COMPUTER GENERATED HOLOGRAMS Optical Sciences 627 W.J. Dallas (Monday, August 23, 2004, 12:38 PM) PART III: CHAPTER ONE DIFFUSERS FOR CGH S

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1 COPUTER GEERATED HOLOGRAS Optical Sciences 67 W.J. Dallas (onday, August 3, 004, 1:38 P) PART III: CHAPTER OE DIFFUSERS FOR CGH S Part III: Chapter One Page 1 of 8 Introduction Hologras for display purposes are generally ade of diffusely reflecting surfaces or transitting objects with a super-iposed diffuser. The reason for applying a diffusing eleent is to spread light about the hologra. Diffusers can be engineered to optiize any if several criteria. Aong these is unifority of the Fourier transfor, band-liiting of the Fourier transfor and spreading inforation redundantly about the Fourier plane. Incidentally spreading light about the Fourier plane is not the sae as spreading inforation. It is possible to design a diffuser that spreads the light alost uniforly, but still allows the inforation associated with local areas in the object to reain clustered in the Fourier transfor. Desirable Properties of the Fourier Spectru Unifority Band Liitation Redundancy Unifority The reason for spreading the light about the Fourier plane is to iniize the range of aplitude inforation that the hologra ust capture. The relevant paraeter in this consideration is the so-called dynaic range. The dynaic range is the ratio of the axiu range to the step-size. It turns out to be then the nuber of steps that can be reproduced in the aplitude. Typical values for a good grey tone laser fil plotter is about 000 steps. For a laser printer, on the other hand, the plotted values are either dark or white, that is a binary plotting range. The second reason for having unifor aplitude is to axiize the diffraction efficiency. The aplitude is noralized to a range (0-1). Perforing this noralization eans that if there is a great variation in the aplitude, especially if a large portion of the aplitudes are sall and a very nuber are large, as is generally the case, for an object without a diffuser, then a sall portion of the light will be diffracted. Another way of stating the fact that the spectru should be unifor is that the object itself would have a delta-like auto-correlation. This relation arises fro the Fourier transfor connection between the Fourierpower spectru and the auto-correlation in an object. Band Liitation Speckle is a grainy-appearance of the reconstructed iage that arises fro high frequencies of the original object being lost through truncation of the Fourier spectru. Because diffusers spread light about the Fourier plane, it is an optiization criteria that is quite iportant that this spreading not be too strong, if it is too strong, light will be spread outside of the area that can be captured by the hologra and therefore speckle will be caused. The speckle cab can also be reduced by reconstruction techniques in addition to the

2 Part III: Chapter One Page of 8 careful design of the original diffuser. The first technique is to replicate the hologra. Replicating the hologra on a regular grid will cause the reconstruction to be point-like. This point-like object will lack speckle because speckle occurs in the regions between the saple points. The iage can then be resoothed using illuination that does not spread the saple valued to great a distance retaining a sort of lupy appearance. The other way is to use partially coherent illuination in the reconstruction because this will iniize the aount of interfacing between the saples. This will give a sooth reconstruction. Redundancy Redundancy in inforation storage can have two significant values. One is that the hologra ay be daaged and still reconstruct an iage. In fact, interferoetric hologras have been deonstrated that can be broken into sall pieces and when the sall pieces reconstructed; the iage is within slightly decreased resolution reconstruction caused by loss of sall copact areas in the hologra. By spreading the inforation around, these burst errors have a reduced effect on the reconstruction quality of the hologra. Diffusers for Interferoetric Hologras ethods for applying diffusers to sooth objects in interferoetric holography use Scotch agic Tape which is a good diffuser or a ground glass in contact with the object. The second ethod is to use a diffuser that is a bleached speckle pattern where the speckle pattern was fored with band-liited illuination. For reflecting objects, the surfaces can be rough or painted with diffusely reflecting paint. There are two classes of diffusers: the so-called general purpose diffusers and the objects specified diffusers. The general purpose diffusers are designed to be applicable to a large class of objects. The coplex aplitude transission of these diffusers is independent of the object being used. The object s specific diffusers, on the other hand, have values that are dependent on the object s aplitude. RADAR Codes There are a class of pulse codes that were developed for ranging applications in TADAR that were used as diffusers for coputer generated hologras. The range of a target is deterined by the transit tie of a pulse sent fro the transitter and detected at the receiver. This pulse ust have two properties that at first appear to be contradictory. First, the pulse ust be narrow, ore precisely; the autocorrelation of the pulse ust approach that of a delta function. Secondly, the pulse power cannot be arbitrarily high. This contradiction is resolved by using pulse codes that are extended and near unifor, but whose autocorrelations are delta like. The conversion for use as diffusers is iediate. A delta autocorrelation eans that the power spectral density is unifor. The biggest difference is that the diffusers are two diensional while RADAR codes are one diensional. Schroeder Codes The Schroeder codes have their origin in radar. The objective is to develop a code for a pulse sequence that when reflected fro a distant object, return it to the receiver and auto-correlate it with itself fors a delta function thereby aking ranging precise. As entioned above, this delta auto-correlation property is equivalent to unifority of the power spectru and therefore these codes have been directly taken fro radar and applied in coputer generated holography. The independent version of the Schroeder code is n φn = π +

3 Part III: Chapter One Page 3 of 8 Object independent Schroeder code We see that the for of the Schroeder code is essentially that if a lens. There is, in fact an easy optic analog looking at the input object being a lens whose focal length is exactly correct to focus a norally incident plane wave on the center of the Fourier transforing lens. The wave coing fro the Fourier transforing lens then is an equivalent of a spherical wave which will give unifor aplitude across the Fourier plane. The proble with this particular diffuser is burst error sensitivity. If an area of the input diffuser (lens) is blocked, there is ore or less a shadow cast into the Fourier plane. The object dependent fro of the Schroeder code is π φ= π α p + α = 0 for = 1 / α = P = u u α Second ter = 0 for odd nuber of points where the second ter equals zero for odd nubers of points, the object dependence is given by the for of the variable p which is Frank Heiiller-Codes This code faily has only general purpose ebers and they are only exact under the discrete Fourier transfor. The phases for the Frank-Heiiller diffusers are generated by [ ] φn = π ασ / + βnτn / α = σ β = n τ σ = = n 1 τn n 1 OD OD There are soe degrees of freedo in generating these diffusers and, in fact, one particular burst error insensitive choice is n φn π ( ασ / ) ( Bnτ / ) π This odified Frank-Heiiller code can be generated by the following forula [D=diffuser coplex aplitude]

4 Part III: Chapter One Page 4 of αβ π i αr bs D α + α, b + β = Trs e r= s= where Trs αβ is chosen so that for each subscript pair (r,s), there is one and only one superscript pair (α,β) for which Trs αβ =1 and is otherwise zero. The unodified Frank-Heiiller code corresponds to Trs αβ = δαrδβ s Algorths cae fro the Butterfly diagra of the FFT in a square root doain. Interlace sapling, individual DFT, ultiply with a linear phase factor. Calabro-Wolf (Hyperbolic) Codes The idea of the square root doain and populating a generating atrix with one delta function per row and per colun can be carried over to generate the Calabro-Wolf codes. Instead of a square-root doain, a transition doain is forulated where the transition doain is id-way through the Fourier transforation in the sense that first rows are transfored, then one is on the transition doain, then the coluns are transfored of this transition atrix to give the full Fourier transfor. The algorith is then the Calabro- Wolf codes existed before this particular algorith. The algorith does give additional flexibility in generating codes in that the delta function phase can be non-zero. Choose S = δ S = U e jk U = S e n ( ) πij/ n nk πi S such that for each j there is one and only one n for which S = 1. Original hyperbolic code odern ethods The Schroeder, Frank-Heiiller and Colabro-Wolf codes are exaples of traditional radar-based diffusers. The ore odern approach is one of optiization. For an optiization approach, it is generally necessary to quantize the diffuser phase so that it can be changed in a certain step-size during the optiization procedure and to choose an optiization criterion in the discrete Fourier transfor. For exaple

5 Siulated Annealing Iterative (Gerchberg-Saxton) Part III: Chapter One Page 5 of 8 Siulated Annealing A siple optiization algorith is as follows: Choose step size in changing φ Choose unifority criterion in DFT, e.g. U n Un U n U n n U U Change φ at one point by step-size Start with soe selection Randoly choose point Randoly choose direction if change ± by one increent DFT Check for unifority If ore unifor, accept change; otherwise reject Etc. for a long tie A slightly ore sophisticated approach is to use an annealing schedule. At the following steps: Accept change with a certain probability even if the change is in the in the wrong direction. Decrease that probability with it iteration nuber e Guarantees global optiu at T n/ kt These additional steps guarantee that a global optiu is attained as the teperature approaches infinity. Unfortunately this also eans the nuber of operations approach infinity. Aong the best ethods of deterining a diffuser for a particular object is the Lese-Jordan-Hirsh (Gerchbert-Saxton) procedure. Frechberg-Saxton (Lese-Jordan-Hirsh) Procedure This algorith was used by Lese, Jordan and Hirsch to design diffusers for Kinofors. They patented the algorith but didn t publish. Gerchberg and Saxton approached the proble fro a different perspective. They cae fro electron icroscopy and applied the ethod to the proble of phase retrieval. That proble is to calculate the coplex aplitude on the basis of inforation of the odulus in the object and the odulus of its Fourier transfor. The algorith starts by assigning a rando phase distribution to the object. It proceeds by ping ponging between direct space and Fourier space and in each space setting the agnitudes to the proper values while retaining the phase fro the iteration cycle. There are iportant

6 Part III: Chapter One Page 6 of 8 odifications to this procedure. The Fienup odification is one of the ost iportant. Instead pf violently returning the aplitudes to their values, a relaxation procedure is used. This relaxation procedure allows the aplitude values to slowly approach their proper values through never, of course, never attaining the values. As in any iterative algorith, it is iportant to look for oscillations. Iproveents can be obtained by averaging values of oscillating cycles. Speckle Suppressions As a siplified explanation of speckle consider the following. The origin of speckle is lost light. If we for an iage using coherent light and ideal, very big, lens there is no speckle. Even when the object is strongly scattering, i.e. diffuses the light. For a lens of finite diaeter, soe of the light is blocked, it is lost. The corresponding points in the iage becoe dark. If we think a little harder about the explanation, we worry about the fact that there is not a clear correspondence between the location of the blocking and the location of dark points in the iage. We ay resort to a statistical view. Blocking the light will cause an overall darkening and a fluctuation that we call speckle. The things we can do to iniize speckle are Band liits the diffuser Sooth the diffuser phase Periodically replicate the CGH Periodically replicate the CGH and reconstruct using a copact source Periodic replication and quasi-onochroatic incoherent source Use a weird irregular replication In order to visualize this weird replication lets use only two CGH s. The reconstructed coplex aplitude will be i 1 φ ; u = Ae u = Ae iφ 1 Let the speckle patterns be S 1 and S. If the hologra for object two is oved off center to the position ξ 0 the reconstructed irradiance is i 1 i I = Ae φ S + Ae φ S e πξ 0 1 i x Why does this direction see weird? Because: rando + rando = rando But ( πξ φ φ ) I = A S + S + S S x+ 1 1 cos 0 1 Assuing a reasonably large separation between these to CGH s the last ter will vary rapidly copared to the saple spacing so the variations will disappear leaving

7 Part III: Chapter One Page 7 of 8 I = A S1 + S This is incoherent averaging. Appendix A: Why does the Gerchberg-Saxton Algorith Work? The fact that the G-S works ay at first see to be a atter of wonder and ystery. What follows is a plausibility arguent relating to why it works. Our approach will be to look at the proble as one of optiization. We will consider the version of the F-S algorith where sall changes are ade in the aplitudes at each iteration. We will find that one optial search technique is actually very siilar to the G- S algorith. Our first ove is to state the proble in ters of the Discrete Fourier Transfor. We liit ourselves to one diension for siplicity. The DFT relation is πin / 1 iφ πin / n = = = = U u e A e e We are seeking phases φ such that with pre-deterined A, the U n attain pre-deterined values. The first ove is to choose a phase distribution in the object and ipress upon it the known aplitude distribution. This initial function will have a DFT with iproper U n. We then seek to change the phases so as to generate a DFT with U n closer to the desired values. We note that changes in the DFT eleents U n are equivalent to changes in the object eleent aplitudes A and in the object eleent phase φ. For sall changes, we can use the differentiation product-rule to obtain the relation πin / 1 iφ πin / n ( φ ) = = U = u e = A + ia e e The change in the object doain is i ( φ ) u = A + i A e φ Let s digress for a oent to reflect on the eaning of this equation. The changes Un are calculated as the differences between the DFT fro out first guess and the corresponding values where we have replaced the aplitudes by the desired values of U. We obtain the object change by using the inverse DFT n 1 u = 1 πin / une n= If we were anipulating the phase we ight attept to calculate the required changes as iφ u e φ = ia A These are precisely the phases given by the G-S ethod when sall changes in the aplitudes and phases are used

8 Part III: Chapter One Page 8 of 8 For out search, however, we wish to choose values for the phase changes that satisfy the constraints of our optiization proble. We do not want to change the object aplitude so we set = 0 giving A u e φ ia iφ WE write and approxiation rather than an equality because the expression on the right ay have an iaginary part that is non-zero. The best we can do is to choose the nearest real value. That value is given by a projection onto the real axis, i.e., by iφ u Re e φ = ia This is the phase change that would best satisfy the given constraints for our search. Our search would proceed by iterating. We would change the object phases by the calculated aount. Perfor a DFT. Adjust the aplitude of the DFT. Calculate the changes in the DFT caused by adjusting the DFT aplitudes to the desired values. Inverse DFT those changes to get the object changes, Use the forula expressed in the previous equation to find the phase changes. Use those changes to calculate the phases to be used in the next round of the iteration. The reason that the sall-increent version of the G-S algorith works is that the values it gives for the phase changes are close to the optial-search phase-change values as long as the fractional aplitude changes are sall copared to the phase changes. Extension of the arguent to two diensions consist in siply adding an extra subscript. The optial phase change would then becoe iφ u Re e φ = ia And now an observation that could becoe an exercise left to the reader. If this plausibility arguent is valid, we would expect a odification of the G-S algorith that uses the optial phase changes of the previous equation to converge in soe way better than the unodified version.

Clustering. Cluster Analysis of Microarray Data. Microarray Data for Clustering. Data for Clustering

Clustering. Cluster Analysis of Microarray Data. Microarray Data for Clustering. Data for Clustering Clustering Cluster Analysis of Microarray Data 4/3/009 Copyright 009 Dan Nettleton Group obects that are siilar to one another together in a cluster. Separate obects that are dissiilar fro each other into