept of the traditional optical microscopy and the laser as the light source. It then uses the computer to process and analyze the acquired digital images of the observed specimens. By continuously scanning the multiple layers of the living cells or tissue slices, the laser confocal scanning microscope can acquire the complete 3D images of each structural layer of a single cell or local structure of a group of cells. Based on the near-field probing theory, the near-field optical scanning microscopy is a new technology developed by using the optical scanning probe needles. It breaks through the limitation of optical diffraction and is able to reach the spatial resolution in the range from 10 nm to 200 nm. By further combining with the related optical spectroscopy, using the near-field optical scanning microscopy creates a new approach to detecting and examining spectrum images of small biological specimen beyond nanometer range. As a result, researchers are able to analyze the single biological atom to date.

4.2. Electron microscopes

There are number of differences between electronic and optical microscopes when applied to pathological images. When using optical microscopes to observe pathological specimens, the observer routinely needs to adjust the micro-focusing system of the microscope and move the specimen to achieve the optimal observation of the tissue structure of the specimen. Using optical microscopes also generates less eye fatigue due to small light simulation to the eyes. However, the magnification power of the optical microscope is much lower than that of the electronic microscope. In addition to the higher magnification power, an electronic microscope can display more detailed tissue structures on the monitor (screen). It also may be able to conduct secondary magnification of the specimen displayed on the screen. Thus, the electronic microscope plays an important role in studying human organ tissue structures and the other sub-nanometer structures.

Due to the short wavelength of the electron beams, the electrical microscopes break through the limitation of the spatial resolution using optical microscopes, which opened a new door for the human eyes to observe the small structures at the molecule and/or atomic level. For example, using electronic microscopes to observe cells, researchers clearly have confirmed the existence of a cell membrane that consists of three thin layers with equal thickness but different density. Two external layers have higher density than the middle (interior) layer. When using electronic microscopes to observe unmyelinated nerve fiber, it was found that the nerve membrane cell of the fiber contained multiple shafts (i.e., 2 to 9). When using electronic microscopes to observe and study muscle structure, researchers discovered two important characteristics of muscle fiber: (1) the muscle fiber depicts the horizontally bright and dark stripes in the regular arrangement (namely the structure of horizontal stripes), and (2) muscle fiber is actually formed by finer fiber units that are perpendicular to the main fiber. Using electronic microscopes to study atomic structure of nucleic acid, observers can identify the linear status inside the nucleic acid atom structure with diameter around 20A°. These examples indicate that the application of electronic microscopes significantly expedites the possibility of exploring the secret of life and also plays an important role in the advance of medicine, as well as the improvement of human health [27].

In pathology, electronic microscopes have showed their importance in many applications. For example, electronic microscopes have been applied to pathological diagnosis of renal biopsy specimens. Renal glomerular disorder is a commonly diagnosed disease in clinical practice and can be classified as primary and secondary types of disorders. In order to more accurately diagnose the pathology of renal glomerular disorder, renal needle biopsy often is used. Besides distinguishing the different patterns of the morphological and tissue elements inside glomeruli, electronic microscopes can be used to observe and detect micro-structural aberration or changes, especially in epithelial cells, mesenterium, base membrane, and interstitial substance, which cannot be observed or detected using the conventional optical microscopes. By analyzing the observed micro-changes, physicians can find whether there are electron-dense deposits and corresponding locations. However, the single electronic microscopic technology also has limitation. In order to make renal needle biopsy more useful and accurate in the pathology diagnosis, the combination of multiple technologies, including the optimal use of conventional optical microscopes, immune-fluorescence microscopes, and electronic microscopes, is needed [23].

Using the electronic microscope not only has improved and enhanced the knowledge of the pathology, but also has provided a reliable foundation for the diagnosis of many diseases. With the rapid advance of science and technology, electronic microscopes have been and will continue be extensively used in pathologic diagnosis. However, single electronic microscopic technology has some limitations in pathologic diagnosis. For example, the field of vision (FOV) is reduced substantially due to the high spatial resolution of the electron microscope. Each FOV typically can only show a single cell or the partial structure of the cell. Meanwhile, observers can only observe still and dead specimens instead of the cells in vivo. Therefore, while electronic microscopes have been used actively in clinical practice, one also needs to combine other technologies such as the quantitative analysis of the morphology of the micro-structural changes using image processing, the nucleic acid hybridization in situ technology, and the apoptotic indices of the observation using electronic microscopes. Meanwhile, the scanning tunnel microscope and the atomic force microscope also have been introduced recently into the pathological area to improve the electronic microscopic technology. With closer connections to computer technology, the observers (i.e., pathologists) more easily and conveniently can use and control electronic microscopes through the connected computer and networking systems.

4.3. Fluorescence microscopes

Irradiated by the short wavelengths of light (e.g., ultraviolet and violet blue light 250 nm–400 nm), some substances can be excited by absorbing the energy and emit a longer wavelengths of light with energy degradation, which includes blue, green, yellow, or red light within the wavelength between 400 nm and 800 nm. This is called photoluminescence [28]. Some specimens naturally can generate fluorescent light when they are illuminated by the light with the proper wavelength. For example, most lipid and protein can emit fluorescent light in blue. This phenomenon is called auto fluorescence or primary fluorescence. However, most substances generate fluorescence light only when they are dyed with fluorochromes and are exposed in the light of short wavelengths.

In the 1930s, florescent staining was applied to investigating and observing the morphology of bacteria, mildew, cells, and fiber. For example, Mycobacterium tuberculosis can be detected in sputum by acid– fast bacteria staining methods. In the 1940s, the fluorescent staining protein technology was invented and applied extensively to the routine clinical immuno-fluorescence staining, which can examine and locate virus, bacteria, mildew, protozoan, parasite, antigen, and antibody in animals and humans. It has been used to investigate the pathogenesis and the cause of diseases such as categorizing and diagnosing glomerular diseases and identifying the relation between human papillomavirus (HPV) and the cervical cancer. The fluorescence microscope, a special tool for detecting fluorescence, has been getting more and more applications in the clinical research and disease diagnosis [29]. The coupling system between a fluorescence microscope and a fluorescence spectrometer has been extensively used in cell biology, biochemistry, physiology, neurobiology, and pathology. These systems can provide qualitative and quantitative analysis for various structures in either living and fixed cells or tissues. Fluorescence microscopes can observe cells and structures either emitting auto fluorescent or stained by the fluorochromes. The light source of a fluorescent microscope is a high-voltage mercury lamp that emits the ultraviolet light with short wavelengths. The fluorescent microscope also is installed with excitation and dichroic filters. The fluorochromes in the specimens are excited by the ultraviolet light and emit light with different colors. This enables one to analyze the distribution and locations of fluorescence inside the cells or structures. Some tissues can emit auto or spontaneous fluorescent light. For example, lipofoscin inside nerve or myocardial cells emit brown fluorescent light; Vitamin A of epithelia cells in hepatic Ito and retinal pigment emit green fluorescent light; and monoamine (e.g., Catecholamine, 5 – HT, Histamine) inside neuroendocrine cells and nerve fibers emits light with different colors under formaldehyde. Quinine and Tetracycline inside the tissues also emit fluorescent light. Some tissues inside the cells can bind with fluorescein to emit fluorescent light. For example, Ethidium bromide and Acridine orange can combine with DNA to measure the content of DNA. The fluorescence microscope also has been widely used in the immunocytochemistry research. By using fluorochrome to mark Isothiocyanate and Rhodamine, the marked antibody directly or indirectly can bind with the corresponding antigen. Thus, clinicians can detect the existence and distribution of antigen.

Since the invention of first fluorescence microscope in 1908, fluorescence detection has been improving constantly. This includes the introduction of high-voltage mercury lamps and the development of the multi-layer coating technology that is able to manufacture the high quality excitation and cut-off filters. With high quality optical components, the fluorescent microscope made a great contribution to the development of immune-fluorescence in the 1980s. In the 1990s, the laser scanning confocal fluorescence microscopy was developed. It measures and determines the fluorescence distribution within cells and tissue biopsies and obtains real-time overlapping phase-contrast and fluorescence images, as well as the primary color images labeled with double fluorescents. Currently, more advanced technologies related to the single-photon fluorescence microscopy, two-photon fluorescence microscopy, 4 pi confocal fluorescence microscopy [30], and full internal reflection fluorescence microscopy [31] (total internal reflectance fluorescence microscopy) have been developed. As a result, using these new tools, the technology and application of fluorescence detection can be applied or implemented in more research and clinical applications in pathology and other biomedical fields.

 

5. Digital microscope imaging

Although microscopy plays a very important role in biology and medicine, the observation is performed with the naked eye under the conventional optical microscopes. This can result in eye fatigue after a relatively long time of continuous observation. In addition, the image information cannot be stored and processed for different image enhancement purposes. To solve these limitations, digital microscopes have been developed and tested. A digital microscope (or imaging system) is an integrated design that combines traditional optical microscope, digital multimedia, and digital processing technology [32, 33]. As shown in Fig. 6, a digital microscope imaging system typically includes three components: Microscopy optical module, data acquisition module, digital image processing, and software control modules. Among these modules, the optical module realizes the function of the microscopic imaging; data acquisition module records the images produced by digital video devices, including CMOS, CCD, digital camera stored in optical module in the digital format, and then transfers these digital images to the computer storage devices through different graphics card interface or USB interface; and the software control module, the core of the whole system, controls the image capture, processing, and measurement in real-time to optimally improve the image quality. The digital images can be monitored in real-time using a color TV or computer monitors. After using a variety of digital image processing methods, digital microscopic imaging systems more sensitively can capture and display the image details. In fact, installation of online image acquisition, processing, and analysis systems have become an important symbol of modern advanced microscopes.

Fig. 6

The configuration of modern digital microscope imaging system.

Due to its significantly technological advantages, microscopic digital imaging technology also has been used and/or integrated in a variety of electronic microscopes. For example, unlike the traditional electron microscopy imaging that uses electronic sensitive floor to generate photographic images, the digital microscope equipped with the CCD imaging system converts electrons that carry image information through electro-optical conversion devices to the light signal and sends the signal to the CCD. After the images are captured with CCD, they can be sent to the computer through an image acquisition card to perform a variety of required image processing procedures (as shown in Fig. 7).

Fig. 7

The working principle of digital microscopic imaging system. 1 – Electron microscope lens; 2 – Fluorescence plate; 3 – Prism; 4 – Prism Block; 5 – Drive screw; 6 – Motor; 7 – Base; 8 – Aperture; ...

Thus, digital microscopic imaging technology is an extension of conventional microscopy. It integrates the microscopy, image acquisition, and computer control and processing technology to manage and control the whole imaging process, including image acquisition, sampling, processing, and data storage. Among these, the image acquisition and processing technology is the core for digital microscopic imaging technology.

Current digital image acquisition devices can be divided into three categories, including the use of (1) analog cameras plus video capture card, (2) consumer grade digital cameras, and (3) professional grade digital cameras. These three devices have their own characteristics and application fields in which professional grade digital cameras are the most popular choice in digital imaging microscopy systems. The introduction of digital imaging technology creates a great opportunity for the post-processing of acquired images. In particular, with the rapid advance of more powerful computers, digital microscopic images can be more effectively and efficiently processed and analyzed.

Image analysis and processing may be custom-designed or may use sophisticated commercial microscopic image processing software packages. Many professional software packages are well designed with variety of useful functions or schemes. They can provide both simple geometric measurement and sophisticated analysis to identify the relationship between the complex geometric structures. By providing the absolute space calibration to ensure the most accurate measurements and the advanced image segmenting technology, these commercial schemes can help observers to better distinguish the overlapping objects, detect the contours and shapes of the small objects, identify similar objects or groups, and categorize or mark the different objects with different colors. Some schemes can be used as advanced analytical tools that use the spectrum diagram, spectral profile, pseudo-color, three-dimensional surface shape, and other methods to manipulate the data and display it in different formats.

The advantage of connecting the microscope to a computer is that it produces digital images. Using these digital images, the research scientists and clinicians can apply various image processing and analysis software to obtain needed experimental data for variety of purposes. For example, using digital microscopes, one can replace the manual process of counting red tide organisms, a very tedious and time-consuming task. The computerized program is able to count the total number of red tide organisms depicted in a sample and classify red tide organisms in different size groups. The magnified red tide organisms can be displayed on the computer screen. As a result, by combining with manual identification, this computerized method greatly reduces the labor intensity of manual count using a conventional optical microscope.

 

6. New development in microscopic imaging technology

With the advance of technology and the special requirements of research projects, many new microscopic imaging methods have been developed and tested that aim to acquire the images with high resolution and large field of depth, as well as in three-dimensional stereoscopic format. As a result, using these new microscopic imaging systems and acquired images, the clinicians are able to conduct dynamic, non-destructive, and non-intervention in vivo (real time) tests in the biomedical research and clinical practice. Following are several new developments in this field.

6.1. Coexistence of a large field depth and high magnification power

In the conventional optical microscope, a large depth of field cannot coexist with high magnification power. As a result, due to the limited depth of field, the optical microscope cannot capture clear images of the targeted small objects located at different depths of the sample surface. The use of a scanning electron microscopy (SEM) or a confocal microscopy can resolve the contradiction between magnification and depth of field. As a result, one can observe clear (sharp) images with larger depth of field.

The SEM uses a very thin electronic beam to scan the sample surface. Electrons produced by scanning are collected by a specially designed detector. The corresponding electrical signals are further delivered to the CRT monitor screen to generate the three-dimensional image of the sample surface. These images also can be captured into the pictures. The resolution of the SEM lies between optical microscopy and transmission electron microscopy (TEM) (i.e., up to 3 nm). Meanwhile, the depth of field of the SEM is a hundred times higher than the optical microscope and at least ten times greater than the TEM. Thus, it is able to acquire a clear picture with a large depth of field.

The confocal microscope is an integrated technology that combines the optical and digital image analysis technique. It is a new microscopic imaging technology specifically useful in acquiring the images of non-planar samples. It can obtain enormous depth of field and exquisite details and also can acquire true-color information that often cannot be obtained using an electron microscope. When using the confocal microscope, one first adjusts the focus to search for and reach the samples distributed in different depth levels, captures all the images distributed in these levels with digital imaging devices, and transfers them to a computer (each of these aspects of the image has a clear location different from other images that record different parts of sample morphology and color information). After data analysis and integration using the special image processing software, the final high quality and clear picture is produced.

6.2. Three-dimensional imaging

The modern microscopy is not only limited to observing the specimen surface. By combining laser and confocal microscopy, researchers and clinicians can acquire and view the images of the cross sections of specimens at different depth levels. Then, using computerized image processing and 3D reconstruction algorithms, the three-dimensional shape profile of specimens can be acquired in high-resolution.

The main principle of laser scanning confocal microscope (LSCM) is to let the laser beam pass through a pinhole to generate a point light source. The laser light passes through the excitation filter and arrives at the beam splitter. Because the beam splitter can reflect excitation light with a shorter wavelength and transmit the light with a longer wavelength, the excitation light is reflected at the beam splitter and transmits through the objective lens. The laser beam scans different focal planes marked by the fluorescent agents under the control of the scanning control device. The excited emission light from the fluorescent marks passes through the original incident light path and directly returns to the beam splitter. After passing through the transmission filter for the specific wavelength, the emission light finally reaches the detection pinhole of the photomultiplier tube (PMT). By converting the PMT current into the digital signals, an image is created and displayed on the computer monitor screen.

Since LSCM can continuously scan the living cells and tissue or cell biopsy samples layer by layer to obtain images at different depth levels, it is also named as “non-destructive optical biopsy”. The distance between two LSCM scanned layers can be less than 0.1 um. Using computerized reconstruction algorithms, one can generate or acquire three-dimensional images that are able to observe or access the sophisticated cytoskeleton, chromosomes, organelles, and cell membrane.

6.3. In vivo detection

Observing and monitoring living cells can be very useful in many research applications, but it is a difficult technical challenge in the microscopic imaging field. To enable researchers or clinicians to observe the detailed structures of the specimens and also dynamically monitor the function of living cells or tissues in real-time using microscopic pathology images, several new technologies have been developed and tested. For example, by applying the specific fluorescence probes to label the material to be observed, the researchers can use LSCM to monitor dynamically the whole process of change in different locations or tissues of the cells after receiving stimulation in real time. LSCM also can be used in many applications such as assaying the changes within the PH of the cells, detect potential changes in membranes, detecting the production of intracellular reactive oxygen, testing the process of drugs into the tissue or cell membrane and locating its position, measuring the fluorescence resonance energy transfer (FRET), and real-time monitoring many other living cells.

 

7. Summary

Pathological imaging technology is a fast growing area of academic investigations, industrial developments and clinical applications. This paper presents a brief review of the fundamental concepts and a discussion on the recent developments and clinical potential of contemporary microscopic imaging techniques. The conventional optical microscopy has been an important tool in clinical pathology. The interface of optical microscopy with digital image acquisition methods combines the power of optical imaging, electronic detection, and computerized analysis. These and other emerging technologies will continue to evolve, enable cellular-, molecular-, and genetic-imaging, as well as tissue imaging with high efficiency and accuracy, to facilitate clinical screening and diagnosis.

 

 

                                                  XXX  .  V0  Optical telescope 

 

An optical telescope is a telescope that gathers and focuses light, mainly from the visible part of the electromagnetic spectrum, to create a magnified image for direct view, or to make a photograph, or to collect data through electronic image sensors.
There are three primary types of optical telescope:

• refractors, which use lenses (dioptrics)
• reflectors, which use mirrors (catoptrics)
• catadioptric telescopes, which combine lenses and mirrors

A telescope's light gathering power and ability to resolve small detail is directly related to the diameter (or aperture) of its objective (the primary lens or mirror that collects and focuses the light). The larger the objective, the more light the telescope collects and the finer detail it resolves.
People use telescopes and binoculars for activities such as observational astronomy, ornithology, pilotage and reconnaissance, and watching sports or performance arts. 


                      
                 Eight-inch refracting telescope at Chabot Space and Science Center  

The telescope is more a discovery of optical craftsmen than an invention of a scientist.^[1][2] The lens and the properties of refracting and reflecting light had been known since antiquity and theory on how they worked were developed by ancient Greek philosophers, preserved and expanded on in the medieval Islamic world, and had reached a significantly advanced state by the time of the telescope's invention in early modern Europe.^[3][4] But the most significant step cited in the invention of the telescope was the development of lens manufacture for spectacles,^[2][5][6] first in Venice and Florence in the thirteenth century,^[7] and later in the spectacle making centers in both the Netherlands and Germany.^[8] It is in the Netherlands in 1608 where the first recorded optical telescopes (refracting telescopes) appeared. The invention is credited to the spectacle makers Hans Lippershey and Zacharias Janssen in Middelburg, and the instrument-maker and optician Jacob Metius of Alkmaar.^[9]
Galileo greatly improved^{[citation needed]} on these designs the following year, and is generally credited as the first to use a telescope for astronomy. Galileo's telescope used Hans Lippershey's design of a convex objective lens and a concave eye lens, and this design is now called a Galilean telescope. Johannes Kepler proposed an improvement on the design^[10] that used a convex eyepiece, often called the Keplerian Telescope.
The next big step in the development of refractors was the advent of the Achromatic lens in the early 18th century,^[11] which corrected the chromatic aberration in Keplerian telescopes up to that time—allowing for much shorter instruments with much larger objectives.
For reflecting telescopes, which use a curved mirror in place of the objective lens, theory preceded practice. The theoretical basis for curved mirrors behaving similar to lenses was probably established by Alhazen, whose theories had been widely disseminated in Latin translations of his work.^[12] Soon after the invention of the refracting telescope Galileo, Giovanni Francesco Sagredo, and others, spurred on by their knowledge that curved mirrors had similar properties as lenses, discussed the idea of building a telescope using a mirror as the image forming objective.^[13] The potential advantages of using parabolic mirrors (primarily a reduction of spherical aberration with elimination of chromatic aberration) led to several proposed designs for reflecting telescopes,^[14] the most notable of which was published in 1663 by James Gregory and came to be called the Gregorian telescope,^[15][16] but no working models were built. Isaac Newton has been generally credited with constructing the first practical reflecting telescopes, the Newtonian telescope, in 1668^[17] although due to their difficulty of construction and the poor performance of the speculum metal mirrors used it took over 100 years for reflectors to become popular. Many of the advances in reflecting telescopes included the perfection of parabolic mirror fabrication in the 18th century,^[18] silver coated glass mirrors in the 19th century, long-lasting aluminum coatings in the 20th century,^[19] segmented mirrors to allow larger diameters, and active optics to compensate for gravitational deformation. A mid-20th century innovation was catadioptric telescopes such as the Schmidt camera, which uses both a lens (corrector plate) and mirror as primary optical elements, mainly used for wide field imaging without spherical aberration.
The late 20th century has seen the development of adaptive optics and space telescopes to overcome the problems of astronomical seeing.

Principles

For specific designs of telescope, see Reflecting telescope, Refracting telescope, and Catadioptric.

The basic scheme is that the primary light-gathering element the objective (1) (the convex lens or concave mirror used to gather the incoming light), focuses that light from the distant object (4) to a focal plane where it forms a real image (5). This image may be recorded or viewed through an eyepiece (2), which acts like a magnifying glass. The eye (3) then sees an inverted magnified virtual image (6) of the object.

Schematic of a Keplerian refracting telescope. The arrow at (4) is a (notional) representation of the original image; the arrow at (5) is the inverted image at the focal plane; the arrow at (6) is the virtual image that forms in the viewer's visual sphere. The red rays produce the midpoint of the arrow; two other sets of rays (each black) produce its head and tail.

Inverted images

Most telescope designs produce an inverted image at the focal plane; these are referred to as inverting telescopes. In fact, the image is both turned upside down and reversed left to right, so that altogether it is rotated by 180 degrees from the object orientation. In astronomical telescopes the rotated view is normally not corrected, since it does not affect how the telescope is used. However, a mirror diagonal is often used to place the eyepiece in a more convenient viewing location, and in that case the image is erect, but still reversed left to right. In terrestrial telescopes such as spotting scopes, monoculars and binoculars, prisms (e.g., Porro prisms) or a relay lens between objective and eyepiece are used to correct the image orientation. There are telescope designs that do not present an inverted image such as the Galilean refractor and the Gregorian reflector. These are referred to as erecting telescopes.

Design variants

Many types of telescope fold or divert the optical path with secondary or tertiary mirrors. These may be integral part of the optical design (Newtonian telescope, Cassegrain reflector or similar types), or may simply be used to place the eyepiece or detector at a more convenient position. Telescope designs may also use specially designed additional lenses or mirrors to improve image quality over a larger field of view.

Characteristics

Design specifications relate to the characteristics of the telescope and how it performs optically. Several properties of the specifications may change with the equipment or accessories used with the telescope; such as Barlow lenses, star diagonals and eyepieces. These interchangeable accessories don't alter the specifications of the telescope, however they alter the way the telescopes properties function, typically magnification, angular resolution and FOV.

Surface resolvability

The smallest resolvable surface area of an object, as seen through an optical telescope, is the limited physical area that can be resolved. It is analogous to angular resolution, but differs in definition: instead of separation ability between point-light sources it refers to the physical area that can be resolved. A familiar way to express the characteristic is the resolvable ability of features such as Moon craters or Sun spots. Expression using the formula is given by the sum of twice the resolving power ${\displaystyle R}$ over aperture diameter ${\displaystyle D}$ multiplied by the objects diameter ${\displaystyle D_{ob}}$ multiplied by the constant ${\displaystyle \Phi }$ all divided by the objects apparent diameter ${\displaystyle D_{a}}$.^[20][21]
Resolving power ${\displaystyle R}$ is derived from the wavelength ${\displaystyle {\lambda }}$ using the same unit as aperture; where 550 nm to mm is given by: ${\displaystyle R={\frac {\lambda }{10^{6}}}={\frac {550}{10^{6}}}=0.00055}$.
The constant ${\displaystyle \Phi }$ is derived from radians to the same unit as the objects apparent diameter; where the Moons apparent diameter of ${\displaystyle D_{a}={\frac {313\Pi }{10800}}}$ radians to arcsecs is given by: ${\displaystyle D_{a}={\frac {313\Pi }{10800}}*206265=1878}$.
An example using a telescope with an aperture of 130 mm observing the Moon in a 550 nm wavelength, is given by: ${\displaystyle F={\frac {{\frac {2R}{D}}*D_{ob}*\Phi }{D_{a}}}={\frac {{\frac {2*0.00055}{130}}*3474.2*206265}{1878}}\approx 3.22}$
The unit used in the object diameter results in the smallest resolvable features at that unit. In the above example they are approximated in kilometers resulting in the smallest resolvable Moon craters being 3.22 km in diameter. The Hubble Space Telescope has a primary mirror aperture of 2400 mm that provides a surface resolvability of Moon craters being 174.9 meters in diameter, or sunspots of 7365.2 km in diameter.

Angular resolution

Ignoring blurring of the image by turbulence in the atmosphere (atmospheric seeing) and optical imperfections of the telescope, the angular resolution of an optical telescope is determined by the diameter of the primary mirror or lens gathering the light (also termed its "aperture").
The Rayleigh criterion for the resolution limit ${\displaystyle \alpha _{R}}$ (in radians) is given by

${\displaystyle \sin(\alpha _{R})=1.22{\frac {\lambda }{D}}}$

where ${\displaystyle \lambda }$ is the wavelength and ${\displaystyle D}$ is the aperture. For visible light (${\displaystyle \lambda }$ = 550 nm) in the small-angle approximation, this equation can be rewritten:

${\displaystyle \alpha _{R}={\frac {138}{D}}}$

Here, ${\displaystyle \alpha _{R}}$ denotes the resolution limit in arcseconds and ${\displaystyle D}$ is in millimeters. In the ideal case, the two components of a double star system can be discerned even if separated by slightly less than ${\displaystyle \alpha _{R}}$. This is taken into account by the Dawes limit

${\displaystyle \alpha _{D}={\frac {116}{D}}}$

The equation shows that, all else being equal, the larger the aperture, the better the angular resolution. The resolution is not given by the maximum magnification (or "power") of a telescope. Telescopes marketed by giving high values of the maximum power often deliver poor images.
For large ground-based telescopes, the resolution is limited by atmospheric seeing. This limit can be overcome by placing the telescopes above the atmosphere, e.g., on the summits of high mountains, on balloon and high-flying airplanes, or in space. Resolution limits can also be overcome by adaptive optics, speckle imaging or lucky imaging for ground-based telescopes.
Recently, it has become practical to perform aperture synthesis with arrays of optical telescopes. Very high resolution images can be obtained with groups of widely spaced smaller telescopes, linked together by carefully controlled optical paths, but these interferometers can only be used for imaging bright objects such as stars or measuring the bright cores of active galaxies.

Focal length and focal ratio

The focal length of an optical system is a measure of how strongly the system converges or diverges light. For an optical system in air, it is the distance over which initially collimated rays are brought to a focus. A system with a shorter focal length has greater optical power than one with a long focal length; that is, it bends the rays more strongly, bringing them to a focus in a shorter distance. In astronomy, the f-number is commonly referred to as the focal ratio notated as ${\displaystyle N}$. The focal ratio of a telescope is defined as the focal length ${\displaystyle f}$ of an objective divided by its diameter ${\displaystyle D}$ or by the diameter of an aperture stop in the system. The focal length controls the field of view of the instrument and the scale of the image that is presented at the focal plane to an eyepiece, film plate, or CCD.
An example of a telescope with a focal length of 1200 mm and aperture diameter of 254 mm is given by: ${\displaystyle N={\frac {f}{D}}={\frac {1200}{254}}\approx 4.7}$
Numerically large Focal ratios are said to be long or slow. Small numbers are short or fast. There are no sharp lines for determining when to use these terms, an individual may consider their own standards of determination. Among contemporary astronomical telescopes, any telescope with a focal ratio slower (bigger number) than f/12 is generally considered slow, and any telescope with a focal ratio faster (smaller number) than f/6, is considered fast. Faster systems often have more optical aberrations away from the center of the field of view and are generally more demanding of eyepiece designs than slower ones. A fast system is often desired for practical purposes in astrophotography with the purpose of gathering more photons in a given time period than a slower system, allowing time lapsed photography to process the result faster.
Wide-field telescopes (such as astrographs), are used to track satellites and asteroids, for cosmic-ray research, and for astronomical surveys of the sky. It is more difficult to reduce optical aberrations in telescopes with low f-ratio than in telescopes with larger f-ratio.

Light-gathering power

 

The light-gathering power of an optical telescope, also referred to as light grasp or aperture gain, is the ability of a telescope to collect a lot more light than the human eye. Its light-gathering power is probably its most important feature. The telescope acts as a light bucket, collecting all of the photons that come down on it from a far away object, where a larger bucket catches more photons resulting in more received light in a given time period, effectively brightening the image. This is why the pupils of your eyes enlarge at night so that more light reaches the retinas. The gathering power ${\displaystyle P}$ compared against a human eye is the squared result of the division of the aperture ${\displaystyle D}$ over the observer's pupil diameter ${\displaystyle D_{p}}$,^[20][21] with an average adult having a pupil diameter of 7mm. Younger persons host larger diameters, typically said to be 9mm, as the diameter of the pupil decreases with age.
An example gathering power of an aperture with 254 mm compared to an adult pupil diameter being 7 mm is given by: ${\displaystyle P=\left({\frac {D}{D_{p}}}\right)^{2}=\left({\frac {254}{7}}\right)^{2}\approx 1316.7}$
Light-gathering power can be compared between telescopes by comparing the areas ${\displaystyle A}$ of the two different apertures.
As an example, the light-gathering power of a 10 meter telescope is 25x that of a 2 meter telescope: ${\displaystyle p={\frac {A_{1}}{A_{2}}}={\frac {\pi 5^{2}}{\pi 1^{2}}}=25}$
For a survey of a given area, the field of view is just as important as raw light gathering power. Survey telescopes such as the Large Synoptic Survey Telescope try to maximize the product of mirror area and field of view (or etendue) rather than raw light gathering ability alone.

Magnification

The magnification through a telescope magnifies a viewing object while limiting the FOV. Magnification is often misleading as the optical power of the telescope, its characteristic is the most misunderstood term used to describe the observable world. At higher magnifications the image quality significantly reduces, usage of a Barlow lens—which increases the effective focal length of an optical system—multiplies image quality reduction.
Similar minor effects may be present when using star diagonals, as light travels through a multitude of lenses that increase or decrease effective focal length. The quality of the image generally depends on the quality of the optics (lenses) and viewing conditions—not on magnification.
Magnification itself is limited by optical characteristics. With any telescope or microscope, beyond a practical maximum magnification, the image looks bigger but shows no more detail. It occurs when the finest detail the instrument can resolve is magnified to match the finest detail the eye can see. Magnification beyond this maximum is sometimes called empty magnification.
To get the most detail out of a telescope, it is critical to choose the right magnification for the object being observed. Some objects appear best at low power, some at high power, and many at a moderate magnification. There are two values for magnification, a minimum and maximum. A wider field of view eyepiece may be used to keep the same eyepiece focal length whilst providing the same magnification through the telescope. For a good quality telescope operating in good atmospheric conditions, the maximum usable magnification is limited by diffraction.

Visual

The visual magnification ${\displaystyle M}$ of the field of view through a telescope can be determined by the telescopes focal length ${\displaystyle f}$ divided by the eyepiece focal length ${\displaystyle f_{e}}$ (or diameter). The maximum is limited by the diameter of the eyepiece.
An example of visual magnification using a telescope with a 1200 mm focal length and 3 mm eyepiece is given by: ${\displaystyle M={\frac {f}{f_{e}}}={\frac {1200}{3}}=400}$

Minimum

There is a lowest usable magnification on a telescope. The increase in brightness with reduced magnification has a limit related to something called the exit pupil. The exit pupil is the cylinder of light coming out of the eyepiece, hence the lower the magnification, the larger the exit pupil. The minimum ${\displaystyle M_{m}}$ can be calculated by dividing the telescope aperture ${\displaystyle D}$ over the exit pupil diameter ${\displaystyle D_{ep}}$.^[22] Decreasing the magnification past this limit cannot increase brightness, at this limit there is no benefit for decreased magnification. Likewise calculating the exit pupil ${\displaystyle D_{ep}}$ is a division of the aperture diameter ${\displaystyle D}$ and the visual magnification ${\displaystyle M}$ used. The minimum often may not be reachable with some telescopes, a telescope with a very long focal length may require a longer-focal-length eyepiece than is possible.
An example of the lowest usable magnification using a 254 mm aperture and 7 mm exit pupil is given by: ${\displaystyle M_{m}={\frac {D}{D_{ep}}}={\frac {254}{7}}\approx 36}$, whilst the exit pupil diameter using a 254 mm aperture and 36x magnification is given by: ${\displaystyle D_{ep}={\frac {D}{M}}={\frac {254}{36}}\approx 7}$

Optimum

A useful reference is:

• For small objects with low surface brightness (such as galaxies), use a moderate magnification.
• For small objects with high surface brightness (such as planetary nebulae), use a high magnification.
• For large objects regardless of surface brightness (such as diffuse nebulae), use low magnification, often in the range of minimum magnification.

Only personal experience determines the best optimum magnifications for objects, relying on observational skills and seeing conditions.

Field of view

Field of view is the extent of the observable world seen at any given moment, through an instrument (e.g., telescope or binoculars), or by naked eye. There are various expressions of field of view, being a specification of an eyepiece or a characteristic determined from and eyepiece and telescope combination. A physical limit derives from the combination where the FOV cannot be viewed larger than a defined maximum, due to diffraction of the optics.

Apparent

Apparent FOV is the observable world observed through an ocular eyepiece without insertion into a telescope. It is limited by the barrel size used in a telescope, generally with modern telescopes that being either 1.25 or 2 inches in diameter. A wider FOV may be used to achieve a more vast observable world given the same magnification compared with a smaller FOV without compromise to magnification. Note that increasing the FOV lowers surface brightness of an observed object, as the gathered light is spread over more area, in relative terms increasing the observing area proportionally lowers surface brightness dimming the observed object. Wide FOV eyepieces work best at low magnifications with large apertures, where the relative size of an object is viewed at higher comparative standards with minimal magnification giving an overall brighter image to begin with.

True

True FOV is the observable world observed though an ocular eyepiece inserted into a telescope. Knowing the true FOV of eyepieces is very useful since it can be used to compare what is seen through the eyepiece to printed or computerized star charts that help identify what is observed. True FOV ${\displaystyle v_{t}}$ is the division of apparent FOV ${\displaystyle v_{a}}$ over magnification ${\displaystyle M}$.^[20][21]
An example of true FOV using an eyepiece with 52° apparent FOV used at 81.25x magnification is given by: ${\displaystyle v_{t}={\frac {v_{a}}{M}}={\frac {52}{81.25}}=0.64^{\circ }}$

Maximum

Max FOV is a term used to describe the maximum useful true FOV limited by the optics of the telescope, it is a physical limitation where increases beyond the maximum remain at maximum. Max FOV ${\displaystyle v_{m}}$ is the barrel size ${\displaystyle B}$ over the telescopes focal length ${\displaystyle f}$ converted from radian to degrees.
An example of max FOV using a telescope with a barrel size of 31.75 mm (1.25 inches) and focal length of 1200 mm is given by: ${\displaystyle v_{m}=B*{\frac {\frac {180}{\pi }}{f}}\approx 31.75*{\frac {57.2958}{1200}}\approx 1.52^{\circ }}$

Observing through a telescope

There are many properties of optical telescopes and the complexity of observation using one can be a daunting task; experience and experimentation are the major contributors to understanding how to maximize one's observations. In practice, only two main properties of a telescope determine how observation differs: the focal length and aperture. These relate as to how the optical system views an object or range and how much light is gathered through an ocular eyepiece. Eyepieces further determine how the field of view and magnification of the observable world change.

Observable world

Observable world describes what can be seen using a telescope, when viewing an object or range the observer may use many different techniques. Understanding what can be viewed and how to view it depends on the field of view. Viewing an object at a size that fits entirely in the field of view is measured using the two telescope properties—focal length and aperture, with the inclusion of an ocular eyepiece with suitable focal length (or diameter). Comparing the observable world and the angular diameter of an object shows how much of the object we see. However, the relationship with the optical system may not result in high surface brightness. Celestial objects are often dim because of their vast distance, and detail may be limited by diffraction or unsuitable optical properties.

Field of view and magnification relationship

Finding what can be seen through the optical system begins with the eyepiece providing the field of view and magnification; the magnification is given by the division of the telescope and eyepiece focal lengths. Using an example of an amateur telescope such as a Newtonian telescope with an aperture ${\displaystyle D}$ of 130 mm (5") and focal length ${\displaystyle f}$ of 650 mm (25.5 inches), one uses an eyepiece with a focal length ${\displaystyle d}$ of 8 mm and apparent field of view ${\displaystyle v_{a}}$ of 52°. The magnification at which the observable world is viewed is given by: ${\displaystyle M={\frac {f}{d}}={\frac {650}{8}}=81.25}$. The true field of view ${\displaystyle v_{t}}$ requires the magnification, which is formulated by its division over the apparent field of view: ${\displaystyle v_{t}={\frac {v_{a}}{M}}={\frac {52}{81.25}}=0.64}$. The resulting true field of view is 0.64°, allowing an object such as the Orion nebula, which appears elliptical with an angular diameter of 65 × 60 arcminutes, to be viewable through the telescope in its entirety, where the whole of the nebula is within the observable world. Using methods such as this can greatly increase one's viewing potential ensuring the observable world can contain the entire object, or whether to increase or decrease magnification viewing the object in a different aspect.

Brightness factor

The surface brightness at such a magnification significantly reduces, resulting in a far dimmer appearance. A dimmer appearance results in less visual detail of the object. Details such as matter, rings, spiral arms, and gases may be completely hidden from the observer, giving a far less complete view of the object or range. Physics dictates that at the theoretical minimum magnification of the telescope, the surface brightness is at 100%. Practically, however, various factors prevent 100% brightness; these include telescope limitations (focal length, eyepiece focal length, etc.) and the age of the observer.
Age plays a role in brightness, as a contributing factor is the observer's pupil. With age the pupil naturally shrinks in diameter; generally accepted a young adult may have a 7 mm diameter pupil, an older adult as little as 5 mm, and a younger person larger at 9 mm. The minimum magnification ${\displaystyle m}$ can be expressed as the division of the aperture ${\displaystyle D}$ and pupil ${\displaystyle p}$ diameter given by: ${\displaystyle m={\frac {D}{d}}={\frac {130}{7}}\approx 18.6}$. A problematic instance may be apparent, achieving a theoretical surface brightness of 100%, as the required effective focal length of the optical system may require an eyepiece with too large a diameter.
Some telescopes cannot achieve the theoretical surface brightness of 100%, while some telescopes can achieve it using a very small-diameter eyepiece. To find what eyepiece is required to get minimum magnification one can rearrange the magnification formula, where it is now the division of the telescope's focal length over the minimum magnification: ${\displaystyle {\frac {F}{m}}={\frac {650}{18.6}}\approx 35}$. An eyepiece of 35 mm is a non-standard size and would not be purchasable; in this scenario to achieve 100% one would require a standard manufactured eyepiece size of 40 mm. As the eyepiece has a larger focal length than the minimum magnification, an abundance of wasted light is not received through the eyes.

Exit pupil

The increase in surface brightness as one reduces magnification is limited; that limitation is what is described as the exit pupil: a cylinder of light that projects out the eyepiece to the observer. An exit pupil must match or be smaller in diameter than one's pupil to receive the full amount of projected light; a larger exit pupil results in the wasted light. The exit pupil ${\displaystyle e}$ can be derived with from division of the telescope aperture ${\displaystyle D}$ and the minimum magnification ${\displaystyle m}$, derived by: ${\displaystyle e={\frac {D}{m}}={\frac {130}{18.6}}\approx 7}$. The pupil and exit pupil are almost identical in diameter, giving no wasted observable light with the optical system. A 7 mm pupil falls slightly short of 100% brightness, where the surface brightness ${\displaystyle B}$ can be measured from the product of the constant 2, by the square of the pupil ${\displaystyle p}$ resulting in: ${\displaystyle B=2*p^{2}=2*7^{2}=98}$. The limitation here is the pupil diameter; it's an unfortunate result and degrades with age. Some observable light loss is expected and decreasing the magnification cannot increase surface brightness once the system has reached its minimum usable magnification, hence why the term is referred to as usable.

These eyes represent a scaled figure of the human eye where 15 px = 1 mm, they have a pupil diameter of 7 mm. Figure A has an exit pupil diameter of 14 mm, which for astronomy purposes results in a 75% loss of light. Figure B has an exit pupil of 6.4 mm, which allows the full 100% of observable light to be perceived by the observer.

Image Scale

When using a CCD to record observations, the CCD is placed in the focal plane. Image scale (sometimes called plate scale) describes how the angular size of the object being observed is related to the physical size of the projected image in the focal plane
${\displaystyle i={\frac {\alpha }{s}},}$
where ${\displaystyle i}$ is the image scale, ${\displaystyle \alpha }$ is the angular size of the observed object, and ${\displaystyle s}$ is the physical size of the projected image. In terms of focal length image scale is
${\displaystyle i={\frac {1}{f}},}$
where ${\displaystyle i}$ is measured in radians per meter (rad/m), and ${\displaystyle f}$ is measured in meters. Normally ${\displaystyle i}$ is given in units of arcseconds per millimeter ("/mm). So if the focal length is measured in millimeters, the image scale is
${\displaystyle i(''/mm)={\frac {1}{f(mm)}}\left[{\frac {180\times 3600}{\pi }}\right].}$
The derivation of this equation is fairly straightforward and the result is the same for reflecting or refracting telescopes. However, conceptually it is easier to derive by considering a reflecting telescope. If an extended object with angular size ${\displaystyle \alpha }$ is observed through a telescope, then due to the Laws of reflection and Trigonometry the size of the image projected onto the focal plane will be
${\displaystyle s=\tan(\alpha )f.}$
Thefore, the image scale (angular size of object divided by size of projected image) will be
${\displaystyle i={\frac {\alpha }{s}}={\frac {\alpha }{\tan(\alpha )f}},}$
and by using the small angle relation ${\displaystyle \tan(a)\approx a}$, when ${\displaystyle a<<1}$ (N.B. only valid if ${\displaystyle a}$ is in radians), we obtain
${\displaystyle i={\frac {\alpha }{\alpha f}}={\frac {1}{f}}.}$

Imperfect images

No telescope can form a perfect image. Even if a reflecting telescope could have a perfect mirror, or a refracting telescope could have a perfect lens, the effects of aperture diffraction are unavoidable. In reality, perfect mirrors and perfect lenses do not exist, so image aberrations in addition to aperture diffraction must be taken into account. Image aberrations can be broken down into two main classes, monochromatic, and polychromatic. In 1857, Philipp Ludwig von Seidel (1821–1896) decomposed the first order monochromatic aberrations into five constituent aberrations. They are now commonly referred to as the five Seidel Aberrations.

The five Seidel aberrations

Spherical aberration 

The difference in focal length between paraxial rays and marginal rays, proportional to the square of the objective diameter.

Coma 

A defect by which points appear as comet-like asymmetrical patches of light with tails, which makes measurement very imprecise. Its magnitude is usually deduced from the optical sine theorem.

Astigmatism 

The image of a point forms focal lines at the sagittal and tangental foci and in between (in the absence of coma) an elliptical shape.

Curvature of Field

The Petzval field curvature means that the image, instead of lying in a plane, actually lies on a curved surface, described as hollow or round. This causes problems when a flat imaging device is used e.g., a photographic plate or CCD image sensor.

Distortion

Either barrel or pincushion, a radial distortion that must be corrected when combining multiple images (similar to stitching multiple photos into a panoramic photo).

Optical defects are always listed in the above order, since this expresses their interdependence as first order aberrations via moves of the exit/entrance pupils. The first Seidel aberration, Spherical Aberration, is independent of the position of the exit pupil (as it is the same for axial and extra-axial pencils). The second, coma, changes as a function of pupil distance and spherical aberration, hence the well-known result that it is impossible to correct the coma in a lens free of spherical aberration by simply moving the pupil. Similar dependencies affect the remaining aberrations in the list.

Chromatic aberrations

Longitudinal chromatic aberration: As with spherical aberration this is the same for axial and oblique pencils.

Transverse chromatic aberration (chromatic aberration of magnification)

Astronomical research telescopes

Two of the four Unit Telescopes that make up the ESO's VLT, on a remote mountaintop, 2600 metres above sea level in the Chilean Atacama Desert.

Optical telescopes have been used in astronomical research since the time of their invention in the early 17th century. Many types have been constructed over the years depending on the optical technology, such as refracting and reflecting, the nature of the light or object being imaged, and even where they are placed, such as space telescopes. Some are classified by the task they perform such as Solar telescopes.

Large reflectors

Nearly all large research-grade astronomical telescopes are reflectors. Some reasons are:

• In a lens the entire volume of material has to be free of imperfection and inhomogeneities, whereas in a mirror, only one surface has to be perfectly polished.
• Light of different colors travels through a medium other than vacuum at different speeds. This causes chromatic aberration.
• Reflectors work in a wider spectrum of light since certain wavelengths are absorbed when passing through glass elements like those found in a refractor or catadioptric.
• There are technical difficulties involved in manufacturing and manipulating large-diameter lenses. One of them is that all real materials sag in gravity. A lens can only be held by its perimeter. A mirror, on the other hand, can be supported by the whole side opposite to its reflecting face.

Comparison of nominal sizes of primary mirrors of some notable optical telescopes

Most large research reflectors operate at different focal planes, depending on the type and size of the instrument being used. These including the prime focus of the main mirror, the cassegrain focus (light bounced back down behind the primary mirror), and even external to the telescope all together (such as the Nasmyth and coudé focus).
A new era of telescope making was inaugurated by the Multiple Mirror Telescope (MMT), with a mirror composed of six segments synthesizing a mirror of 4.5 meters diameter. This has now been replaced by a single 6.5 m mirror. Its example was followed by the Keck telescopes with 10 m segmented mirrors.
The largest current ground-based telescopes have a primary mirror of between 6 and 11 meters in diameter. In this generation of telescopes, the mirror is usually very thin, and is kept in an optimal shape by an array of actuators (see active optics). This technology has driven new designs for future telescopes with diameters of 30, 50 and even 100 meters.

Harlan J. Smith Telescope reflecting telescope at McDonald Observatory, Texas

Relatively cheap, mass-produced ~2 meter telescopes have recently been developed and have made a significant impact on astronomy research. These allow many astronomical targets to be monitored continuously, and for large areas of sky to be surveyed. Many are robotic telescopes, computer controlled over the internet (see e.g. the Liverpool Telescope and the Faulkes Telescope North and South), allowing automated follow-up of astronomical events.
Initially the detector used in telescopes was the human eye. Later, the sensitized photographic plate took its place, and the spectrograph was introduced, allowing the gathering of spectral information. After the photographic plate, successive generations of electronic detectors, such as the charge-coupled device (CCDs), have been perfected, each with more sensitivity and resolution, and often with a wider wavelength coverage.
Current research telescopes have several instruments to choose from such as:

• imagers, of different spectral responses
• spectrographs, useful in different regions of the spectrum
• polarimeters, that detect light polarization.

The phenomenon of optical diffraction sets a limit to the resolution and image quality that a telescope can achieve, which is the effective area of the Airy disc, which limits how close two such discs can be placed. This absolute limit is called the diffraction limit (and may be approximated by the Rayleigh criterion, Dawes limit or Sparrow's resolution limit). This limit depends on the wavelength of the studied light (so that the limit for red light comes much earlier than the limit for blue light) and on the diameter of the telescope mirror. This means that a telescope with a certain mirror diameter can theoretically resolve up to a certain limit at a certain wavelength. For conventional telescopes on Earth, the diffraction limit is not relevant for telescopes bigger than about 10 cm. Instead, the seeing, or blur caused by the atmosphere, sets the resolution limit. But in space, or if adaptive optics are used, then reaching the diffraction limit is sometimes possible. At this point, if greater resolution is needed at that wavelength, a wider mirror has to be built or aperture synthesis performed using an array of nearby telescopes.
In recent years, a number of technologies to overcome the distortions caused by atmosphere on ground-based telescopes have been developed, with good results. See adaptive optics, speckle imaging and optical interferometry.

 

 

 

                                     XXX  .  V00  Amateur telescope making 

 

Amateur telescope making is the activity of building telescopes as a hobby, as opposed to being a paid professional. Amateur telescope makers (sometimes called ATMs) build their instruments for personal enjoyment of a technical challenge, as a way to obtain an inexpensive or personally customized telescope, or as a research tool in the field of astronomy. Amateur telescope makers are usually a sub-group in the field of amateur astronomy.

 

  

A 22" Newtonian reflector sits in front of the clubhouse at Stellafane, home of the Springfield Telescope Makers  

 

 

Beginnings

Ever since Galileo took a Dutch invention and adapted it to astronomical use, astronomical telescope making has been an evolving discipline. Many astronomers after the time of Galileo built their own telescopes out of necessity, but the advent of amateurs in the field building telescopes for their own enjoyment and education seems to have come into prominence in the 20th century.
Before the advent of modern mass-produced telescopes the price of even a modest instrument was often beyond the means of an aspiring amateur astronomer. Building your own was the only economical method to obtain a suitable telescope for observing. There were many published works that helped spark an interest in building telescopes such as Irish telescope maker Rev. W. F. A. Ellison's 1920 book "The Amateur's Telescope".
In the United States in the early 1920s articles in Popular Astronomy by Russell W. Porter and in Scientific American by Albert G. Ingalls featuring Porter and the Springfield Telescope Makers^[1] helped expand interest in the hobby. There was so much public interest Ingalls began a regular column for Scientific American on the subject (spawning that publications The Amateur Scientist column) and later compiled into three books titled Amateur Telescope Making Vol. 1-3. These had a large readership of enthusiast (sometimes called "telescope nuts"^[2]) constructing their own instruments. Between 1933 and 1990, Sky and Telescope magazine ran a regular column called "Gleanings for ATMs" edited by Earle Brown, Robert E. Cox & Roger Sinnott. The ready supply of surplus optical components after World War 2 and later Sputnik and the space race also greatly expanded the hobby.

Common amateur designs

A 6” (15cm) Newtonian reflector built by a school student on display at Stellafane

Although the types of telescopes that amateurs build vary widely, including Refractors, Schmidt–Cassegrains and Maksutovs, the most popular telescope design is the Newtonian reflector,^[3] described by Russell W. Porter as “The Poor Man's Telescope”. The Newtonian has the advantage of being a simple design that allows for maximum size for the minimum expense. And since the design employs a single front surface mirror as its objective it only has one surface that has to be ground and polished, as opposed to three for the Maksutov and four for the refractor and the Schmidt–Cassegrain. Typically a Newtonian telescope of 6” or 8" (15 or 20 cm) aperture is a standard starter project, constructed as a club project or by individuals working from books or from plans found on the Internet.

Mirror making

Grinding a mirror using an abrasive and a smaller tool over 300 mm mirror ("ATM Korenica 2006", in Korenica, Croatia)

Since the Newtonian reflector is the most common telescope built by amateur telescope makers, large sections of the literature on the subject are devoted to fabrication of the primary mirror. The mirror has to be carefully ground, polished and figured to an extremely accurate shape, usually a paraboloid. Telescopes with high focal ratios may use spherical mirrors since the difference in the two shapes is insignificant at those ratios. The tools used to achieve this shape are surprisingly simple, consisting of a similarly sized glass tool, a series of finer abrasives, and a polishing pitch lap made from a type of tree sap. Through a whole series of random strokes the mirror naturally tends to become spherical in shape. At that point, a variation in polishing strokes is typically used to create and perfect the desired paraboloidal shape.

Optical blank casting

In order to grind telescope optics into an accurate shape, a highly versatile material must be used such as glass. The amateur casting process is usually done through a temperature controlled glass fusing kiln, in which layers or chunks of glass (usually plate or borosilicate glass) are stacked on top of or around each other within a round mold specified to aperture size. After the melting point is reached, the glass fuses together and then the annealing process begins. Proper annealing procedures ensure that internal stresses are relieved within the glass, which reduces risk of failure for the newly cast blank.

Foucault test

The equipment most amateurs use to test the shape of the mirrors, a Foucault knife-edge test, is, like the tools used to create the surface, simple to fabricate. At its most basic it consists of a light bulb, a piece of tinfoil with a pinhole in it, and a razorblade.

Foucault test setup to measure a mirror

Parabolic mirror showing Foucault shadow patterns made by knife edge inside radius of curvature R (red X), at R and outside R.

After the mirror is polished out it is placed vertically in a stand. The Foucault tester is set up at a distance close to the mirror's radius of curvature. The tester is adjusted so that the returning beam from the pinhole light source is interrupted by the knife edge. Viewing the mirror from behind the knife edge shows a pattern on the mirror surface. If the mirror surface is part of a perfect sphere, the mirror appears evenly lighted across the entire surface. If the mirror is spherical but with defects such as bumps or depressions, the defects appear greatly magnified in height. If the surface is paraboloidal, the mirror looks like a doughnut or lozenge. It is possible to calculate how closely the mirror surface resembles a perfect paraboloid by placing a special mask over the mirror and taking a series of measurements with the tester. This data is then reduced and graphed against an ideal parabolic curve.
Some amateur telescope makers use a similar test called a Ronchi test that replaces the knife edge with a grating comprising several fine parallel wires or an etching on a glass plate. Other tests used include the Gaviola or Caustic test which can measure mirrors of fast f/ratio more accurately, and home-brew Interferometric testing made possible in recent years by affordable lasers, digital cameras (such as webcams), and computers.

Aluminizing or "silvering" the mirror

Once the mirror surface has the correct shape a very thin coating of a highly reflective material is added to the front surface.
Historically this coating was silver. Silvering was put on the mirror chemically, typically by the mirror maker or user. Silver coatings have higher reflectivity than aluminum but corrode quickly and need replacing after a few months.
Since the 1950s most mirror makers have an aluminum coating applied by a thin-film deposition process (work that has to be done by a firm specializing in the process). Modern coatings usually consist of an aluminum layer overcoated with protective transparent compounds.
The mirror is aluminized by placing it in a vacuum chamber with electrically heated tungsten or nichrome coils that can evaporate aluminum.^[4] In a vacuum, the hot aluminum atoms travel in straight lines. When they hit the surface of the mirror, they cool and stick. Some mirror coating shops then evaporate a layer of quartz onto the mirror, whereas others expose it to pure oxygen or air in an oven so that the mirror will form a tough, clear layer of aluminum oxide.

Telescope design

A large fork mounted telescope and several other amateur built telescopes on display at Stellafane

The telescopes amateur telescope makers build range from backyard variety to sophisticated instruments that make meaningful contributions to the field of astronomy. Instruments built by amateurs have been employed in planetary study, astrometry, photometry, comet and asteroid discovery to name just a few. Even the “hobbyist” end of the field can break down into several distinct categories such as: observing deep sky objects, observing the planets, solar observing, lunar observation, and astrophotography of all those classes of objects. Therefore, the design, size, and construction of the telescopes vary as well. Some amateur telescope makers build instruments that, while looking crude, are wholly suited to the purpose they are designed for. Others may strive for a more aesthetic look with high levels of mechanical “finish”. Since some amateur telescope makers do not have access to high-precision machining equipment, many elegant designs such as the Poncet Platform, Crayford focuser, and the Dobsonian telescope have evolved, which achieve functionality and stability without requiring precision machining.
The difficulty of construction is another factor in an amateur’s choice of project. For a given design the difficulty of construction grows roughly as the square^{[citation needed]} of the diameter of the objective. For example, a Newtonian telescope of 4 inches (100 mm) aperture is a moderately easy science fair project. A 6-to-8-inch (150 to 200 mm) Newtonian is considered a good compromise size since construction is not difficult and results in an instrument that would be expensive to purchase commercially. A 12-to-16-inch (300 to 410 mm) reflecting telescope is difficult, but still within the ability of the average amateur who has had experience building smaller instruments. Amateurs have constructed telescopes as large as 1 metre (39 in) across, but usually small groups or astronomy clubs take on such projects.

 

An amateur built Dobsonian telescope of moderate size

 

A 180 mm Texereau standard telescope 

 

 

 

      XXX  .  V000   What does it mean by resolving power of a microscope and telescope? 

 

Resolving power, as the word implies means the ability to distinguish between two distinct light sources as two distinct light sources. Under normal condition, the resolution limit is reached by diffraction of light a.k.a. the bending of light around corner.

So, basically, (a) has resolved the light sources pretty well, while (c) has unable to even register the two light sources as distinct. (b) on the other hand has intermediate resolving power and will produce image which will be 'blur'. 

A real life example looks something like this,

The atmospheric disturbances experienced by land based telescopes limit the resolving power of such telescope, while Hubble is limited by the theoretical limit of resolution limited by the wavelength of incident light.

Theoretical limit to resolution

is expressed by Rayleigh formula,

For everything remaining same, the wavelength of light impacts resolving power linearly i.e. larger the wavelength of light, higher the resolving power will be. This is the reason, why you use blue light (450–495 nm) for higher resolution in a light microscope instead of white light (570–590 nm). If you want even higher resolving power, you can go for electron microscope which used electron (2.5 pm at 100keV) to 'view' the objects.

 

 

                                       XXX  .  V0000  Microscope instrument 

 

Microscope, instrument that produces enlarged images of small objects, allowing the observer an exceedingly close view of minute structures at a scale convenient for examination and analysis. Although optical microscopes are the subject of this article, an image may also be enlarged by many other wave forms, including acoustic, X-ray, or electron beam, and be received by direct or digital imaging or by a combination of these methods. The microscope may provide a dynamic image (as with conventional optical instruments) or one that is static (as with conventional scanning electron microscopes).

A compound microscope.Comstock Images/Jupiterimages

The magnifying power of a microscope is an expression of the number of times the object being examined appears to be enlarged and is a dimensionless ratio. It is usually expressed in the form 10× (for an image magnified 10-fold), sometimes wrongly spoken as “ten eks”—as though the × were an algebraic symbol—rather than the correct form, “ten times.” The resolution of a microscope is a measure of the smallest detail of the object that can be observed. Resolution is expressed in linear units, usually micrometres (μm).
The most familiar type of microscope is the optical, or light, microscope, in which glass lenses are used to form the image. Optical microscopes can be simple, consisting of a single lens, or compound, consisting of several optical components in line. The hand magnifying glass can magnify about 3 to 20×. Single-lensed simple microscopes can magnify up to 300×—and are capable of revealing bacteria—while compound microscopes can magnify up to 2,000×. A simple microscope can resolve below 1 micrometre (μm; one millionth of a metre); a compound microscope can resolve down to about 0.2 μm.
Images of interest can be captured by photography through a microscope, a technique known as photomicrography. From the 19th century this was done with film, but digital imaging is now extensively used instead. Some digital microscopes have dispensed with an eyepiece and provide images directly on the computer screen. This has given rise to a new series of low-cost digital microscopes with a wide range of imaging possibilities, including time-lapse micrography, which has brought previously complex and costly tasks within reach of the young or amateur microscopist.
Other types of microscopes use the wave nature of various physical processes. The most important is the electron microscope, which uses a beam of electrons in its image formation. The transmission electron microscope (TEM) has magnifying powers of more than 1,000,000×. TEMs form images of thin specimens, typically sections, in a near vacuum. A scanning electron microscope (SEM), which creates a reflected image of relief in a contoured specimen, usually has a lower resolution than a TEM but can show solid surfaces in a way that the conventional electron microscope cannot. There are also microscopes that use lasers, sound, or X-rays. The scanning tunneling microscope (STM), which can create images of atoms, and the environmental scanning electron microscope (ESEM), which generates images using electrons of specimens in a gaseous environment, use other physical effects that further extend the types of objects that can be examined.

Transmission electron microscope (TEM).Encyclopædia Britannica, Inc.

History of optical microscopes

The concept of magnification has long been known. About 1267 English philosopher Roger Bacon wrote in Perspectiva, “[We] may number the smallest particles of dust and sand by reason of the greatness of the angle under which we may see them,” and in 1538 Italian physician Girolamo Fracastoro wrote in Homocentrica, “If anyone should look through two spectacle glasses, one being superimposed on the other, he will see everything much larger.”
Three Dutch spectacle makers—Hans Jansen, his son Zacharias Jansen, and Hans Lippershey—have received credit for inventing the compound microscope about 1590. The first portrayal of a microscope was drawn about 1631 in the Netherlands. It was clearly of a compound microscope, with an eyepiece and an objective lens. This kind of instrument, which came to be made of wood and cardboard, often adorned with polished fish skin, became increasingly popular in the mid-17th century and was used by the English natural philosopher Robert Hooke to provide regular demonstrations for the new Royal Society. These demonstrations commenced in 1663, and two years later Hooke published a folio volume titled Micrographia, which introduced a wide range of microscopic views of familiar objects (fleas, lice, and nettles among them). In this book he coined the term cell.

Robert Hooke's drawings of the cellular structure of cork and a sprig of sensitive plant from Micrographia (1665).From Micrographia, by Robert Hooke, 1665

Hidden in the unnumbered pages of Micrographia’s preface is a description of how a single high-powered lens could be made into a serviceable microscope, and it was using this design that the Dutch civil servant Antonie van Leeuwenhoek began his pioneering observations of freshwater microorganisms in the 1670s. He made his postage-stamp-sized microscopes by hand, and the best of them could resolve details around 0.7 μm. His fine specimens discovered in excellent condition at the Royal Society more than three centuries later prove what a great technician he was. Using his simple microscope, Leeuwenhoek effectively launched microbiology in 1674, and single-lensed microscopes remained popular until the 1850s. In 1827 they were used by Scottish botanist Robert Brown to demonstrate the ubiquity of the cell nucleus, a term he coined in 1831.

Microscope made by Antonie van Leeuwenhoek.Photos.com/Thinkstock

Simple microscopes using single lenses can generate fine images; however, they can also produce spurious colours due to chromatic aberration, in which different wavelengths of light do not come to the same focus. The aberrations were worse in the compound microscopes of the time, because the lenses magnified the aberrations at least as much as they magnified the images. Although the compound microscopes were beautiful objects that conferred status on their owners, they produced inferior images. In 1733 the amateur English optician Chester Moor Hall found by trial and error that a combination of a convex crown-glass lens and a concave flint-glass lens could help to correct chromatic aberration in a telescope, and in 1774 Benjamin Martin of London produced a pioneering set of colour-corrected lenses for a microscope.

A 17th-century compound microscope.Golub Collection—University of California, Berkeley/Steven Ruzin, Curator

The appearance of new varieties of optical glasses encouraged continued development of the microscope in the 19th century, and considerable improvements were made in understanding the geometric optics of image formation. The concept of an achromatic (non-colour-distorting) microscope objective was finally introduced in 1791 by Dutch optician Francois Beeldsnijder, and the English scientist Joseph Jackson Lister in 1830 published a work describing a theoretical approach to the complete design of microscope objectives. The physics of lens construction was examined by German physicist Ernst Abbe. In 1868 he invented an apochromatic system of lenses, which had even better colour correction than achromatic lenses, and in 1873 he published a comprehensive analysis of lens theory. Light microscopes that were produced in the closing quarter of the 19th century reached the effective limits of optical microscopy. Subsequent instruments, such as phase-contrast microscopes, interference microscopes, and confocal microscopes, solved specific problems that had arisen during the study of specimens such as living cells.

The simple microscope

Principles

The simple microscope consists of a single lens traditionally called a loupe. The most familiar present-day example is a reading or magnifying glass. Present-day higher-magnification lenses are often made with two glass elements that produce a colour-corrected image. They can be worn around the neck packaged in a cylindrical form that can be held in place immediately in front of the eye. These are generally referred to as eye loupes or jewelers’ lenses. The traditional simple microscope was made with a single magnifying lens, which was often of sufficient optical quality to allow the study of microscopical organisms including Hydra and protists.

A lens magnifying an object.Encyclopædia Britannica, Inc.

Magnification

It is instinctive, when one wishes to examine the details of an object, to bring it as near as possible to the eye. The closer the object is to the eye, the larger the angle that it subtends at the eye, and thus the larger the object appears. If an object is brought too close, however, the eye can no longer form a clear image. The use of the magnifying lens between the observer and the object enables the formation of a “virtual image” that can be viewed in comfort. To obtain the best possible image, the magnifier should be placed directly in front of the eye. The object of interest is then brought toward the eye until a clear image of the object is seen.
Without lenses, the highest possible magnification is when the object is brought to the closest position at which a clear virtual image is observed. For many people, this image distance is about 25 cm (10 inches). As people age, the nearest point of distinct vision recedes to greater distances, thus making a magnifier a useful adjunct to vision for older people.
The magnifying power, or extent to which the object being viewed appears enlarged, and the field of view, or size of the object that can be viewed, are related by the geometry of the optical system. A working value for the magnifying power of a lens can be found by dividing the least distance of distinct vision by the lens’ focal length, which is the distance from the lens to the plane at which the incoming light is focused. Thus, for example, a lens with a least distance of distinct vision of 25 cm and a focal length of 5 cm (2 inches) will have a magnifying power of about 5×.
If the diameter of the magnifying lens is sufficient to fill or exceed the diameter of the pupil of the eye, the virtual image that is viewed will appear to be of substantially the same brightness as the original object. The field of view of the magnifier will be determined by the extent to which the magnifying lens exceeds this working diameter and also by the distance separating the lens from the eye. The clarity of the magnified virtual image will depend upon the aberrations present in the lens, its contour, and the manner in which it is used.

Aberration

Various aberrations influence the sharpness or quality of the image. Chromatic aberrations produce coloured fringes about the high-contrast regions of the image, because longer wavelengths of light (such as red) are brought to focus in a plane slightly farther from the lens than shorter wavelengths (such as blue). Spherical aberration produces an image in which the centre of the field of view is in focus when the periphery may not be and is a consequence of using lenses with spherical (rather than nonspherical, or aspheric) surfaces. Distortion produces curved images from straight lines in the object. The type and degree of distortion visible is intimately related to the possible spherical aberration in the magnifier and is usually most severe in high-powered lenses.

Chromatic aberration. Different wavelengths of light have different focal points.Encyclopædia Britannica, Inc.

Spherical aberration. Light rays form a circular cross section that varies with distance along the optical axis; the smallest size is known as the circle of least confusion. The image with the least spherical aberration is found at this distance.Encyclopædia Britannica, Inc.

Two common types of distortion. In barrel distortion (left), magnification decreases with distance from the centre of the image; in pincushion distortion (right), magnification increases with distance.Encyclopædia Britannica, Inc.

The aberrations of a lens increase as the relative aperture (i.e., the working diameter divided by the focal length) of the lens is increased. Therefore, the aberrations of a lens whose diameter is twice the focal length will be worse than those of a lens whose diameter is less than the focal length. There is thus a conflict between a short focal length, which permits a high magnifying power but small field of view, and a longer focal length, which provides a lower magnifying power but a larger linear field of view. (Leeuwenhoek’s high-powered lenses of the 1670s had a focal length—and thus a working distance—of a few millimetres. This made them difficult to use, but they provided remarkable images that were not bettered for a century.)

Types of magnifiers

There are several types of magnifiers available. The choice of an optical design for a magnifier depends upon the required power and the intended application of the magnifier.
For low powers, about 2–10×, a simple double convex lens is applicable. (Early simple microscopes such as Leeuwenhoek’s magnified up to 300×.) The image can be improved if the lens has specific aspheric surfaces, as can be easily obtained in a plastic molded lens. A reduction of distortion is noted when an aspheric lens is used, and the manufacture of such low-power aspheric plastic magnifiers is a major industry. For higher powers of 10–50×, there are a number of forms for magnifiers in which the simple magnifier is replaced by a compound lens made up of several lenses mounted together.
A direct improvement in the distortion that may be expected from a magnifier can be obtained by the use of two simple lenses, usually plano-convex (flat on one side, outward-curved on the other, with the curved surfaces facing each other). This type of magnifier is based upon the eyepiece of the Huygenian telescope, in which the lateral chromatic aberration is corrected by spacing the elements a focal length apart. Since the imaging properties are provided and shared by two components, the spherical aberration and the distortion of the magnifier are greatly reduced over those of a simple lens of the same power.
A Coddington lens combines two lens elements into a single thick element, with a groove cut in the centre of the element to select the portion of the imaging light with the lowest aberrations. This was a simple and inexpensive design but suffers from the requirement that the working distance of the magnifier be very short.
More-complex magnifiers, such as the Steinheil or Hastings forms, use three or more elements to achieve better correction for chromatic aberrations and distortion. In general, a better approach is the use of aspheric surfaces and fewer elements.
Mirrors are also used. Reflecting microscopes, in which the image is magnified through concave mirrors rather than convex lenses, were brought to their peak of perfection in 1947 by British physicist C.R. Burch, who made a series of giant instruments that used ultraviolet rays. There is no chromatic aberration using a reflector, and distortion and spherical aberration are controlled through the introduction of a carefully contoured aspheric magnifying mirror. Present-day reflecting microscopes are confined to analytical instruments using infrared rays.

The compound microscope

The limitations on resolution (and therefore magnifying power) imposed by the constraints of a simple microscope can be overcome by the use of a compound microscope, in which the image is relayed by two lens arrays. One of them, the objective, has a short focal length and is placed close to the object being examined. It is used to form a real image in the front focal plane of the second lens, the eyepiece or ocular. The eyepiece forms an enlarged virtual image that can be viewed by the observer. The magnifying power of the compound microscope is the product of the magnification of the objective lens and that of the eyepiece.
In addition to these two lens arrays, a compound microscope consists of a body tube, in which the lenses can be housed and kept an appropriate distance apart; a condenser lens that lies beneath the specimen stage and focuses light upon the specimen; and an illumination system, which either transmits light through or reflects light from the object being examined. A method for focusing the microscope, usually with coarse and fine focusing controls, must also be provided.
The basic form of a compound microscope is monocular: a single tube is used, with the objective at one end and a single eyepiece at the other. In order to permit viewing with two eyes and thereby increase comfort and acuity, a single objective can be employed in a binocular tube fitted with a matched pair of eyepieces; beam-splitting prisms are used to send half of the light from the image formed by the objective to each eye. These prisms are mounted in a rotating mechanical assembly so that the separation between the eyepieces can be made to match the required interpupillary distance for the observer. A true stereoscopic microscope is configured by using two objectives and two eyepieces, enabling each eye to view the object separately, making it appear three-dimensional.

Optics

There are some obvious geometric limitations that apply to the design of microscope optics. The attainable resolution, or the smallest distance at which two points can be seen as separate when viewed through the microscope, is the first important property. This is generally set by the ability of the eye to discern detail, as well as by the basic physics of image formation.
The eye’s ability to discern detail is determined by several factors, including the level of illumination and the degree of contrast between light and dark regions on the object. Under reasonable light conditions, a normal eye with good visual acuity is capable of seeing two high-contrast points if they subtend a visual angle of at least one arc minute in size. Thus, at a nominal viewing distance of 25 cm (10 inches), the points must be at least 0.1 mm (0.004 inch) apart for the eye to see them as separate. With a simple magnifier of 10×, an observer can see two points separated by perhaps 0.01 mm (0.0004 inch); and with a compound microscope magnifying 100×, one might expect the observer to be able to distinguish two points only 0.001 mm (0.00004 inch) apart. However, an additional complication arises for the high magnifications encountered in a compound microscope. When the dimensions to be resolved approach the wavelength of light, consideration must be given to the effect of diffraction upon the eye’s ability to resolve details upon objects (see below The theory of image formation).
The resolution and the light-collecting capability of the microscope are determined by the numerical aperture (N.A.) of the objective. The N.A. is defined as the sine of half the angle of the cone of light from each point of the object that can be accepted by the objective multiplied by the refractive index (R.I.) of the medium in which the object is immersed. Thus, the N.A. increases as the lens becomes larger or the R.I. increases. Typical values for microscope objective N.A.’s range from 0.1 for low-magnification objectives to 0.95 for dry objectives and 1.4 for oil-immersion objectives. A dry objective is one that works with the air between the specimen and the objective lens. An immersion objective requires a liquid, usually a transparent oil of the same R.I. as glass, to occupy the space between the object and the front element of the objective.
The limit of resolution is set by the wavelength of light and the N.A. The resolution can be improved either by increasing the N.A. of a lens or by using light with a shorter wavelength. In an immersion objective, the effective wavelength of the light is reduced by the index of refraction of the media within which the object being examined resides. The use of immersion imaging techniques in microscopy improves the resolution capabilities of the microscope.

Mechanical components

The microscope body tube separates the objective and the eyepiece and assures continuous alignment of the optics. It is a standardized length, anthropometrically related to the distance between the height of a bench or tabletop (on which the microscope stands) and the position of the seated observer’s eyes. It is typically fitted with a rotating turret that permits objectives of different powers to be interchanged with the assurance that the image position will be maintained. Traditionally, the length of the body tube has been defined as the distance from the upper end of the objective to the eyepiece end of the tube.

Compound microscope.Encyclopædia Britannica, Inc.

A standard body-tube length of 160 mm (6.3 inches) has been accepted for most uses. (Metallographic microscopes have a 250-mm [10-inch] body tube.) Microscope objectives are designed to minimize aberrations at the specified tube length. Use of other distances will affect the aberration balance for high-magnification objectives. Therefore, focusing of the traditional microscope requires moving the objective, the tube, and the eyepiece as a rigid unit. To achieve this, the entire tube is fitted with a rack-and-pinion mechanism that allows it, together with the objective and the eyepiece, to be moved toward or away from the specimen.
The specimen is usually mounted on a glass slide. Routine microscope slides were fixed at 3 × 1 inches during the Victorian era and are still produced at the metric equivalent of those dimensions (7.5 × 2.5 cm) today. The specimen, usually immersed in a material with an R.I. that matches that of the slide, is covered with a thin cover slip. The mechanical stage on which the slide lies is fitted with a pair of controls featuring a rack-and-pinion arrangement. This permits the glass slide to be moved across the stage in two directions, so that different areas of the specimen can be examined. Computer-controlled microscopes track the position of the slide and can return to designated areas of the specimen when required to do so.
The accuracy with which the focusing and the movement of the slide have to be maintained increases as the depth of focus of the objective decreases. For high-N.A. objectives, this depth of focus can be as small as 1 or 2 μm, which means that the mechanical components must provide stable motion at even smaller increments.
Several approaches have been introduced to achieve such precise stable motion at reasonable cost. Some designers have eliminated the sliding mechanism of the body tube, incorporating adjustments for the vertical movement needed for focusing, as well as the lateral motion of the object, in a single mechanical system. An alternative approach has been to mount a relay objective doublet of 160 mm (6.3 inches) focal length into the lower end of the tube. This tube lens is designed to accept light from an image created by the objective at infinity. The objective itself is designed to have aberrations corrected for an infinite image distance. An advantage of this approach is that, since the relayed image is at infinity, the microscope objective itself, a very lightweight component, can be moved to effect focusing without upsetting the correction of aberrations.
In some microscopes the eyepiece is designed as a portion of a zoom lens, which permits continuous variation of the magnification over a limited range without loss of focus. Such microscopes are widely used in industry.

The illumination system

The illumination system of the standard optical microscope is designed to transmit light through a translucent object for viewing. In a modern microscope it consists of a light source, such as an electric lamp or a light-emitting diode, and a lens system forming the condenser.
The condenser is placed below the stage and concentrates the light, providing bright, uniform illumination in the region of the object under observation. Typically, the condenser focuses the image of the light source directly onto the plane of the specimen, a technique called critical illumination. Alternatively, the image of the source is focused onto the condenser, which is in turn focused onto the entrance pupil of the microscope objective, a system known as Köhler illumination. The advantage of the latter approach is that nonuniformities in the source are averaged in the imaging process. To obtain optimal use of the microscope, it is important that the light from the source both covers the object and fills the entrance aperture of the objective of the microscope with light.
Early microscopes had as their condenser a single lens, which was fixed in the end of the instrument facing the lamp (as in barrel microscopes) or mounted below the stage (as in the Bancks microscopes used by Robert Brown, Charles Darwin, and others). More-complex designs followed, their development driven by the peculiarly English obsession of observing fine details on diatom frustules. Achromatic condensers followed, but they are more troublesome to use because they need precise focusing, and the working distance is short.
Apart from condensers that are matched to specialized objectives (such as phase-contrast systems), others are available for specific applications. Thus, the dark-ground, or dark-field, condenser illuminates specimens against a black background and is eminently applicable to the observation of structures such as bacteria and flagellated cells in water. The use of colour filters, pioneered in the closing years of the 19th century by British microscopist Julius Rheinberg and now known as Rheinberg illumination, allows one to practice a form of dark-ground microscopy in which the background and the specimen are in contrasting colours. Although this technique is of no diagnostic benefit, the results can be spectacularly beautiful.

The objective

The optics of the microscope objective are defined by the focal length, N.A., and field of view. Objectives that have been corrected for aberrations are further defined by the wavelength requirements and the tube length of the microscope.
Manufacturers provide objective lenses with standard magnifications usually ranging from 2 to 100×. The focal length of the objective is inversely proportional to the magnification and, in the majority of modern microscopes, equals the tube length (usually 160 mm [6.3 inches]) divided by the magnification. The field of view of the eyepiece is usually set to be a standard size of about 20 mm (0.8 inch) diameter. The field of view of the objective is then set to range from 10 mm (0.4 inch) for an objective with a magnifying power of 2× to 0.2 mm (0.008 inch) for an objective with a magnification of 100×. As a result, the angular field of view is about 7° for all objectives.

Aberration correction

The N.A. and the complexity of the objective increase as the magnification increases. Low-power objectives, of order 2–5×, are generally two-element lenses. Ordinary crown glass and flint glass (optical glasses with, respectively, relatively low and high R.I.’s) can be used to correct for spherical and chromatic aberration.
For objectives with magnifying powers of 10×, the required N.A. increases to 0.25, and a more complex type of lens is required. Most microscope objectives of this magnification use a separated pair of doublets that share the refractive power. The correction of spherical aberration is readily achieved, but residual chromatic aberration is obtained when normal optical glasses are used for the lens elements. For most optical applications this is not important, but for critical high-magnification objectives (magnifications greater than 25×) this aberration is visible as chromatic blur. The correction of this residual aberration is achieved through the use of special optical glasses whose dispersion properties vary from normal glasses. There are only a few such glasses or crystalline materials that are useful for this purpose. Objectives that use these special glasses are called apochromats and were first produced commercially by Abbe in the 1870s.
Conventional objectives do not produce a flat image surface. The field curvature is generally of little importance in the visual use of the microscope because the eye has a reasonable accommodative capability when examining the image. Field curvature is a problem for imaging systems, however. Special objectives with flat-field lenses have been designed for these systems.

High-power objectives

High-power objectives pose several design problems. Because the focal length of an objective decreases as the N.A. and magnifying power increase, the working distance, or distance from the front of the objective to the top of the slide, is shorter for higher-power objectives. The need to use additional elements in the lens system for high magnifications further shortens the working distance to only 10 to 20 percent of the focal length. Thus, a 40× objective of 4-mm (0.2-inch) focal length may have a working distance of less than 0.4 mm (0.02 inch), so objectives with an increased working distance have been designed. These use a negative lens element between the object and the eyepiece, which has the added attraction of providing some field flattening as well. These objectives are especially of value in use with video systems.
In objectives with magnifying powers of 25× or greater, meniscus-shaped aplanatic elements are designed into the microscope objective in the space between the object and the pairs of doublets that carry out the relayed imaging. These aplanatic components have the property of converging the light without adding spherical aberration to the image and provide an increase in the N.A. without introducing significant aberration.
The highest-power microscope objective available is the immersion objective. When this type of objective is used, a drop of oil must be placed between the object on the microscope slide and the objective. The oil used has an R.I. that matches that of the glass in the first component of the objective.
The first component of immersion objectives is generally a hyper-hemisphere (a small optical surface shaped like a hemisphere but with a boundary curve exceeding 180°), which acts as an aplanatic coupler between the slide and the rest of the microscope objective. An immersion objective with a high N.A. typically consists of a hyper-hemisphere followed by one or two aplanatic collectors and then two or more sets of doublets. Such objectives are made with magnifying powers greater than 50×, the extreme being about 100×.
Water-immersion lenses are also available. These use water as an immersion liquid and allow biologists to examine specimens in a watery medium without the burden of a cover slip confining the living organisms.

Depth of focus

The large N.A. of a microscope objective restricts the focusing requirements of the objective. The depth of focus is shown in the table as the accuracy with which the focal plane must be located in a direction along the axis of the microscope optics in order that the highest possible resolution can be obtained.

Maximum resolving power and depth of focus for a visual microscope
objective focal length (mm)	numerical aperture (N.A.)	maximum useful magnification in compound microscope	maximum resolution on object (mm)	objective depth of focus (mm)
32	0.10	100×	0.0025	0.025
16	0.25	250×	0.001	0.0038
8	0.50	500×	0.0005	0.00086
4	0.95	1,000×	0.00026	0.00024
3	1.38	1,500× (oil immersion)	0.00018	0.00010

The eyepiece

The eyepiece is selected to examine the relayed image under conditions that are comfortable for the viewer. The magnifying power of the eyepiece generally does not exceed 10×. The field of view is then about 40° total, a convenient value for a relatively simple optical design. The observer places the eye at the exit pupil of the eyepiece, the point at which the light rays leaving the eyepiece come together. In most cases an eye relief (or distance from the exit pupil to the last element of the eyepiece) of about 1 cm is desirable. Too short an eye relief makes viewing difficult for observers who wear corrective eyeglasses.

Image capture

The objectives described above are usually intended to project an image through an eyepiece for direct viewing by an observer. The use of a photographic recording method permits the capture of a real image in a film holder or digital imaging system without an eyepiece lens. One approach is to remove the eyepiece and place the film holder or digital camera in the focal plane of the eyepiece, thus intercepting the image from the objective directly. A better approach is to use a specifically designed projection eyepiece, which can be adjusted to provide the appropriate magnification coupling the image to the film. Such an eyepiece can incorporate a change in the chromatic aberration correction to accommodate the requirements of the image-capture system.
Increasingly prevalent today is the use of an electronic detector such as a complementary metal-oxide semiconductor (CMOS) or charge-coupled device (CCD) chip to capture the magnified image as a digital signal. This signal can be transmitted to a computer and translated into an image on the monitor. Software allows the user to take single pictures, moving video sequences, or time-lapse sequences at the click of a mouse. These may be saved for conventional viewing, and image processing can be used to enhance the result. Analysis of area and particle size and distribution is easily done by conventional analytical means once the images have been digitally captured. The production of computer presentations, transmission via e-mail, and ease of printing are benefits that digital imaging brings to the modern microscopist.

 

 

The theory of image formation

The objective collects a fan of rays from each object point and images the ray bundle at the front focal plane of the eyepiece. The conventional rules of ray tracing apply to the image formation. In the absence of aberration, geometric rays form a point image of each object point. In the presence of aberrations, each object point is represented by an indistinct point. The eyepiece is designed to image the rays to a focal point at a convenient distance for viewing the image. In this system, the brightness of the image is determined by the sizes of the apertures of the lenses and by the aperture of the pupil of the eye. The focal length and resulting magnification of the objective should be chosen to attain the desired resolution of the object at a size convenient for viewing through the eyepiece. Image formation in the microscope is complicated by diffraction and interference that take place in the imaging system and by the requirement to use a light source that is imaged in the focal plane.

Image formation in a microscope, according to the Abbe theory. Specimens are illuminated by light from a condenser. This light is diffracted by the details in the object plane: the smaller the detailed structure of the object, the wider the angle of diffraction. The structure of the object can be represented as a sum of sinusoidal components. The rapidity of variation in space of the components is defined by the period of each component, or the distance between adjacent peaks in the sinusoidal function. The spatial frequency is the reciprocal of the period. The finer the details, the higher the required spatial frequency of the components that represent the object detail. Each spatial frequency component in the object produces diffraction at a specific angle dependent upon the wavelength of light. Here, for example, a specimen with structure that has a spatial frequency of 1,000 lines per millimetre produces diffraction with an angle of 33.6°. The microscope objective collects these diffracted waves and directs them to the focal plane, where interference between the diffracted waves produces an image of the object. Encyclopædia Britannica, Inc.

The modern theory of image formation in the microscope was founded in 1873 by the German physicist Ernst Abbe. The starting point for the Abbe theory is that objects in the focal plane of the microscope are illuminated by convergent light from a condenser. The convergent light from the source can be considered as a collection of many plane waves propagating in a specified set of directions and superimposed to form the incident illumination. Each of these effective plane waves is diffracted by the details in the object plane: the smaller the detailed structure of the object, the wider the angle of diffraction.
The structure of the object can be represented as a sum of sinusoidal components. The rapidity of variation in space of the components is defined by the period of each component, or the distance between adjacent peaks in the sinusoidal function. The spatial frequency is the reciprocal of the period. The finer the details, the higher the required spatial frequency of the components that represent the object detail. Each spatial frequency component produces diffraction at a specific angle dependent upon the wavelength of light. As an example, spatial frequency components having a period of 1 μm would have a spatial frequency of 1,000 lines per millimetre. The angle of diffraction for such a component for visible light with a wavelength of 550 nanometres (nm; 1 nanometre is 10⁻⁹ metre) will be 33.6°. The microscope objective collects these diffracted waves and directs them to an image plane, where interference between the diffracted waves produces an image of the object.
Because the aperture of the objective is limited, not all the diffracted waves from the object can be transmitted by the objective. Abbe showed that the greater the number of diffracted waves reaching the objective, the finer the detail that can be reconstructed in the image. He designated the term numerical aperture (N.A.) as the measure of the objective’s ability to collect diffracted light and thus also of its power to resolve detail. On this basis it is obvious that the greater the magnification of the objective, the greater the required N.A. of the objective. The largest N.A. theoretically possible in air is 1.0, but optical design constraints limit the N.A. that can be achieved to around 0.95 for dry objectives.
For the example above of a specimen with a spatial frequency of 1,000 lines per millimetre, the required N.A. to collect the diffracted light would be 0.55. Thus, an objective of 0.55 N.A. or greater must be used to observe and collect useful data from an object with details spaced 1 μm apart. If the objective has a lower N.A., the details of the object will not be resolved. Attempts to enlarge the image detail by use of a high-power eyepiece will yield no increase in resolution. This latter condition is called empty magnification.
The wavelength of light is shortened when it propagates through a dense medium. In order to resolve the smallest possible details, immersion objectives are able to collect light diffracted by finer details than can objectives in air. The N.A. is multiplied by the index of refraction of the medium, and working N.A.’s of 1.4 are possible. In the best optical microscopes, structures with spatial frequency as small as 0.4 μm can be observed. Note that the single lenses made by Leeuwenhoek have been shown to be capable of resolving fibrils only 0.7 μm in thickness.

Specialized optical microscopes

The basic form of the optical microscope is modified by designers for expediency, and a range of special adaptations is available for specific purposes. Some have been designed for ergonomics and others for ease of access to components, while some have been aimed at a specific age group and others at a clearly defined purpose. The largest optical microscope put into production, the Burch reflector made in 1947, weighed 200 kg (440 pounds), while the smallest microscope—a single-lens instrument made by British optician Horace Dall in 1950—weighed no more than 24 grams (0.8 ounce). Even smaller were the diminutive instruments made by Leeuwenhoek, which typically weighed less than 15 grams (0.5 ounce). The most successful commercially available small microscope was designed by British doctor John McArthur in 1958, and McArthur microscopes have been produced by several manufacturers since. Some specialized microscopes intended for handheld use (e.g., the Microwatcher made in 1989) incorporate the illuminator and lens systems into a single unit. More recently, small digital microscopes have been introduced.

Inverted microscopes

For some special purposes, notably the examination of cell cultures, it is more practical if the microscope is mounted upside down. In this form of microscope, the inverted microscope, the light source and condenser are situated uppermost and direct light down through the stage. The objective is set with its front element uppermost, and the eyepieces are angled upward so that the observer can study specimens that are still in their watery medium. Inverted microscopes are important in biology and medical research.

Nikon TE2000 Inverted Research MicroscopeNikon TE2000 Inverted Research Microscope, the first to utilize the infinity space.PRNewsFoto/Nikon Technologies Inc./AP Images

Stereoscopic microscopes

Binocular stereomicroscopes are a matched pair of microscopes mounted side by side with a small angle between the optical axes. The object is imaged independently to each eye, and the stereoscopic effect, which permits discrimination of relief on the object, is retained. The effect can be exaggerated by proper choice of the design parameters for the microscopes. For practical reasons, the magnifying power of such instruments is usually in the range of 5–250×. Such microscopes are important in any work in which fine adjustment of tools or devices is to be made. For example, the stereomicroscope is often used in biological laboratories for dissection of subjects and in the operating room for microsurgical procedures. Moderate-power stereomicroscopes are even more widely used in the electronics manufacturing industry, where they enable technicians to monitor the bonding of leads to integrated circuits.

Polarizing microscopes

Polarizing microscopes are conventional microscopes with additional features that permit observation under polarized light. The light source of such an instrument is equipped with a polarizing filter, the polarizer, so that the light it supplies is linearly polarized (i.e., the light waves vibrate in a given direction rather than randomly in all directions as in ordinary light). When this linearly polarized light passes through the object under examination, it may be unaffected or, if the object is birefringent, it may be split into two beams with different polarizations. A second filter, a polarization analyzer, is fitted to the eyepiece, where it blocks out all but one polarization of the light. The analyzer can be rotated to obtain maximum contrast in the image, and so the direction of polarization of the light transmitted through the object can be determined. The eyepiece can also be equipped with a polarization retarder, which shifts the phase of the light between selected polarization directions and which can be rotated to measure the amount of elliptic polarization produced by the specimen.
Many precautions must be taken in the design and construction of a polarizing microscope to avoid using optical components that introduce undesirable polarization retardation in the beam of light after it has left the object. There is a basic limitation placed upon the use of objectives with high N.A.’s wherein the necessary high angles of incidence on the surface produce some depolarization. Specialized microscope objectives that minimize this effect have been designed and produced. Polarizing microscopes are primarily used to examine the nature of crystals in geologic samples and to analyze the details of birefringence and stress in biological structures. They have been of crucial importance in the detection and monitoring of asbestos fibres.

Metallographic microscopes

Metallographic microscopes are used to identify defects in metal surfaces, to determine the crystal grain boundaries in metal alloys, and to study rocks and minerals. This type of microscope employs vertical illumination, in which the light source is inserted into the microscope tube below the eyepiece by means of a beam splitter. Light shines down through the objective and is focused through the objective onto the specimen. The light reflected or scattered back to the objective is then imaged back at the eyepiece. In this manner, opaque objects such as metals can be examined under the microscope. Such systems also have applications in forensic science and diagnostic microscopy.

Reflecting microscopes

Microscopes of this type feature reflecting rather than refracting objectives. They are used to carry out microscopy over a wide range of visible light and especially in the ultraviolet or infrared regions, where conventional optical glasses do not transmit. The reflecting microscope objective usually consists of two components: a relatively large, concave primary mirror and a smaller, convex secondary mirror, which is located between the primary mirror and the object and serves to relay the image from the primary mirror to the focal plane of the eyepiece. Although reflecting objectives do not have chromatic aberration, they need to be corrected for spherical aberrations, either by using aspheric reflecting components or by adding appropriate refracting lenses.

Phase-contrast microscopes

Many biological objects of interest consist of cell structures such as nuclei that are almost transparent; they transmit as much light as the mounting medium that surrounds them does. Because there is no colour or transmission contrast in such an object, it is not possible to observe the structure using a conventional optical microscope.

Phase-contrast microscopePhase-contrast microscope.Encyclopædia Britannica, Inc.

However, the R.I. of the cell structure varies slightly from the surrounding material, and it is possible to exploit this difference. The propagation of light through such an object provides a change in the optical path across the object, as well as a resulting shift in the phase of the light that has passed through the structure of interest relative to light passing around the structure. This phase-shift information can be used to form a visible image if it is converted into intensity variations that are detectable by the observer. The Dutch physicist Frits Zernike developed a method for doing just that in 1934. (He won the Nobel Prize for Physics in 1953 for his invention.) In a phase-contrast microscope the phase difference between light that is diffracted by a specimen and light that is direct and undeflected is one-quarter of a wavelength or less. By placing an appropriate mask in the back focal plane of the objective to provide selective filtering of the diffracted light, Zernike increased this phase difference by another quarter wavelength. Waves that differ in phase by half a wavelength cancel one another. In places in the phase image where this occurs, no light is transmitted. As a result, phase differences caused by variations in the specimen appear as intensity variations in the image.
There are several approaches to achieving a good-quality image by this technique. One of the most common is the use of an annular light source imaged onto an annular mask in the back focal plane. Other techniques use edges, small dots, or other combinations of source shape and mask shape.

Interference microscopes

Although all optical microscopes in the strict sense create images by diffraction, interference microscopy creates images using the difference between an interfering beam unmodified by the specimen and an otherwise identical beam that illuminates it. A beam splitter divides light into two paths, one of which passes through the specimen while the other bypasses it. When the two beams are combined, the resulting interference between them reveals the structure of the specimen. The first successful system, invented by British microscopist Francis Smith and French physicist Maurice Françon in 1947, used quartz lenses to produce reference and image-forming beams that were perpendicularly polarized. Although this worked well for continuous specimens, in the case of particulates it was better to have the reference beam pass through a bare area of the specimen preparation, and by 1950 the use of half-silvered surfaces and slightly tapering slides allowed polarized light to be dispensed with.
Meanwhile, differential interference contrast (DIC) was developed by Polish-born French physicist Georges Nomarski in 1952. A beam-splitting Wollaston prism emits two beams of polarized light that are plane-polarized at right angles to each other and that slightly diverge. The rays are focused in the back plane of the objective, where they pass through a composite prism that is isotropic at the midpoint, with an increasing optical path difference away from the midpoint. The background colour of the image depends on the setting of the prism, which can be slid longitudinally to produce a spectrum of colours that vary through the spectrum to black. The sensitivity is greatest in the middle position, but the colour contrast is greatest when a strong background tint is selected. More-recent developments include asymmetrical illumination contrast and modulation contrast, which exploit offset or oblique illumination.

Confocal microscopes

The field of view of a microscope is limited by the geometric optics and by the ability to design optics that provide a constant aberration correction over a large field of view. If a scanning arrangement is used, the objective can be used over a continuous series of small fields and the results used to build up an image of a larger region.

confocal microscopeEncyclopædia Britannica, Inc.

The concept has been harnessed in the confocal scanning microscope. Confocal microscopy’s main feature is that only what is in focus is detected, and anything out of focus appears as black. This is achieved by focusing the light source, usually a laser, to a point and detecting the image through a pinhole. Since only light from the focused point contributes to the final image in a confocal system, it is particularly useful for the eludication of fine and three-dimensional structures of biological specimens.
In a laser scanning confocal microscope (LSCM), the focal point of a laser is scanned across a specimen to build up a two-dimensional optical section. Three-dimensional images can be reconstructed by taking a series of two-dimensional images at different focal depths in the specimen (known as a Z-series). Argon and krypton/argon lasers are commonly used, and a multiphoton system has been designed in which an argon laser excites a synthetic titanium-sapphire. A range of colours can thus be utilized.
Specimens are distinguished by fluorescence. Some components fluoresce spontaneously, but most are stained with dyes that fluoresce under the rays from a laser. Scanning rates can be high—in some systems a line of image is scanned every 0.488 millisecond (one millisecond is one thousandth of a second)—so a sequence of images showing how a specimen changes over time can be easily assembled. Methods like these are used to study the behaviour of living cells.

Ultraviolet microscopes

Ultraviolet (UV) microscopy was developed in the early 20th century by the German scientists August Köhler and Moritz von Rohr. Because of the shorter wavelength of UV light, higher resolution was possible, but the opacity of conventional glass lenses to these wavelengths necessitated the use of either a reflecting microscope or specially made quartz lenses. UV microscopes became most widely used for fluorescent microscopy, in which the UV caused microscope stains to fluoresce. In modern microscopy, lamps in the range from deep blue to near UV are more often used for this purpose. However, the interest in UV led to the recognition that the electron beam could be used as an illuminant of very short wavelength, and it was this that gave rise to interest in the electron microscope.

 

 

 

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The Magnifying Glass

A magnifying glass is a convex lens that lets the observer see a larger image of the object being observed.

Learning Objectives

Key Takeaways

Key Points

• The magnification of a magnifying glass depends upon where it is placed between the user’s eye and the object being viewed and upon the total distance between eye and object.
• The magnifying power is the ratio of the sizes of the images formed on the user’s retina with and without the lens.
• The highest magnifying power is obtained by putting the lens very close to the eye and moving both the eye and the lens together to obtain the best focus.

Key Terms

• lens: an object, usually made of glass, that focuses or defocuses the light that passes through it
• diopter: a unit of measure of the power of a lens or mirror, equal to the reciprocal of its focal length in meters. Myopia is diagnosed and measured in diopters
• convex: curved or bowed outward like the outside of a bowl or sphere or circle

A magnifying glass is a convex lens that lets the observer see a larger image of the object under observation. The lens is usually mounted in a frame with a handle, as shown below.

Magnifying Glass: A magnifying glass is a convex lens that lets the observer see a larger image of the object under observation.

The magnification of a magnifying glass depends upon where the instrument is placed between the user’s eye and the object being viewed and upon the total distance between eye and object. The magnifying power is the ratio of the sizes of the images formed on the user’s retina with and without the magnifying glass. When not using the lens, the user would typically bring the object as close to the eye as possible without it becoming blurry. (This point, known as the near point, varies with age. In a young child its distance can be as short as five centimeters, while in an elderly person its distance may be as long as one or two meters. ) Magnifiers are typically characterized using a “standard” value of 0.25m.
The highest magnifying power is obtained by putting the lens very close to the eye and moving both the eye and the lens together to obtain the best focus. When the lens is used this way, the magnifying power can be found with the following equation:
${\text{MP}}_0=\frac{1}{4}\cdot \Phi +1$
where $\Phi$ = optical power. When the magnifying glass is held close to the object and the eye is moved away, the magnifying power is approximated by:
${\text{MP}}_0=\frac{1}{4}\cdot \Phi$
Typical magnifying glasses have a focal length of 25cm and an optical power of four diopters. This type of glass would be sold as a 2x magnifier, but a typical observer would see about one to two times magnification depending on the lens position.
The earliest evidence of a magnifying device was Aristophanes’s “lens” from 424 BC, a glass globe filled with water. (Seneca wrote that it could be used to read letters “no matter how small or dim.”) Roger Bacon described the properties of magnifying glasses in the 13th century, and eyeglasses were also developed in 13th-century Italy.

The Camera

Cameras are optical devices that allow a user to record an image of an object, either on photo paper or digitally.

Learning Objectives

Key Takeaways

Key Points

• Cameras work very similarly to how the human eye works. The iris is similar to the lens; the pupil is similar to the aperture; and the eyelid is similar to the shutter.
• Cameras are a modern evolution of the camera obscura. The camera obscura was a device used to project images.
• The most important part of a camera is the lens, which allows the image to be magnified and focused. This can be done manually on some cameras and automatically on newer cameras.
• Movie cameras work by taking many pictures each second and then showing each image in order very quickly to give the effect that the pictures are moving. This is where the name “movie” comes from.

Key Terms

• shutter speed: The duration of time for which the shutter of a camera remains open when exposing photographic film or other photosensitive material to light for the purpose of recording an image

What is a Camera?

A camera is a device that allows you to record images, either on film or digitally. Cameras can record images as well as movies; movies themselves got their name from moving pictures. The word camera comes from the Latin phrase camera obscura, which means “dark chamber. ” The camera obscura was an early instrument for projecting images from slides. The camera that you use today is an evolution of the camera obscura.
A camera is usually comprised of an opening, or aperture, that allows light to enter into a hollow area and a surface that records the light at the other end. In the 20^th century, these images would be stored on photographic paper that then had to be developed, but now most cameras store images digitally.

How does a Camera Work?

Cameras have a lot of components that allow them to work. Let’s look at them one at a time.

The Lens

The camera lens allows the light to enter into the camera and is typically convex. There are many types of lenses that can be used, each for a different type of photography. There are lenses for close-ups, for sports, for architecture, and for portraits.
The two major features of a lens are focal length and aperture. The focal length determines the magnification of the image, and the aperture controls the light intensity. The f-number on a camera controls the shutter speed. This is the speed at which the shutter, which acts as its “eyelid,” opens and closes. The larger the aperture, the smaller the f-number must be in order to get the shutter opened and closed fully. The time it takes to open and close the shutter is called the exposure. shows an example of two lenses of the same size but with different apertures.

Focus

Some cameras have a fixed focus, and only objects of a certain size at a certain distance from the camera will be in focus. Other cameras allow you to manually or automatically adjust the focus. shows a picture taken with a camera with manual focus; this allows the user to determine which objects will be in focus and which will not. The range of distance within which objects appear sharp and clear is called the depth of field.

Exposure

The aperture controls the intensity of the light entering the camera, and the shutter controls the exposure — the amount of time that the light is allowed into the camera.

Shutter

The shutter is what opens and closes to allow light through the aperture. The speed at which it opens and closes is called the f-number. For a larger aperture, the f-number is generally small for a quick shutter speed. For a smaller aperture, the f-number is larger, allowing for a slower shutter speed.

The Compound Microscope

A compound microscope is made of two convex lenses; the first, the ocular lens, is close to the eye, and the second is the objective lens.

Learning Objectives

Key Takeaways

Key Points

• A compound microscope uses multiple lenses to create an enlarged image that is easier for a human eye to see; this is due to the fact that the final image is farther away from the observer, and therefore the eye is more relaxed when viewing the image.
• The object is placed just beyond to focal length of the objective lens. An enlarged image is then captured by the objective lens, which acts as the object for the ocular lens. The ocular lens is closer to the new image than its focal length, causing it to act as a magnifying glass.
• Since the final image is just a multiple of the size of the first image, the final magnification is a product of both magnifications from each lens.

Key Terms

• chromatic aberration: an optical aberration, in which an image has colored fringes, caused by differential refraction of light of different wavelengths

A compound microscope uses multiple lenses to magnify an image for an observer. It is made of two convex lenses: the first, the ocular lens, is close to the eye; the second is the objective lens.
Compound microscopes are much larger, heavier and more expensive than simple microscopes because of the multiple lenses. The advantages of these microscopes, due to the multiple lenses, are the reduced chromatic aberrations and exchangeable objective lenses to adjust magnification.
shows a diagram of a compound microscope made from two convex lenses. The first lens is called the objective lens and is closest to the object being observed. The distance between the object and the objective lens is slightly longer than the focal length, f₀. The objective lens creates an enlarged image of the object, which then acts as the object for the second lens. The second or ocular lens is the eyepiece. The distance between the objective lens and the ocular lens is slightly shorter than the focal length of the ocular lens, f_e. This causes the ocular lens to act as a magnifying glass to the first image and makes it even larger. Because the final image is inverted, it is farther away from the observer’s eye and thus much easier to view.

 

Diagram of a Compound Microscope: This diagram shows the setup of mirrors that allow for the magnification of images.

Since each lens produces a magnification that multiplies the height of the image, the total magnification is a product of the individual magnifications. The equation for calculating this is as follows:
$\text{m}={\text{m}}_{\text{o}}{\text{m}}_{\text{e}}$
where m is total magnification, m_o is objective lens magnification, m_e is ocular lens magnification.

The Telescope

The telescope aids in observation of remote objects by collecting electromagnetic radiation, such as visible light.

Learning Objectives

Key Takeaways

Key Points

• Until the invention of silver-backed mirrors, refractive mirrors were the standard for use in telescopes. This was because of the highly corrosive nature of the metals used in older mirrors. Since then, reflective mirrors have replaced refractive mirrors in astronomy.
• There are three main types of optical telescopes: refractive, reflective, and catadioptric.
• Refractive telescopes, such as the one invented by Galileo, use an objective lens and an eyepiece. The image is focused at the focal point and allows the observer to see a brighter, larger image than they would with their own eye.
• Reflective telescopes use curved mirrors that reflect light to form an image. Sometimes a secondary mirror redirects the image to an eyepiece. Other times the image is recorded by a sensor and observed on a computer screen.
• Catadioptric telescopes combine mirrors and lenses to form an image. This system has a greater degree of error correction than other types of telescopes. The combination of reflective and refractive elements allows for each element to correct the errors made by the other.

Key Terms

• chromatic aberration: an optical aberration, in which an image has colored fringes, caused by differential refraction of light of different wavelengths
• spherical aberration: a type of lens aberration that causes blurriness, particularly away from the center of the lens
• achromatic: free from color; transmitting light without color-related distortion

The telescope aids in observation of remote objects by collecting electromagnetic radiation, such as x-rays, visible light, infrared, and submillimeter rays. The first telescopes were invented in the Netherlands in the 1600s and used glass lenses. Shortly after, people began to build them using mirrors and called them reflecting telescopes.

History

The first telescope was a refracting telescope made by spectacle makers in the Netherlands in 1608. In 1610, Galileo made his own improved design. After the refracting telescope was invented, people began to explore the idea of a telescope that used mirrors. The potential advantages of using mirrors instead of lenses were a reduction in spherical aberrations and the elimination of chromatic aberrations. In 1668, Newton built the first practical reflecting telescope. With the invention of achromatic lenses in 1733, color aberrations were partially corrected, and shorter, more functional refracting telescopes could be constructed. Reflecting telescopes were not practical because of the highly corrosive metals used to make mirrors until the introduction of silver-coated glass mirrors in 1857.

Types of Telescopes

Refracting Telescopes

Schematic of Keplerian Refracting Telescope: All refracting telescopes use the same principles. The combination of an objective lens 1 and some type of eyepiece 2 is used to gather more light than the human eye is able to collect on its own, focus it 5, and present the viewer with a brighter, clearer, and magnified virtual image 6.

The figure above is a diagram of a refracting telescope. The objective lens (at point 1) and the eyepiece (point 2) gather more light than a human eye can collect by itself. The image is focused at point 5, and the observer is shown a brighter, magnified virtual image at point 6. The objective lens refracts, or bends, light. This causes the parallel rays to converge at a focal point, and those that are not parallel converge on a focal plane.

Reflecting Telescopes

Reflecting telescopes, such as the one shown in, use either one or a combination of curved mirrors that reflect light to form an image. They allow an observer to view objects that have very large diameters and are the primary type of telescope used in astronomy. The object being observed is reflected by a curved primary mirror onto the focal plane. (The distance from the mirror to the focal plane is called the focal length. ) A sensor could be located here to record the image, or a secondary mirror could be added to redirect the light to an eyepiece.

Catadioptric Telescopes

Catadioptric telescopes, such as the one shown in, combine mirrors and lenses to form an image. This system has a greater degree of error correction than other types of telescopes. The combination of reflective and refractive elements allows for each element to correct the errors made by the other.

X-Ray Diffraction

The principle of diffraction is applied to record interference on a subatomic level in the study of x-ray crystallography.

Learning Objectives

Key Takeaways

Key Points

• Diffraction is what happens when waves encounter irregularities on a surface or object and are caused to interfere with each other, either constructively or destructively.
• The Bragg law pertains to applying the laws of diffraction to crystallography in order to obtain precise images of the lattice structures in atoms.
• The x-ray diffractometer is the machine used to scan the object by shooting a wave at it and recording the interference it encounters.
• Most XRDs are equipped with a Soller slit, which acts like a polarizer for the incident beam. It makes sure that the incident beam being recorded is perfectly parallel to the object being analyzed.

Key Terms

• constructive interference: Occurs when waves interfere with each other crest to crest and the waves are exactly in phase with each other.
• crystallography: The experimental science of determining the arrangement of atoms in solids.
• destructive interference: Occurs when waves interfere with each other crest to trough (peak to valley) and are exactly out of phase with each other.

X-ray diffraction was discovered by Max von Laue, who won the Nobel Prize in physics in 1914 for his mathematical evaluation of observed x-ray diffraction patterns.
Diffraction is the irregularities caused when waves encounter an object. You have most likely observed the effects of diffraction when looking at the bottom of a CD or DVD. The rainbow pattern that appears is a result of the light being interfered by the pits and lands on the disc that hold the data. shows this effect. Diffraction can happen to any type of wave, not just visible light waves.

Bragg Diffraction

In x-ray crystallography, the term for diffraction is Bragg diffraction, which is the scattering of waves from a crystalline structure. William Lawrence Bragg formulated the equation for Bragg’s law, which relates wavelength to the angle of incidence and lattice spacing. Refer to for a diagram of the following equation: $\text{n}\lambda =2\text{d}\sin \left(\theta \right)$

• n – numeric constant known as the order of the diffracted beam
• λ – wavelength
• d – distance between lattice planes
• θ – angle of diffracted wave

The waves will experience either constructive interference or destructive interference. Similarly, the x-ray beam that is diffracted off a crystal will have some parts that have stronger energy, and others that lose energy. This depends on the wavelength and the lattice spacing.

The X-ray Diffractometer

The XRD machine uses copper metal as the element for the x-ray source. Diffraction patterns are recorded over an extended period of time, so it is very important that the beam intensity remains constant. Film used to be used to record the data, but that was inconvenient because it had to be replaced often. Now the XRD machines are equipped with semiconductor detectors. These XRD machines record images in two ways, either continuous scans or step scanning. In continuous scans, the detector moves in circular motions around the object, while a beam of x-ray is constantly shot at the detector. Pulses of energy are plotted with respect to diffraction angle. The step scan method is the more popular method. It is much more efficient than continuous scans. In this method, the detector collects data at a single fixed angle at a time. To ensure that the incident beam is continuous, XRD machines are equipped with a Soller slit. This acts like polarized sunglasses by organizing random x-ray beams into a stack of neatly arranged waves parallel to the plane of the detector.

X-Ray Imaging and CT Scans

Radiography uses x-rays to view material that cannot be seen by the human eye by identifying areas of different density and composition.

Learning Objectives

Key Takeaways

Key Points

• Radiography uses x-rays to take pictures of materials with in an object that can not be seen. They shoot x-ray beams through the object, and collect the rays on film or a detector on the other side. Some of the rays are absorbed into the denser materials, and this is how the image is produced.
• X-ray radiographs take images of all the materials within an object superimposed on each other.
• The traditional, superimposed images can be helpful for a number of applications, but CT scans enable the observer to see just the desired sections of a material.
• Modern CT scans can even take all the slices, or layers, and arrange them into a three-dimensional representation of the object.

Key Terms

• radiography: The use of X-rays to view a non-uniformly composed material such as the human body.
• tomography: Imaging by sections or sectioning.
• superimposed: Positioned on or above something else, especially in layers

Overview

X-ray imaging, or radiography, used x-rays to view material within the body that cannot be seen by the human eye by identifying areas of different density and composition. CT Scans use the assistance of a computer to take this information, and generate 3 dimensional images.

X-ray Imaging

X-ray radiographs are produced by projecting a beam of X-rays toward an object, in medical cases, a part of the human body. Depending on the physical properties of the object (density and composition), some of the X-rays can be partially absorbed. The portion of the rays that are not absorbed then pass through the object and are recorded by either film or a detector, like in a camera. This provides the observer with a 2 dimensional representation of all the components of that object superimposed on each other. shows an image of a human elbow.

X-Ray Radiography: Radiography of the knee in a modern X-ray machine.

Tomography

Tomography refers to imaging by sections, or sectioning. demonstrates this concept. The three-dimensional image is broken down into sections. (S₁) shows a section from the left and (S₂) shows a section from the right.

CT Scans

CT scans, or computed tomography scans use a combination of X-ray radiography and tomography to produce slices of areas of the human body. Doctors can analyze the area, and based on the ability of the material to block the X-ray beam, understand more about the material. shows a CT Scan of a human brain. Doctors can cross reference the images with known properties of the same material and determine if there are any inconsistencies or problems. Although generally these scans are shown as in, the information recorded can be used to create a 3 dimensional image of the area. shows a three dimensional image of a brain that was made by compiling CT Scans.

Specialty Microscopes and Contrast

Microscopes are instruments that let the human eye see objects that would otherwise be too small.

Learning Objectives

Key Takeaways

Key Points

• For better resolution, it is important that there is a lot of contrast between the image and the background.
• Microscopes are classified by what interacts with the object, such as light or electrons. They are also classified by whether they take images by scanning a little at a time or by taking images of the entire object at once.
• Some common types of specialty microscopes are scanning electron micrscopes (SEM), transmission electron microscopes (TEM), both of which are electron microscopes, and atomic force microscopes (ATM) which is a scanning probe microscope.

Key Terms

• contrast: A difference in lightness, brightness and/or hue between two colors that makes them more or less distinguishable

Microscopes are instruments that let the human eye see objects that would otherwise be too small. There are many types of microscopes: optical microscopes, transmission electron microscopes, scanning electron microscopes and scanning probe microscopes.

Microscope Classes

One way to group microscopes is based on how the image is generated through the microscope. Here are three ways we can classify microscopes:
1. ) Light or Photon – optical microscopes
2. ) Electrons – electron microscopes
3. ) Probe – scanning probe microscopes.
Microscopes can also be classified based on whether they analyze the sample by scanning a point at a time (scanning electron microscopes), or by analyzing the entire sample at once (transmission electron microscopes).

Types of Microscopes

• In optical microscopes, the better the contrast between the image and the surface it is being viewed on, the better the resolution will be to the viewer. There are many illumination techniques to generate improved contrast. These techniques include “dark field” and “bright field.” With the dark field technique the light is scattered by the object and the image appears to the observer on a dark background. With the bright field technique the object is illuminated from below to increase the contrast in the image seen by viewers.
• Transmission Electron Microscope: The TEM passes electrons through the sample, and allows people to see objects that are normally not seen by the naked eye. A beam of electrons is transmitted through an ultra thin specimen, interacting with the specimen as it passes through. This interaction forms an image that is magnified and focused onto an imaging device.
• Scanning Electron Microscopes: Referred to as SEM, these microscopes look at the surface of objects by scanning them with a fine electron beam. The electron beam of the microscope interacts with the electrons in the sample and produces signals that can be detected and have information about the topography and composition.
• Atomic Force Microscopy: The AFM is a scanning probe type of microscopy with very high resolution and is one of the foremost tools for imaging at the nanoscale. The mechanical probe feels the surface with a cantilever with a sharp tip. The deflection of the tip is then measured using a laser spot that is reflected from the surface of the cantilever.

Limits of Resolution and Circular Aperatures

In optical imaging, there is a fundamental limit to the resolution of any optical system that is due to diffraction.

Learning Objectives

Key Takeaways

Key Points

• Since effects of diffraction become most prominent for waves whose wavelength is roughly similar to the dimensions of the diffracting objects, the wavelength of the imaging beam sets a fundamental limit on the resolution of any optical system.
• The Abbe diffraction limit for a microscope is given as $\text{d}=\frac{\lambda }{2\left(\text{n}\sin \theta \right)}$ .
• Since the diffraction limit is proportional to wavelength, to increase the resolution, shorter wavelengths can be used such as UV and X-ray microscopes.

Key Terms

• diffraction: The bending of a wave around the edges of an opening or an obstacle.
• nanostructure: Any manufactured structure having a scale between molecular and microscopic.
• aperture: The diameter of the aperture that restricts the width of the light path through the whole system. For a telescope, this is the diameter of the objective lens (e.g., a telescope may have a 100 cm aperture).

The resolution of an optical imaging system (e.g., a microscope, telescope, or camera) can be limited by factors such as imperfections in the lenses or misalignment. However, there is a fundamental maximum to the resolution of any optical system that is due to diffraction (a wave nature of light). An optical system with the ability to produce images with angular resolution as good as the instrument’s theoretical limit is said to be diffraction limited.
For telescopes with circular apertures, the size of the smallest feature in an image that is diffraction limited is the size of the Airy disc, as shown in. As one decreases the size of the aperture in a lens, diffraction increases and the ring features from diffraction become more prominent. Similarly, when imaged objects get smaller, features from diffraction begin to blur the boundary of the object. Since effects of diffraction become most prominent for waves whose wavelength is roughly similar to the dimensions of the diffracting objects, the wavelength of the imaging beam sets a fundamental limit on the resolution of any optical system.

Airy Disk: Computer-generated image of an Airy disk. The gray scale intensities have been adjusted to enhance the brightness of the outer rings of the Airy pattern.

The Abbe Diffraction Limit for a Microscope

The observation of sub-wavelength structures with microscopes is difficult because of the Abbe diffraction limit. In 1873, Ernst Abbe found that light, with wavelength λ, traveling in a medium with refractive index n, cannot be converged to a spot with a radius less than:
$\text{d}=\frac{\lambda }{2\left(\text{n}\sin \theta \right)}$ .
The denominator $\text{n}\sin \theta$ is called the numerical aperture and can reach about 1.4 in modern optics, hence the Abbe limit is roughly d=λ/2. With green light around 500 nm, the Abbe limit is 250 nm which is large compared to most nanostructures, or biological cells with sizes on the order of 1μm and internal organelles which are much smaller. Using a 500 nm beam, you cannot (in principle) resolve any features with size less than around 250 nm.

Improving Resolution

To increase the resolution, shorter wavelengths can be used such as UV and X-ray microscopes. These techniques offer better resolution but are expensive, suffer from lack of contrast in biological samples and may damage the sample. There are techniques for producing images that appear to have higher resolution than allowed by simple use of diffraction-limited optics. Although these techniques improve some aspect of resolution, they generally involve an enormous increase in cost and complexity. Usually the technique is only appropriate for a small subset of imaging problems.

Aberrations

An aberration, or distortion, is a failure of rays to converge at one focus because of limitations or defects in a lens or mirror.

Learning Objectives

Key Takeaways

Key Points

• There are many types of aberrations, including chromatic, spherical, comatic, astigmatism, and barrel distortion.
• Chromatic aberrations occur due to the fact that lenses have different refractive indexes for different wavelengths, and therefore colors. These aberrations occur right on the edges of images between light and dark areas of the picture.
• Mirrors do not have chromatic aberrations because they do not rely on the index of refraction, but rather the index of reflection, which is independent of wavelength.
• Comatic aberrations are due to imperfections in lenses and cause the point source to be off-center. This can cause images to appear pear-shaped, or cause images to have tails, as with comets.

Key Terms

• distortion: (optics) an aberration that causes magnification to change over the field of view.
• refraction: Changing of a light ray’s direction when it passes through variations in matter.
• aberration: The convergence to different foci, by a lens or mirror, of rays of light emanating from one and the same point, or the deviation of such rays from a single focus; a defect in a focusing mechanism that prevents the intended focal point.

The Basics of Aberrations

An aberration is the failure of rays to converge at one focus because of limitations or defects in a lens or mirror. Basically, an aberration is a distortion of an image due to the fact that lenses will never behave exactly according to the way they were modeled. Types of aberrations vary due to the size, material composition, or thickness of a lens, or the position of an object.

Chromatic Aberration

A chromatic aberration, also called achromatism or chromatic distortion, is a distortion of colors. This aberration happens when the lens fails to focus all the colors on the same convergence point. This happens because lenses have a different index of refraction for different wavelengths of light. The refractive index decreases with increasing wavelength. These aberrations or distortions occur on the edges of color boundaries between bright and dark areas of an image. Since the index of refraction of lenses depends on color or wavelength, images are produced at different places and with different magnifications for different colors. shows chromatic aberration for a single convex lens. Since violet rays have a higher refractive index than red, they are bent more and focused closed to the lens. shows a two-lens system using a diverging lens to partially correct for this, but it is nearly impossible to do so completely.
The law of reflection is independent of wavelength, and therefore mirrors do not have this problem. This is why it is advantageous to use mirrors in telescopes and other optical systems.

Comatic Aberration

A comatic aberration, or coma, occurs when the object is off-center. Different parts of a lens of a mirror do not refract or reflect the image to the same point, as shown in. They can also be result of an imperfection in the lens or other component and result in off- axis point sources. These aberrations can cause objects to appear pear-shaped. They can also cause stars to appear distorted or appear to have tails, as with comets.

Other Aberrations

Spherical aberrations are a form of aberration where rays converging from the outer edges of a lens converge to a focus closer to the lens, and rays closer to the axis focus further. Astigmatisms are also a form of aberration in the lenses of the eyes where rays that propagate in two perpendicular planes have different foci. This can eventually cause a monochromatic image to distort vertically or horizontally. Another aberration or distortion is a barrel distortion where image magnification decreases with the distance from the optical axis. The apparent effect is that of an image which has been mapped around a sphere, like in a fisheye lens.

 

Superposition and Interference

Conditions for Wave Interference: Reflection due to Phase Change

Interference is a phenomenon in which two waves superimpose to form a resultant wave of greater or lesser amplitude.

Learning Objectives

Contrast the effects of constructive and destructive interference

Key Takeaways

Key Points

• When two or more waves are incident on the same point, the total displacement at that point is equal to the vector sum of the displacements of the individual waves.
• Light exhibits wave characteristics in various media as well as in a vacuum. When light goes from a vacuum to some medium, like water, its speed and wavelength change, but its frequency f remains the same.
• When light is reflected off a medium with a higher index of refraction, crests get reflected as troughs and troughs get reflected as crests. In other words, the wave undergoes a 180 degree change of phase upon reflection, and the reflected ray “jumps” ahead by half a wavelength.

Key Terms

• coherent: Of waves having the same direction, wavelength and phase, as light in a laser.
• plane wave: A constant-frequency wave whose wavefronts (surfaces of constant phase) are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector.

Conditions for Wave Interference

Interference is a phenomenon in which two waves superimpose to form a resultant wave of greater or lesser amplitude. Its effects can be observed in all types of waves (for example, light, acoustic waves and water waves). Interference usually refers to the interaction of waves that are correlated (coherent) with each other because they originate from the same source, or they have the same or nearly the same frequency. When two or more waves are incident on the same point, the total displacement at that point is equal to the vector sum of the displacements of the individual waves. If a crest of one wave meets a crest of another wave of the same frequency at the same point, then the magnitude of the displacement is the sum of the individual magnitudes. This is constructive interference and occurs when the phase difference between the waves is a multiple of 2π. Destructive interference occurs when the crest of one wave meets a trough of another wave. In this case, the magnitude of the displacements is equal to the difference in the individual magnitudes, and occurs when this difference is an odd multiple of π. Examples of constructive and destructive interference are shown in. If the difference between the phases is intermediate between these two extremes, then the magnitude of the displacement of the summed waves lies between the minimum and maximum values.

Wave Interference: Examples of constructive (left) and destructive (right) wave interference.

 

 

Interference of Plane Waves: Geometrical arrangement for two plane wave interference.

$\Delta \phi =\frac{2\pi \text{d}}{\lambda }=\frac{2\pi \text{x}\sin \theta }{\lambda }$
Constructive interference occurs when the waves are in phase, or
[latex]\frac{\text{x} \sin {\theta}}{\lambda} = 0, \pm 1, \pm 2,…[/latex]
Destructive interference occurs when the waves are half a cycle out of phase, or
[latex]\frac{\text{x} \sin {\theta}}{\lambda} = \pm \frac{1}{2}, \pm \frac{3}{2},…[/latex]

Reflection Due to Phase Change

Light exhibits wave characteristics in various media as well as in a vacuum. When light goes from a vacuum to some medium (like water) its speed and wavelength change, but its frequency f remains the same. The speed of light in a medium is v = c/n, where n is the index of refraction. For example, water has an index of refraction of n = 1.333. When light is reflected off a medium with a higher index of refraction, crests get reflected as troughs and troughs get reflected as crests. In other words, the wave undergoes a 180 degree change of phase upon reflection, and the reflected ray “jumps” ahead by half a wavelength.

Air Wedge

An air wedge is a simple interferometer used to visualize the disturbance of the wave front after propagation through a test object.

Learning Objectives

Describe how an air wedge is used to visualize the disturbance of a wave front after proagation

Key Takeaways

Key Points

• An air wedge interferometer consists of two optical glass wedges (~2-5 degrees), pushed together and then slightly separated from one side to create a thin air-gap wedge.
• The air gap between the two glass plates has two unique properties: it is very thin (micrometer scale) and has perfect flatness.
• To minimize image aberrations of the resulting fringes, the angle plane of the glass wedges has to be placed orthogonal to the angle plane of the air wedge.

Key Terms

• orthogonal: Of two objects, at right angles; perpendicular to each other.
• interferometer: Any of several instruments that use the interference of waves to determine wavelengths and wave velocities, determine refractive indices, and measure small distances, temperature changes, stresses, and many other useful measurements.

Air Wedge

An air wedge is one of the simplest designs of shearing interferometers used to visualize the disturbance of the wave front after propagation through a test object. An air wedge can be used with nearly any light source, including non-coherent white light. The interferometer consists of two optical glass wedges (~2-5 degrees), pushed together and then slightly separated from one side to create a thin air-gap wedge. An example of an air wedge interferometer is shown in.

Air Wedge: Example of air wedge interferometer

The air gap between the two glass plates has two unique properties: it is very thin (micrometer scale) and has perfect flatness. Because of this extremely thin air-gap, the air wedge interferometer has been successfully used in experiments with femto-second high-power lasers.
An incident beam of light encounters four boundaries at which the index of refraction of the media changes, causing four reflected beams (or Fresnel reflections ) as shown in. The first reflection occurs when the beam enters the first glass plate. The second reflection occurs when the beam exits the first plate and enters the air wedge, and the third reflection occurs when the beam exits the air wedge and enters the second glass plate. The fourth beam is reflected when it encounters the boundary of the second glass plate. The air wedge angle, between the second and third Fresnel reflections, can be adjusted, causing the reflected light beams to constructively and destructively interfere and create a fringe pattern. To minimize image aberrations of the resulting fringes, the angle plane of the glass wedges has to be placed orthogonal to the angle plane of the air-wedge.

Light Reflections Inside an Air Wedge Interferometer: Beam path inside of air wedge interferometer

Newton’s Rings

Newton’s rings are a series of concentric circles centered at the point of contact between a spherical and a flat surface.

Learning Objectives

Apply Newton’s rings to determine light characteristics of a lens

Key Takeaways

Key Points

• When viewed with monochromatic light, Newton’s rings appear as alternating bright and dark rings; when viewed with white light, a concentric ring pattern of rainbow colors is observed.
• If the path length difference between the two reflected light beams is an odd multiple of the wavelength divided by two, λ/2, the reflected waves will be 180 degrees out of phase and destructively interfere, causing a dark fringe.
• If the path length difference is an even multiple of λ/2, the reflected waves will be in-phase with one another. The constructive interference of the two reflected waves creates a bright fringe.

Key Terms

• wavelength: The length of a single cycle of a wave, as measured by the distance between one peak or trough of a wave and the next; it is often designated in physics as λ, and corresponds to the velocity of the wave divided by its frequency.
• lens: an object, usually made of glass, that focuses or defocuses the light that passes through it
• monochromatic: Describes a beam of light with a single wavelength (i.e., of one specific color or frequency).

Newton’s Rings

In 1717, Isaac Newton first analyzed an interference pattern caused by the reflection of light between a spherical surface and an adjacent flat surface. Although first observed by Robert Hooke in 1664, this pattern is called Newton’s rings, as Newton was the first to analyze and explain the phenomena. Newton’s rings appear as a series of concentric circles centered at the point of contact between the spherical and flat surfaces. When viewed with monochromatic light, Newton’s rings appear as alternating bright and dark rings; when viewed with white light, a concentric ring pattern of rainbow colors is observed. An example of Newton’s rings when viewed with white light is shown in the figure below.

Newton’s Rings in a drop of water: Newton’s rings seen in two plano-convex lenses with their flat surfaces in contact. One surface is slightly convex, creating the rings. In white light, the rings are rainbow-colored, because the different wavelengths of each color interfere at different locations.

The light rings are caused by constructive interference between the light rays reflected from both surfaces, while the dark rings are caused by destructive interference. The outer rings are spaced more closely than the inner ones because the slope of the curved lens surface increases outwards. The radius of the Nth bright ring is given by:
${\text{r}}_{\text{N}}={\left[\left(\text{N}-\frac{1}{2}\lambda \text{R}\right)\right]}^{1/2}$
where N is the bright-ring number, R is the radius of curvature of the lens the light is passing through, and λ is the wavelength of the light passing through the glass.
A spherical lens is placed on top of a flat glass surface. An incident ray of light passes through the curved lens until it comes to the glass-air boundary, at which point it passes from a region of higher refractive index n (the glass) to a region of lower n (air). At this boundary, some light is transmitted into the air, while some light is reflected. The light that is transmitted into the air does not experience a change in phase and travels a a distance, d, before it is reflected at the flat glass surface below. This second air-glass boundary imparts a half-cycle phase shift to the reflected light ray because air has a lower n than the glass. The two reflected light rays now travel in the same direction to be detected. As one gets farther from the point at which the two surfaces touch, the distance d increases because the lens is curving away from the flat surface.

Formation of Interference Fringes: This figure shows how interference fringes form.

If the path length difference between the two reflected light beams is an odd multiple of the wavelength divided by two, λ/2, the reflected waves will be 180 degrees out of phase and destructively interfere, causing a dark fringe. If the path-length difference is an even multiple of λ/2, the reflected waves will be in phase with one another. The constructive interference of the two reflected waves creates a bright fringe.

Diffraction

Huygens’ Principle

Huygens’s Principle states that every point on a wavefront is a source of wavelets, which spread forward at the same speed.

Learning Objectives

Formulate Huygens’s Principle

Key Takeaways

Key Points

• Diffraction is the concept that is explained using Huygens’s Principle, and is defined as the bending of a wave around the edges of an opening or an obstacle.
• This principle can be used to define reflection, as shown in the figure. It can also be used to explain refraction and interference. Anything that experiences this phenomenon is a wave. By applying this theory to light passing through a slit, we can prove it is a wave.
• The principle can be shown with the equation below: s=vt s – distance v – propagation speed t – time Each point on the wavefront emits a wave at speed, v. The emitted waves are semicircular, and occur at t, time later. The new wavefront is tangent to the wavelets.

Key Terms

• diffraction: The bending of a wave around the edges of an opening or an obstacle.

Overview

The Huygens-Fresnel principle states that every point on a wavefront is a source of wavelets. These wavelets spread out in the forward direction, at the same speed as the source wave. The new wavefront is a line tangent to all of the wavelets.

Background

Christiaan Huygens was a Dutch scientist who developed a useful technique for determining how and where waves propagate. In 1678, he proposed that every point that a luminous disturbance touches becomes itself a source of a spherical wave. The sum of the secondary waves (waves that are a result of the disturbance) determines the form of the new wave. shows secondary waves traveling forward from their point of origin. He was able to come up with an explanation of the linear and spherical wave propagation, and derive the laws of reflection and refraction (covered in previous atoms ) using this principle. He could not, however, explain what is commonly known as diffraction effects. Diffraction effects are the deviations from rectilinear propagation that occurs when light encounters edges, screens and apertures. These effects were explained in 1816 by French physicist Augustin-Jean Fresnel.

Straight Wavefront: Huygens’s principle applied to a straight wavefront. Each point on the wavefront emits a semicircular wavelet that moves a distance s=vt. The new wavefront is a line tangent to the wavelets.

Huygens’s Principle

Figure 1 shows a simple example of the Huygens’s Principle of diffraction. The principle can be shown with the equation below:
$\text{s}=\text{vt}$ ,
where s is the distance, v is the propagation speed, and t is time.
Each point on the wavefront emits a wave at speed, v. The emitted waves are semicircular, and occur at t, time later. The new wavefront is tangent to the wavelets. This principle works for all wave types, not just light waves. The principle is helpful in describing reflection, refraction and interference. shows visually how Huygens’s Principle can be used to explain reflection, and shows how it can be applied to refraction.

Huygens’s Refraction: Huygens’s principle applied to a straight wavefront traveling from one medium to another where its speed is less. The ray bends toward the perpendicular, since the wavelets have a lower speed in the second medium.

Reflection: Huygens’s principle applied to a straight wavefront striking a mirror. The wavelets shown were emitted as each point on the wavefront struck the mirror. The tangent to these wavelets shows that the new wavefront has been reflected at an angle equal to the incident angle. The direction of propagation is perpendicular to the wavefront, as shown by the downward-pointing arrows.

Examples

This principle is actually something you have seen or experienced often, but just don’t realize. Although this principle applies to all types of waves, it is easier to explain using sound waves, since sound waves have longer wavelengths. If someone is playing music in their room, with the door closed, you might not be able to hear it while walking past the room. However, if that person where to open their door while playing music, you could hear it not only when directly in front of the door opening, but also on a considerable distance down the hall to either side. is a direct effect of diffraction. When light passes through much smaller openings, called slits, Huygens’s principle shows that light bends similar to the way sound does, just on a much smaller scale. We will examine in later atoms single slit diffraction and double slit diffraction, but for now it is just important that we understand the basic concept of diffraction.

Diffraction

As we explained in the previous paragraph, diffraction is defined as the bending of a wave around the edges of an opening or an obstacle.

Young’s Double Slit Experiment

The double-slit experiment, also called Young’s experiment, shows that matter and energy can display both wave and particle characteristics.

Learning Objectives

Explain why Young’s experiment more credible than Huygens’

Key Takeaways

Key Points

• The wave characteristics of light cause the light to pass through the slits and interfere with each other, producing the light and dark areas on the wall behind the slits. The light that appears on the wall behind the slits is partially absorbed by the wall, a characteristic of a particle.
• Constructive interference occurs when waves interfere with each other crest-to-crest and the waves are exactly in phase with each other. Destructive interference occurs when waves interfere with each other crest-to-trough (peak-to-valley) and are exactly out of phase with each other.
• Each point on the wall has a different distance to each slit; a different number of wavelengths fit in those two paths. If the two path lengths differ by a half a wavelength, the waves will interfere destructively. If the path length differs by a whole wavelength the waves interfere constructively.

Key Terms

• constructive interference: Occurs when waves interfere with each other crest to crest and the waves are exactly in phase with each other.
• destructive interference: Occurs when waves interfere with each other crest to trough (peak to valley) and are exactly out of phase with each other.

The double-slit experiment, also called Young’s experiment, shows that matter and energy can display both wave and particle characteristics. As we discussed in the atom about the Huygens principle, Christiaan Huygens proved in 1628 that light was a wave. But some people disagreed with him, most notably Isaac Newton. Newton felt that color, interference, and diffraction effects needed a better explanation. People did not accept the theory that light was a wave until 1801, when English physicist Thomas Young performed his double-slit experiment. In his experiment, he sent light through two closely spaced vertical slits and observed the resulting pattern on the wall behind them. The pattern that resulted can be seen in.

Young’s Double Slit Experiment: Light is sent through two vertical slits and is diffracted into a pattern of vertical lines spread out horizontally. Without diffraction and interference, the light would simply make two lines on the screen.

Wave-Particle Duality

The wave characteristics of light cause the light to pass through the slits and interfere with itself, producing the light and dark areas on the wall behind the slits. The light that appears on the wall behind the slits is scattered and absorbed by the wall, which is a characteristic of a particle.

Young’s Experiment

Why was Young’s experiment so much more credible than Huygens’? Because, while Huygens’ was correct, he could not demonstrate that light acted as a wave, while the double-slit experiment shows this very clearly. Since light has relatively short wavelengths, to show wave effects it must interact with something small — Young’s small, closely spaced slits worked.
The example in uses two coherent light sources of a single monochromatic wavelength for simplicity. (This means that the light sources were in the same phase. ) The two slits cause the two coherent light sources to interfere with each other either constructively or destructively.

Constructive and Destructive Wave Interference

Constructive wave interference occurs when waves interfere with each other crest-to-crest (peak-to-peak) or trough-to-trough (valley-to-valley) and the waves are exactly in phase with each other. This amplifies the resultant wave. Destructive wave interference occurs when waves interfere with each other crest-to-trough (peak-to-valley) and are exactly out of phase with each other. This cancels out any wave and results in no light. These concepts are shown in. It should be noted that this example uses a single, monochromatic wavelength, which is not common in real life; a more practical example is shown in.

Practical Constructive and Destructive Wave Interference: Double slits produce two coherent sources of waves that interfere. (a) Light spreads out (diffracts) from each slit because the slits are narrow. These waves overlap and interfere constructively (bright lines) and destructively (dark regions). We can only see this if the light falls onto a screen and is scattered into our eyes. (b) Double-slit interference pattern for water waves are nearly identical to that for light. Wave action is greatest in regions of constructive interference and least in regions of destructive interference. (c) When light that has passed through double slits falls on a screen, we see a pattern such as this.

Theoretical Constructive and Destructive Wave Interference: The amplitudes of waves add together. (a) Pure constructive interference is obtained when identical waves are in phase. (b) Pure destructive interference occurs when identical waves are exactly out of phase (shifted by half a wavelength).

The pattern that results from double-slit diffraction is not random, although it may seem that way. Each slit is a different distance from a given point on the wall behind it. For each different distance, a different number of wavelengths fit into that path. The waves all start out in phase (matching crest-to-crest), but depending on the distance of the point on the wall from the slit, they could be in phase at that point and interfere constructively, or they could end up out of phase and interfere with each other destructively.

Diffraction Gratings: X-Ray, Grating, Reflection

Diffraction grating has periodic structure that splits and diffracts light into several beams travelling in different directions.

Learning Objectives

Describe function of the diffraction grating

Key Takeaways

Key Points

• The directions of the diffracted beams depend on the spacing of the grating and the wavelength of the light so that the grating acts as the dispersive element.
• Gratings are commonly used in monochromators, spectrometers, wavelength division multiplexing devices, optical pulse compressing devices, and other optical instruments.
• Diffraction of X-ray is used in crystallography to produce the three-dimensional picture of the density of electrons within the crystal.

Key Terms

• diffraction: The bending of a wave around the edges of an opening or an obstacle.
• interference: An effect caused by the superposition of two systems of waves, such as a distortion on a broadcast signal due to atmospheric or other effects.
• iridescence: The condition or state of being iridescent; exhibition of colors like those of the rainbow; a prismatic play of color.

Diffraction Grating

A diffraction grating is an optical component with a periodic structure that splits and diffracts light into several beams travelling in different directions. The directions of these beams depend on the spacing of the grating and the wavelength of the light so that the grating acts as the dispersive element. Because of this, gratings are often used in monochromators, spectrometers, wavelength division multiplexing devices, optical pulse compressing devices, and many other optical instruments.
A photographic slide with a fine pattern of purple lines forms a complex grating. For practical applications, gratings generally have ridges or rulings on their surface rather than dark lines. Such gratings can be either transmissive or reflective. Gratings which modulate the phase rather than the amplitude of the incident light are also produced, frequently using holography.
Ordinary pressed CD and DVD media are every-day examples of diffraction gratings and can be used to demonstrate the effect by reflecting sunlight off them onto a white wall. (see ). This is a side effect of their manufacture, as one surface of a CD has many small pits in the plastic, arranged in a spiral; that surface has a thin layer of metal applied to make the pits more visible. The structure of a DVD is optically similar, although it may have more than one pitted surface, and all pitted surfaces are inside the disc. In a standard pressed vinyl record when viewed from a low angle perpendicular to the grooves, one can see a similar, but less defined effect to that in a CD/DVD. This is due to viewing angle (less than the critical angle of reflection of the black vinyl) and the path of the light being reflected due to being changed by the grooves, leaving a rainbow relief pattern behind.

Readable Surface of a CD: The readable surface of a Compact Disc includes a spiral track wound tightly enough to cause light to diffract into a full visible spectrum.

Some bird feathers use natural diffraction grating which produce constructive interference, giving the feathers an iridescent effect. Iridescence is the effect where surfaces seem to change color when the angle of illumination is changed. An opal is another example of diffraction grating that reflects the light into different colors.

X-Ray Diffraction

X-ray crystallography is a method of determining the atomic and molecular structure of a crystal, in which the crystalline atoms cause a beam of X-rays to diffract into many specific directions. By measuring the angles and intensities of these diffracted beams, a crystallographer can produce a three-dimensional picture of the density of electrons within the crystal. From this electron density, the mean positions of the atoms in the crystal can be determined, as well as their chemical bonds, their disorder and various other information.
In an X-ray diffraction measurement, a crystal is mounted on a goniometer and gradually rotated while being bombarded with X-rays, producing a diffraction pattern of regularly spaced spots known as reflections (see ). The two-dimensional images taken at different rotations are converted into a three-dimensional model of the density of electrons within the crystal using the mathematical method of Fourier transforms, combined with chemical data known for the sample.

Reflections in Diffraction Patterns: Each dot, called a reflection, in this diffraction pattern forms from the constructive interference of scattered X-rays passing through a crystal. The data can be used to determine the crystalline structure.

Single Slit Diffraction

Single slit diffraction is the phenomenon that occurs when waves pass through a narrow gap and bend, forming an interference pattern.

Learning Objectives

Formulate the Huygens’s Principle

Key Takeaways

Key Points

• The Huygens’s Principle states that every point on a wavefront is a source of wavelets. These wavelets spread out in the forward direction, at the same speed as the source wave. The new wavefront is a line tangent to all of the wavelets.
• If a slit is longer than a single wavelength, think of it instead as a number of point sources spaced evenly across the width of the slit.
• Downstream from a slit that is longer than a single wavelength, the light at any given point is made up of contributions from each of these point sources. The resulting phase differences are caused by the different in path lengths that the contributing portions of the rays traveled from the slit.

Key Terms

• diffraction: The bending of a wave around the edges of an opening or an obstacle.
• monochromatic: Describes a beam of light with a single wavelength (i.e., of one specific color or frequency).

Diffraction

As we explained in a previous atom, diffraction is defined as the bending of a wave around the edges of an opening or obstacle. Diffraction is a phenomenon all wave types can experience. It is explained by the Huygens-Fresnel Principle, and the principal of superposition of waves. The former states that every point on a wavefront is a source of wavelets. These wavelets spread out in the forward direction, at the same speed as the source wave. The new wavefront is a line tangent to all of the wavelets. The superposition principle states that at any point, the net result of multiple stimuli is the sum of all stimuli.

Single Slit Diffraction

In single slit diffraction, the diffraction pattern is determined by the wavelength and by the length of the slit. Figure 1 shows a visualization of this pattern. This is the most simplistic way of using the Huygens-Fresnel Principle, which was covered in a previous atom, and applying it to slit diffraction. But what happens when the slit is NOT the exact (or close to exact) length of a single wave?

Single Slit Diffraction – One Wavelength: Visualization of single slit diffraction when the slit is equal to one wavelength.

A slit that is wider than a single wave will produce interference -like effects downstream from the slit. It is easier to understand by thinking of the slit not as a long slit, but as a number of point sources spaced evenly across the width of the slit. This can be seen in Figure 2.

Single Slit Diffraction – Four Wavelengths: This figure shows single slit diffraction, but the slit is the length of 4 wavelengths.

To examine this effect better, lets consider a single monochromatic wavelength. This will produce a wavefront that is all in the same phase. Downstream from the slit, the light at any given point is made up of contributions from each of these point sources. The resulting phase differences are caused by the different in path lengths that the contributing portions of the rays traveled from the slit.
The variation in wave intensity can be mathematically modeled. From the center of the slit, the diffracting waves propagate radially. The angle of the minimum intensity (θ_min) can be related to wavelength (λ) and the slit’s width (d) such that:
$\text{d}\sin {\theta }_{\text{min}}=\lambda$ .
The intensity (I) of waves at any angle can also be calculated as a relation to slit width, wavelength and intensity of the original waves before passing through the slit:
$\text{I}\left(\theta \right)={\text{I}}_0(\frac{\sin \left(\pi \text{x}\right)}{\pi \text{x}}{)}^2$ ,
where x is equal to:
$\frac{\text{d}}{\lambda }\sin \theta$ .

The Rayleigh Criterion

The Rayleigh criterion determines the separation angle between two light sources which are distinguishable from each other.

Learning Objectives

Explain meaning of the Rayleigh criterion

Key Takeaways

Key Points

• Diffraction plays a large part in the resolution at which we are able to see things. There is a point where two light sources can be so close to each other that we cannot distinguish them apart.
• When two light sources are close to each other, they can be: unresolved (i.e., not able to distinguish one from the other), just resolved (i.e., only able to distinguish them apart from each other), and a little well resolved (i.e., easy to tell apart from one another).
• In order for two light sources to be just resolved, the center of one diffraction pattern must directly overlap with the first minimum of the other diffraction pattern.

Key Terms

• diffraction: The bending of a wave around the edges of an opening or an obstacle.
• resolution: The degree of fineness with which an image can be recorded or produced, often expressed as the number of pixels per unit of length (typically an inch).

Resolution Limits

Along with the diffraction effects that we have discussed in previous atoms, diffraction also limits the detail that we can obtain in images. shows three different circumstances of resolution limits due to diffraction:

Resolution Limits: (a) Monochromatic light passed through a small circular aperture produces this diffraction pattern. (b) Two point light sources that are close to one another produce overlapping images because of diffraction. (c) If they are closer together, they cannot be resolved or distinguished.

• (a) shows a light passing through a small circular aperture. You do not see a sharp circular outline, but a spot with fuzzy edges. This is due to diffraction similar to that through a single slit.
• (b) shows two point sources close together, producing overlapping images. Due to the diffraction, you can just barely distinguish between the two point sources.
• (c) shows two point sources which are so close together that you can no longer distinguish between them.

This effect can be seen with light passing through small apertures or larger apertures. This same effect happens when light passes through our pupils, and this is why the human eye has limited acuity.

Rayleigh Criterion

In the 19th century, Lord Rayleigh invented a criteria for determining when two light sources were distinguishable from each other, or resolved. According to the criteria, two point sources are considered just resolved (just distinguishable enough from each other to recognize two sources) if the center of the diffraction pattern of one is directly overlapped by the first minimum of the diffraction pattern of the other. If the distance is greater between these points, the sources are well resolved (i.e., they are easy to distingiush from each other). If the distance is smaller, they are not resolved (i.e., they cannot be distinguished from each other). The equation to determine this is:
$\theta =1.22\frac{\lambda }{\text{D}}$
θ – angle the objects are separated by, in radian λ – wavelength of light D – aperture diameter. shows this concept visually. This equation also gives the angular spreading of a source of light having a diameter D.

Rayleigh Criterion: (a) This is a graph of intensity of the diffraction pattern for a circular aperture. Note that, similar to a single slit, the central maximum is wider and brighter than those to the sides. (b) Two point objects produce overlapping diffraction patterns. Shown here is the Rayleigh criterion for being just resolvable. The central maximum of one pattern lies on the first minimum of the other.

Thin Film Interference

Thin film interference occurs when incident light waves reflected by the different layers of a thin film interfere and form a new wave.

Learning Objectives

Describe process that leads to the thin film interference

Key Takeaways

Key Points

• When the incident light reaches the thin film, it is partially reflected and passed through to the bottom layer of the film. The difference in refractive indices of the air and film causes the light to change direction and interfere with the reflected portion of the ray when it emerges.
• This interference can be constructive, producing bright colors, or destructive, producing darker colors.
• Interference will be constructive if the optical path difference is equal to an integer multiple of the wavelength of light.

Key Terms

• incident ray: The ray of light that strikes the surface.
• interference: An effect caused by the superposition of two systems of waves, such as a distortion on a broadcast signal due to atmospheric or other effects.
• wavelength: The length of a single cycle of a wave, as measured by the distance between one peak or trough of a wave and the next; it is often designated in physics as λ, and corresponds to the velocity of the wave divided by its frequency.

Thin Film Interference

This is a phenomenon that occurs when incident rays reflected by the upper and lower boundaries of a thin film interfere with one another and form a new wave. A material is considered to be a thin film if its thickness is in the sub-nanometer to micron range, e.g. a soap bubble. Studying the new wave can shed light on properties of the film, such as thickness or refractive index. Interference effects are most prominent when the light interacts with something that has a size similar to its own wavelength. The thickness of a thin film is a few times smaller than the wavelength of the light, λ. Color is indirectly associated with wavelength. The interference ratio of wavelength to size of the object causes the appearance of colors.

Thin Film Interference: In this video I continue with my tutorials on Electromagnetism to Optics which is pitched at university undergraduate level. I have intended for a long time to record videos which describe the transition made from classical electromagnetism to optics. In many respects these videos will cover ‘wave’ optics. I devote much time to discussing the complex exponential representation of waves, Maxwell’s Equations, the wave equation etc.Specifically here, I derive the formula for the optical path difference and the phase difference for a ‘wave’ of light propagating through a thin film. This expression can be used for anti reflective coatings. The phase difference is the product of the optical path differene and the wave vector k.I hope it’s of use!! Thank you for watching and I hope that this matches your requirements.

Thin Film Interference in Oil: Thin film interference can be seen in this oil slick.

Examples of Thin Film Interference

You have probably witnessed thin film interference in your every day life and just not realized it. Whenever you see the bright, rainbow like colors in oil floating in water, shown in, this is thin film interference. The colors that appear in bubbles that kids play with are also a result of thin film interference. Thin film interference can have commercial applications, such as anti- reflection coatings and optical filters.

How it Works

shows a diagram of how thin film interference works. As light strikes the surface of a film it is either transmitted or reflected at the upper surface. Light that is transmitted reaches the bottom surface and may once again be transmitted or reflected. The light reflected from the upper and lower surfaces will interfere. The degree of constructive or destructive interference between the two light waves is dependent upon the difference in their phase. This difference is dependent upon the thickness of the film layer, the refractive index of the film, and the angle of incidence of the original wave on the film. Additionally, a phase shift of 180° or $\pi$ radians may be introduced upon reflection at a boundary depending on the refractive indices of the materials on either side said boundary. This phase shift occurs if the refractive index of the medium the light is travelling through is less than the refractive index of the material it is striking. In other words, if ${\text{n}}_1<{\text{n}}_2$ and the light is travelling from material 1 to material 2, then a phase shift will occur upon reflection. The pattern of light that results from this interference can appear either as light and dark bands or as colorful bands depending upon the source of the incident light.

Light on a Thin Film: Light incident on a thin film. Demonstration of the optical path length difference for light reflected from the upper and lower boundaries.

Interference will be constructive if the optical path difference is equal to an integer multiple of the wavelength of light:
$2{\text{n}}_2\text{d}\cos \left({\theta }_2\right)=\text{m}\lambda$
where $\text{m}$ is the integer, $\text{d}$ is the thickness of the film, and $\lambda$ is the wavelength of light. However, this condition may change if phase shifts occur upon reflection.

Polarization By Passing Light Through Polarizers

Polarization is the attribute that wave oscillations have a definite direction relative to the direction of propagation of the wave.

Learning Objectives

Discuss polarization of electromagnetic waves

Key Takeaways

Key Points

• The direction of polarization is parallel to the direction of the electric field associated with an electromagnetic wave, which includes light waves.
• By employing the polarization properties of waves, companies like Polariod how been able to produce materials that filter out unwanted waves of light, and minimize light intensity.
• Polarization materials can be oriented at different angles to produce different effects. The new intensity of light after passing through these materials can be found using the following formula: I=I₀cos2θ.

Key Terms

• oscillate: To swing back and forth, especially if with a regular rhythm.

Definition of Polarization
As we discusses in previous atoms, light waves are a type of electromagnetic waves, in the visible spectrum. These electromagnetic (EM) waves are transverse waves. Figure 1 demonstrates that a transverse wave is one oscillates perpendicular to the direction of the energy transfer. If the wave is traveling from left to right, it is oscillating up and down. Polarization is the property of waves that allow them to oscillate in more than one direction, but that direction is relative to that of the direction the wave is traveling in. For an EM wave, the direction of polarization is the direction parallel to the electric field. In Figure 2 you can see that the EM and magnetic fields are perpendicular to the path of travel. Since the direction of polarization is parallel to the electric field, you can consider the blue arrows to be the direction of polarization.

Figure 2: An EM wave, such as light, is a transverse wave. The electric and magnetic fields are perpendicular to the direction of propagation.

Figure 1: Transverse Waves

How it Works
Now, to examine the effects of passing light through a polarizer, lets look at Figure 3. shows a stationary vertical slot, which will act as a polarizer, and two waves traveling in the same direction, but one is oscillating vertically,a, (and therefore vertically polarized), and the other, b, horizontally. What happens to these waves as they pass through the polarizer?When wave a, the vertically oscillating wave is passed through the vertically polarized slot, nothing happens. The wave passes through untouched or manipulated. When wave b, the horizontally oscillating wave is passed through, the vertically polarized slot blocks the wave, and it is not passed through at all. Now that we understand the concept of polarization, and how it works, how can we apply this to make it useful? Look at Figure 4. The image on the left is full of glare, which makes it hard to see what we are looking at. The image on the right was taken with a polarized lens, so you only see the image, and none of the annoying glare. How this works is diagramed in Figure 5. Many light sources are unpolarized, and are comprised of many waves in all possible directions. Polarized lenses only allow one direction of light to pass through, minimizing the unwanted aspects of the light rays, such as glare. Simply, passing light through a polarized material changes the intensity of the light.

Figure 5: A polarizing filter has a polarization axis that acts as a slit passing through electric fields parallel to its direction. The direction of polarization of an EM wave is defined to be the direction of its electric field.

Figure 4: These two photographs of a river show the effect of a polarizing filter in reducing glare in light reflected from the surface of water. Part (b) of this figure was taken with a polarizing filter and part (a) was not. As a result, the reflection of clouds and sky observed in part (a) is not observed in part (b). Polarizing sunglasses are particularly useful on snow and water. (credit: Amithshs, Wikimedia Commons)

Figure 3: Example of passing light through a polarizer

Intensity
Lets call the angle between the direction of polarization and the axis of the polarization filter θ. The original intensity of the light before it is passed through a filter is denoted by I₀. In order to find the new intensity of light after traveling through the material is shown by the following equation:
I=I₀cos2θ
If you pass light through two polarizing filters, you will get varied effects of polarization. If the two filters are oriented exactly perpendicular to each other, no light will pass through at all. If they are exactly parallel to each other, there will be no additional affect from the additional filter.

Polarization By Scattering and Reflecting

Unpolarized light can be polarized artificially, as well as by natural phenomenon like reflection and scattering.

Learning Objectives

Calculate angle of reflection of complete polarization from indices of refraction

Key Takeaways

Key Points

• When unpolarized light hits a reflective surface, the vertically polarized aspects are refracted into the surface. The horizontally polarized aspects are reflected off the surface and the light is now perceived as partially polarized.
• When light is reflected, there is an angle at which this light is completely polarized. This is called Brewster’s angle, after the Scottish physicst who discovered the law.
• Unpolarized light can also become polarized when it is scattered in air (also known as Rayleigh scattering). This occurs due to the fact that the EM waves cause the electron in air to vibrate, producing radiation and causing polarization of the light.

Key Terms

• index of refraction: For a material, the ratio of the speed of light in vacuum to that in the material.
• polarization: The production of polarized light; the direction in which the electric field of an electromagnetic wave points.

Polarization by Reflection

In the previous atom we discussed how polarized lenses work. In the case of polarized sunglasses, for example, when you look through them, reflected light is not entirely filtered out; reflected light can be slightly polarized by the reflection process (as shown in ). Most light sources produce unpolarized light. When light hits a reflective surface, the vertically polarized aspects of that light are refracted at that surface. The reflected light is more horizontally polarized. To better remember this, we can think of light as an arrow and the reflective surface as a target. If the arrow hits the target perpendicularly (vertically polarized), it is going to stick in the target (be refracted into the surface). If the arrow hits the target on its side (horizontally polarized) then it will bounce right off (be reflected).

Polarization by Reflection: Unpolarized light has equal amounts of vertical and horizontal polarization. After interaction with a surface, the vertical components are preferentially absorbed or refracted, leaving the reflected light more horizontally polarized. This is akin to arrows striking on their sides bouncing off, whereas arrows striking on their tips go into the surface.

Since the light is split into two, and part of it is refracted, the amount of polarization to the reflected light depends on the index of refraction of the reflective surface. We can use the following equation to determine the angle of reflection at which light will be completely polarized:
$\tan {\theta }_{\text{b}}=\frac{{\text{n}}_2}{{\text{n}}_1}$
where: θ_b = angle of reflection of complete polarization (also known as Brewster’s angle); n₁ = index of refraction of medium in which reflected light will travel; and n₂ = index of refraction of medium by which light is reflected.

Polarization by Scattering

Just as unpolarized light can be partially polarized by reflecting, it can also be polarized by scattering (also known as Rayleigh scattering; illustrated in ). Since light waves are electromagnetic (EM) waves (and EM waves are transverse waves) they will vibrate the electrons of air molecules perpendicular to the direction in which they are traveling. The electrons then produce radiation (acting like small antennae) that is polarized perpendicular to the direction of the ray. The light parallel to the original ray has no polarization. The light perpendicular to the original ray is completely polarized. In all other directions, the light scattered by air will be partially polarized.

Polarization by Scattering: Also known as Rayleigh scattering. Unpolarized light scattering from air molecules shakes their electrons perpendicular to the direction of the original ray. The scattered light therefore has a polarization perpendicular to the original direction and none parallel to the original direction.

Scattering of Light by the Atmosphere

Rayleigh scattering describes the air’s gas molecules scattering light as it enters the atmosphere; it also describes why the sky is blue.

Learning Objectives

Describe wave-particle relationship that leads to Rayleigh scattering and apply it to explain common phenomena

Key Takeaways

Key Points

• The phenomenon of light scattering is called Rayleigh scattering; it can happen to any electromagnetic waves. It only occurs when waves encounter particles that are much smaller than the wavelengths of the waves.
• The amount that light is scattered is inversely proportional to the fourth power of the light wavelength. For this reason, shorter-wavelength light like greens and blues scatter more easily than longer wavelengths like yellows and reds.
• As you look closer to the sky’s light source, the sun, the light is scattered less and less because the angle between the sun and the scattering particles approaches 90 degrees. This is why the sun has a yellowish color when we look at it from Earth while the rest of the sky appears blue.
• In outer space, where there is no atmosphere and therefore no particles to scatter the light, the sky appears black and the sun appears white.
• During a sunset, the light must pass through an increased volume of air. This increases the scattering effect, causing the light in the direct path of the observer to appear orange rather than blue.

Key Terms

• electromagnetic radiation: radiation (quantized as photons) consisting of oscillating electric and magnetic fields oriented perpendicularly to each other, moving through space
• polarizability: The relative tendency of a system of electric charges to become polarized in the presence of an external electric field

Rayleigh Scattering

Rayleigh scattering is the elastic scattering of waves by particles that are much smaller than the wavelengths of those waves. The particles that scatter the light also need to have a refractive index close to 1. This law applies to all electromagnetic radiation, but in this atom we are going to focus specifically on why the atmosphere scatters the visible spectrum of electromagnetic waves, also known as visible light. In this case, the light is scattered by the gas molecules of the atmosphere, and the refractive index of air is 1.
Rayleigh scattering is due to the polarizability of an individual molecule. This polarity describes how much the electric charges in the molecule will vibrate in an electric field. The formula to calculate the intensity of the scattering for a single particle is as follows:
$\text{I}={\text{I}}_0\frac{8{\pi }^4{\alpha }^2}{{\lambda }^4{\text{R}}^2}(1+{\cos }^2\theta )$
where I is the resulting intensity, I₀ is the original intensity, α is the polarizability, λ is the wavelength, R is the distance to the particle, and θ is the scattering angle.
While you will probably not need to use this formula, it is important to understand that scattering has a strong dependence on wavelength. From the formula, we can see that a shorter wavelength will be scattered more strongly than a longer one.( The longer the wavelength, the larger the denominator, and from algebra we know that a larger denominator in a fraction means a smaller number. )

Why is the Sky Blue?

As we just learned, light scattering is inversely proportional to the fourth power of the light wavelength. So, the shorter the wavelength, the more it will get scattered. Since green and blue have relatively short wavelengths, you see a mixture of these colors in the sky, and the sky appears to be blue. When you look closer and closer to the sun, the light is not being scattered because it is approaching a 90-degree angle with the scattering particles. Since the light is being scattered less and less, you see the longer wavelengths, like red and yellow. This is why the sun appears to be a light yellow color.

Why are Sunsets Colorful?

shows a sunset. We know why the sky is blue, but why are there all those colors in a sunset? The reddening that occurs near the horizon is because the light has to pass through a significantly higher volume of air than when the sun is high in the sky. This increases the Rayleigh scattering effect and removes all blue light from the direct path of the observer. The remaining unscattered light is of longer wavelengths and so appears orange.

Sunset: A gradient of colors in the sky during sunset

Dispersion of the Visible Spectrum

Dispersion is the spreading of white light into its full spectrum of wavelengths; this phenomenon can be observed in prisms and rainbows.

Learning Objectives

Describe process of dispersion

Key Takeaways

Key Points

• Dispersion is a side effect of the law of refraction. As refraction angles depend on wavelength, when light enters a medium with a different index of refraction, it can be dispersed, like a prism.
• The dispersion of white light can often cause the refracted light to be observed is in order of either increasing or decreasing wavelength, causing a rainbow effect.
• As you can see from the visible spectrum, there are some colors that the brain perceives that are not included. This is because some colors are a mixture of different wavelengths, like pink and magenta.

Key Terms

• refraction: Changing of a light ray’s direction when it passes through variations in matter.
• reflection: the property of a propagated wave being thrown back from a surface (such as a mirror)
• dispersion: The separation of visible light by refraction or diffraction.

The Visible Spectrum

Within the electromagnetic spectrum, there is only a portion that is visible to the human eye. Visible light is the range of wavelengths of electromagnetic radiation that humans can see. For a typical human eye, this ranges from 390 nm to 750 nm. shows this range and the colors associated with it:

The Visible Spectrum: Visible Spectrum, represented linearly

• Violet: 380-450 nm
• Blue: 450-495 nm
• Green: 495-570 nm
• Yellow: 570-590 nm
• Orange: 590-620 nm
• Red: 620-750 nm

As you can see from, these are the colors of a rainbow and it is no coincidence.

Dispersion

Dispersion is the spreading of white light into its full spectrum of wavelengths. How does this happen? The index of refraction is different for every medium that light travels through, as we learned in previous atoms. When a light ray enters a medium with a different index of refraction, the light is dispersed, as shown in with a prism. When white light enters the prism, it spreads. Since the index of refraction varies with wavelength, the light refracts at different angles as it exits, causing the exiting light rays to appear as a rainbow, or as a sequence of decreasing wavelengths, from red to violet.

Light and a Glass Prism: (a) A pure wavelength of light falls onto a prism and is refracted at both surfaces. (b) White light is dispersed by the prism (shown exaggerated). Since the index of refraction varies with wavelength, the angles of refraction vary with wavelength. A sequence of red to violet is produced, because the index of refraction increases steadily with decreasing wavelength.

This same principle can be applied to rainbows. Refer to. Rainbows are not only caused by refraction, like prisms, but also reflection. Light enters a drop of water and is reflected from the back of the droplet. The light is refracted once as it enters the drop, and again as is exits the drop. In water, the refractive index varies with wavelength, so the light is dispersed.

Light and a Water Droplet: Part of the light falling on this water drop enters and is reflected from the back of the drop. This light is refracted and dispersed both as it enters and as it leaves the drop.

Applications of Wave Optics

Enhancement of Microscopy

Microscopy helps us view objects that cannot be seen with the naked eye.

Learning Objectives

Compare optical and electron microscopy

Key Takeaways

Key Points

• In optical microscopy, light reflected from an object passes through the microscope’s lenses; this magnifies the light. The resultant, magnified image is then seen by the eye. This technique has many limitations but can be enhanced in various ways to create more contrast.
• The transmission electron microscope (TEM) sends an electron beam through a thin slice of a specimen. The electron is then transmitted onto photographic paper or a screen. Since electron beams have a much smaller wavelength than traditional light, the resolution of this image is much higher.
• The scanning electron microscope (SEM) shows details on the surface of a specimen and produces a three-dimensional view by scanning the specimen.

Key Terms

• microscopy: using microscopes to view objects that cannot be seen with the naked eye
• contrast: A difference in lightness, brightness and/or hue between two colors that makes them more or less distinguishable

Microscopes are used to view objects that cannot be seen with the naked eye. In this section we will discuss both optical and electron microscopy.

Optical Microscopy

You have probably used an optical microscope in a high school science class. In optical microscopy, light reflected from an object passes through the microscope’s lenses; this magnifies the light. The resultant, magnified image is then seen by the eye. Although this type of microscopy has many limitations, there are several techniques that use properties of light and optics to enhance the magnified image:

• Bright field: This technique increases the contrast by illuminating the surface on which the objects sit from below.
• Oblique illumination: This technique illuminates the object from the side, giving it a three-dimensional appearance and highlighting features that would otherwise not be visible.
• Dark field: This technique is good for improving the contrast of transparent objects. A carefully aligned light source minimizes the un-scattered light entering the object plane and so only collects the light that is scattered by the object itself.
• Dispersion staining: This results in a colored image of a colorless object; it does not actually require that the object be stained.
• Phase contrast: This uses the refractive index of an object to show differences in optical density as a difference in contrast. provides a demonstration of this technique.

Electron Microscopy

Electron microscopes use electron beams to achieve higher resolutions than are possible in optical microscopy. Two kinds of electron microscopes are:

• Transmission electron microscope (TEM): The TEM sends an electron beam through a thin slice of a specimen. The electron interacts with the specimen and is then transmitted onto photographic paper or a screen. Since electron beams have a much smaller wavelength than traditional light, the resolution of the resulting image is much higher.
• Scanning electron microscope (SEM): The SEM shows details on the surface of a specimen and produces a three-dimensional view by scanning the specimen. shows an SEM image of pollen.

The Spectrometer

A spectrometer uses properties of light to identify atoms by measuring wavelength and frequency, which are functions of radiated energy.

Learning Objectives

Compare design and function of early and modern spectrometers

Key Takeaways

Key Points

• The source is placed in front of a mirror, which reflects the light emitted from that object onto a diffraction grating. This grating then disperses the emitted light to anther mirror which spreads the different resultant wavelengths and reflects them onto a detector which records the findings.
• Early forms of spectrometers were simple prisms, but modern spectrometers are automated by a computer and can record a much broader range of frequencies.
• Spectrometers are used in spectroscopy. Spectroscopy studies the interaction between matter and radiated energy. This radiated energy is a function of wavelength and frequency. Every type of atom has its own frequency.

Key Terms

• incandescence: Incandescence is the emission of light (visible electromagnetic radiation) from a hot body as a result of its temperature.

The Spectrometer

A spectrometer is an instrument used to intensely measure light over a specific portion of the electromagnetic spectrum, to identify materials. The instrument produces lines, much like those produced from diffraction grating as covered in a previous atom, and then measures the wavelengths and intensities of those lines.
shows a diagram of how a spectrometer works. The source is placed in front of a mirror, which reflects the light emitted from that object onto a diffraction grating. This grating then disperses the emitted light to anther mirror which spreads the different resultant wavelengths and reflects them onto a detector which records the findings. This type of instrument is used in spectroscopy.

Spectrometer Diagram: This diagram shows the light pathways in a spectrometer.

Spectroscopy

Spectroscopy studies the interaction between matter and radiated energy. This radiated energy is a function of wavelength and frequency. Every type of atom has its own frequency. When the spectrometer produces a reading, the observer can then use spectroscopy to identify the atoms and therefore molecules that make up that object.

Spectroscopes

Spectroscopes are used in a variety of fields, such as astronomy and chemistry. They use a diffraction grating, movable slit, and a photodetector. All of these elements are controlled by a computer, which records the findings. A material is heated to incandescence and it emits a light that is characteristic of its atomic makeup. Each atom has its own spectroscopic ‘fingerprint’. In you can see a very simple spectroscope based on a prism. As another example, Sodium produces a double yellow band.

A simple spectroscope: A very simple spectroscope based on a prism

The Michelson Interferometer

The Michelson interferometer is the most common configuration for optical interferometry.

Learning Objectives

Explain how the Michelson interferometer works

Key Takeaways

Key Points

• Interferometry refers to techniques that use superimposed waves to extract information about the waves.
• Michelson interferometer works by splitting a beam of light into two paths, bouncing them back and recombining them to create an interference pattern. To create interference fringes on a detector, the paths may be different lengths or composed of different materials.
• The best known application of the Michelson interferometer is the Michelson-Morley experiment, the unexpected null result of which was an inspiration for special relativity.

Key Terms

• special relativity: A theory that (neglecting the effects of gravity) reconciles the principle of relativity with the observation that the speed of light is constant in all frames of reference.
• superimposed: Positioned on or above something else, especially in layers
• interference: An effect caused by the superposition of two systems of waves, such as a distortion on a broadcast signal due to atmospheric or other effects.

Interferometry

Before we can discuss the Michelson Interferometer, it is important we first understand interferometry—which refers to techniques that use superimposed waves to extract information about the waves. More simply, it uses the interference these waves experience to make accurate measurements of the waves. It is used in many areas of science, such as astronomy, engineering, oceanography, physics, and fiber optics.
Popular applications of interferometry in industry include the measurement of small displacements, refractive index changes, and surface irregularities. As shown in previous atoms, when two waves with the same frequency combine, the resulting pattern is determined by the phase difference between the two. Constructive interference occurs when the waves are in phase, and destructive interference occurs when they are out of phase. Interferometry uses this principle to combine waves and study the resulting wave in order to obtain information about the original state of the waves.

The Michelson Interferometer

The most common tool in interferometry, the Michelson Interferometer, shown in Figure 1, was invented by Albert Abraham Michelson, the first American to win a Nobel Prize for science. The interferometer works by splitting a beam of light into two paths, bouncing them back and then recombining them to create an interference pattern. To create interference fringes on a detector (see Figure 2 ), the paths may be different lengths or composed of different materials.

Fringes in a Michelson Interferometer: Colored and monochromatic fringes in a Michelson interferometer: (a) White light fringes where the two beams differ in the number of phase inversions; (b) White light fringes where the two beams have experienced the same number of phase inversions; and (c) Fringe pattern using monochromatic light (sodium D lines).

Michelson Interferometer: A Michelson Interferometer.

Figure 3 shows a diagram of how a Michelson Interferometer works. M₁ and M₂ are two highly polished mirrors, S is the light source, M is a half silvered mirror that acts as a beam splitter when the light hits the surface, and C is a point on M that is partially reflective. When the beam S hits this point on M it is split into two beams. One beam is reflected in the direction of A and the other is transmitted through the surface of M to the point B. A and B are both points on the highly polished (and therefore reflective) mirrors M₁ and M₂. When the beams hit these points, they are then reflected back to point C’, where they recombine to produce an interference pattern. At point E, the interference pattern produced at point C’ is visible to an observer.

Figure 3: This diagram of a Michelson Interferometer shows the path that the light waves travels in the instrument.

Applications

The Michelson Interferometer has been used for the detection of gravitational waves, as a tunable narrow band filter, and as the core of Fourier transform spectroscopy. It has played an important role in studies of the upper atmosphere, revealing temperatures and winds (employing both space-borne and ground-based instruments) by measuring the Doppler widths and shifts in the spectra of airglow and aurora. The best known application of the Michelson Interferometer is the Michelson-Morley experiment—a failed attempt to demonstrate the effect of the hypothetical ” aether wind” on the speed of light. Their experiment left theories of light based on the existence of a luminiferous aether without experimental support, and served ultimately as an inspiration for special relativity.

LCDs

Liquid crystal displays use liquid crystals which do not emit light, but use the light modulating properties of the crystals.

Learning Objectives

Explain how liquid crystal displays produce images and discuss their benefits and deficiencies

Key Takeaways

Key Points

• LCDs use an electric field to arrange the liquid crystals into the desired pattern, and then pass light through these layers to produce an image on the screen.
• LCDs can be used to display an arbitrary image made up of tiny fixed pixels, or can be used to display a fixed image, as in on a digital clock.
• A twisted nematic display is the most common LCD in use. This type of display is on calculators, digital watches, and clocks. When no electric field is applied, the molecules are twisted, and let some light through. When the field is applied, they untwist, blocking the light and are seen as black.

Key Terms

• LCD: a liquid crystal display.
• helical: In the shape of a helix, twist.
• nematic: Describing the structure of some liquid crystals whose molecules align in loose parallel lines.

LCDs

LCD stands for a liquid crystal display. The liquid crystals themselves do not emit light, but the display uses the light modulating properties of the crystals. LCDs can be used to display arbitrary images, such as in a computer monitor or television, by using a large number of very small pixels, or they can be used to display fixed images, like a digital clock, such as in.

Digital Clock: A digital clock which uses LCD to either hide or display fixed images.

Unlike the newer cathode ray tube (CRT) and plasma displays, LCDS do not use phosphors. For this reason they do not suffer image burn-in. They do however suffer image persistence. Image burn-in occurs when an image is displayed so many times, or for so long, that an outline of image can be seen even when the display is turned off. Image persistence is similar, but the outline fades away shortly after the display is turned off and is not permanent.
LCD displays are made up of numerous layers. A typical layer is diagrammed in. Each pixel of an LCD consists of a layer of molecules aligned between two transparent electrodes and two polarizing films, and the actual liquid crystals are between these polarizing filters. The light passes through the first filter, and is blocked by the second. The electrodes are used to align the crystals in a particular direction, which produces the image seen on the screen. The crystals do not emit any light, but rather give the light a specific shape to be emitted in.

Layers of LCD Displays: Polarizing filter film with a vertical axis to polarize light as it enters.Glass substrate with ITO electrodes. The shapes of these electrodes will determine the shapes that will appear when the LCD is turned on. Vertical ridges etched on the surface are smooth.Twisted nematic liquid crystal.Glass substrate with common electrode film (ITO) with horizontal ridges to line up with the horizontal filter.Polarizing filter film with a horizontal axis to block/pass light.Reflective surface to send light back to viewer. (In a backlit LCD, this layer is replaced with a light source. )

Twisted Nematic Devices

The twisted nematic device is the most common LCD application. When no electric field is being applied, the surface alignment directions at the electrodes are perpendicular to each other. The molecules arrange themselves in a helical structure (twisted structure). Some light is able to pass through and some is not, so the result is that the screen appears gray. When the electric field is applied, the crystals in the center layer untwist, and the light is completely blocked from passing through and those pixels will appear black.

Using Interference to Read CDs and DVDs

Optical discs are digital storing media read in an optical disc drive using laser beam.

Learning Objectives

Explain how information is stored on the optical disks

Key Takeaways

Key Points

• The iridescent layer of the disc is imprinted with tiny pits and lands. Pits scatter light when illuminated, and produce a reading of 0; lands reflect light back and produce a reading of 1.
• The optical disc drive records the 0 and 1 readings and translates them into binary data which is used to relay whatever information is recorded on the disc.
• Rainbow pattern on the back of a CD is due diffraction of the reflected light by pits.

Key Terms

• binary data: Data which can take on only two possible values, traditionally termed 0 and 1.
• pit: An imprint on an optical disc that scatters light when illuminated.
• land: A flat area on an optical disc that reflects light when illuminated.

Overview

Compact disks (CDs) and digital video disks (DVDs) are examples of optical discs. They are read in an optical disc drive which directs a laser beam at the disc. The reader then detects whether the beam has been reflected or scattered.

Function of Digital Discs

Optical discs are digital storing media. They can store music, files, movies, pictures etc.. These discs are flat, usually made of aluminum, and have microscopic pits and lands on one of the flat surfaces (as shown in ). The information on these discs are read by a computer in the form of binary data. First, a laser beam is shot at the disc. If the beam hits a land, it gets reflected back and is recorded as a value of 1. If the beam hits a pit, it gets scattered and is recorded as a value of zero.

Early Version of an Optical Disc: In this early version of an optical disc, you can see the pits and lands which either reflect back light or scatter it.

These microscopic pits and lands cover the entire surface of the disc in a spiral path, starting in the center and working its way outward. The data is stored either by a stamping machine or laser and is read when the data is illuminated by a laser diode in the disc drive. The disc spins at a faster speed when it is being read in the center track, and slower for an outer track. This is because the center tracks are smaller in circumference and therefore can be read quicker.
These pits also act as slits and cause the light to be diffracted as it is reflected back, which causes an iridescent effect. This explains the rainbow pattern that you see on the back of a CD, as shown in.

Compact Disc: The bottom surface of a compact disc showing characteristic iridescence.

Introduction

Gallilean-Newtonian Relativity

Galilean invariance or Galilean relativity states that the laws of motion are the same in all inertial (or non-accelerating) frames.

Learning Objectives

Explain why the Galilean invariance didn’t work in Maxwell’s equations

Key Takeaways

Key Points

• Galilean invariance states that Newton’s laws hold in all inertial frames.
• Newtonian mechanics assumes that there exists an absolute space and that time is universal.
• Albert Einstein’s central insight in formulating special relativity was that, for full consistency with electromagnetism, mechanics must also be revised such that Lorentz invariance (introduced later) replaces Galilean invariance.

Key Terms

• Lorentz invariance: First introduced by Lorentz in an effort to explain how the speed of light was observed to be independent of the reference frame, and to understand the symmetries of the laws of electromagnetism.
• absolute space: A concept introduced by Newton that assumes space remains always similar and immovable.

Galilean invariance or Galilean relativity states that the laws of motion are the same in all inertial (or non-accelerating) frames. Galileo Galilei first described this principle in 1632 using the example of a ship travelling at constant velocity, without rocking, on a smooth sea; any observer doing experiments below the deck would not be able to tell whether the ship was moving or stationary. The fact that the Earth orbits around the sun at approximately 30 km/s offers a somewhat more dramatic example, though it is technically not an inertial reference frame.
Specifically, the term Galilean invariance today usually refers to this principle as applied to Newtonian mechanics—that is, Newton’s laws hold in all inertial frames. In this context it is sometimes called Newtonian relativity. Among the axioms from Newton’s theory are:

• There exists an absolute space in which Newton’s laws are true. An inertial frame is a reference frame in relative uniform motion to absolute space.
• All inertial frames share a universal (or absolute) time.

Derivation
Galilean relativity can be shown as follows. Consider two inertial frames S and S’. A physical event in S will have position coordinates r = (x, y, z) and time t; similarly for S’. By the second axiom above, one can synchronize the clock in the two frames and assume t = t‘. Suppose S’ is in relative uniform motion to S with velocity v. Consider a point object whose position is given by r = r(t) in S. We see that
${\text{r}}^{\prime }\left(\text{t}\right)=\text{r}\left(\text{t}\right)-\text{vt}.$
This transformation of variables between two inertial frames is called Galilean transformation. Now, the velocity of the particle is given by the time derivative of the position:

Galilean Invariance: Newtonian mechanics is invariant under a Galilean transformation between observation frames (shown). This is called Galilean invariance.

${\text{u}}^{\prime }\left(\text{t}\right)=\frac{\text{d}}{\text{dt}}{\text{r}}^{\prime }\left(\text{t}\right)=\frac{\text{d}}{\text{dt}}\text{r}\left(\text{t}\right)-\text{v}=\text{u}\left(\text{t}\right)-\text{v}.$
Another differentiation gives the acceleration in the two frames:
${\text{a}}^{\prime }\left(\text{t}\right)=\frac{\text{d}}{\text{dt}}{\text{u}}^{\prime }\left(\text{t}\right)=\frac{\text{d}}{\text{dt}}\text{u}\left(\text{t}\right)-0=\text{a}\left(\text{t}\right).$
It is this simple but crucial result that implies Galilean relativity. Assuming that mass is invariant in all inertial frames, the above equation shows that Newton’s laws of mechanics, if valid in one frame, must hold for all frames. But it is assumed to hold in absolute space, therefore Galilean relativity holds.

Issues

Both Newtonian mechanics and the Maxwell’s equations were well established by the end of the 19^th century. The puzzle lied in the fact that the Galilean invariance didn’t work in Maxwell’s equations. That is, unlike Newtonian mechanics, Maxwell’s equations are not invariant under a Galilean transformation. Albert Einstein’s central insight in formulating special relativity was that, for full consistency with electromagnetism, mechanics must also be revised, such that Lorentz invariance (introduced later) replaces Galilean invariance. At the low relative velocities characteristic of everyday life, Lorentz invariance and Galilean invariance are nearly the same, but for relative velocities close to that of light they are very different.

Einstein’s Postulates

Special relativity is based on Einstein’s two postulates: the Principle of Relativity and the Principle of Invariant Light Speed.

Learning Objectives

Identify two postulates forming the foundation of special relativity

Key Takeaways

Key Points

• The special theory of relativity explores the consequences of the invariance of c with the assumption that the laws of physics are the same in all inertial frames of reference.
• Einstein’s first postulate says that the laws of physics are the same and can be stated in their simplest form in all inertial frames of reference. It means that there is no preferred frame and, therefore, no absolute motion.
• The speed of light c is a constant, independent of the relative motion of the source and observer.

Key Terms

• Maxwell’s equations: A set of equations describing how electric and magnetic fields are generated and altered by each other and by charges and currents.
• luminiferous aether: Light-bearing aether; the postulated medium for the propagation of light.

In the late 19^th century, the Newtonian mechanics was considered to be valid in all inertial frames of reference, which are moving at a constant relative velocity with respect to each other. (See our previous lesson on “Galilean-Newtonian Relativity. “) One issue, however, was that another well-established theory, the laws of electricity and magnetism represented by Maxwell’s equations, was not “invariant” under Galilean transformation—meaning that Maxwell’s equations don’t maintain the same forms for different inertial frames. In his “Special Theory of Relativity,” Einstein resolved the puzzle and broadened the scope of the invariance to extend the validity of all physical laws, including electromagnetic theory, to all inertial frames of reference.

Albert Einstein: Albert Einstein, a true pioneer of modern physics. His work on relativity, gavity, quantum mechanics, and statistical physics revolutionized physics.

Einstein’s Postulates

With two deceptively simple postulates and a careful consideration of how measurements are made, Einstein produced the theory of special relativity.
1. The Principle of Relativity: The laws of physics are the same and can be stated in their simplest form in all inertial frames of reference.
This postulate relates to reference frames. It says that there is no preferred frame and, therefore, no absolute motion.
2. The Principle of Invariant Light Speed: The speed of light c is a constant, independent of the relative motion of the source and observer.
The laws of electricity and magnetism predict that light travels at c = 2.998×10⁸ m/s in a vacuum, but they do not specify the frame of reference in which light has this speed. Physicists assumed that there exists a stationary medium for the propagation of light, which they called ” luminiferous aether. ” In 1887, Michelson and Morley attempted to detect the relative motion of the Earth through the stationary luminiferous aether, but their negative results implied the speed of light c is independent of the motion of the source relative to the observer. Einstein accepted the result of the experiment and incorporated it in his theory of relativity.
This postulate might sound easy to accept, but it is rather counterintuitive. Imagine that you can throw a baseball at a speed v (relative to you). If you are on a train moving at a speed V and throw a ball in the direction of the train’s movement, the baseball will travel at a speed v+V for an observer stationary on the ground.
Now, instead of a baseball, let’s say you have a laser pointer. You turn on the laser pointer while you are on a moving train. What would be the speed of light from the laser pointer for a stationary observer on the ground? Our intuition says that it should be c+V. However, Einstein says that it should be only c!

The Speed of Light

The speed of light in vacuum is a universal physical constant crucial to many areas of physics.

Learning Objectives

Discuss the invariance of the speed of light and identify the value of that speed in vacuum

Key Takeaways

Key Points

• The speed at which light waves propagate in vacuum is independent both of the motion of the wave source and of the inertial frame of reference of the observer. This invariance of the speed of light was postulated by Einstein in 1905 in his work on special relativity.
• The value of the speed of light is 299,792,458 m/s; this is a precise known value because the length of the meter is itself derived from this constant and the international standard for time.
• c is the maximum speed at which all energy, matter, and information in the universe can travel.

Key Terms

• special relativity: A theory that (neglecting the effects of gravity) reconciles the principle of relativity with the observation that the speed of light is constant in all frames of reference.
• luminiferous aether: Light-bearing aether; the postulated medium for the propagation of light.
• Lorentz transformation: a transformation relating the spacetime coordinates of one frame of reference to another in special relativity

The speed of light in vacuum, commonly denoted c, is a universal physical constant that is crucial to many areas of physics. Its value is 299,792,458 m/s; this is a precise known value because the length of the meter is itself derived from this constant and the international standard for time. According to special relativity, c is the maximum speed at which all energy, matter, and information in the universe can travel. It is the speed at which all massless particles and associated fields (including electromagnetic radiation such as light) travel in vacuum. It is also the speed of gravity (i.e., of gravitational waves) predicted by current theories. Such particles and waves (including light) travel at c regardless of the motion of the source or the inertial frame of reference of the observer. In the theory of relativity, c interrelates space and time in the Lorentz transformation; it also appears in the famous equation of mass-energy equivalence: E = mc².

Sunlight’s Flight to Earth: Sunlight takes about 8 minutes and 19 seconds to reach the earth (based on the average distance between the sun and the earth)

First Measurement

The first quantitative estimate of the speed of light was made in 1676 by Rømer. From the observation that the periods of Jupiter’s innermost moon (Io) appeared to be shorter when Earth was approaching Jupiter than when it was moving away, he concluded that light travels at a finite speed. He estimated that it takes light 22 minutes to cross the diameter of Earth’s orbit. Christiaan Huygens combined this estimate with an estimate for the diameter of the Earth’s orbit to obtain an estimate of the speed of light of 220,000 km/s, 26 percent lower than the actual value.

Fundamental Role in Physics

The speed at which light waves propagate in vacuum is independent both of the motion of the wave source and of the inertial frame of reference of the observer. This invariance of the speed of light was postulated by Einstein in 1905 after being motivated by Maxwell’s theory of electromagnetism and the lack of evidence for “luminiferous aether”; it has since been consistently confirmed by many experiments.

Consequences of Special Relativity

Simultaneity

The relativity of simultaneity is the concept that simultaneity is not absolute, but depends on the observer’s reference frame.

Learning Objectives

Formulate conclusions of the theory of special relativity, noting the assumptions that were made in deriving it

Key Takeaways

Key Points

• According to the theory of special relativity, it is impossible to say in an absolute sense whether two distinct events occur at the same time if those events are separated in space.
• A mathematical form of the relativity of simultaneity was introduced by Hendrik Lorentz and physically interpreted by Henri Poincaré. The conceptions were based on the aether as a preferred but undetectable frame of reference.
• Albert Einstein deduced the failure of absolute simultaneity from two assumptions: 1) the principle of relativity; 2) the constancy of the speed of light detected in empty space, independent of the relative motion of its source.

Key Terms

• aether: A space-filling substance or field, thought to be necessary as a transmission medium for the propagation of electromagnetic or gravitational forces.
• special relativity: A theory that (neglecting the effects of gravity) reconciles the principle of relativity with the observation that the speed of light is constant in all frames of reference.

The relativity of simultaneity is the concept that simultaneity–whether two events occur at the same time–is not absolute, but depends on the observer’s frame of reference.
According to the theory of special relativity, it is impossible to say in an absolute sense whether two distinct events occur at the same time if those events are separated in space, such as a car crash in London and another in New York. The question of whether the events are simultaneous is relative: in some reference frames the two accidents may happen at the same time, in other frames (in a different state of motion relative to the events) the crash in London may occur first, and still in other frames, the New York crash may occur first. If the two events are causally connected (“event A causes event B”), then the relativity of simultaneity preserves the causal order (i.e. “event A causes event B” in all frames of reference).
If we imagine one reference frame assigns precisely the same time to two events that are at different points in space, a reference frame that is moving relative to the first will generally assign different times to the two events. This is illustrated in the ladder paradox, a thought experiment which uses the example of a ladder moving at high speed through a garage.
A mathematical form of the relativity of simultaneity (“local time”) was introduced by Hendrik Lorentz in 1892, and physically interpreted (to first order in v/c) as the result of a synchronization using light signals by Henri Poincaré in 1900. However, both Lorentz and Poincaré based their conceptions on the aether as a preferred but undetectable frame of reference, and continued to distinguish between “true time” (in the aether) and “apparent” times for moving observers.
In 1905, Albert Einstein abandoned the (classical) aether and emphasized the significance of relativity of simultaneity to our understanding of space and time. He deduced the failure of absolute simultaneity from two stated assumptions: 1) the principle of relativity–the equivalence of inertial frames, such that the laws of physics apply equally in all inertial coordinate systems; 2) the constancy of the speed of light detected in empty space, independent of the relative motion of its source.

Observer Standing on the Platform: Reference frame of an observer standing on the platform (length contraction not depicted).

Observer Onboard the Train: The train-and-platform experiment from the reference frame of an observer onboard the train.

Time Dilation

Time dilation is an actual difference of elapsed time between two events as measured by observers moving relative to each other.

Learning Objectives

Explain why time dilation can be ignored in daily life

Key Takeaways

Key Points

• Time dilation effects are extremely small for speeds below 1/10 the speed of light and can be safely ignored at daily life.
• Time dilation effects become important when an object approaches speeds on the order of 30,000 km/s (1/10 the speed of light).
• The formula for determining time dilation is: $\Delta {\text{t}}^{\prime }=\frac{\Delta \text{t}}{\sqrt{1-{\text{v}}^2/{\text{c}}^2}}$ .

Key Terms

• speed of light: the speed of electromagnetic radiation in a perfect vacuum: exactly 299,792,458 meters per second by definition
• time dilation: The slowing of the passage of time experienced by objects in motion relative to an observer; measurable only at relativistic speeds.
• Lorentz factor: The factor, used in special relativity, to calculate the degree of time dilation, length contraction and relativistic mass of an object moving relative to an observer.

Time dilation is an actual difference of elapsed time between two events as measured by observers either moving relative to each other.
For instance, two rocket ships (A and B) speeding past one another in space would experience time dilation. If they somehow had a clear view into each other’s ships, each crew would see the others’ clocks and movement as going too slowly. That is, inside the frame of reference of Ship A, everything is moving normally, but everything over on Ship B appears to be moving slower (and vice versa).
From a local perspective, time registered by clocks that are at rest with respect to the local frame of reference (and far from any gravitational mass) always appears to pass at the same rate. In other words, if a new ship, Ship C, travels alongside Ship A, it is “at rest” relative to Ship A. From the point of view of Ship A, new Ship C’s time would appear normal too.
The formula for determining time dilation is: $\Delta {\text{t}}^{\prime }=\gamma \Delta \text{t}=\frac{\Delta \text{t}}{\sqrt{1-{\text{v}}^2/{\text{c}}^2}}$
where Δt is the time interval between two co-local events (i.e. happening at the same place) for an observer in some inertial frame (e.g. ticks on his clock), this is known as the proper time, Δt’ is the time interval between those same events, as measured by another observer, inertially moving with velocity v with respect to the former observer, v is the relative velocity between the observer and the moving clock, c is the speed of light, and $\gamma =\frac{1}{\sqrt{1-{\text{v}}^2/{\text{c}}^2}}$
is the Lorentz factor. Thus the duration of the clock cycle of a moving clock is found to be increased: it is measured to be “running slow”. Note that for speeds below 1/10 the speed of light, Lorentz factor is approximately 1. Thus, time dilation effects and extremely small and can be safely ignored in a daily life. They become important only when an object approaches speeds on the order of 30,000 km/s (1/10 the speed of light).

Lorentz Factor: Lorentz factor as a function of speed (in natural units where c = 1). Notice that for small speeds (less than 0.1), γ is approximately 1.

Effects of Time Dilation: The Twin Paradox and the Decay of the Muon

The twin paradox is a thought experiment: one twin makes a journey into space and returns home to find that twin remained aged more.

Learning Objectives

Explain the twin paradox within the standard framework of special relativity

Key Takeaways

Key Points

• From a naive application of time dilation, each twin should paradoxically find the other to have aged more slowly.
• The scenario is resolved within the standard framework of special relativity: the twins are not equivalent, the space twin experiences additional, asymmetrical acceleration when switching direction to return home.
• Twin paradox is useful as a demonstration that special relativity is self-consistent.

Key Terms

• special relativity: A theory that (neglecting the effects of gravity) reconciles the principle of relativity with the observation that the speed of light is constant in all frames of reference.
• time dilation: The slowing of the passage of time experienced by objects in motion relative to an observer; measurable only at relativistic speeds.

The twin paradox is a thought experiment in special relativity involving identical twins, one of whom makes a journey into space in a high-speed rocket and returns home to find that the twin who remained on Earth has aged more.
This occurs because special relativity shows that the faster one travels, the slower time moves for them.
This result appears puzzling because each twin sees the other twin as traveling, and so, according to a naive application of time dilation, each should paradoxically find the other to have aged more slowly. In other words, from the perspective of the rocketship, the earth is traveling away from the ship and from the perspective of the earth, the rocket is traveling away.
However, this scenario can be resolved within the standard framework of special relativity (because the twins are not equivalent; the space twin experienced additional, asymmetrical acceleration when switching direction to return home), and therefore is not a paradox in the sense of a logical contradiction.
The Earth and the ship are not in a symmetrical relationship: regardless of whether we view the situation from the perspective of the Earth or the ship, the ship experiences additional acceleration forces. The ship has a turnaround in which it accelerates and changes direction whereas the earth does not. Since there is no symmetry, it is not paradoxical if one twin is younger than the other. Nevertheless twin paradox is useful as a demonstration that special relativity is self-consistent.
In the spacetime diagram, drawn for the reference frame of the Earth-based twin, that twin’s world line coincides with the vertical axis (his position is constant in space, moving only in time). On the first leg of the trip, the second twin moves to the right (black sloped line); and on the second leg, back to the left. Blue lines show the planes of simultaneity for the traveling twin during the first leg of the journey; red lines, during the second leg. Just before turnaround, the traveling twin calculates the age of the Earth-based twin by measuring the interval along the vertical axis from the origin to the upper blue line. Just after turnaround, if he recalculates, he’ll measure the interval from the origin to the lower red line. In a sense, during the U-turn the plane of simultaneity jumps from blue to red and very quickly sweeps over a large segment of the world line of the Earth-based twin. The traveling twin reckons that there has been a jump discontinuity in the age of the Earth-based twin.

Spacetime Diagram of the Twin Paradox: Spacetime diagram of the twin paradox. Time is relative, but both twins are not equivalent (the ship experiences additional acceleration to changes the direction of travel).

Length Contraction

Objects that are moving undergo a length contraction along the dimension of motion; this effect is only significant at relativistic speeds.

Learning Objectives

Explain why length contraction can be ignored in daily life

Key Takeaways

Key Points

• Length contraction is negligible at everyday speeds and can be ignored for all regular purposes.
• Length contraction becomes noticeable at a substantial fraction of the speed of light with the contraction only in the direction parallel to the direction in which the observed body is travelling.
• An observer at rest viewing an object travelling very close to the speed of light would observe the length of the object in the direction of motion as very near zero.

Key Terms

• speed of light: the speed of electromagnetic radiation in a perfect vacuum: exactly 299,792,458 meters per second by definition

Length contraction is the physical phenomenon of a decrease in length detected by an observer of objects that travel at any non-zero velocity relative to that observer. Length contraction arises due to the fact that the speed of light in a vacuum is constant in any frame of reference. By taking this into account, as well as some geometrical considerations, we will show how perceived time and length are affected.

Examples

Example 1

Let us imagine a simple clock system that consists of two mirrors A and B in a vacuum. A light pulse bounces between the two mirrors. The separation of the mirrors is L, and the clock ticks once each time the light pulse hits a given mirror. Now imagine that the clock is at rest. The time that it will take for the light pulse to go from mirror A to mirror B and then back to mirror A be can be described by:
$\Delta \text{t}=\frac{2\text{L}}{\text{c}}$
where c is the speed of light. Now imagine that the clock is moving in the horizontal direction relative to a stationary observer. The light pulse is emitted from mirror A. To the stationary observer, it appears that the light pulse has a longer path to travel because by the time the light reaches mirror B the clock has already moved somewhat in the horizontal direction. This is the same case for the light pulse on its way back. The stationary observer will perceive that it will take the light a total of:

Geometry for a Clock at Rest: This illustrates the path that light must traverse when the clock is at rest.

$\Delta {\text{t}}^{\prime }=\frac{2\text{D}}{\text{c}}$
to traverse its path. We can see that D is longer than L, so that means that.

Geometry for a Moving Clock: This illustrates the path that light must traverse when the clock is moving from the perspective of a stationary observer.

Example 2

We have established that in a frame of reference that is moving relative to the clock (the stationary observer is moving in the clock’s frame of reference), the clock appears to run more slowly. Now let us imagine that we want to measure the length of a ruler. This time let us imagine that you are moving with velocity v. You can mathematically determine the length of the ruler in your frame of reference (L‘) by multiplying your velocity (v) by the time that you perceive that it takes you to pass by the ruler (t‘). Expressing this in equation form, L‘ = vt‘. Now, if someone in the ruler’s rest frame wanted to determine the length of the ruler, they could do the following. They could mathematically determine the length of the ruler in their frame of reference (L) by multiplying your velocity (v) by the time that they perceive that it takes you to pass by the ruler (t). This is expressed in the following equation: L = vt. Just as in the clock explanation, the ruler appears to be moving in your frame of reference, so t will be longer than t‘ (your time interval). Consequently, the length of the ruler will appear to be shorter in your frame of reference (the phenomenon of length contraction occurred).
The effect of length contraction is negligible at everyday speeds and can be ignored for all regular purposes. Length contraction becomes noticeable at a substantial fraction of the speed of light (as illustrated in ) with the contraction only in the direction parallel to the direction in which the observed body is travelling.

Observed Length of an Object: Observed length of an object at rest and at different speeds

Example 3

For example, at a speed of 13,400,000 m/s (30 million mph,.0447c), the length is 99.9 percent of the length at rest; at a speed of 42,300,000 m/s (95 million mph, 0.141c), the length is still 99 percent. As the magnitude of the velocity approaches the speed of light, the effect becomes dominant.The mathematical formula for length contraction is:
$\text{L}=\frac{{\text{L}}_0}{\gamma \left(\upsilon \right)}={\text{L}}_0\sqrt{1-\frac{{\upsilon }^2}{{\text{c}}^2}}$
where L₀ is the proper length (the length of the object in its rest frame); L is the length observed by an observer in relative motion with respect to the object; v is the relative velocity between the observer and the moving object; c is the speed of light; and the Lorentz factor is defined as:
$\gamma \left(\upsilon \right)=\frac{1}{\sqrt{1-{\text{v}}^{{}^2}/{\text{c}}^2}}$
In this equation it is assumed that the object is parallel with its line of movement. For the observer in relative movement, the length of the object is measured by subtracting the simultaneously measured distances of both ends of the object. An observer at rest viewing an object traveling very close to the speed of light would observe the length of the object in the direction of motion as very close to zero.

Relativistic Quantities

Relativistic Addition of Velocities

A velocity-addition formula is an equation that relates the velocities of moving objects in different reference frames.

Learning Objectives

Express velocity-addition formulas for objects at speeds much less and approaching the speed of light

Key Takeaways

Key Points

• When two objects are moving slowly compared to speed of light, it is accurate enough to use the vector sum of velocities: $\text{s}=\text{u}+\text{v}$ .
• As the velocity increases towards the speed of light, the vector sum of velocities is replaced with: $\text{s}=\frac{\text{v}+\text{u}}{1+\text{vu}/{\text{c}}^2}$ .
• Composition law for velocities gave the first test of the kinematics of the special theory of relativity when, using a Michelson interferometer, Hyppolite Fizeau measured the speed of light in a fluid moving parallel to the light.

Key Terms

• special relativity: A theory that (neglecting the effects of gravity) reconciles the principle of relativity with the observation that the speed of light is constant in all frames of reference.
• interferometer: Any of several instruments that use the interference of waves to determine wavelengths and wave velocities, determine refractive indices, and measure small distances, temperature changes, stresses, and many other useful measurements.
• speed of light: the speed of electromagnetic radiation in a perfect vacuum: exactly 299,792,458 meters per second by definition

A velocity-addition formula is an equation that relates the velocities of moving objects in different reference frames.
As Galileo Galilei observed in 17th century, if a ship is moving relative to the shore at velocity $\text{v}$ , and a fly is moving with velocity $\text{u}$ as measured on the ship, calculating the velocity of the fly as measured on the shore is what is meant by the addition of the velocities $\text{v}$ and $\text{u}$ . When both the fly and the ship are moving slowly compared to speed of light, it is accurate enough to use the vector sum $\text{s}=\text{u}+\text{v}$ where $\text{s}$ is the velocity of the fly relative to the shore.
According to the theory of special relativity, the frame of the ship has a different clock rate and distance measure, and the notion of simultaneity in the direction of motion is altered, so the addition law for velocities is changed.
Since special relativity dictates that the speed of light is the same in all frames of reference, light shone from the front of a moving car can’t go faster than light from a stationary lamp. Since this is counter to what Galileo used to add velocities, there needs to be a new velocity addition law.
This change isn’t noticeable at low velocities but as the velocity increases towards the speed of light it becomes important. The addition law is also called a composition law for velocities. For collinear motions, the velocity of the fly relative to the shore is given by the following equation:
$\text{s}=\frac{\text{v}+\text{u}}{1+\text{vu}/{\text{c}}^2}$ .
Composition law for velocities gave the first test of the kinematics of the special theory of relativity. Using a Michelson interferometer, Hyppolite Fizeau measured the speed of light in a fluid moving parallel to the light in 1851. The speed of light in the fluid is slower than the speed of light in vacuum, and it changes if the fluid is moving along with the light. The speed of light in a collinear moving fluid is predicted accurately by the collinear case of the relativistic formula.

Setup of the Fizeau Experiment: A light ray emanating from the source S’ is reflected by a beam splitter G and is collimated into a parallel beam by lens L. After passing the slits O1 and O2, two rays of light travel through the tubes A1 and A2, through which water is streaming back and forth as shown by the arrows. The rays reflect off a mirror m at the focus of lens L’, so that one ray always propagates in the same direction as the water stream, and the other ray opposite to the direction of the water stream. After passing back and forth through the tubes, both rays unite at S, where they produce interference fringes that can be visualized through the illustrated eyepiece. The interference pattern can be analyzed to determine the speed of light traveling along each leg of the tube.

Relativistic Momentum

Relativistic momentum is given as $\gamma {\text{m}}_0\text{v}$ where ${\text{m}}_0$ is the object’s invariant mass and $\gamma$ is Lorentz transformation.

Learning Objectives

Compare Newtonian and relativistic momenta for objects at speeds much less and approaching the speed of light

Key Takeaways

Key Points

• Newtonian physics assumes that absolute time and space exist outside of any observer, resulting in a prediction that the speed of light can vary from one reference frame to another.
• In the theory of special relativity, the equations of motion do not depend on the reference frame while the speed of light (c) is invariant.
• Within the domain of classical mechanics, relativistic momentum closely approximates Newtonian momentum. At low velocity $\gamma {\text{m}}_0\text{v}$ is approximately equal to ${\text{m}}_0\text{v}$ , the Newtonian expression for momentum.

Key Terms

• Galilean transformation: a transformation used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics.
• Lorentz transformation: a transformation relating the spacetime coordinates of one frame of reference to another in special relativity
• special relativity: A theory that (neglecting the effects of gravity) reconciles the principle of relativity with the observation that the speed of light is constant in all frames of reference.

Relativistic Momentum

Newtonian physics assumes that absolute time and space exist outside of any observer. This gives rise to Galilean relativity, which states that the laws of motion are the same in all inertial frames. It also results in a prediction that the speed of light can vary from one reference frame to another. However, this is contrary to observation. In the theory of special relativity, Albert Einstein keeps the postulate that the equations of motion do not depend on the reference frame, but assumes that the speed of light c is invariant. As a result, position and time in two reference frames are related by the Lorentz transformation instead of the Galilean transformation.

Albert Einstein: Albert Einstein in 1921

Consider, for example, a reference frame moving relative to another at velocity v in the x direction. The Galilean transformation gives the coordinates of the moving frame as
${\text{t}}^{\prime }=\text{t}$
${\text{x}}^{\prime }=\text{x}-\text{vt}$
while the Lorentz transformation gives
${\text{t}}^{\prime }=\gamma (\text{t}-\frac{\text{vx}}{{\text{c}}^2})$
${\text{x}}^{\prime }=\gamma (\text{x}-\text{vt})$
where $\gamma$ is the Lorentz factor:
$\gamma =\frac{1}{\sqrt{1-(\text{v}/\text{c}{)}^2}}$ .
Conservation laws in physics, such as the law of conservation of momentum, must be invariant. That is, the property that needs to be conserved should remain unchanged regardless of changes in the conditions of measurement. This means that the conservation law needs to hold in any frame of reference. Newton’s second law [with mass fixed in the expression for momentum (p=m*v)], is not invariant under a Lorentz transformation. However, it can be made invariant by making the inertial mass m of an object a function of velocity:
$\text{m}=\gamma {\text{m}}_0$
where ${\text{m}}_0$ is the object’s invariant mass.
The modified momentum,
$\text{p}=\gamma {\text{m}}_0\text{v}$
obeys Newton’s second law:
$\text{F}=\frac{\text{dp}}{\text{dt}}$ .
It is important to note that for speeds much less than the speed of light, Newtonian momentum and relativistic momentum are approximately the same. As one approaches the speed of light, however, relativistic momentum becomes infinite while Newtonian momentum continues to increases linearly. Thus, it is necessary to employ the expression for relativistic momentum when one is dealing with speeds near the speed of light.

Relativistic and Newtonian Momentum: This figure illustrates that relativistic momentum approaches infinity as the speed of light is approached. Newtonian momentum increases linearly with speed.

Relativistic Energy and Mass

In special relativity, as the object approaches the speed of light, the object’s energy and momentum increase without bound.

Learning Objectives

Evaluate possibility for an object to travel at the speed of light

Key Takeaways

Key Points

• In special relativity, an object that has a mass cannot travel at the speed of light.
• Relativistic mass is defined as ${\text{m}}_{\text{rel}}=\frac{\text{E}}{{\text{c}}^2}$ and can be viewed as the proportionality constant between the velocity and the momentum.
• Relativistic energy is connected with rest mass via the following equation: ${\text{E}}_{\text{r}}=\sqrt{({\text{m}}_0{\text{c}}^2{)}^2+(\text{pc}{)}^{{}^2}}$ .

Key Terms

• Lorentz factor: The factor, used in special relativity, to calculate the degree of time dilation, length contraction and relativistic mass of an object moving relative to an observer.
• rest mass: the mass of a body when it is not moving relative to an observer
• special relativity: A theory that (neglecting the effects of gravity) reconciles the principle of relativity with the observation that the speed of light is constant in all frames of reference.

Relativistic Energy and Mass

In special relativity, an object that has a mass cannot travel at the speed of light. As the object approaches the speed of light, the object’s energy and momentum increase without bound. Relativistic corrections for energy and mass need to be made because of the fact that the speed of light in a vacuum is constant in all reference frames. The conservation of mass and energy are well-accepted laws of physics. In order for these laws to hold in all reference frames, special relativity must be applied. It is important to note that for objects with speeds that are well below the speed of light that the expressions for relativistic energy and mass yield values that are approximately equal to their Newtonian counterparts.

Relativistic and Newtonian Kinetic Energy: This figure illustrates how relativistic and Newtonian Kinetic Energy are related to the speed of an object. The relativistic kinetic energy increases to infinity when an object approaches the speed of light, this indicates that no body with mass can reach the speed of light. On the other hand, Newtonian kinetic energy continues to increase without bound as the speed of an object increases.

Relativistic Mass

Relativistic mass was defined by Richard C. Tolman pictured left of Albert Einstein here in 1934 as which holds for all particles, including those moving at the speed of light. For a slower than light particle, a particle with a nonzero rest mass, the formula becomes where is the rest mass and is the Lorentz factor. The Lorentz factor is equal to: $\gamma =\frac{1}{\sqrt{1-{\text{v}}^2/{\text{c}}^2}}$ , where v is the relative velocity between inertial reference frames and c is the speed of light.

Richard C. Tolman and Albert Einstein: Richard C. Tolman (1881 – 1948) with Albert Einstein (1879 – 1955) at Caltech, 1932

When the relative velocity is zero, is simply equal to 1, and the relativistic mass is reduced to the rest mass. As the velocity increases toward the speed of light (c), the denominator of the right side approaches zero, and consequently approaches infinity.
In the formula for momentum the mass that occurs is the relativistic mass. In other words, the relativistic mass is the proportionality constant between the velocity and the momentum.
While Newton’s second law remains valid in the form the derived form is not valid because in is generally not a constant.

Relativistic Energy

Relativistic energy (${\text{E}}_{\text{r}}=\sqrt{({\text{m}}_0{\text{c}}^2{)}^2+(\text{pc}{)}^{{}^2}}$ ) is connected with rest mass via the following equation: $\text{m}=\frac{\sqrt{({\text{E}}^2-(\text{pc}{)}^{{}^2}}}{{\text{c}}^2}$ . Here the term represents the square of the Euclidean norm (total vector length) of the various momentum vectors in the system, which reduces to the square of the simple momentum magnitude, if only a single particle is considered. This equation reduces to when the momentum term is zero. For photons where the equation reduces to.
Today, the predictions of relativistic energy and mass are routinely confirmed from the experimental data of particle accelerators such as the Relativistic Heavy Ion Collider. The increase of relativistic momentum and energy is not only precisely measured but also necessary to understand the behavior of cyclotrons and synchrotron, which accelerate particles to near the speed of light.

Matter and Antimatter

Antimatter is composed of antiparticles, which have the same mass as particles of ordinary matter but opposite charge and quantum spin.

Learning Objectives

Describe properties of antiparticles

Key Takeaways

Key Points

• The end result of antimatter meeting matter is a release of energy proportional to the mass, as shown in the mass-energy equivalence equation, E = mc².
• Although almost all matter observable from the Earth seems to be made of matter rather than antimatter, antimatter may still exist in relatively large amounts in far-away galaxies.
• No explanation for the apparent asymmetry of matter and antimatter in the visible universe exists.

Key Terms

• annihilation: the process of a particle and its corresponding antiparticle combining to produce energy
• positron: The antimatter equivalent of an electron, having the same mass but a positive charge.
• antimatter: matter that is composed of the antiparticles of those that constitute normal matter

Antimatter is material composed of antiparticles, which have the same mass as particles of ordinary matter but have opposite charge and quantum spin. Antiparticles bind with each other to form antimatter in the same way that normal particles bind to form normal matter. For example, a positron (the antiparticle of the electron, with symbol e⁺) and an antiproton (symbol p^–) can form an antihydrogen atom. Furthermore, mixing matter and antimatter can lead to the annihilation of both, in the same way that mixing antiparticles and particles does. This gives rise to high-energy photons (gamma rays) and other particle-antiparticle pairs. The end result of antimatter meeting matter is a release of energy proportional to the mass, as shown in the mass-energy equivalence equation, E = mc².

Antihydrogen and Hydrogen Atoms: Antihydrogen consists of an antiproton and a positron; hydrogen consists of a proton and an electron.

Almost all matter observable from the earth seems to be made of matter rather than antimatter. If antimatter-dominated regions of space existed, the gamma rays produced in annihilation reactions along the boundary between matter and antimatter regions would be detectable.
Antimatter may still exist in relatively large amounts in far-away galaxies due to cosmic inflation in the primordial time of the universe. Antimatter galaxies, if they exist, are expected to have the same chemistry and absorption and emission spectra as normal-matter galaxies, and their astronomical objects would be observationally identical, making them difficult to distinguish from normal-matter galaxies.
There is considerable speculation as to why the observable universe is apparently composed almost entirely of matter (as opposed to a mixture of matter and antimatter), whether there exist other places that are almost entirely composed of antimatter instead, and what sorts of technology might be possible if antimatter could be harnessed. At this time, the apparent asymmetry of matter and antimatter in the visible universe is one of the greatest unsolved problems in physics.

Relativistic Kinetic Energy

Relativistic kinetic energy can be expressed as: ${\text{E}}_{\text{k}}=\frac{{\text{mc}}^2}{\sqrt{1-\left(\text{v}/\text{c}{)}^2\right)}}-{\text{mc}}^2$ where $\text{m}$ is rest mass, $\text{v}$ is velocity, $\text{c}$ is speed of light.

Learning Objectives

Compare classical and relativistic kinetic energies for objects at speeds much less and approaching the speed of light

Key Takeaways

Key Points

• Relativistic kinetic energy equation shows that the energy of an object approaches infinity as the velocity approaches the speed of light. Thus it is impossible to accelerate an object across this boundary.
• Kinetic energy calculations lead to the mass -energy equivalence formula: ${\text{E}}_{\text{rest}}={\text{E}}_0={\text{mc}}^2$ .
• At a low speed ($\text{v}<<\text{c}$ ), the relativistic kinetic energy may be approximated by the classical kinetic energy. Thus, the total energy can be partitioned into the energy of the rest mass plus the traditional Newtonian kinetic energy at low speeds.

Key Terms

• special relativity: A theory that (neglecting the effects of gravity) reconciles the principle of relativity with the observation that the speed of light is constant in all frames of reference.
• classical mechanics: All of the physical laws of nature that account for the behaviour of the normal world, but break down when dealing with the very small (see quantum mechanics) or the very fast or very heavy (see relativity).
• Lorentz factor: The factor, used in special relativity, to calculate the degree of time dilation, length contraction and relativistic mass of an object moving relative to an observer.

In classical mechanics, the kinetic energy of an object depends on the mass of a body as well as its speed. The kinetic energy is equal to the mass multiplied by the square of the speed, multiplied by the constant 1/2. The equation is given as:
${\text{E}}_{\text{k}}=\frac{1}{2}{\text{mv}}^2$ ,
where $\text{m}$ is the mass and $\text{v}$ is the speed (or the velocity) of the body.
The classical kinetic energy of an object is related to its momentum by the equation:
${\text{E}}_{\text{k}}=\frac{{\text{p}}^2}{2\text{m}}$ ,
where $\text{p}$ is momentum.
If the speed of a body is a significant fraction of of the speed of light, it is necessary to employ special relativity to calculate its kinetic energy. It is important to know how to apply special relativity to problems with high speed particles. In special relativity, we must change the expression for linear momentum. Using $\text{m}$ for rest mass, $\text{v}$ and $\nu$ for the object’s velocity and speed respectively, and $\text{c}$ for the speed of light in vacuum, the relativistic expression for linear momentum is:

Time Magazine – July 1, 1946: The popular connection between Einstein, E = mc2, and the atomic bomb was prominently indicated on the cover of Time magazine (July 1946) by the writing of the equation on the mushroom cloud itself.

$\text{p}=\text{m}\gamma \text{v}$ , where $\gamma$ is the Lorentz factor:
$\gamma =\frac{1}{\sqrt{1-(\text{v}/\text{c}{)}^2}}$ .
Since the kinetic energy of an object is related to its momentum, we intuitively know that the relativistic expression for kinetic energy will also be different from its classical counterpart. Indeed, the relativistic expression for kinetic energy is:
${\text{E}}_{\text{k}}=\frac{{\text{mc}}^2}{\sqrt{1-\left(\text{v}/\text{c}{)}^2\right)}}-{\text{mc}}^2$ .
The equation shows that the energy of an object approaches infinity as the velocity $\text{v}$ approaches the speed of light $\text{c}$ . Thus it is impossible to accelerate an object across this boundary.
The mathematical by-product of this calculation is the mass-energy equivalence formula (referred to in ). The body at rest must have energy content equal to:
${\text{E}}_{\text{rest}}={\text{E}}_0={\text{mc}}^2$ .
The general expression for the kinetic energy of an object that is not at rest is:
$\text{KE}={\text{mc}}^2-{\text{m}}_0{\text{c}}^2$ , where m is the relativistic mass of the object and m₀ is the rest mass of the object.
At a low speed ($\text{v}<<\text{c}$ ), the relativistic kinetic energy may be approximated well by the classical kinetic energy. We can show this to be true by using a Taylor expansion for the reciprocal square root and keeping first two terms of the relativistic kinetic energy equation. When we do this we get:
${\text{E}}_{\text{k}}\approx \text{m}{\text{c}}^2\left(1+\frac{1}{2}{\text{v}}^2/{\text{c}}^2\right)-\text{m}{\text{c}}^2=\frac{1}{2}\text{m}{\text{v}}^2$ .
Thus, the total energy can be partitioned into the energy of the rest mass plus the traditional classical kinetic energy at low speeds.

Implications of Special Relativity

Shifting the Paradigm of Physics

Special relativity changed the way we view space and time and showed us that time is relative to an observer.

Learning Objectives

Formulate major changes in the understanding of time, space, mass, and energy that were introduced by the theory of Special Relativity

Key Takeaways

Key Points

• Time is relative.
• Lengths of moving objects are different than if their lengths were measured in a co-moving frame.
• Time and space are not independent.

Key Terms

• special relativity: A theory that (neglecting the effects of gravity) reconciles the principle of relativity with the observation that the speed of light is constant in all frames of reference.
• time dilation: The slowing of the passage of time experienced by objects in motion relative to an observer; measurable only at relativistic speeds.
• length contraction: Observers measure a moving object’s length as being smaller than it would be if it were stationary.

The theory of Special Relativity and its implications spurred a paradigm shift in our understanding of the nature of the universe, the fundamental fabric of which being space and time. Before 1905, scientists considered space and time as completely independent objects. Time could not affect space and space could not affect time. After 1905, however, the Special Theory of Relativity destroyed this old, but intuitive, view. Specifically, Special Relativity showed us that space and time are not independent of one another but can be mixed into each other and therefore must be considered as the same object, which we shall denote as space-time. The consequences of space/time mixing are:

• time dilation
• and length contraction.

Other important consequences which will be discussed in another section are

• Relativity of Simultaneity (for certain events, the sequence in which they occur is dependent on the observer)
• Nothing can move faster than the speed of light (we shall denote the value of the speed of light as $\text{c}$ )

Why is there a mixing between space and time? In order to examine this we must know the founding principles of relativity. They are:

1. The Principle of Relativity: The laws of physics for observers which are not accelerating relative to one another should be the same.
2. The Principle of Invariant Light Speed: All observers, moving at constant speed, measure the same speed of light regardless of how fast they are moving.

If one accepts the second principle as fact then it immediately follows that space and time are not independent. Why? Let us look at the experiment in in which there is a light source, a fixed observer, and a rocket ship moving toward the light source. The two observers are related by a coordinate , or space-time, transformation. The second principle then tells us that no matter how fast the rocket is moving, both observers must measure the same light speed emanating from the source. The only way this can happen is if the clock of the rocket observer is ticking at a different rate than the stationary observer. How much different? This can be expressed in the time dilation equation:

Measuring Light: A stationary observer will measure the same speed of light as an observer who is moving in a rocket ship even if that rocket is moving close to light speed.

${\text{t}}_{\text{s}}=\frac{{\text{t}}_{\text{m}}}{\sqrt{1-\left(\frac{{\text{v}}^2}{{\text{c}}^2}\right)}}$
Where ${\text{t}}_{\text{s}}$ is the time elapsed for the stationary observer, ${\text{t}}_{\text{m}}$ is the time elapsed for the moving observer, and $\text{v}$ is the velocity of the rocket as measured from the stationary frame. One can then see that as $\text{v}\to \text{c}$ then the time elapsed in the stationary frame goes to infinity. To place some numbers, let $\text{v}=0.99986\text{c}$ , and ${\text{t}}_{\text{m}}=1\text{sec}$ . The time elapsed in the stationary frame is then one hour. Thus for every second lived in the rocket, the stationary man lives one hour!
One of the more radical results of time dilation is the so-called “twin paradox. ” The twin paradox is a thought experiment in special relativity that involves identical twins. One of the twins goes on a journey into space in a rocket that has a velocity near the speed of light. Upon returning home the twin finds that the twin that remained on Earth has aged more. This consequence altered the perception that aging is necessarily constant.
The square root factor in the time dilation equation is very important and we denote it as:
This factor shows up frequently in special relativity. For example the length contraction formula is:
$\text{L}={\text{L}}_0\sqrt{1-\left(\frac{\text{v}}{\text{c}}\right)}=\frac{{\text{L}}_0}{\gamma }$
where ${\text{L}}_0$ is the rest length, the length of an object measured in the co-moving frame of the object, and $\text{L}$ is the length of the object as measured by the observer who sees the object moving at speed $\text{v}$ . In our rocket example, the stationary observer measures the length of the rocket as being less than what someone who was moving with the rocket would measure. This altered the perception that the length of an object would appear the same regardless of the reference frame of the observer.
Another radical finding that was made possible by the discovery of special relativity is the equivalence of energy and mass. Combined with other laws of physics, the postulates of special relativity predict that mass and energy are related by: $\text{E}={\text{mc}}^2$ , where c is the speed of light in vacuum. This altered the perception that mass and energy are completely different things and paved the way for nuclear power and nuclear weapons.
As a final note it is important to note how big the speed of light is compared to every day life. The speed of light is:
$\text{c}=3\times {10}^8\text{m}/\text{s}$ .
Thus in every day life $\gamma \approx 1$ and we do not experience significant time dilation or length contraction. If we did, life would be very different.

Four-Dimensional Space-Time

We live in four-dimensional space-time, in which the ordering of certain events can depend on the observer.

Learning Objectives

Formulate major results of special relativity

Key Takeaways

Key Points

• We live in a four-dimensional universe; the first three dimensions are spatial, and the fourth is time.
• The coordinate system of physical observers are related to one another via Lorentz transformations.
• Nothing can travel faster than the speed of light.

Key Terms

• line element: An invariant quantity in special relativity
• Lorentz transformation: a transformation relating the spacetime coordinates of one frame of reference to another in special relativity
• relativity of simultaneity: For space-like separated space-time points, the time-ordering between events is relative.

Working in Four Dimensions

Let us examine two observers who are moving relative to one another at a constant velocity. We shall denote them as observer A and observer A’. Observer A sets up a space-time coordinate system (t, x, y, z); similarly, A’ sets up his own space-time coordinate system (t’, x’, y’, z’). (See for an example. ) Therefore both observers live in a four-dimensional world with three space dimensions and one time dimension.

Two Coordinate Systems: Two coordinate systems in which the primed frame moves with velocity v with respect to the unprimed frame

You should not find it odd to work with four dimensions; any time you have to meet your friend somewhere you have to tell him four variables: where (three spatial coordinates) and when (one time coordinate). In other words, we have always lived in four dimensions, but so far you have probably thought of space and time as completely separate.

The Movement of Light in Four Dimensions

Back to our example: let us assume that at some point in space-time there is a beam of light that emerges. Both observers measure how far the beam has traveled at each point in time and how long it took to travel to travel that distance. That is to say, observer A measures:
$(\Delta \text{t},\Delta \text{x},\Delta \text{y},\Delta \text{z})$
where, for example, $\Delta \text{t}=\text{t}-{\text{t}}_0$ ; t is the time at which the measurement took place; and t₀ is the time at which the light was turned on.
Similarly, observer A’ measures:
$(\Delta {\text{t}}^{\prime },\Delta {\text{x}}^{\prime },\Delta {\text{y}}^{\prime },\Delta {\text{z}}^{\prime })$
where we set up the system such that both observers agree on $({\text{t}}_0,{\text{x}}_0,{\text{y}}_0,{\text{z}}_0)$ . Due to the invariance of the speed of light both observers will agree on:
$\frac{\Delta {\text{X}}^2+\Delta {\text{Y}}^2+\Delta {\text{Z}}^2}{\Delta {\text{T}}^2}={\text{c}}^2\to 0=-{\text{c}}^2\Delta {\text{T}}^2+\Delta {\text{X}}^2+\Delta {\text{Y}}^2+\Delta {\text{Z}}^2$
where $(\text{T},\text{X},\text{Y},\text{Z})$ refers to the coordinates in either frame. Therefore there is a specific rule that all light paths must follow. For general events we can define the quantity:
${\text{s}}^2=-{\text{c}}^2\Delta {\text{t}}^2+\Delta {\text{x}}^2+\Delta {\text{y}}^2+\Delta {\text{z}}^2$
This is known as the line element and is the same for all observers. (Using the principles of relativity, you can prove this for general separations, not just light rays). The set of all coordinate transformations that leave the above quantity invariant are known as Lorentz Transformations. It follows that the coordinate systems of all physical observers are related to each other by Lorentz Transformations. (The set of all Lorentz transformations form what mathematicians call a group, and the study of group theory has revolutionized physics). For the coordinate transformation in, the transformations are:
${\text{t}}^{\prime }=\gamma (\text{t}-\text{vx}/{\text{c}}^2){\text{x}}^{\prime }=\gamma (\text{x}-\text{vt}){\text{y}}^{\prime }=\text{y}{\text{z}}^{\prime }=\text{z}$
where
$\gamma =\frac{1}{\sqrt{1-(\text{v}/\text{c}{)}^2}}$
We define the separation between space-time points as follows:
${\text{s}}^2>0:\text{space-like}$
${\text{s}}^2<0:\text{time-like}$
${\text{s}}^2=0:\text{null}$
We separate these events because they are very different. For example, for a space-like separation you can always find a coordinate transformation that reverses the time-ordering of the events (try to prove this for the example in ). This phenomenon is known as the relativity of simultaneity and may be counterintuitive.

Space-Like Space-Time Separations

Let us examine two car crashes, one in New York and one in London, such that they occur in the same time in one frame. In this situation, the space-time separation between the two events is space-like. The question of whether the events are simultaneous is relative: in some reference frames the two accidents may happen at the same time; in other frames (in a different state of motion relative to the events) the crash in London may occur first; and in still other frames the New York crash may occur first. (In practice, this does not affect us because the frames that switch the order of the events must move at unreachably high speeds. )

Time-Like and Null Space-Time Separations

Events that are time-like or null do not share this property, and therefore there is a causal ordering between time-like events. In other words, if two events are time-like separated, then they can effect each other. The reason is that if two space-time points are time-like or null separated, one can always send a light signal from one point to another.

Special Relativity

Finally, let’s discuss an important result of special relativity — that the energy $\text{E}$ of an object moving with speed $\text{v}$ is:
$\text{E}=\frac{{\text{m}}_0{\text{c}}^2}{\sqrt{1-(\text{v}/\text{c}{)}^2}}=\text{m}{\text{c}}^2$
where ${\text{m}}_0$ is the mass of the object at rest and $\text{m}=\gamma {\text{m}}_0$ is the mass when the object is moving. The above formula immediately shows why it is impossible to travel faster than the speed of light. As $\text{v}\to \text{c}$ , $\text{m}\to \infty$ , and it takes an infinite amount of energy to accelerate the object further.

The Relativistic Universe

Gravity is a geometrical effect in which a metric matrix plays a special role, and the motion of objects are altered by curved space.

Learning Objectives

Identify factors that affect motion near massive objects

Key Takeaways

Key Points

• The metric matrix can be used to calculate the curvature of space-time.
• Curvature can be related to energy and momentum via the Einstein equations.
• Due to the curvature of space-time, the motion of objects are altered near massive objects.

Key Terms

• general relativity: A theory extending special relativity and uniformly accounting for gravity and accelerated frames of reference, postulating that space-time curves in the presence of mass.
• Minkowski space: A four dimensional flat space-time. Because it is flat, it is devoid of matter.
• metric: A metric, or distance function, is a function which defines a distance between elements of a set.

Relativity

Special relativity indicates that humans live in a four-dimensional space-time where the ‘distance’ [latex]\text{s}[/latex] between points in space-time can be regarded as:
${\text{s}}^2=-{\text{c}}^2\Delta {\text{t}}^2+\Delta {\text{x}}^2+\Delta {\text{y}}^2+\Delta {\text{z}}^2={\text{X}}^{\text{T}}\eta \text{X}$
The last equation is a matrix relation in which ${}^{\text{T}}$ denotes transpose (for vectors ${\text{A}}^{\text{T}}\text{B}=\text{A}\cdot \text{B}$ ), $\text{X}$ is the vector $(\text{c}\Delta \text{t},\Delta \text{x},\Delta \text{y},\Delta \text{z})$ , and $\eta$ is the matrix:
$\eta =\left(\begin{array}{cccc}-1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$
A matrix that goes in between two vectors to give a length is called a metric. In mathematics, a metric or distance function is a function which defines a distance between elements of a set. In this case, the set is the space-time and the elements are points in that space-time. A space-time with the $\eta$ metric is called Minkowski space and $\eta$ is the Minkowski metric.
Four-dimensional Minkowski space-time is only one of many different possible space-times (geometries) which differ in their metric matrix. An arbitrary metric matrix can be denoted as $\text{g}$ , raising questions as to what space-times with different metrics represent. In 1916, Einstein found the importance of these space-times in his theory of general relativity.

General Relativity

General relativity, or the general theory of relativity, is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics. General relativity generalizes special relativity and Newton’s law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or space-time. In particular, the curvature of space-time is directly related to the energy and momentum of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of partial differential equations.
$\text{g}\leftrightarrow \text{curvature}\leftrightarrow \text{Energy and Momentum}$
People can use the metric to calculate curvature and then use the Einstein field equations to relate the curvature to the energy and momentum of the space-time. Going in the reverse order, energy and momentum affect the curvature and the space-time. Thus, energy and momentum curves space-time. Minkowski space is the special space devoid of matter, and as a result, it is completely flat. The precise definition of curvature requires knowledge of advanced mathematics, but an intuitive way to understand it is that the definition of a straight line changes in curved spacetime.
A pictorial example is shown in the figure below, where anything that is near the Earth has its motion altered due to the curved space-time. This altering of motion affects satellites, the moon, and even humans. If the space around the Earth were flat, humans would simply float away if they jumped upwards. Since the Earth alters the space-time, humans are pulled toward the Earth. In essence, gravity is a geometrical effect.

Curvature of space-time: The massive Earth is altering the curvature of space-time.