COMPUTER GENERATED HOLOGRAMS Optical Sciences W.J. Dallas PART II: CHAPTER ONE HOLOGRAPHY IN A NUTSHELL

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1 What is a Hologram? Holography in a Nutshell: Page 1 o 1 C:\_Dallas\_Courses\3_OpSci_67 8\ MsWord\_TheCgh\1_MSWord\_1 Holography.doc Version: Wednesday, September 4, 8, 8: AM COMPUTER GENERATED HOLOGRAMS Optical Sciences W.J. Dallas PART II: CHAPTER ONE HOLOGRAPHY IN A NUTSHELL The word "hologram" is assembled rom Greek words approximating the meaning "entire recording". Entire in this context is meant to distinguish the hologram rom the photograph. A photograph records only the irradiance or strength o an incident wave whereas the hologram records not only the strength o the wave at each point on its surace but also the direction that the wave is propagating at that point. The direction o propagation is equivalent o knowledge o the phase o the wave across the recording medium. The gradient o the phase gives the propagation direction. Selected Events in the History o Holography The ollowing are a ew milestones in the development o computer generated and intererometric holography. 1. Bragg (Nature, 1939). In this experiment, an x-ray diractogram was recorded o a crystal. The x- ray diractogram is in essence a Fourier transorm o the crystal structure. Bragg reasoned that or a center symmetric structure, the diractogram would be real so that recording the x-rays with ilm would not lose the diractogram phase. He then synthesized a hologram by drilling holes at the diraction peak locations in a sheet o brass. Illuminating with a ilter mercury arc lamp, the wave propagation implemented an inverse Fourier transorm and so the atoms in the crystal structure were reconstructed. Because o the wave length dierence between x-rays and optical waves, a strong magniication was achieved.. Gabor (Nature, 1948). Gabor actually named the hologram. He was attempting to develop a method or recording and reconstructing electron micrographs. In the process, what we now call the "on axis intererometric hologram" was born. He imaged a low contrast object that was well described as the sum o two waves: the background and modulation. The hologram was ormed by the intererence o these waves. 3. Rogers (Proceedings o the Royal Society o Edinburgh, ). Rogers reasoned that or simple objects, or instance a wire, the intererence pattern or an on-axis hologram could be analytically calculated and the synthetic hologram drawn by hand. He actually abricated the holograms and reconstructed the images. 4. Lohmann (Optica Acta, 1956). With single-sideband holography Lohmann combined communications theoretical and physical views o optics to address the twin-image problem o Gabor holograms. 5. Leith and Upatnieks (Journal o the Optical Society o America, 1961) invented the o-axis hologram. With the advent o the laser and the invention o this technique, very high quality holograms began to capture the imagination o the scientiic and popular world.

2 Waves Holography in a Nutshell: Page o 1 C:\_Dallas\_Courses\3_OpSci_67 8\ MsWord\_TheCgh\1_MSWord\_1 Holography.doc Version: Wednesday, September 4, 8, 8: AM 6. Lohmann and Brown (Applied Optics, 1966) invented the binary computer-generated hologram. Here the newly increased power o computers and the recently invented ast Fourier transorm algorithm were combined with a calculation method utilizing detour phase to create holograms that had transmittance o only zero and one across their suraces. 7. MacGovern and Wyant (Applied Optics, 1971) applied the computer generate hologram to testing optical elements. This CGH application has been the most successul and widely used. We consider monochromatic wave ields, those where the electrical ield can be expressed as ( x, y,z,t ) = s ( x, y,z) e π ν E E We will generally consider a scalar which may be interpreted as one component o the E-ield with the time variation removed. This scalar we call the complex amplitude u which is related to the electric ield by i t π ν E x x, y,z,t = u x, y,z e The complex amplitude obeys the wave equation. We will be using the source-driven orm o the wave equation ( + k ) u = s where k is the magnitude o the wave vector. We consider below three simple types o complex amplitude waves: the plane wave, the spherical wave and the pinhole wave. These waves are useul because general wave ields can be decomposed into sums o any one o those three. Plane Wave The plane wave has wave ronts or equiphase suraces which are planes. These planes are parallel and equally spaced by the wave length. The mathematical expression is: i t Spherical Wave i k ir πi ρi r πi ( ξ x + η y + ζ z) u p x, y, z = e = e = e The spherical wave has equiphase suraces which are concentric spheres equally spaced by a wave length. The expression is: i k r πi ρ r e e u s ( x, y, z ) = = r r Pinhole Wave Another important wave is that which propagates rom a pinhole in an ininite plane. It is known as the Huygen s wavelet, but we just reer to it as the pinhole wave. The pinhole wave's complex amplitude is

3 Holography in a Nutshell: Page 3 o 1 C:\_Dallas\_Courses\3_OpSci_67 8\ MsWord\_TheCgh\1_MSWord\_1 Holography.doc Version: Wednesday, September 4, 8, 8: AM u ikr ikr ph 1 e x, y, z = = 1 ik 1 z 1 i e kz π z r e π r r π r ikr The second term, 1, decays very rapidly away rom the pinhole plane so it can be ignored. A quick r explanation o why the pinhole wave is properly named can be seen rom looking at the waveronts o the spherical wave in the plane z. These waveronts are all pointing along the plane except at point (x=,y=). The "z" derivative o the wave is then "" in that plane except at the point (,). In other words it describes the wave propagating away rom an illuminated pinhole. To see that this wave actually satisies the wave equation to the right o the plane, we simply substituted in the wave equation, exchange the derivative order and see that it does satisy the wave equation. The ollowing acts will be very useul to us in our considerations o optical propagation. Because we are working with linear shit-invariant systems they can be described by point spread unctions, i.e., ps 's. The pinhole wave is the ree-space ps or wave propagation rom the x-y plane into the right hal-space. The Parabolic Approximation It is convenient when considering propagation pinhole waves to make an approximation. This approximation is to replace the spherical waveronts by paraboloids. In Appendix A we explain the approximation and simpliications that result in the ollowing expression or the pinhole wave radiating rom the origin and traveling in the z-direction. iπ ( x + y ) λ z u ph( x, y,z ) = e Optical Elements We will be working with complex-amplitude transmitting thin elements. These elements will be characterized by a complex-amplitude transmittance. This transmittance describes the element's action on a general wave. I we suppose that the element is located in the plane "z = ", then an incident wave slightly to the let o the element will be converted to an exiting wave slightly to the right o the element in the manner: + u x, y, = u x, y, t x, y The Neutral-Density Filter The thin neutral-density ilter is characterized by a constant amplitude attenuation The Plane-Parallel Plate t nd_ ilter ( x, y ) = A The thin plane-parallel plate is characterized by a constant phase shit over its entire surace. The expression or the plate's complex amplitude transmittance is t i plate x, y = e φ The Prism A prism is a thin element which has a phase shit that varies linearly with position. The expression or the prism's complex amplitude transmittance is

4 Holography in a Nutshell: Page 4 o 1 C:\_Dallas\_Courses\3_OpSci_67 8\ MsWord\_TheCgh\1_MSWord\_1 Holography.doc Version: Wednesday, September 4, 8, 8: AM i x + t y prism x, y = e π ξ η where ξ and η are constants. The Lens The lens is a ocusing element. It essentially converts one pinhole wave into a second pinhole wave. It is considered to have, in the parabolic approximation, a quadratic phase actor. The complex-amplitude transmittance o a lens located on-axis is Diraction Gratings iπ ( x +y ) λ tlens x, y = e ± power Diraction gratings are periodic. The structure internal to a period, the groove shape, is quite arbitrary. We consider only two types o gratings: the cosine grating and the square-wave grating. The cosine grating has complex amplitude transmittance: t A 1 π π x, y = A 1 + π ( ( ξ x + η y ) = A + A e + A i ξ x + η y i ξ x + η y cosine grating cos 1 A 1e The square-wave grating has binary transmittance, that is it transmits either none or all o the light depending on where the light is incident on its surace. The expression or the square-wave grating is, x-m Sq ( a; x ) = rect = a sinc ( ma ) e m=- a m=- 1 1 or x < 1 1 sin ( π x) rect x = or x = sinc x =. π x 1 or x > where and The Ronchi ruling is a special case o a square-wave grating where the duty cycle, a, the ratio o the slit width to the period is one-hal so that the opaque and clear stripes have equal width. The Optical Fourier Transorm A lens is a Fourier transormer, it takes a complex amplitude distribution in its ront ocal plane and produces the Fourier transorm o that distribution in its back ocal plane. See Figure 1 or a graphical rendition o the arrangement described. The explanation o this surprising act can be broken into two steps. The irst step is to look at the direction o propagation o a plane wave as a unction o the distance between the intersections o its peaks and the input plane. This relation is λ sin θ = d Note that the spatial requency o the plane wave in the input plane is just the reciprocal o the peak πimx )

5 spacing: Holography in a Nutshell: Page 5 o 1 C:\_Dallas\_Courses\3_OpSci_67 8\ MsWord\_TheCgh\1_MSWord\_1 Holography.doc Version: Wednesday, September 4, 8, 8: AM 1 sinθ ξ = = d λ

6 Holography in a Nutshell: Page 6 o 1 C:\_Dallas\_Courses\3_OpSci_67 8\ MsWord\_TheCgh\1_MSWord\_1 Holography.doc Version: Wednesday, September 4, 8, 8: AM The second stage is to calculate the position o the point in the output plane that the plane wave is ocused to. This calculation needs only the simplest trigonometry, x tanθ = and sinθ tanθ thereore sinθ tanθ x ξ = = λ λ λ One common source o conusion is the relation between the position coordinates in the back ocal plane and the spatial requency components o the object. The spatial requencies are expressed in the reduced, x y is the actual position in the back ocal plane, λ is the wavelength o light and coordinates ( ξ η ) where (, ) is the ocal length o the lens. The reduced coordinates are x y ξ= η= λ λ We will concentrate our attention on the telecentric imaging system or " 4 " imaging system. This imaging system consists o two lenses o equal ocal length separated by twice that ocal length. The input plane is one ocal length in ront o the irst lens and the output plane is one ocal length behind the second lens. An intermediate plane midway between the lenses is the Fourier plane. Recording Media Characteristics There are two recording media that will be important to us: monochrome photographic ilm and color slide ilm. Linear T-E Monochrome Film The monochrome photographic ilms used or recording intererometric holograms and gray-tone CGH's have linear transmission versus exposure, T-E, curves. The linear T-E operating region o a photographic material is generally ound in the toe portion o the Hurter-Driield curve. See Figure or a stylized comparison o the regions. The advantage o using a T-E expression or the photographic processing is that the irradiance appears simply multiplied by a constant in the transmission. High-Contrast Monochrome Film The monochrome photographic ilms used or recording binary CGH's generally have a high-contrasth-d curve. The high contrast acilitates elimination o unwanted gray-tones. Color Slide Film Consider color transparency slide ilm. Slide ilm consists o our layers: three color layers and one blue blocking layer. For our discussions, only the color absorbing layers are important. The layers are tuned to absorb preerentially red, green or blue light. Ater photographic processing, each o these layers modiies only the corresponding spectral band. So, or instance, i the ilm is unexposed, no light is transmitted. I

7 Holography in a Nutshell: Page 7 o 1 C:\_Dallas\_Courses\3_OpSci_67 8\ MsWord\_TheCgh\1_MSWord\_1 Holography.doc Version: Wednesday, September 4, 8, 8: AM the ilm is exposed only in red light, then only red light is transmitted. I it is exposed only in a mixture o red and green light, then only red and green light will be transmitted and the color range will be between red and green passing through cyan. I we consider using red light, or instance a helium neon laser at 63.8 nm, then it is easy to see that we can control the amplitude transmitted through the slide ilm by exposing it to a red image. What is not so clear is that the exposure to red light causes not only an amplitude change, but also a phase change because the thickness o the ilm will vary according to exposure and the index o reraction. We can use this eect to good advantage by using the other layers to compensate in this phase variation and, in addition, supply a desired phase distribution. The Hologram as a Distorted Diraction Grating The underpinnings o holography actually came rom ruled diraction gratings. A grating ruling machine draws an enormous number o very straight parallel lines. When, however, the ruling engine is not perect, these lines are also not perect. They may not be straight, they may not be parallel. Imperections in gratings create what is known as "grating ghosts" in the diraction pattern. The grating ghosts are a deviation o the diraction pattern rom that o pure points to blurred-out points. A hologram can be viewed as a grating that has been deliberately distorted in order to generate careully deined ghosts. The ghosts are the reconstructed images. Fringe Distortions in Intererometric Holograms It is straight orward to include the distortions in discussing intererometric holograms. Consider an intererometric grating that is generated by intererence o two plane waves on a photographic medium. See Figure 3 or an illustration o the grating=s ormation, and diraction properties. The waves are designated as the reerence wave and the object wave. The irradiance pattern is as ollows: * * r r r o r o I = u+ u = u + u + uu + uu u = 1 ; u = e r πξ i x ( πξ ) I = 1+ cos x Photographic processing yields a grating with complex amplitude transmittance: ( πξ ) T =c c1i = c c 1 c1cos x Now consider a reerence wave which is a not plane wave but a second wave we will call the object wave that has the ollowing amplitude and phase distortions φ π ξ i x,y i x u = A x, y e e The complex amplitude transmittance o the processed photographic material has the orm: T = c c1 1+ A ( x, y ) c1 A( x, y ) cos πξ x + φ( x, y ) As we can see, the perectly regular cosine ringes o the undistorted grating have now been changed. First, there is an amplitude variation but second, we see that the positions o the ringes have now been changed. These changes are closely related to the phase. On reconstruction, we have a number o terms which naturally separate themselves by viewing angle. Thereore we can see a reconstruction o the object wave's

8 Holography in a Nutshell: Page 8 o 1 C:\_Dallas\_Courses\3_OpSci_67 8\ MsWord\_TheCgh\1_MSWord\_1 Holography.doc Version: Wednesday, September 4, 8, 8: AM x, y arguments we have complex amplitude when viewing at a certain angle. Suppressing the ξ ut= ( c c ) c A c Ae e c Ae e r iφ πi x iφ πiξx ur = 1, the reconstruction wave The terms on the right side o this equation correspond to various images. The irst term corresponds to the on-axis point sometimes known as the DC Spike. The second term is auto-correlation term. The third term corresponds to the desired reconstruction. We notice it is located o-axis. The ourth term corresponds to the twin image. This image is located axially opposed (on the opposite side o the zero order) to the desired reconstruction. Binary Holograms A binary hologram is black and white, really opaque and transparent. We can extend the analogy between intererometic gratings and holograms to binary holograms, considering the process o hard clipping. An ideal cosine grating which is hard clipped with the clipping levels set to its average value, becomes a square- wave grating. We know that the eect o this process is to generate higher diraction orders but not to distort the originally present orders. This analogy carries through as we can expand the groove shape o the grating in a Fourier series and substitute in the more complicated argument. The irst harmonic contains the reconstructed wave ront that we desire. Incidentally the mathematical procedure we have just applied is known as "Generalized Harmonic Analysis". Detour Phase There is an alternate and intuitive way to see how the phase is inluenced by shits in the ringe position. I we imagine a grating diracting a normally incident plane wave, the waves headed to the various diraction orders are plane waves. We now concentrate our attention on the wave traveling toward the irst order. All o the slits in the grating are lined up so that they are in phase with that wave. Now suppose we translate one o the ringes, thereby changing the ringe spacing locally. The ringe has now been moved nearer or urther rom the diraction order. This change in distance must correspond to a change in phase. The wave has been detoured rom its original path hence the term detour phase. Figure 4 illustrates this behavior. Diusers The diuser is an important but invisible component that is incorporated in the computer-generated hologram. When we speciy the object, generally we are concerned only with the irradiance distribution. An arbitrary phase pattern can be superimposed without aecting this irradiance. A phase, such as a random phase or an optimized diuser phase distribution, has proound eects on the computer generated hologram. For instance, the random phase will spread the light to more distant corners o the hologram. It will improve the hologram's diraction eiciency and it will distribute the image inormation around the hologram so that it lends a burst-error insensitivity to the hologram. When a diuser is introduced into the object, it not only has the positive eects described but also causes some negative results. These negative results come rom the act that two imaging systems have stops and the diuser causes some light to scatter so that it is blocked by these stops. In the simplest case, one may consider the stop to be in the Fourier plane so that a low pass spatial iltering process takes place. This low

9 Holography in a Nutshell: Page 9 o 1 C:\_Dallas\_Courses\3_OpSci_67 8\ MsWord\_TheCgh\1_MSWord\_1 Holography.doc Version: Wednesday, September 4, 8, 8: AM pass process causes high requency details to disappear. I we consider the case o a uniorm diuser, the image without a stop would simply be uniorm. With a stop, the missing high requency inormation causes dark structures to appear in the image. These dark structures resemble a plate o black spaghetti. This structure is known as speckle. In act, it is easy to calculate the spatial requency o these structures rom the size o the stop. For simplicity let us limit ourselves to one dimension. Consider the stop to have an opening o length x. The bandwidth o the passed spatial requencies, calculated rom the reduced coordinate relations, is and the speckle structure has spatial requencies x ξ = λ Phase-Only Holograms > x ξ λ Just as the name implies, the phase only hologram records not the complex amplitude, but only the phase o the wave at the hologram. The amplitude is ignored or destroyed. For our considerations, the complex amplitude transmittance o a phase only hologram corresponding to the wave complex amplitude is just i (, ) ( ξη ) A, e φ ξη i (, ) e φ ξη There is a distinction between a phase only hologram and a phase hologram. The phase only hologram records only the phase o the object wave at the hologram. The phase hologram, on the other hand, reers to the entire transmittance o the hologram so that inormation is coded into a phase medium. In act, amplitude and phase o the object wave can be stored in a phase hologram. Depth Eects Holograms are most amous or their ability to reconstruct three-dimensional images. One simple way o looking at this 3-D reconstruction ability is to consider an object consisting o a single point. This point can be reconstructed i the hologram contains the transmittance o a lens. With a plane wave incident on the hologram, a point will be constructed at the ocal length o this lens. I we then consider a sel-luminous object, one which is built up rom several point images, the corresponding hologram can be looked at as the sum o lens transmittances. This is one way to build in the 3-D eect. Another, slightly dierent, method is to look at the hologram transmittance which would be necessary to reconstruct an entire plane and then multiply this unction by a lens in order to pull the reconstruction out o the Fourier plane. In any case, the chie element responsible or incorporating depth eects into holograms is the lens.

10 The pinhole wave is Holography in a Nutshell: Page 1 o 1 C:\_Dallas\_Courses\3_OpSci_67 8\ MsWord\_TheCgh\1_MSWord\_1 Holography.doc Version: Wednesday, September 4, 8, 8: AM APPENDIX A DETAILS OF THE PARABOLIC OR PARAXIAL APPROXIMATION u 1 1 ( x, y, z ) = ik z 1 ikz e π r r π r ikr ph e The magnitude o the radius vector in the exponent is replaced by its irst-order Taylor expansion. The radius is ( ) ( y ) ( ) ikr r = x x + y + z z We begin by actoring out the "z" distance rom under the radical and noting that when the "z" distance is large compared to the "x" and "y" distances, the square root can be expanded using the standard Taylor series expansion. The standard Taylor expansion or one-dimension is the irst order approximation is For our application the irst-order expansion is Simple substitution gives us the inal orm ( x x ) + ( y y ) ( z z ) r = ( z z ) 1 + n=! ( n ) ( x ) ( x) = x x n (x) x + x x x r ( z z ) + x + x ( x x ) + ( y y ) ( z z ) n The in the denominator is simply replaced by its zero order Taylor expansion which is just (z z ). The r resulting parabolic approximation or the pinhole wave is

11 Holography in a Nutshell: Page 11 o 1 C:\_Dallas\_Courses\3_OpSci_67 8\ MsWord\_TheCgh\1_MSWord\_1 Holography.doc Version: Wednesday, September 4, 8, 8: AM iπ 1 i k ( z z ) ( x x ) + ( y y ) ( ) z uph x, y, z z λ e e i z z λ ( ) Now we will make a simpliication that we will use repetedly. Because the actor 1 i k z z ( ) iλ z z e ( ) is a constant in any given plane we will supress it. The parabolic approximation o the pinhole wave then becomes ( ) u ph x, y, z z = e iπ λz ( x x ) + ( y y ) The pinhole wave radiating rom the coordinate system's origin is u ph x, y, z = e iπ λ z ( x + y )

12 FIGURES Holography in a Nutshell: Page 1 o 1 C:\_Dallas\_Courses\3_OpSci_67 8\ MsWord\_TheCgh\1_MSWord\_1 Holography.doc Version: Wednesday, September 4, 8, 8: AM Optical Fourier Transorm PHOTOGRAPHIC FILM CHARACTERISTICS D Hurter-Driield (H&D) Transmission vs Exposure (T-E) T x θ d θ λ d = sin λ ξ 1 = = d sin λ sin x = tan θ = = 1 sin θ 1 λξ For small angles: x Linear T-E log E Linear T-E E II_1_OpticalFT Figure 1 Figure II_1_FilmCharacteristics INTERFEROMETRIC FOURIER HOLOGRAM OF A SINGLE POINT Recording Plane Wave DETOUR PHASE Displaced Aperture Reconstruction W a v e P ropagation D irection or the 1 D iraction O r d e r s t Grating II_1_PointHologram Figure 3 Figure 4 II-1_DetourPhase