II.  The Physical Basis of Holography

The Physical Model

To make holography accessible to a general audience with widely varied backgrounds, a physical model is useful. Just as chemists use sticks and balls to help them visualize the structure of molecules, our model will allow us to visualize, and thus “understand,” the physical characteristics of holograms without using advanced mathematics.

Two-Source Interference

In two dimensions (the plane of this paper), the pattern of waves from a stationary source generating waves at constant frequency (and wavelength) is a set of concentric circles (see Figure 3-2 [a] and [b] in Module 1-3, Basic Geometrical Optics, as an example of water waves). The distance between any two adjacent circles is one wavelength (l). Let each circle represent the crest of a wave. Halfway between any two waves is the trough. The pattern shown in Figure 10-1 represents a snapshot of such a wave pattern.

To simulate the interference pattern caused by two point sources emitting waves at the same frequency and amplitude, let’s make a transparent photocopy of the set of concentric circles shown in Figure 10-1. Then let’s place the copy on top of Figure 10-1, move it around, and observe the results. A typical pattern is shown in Figure 10-2.

Figure 10-1  A two-dimensional “snapshot” of wave fronts from a constant-frequency point source at the center. The radial distance from one line to the next is one wavelength (l).

Figure 10-2  The two-dimensional interference pattern caused by two point sources of waves of constant frequency. Each set of waves is moving away from its source at the same constant speed. Nevertheless, the overall interference pattern remains constant in time. The two point sources creating this pattern can be seen near the center, along a horizontal direction, about four centimeters apart.

The bright (white) areas represent constructive interference because the crests from both sources—as well as the troughs from both sources—coincide, causing the waves to go up and down with twice the amplitude of each wave alone. The dark  (black) areas represent destructive interference because the crest of one wave encounters the trough of the other wave, thus causing a cancellation of wave amplitude at that point. For water waves, the center of each dark area represents perfect and permanent calm in spite of the fact that waves from the two sources are passing through the dark area at all times. For sound waves, the same areas would represent regions of absolute silence.

A Trace of the Maxima from Two Point Sources

Figure 10-3 represents a trace of the constructive interference maxima—the white regions—observed in Figure 10-2. Here, S and S¢ denote the locations of the centers of the two sources. At precisely the midpoint between S and S¢ is a straight line OO¢. At any point along this line, waves arriving from the two synchronized sources meet exactly in phase (or have zero phase difference), since they have traveled the same distance. This is called the zeroth order of constructive interference. For all points on the first curved line PP¢(a hyperbola) at the right of the zeroth-order line, waves from S ¢ travel a distance exactly one wavelength more than waves from S. Thus, this line represents the location of the first order of constructive interference. Similarly, the first curved line at the left of the zeroth order is also a first-order constructive interference pattern.

Figure 10-3  A computer trace of the locations of interference maxima on a plane containing the two point sources S and S¢

Thus, the tenth curves at the left and right of the zeroth order are tenth orders of constructive interference. Any point on these curves has a difference in distance from S and S ¢ equal to 10l.

Exactly halfway between consecutive orders of interference maxima are hyperbolas (not drawn in Figure 10-3), which represent the minima, where the wave amplitude is always zero in spite of the fact that waves from two sources of disturbance are continuously passing through. In other words, waves meeting at any point along any line in Figure 10-3 are in phase. And waves meeting at any point along lines halfway between the constructive interference lines are out of phase and result in zero amplitude.

Now, suppose we imagine the interference pattern from S and S ¢ as it exists in space, that is, in three dimensions. Figure 10-4 represents a cross section of this three-dimensional interference pattern. If the pattern were to spin around the x-axis, one would observe a set of hyperboloidal surfaces. The zeroth order (m = 0) is a flat plane, and all other orders (m = 1 and higher) are smooth surfaces of varying curvatures. On the x-y and x-z planes are interference patterns like those shown in Figure 10-3, a set of hyperbolas. On the y-z plane, as shown in Figure 10-4, the pattern is a set of concentric circles.

Figure 10-4  A three-dimensional interference pattern of waves from two point sources S and S'. The constructive interference order numbers, m, are indicated for the first several maxima.

Using the transparency of circles of Figure 10-1 that we made earlier, and placing it again on top of the original set (Figure 10-1), we can demonstrate how uniquely different interference patterns are formed—corresponding to each unique location of S versus S ¢. Observing the pattern along the axial direction (line through S and S ¢) reveals the concept of Michelson interferometry. Far above and below S and S ¢, Young’s double-slit interference pattern is recreated. Changing the distance between S and S ¢ shows how the pattern changes correspondingly. For example, as the separation between S and S ¢ increases, the interference fringes become more dense, i.e., more maxima and minima per unit distance. This is measured in terms of spatial frequency, or cycles per millimeter. Conversely, as the distance between S and S ¢ decreases, the spatial frequency of the interference pattern decreases—less cycles per millimeter.

The Physical Model

Some interesting characteristics of hyperboloids are represented in Figure 10-5. Think of the separate hyperboloids as the three-dimensional surface traced out when Figure 10-3 is rotated about an axis through the points SS ¢. Imagine that all the hyperboloidal surfaces are mirrors. Take the zeroth-order “mirror” OO¢, which perpendicularly bisects the line SS ¢ joining the two sources. In three dimensions, this is a flat mirror. Each ray from point S, striking the hyperboloidal surface (mirrors) at m = 2, m = 1, m = 0, m = –1, and m = –2, as shown, reflects from these surfaces (mirrors) in a direction such that the reflected ray appears to come from point S¢. Two such rays from S are shown in Figure 10-5, one up and to the left (labeled a), the other down and to the left (labeled b). The reflected rays are labeled a0, a1, a2, a–1, a–2, and b2 and b1, in accordance with the appropriate ray from S and the hyperboloid from which they reflect. Thus, S ¢ is the virtual image of S, for any and all of the surfaces. The reverse is true for light from S ¢ incident on the surfaces, for which S would then be the virtual image.

Figure 10-5  Light from S is reflected by any part of any hyperboloidal surface (mirror) in a direction such that it appears to be originating from S¢.

With this physical model in mind, we are now ready to explain all the important characteristics of holograms recorded in a medium such as photographic emulsion that has a thickness of about 6 to 7 micrometers (µm).

Applications of the Model

Creating the Virtual Image

Figure 10-6 shows the optical case of two-beam interference in three-dimensional space. Assume that the light from the two sources (S and S ¢) is directed at a recording medium such as a silver halide photographic emulsion (“holoplate”) at a position as shown. The flat rectangle in the figure represents the top edge of the holoplate. Since the typical thickness of these emulsions is in the vicinity of 6 or 7 µm and the wavelength of laser light used to record holograms is 0.633 µm (HeNe laser), this thickness is approximately 10l. The interference pattern recorded inside the emulsion represents sections of hyperboloidal surfaces of many different orders of m. These are, of course, sections of the hyperboloidal surfaces that we have been describing.

Observe carefully the orientation of the “mirrors” formed inside the emulsion. The “mirrors” on the left side lean toward the right, those on the right side lean toward the left, and those in the center are perpendicular to the plane of the holograms. In precise terms, the plane of each “mirror” bisects the angle formed between rays from S and S¢.

Figure 10-6  Light from S interferes with light from S¢ and produces a three-dimensional interference pattern inside a “thick” medium such as photographic emulsion (the “holoplate”).

The exposed and developed emulsion (holoplate) is called a hologram. Within the hologram, the recorded silver surfaces are partially reflecting—as well as partially transmitting and absorbing. If we replace the hologram in its original position during the recording, take away S¢, and illuminate the hologram with S alone, as shown in Figure 10-7, all the reflected rays will appear to originate from S¢. An observer would see these reflected rays as if they all came from S¢. In other words, the virtual image of S¢ has been created.

Figure 10-7  When the developed emulsion (hologram) is illuminated by S alone, the virtual image of S¢ is observed.

We can arbitrarily call the light from source S a reference beam and from source S ¢ an object beam. If more than one point source is located in the vicinity of S ¢, each source will form a unique hyperboloid set with source S and the film will record all of them. When the processed hologram is illuminated, with source S only, each set will reflect light in such a way as to recreate the virtual image of all its object points.

If we replace point source S ¢ with a three-dimensional scene (or object) illuminated by light having the same constant frequency as the reference beam, each point on its surface (S1¢ and S2¢, for example) creates a unique hyperboloid set of patterns with S inside the emulsion. Thus, we have a hologram of a three-dimensional object (Figure 10-8). When the hologram is illuminated by S, each set of hyperboloidal mirrors recreates a virtual image of each point (S1¢, S2¢, etc.), so that a complete, three-dimensional virtual image of the object is reconstructed.

Figure 10-8  A hologram of a three-dimensional object can be considered as a superposition of many individual holograms of points on an object.

A general statement of the model can now be given as follows:

Creating a Real Image

Take the hologram from Figure 10-7 and illuminate it in a “backward direction” by focusing a beam of light back toward S (Figure 10-9). The reflected light from our hyperboloidal mirrors will focus at S ¢ so that, if a projection screen were present there, we would have a real image of the original 3-D object. This can be done also with the hologram formed in Figure 10-8. The real image in this case will appear on the screen as a two-dimensional image of our original object. Depending on the location of the screen, different slices of this scene will come into focus.

Figure 10-9  If a hologram is illuminated from the “front,” exactly backward toward S, the hologram will reflect the light behind the hologram to form a real image of S¢ on a screen.

Redundancy

If a transmission hologram is broken into pieces, each piece will give a complete perspective of the original scene. This can be understood from Figure 10-10. Imagine that the holoplate were half or a small fraction of its original size shown earlier in Figure 10-8. Since every elementary volume in the hologram was formed with light from a complete perspective of the scene, each of these elementary volumes will produce a complete perspective. In other words, the size of the holoplate used to make a hologram is independent of the size of the scene. A large hologram can be considered as the sum of many smaller holograms.

Figure 10-10  Each piece of a hologram can recreate a complete view of the object. A large hologram can be considered as a collection of many smaller holograms.

For the purpose of projecting a real image on a screen with a laser beam, it is desirable to select only a narrow area by using an undiverged beam so that the area covered does not exceed a few millimeters in diameter. In this case, the real image consists of rays at small angles relative to one another. This increases the depth of field, allowing us to have a focused image over a long distance along the beam paths that form the real image. Many laws of geometric optics operate here, i.e., aperture, depth of field, and resolution.

Dynamic Range

Not only the locations of all points on an object are reproduced in the hologram, but their relative intensities as well. Consider two spots S1¢ and S2¢ on the object, with S1¢ having the same intensity as S and S2¢ being less intense than S (Figure 10-11). In the areas where the light waves from S are in phase with those from S1¢, the total amplitude is doubled. Where they are out of phase, the total amplitude is zero. The same cannot occur between waves from S and S2¢. The result is that the interference pattern formed between S and S1¢ has a higher contrast than those formed between S and S2¢, and results in having “mirrors” with higher reflectivity. In other words, S1¢ simply makes a brighter hologram than S2¢.

Figure 10-11  A bright spot S1¢ creates a brighter hologram with S than does a dim spot S2¢.

Thus, a hologram can recreate images with intensity levels varying from almost complete darkness to glares. The holographic image of a diamond, for example, can be seen to “sparkle.” The hologram is said to have a large dynamic range.

By contrast, a photograph (or a TV picture) cannot reproduce glares or sparkles. The brightest part of such an image is the diffused white color of paper (or screen).

“Noise”

As is true in all information-transmission systems (radio, TV, recording a picture, etc.), the output always has background noise added to it. Similarly, a copy of a photograph is never as high in quality as the original.

Besides the so-called grain “noise” of photographic emulsion, which is caused by the scattering of light by the silver grains, another source is called intermodulation noise. In Figure 10-11, in addition to the interference patterns formed between S and S1¢, and S and S2¢, there is a pattern formed between S1¢ and S2¢ (not shown). The latter pattern forms a set of hyperboloids throughout the volume of the emulsion but of much lower spatial frequency (fewer lines per millimeter across the surface of the film due to the close proximity between S1¢ and S2¢). These extra “mirrors” direct light in arbitrary directions when the hologram is illuminated with S alone, and create a background of “fog.”

When the scene consists of a three-dimensional object, each pair of points on the object creates a set of unwanted interference patterns. With a small object located far from the hologram, the range of angles between pairs of points (S ¢ and S ²) on the object is small and the noise problem is not severe. In the extreme case where the object is simply a point source S ¢, there is no intermodulation noise at all.

On the other hand, a large object situated near the hologram can create severe noise problems because every pair of points on it creates a set of unwanted interference patterns. Referring to Figure 10-8, light from S1¢ and S2¢ will interfere throughout the emulsion. The larger the object and the closer it is to the hologram, the higher the spatial frequency of its interference patterns (i.e., more mirrors created), along with the increase in the severity of the noise problem.

Beam Ratio

To help minimize the effects of intermodulation noise, practical holography requires that the reference beam be of higher intensity than light from any point on the object. In practice, the intensity ratio—as measured (at the location of the holoplate) by using a diffuser in front of a light meter, between the reference and object beams—varies from 1:1 to l0:1 for transmission holograms, the type so far under discussion. This allows the “mirrors” to form due to light from the reference beam and points on the object to be generally of higher reflectivity than those formed due to light from any pair of points on the object. Also, the noise can be further minimized by having the smallest angle between the reference beam and any object beam (qR in Figure 10-10) be greater than the largest angle formed by a pair of points on the object (qO in Figure 10-10). This ensures that the minimum spatial frequency formed by the object and the reference beam is greater than the spatial frequency of the noise. When the hologram is illuminated, the intermodulation noise is diverted to angles smaller than the signal. In this way, even though we cannot eliminate the noise, we can isolate it. In engineering terms, we have spatially isolated the frequency domains of the signal from the noise. In doing so, we have increased the signal-to-noise ratio, i.e., increased the quality of the hologram.

Multiple Scenes

One of the most dramatic features of a hologram is that it is capable of recording more than one independent scene in the same volume of space and then displaying them independently by changing the angle between the hologram and the reference beam. This phenomenon was discovered by a father and son team (William Henry and William Lawrence Bragg) through the study of X-ray diffraction by crystals. They were awarded the 1915 Nobel Prize in physics.

To understand what follows, it is necessary to review the subject of thin-film interference (Module 1-4, Basic Physical Optics) and be able to explain what causes the beautiful colors when diffused white light is reflected by soap bubbles.

Figure 10-12 shows a realistic cross-sectional view of the interference pattern formed in a microscopically small piece of hologram created using the configuration shown in Figure 10-5. (For simplicity, Snell refraction [Module 1-3, Basic Geometrical Optics] has been ignored by assuming that the hologram is sandwiched between material with the same index of refraction as the emulsion.) The hyperboloidal surfaces inside the thin emulsion layer approach flat planes and are perpendicular to the film surface, like venetian blinds in the “open” position. Because of the large angle between rays from S and S ¢ (about 90°), the separation between “mirrors” is about l, much smaller than the emulsion thickness of 10l. Here, l represents the wavelength of light inside the medium.

Figure 10-12  A microscopic cross section of a hologram. The thickness of the hologram far exceeds the distance between interference orders, and light from S undergoes multiple reflections.

When the hologram is viewed with S alone, the light undergoes multiple reflection by successive planes when it penetrates the film. However, because of the inherent characteristics of this family of hyperboloid “mirrors” (recall Figure 10-5), each successive reflection will have a precise phase shift of 2p because the optical path is increased by precisely the distance of one wavelength. All the reflected waves are precisely in phase and, therefore, add in amplitude, resulting in a strongest possible wave front representing the object beam. If the angle of incidence of S is deviated from the original reference beam relative to the hologram, all the multiply reflected beams will have phase differences other than 2p and the resultant reflected wave will be much lower in intensity, even zero.

Quantitatively, we can state that, in the case of correct illumination, the absolute values of all the amplitudes—a1, a2, ¼ an, of successive reflections—add directly and the intensity is I = (|a1| + |a2| + ¼ + |an|)2. In the case of misaligned illumination, the phase shift of each successive reflection is different from 2p and I¢ = (1 + 2 + ¼ + n)2, so that I > I ¢ always.

In practice, when the illuminating angle is significantly different from the correct angle, no image can be seen.

This phenomenon—Bragg diffraction—is of great historical significance. Using a beam of X rays of known wavelength, directing it at a crystal such as rock salt, and studying the angles of maximum diffraction, the distance between the atomic layers was measured. In turn, using crystals with known atomic spacing, X-ray wavelengths were accurately measured. This allowed scientists to study the inner workings of the atom.

In modern photonics, this principle is used to design multilayer thin films on mirror surfaces such as those used in laser cavities, to make optical fibers that selectively transmit and reflect chosen wavelengths, and to fabricate exotic crystal for massive holographic memories.

To create multiple-scene holograms (see the laboratory for making two-channel transmission holograms), one exposes the film with object O1, stops the exposure, changes to a second object O2, changes the angle between the reference beam and the film, and exposes a second time. Generally, each exposure time is one-half that of the exposure for a single-scene hologram, assuming no great change in object brightness. The resultant pattern recorded in the processed film is the equivalence of the superposition of two independent sets of hyperboloids, each corresponding to a given scene. During reconstruction, depending on the orientation of the hologram with respect to the reference beam, the wave front of one or the other scene can be recreated. This is true for both the real and the virtual images.

Depending on the thickness of the emulsion, the sizes of O1 and O2, and their proximity to the film, different degrees of success can be achieved in recording multiple images with minimum “cross-talk” (image overlap).

It is interesting to compare time-domain versus space-domain in information systems. A long song takes a long time to sing; a long written story requires a thick book (large volume) to record.

White-Light Reflection Holograms

Gabriel Lippmann received the 1908 Nobel Prize in physics for his discovery of recording color photographs on a “black-and-white” silver halide emulsion. Today, we use a similar “Lippmann emulsion” for recording holograms.

Figure 10-13 shows a photographic camera in which the lens focuses a colorful object on the recording emulsion several micrometers in thickness. The back of the emulsion is in physical contact with a pool of mercury, which serves as a mirror and reflects back all light incident on it.

Figure 10-13  Lippmann photographs are recorded in a camera in which mercury is used to reflect the light back through the emulsion to create a standing wave for each color of light.

Suppose that the image of the petal of a red rose falls in a particular region of the emulsion. The red light traverses the emulsion and is reflected back through it. The incident red light interferes with the reflected red light and forms a standing wave in which the distance between adjacent maxima (or minima) is 1/2l. In another area of the emulsion, a green leaf is focused there, forming an image. The green light similarly forms a standing wave pattern throughout the emulsion. Because green light has a shorter wavelength than red light, its interference pattern is of a higher spatial frequency. Another, still higher-frequency, pattern may be formed elsewhere where a patch of the blue sky is focused.

After the photograph is developed, layers of silver remain in all locations where the maxima of all the interference patterns are located. When illuminated with a patch of white light, each area of the photograph reflects—by constructive interference—a color corresponding to the original focused image. The result is a full-color photograph recorded in a colorless emulsion.

With the availability of highly coherent laser light and an improved Lippmann emulsion, we can use the same concept to record three-dimensional scenes in color.

Figure 10-14 shows again the overall interference pattern between point sources of coherent lights S and S ¢. Note that the region between S and S ¢ is occupied by standing waves, where the distance between adjacent maxima (or minima) is 1/2l. If we place a 10l-thick emulsion midway between the sources (where MN and PQ are surfaces at the opposite sides of the emulsion) and record the interference pattern, we will have 20 hyperboloidal surfaces throughout the hologram. When illuminated by S alone, a virtual image of S ¢ is created—and vice versa—as discussed earlier.

Recall that the interference patterns recorded in a transmission hologram consist of slivers of hyperboloidal “mirrors” (Figure 10-6) that are perpendicular to the plane of the emulsion, like venetian blinds in the “open” position. In a reflection hologram (Figure 10-14), the “mirrors” are parallel to the plane of the emulsion. Furthermore, the latter has twenty mirrors—for a 10l-thick emulsion, as explained above—whereas the other has only four (Figure 10-12) when each is being illuminated by its reference beam.

Figure 10-14  Reflection holograms are recorded in the region between the reference and object beams where standing waves are created. MN and PQ are surfaces on opposite sides of the emulsion. (Only a portion of the “mirror” surfaces is shown between planes MN and PQ.)

The result is that the reflection hologram can be viewed with incandescent light located at S, due to the fact that the additional sixteen “mirrors” strongly select the same wavelength of the laser that recorded them.

Figure 10-15 shows the simplest method of making holograms: A solid object is placed in contact with the holoplate, with the emulsion side facing the object. Light spreading from a diode laser (located at point S) is allowed to expose the emulsion. Light that passes through the holoplate illuminates the object. Each point on the object scatters light back through the emulsion and interferes with the direct light from S, creating a set of unique hyperboloidal “mirrors” throughout the volume of the hologram.

Figure 10-15  Simplest method of making a hologram: The spreading light from a diode laser interferes with the light scattered back by the object and forms standing waves throughout the volume of the recording medium.

When the developed hologram is illuminated by a point source of incandescent light, each set of hyperboloidal “mirrors” reconstructs a virtual image of a point on the object. Together, all the sets of “mirrors” reconstruct the entire three-dimensional image in a single color.

To make full-color holograms, light from three lasers (red, green, and blue) is combined into one single beam and the hologram is recorded in the same way as above. However, a special development process is required to ensure that there is no shrinkage in the emulsion.

At this point, let’s compare the main differences between transmission and reflection holograms: transmission holograms are made with both reference and object beams on the same side; the interference patterns recorded are like venetian blinds and are more or less perpendicular to the surface of the hologram. On the other hand, reflection holograms are made with the reference beam on the opposite side of the object beam. Its hyperboloidal fringes are more or less parallel to the surface of the hologram.

Since the entire object is on the same side of any hologram, any pair of points on it creates unwanted intermodulation noise by forming a transmission type of interference pattern. Therefore, when a reflection hologram is illuminated, the intermodulation noise is transmitted to the side away from the observer, while the image is reflected toward the observer. Thus, in making reflection holograms, the best beam ratio is one-to-one because we can ignore the intermodulation noise.

Holographic Interferometry

Consider a “double exposure” as represented by Figure 10-16a. Here the object is located at S1¢ during the first exposure. It is then moved to S2¢, a distance of the order of a few l’s, for the second exposure. The resultant hologram is a superposition of two sets of hyperboloids that coincide on the left side of the hologram but fall in between one another on the right side.

Figure 10-16a  A double-exposure hologram of S¢ in two locations. The two interference patterns are coincident at the left of the hologram but are anticoincident at the right side.

When the virtual image of the hologram is reconstructed by S, a bright point located in the vicinity of the original object is seen from the left side of the hologram. As the viewer moves toward the right side of the hologram, the object S ¢ disappears! The reason is that, at the right side, the two superposed interference patterns cancel each other when the maxima of one set fall in the positions of the minima of the other set.

If the object point had been displaced differently (from S1¢ to S3¢, as shown in Figure 10-16b), its interference pattern would be shifted in such a way that the two patterns are anticoincident on the left side of the hologram but coincident on the right side, and the situation then would be reversed. (Note: A comparison of Figures 10-16a and 10-16b will not disclose the small difference between positions S1¢, S2¢ in Figure 10-16a and S1¢, S3¢ in Figure 10-16b. Nevertheless, the two sets of positions are sufficiently different to interchange the coincident and anticoincident characteristics of the holoplate.)

Figure 10-16b  A double-exposure hologram of S¢ in two different locations. The two interference patterns are anticoincident at the left of the hologram but coincident at the right side.

Therefore, when a rigid object is slightly moved (displaced) between the two exposures, a set of straight black lines (fringes) will be superimposed on the image of the object. Each point on any given fringe will displace the same distance. On the other hand, if the object is deformed (such as a “live” plant having grown between exposures), complex curved fringes will be observed.

All points on a given fringe represent locations with the same displacement.

“Time-averaged” holographic interferometry is a single-exposure hologram of an object under normal mode vibration. The antinodal areas are moving, so the interference patterns are “smeared out.” The nodal areas have not moved; thus they appear bright. This renders vibration modes directly visible. (This technique has been used in the study of musical instruments.)

“Real-time” holographic interferometry involves making a hologram, processing it, relocating it in the position where it was made, and observing the virtual image superposed on the real object. If the object is moved or deformed (such as a growing object), the wave front of the image interferes with light from the real object, and an interference pattern can be observed in real time.

Holographic interferometry encompasses an entire discipline in mechanical engineering and is applied widely to solve myriad technical problems, mostly in the area called nondestructive testing.

Coherence Length (temporal coherence)

One of the basic techniques in making good holograms is to ensure that the optical path of the object and the reference beams are equalized well within the coherence length of the laser light. Beginning from the output of the laser, trace the total distance traveled by the reference beam to the holoplate. Now do the same for the object beam. The difference in the two distances must not exceed the coherence length. So, what is this coherence length?

In general, lasers operating without internal etalons emit simultaneously more than one frequency (see Module 1-5, Lasers). For example, the low-cost helium-neon lasers emit typically two to three different frequencies simultaneously.

When a hologram is recorded by using a two-frequency laser, one can think of it as the sum of two holograms being made simultaneously, one at each frequency. The higher-frequency interference pattern is caused by light with a shorter wavelength, thus it has more interference orders over any given space. Figure 10-17 is the superposition (summation) of two independent sets of interference patterns, one caused by S1 and S1¢ of one frequency and the other caused by S2 and S2¢ of a slightly higher frequency.

Figure 10-17  A superposition of two independent sets of interference patterns, each caused by light of slightly different wavelength from the other. The smudge-like area is caused by the anticoincidence of the two patterns; no hologram can be recorded there.

Near the zeroth order OO¢ in Figure 10-17 (the flat plane that is midway between the two interference point sources, the two sets of interference patterns coincide. Near this region, light of any color can produce distinct interference patterns. Lippmann photography is based on this fact. If a holoplate is located in this area, a good hologram can be made. On the other hand, notice the two dark, smudge-like areas, one on each side of the zeroth order. This is caused by the fact that the maxima of one pattern are located at or near the minima of the other pattern. If the holoplate is located in this region, the hologram will be very dim or not exist at all.

The coherence length of this two-frequency laser can be defined verbally as the difference in distances travel by light from points S1, S2 and S1¢, S2¢ to the center of either dark area. For a typical HeNe laser, this distance is about 30 cm.

Notice (Figure 10-17) that, if we increase the difference in the optical paths, the two patterns become coincident again.

For lasers with internal etalons or for highly stabilized diode lasers, the output has only one frequency and the coherence length is many meters. When using such lasers for making holograms, one does not need to equalize optical paths between reference and object beams.

Thin holograms

As is true with all “models,” the physical model for holography must break down at a limit (like the stick-and-ball model for molecules). Our limit is the “thin” hologram.

A hologram is considered “thin” when the separation between successive hyperboloidal surfaces exceeds the thickness of the recording medium. At this point, the model breaks down because there no longer is a set of hyperboloidal surfaces to be recorded. For example, embossed holograms (baseball cards and VISA/MC cards) are made by stamping a set of two-dimensional interference patterns on a sheet of plastic. These holograms cannot reproduce true color because there is no Bragg diffraction.

Then how does one explain the theory of a thin hologram?

A thin hologram is made by recording a two-dimensional interference pattern between a reference beam and light from the object on the surface of a recording medium (Figure 10-18). If the location of the holoplate is sufficiently far from S and S ¢, the two-dimensional interference pattern formed at the surface of the holoplate consists of Young’s double-slit interference patterns (Module 1-4, Basic Physical Optics). The pattern consists of a set of straight dark and bright parallel lines whose spatial frequency depends on the distance of S from S ¢ (slit separation) and the distance the hologram is from S and S ¢ (screen distance).

Figure 10-18  A thin hologram is made by recording a two-dimensional interference pattern on the surface of a recording medium. The result is a diffraction grating.

We learned previously (Module 1-4, Basic Physical Optics) also that a set of parallel dark and transparent lines is called a diffraction grating. When we are making a thin hologram, light from each point (S ¢) on the object interferes with S and forms a two-dimensional diffraction-grating-like pattern on each small area of the hologram. Therefore, each small area of a thin hologram is a summation of diffraction gratings formed by the interference of light from each point on the object with the reference beam. The branch of mathematics used for quantitative study of holography is called Fourier transformation, and the formation of a hologram is a process called Fourier synthesis.

When the hologram is illuminated by S, each grating on each elementary area diffracts light to recreate the wave fronts of an object point (a process of Fourier analysis). If white light is used to illuminate this hologram, each different wavelength will be diffracted to a different angle and the image is smeared into a continuous spectrum. Thus, the transmission holograms discussed previously are best viewed with laser light.

In reality, most holograms are “quasi-thick” and “quasi-thin,” requiring a combination of physical explanations.