III Computer-Generated Holograms: Techniques and Applications

E. WOLF, PROGRESS IN OPTICS XVI @ NORTH-HOLLAND 1978

111

COMPUTER-GENERATED HOLOGRAMS TECHNIQUES AND APPLICATIONS BY

WAI-HON LEE Xerox Palo Alto Research Center 3333 Coyote Hill Road, California 94304, U.S.A.

CONTENTS PAGE

9 1. INTRODUCTION . . . . . . . . . . . . . . . . 121 § 2. TECHNIQUES FOR

ATED HOLOGRAMS

8 3. QUANTIZATION HOLOGRAMS

MAKING COMPUTER-GENER-

. . . . . . . . . . . . . . 126

IN

COMPUTER-GENERATED

. . . . . . . . . . . . . . . . . 168

§ 4. APPLICATIONS OF COMPUTER-GENERATED HOLO-

GRAMS. . . . . . . . . . . . . . . . . . . . 173

9 5. SUMMARY AND COMMENTS . . . . . . . . . . . 227 REFERENCES

. . . . . . . . . . . . . . . . . . . 229

0 1. Introduction Computer-generated holograms, synthetic holograms and computer holograms are terms used to refer to a class of holograms which are produced as graphical output from a digital computer. Given a mathematical description of a wavefront or an object represented by an array of points, the computer can calculate the amplitude transmittance of the hologram and display the result on a CRT or plot it on paper. Just as for conventional holograms, computer-generated holograms can be classified as image holograms, Fourier transform holograms or Fresnel holograms, depending on the relationship between the object and the complex wavefront recorded in the hologram. Using photoreduced copy of the graphical output from a computer as holograms is only one of many things that distinguishes computergenerated holograms from conventional ones. When the digital computer is used to calculate the transmittance function of a hologram, the object wavefront is just a mathematical description inside the computer. In practice, it may not even be realizable with optical components. Consequently, with the digital computer we can create optical elements that cannot be fabricated by conventional methods. Another distinction between a computer-generated hologram and a conventional hologram is in the way that the complex wavefront is recorded. In off-axis reference beam holograms as developed by LEITH and UPATNIEKS [1962], the amplitude transmittance of a hologram recorded under ideal conditions is proportional to t(x, y) = 1ReJZmax +A(x, y)eJ'p(x,Y)12

In eq. (1.11, ReJzmrxrepresents the tilted reference wave and A(x, p)eJs(x*y) the object wave. t ( x , y) is the resulting intensity variation of the interference pattern between the two waves. In computer-generated holograms, the transmittance of the hologram and the object wave is not 121

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restricted to the relationship specified by eq. (1.1). In fact, most of the work in computer-generated holograms has dealt with the problem of coding the complex object wavefront for convenient production on computer graphic devices. Coding as used here means the conversion of a complex valued function into a real, nonnegative function in such a way that the complex valued function can be retrieved intact by optical means at a later stage. Over the last decade, because of the interest in coherent optics and the availability of digital computers as scientific tools, this special field of using computer and graphic devices to make holograms has attracted the attentions of many researchers, and has resulted in many publications in the scientific journals. The references at the end of this chapter contain a listing of the publications that are related to computer-generated holograms. Studies on computer-generated holograms can be roughly divided into three main categories: (1) Coding techniques, (2) Applications, and (3) Techniques for improving the quality of computer-generated holograms. In the third category are problems such as that of finding the best random phase code or algorithm for reducing the dynamic range in the Fourier transform of an image (AKAHORI [1973], ALLEBACH and LIU [1975], DALLAS[1973a], GABELand LIU [1970], GABEL[1975], GALLAGHER and LIU [1973], GALLAGHER [1974]), and the problem of quantization noise in the hologram (ANDERSON and HUANG[19691, GOODMAN and SILVESTRI [1970], DALLAS[1971], DALLASand LOHMANN [1972], NAIDU[19751). Quantization noise occurs in computer generated holograms because the computer graphic devices have limited gray levels and a limited number of addressable locations in their outputs. Depending on the techniques used to make the computer-generated holograms, quantization will limit accuracy in reconstructing the phases, the amplitudes or both of the desired wavefronts. The coding of complex wavefronts to make computer-generated holograms was first demonstrated by BROWN and LOHMANN [19661 with their detour phase hologram. An interesting aspect of their technique is that the computer-generated hologram is made without explicit use of a reference wave or a bias. Also, their holograms have only two levels of amplitude transmittance (0 or 1). This makes the holograms very easy to make, and they can be copied many times without degradation. Because computer-guided plotters are available in most computing centers, Brown and Lohmann’s method has been widely used for making binary

111, § 11

INTRODUCITON

I23

computer-generated holograms. Their approach to the coding problem is evident in the many techniques that developed afterwards. A different coding method for making computer-generated holograms was later described by BURCH [1967]. Since the term A*(x, y) in eq. (1.1) does not contribute to the reconstruction of the object wavefront, Burch suggested that the computer calculate only the values of the last term in t ( x , y ) at regular sampling intervals. A constant bias would then be added to all the samples to make them nonnegative. Because the data samples in the computer-generated holograms are all equally spaced, Burch's holograms can be recorded from CRT displays. HUANCand PRASADA [1966] suggested a similar modification of eq. (1 1) for making computergenerated holograms. Another type of computer-generated holograms is the kinoform (LESEM, HIRSCH and JORDAN [1967, 1968, 1969, 1970]), which uses the relief images recorded on film to record the phase variations of the complex wavefronts calculated by the computer. The relief heights of the kinoforms are proportional to the residues of' the phase variations after taken modulo 2 m As a result, phase variations in kinoforms are restricted to a range of [0,27~].Because kinoforms are made without using a carrier frequency, wavefronts reconstructed from kinoforms are centered on the optical axis. When properly made, kinoforms can have a diffraction efficiency of 100%. However, the kinoform, as originally conceived, cannot record the amplitude variation of the wavefront. LEE [1970a] recognized that if two real functions were sampled at periodic intervals and if there were a delay in the sampling of one of the functions, a constant phase difference in the Fourier transforms of the two functions could be created. Therefore, by combining four real nonnegative functions that have been sampled with quadrature phase delays, one can produce the Fourier transform of the desired complex valued function in the Fourier transform plane. Complex valued functions can also be decomposed into three positive components with phase differences of 120" (BURCKHARDT [1970]). Lee's method, like Burch's, is designed for displaying computer-generated holograms on a CRT. KIRKand JONES [1971] described a method €or incorporating amplitude information into kinoforms. Their method requires the use of a carrier frequency which results in a reduction in the efficiency of the kinoform. CHU,FIENUP and GOODMAN [1973] used Kodachrome slide transparency film to make kinoforms that can have both phase and amplitude variations. They called their computer-generated holograms ROACH for

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“Referenceless On- Axis Complex Holograms”. Another method of modifying the kinoform to record the amplitude information was also described by CHUand FIENW[1974]. The most recent technique for making computer-generated holograms, developed by LEE [1974], is for making image holograms of wavefronts having only phase variations. This type of hologram is similar to an interferogram and is made by finding the location of the fringes corresponding to the interferograms and plotting them on paper. WYANTand BENNETT [19721 used a similar approach to generate aspheric wavefronts for testing optical surfaces. However, they used a ray tracing program for locating the fringes for holograms. These various techniques for making computer-generated holograms are summarized in Table 1. Except for the kinoform-type holograms which are already phase-relief recordings, amplitude-type holograms can be converted into phase recordings by bleaching the developed holograms. Applications of computer-generated holograms can be divided into five areas: (1) 3-D image display, (2) optical data processing, ( 3 ) interferometry, (4) optical memories, and (5) laser beam scanning. Some of the early work in computer-generated holograms was on the problem of computing the wavefronts for three dimensional objects (WATERS [1966, 19681). This problem is far more complicated than the coding problem. Most of the computer graphic devices are rather limited in resolution and cannot be used to display holograms containing high degrees of complexity. For displaying three dimensional objects generated by computer, the method developed by KING, NOLL and BERRY [1970] is an attractive alternative. Recently, YATAGAI [1974, 19761 used the same method to make mosaic computer-generated holograms for 3-D image display. In the early development of computer-generated holograms, their application to matched filtering, code translation and spatial frequency filtering was investigated quite extensively (BROWN, LOHMANN and PARIS [1966], LOHMANN, PARISand WERLICH[1967], LOHMANN and PARIS [19671). More recently, computer-generated holograms have been used as matched filters for processing synthetic aperture radar data (LEE and

TABLE1 Different types of computer-generated holograms Type of hologram

On-axis reconstruction

Amplitude transmittance

Other features

References

Detour phase holograms

No

Binary

Amplitude and phase of wavefront are coded separately

BROWNand IDHMANN [1966, 19691

Modified off -axis reference beam holograms

No

Gray levels

Require reference beam and bias

BURCH[1967], HUANCand PRASADA [1966], LEE[1970]

Kinoforms

Yes

Constant

Wavefronts are recorded as surface relief on film

LESEM,HIRSCH and JORDAN [19671, KIRKand JONES[19711, CHU,FIENUP and GOODMAN [ 19731, CHUand ~ ~ E N U[:97Sj F

Computergenerated interferogram

No

Binary

Hologram has many fringes similar to in terferograms

LEE [ 19741

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GREER[19741). Deblurring experiments using computer-generated holograms were reported by CAMPBELL, WECKSUNG and MANSFIELD [19741. The potential of computer-generated hologram in interferometry was [1969]. Experiments demonstrating the use of pointed out by PASTOR computer-generated holograms in interferometers were reported by MACGOVERN and WYANT [1971] and WYANTand BENNETT[1972]. Inclusion of computer-generated holograms in interferometers has become an established technique for testing optical surfaces (BIRCHand GREEN[19721, ICHIOKA and LOHMANN [1972], FERCHER and KRIESE [1972], TAKAHASHI, KONNOand KAWAI[1974], SIROHI,BLUMEand ROSENBRUCH [1976]). Recently, BRYNGDAHL [ 19731 used computer-generated holograms to provide reference wavefronts for displaying interferograms with radial and circular fringes. Computer-generated holograms have also been used in a shearing interferometer (BRYNGDAHL and LEE[1974]). One dimensional computer-generated holograms have been used in optical data storage (KOZMA,LEE and PETERS[1971], KOZMA[1973]). Fourier transforms of segments of the digital data are calculated in real time and the holograms are recorded on film using a scanning laser beam recorder. This solves the problem of having to use page composers and elaborate optics to form the Fourier transform holograms. Computer-generated holograms can be viewed as variable-frequency gratings. By controlling the frequency variations of the hologram, the hologram can be used to deflect the laser beam in any desired pattern (BRYNGDAHL and LEE [1975, 19761, BRYNGDAHL [1975]). Raster scans or two dimensional scan patterns could be produced by moving a computergenerated hologram across the laser beam. We will review the different techniques for making computer-generated holograms and the five areas of applications of computer-generated holograms. The effects of quantization on reconstructed wavefronts from computer-generated holograms will also be discussed. § 2. Techniques for Making Computer-Generated Holograms 2.1. DETOUR PHASE HOLOGRAMS

In 1966, BROWNand LOHMANN described a detour phase method for making binary computer-generated holograms for complex spatial filtering. Their holograms are distinguished by three unique properties: (1)the transmission of the hologram is binary, (2) the hologram can record both

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the amplitude and phase of any complex valued function and (3) the hologram is not recorded with the explicit use of a carrier frequency o r a bias as in the off -axis reference beam holograms. To make a binary hologram of this type, the wavefront represented by the complex valued function, A(x, y)e’s(x,y),is first sampled at equally spaced intervals in accordance with the sampling theorem. That is, the sampling distance must be smaller than 1/U where U is the spatial bandwidth of the wavefront in the direction of sampling. In plotting the hologram the paper is divided into equally spaced cells. Rectangular apertures are drawn inside each cell. Each aperture is determined by three parameters: its height, h,,,, its width, w,,, and its center with respect to the center of the cell, c,,. Figure 1 shows one of the sampling cells in the detour phase hologram. The index nm indicates the relative location of the cell in the hologram plane. The parameters of the aperture are selected as follows:

and

:

y=md ._ _ _ - _ _

w

:“I I

hnrn

-.;___..-._ __ __

I 1

Cnl?

x =nd,

-

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COMPUTER-GENERATED HOLOGRAMS: TECHNIQUES AND APPLICATIONS

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with many small apertures is plotted and recorded on photographic film, it creates an amplitude transmittance on the film given by

The function p ( x ) is given by

P(X> =

[XIS+

1 0

for otherwise.

(2.3)

That this binary hologram works can be verified by putting it in the optical system in Fig. 2. The hologram is illuminated by a collimated laser beam. The Fourier transform of the desired wavefront occurs in an off-axis region on the Fourier transform (back focal) plane of lens L,. The aperture mask shown in Fig. 2 passes only one diffracted wave from the hologram through the optical system. Lens L2 performs the inverse Fourier transformation to produce the wavefront A (x,y)eJ'+'p(x,y , at the back focal plane of lens L2. The lens system in Fig. 2 is a telecentric system. Without the aperture mask, the optical system images the hologram one to one at the back focal plane of lens L,. The aperture mask in the optical system converts it to a bandpass system. The wavefront at the is the bandpassed output of the diffracted back focal plane of the lens waves from the hologram. This property of the binary hologram can be demonstrated analytically by examining the Fourier transform, Tl(u,v), of the function t,(x, y):

Tl(u, v) = =

I

m

m

-m

-m

t,(x,

y)e-12T(ux+uy) d x dY

c c G,,(u, exp {-j2.rr(ndxu + mdyv)>, n

m

V)

(2.4)

Fig. 2. Optical system for reconstructing wavefronts from computer-generated holograms.

I n , § 21

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MAKING COMPUTER-GENERATED HOLOGRAMS

where Gnm(u, u

sin T W U sin m k m v

)

=

TU

p

m

exp (-~~Tc,,,u).

If the coefficients, G,,,(u, v ) , are independent of u and u, the function T,(u, v) is a Fourier series expansion of a periodic function. To continue, we expand the terms in G,,,(u, u ) that are dependent on the index, n, m, about u, = k/d, and v, = 0, the center location of the aperture in Fig. 2. As will be seen, the spatial frequency, uc, is equivalent to the carrier frequency in the off-axis reference beam holograms. The function Gnm(u,v), in a region where l u - u , l ~ l / d , and lu1<1/2d,,, can be expanded as

x { l + ( m h , m ) 2 / 6 - j 2 ~ c n m ( ~ - ~. c ) +(2.5) ~ .}a

Substituting eq. (2.5) into eq. (2.4) yields

xexp{-j2T(ndxu +md,,v)}

xexp{-j2.rr(ndxu+rn~u)},

(2.6)

where

Q,,, = ( m d , , A , m ) 2 / 6 - j 2 r r d x ~ n m-( uu,)/M. Here we have neglected the higher-order terms of Qnm. Except for the noise term, Q,,,, in the Fourier series, the function Tl(u,u ) in the region about u, and v, is just the Fourier transform of the sampled complex wavefront when M is equal to k. Because the center of the aperture in Fig. 2 is proportional to k, the parameter M in eq. (2.1) provides a means for controlling the center location of the reconstructed wave in the Fourier transform plane. Hence, M/d, is equivalent to the carrier frequency, a,in an off-axis reference beam hologram. The severity of the noise term, Q,, on the reconstructed wavefront can be estimated. Suppose that the bandwidths of the wavefront A ( x , y)ej'@(x,y)along the coordinates u and u are U and V, respectively.

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The maximum value of the noise term, Q,,,, is given by by setting Iu -u, I = U/2 and 21 = V/2.Even if V d , = 1, the maximum value in the first term of Q,, is about 0.4. On the other hand, for k = 1 and Jqnrnl = T,the second term is about 1.5 with Ud, = 1. Hence, the predominant noise in the reconstructed wavefront is caused by the method used for coding the phase of the wavefront. As evident in eq. (2.7), the noise Q,, can be reduced by keeping the values of Vd, and Ud, much less than 1. In other words, the sampling rate for digitizing the wavefront should be higher than the minimum rate required by the sampling theorem if this type of coding techniques is used to record the wavefront. Some of the properties of this coding technique can be illustrated by the following three Fourier transform holograms. The object recorded in the holograms is the Chinese character “light” which is represented by the nonzero elements in a 32x32 matrix. Before computing the discrete Fourier transform of the matrix, each of the elements in the matrix is multiplied by a random phase factor to help reduce the dynamic range in the transform. After the transform has been computed, the detour phase coding technique is applied to all the elements in the transform to produce the binary hologram. The holograms in Fig. 3(a) and (c) are recorded by using M = 1 and M = 1.5, respectively. These two holograms illustrate the selection of the effective carrier frequency of the hologram. With M = 1.5 the image formed from the wavefront reconstructed from the hologram in Fig. 3(c) is farther away from the optical axis than the image resulting from the hologram with M = 1. Equation (2.7) indicates that the noise will be less when M is larger than 1. However, this will not happen if the number of phase quantization levels in the hologram with larger M is not increased by the same factor. Since the holograms in Fig. 3(a) and (c) are made by using the same number of quantization levels, the images are of comparable quality. The noise can be reduced significantly by making Ud, less than 1. In Fig. 3(e), the number of samples of the hologram along the x coordinate is increased to 64 by using dJ2 as the sampling interval. The hologram is made with M = 1. The reduction in the sampling interval automatically puts the hologram on a higher carrier frequency. The increase in carrier frequency and the reduction in noise level can be clearly noted in Fig. 3(f). This agrees with the analytical result in eq. (2.7). BROWNand LOHMANN described an improvement to their original

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131

Fig. 3. Examples of detour phase holograms. The holograms in (a) and (c) have 3 2 x 3 2 samples and the hologram in (e) has 32 x 64 samples. The hologram in (c) is phase encoded to produce an image at higher spatial frequencies. The images reconstructed from the holograms are shown to the right of the holograms.

132

(b)

Fig. 4. Improved detour phase hologram and the reconstructed image. (Courtesy of Brown and Lohmann.)

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detour phase technique in 1969. Rather than using the phase at the center of the sampling cell to determine c,, they put the aperture inside the cell so that its x coordinate satisfies the equation

2rrxldx - q ( x , y ) = 2rrn,

(2.8)

where n is an integer. This technique is similar to that used by LEE[19741 to determine the location of fringes in computer-generated interferograms. Justification of this technique will be discussed in D 2.4. With this improvement, holograms can be made without the phase-dependent noise in the reconstructed wavefronts. Figure 4 shows a hologram with 1 2 8 ~ 128 samples made by this improved technique. It can be seen that the image produced by this hologram does not have the noisy background of the reconstructed images in Fig. 3 .

2.1.1. Other coding techniques

BROWNand LOHMA" describe two additional coding techniques in their 1966 paper. For both of these, the height of the aperture in the hologram is kept constant. The coding of the amplitude as well as the phase is done by means of the widths and the centering of the apertures along the x coordinates. These two schemes do not use the additional degree of freedom provided by the y coordinate. The different methods of putting the apertures inside the sampling cells will change the Fourier coefficient G,,(u, v ) in eq. (2.4). In Brown and Lohmann's second scheme, for example, the apertures inside the cells are defined by h,,

h,

w,,

sin-' A,,

= d, ~rr

and

(2.9)

An example of this coding scheme is illustrated in Fig. 5(a). The Fourier coefficient, Gn,(u, v ) , in eq. (2.3) is simply the Fourier transform of the aperture in the (n , m) cell. For the aperture shown in Fig. 5(a), sin rrhv sin rrwn,u exp ( - - ~ ~ T C , , , U ) . (2.10) Gnm(U,V ) = WV

5-U

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Fig. 5 . The apertures in (a) and (b) illustrate two other encoding methods for making detour phase holograms.

As can be seen, all the codings are along the u coordinate. The constant height of the aperture produces a window effect in the Fourier transform of the hologram. To see how effective this coding scheme is, the function G,,(u, v ) is expanded about u, = l/d, and v,= 0 to give

where

In the previous scheme the error caused by the amplitude coding occurs as a second order term in v in the reconstructed wave. However, in this scheme it has an effect on tlie first order term in the expansion of G",(% v ) . Another scheme described by Brown and Lohmann is shown in Fig. 5(b). Two apertures with the same width and height are used inside each cell to encode the amplitude and phase information in the hologram. The centroid of the two apertures is shifted by an amount c,, as in the previous two schemes for encoding the phase information. To encode the amplitude information the spacing z,, between the two apertures is set by cos-' A,,,,, (2.12) z,, = dx rr

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MAKING COMPUTER-GENERATED HOLOGRAMS

For this coding scheme the Fourier coefficients G,,(u,

v ) is

sin nhv sin mwu G,,(u, v)=___ cos I T Z , , , ~ exp (-j2.ncn,u) TV

7TU

where

Q,,,,

= ( - j 2m,, - m,,tan( m,,/d,

))(u - u,)

+ ...

,

The noise Q,,, in this scheme is very similar to that of the previous case. Both affect the first order term in the expansion of G,,,(u, v). In this regard these two schemes generate more noise in the reconstruction than the first detour phase scheme. HASKELL [1973] further generalized this type of encoding technique. He divided each of the sampling cells into KX K subcells. The transmittance of each subcell designated by bkl has a value of either 0 or 1.This type of sampling cell produces Fourier coefficient: G,,(u,

v) =sin ( r d p / K )sin (nd,,u/K)/(n2uv) x

C 1bkl exp{-j2.rr(kdXu + Z V ~ ) / K } . (2.14) I

k

If the exponential inside the double summation is expanded about u, = l / d x and v,= 0, G,,(u, v ) is approximately equal to G,,(u, v)-sin (ndxu/K)sin ( I T ~ v / K ) / ( I T ’ u v )

x

1 1

k

bkl (n, rn)e-j2mWK{ 1 - J 2 r k d x ( u - u , ) / K - j 2 ~ Z ~ v / K +* ..}. (2.15)

By neglecting the u, u dependent terms inside the summation, the double summation in eq. (2.15) can be written as (2.16) where &(n, m ) has values between 0 and K. To make the binary hologram, it is necessary to solve for values o f &(n, rn) such that eq. (2.16) will equal A,, exp (-jq,,). This is generally a difficult problem. One alternative is to tabulate all the possible complex numbers that can be generated with these K x K subcells and store them in the computer. When one wants to make this kind of binary hologram, the coding for

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each cell can be obtained by looking up the values of B,(n, rn) in the table. In practice, the effect of the u, v dependent term in eq. (2.15) cannot be neglected. These terms will produce noise in the reconstructed wavefront. 2.2. MODIFIED OFF-AXIS REFERENCE BEAM HOLOGRAMS

2.2.1. BURCH'S method [1967] Computer-generated holograms could be made from the transmittance function t ( x , y ) given in eq. (1.1). Because the function t ( x , y ) is already real and nonnegative, the computer simply calculates its values at discrete intervals and produces a graphical output of the function on a CRT display or other device. The sampling rate for the function t(x, y) is proportional to the total bandwidth of the function. If a transparency with transmittance t ( x , y) is put in the optical system in Fig. 2, the Fourier transform of the transparency will have values in the region shown in Fig. 6. The two rectangular regions centering at u = f a contain the Fourier and its complex conjutransforms of the complex function A(x, y)eJ'p(x.y) gate. Each occupies an area U XV in the u, v plane, where U and V are, respectively, the spatial bandwidths of the function A (x,y)eJvCx, y , along the two coordinates. The circle in the center in Fig. 6 is an impulse function resulting from the constant bias R 2 in t ( x , y). The rectangle in the center represents the spatial frequencies from the term A2(x, y ) in the v

Fig. 6 . Spatial frequency distribution in an otf-axis reference beam hologram. The two rectangular regions centering at +a are the spatial frequencies from the two conjugate waves reconstructed from the hologram.

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137

function t(x, y). To avoid overlapping of these terms in the frequency plane, the carrier frequency, a, must be equal to at least 1.5U. The highest frequencies of the function t(x, y) along the two coordinates are, therefore, equal to 2 U and V, respectively. Hence, the sampling intervals for the function t(x, y) must be smaller than d

1 4u

=-

and

d

1 2v

=-

(2.17)

In conventional holograms, the form of the function t(x, y ) is determined by the interferometric recording technique and cannot be changed. This is not so for computer-generated holograms. The function t(x, y) can [1967] observed be modified to produce a number of advantages. BURCH that the term A2(x, y) is not important to the reconstruction of the desired wavefront A (x, y)eJqpcx, y ) . Moreover, its presence increases the bandwidth requirement of the hologram along the u coordinate. Burch suggested, then, that, in making computer-generated holograms, the amplitude transmittance of the hologram should be

Because of the absence of the A2(x, y) term in t(x, y) in eq. (2.18), the carrier frequency a can be as low as Ul2 and the sampling intervals become 1 2u

d =-

and

1 4=-V'

(2.19)

A small section of a computer-generated hologram made according to eq. (2.18) is shown in Fig. 7(a); the image'obtained from the hologram is shown in Fig. 7(b). The hologram is a Fourier transform hologram with 5 12 x 5 12 samples. The original data samples have been multiplied by a pseudo random phase factor to achieve the effect of having a diffuser in contact with the object. This hologram is recorded directly onto film with a microdensitometer. The spacing between the data points in the hologram is about 25 p,m. The recording spot size is about 20 p,m. The speckles in the reconstruction are the result of the random phase used in the construction of the hologram.

Fig. 7. (a) is an enlargement of a small section of a computer-generated hologram made by Burch’s method. The image reconstructed from the complete hologram is shown in (b). (Courtesy of Campbell, Wecksung and Mansfield.)

2.2.2. HUAM,and PRASADA’S method [1966] Huang and Prasada suggested another modification on eq. (1.1) for making sampled off-axis holograms. They proposed the use of the following amplitude transmittance for the hologram t(x, Y ) = O . ~ A ( X Y,) ( ~ + C O S [ ( ~ . ~ ~ C y)]}. ~ X - C ~ ( X(2.20) , Because the bias term in eq. (2.20) is not a constant, this method does not have the bandwidth reduction advantage of Burch’s hologram. Also, since the average transmittance of this hologram is proportional to A(x, y), the hologram is very sensitive to the nonlinearity of the recording medium. The advantage of this method is that the contrast in the hologram is high and independent of the fluctuation in A(x, y). Generating computer holograms by sampling the amplitude transmittance of the holograms at regular intervals such as in all these off-axis reference beam holograms makes them highly suitable for display on a CRT. However, this does not limit these methods to use with CRT displays alone. Figure 8(a) shows a sampled Fourier transform hologram produced on a computer-guided plotter. The amplitude transmittance of the hologram is computed by using Huang and Prasada’s method. Since each data sample is already a real and nonnegative number, the only consideration left in making such a binary hologram is the recording of the amplitude information on the hologram. In the hologram shown in Fig. 8(a), the sampled values of the hologram are recorded by the heights

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139

Fig. 8. (a) is a binary hologram made by using Huang and Prasada’s method. The image reconstructed from the hologram (b) shows the Chinese characters for “1976”.

of the apertures inside the sampling cells. The apertures are all equally spaced in the hologram. The objects used for making this hologram are the Chinese characters for “1976”. Each character consists of 16 x 16 points. The hologram contains a total of 64x64 samples. Four times more samples along the x-axis are used to record the complex Fourier transform of the object because of the carrier frequency. The image obtained from the hologram is shown in Fig. 8(b). The multiple images in the reconstruction are the result of the discrete nature of the hologram; they are not caused by the nonlinearity of the recording material.

2.2.3. LEE’Sdelayed sampling method [ 19701 Another modification of the off-axis reference beam hologram was suggested by LEE[1970]. His technique is somewhat similar to the detour phase technique. The phase information of the complex wavefront is encoded by the positions of the samples in the hologram. Also, in one interpretation his holograms can be considered as recorded without the explicit use of a carrier frequency or bias. The principle behind Lee’s method comes from one special property of the sampled function which we shall discuss briefly here. Suppose that f\(x) is the sampled function of f(x). Then f s ( x ) can be written as f<(x)=

n

f ( n d x+ E

) ~ ( x nd,

-E ) ,

(2.21)

where 6(x) is the Dirac delta function and d , is the sampling interval. The

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parameter E is a small delay in the positions of the sampling pulses with respect to the origin of the function f(x). If f(x) is a bandlimited function, it can be recovered from f s ( x ) by passing f , ( x ) through a low pass filter. In that case, the delay in the sampling has no effect on the reconstruction of the function f ( x ) from its sampled function. However, if fs(x) is passed through a bandpass filter with the impulse response (2.22)

the output is

fl(x) = = exp

c n

fhd,

+E )

sin [ ~ ( x- 4 - &)/&I exp C j 2 4 x - nd, 7~ (X - d, - E )

f(nd, + E )

(j2mx/dx)exp (j27rs/dx) n

sin [ r ( x - d,

-

-

E)/d,}

E)/d, J

. (2.23)

Tr(X-4-E)

The summation in eq. (2.23) is just the low-pass output of fs(x) and is therefore equal to f(x). We can write fi(x) as

fib> = exp (J2mx/dX) exp ( J 2 m / d J f ( x ) .

(2.24)

Note that there is a constant phase in fi(x) given by 27rsld,. This result indicates that a constant phase can be added to a function by first sampling the function with certain delay E and then by passing the sampled function through a bandpass filter. With this observation, it is clear that if any complex valued function, A (x, y)eJwp(x, y), is decomposed into component functions which are real and nonnegative, then the phase information of each of the component functions can be encoded by sampling the function with proper delay E. The complex function thus encoded can be reconstructed by bandpass filtering. One way to decompose the complex function into real, nonnegative components is to represent the complex function as follows: A (x,y)eJ*

(X.

Y)

=

{fr+(x,y>-f,-(x,

y>)+jcf,+(x, Y>-fi-(x, Y ) } . (2.25)

The functions on the right-hand side of eq. (2.25) are, in order, the positive and negative portions of the real and imaginary parts of the complex function. The phase information of these four functions can be recorded by sampling each function with delays, E , given by 0, dJ4, d,/2 and 3dJ4. The smallest value of E is d,/4, this is 1/4 of the sampling

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interval. Therefore, the complex function must be sampled at a rate equal to four times the spatial bandwidth of the function along the x coordinate. This is in contrast to the double bandwidth required by Burch’s method. The transmittance of the hologram resulting from this method is equal

to

Because each of the component functions is real and nonnegative, the function t(x, y) in eq. (2.26) is also real and nonnegative. Recently, CAMPBELL, WECKSUNG and MANSFIELD [19741 observed that Lee’s method could be interpreted as a modified off-axis reference beam hologram with amplitude transmittance function given by t(x, y ) = A ( x , Y){COS [2raX-cP(X7

Y)I+ICOS[ ~ T ~ ~ x - vy)II>, (x,

(2.27)

where a = Ud,. The bias term in such a hologram is the function A(x, y) (cos[ ~ T C Y X q ( x , y)]l. The equivalence of the transmittance function in eq. (2.26) and eq. (2.27) can be demonstrated by sampling the function t(x, y) in eq. (2.27) at locations x = nd, + kdJ4 and y = md,, where k = 0, 1 , 2, 3 and showing that its value at these locations are the same as the values given by eq. (2.26). For example, the sampled value of fi+(x,y ) at x = nd, + dJ4 and y = md, is equal to fi+(ndx+ dJ4, mdy) = O.SA(nd, + dx/4,m&){sin cp(nd, + dJ4, m4)+ lsin q(nd, + dJ4, rnd,,)l>. (2.28)

On the other hand, the sampled value of t(x, y) in eq. (2.27) is equal to t(&

+ &/4, m 4 ) = O.SA(n4+dJ4,m4) x{cos[2~/4-cp(nd,+dJ4,m4)I+(cos [ 2 ~ / 4 - c p ( n d+dx/4, , md,)lI} = fi+(ndx+

m4).

(2.29)

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Fig. 9. (a) Original continuous-tone picture. (b) Enlargement of a small section of the hologram made by Lee's method. (c)The image reconstructed from the hologram in (b).

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The equivalence of the two functions at other sampling locations can be shown in a similar way. With this interpretation o f Lee’s method it can be shown that by choosing CY to be 2/3dx, the transmittance of the hologram will be equal to that suggested by BURCKHARDT [1970] for reducing the sampling rate in the Lee hologram. In this case, the complex function is decomposed into only three components rather than four components as used by Lee. An example of Lee’s hologram is shown in Fig. 9. The object for this Fourier transform hologram is the continuous tone picture with 128 X 128 pixels as shown in Fig. 9(a). The Fourier transform of the picture has 128x 5 12 samples. Four times more samples are used along one direction than the other as required by this coding technique. The value of the transmittance of the hologram is quantized into 4 levels and is recorded by the different heights of the narrow bars in the hologram. This creates a hologram having 512 x 5 12 samples. Figure 9(b) is a magnified portion of the hologram. The reconstructed image is shown in Fig. 9(c). The speckles in the image are due to the random phase factor applied to the original picture for reducing the dynamic range in the Fourier transform of the picture. 2.3. KINOFORMS

2.3.1. Fourier transform-type kinoforms The kinoform is a different form of computer generated hologram. It is not recorded as amplitude transmittance on film as are the previous types of computer-generated holograms but as relief patterns on film. Most of the computer-generated holograms discussed so far rely on a diffraction effect to reconstruct a complex wave field. The kinoform, however, like a Fresnel lens, changes the phase of the illuminating wave by its thickness variation. The kinoforms made by LESEM,HIWCHand JORDAN [1967], were Fourier transform type holograms. To make such a kinoform, the discrete Fourier transform of an object is first calculated by the computer. The phase angles of the complex samples of the Fourier transform are determined and the amplitude of the transform is set equal to 1. When the kinoform is used to display the image of an object, a pseudorandom phase array is used with the original object to reduce the effect of losing the amplitude information in the kinoform. The phase angles of the

Fourier transform are obtained by taking the arctangent of the ratio bJb, where bi and b, are the imaginary and real part of the complex sample of the transform. The angles determined this way have values between - 7 ~ to 7~ radians. The variations of the phase of the transform are then displayed on a CRT and recorded on film as an intensity variation. An example of the kinoform obtained up to this point is shown in Fig. lO(a). The exposed film after development then goes through a bleaching process to create a relief pattern on the film emulsion. When the kinoform is inserted in the optical system shown in Fig. 2, an image of the digital picture will be reconstructed at the back focal plane of lens L1. Because the amplitude variation is not recorded, the reconstructed image tends to be noisy. Moreover, if the relief height in the kinoform is not matched to the phase variation of the Fourier transform of the object, there will be, as can be observed in Fig. 10(b), a focused spot in the center of the reconstructed image.

2.3.2. Phase Fresnel lens

A similar procedure for creating a phase image has been described by MIYAMOTO [1961] for making the phase Fresnel lens. For this application, the wavefront to be recorded has only phase variation. Hence, the kinoform technique can be used to record this type of wavefront without error. Recently, BRYNGDAHL [1973] used the kinoform technique to record helicoid wavefronts for use in an interferometer. The phase variation in a helicoid wavefront is linear in the azimuthal direction as shown in Fig. ll(a). The interferogram in Fig. l l ( b ) and (c) illustrates the phase variation of the wavefront reconstructed from the kinoform. The interferograms are the interference pattern between the reconstructed wave from the kinoform and two different phase objects. The spokes in the interferograms are the constant-phase contours of the phase variation recorded in the kinoform. How well does the kinoform work is dependent on the precise control of the relief heights on the developed film emulsion. If the relief heights in the kinoform produce phase variations exceeding 277 rad., multiple waves will be generated by the kinoform. The bright spot in the center of the reconstructed image in Fig. 10(b) is the result of the phase mismatch in the kinoform. This phase mismatch problem of the kinoform can be studied by modelling the phase recording process in the kinoform as a nonlinear

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Fig. 10. (a) Phase variation for making a kinoform. (b) Image reconstructed from the kinoform with the phase variation in (a). (Courtesy of Lesem, Hirsch and Jordan.)

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Fig. 11. The phase variation in (a) is made into a kinofonn. The interferograms (b) and ( c ) are obtained by using the wavefront from the kinofom as a reference wave.

operation on the phase function. The nonlinear limiter characterizing the kinoform recording process is shown in Fig. 12(a) with Z being the input to the limiter and 2' as the output. 2' is linear in 2 for a short interval and its value is limited to the range 0 to 1+ p. Figure 12(b) shows a phase function q ( x ) and the resulting phase variation after it passes through the nonlinear limiter. The constant p is the parameter determining the amount of phase mismatch in the kinoform. If the phase variation of the original wavefront is Z, the wavefront reconstructed from the kinoform is equal to eJz'. Since 2' is a periodic

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\ (b)

Fig. 12. A nonlinear limiter with the input-output relationship as shown in (a) can convert a continuous-phase function into a discrete valued function such as the one shown in (b).

function in 2, we can expand eJZ’in a Fourier series:

where

(2.31)

Equation (2.30) is valid for any real function 2. Therefore, we can substitute &, y) for 2 in eq. (2.30) and get (2.32) Note that if p = 0, the only nonzero term in eJZ’is c l . Hence ejZ’

-e

j d x , Y)

(2.33)

as desired. For /3 an integer, the nonzero term in eq. (2.30) is shifted to n = 1+ p. This produces a wavefront in the reconstruction with IZ times more phase variation. However, if p is not equal to an integer, there will be more than one nonzero term in elz’. The n == 0 term in eq. (2.30) will cause a bright spot such as the one shown in Fig. 10(b) to occur in the

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reconstruction. One way to solve the phase mismatch problem of the kinoform is to add a linear phase term to the phase variation before it is recorded in the kinoform. This is similar to the use of a carrier frequency in the off-axis reference beam hologram. With this linear phase term the higher order diffracted waves generated by the mismatch of the relief heights in the kinoform will propagate in different directions and can be separated in the reconstruction process. If there is perfect phase matching in the kinoform, the addition of the linear phase term will not affect the diffraction efficiency of the kinoform. It will change only the direction of propagation of the reconstructed wave.

2.3.3. Extensions of kinoform technique Since kinoforms cannot record the amplitude variation of the wavefronts, the natural extension to the kinoform technique is to add an extra function to the phase variation so that in the reconstruction this extra term in the phase can modify the amplitude of the incident wavefront. To introduce amplitude variation to the reconstructed wavefront, light has to be taken away from the wavefront. This results in a kinoform with less light efficiency. A phase function that can modify the phase and the amplitude of an incident wave is (KIRKand JONES [1971])

qo,(x,y) = a(x, y ) cos 2TffX+ d x ,Y),

(2.34)

where a ( x , y) is a function of the amplitude of the wavefront. With a plane wave incident on the kinoform with q l ( x , y ) as its phase variation, the waves emerging from the kinoform are terms in the expansion of eiv,(x.

Y).

eIm,(*.Y)=e"P'l.L)

C b , ~ , [ a ( xy)] , cos 2 ~ n a x ,

(2.35)

n

where J,(x) is the Bessel function and b,, is equal to 1 for n = 0 and 2 for nfO. Consequently, if a(x, y) is such that

the diffracted wave from the n = 0 term in eq. (2.36)will have the proper

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amplitude and phase information. The wavefront can also be reconstructed from the higher order diffracted waves of this kinoform provided that

J J a ( x , y ) l = A(x, Y ) .

(2.3'7)

Another way to introduce amplitude control to the kinoform is to use an additional layer of emulsion. Some photographic films, such as Kodachrome 11, have more than one layer of emulsion. The different layers of the emulsion can be exposed independently by light of different wavelengths. After the film is processed and is illuminated with monochromatic light, one layer can be made to modify the amplitude of the incoming wave; while other layers which are transparent can cause phase shifts of the incoming wave. CHU,FIENUP and GOODMAN [1973] used this technique to produce kinoforms that can record both the amplitude and phase of the wavefronts. To make this type of multiemulsion kinoform which is called ROACH (Referenceless -On-Axis Complex Hologram) by its inventors, the film is first exposed to the brightness variation corresponding to the amplitude of the wavefront through a red filter. The phase variation will be exposed to the film through a blue-green filter to generate a relief pattern. If this transparency after development is illuminated by the red light from a helium-neon laser, the layer exposed to red light will modulate the amplitude of the incident wave and the layers exposed to blue-green light will modulate the phase. C m and FIENW[1974] also proposed two other methods that use parity sequence to make kinoforms that can have amplitude control with a single layer of emulsion. In one of their methods, the phase recorded in the kinoform is equal to (2.38) The phase angle O(x, y) will be determined by the amplitude of the wavefront. This particular kinoform will produce a wavefront given by - ejv(x.

Y )

{cos O(x, y ) + j sin O(x, y)}.

(2.39)

Hence, if cos O(x, y ) = A ( x , y), the first term in eq. (2.39) will be the desired complex wavefront. To determine O ( x , y) we assume that the maximum value of A(x, y) is normalized to 1. Although eq. (2.39) contains the correct wavefront, it also contains a term that has an

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improper amplitude variation. To get rid of the second term, Chu and Fienup used a spatial multiplexing technique to combine two functions in a single kinoform. One of the functions is, simply, f,(x, y ) in eq. (2.39); the second is f 2 ( x , y ) =e ~ ~ v ) - ed (x,~ Y)I , -

eJ*P(X,

Y)

[cos O(x, y ) -j sin O(x, y ) ] .

(2.40)

It is clear that A ( x , y)eJqP(x,y) is equal to O.5{f1(x,y ) + f 2 ( x , y)}. These two functions are combined in the kinoform by means of the delay sampling technique discussed in 0 2.2.3. The sampled function f ( x , y) used to make the kinoform is

Because f i ( x , y) and f2(x, y ) have only phase variation, f ( x , y) has constant amplitude and is suitable for recording by using the kinoform technique. In the reconstruction process, when f(x, y) is passed through a low-pass filter with impulse response, g ( x , Y ) = (sin ~x/d,)(d,/.rrx)(sin ~ y / d , , ) ( d , , / ~ y ) ,

(2.42)

the output from the filter will be

The output, fo(x, y), is the desired complex wave field. The delay sampling technique used in 52.2.3 not only combines four functions in a hologram, but also generates the proper phase shifts for each one in the reconstruction by means of a bandpass filter. However, the delay sampling technique used here simply combines two functions for recording in the kinoform. The phase shift due to the delay in the sampling is not used in the reconstruction. Figure 13 shows such a kinoform and the image reconstructed from it. Note that the noise from the second term in the function f i ( x , y) is not

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Fig. 13 The phase variation shown in (a) is generated by using the parity sequence method which can add amplitude variation to the reconstructed wave. The reconstructed image is shown in (b). (Courtesy of Chu and Fienup.)

present in the low-frequency region where the reconstructed image is located. However, the bright spot in the center of the reconstruction indicates that there is a phase mismatch in the kinoform. If the function f2(x, y ) used in making the kinoform is f2(x, y ) = -el*(%Y)-e(X. Y ) (2.44) the correct wavefront will occur in the first diffracted order similar to Lee's hologram discussed in Q 2.2.3. In this case, the phase recorded by the delay sampling is used to cancel the second term in the function fl(x, y). Another phase function suggested by CHUand FIENUP[1974] is of the form f(x, y)=f1(x, y>+f2(x-L, Y ) .

(2.45)

The functions fi(x, y) and f2(x, y) are given in eqs. (2.39) and (2.40). The parameter L is the extent of the function fl(x, y') along the x-direction. It is used to make the functions fi(x, y) and f2(x, y) nonoverlapping in recording the kinoform. If F,(u, u ) and F,(u, u ) are the Fourier transforms of eJ'4(X,Y) cos O(x, y) and e"P("3Y)sin O(x, y), the Fourier transform of f(x, y) in eq. (2.45) can be written as F ( u , u ) = F,(u, u ) cos 2 r L u + F,(u, u ) sin 27~Lu.

(2.46)

Therefore, when the function F(u, u ) is sampled at u = n/L, its value is equal to F,( u, v), the Fourier transform of the desired complex wavefront. This technique is useful in making Fourier transform type kinoforms. In

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general this method is not as attractive as the first one discussed because of the difficulty in separating the functions F,(u,v) and Fi(u,v) in the reconstruction process. 2.4. COMPUTER-GENERATED INTERFEROGRAMS

2.4.1. Generation of binary holograms In many applications, the wavefronts to be recorded in the holograms have only phase variations. When these wavefronts are recorded as image holograms, they are similar to interferograms (BRYNGDAHL and LOHMANN [1968]). If a wavefront with phase variation only is recorded as an off-axis reference beam hologram, the amplitude transmittance of the hologram is t(x, y ) = 0.5{1 +COS

[ ~ T C Z X - P(X, y)]}.

(2.47)

In eq. (2.47) the function t(x, y ) has its maximum values at locations where 2rrax - q(x, y ) = 2mn,

(2.48)

and its minimum values at locations where 2 ~ ax q(x, y) = 2 ~ ( +$). n

(2.49)

Either eq. (2.48) or eq. (2.49) defines the location of the fringes in the hologram. The contrast of the fringes in the hologram can be enhanced by using the nonlinearity of the photographic film in the recording process to produce a binary fringe pattern. The fact that binary interferograms can be obtained by using the nonlinearity of photographic film suggests that binary computer-generated holograms can also be obtained by passing the function cos [27rax - q(x, y)] through a nonlinear limiter simulated by the computer during the computation. The desired nonlinear operation on the sinusoidal signal is shown in Fig. 14(a). For any input, the output of the limiter is either 0 or 1. The bias, cos Tq, is added to the input signal to control the width of the fringes in the binary hologram. The relationship between the output and the input of this nonlinear limiter can be analyzed by assuming that the input function is cos ~ T which Z produces at the output of the limiter the periodic function shown in Fig. 14(b). The width of the rectangular pulses in the figure is equal to q. Hence, the

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(b)

Fig. 14. Nonlinear limiter for generating binary holograms

output function h ( z ) can be expanded as a Fourier series:

(2.50) The output of the limiter with cos [27rax - q ( x , y)] as input is obtained by substituting 2 ~ a x cp(x, y ) for 2rr.Z and q(x, y ) for q into eq. (2.50) to get

(2.51) In eq. (2.51) it is assumed that the bias function in Fig. 14(a) is a space-variant function. By selecting the function q(x, y) in such a way that A(x, y) =sin rrq(x, y), the m = -1 term in eq. (2.51) produces the wavefront A(x, y)eJq(x, y ) . Therefore, binary holograms can record both the amplitude and phase information of a wavefront without resorting to approximations such as those used in the detour phase hologram. For wavefronts that have constant amplitude, the parameter q can be used to determine the diffraction efficiency of the hologram in the reconstruction. When q is 0.5, all the even terms in eq. (2.51), except the m = 0 term, will disappear. This particular value of q allows the hologram to have a diffraction efficiency of 10% at its first diffracted order. Still

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higher diffraction efficiency (40%) can be achieved by bleaching the hologram to convert it to a phase relief hologram. From Fig. 14(a) the hologram function h(x, y ) is equal to 1 if cos[2rrax-cp(x, y)]>cos 7rq.

(2.52)

This can also be written -qrr

< 27rax - q ( x , y) + 27rn < q7r,

(2.53)

where n is an integer. Equation (2.53) is the most general equation for making binary computer-generated holograms. When the wavefronts have only phase variation, it is easier to plot holograms with narrow fringes. This means that the value of q should be set to 0 for making a hologram with narrow fringes. For q = 0 eq. (2.53) becomes

2rrax - q ( x , y) = 2rrn.

(2.54)

This simplified equation is used below to discuss the procedure for making this type of binary hologram.

2.4.2. Considerations in making binary holograms The first consideration in making the hologram is the selection of the carrier frequency, a. Because there are many diffracted waves reconstructed from the binary hologram, it is important to use a sufficiently high carrier frequency to separate the first order diffracted wave from the higher-order waves.. From eq. (2.5 l), the spatial frequencies of the different diffracted orders in the x-direction are given by

%(X, y > = m b -(1/27r)dq(x, Y ) / W ,

(2.55)

where m indicates the diffracted orders. Similarly, the spatial frequencies in the y-direction for the diffracted waves are given by u,(x, Y)

= - (m/27r)fwix,YYdY.

(2.56)

It can be seen from eqs. (2.55) and (2.56) that the spatial frequency bandwidth along both directions increases linearly with m. The higherorder diffracted wave occupies a larger region in the frequency plane than the lower order diffracted wave. Because the spatial frequencies along the y-direction are independent of a, they will have no influence on the selection of the camer frequency a.

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Suppose that the spatial frequencies along the x-direction are bounded between - U , and U,. To avoid overlapping between the first- and second-order diffracted waves in the frequency plane, the carrier frequency, a,must satisfy a

+ u,<2ff - 2 u , ,

(2.57)

or, ff

>u2+2u,.

In eq. (2.57), cy + U, is the highest spatial frequency in the first-order diffracted wave and 2 a - 2 U , is the lowest spatial frequency from the second-order diffracted wave. Equation (2.57) is simply the mathematical statement for nonoverlapping between the first- and second-order diffracted waves. This condition also guarantees that the spatial frequencies from the higher-order waves will not extend to the spatial frequency region of the first-order diffracted wave. It is easy to show that the value of a determined by eq. (2.57) also satisfies the following inequality ff

+ u,< m(ff - U,),

(2.58)

where the righthand side of the inequality is just the lowest spatial frequency for the mth-order diffracted wave. Many wavefronts of interest have even symmetry about the y axis. This gives rise to symmetric distribution of the spatial frequencies of the wavefront about the u axis in the frequency plane. Therefore, the bounds on the spatial frequencies designated previously as UI and U, are the same and can be set equal to half of the bandwidth, U, of the wavefront along the x-direction. For these wavefronts, the condition given in eq. (2.56) for the carrier frequency a becomes a > 1.5 U.

(2.59)

This carrier frequency is three times higher than that needed for a Burch hologram. This is the price of recording the hologram in a binary format. When q = 1/2, the second diffracted order disappears because of the coefficient in eq. (2.51). This sets the third-order diffracted wave next to the first-order diffracted wave. This permits the condition on a to be somewhat relaxed, as can be seen in the following inequality a >33u,+ UJ.

(2.60)

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For a symmetric wavefront where U , = U,= U/2, eq. (2.60) becomes a>

u.

(2.61)

It is more time-consuming to make binary holograms with q = 0 . 5 (fringe widths equal to half of the fringe spacing). However, it does allow better utilization of the bandwidth of the hologram and produces a hologram with higher diffraction efficiency. The selection of the carrier frequency according to eq. (2.59) or eq. (2.61) also helps to reduce the complexities in locating the fringes in the hologram. Since the fringes are found by solving eq. (2.54) for some integer n, this is similar to finding the constant-value contour for an arbitrary function. But with the term 27mx on the left-hand side of eq. (2.541, the function is a monotonically increasing function in x. This property of the function on the left-hand side of the eq. (2.54), now designated by g(x, y), can be demonstrated by proving that

g(x1, y)-g(x*, Y)>O

(2.62)

for xl>x,. The left-hand side of eq. (2.62) is equal to

When x1- x2= A is small but positive, eq. (2.63) can be written as

g(xi, Y > - ~ ( x ,~, ) = 2 m A [ a- ( ~ / ~ T ) ~ c Py)laxI. (x,

(2.64)

By virtue of selection of the carrier frequency, the right-hand side of eq. (2.64) is always positive and this proves the inequality in eq. (2.62). The condition given in eq. (2.62) restricts the fringes, with the larger fringe index n, always on the right-hand side of the fringes with the lower index. As a result, once a fringe in the hologram is found, its location can be used as the starting point for searching for the next fringe. The inequality in eq. (2.62) also guarantees that the fringes in the hologram plane will not form closed loops. After selection of the carrier frequency, a, the next step in generating the computer hologram is to solve eq. (2.54) for the locations of the fringes. In general it is difficult to find an analytical expression that gives the location of y in terms of the coordinate x and the fringe index n. Therefore, the fringe locations are found by substituting the two coordinates successively into eq. (2.54) and testing to see whether they satisfy the equation. The spacing between the discrete points along the xdirection is given by T / M where T is l / a , the grating period of the

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hologram, and M is an integer. The accuracy in locating the fringe is determine by TIM. The spacing between the points along the y-direction is equal to d,, = 1IV. V is the spatial bandwidth of the wavefront in the y-direction. A t these discrete locations where 3: = k,T/M and y = k,,dy eq. (2.54) is equal to

2~k,lM-cp(k,T/M, k,,d,,) = = 2 ~ n .

(2.65)

Multiplying both sides of eq. (2.65) by M / ~ Treduces the equation to

k , -(M/2rr)cp(kXT/M,k,d,)

= nM.

(2.66)

If modulo M is applied to eq. (2.66), the right-hand side of eq. (2.66) will be 0 because it is proportional to M. Thus, eq. (2.66) is simplified to , =O. Mod,{kx - ( M / ~ T ) v O ( ~ , T I Mk,,dy)}

(2.67)

By the use of residue arithmetic the fringe index II has been removed from eq. (2.66). Now instead of finding the pairs [k,, k,,]that can satisfy eq. (2.66) for a given value of n, it is only necessary to find the ones that make the residue of the left-hand side of eq. (2.66) equal to 0. This simplifies the computer programming involved in finding the fringes. There are two ways to plot binary holograms. Both require only a small number of memory locations for storing the computed data related to the hologram. In one method, a line segment with length d, is drawn at location ( k x ,k,,)where the condition in eq. (2.67) is met. In this case there is no need for storing computed data in the computer memory. Each point will be plotted as it is found. There is also no need to know the fringe index for the fringe points, as was pointed out earlier. However, this method requires longer plotting time because the pen has to move up and down for each line segment plotted. To solve this problem the successive points found along the y-directions are stored in the memory. The fringe index for each of these points is then calculated by substituting its location back into eq. (2.66). All the points with the same fringe index are then connected to form one fringe in the hologram. Once the first fringe has been found and plotted, the computer program will look for fringe points on the right-hand side of the previous fringe because of the property in eq. (2.62). This procedure is repeated until the whole hologram is plotted. In using this method, only one fringe line has to be in the computer memory at a time. The number of storage locations needed is equal to the space bandwidth product of the wavefront along the y direction.

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2.4.3. Examples In this section, four different holograms are used to illustrate some of the properties of computer-generated interferograms. Some of the phase variations recorded in the holograms can be found in optical components such as a Fresnel lens, axicon, etc. For example, the spherical wavefronts in the first example can be obtained from a zone plate, or a lens. The conical wavefront in the second example is that of an axicon. The third hologram is equivalent to a Schmidt plate (LINFOOT [19S8]) used to correct the spherical aberrations in a spherical mirror. The last example is somewhat unconventional. The phase variation is linear in the azimuthal direction. This example demonstrates that a computer-generated interferogram can indeed produce phase variation that would be difficult to obtain with other fabrication techniques. A. Spherical wavefronts A spherical wavefront has phase variation given by

(2.68) where r 2 = x 2 + y 2 . The focal distance of the wavefront is F, and the wavelength of the laser illumination is A. The phase variation of a thin lens can be approximated by the phase variation in eq. (2.68). This wavefront has spatial frequencies along the x-direction and y-direction given by (2.69) The spatial frequencies of this wavefront are dependent on the location of the wavefront in a linear way. The maximum frequencies, as can be seen in eq. (2.69), occur at the boundary of the wavefront. Therefore, the spatial frequencies of the wavefront along the x-direction are bounded between -D/(2hF) and D/(2hF), where D is the lateral extent of the wavefront. Suppose that a hologram with a diameter of S m m is made to record the spherical wavefront with a focal distance of 8 m . The bandwidth of this wavefront along both axes is about 10 lines/mm €or h = 632.8 nm. To satisfy eq. (2.59), the carrier frequency, a,for the hologram is selected as 20 lines/mm. The average number of fringes in the hologram is given by N+= Da

= 100.

(2.70)

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MAKING COMPUTER-GENERATED HOLOGRAMS

159

The maximum phase deviation of the wavefront at the edge of this hologram measured in 27r units (or number of wavelengths) can be obtained from eq. (2.68) and is equal to (2.71) In eq. (2.71) the relationship a = 2D/(AF) has been used. As will be found in other examples as well the maximum phase variation in the hologram is linearly proportional to the number of fringes in the hologram. The constant of proportionality between the number of fringes and the maximum phase deviation is strongly dependent on the maximum gradient (spatial frequency) in the wavefront. Figure 15(a) shows the computer-generated interferogram of the spherical wavefront. The original size of the hologram is 25.4 cm x 25.4 cm. It is plotted on a Hewlett-Packard 7202A graphic plotter. Plotting time and computation time is a total of about 20 minutes. The plot is photoreduced to about 5 mm o n the side onto Kodak 649F film. The Fourier transform of the hologram is shown in Fig. 15(b). The spectra of three different diffracted waves on either side of the optical axis can be observed. Because the spatial frequencies of this wavefront are linearly dependent on the coordinates of the wavefront, the spectrum of the wavefront takes on the shape of the hologram. The first two orders of the diffracted waves are separated as expected from the selection of the carrier frequency. The phase magnification in the higher order diffracted waves makes the spatial frequency bandwidth of the higher order difbacted wave larger. The bandwidth of the second-order diffracted wave is doubled along both coordinates so that the spatial frequencies occupy an area four times that of the first-order wave. The hologram in Fig. 15(a) is an image hologram of the phase variation in eq. (2.68). To show that the reconstructed wave from the hologram is indeed a spherical wave, we obtain an interferogram of the reconstructed wave. The computer-generated hologram is put at plane CGH in the optical system in Fig. 16. The regular grating in the optical system serves as a beam splitter to provide two illumination beams for the hologram. The frequency of the regular grating is matched to that of the hologram. With the two plane waves incident on the hologram many diffracted waves are reconstructed from the hologram. But only two of the first-order conjugate waves reconstructed are propagated along the optical axis. The

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Fig. 15. (a) Computer-generated hologram of a spherical wavefront. (b) The Fraunhofer diffraction pattern of the hologram in (a). (c) The interference pattern of the reconstructed wave from the hologram with a plane reference beam.

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Grating

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MAKING COMPUTER-GENERATED HOL.OGRAMS

L,

M,

CGH

L,

L,

M*

L,

I

Fig. 16. Optical system for obtaining the interferogram in Fig. 15(c)

second aperture mask in the optical system lets these two waves pass to form the interferogram at plane I. The intensity variation of the interference pattern is given by I(x, y)=c0sZq(x, y)=;{l+coS2q(x, y)}.

(2.72)

The interferogram shows twice the amount of the phase variation recorded in the hologram. The fringes in the interferogram are the constant phase contours of the wavefront. The interferogram in Fig. 15(c) is similar to a Fresnel zone plate. Therefore, this method can indeed generate holograms of wavefronts which are given to the computer only as mathematical descriptions.

B. Conical wavefronts A plane wave after passing through an axicon (MCLEOD[1954]) has a cone-shape wavefront. The phase variation of such a wavefront can be described by q ( x , y) = 2rrrlro.

(2.73)

r, determines the gradients of the wavefront along the radial direction. The bandwidth U of this wavefront along the x-direction is equal to 2/r0. Using a carrier frequency which is twice the bandwidth of the wavefront gives the following relationship between r, and a : (Y

= 4/ro.

(2.74)

The hologram shown in Fig. 17(a) is made by using a = 20 lineslmm to give an average of 100 fringes in a 5 mm-wide hologram. The maximum phase deviation of the conical wavefront recorded in this hologram is N,

= $D/r,,= iNf =

12.25.

(2.75)

The constant slope of the conical wavefront allows more phase variation to be recorded in this hologram than in the previous hologram, even

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though the number of fringes in the holograms are the same. Since the slope of the wavefront along the radial direction is constant, the Fourier transforms of the diffracted waves from the hologram shown in Fig. 17(b) are circles. The interferogram of this wavefront (see Fig. 17(c)) consists of equally spaced concentric circles. C. Aspheric wavefronts One of the phase variations for an aspheric wavefront is given by q ( x , y ) = - ( g . r r l A F ) ( ~ ~ + y * ) + ( 2 ~ / 8 A F ~ ) ( x * + y ~ ) (2.76) ~,

where g = 1/16(D/F)2.F is the focal length of the spherical mirror having this amount of spherical aberration and D is the diameter of the mirror. The factor g is chosen so that the corrected wavefront will have the same phase variation as the spherical wave at the center and the circumference of the mirror. That is, q ( x , y) is zero at both r = 0 and r = D/2. This hologram can be used to correct the spherical aberration of a mirror with focal length F. The hologram shown in Fig. 18(a) is made with parameters D = 10 mm, F = 75 mm and A = 632.8 nm. The carrier frequency for this hologram is again 20 lines/mm to give 100 fringes in the hologram. The Fourier transform of the wavefront shown in Fig. 18(b) is similar to the focused spot of a lens with the same amount of spherical aberration. Because the phase varies as r4, the spatial frequencies at the four corners of the hologram are very high. These frequencies produce the star-shape in the Fourier transform. The phase variation of the reconstructed wavefront is shown in the interferogram in Fig. 18(c). The maximum phase variation at r = 0124 is

N,= 1.17.

(2.77)

Therefore, the same bandwidth that is used to record the spherical wave and the conical wave can only record a few wavelength variations in the corrector plate because of the large gradient of the wavefront at the circumference of the hologram. D. Helical wavefronts A helical wavefront has linear phase variation along the azimuthal direction: q ( x , Y ) = 27T%/%O,

where % =tan-' y l x .

(2.78)

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163

Fig. 17. (a) Computer-generated hologram of a conical wavefront. (b) The Fraunhofer diffraction pattern of the hologram in (a). (c) The interference pattern of the reconstructed wave from the hologram with a plane reference beam.

164

Fig. 18. (a) Computer-generated hologram of an aspheric wavefront. (b) The Fraunhofer diffraction pattern of the hologram in (a). (c) The interference pattern of the reconstructed wave from the hologram with a plane reference beam.

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165

Fig. 19. (a) Computer-generated hologram of a helical wavefront. (b) The Fraunhofer diffraction pattern of the hologram in (a). (c) The interference pattern of the reconstructed wave from the hologram with a plane reference beam.

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COMPUTER-GENERATED HOLOGRAMS TECHNIQUES AND APPLICATIONS

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Phase variation of this type has not been made with conventional fabrication technique. The spatial frequency of this wavefront along the x -direction is -sin 0 (2.79) u,(r, 8)=-.

re,

The spatial frequency is inversely proportional to the radius r. Obviously, a fixed carrier frequency cannot satisfy eq. (2.59) for all radii. To get around this problem a small region in the center of the hologram is not plotted with fringes. The radius of that region must be larger than 1 . 5 ~ ~ 0 , so that the carrier frequency can satisfy eq. (2.59). The hologram in Fig. 19 is made with O0 = 2 ~ / 1 and 6 cx = 20 lines/mm. The diameter of the hole in the center of the hologram after photoreduction is about 0.1mm. The hologram contains about 100 fringes. The Fourier transform of the hologram is shown in Fig. 19(b). The fine structures in the higher spatial frequencies come from the region near the origin of the hologram. The outer circumference of the hologram contains all the low spatial frequencies of the wavefront. The interferogram of this wavefront shown in Fig. 19(c) consists of spokes along the radial direction. This interferogram is similar to the interferogram obtained from the kinoform in Fig. 11. 2.4.4, Generalization of computer-generated interferograms We have discussed how a space variant bias function coupled with a nonlinear limiter can produce a binary hologram that can modify the amplitude as well as the phase of a wavefront passing through the hologram. The amplitude of the diffracted wave from the hologram is recorded by the widths of the fringes in the hologram and the phase is recorded in the positions of the fringes. Both pieces of information are recorded along the x coordinate. To better utilize the two-dimensional characteristics of the hologram, it is better to record the amplitude information along the y -direction instead of the x-direction. A method for producing such a binary hologram is outlined in Fig. 20. In the lower branch of the figure, the phase information is still encoded in the same way as shown in Fig. 14(a) with the exception that the bias in Fig. 20 is now a constant q. In the upper branch a sinusoidal function with frequency y is used as a carrier and the amplitude variation A(x, y ) takes the place of the function q ( x , y ) . From previous discussions the output

111, § 21

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MAKING COMPUTER-GENERATED HOLOGRAMS

(x)

COS~TA(X,YI

hkv1 =

qk"

h,bw 1

=

h,(x, y ) from the upper branch is

The first term in eq. (2.80) is the amplitude variation A(x, y). The output from the lower branch in Fig. 20 is the same as eq. (2.51) with q a constant. The function h(x, y) is the product of h,(x, y) and h,(x, y). By multiplying the right-hand side of eq. (2.80) with the right-hand side of eq. (2.51), one of the terms in the product h(x, y) with n = 0 and m = -1 is equal to A(x, y)exp{j[2rrax+cp(x, y)]}. In this generalization of the computer-generated hologram, the amplitude is recorded in a way similar to the halftone technique used in printing. The amplitude variation is controlled by the dot size or, in this case, the length of the line segments in the interferogram. When the amplitude of the wavefront is approximately constant inside each period of the sinusoidal carrier sin 2rryy and is equal to the sampled value of the wavefront at the center of each period, this generalized method is the same as the method used by Brown and Lohmann in their improved detour phase technique. In this regard, their improved method can indeed solve the many problems associated with the approximations in their original holograms.

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D 3. Quantizations in Computer-Generated Holograms Before making a computer-generated hologram, the function representing the amplitude transmittance of the hologram or the complex wavefront itself has to be digitized by the computer. The digitization involves first sampling the function at regular intervals and then quantizing each of the sampled values into a finite number of levels. The sampling process, if carried out at a sufficiently high rate, will incur no loss of information to the original function. However, when the sampled values are quantized, irreducible errors are added to the original function. The errors which are due to quantization are often regarded as additive noise. In computer-generated holograms,the quantization noise limits the accuracy of the wavefront reconstructed from the hologram. It also reduces the detectability of digital data or images stored in the computergenerated holograms (POWERS and GOODMAN [ 19751). The term quantization has also been used to mean the conversion of a continuous function into a discrete valued function (GOODMAN and SILVESTRI [1970], DALLAS [1971a and b], DALLAS and LOHMANN 119721).A continuous function will become a discrete-valued function if it is passed through a nonlinear limiter with an input-output characteristic such as that shown in Fig. 2 1. The distinction between a quantized function and a discrete valued function is illustrated in Fig. 22. The digitized function is shown as the solid step function which is obtained by first sampling the continuous curve at the center of the sampling cell and then assigning its sampled values into one of ten levels. The transitions in the digitized function occurs at x = n. On the other hand, if the same curve is passed through the nonlinear limiter in Fig. 21, the resulting discrete function is the dotted step function in Fig. 22. The transition of the discrete valued function does not occur at x = n. Instead it occurs whenever cp(x)= 2 ~ n / 1 0 where , n is an integrer. It can be seen that the discrete-valued Z

Fig. 21. Input-utput

characteristic of a limiter for generating discrete valued functions

111, 0 31

QUANTIZATIONS IN COMPUTER-GENERATED HOLOGRAMS

169

Fig. 22. The solid step function is a quantized function of the continuous curve. The dotted step function is a discrete valued function derived from the continuous function by using the limiter in Fig. 21.

function generated by passing through the limiter approximates the original function better. This special interpretation of quantization, as it turns out, is useful for studying kinoforms made by depositing a finite number of thin films on a substrate (D’AuRIA, HUIGINARD, ROYand Spnz [1972]). As will be shown in the following section, the discrete phase kinoform has the same properties as the continuous-phase kinoform.

3.1. DISCRETE PHASE KINOFORMS

Discrete-phase kinoforms are made from a phase function which has been passed through the limiter with the input-output relationship shown in Fig. 21. The step size in the staircase function is generally equal to l/Na, where N, is the total number of quantization levels. As N, approaches infinity, the quantization limiter becomes the limiter shown in Fig. 12(a). Therefore, the procedure used in 0 2.3.2 to derive the output function of a nonlinear limiter is applicable here. From Fig. 21 it is clear that the output 2’ of the limiter is a periodic function in Z, and so is the function g(Z)=exp (j27rZ’). The function g(Z) can be expanded into a Fourier series:

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COMPUTER-GENERATED HOLOGRAMS: TECHNIQUES AND APPLICATIONS

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where

The summation in the Fourier coefficient c, is equal to 1 when (1- m ) / N , is equal to an integer and 0 for other values of m. As a result, c, can be written as c, =

where (l-m)/N,=

sin n ( n - lIiV,) n ( n - l/NJ '

(3.3)

a. Thus, the function g ( Z ) is equal to

The wavefront of a discrete phase kinoform of the phase function q ( x , y ) is obtained by substituting q ( x , y) for 27rZ in eq. (3.4):

The n = 0 term in the wavefront of the discrete-phase kinoform is equal to e j p k y ) , which is the same as in the continuous-phase kinoform. Because of the presence of additional diffracted waves in eq. ( 3 . 9 , the light efficiency of the discrete phase kinoform cannot be as high as the continuous-phase kinoform. Moreover, when the higher order waves from the kinoform cannot be neglected from eq. (3.9, a linear phase term should be added to q ( x , y ) so that the higher order waves from the kinoform will not propagate in the same direction as the 0th-order wave. As N, approaches infinity, all the Fourier coefficients c,, except c ~ , become zero. The discrete-phase kinoform becomes identical to the continuous-phase kinoform. Phase mismatch can happen in discrete phase kinoforms, especially when the kinoform is designed for use in one wavelength and is actually

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171

used at a different wavelength. Or, the thin film layers that make up the phase variation in the kinoform do not add up to produce the correct amount of phase shift on the incident laser illumination. When this happens, the Fourier coefficients c, in eq. (3.2) should be modified by replacing ( 1 - m ) by ( l + p - m ) : 1 sin mT/Na 1 c-=N, 1 m T / ~ a

=-

1

Na

-eJ2n(l+6-tn)

-ej2dI+P-m)JNA

exp ( j ~ ( lp+- m ) ( l - l/Na)} X

sin(rnT/Na) sin . r r ( l + p - m ) . rnr/Na v (1 + P - m ) / N ,

sin

(3.6)

It can be shown that the coefficient ch approaches the coefficient given in eq. (2.31) as N, becomes infinite. ct, is again reduced to the coefficients in eq. (3.3) when p = 0. 3.2. QUANTIZATION NOISE

The effect of quantization on a function can be modelled as an additive noise n(x, y ) with a probability density function given by (3.7)

The amplitude of the noise is uniformly distributed between *l/(2Na). The severity of the effect of the quantization on the wavefront reconstructed from a quantized computer-generated hologram can be measured in terms of the mean square errors in the reconstructed wavefront. For example, the amplitude transmittance of an off-axis reference beam hologram after quantization can be written as t'(x,

Y) = t(x, Y ) + n(x, Y ) .

(3.8)

The mean square error in the reconstruction (attributed to quantization) is simply equal to MSE = E[n2(x, y)] = 1 / ( 1 2 N 3 .

(3.9)

E[. . .] in eq. (3.9) indicates the statistical average of the function inside the bracket. For this type of computer generated hologram the mean square error is independent of t ( x , y). However, when the amplitude and

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phase of the wavefront are recorded separately, as in the detour phase holograms or the modified kinoforms, the wavefront reconstructed from a hologram in which quantization has taken place is given by f(x, Y ) = [ A ( ~ ,) + n , ( x~)Iexp{j[cp(x, , y)+n2(x,

v>lI.

(3.10)

nl(x, y) and n,(x, y ) are, respectively, the quantization noise in the amplitude and phase of the wavefront. The mean square error in the reconstructed wave is MSE = E[(A(x, y)eJwp(x, y’ - { A(x, Y ) + n ,(x,Y )I exp {j[cp(x, Y) + n 2 k Y )1>1’1

=E[ni(x, Y ) ’ + ~ A ( x ,y){A(x, ~>+fin,(x, Y ) I {-COS ~ nz(x3 Y)II.

(3.11)

The probability density function pn(a) of the random noise, n,(x, y), is given in eq. (3.7). The phase quantization noise, n,(x, y), is also assumed to be uniformly distributed with the density function (3.12) where N, is the number of phase quantization levels. The mean square error MSE in eq. (3.11) then becomes

MSE= 1/(12N3+2A2(x, y)[l-(N,/~) sin (TIN,)].

(3.13)

The mean square error in the quantized wavefront has two parts. One is due entirely to the amplitude quantization. Its value is equal to that in the off-axis reference beam hologram. The second part is due to the phase quantization alone. The effect of phase quantization is coupled with the amplitude variation of the wavefront and is largest when A(x, y ) = 1. As N , becomes infinite, the second term in eq. (3.13) becomes zero. The mean square error becomes indentical to that of the off-axis reference beam hologram.

3.3. PHASE ERROR FROM QUANTIZATION

In many applications, only phase information is recorded in computergenerated holograms. In those cases, it is useful to determine the effect of quantization on the accuracy of the phase variation reconstructed from the hologram. When the phase variation is recorded as a computer generated interferogram as discussed in 0 2.4, the inaccuracy in the

111, § 41

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APPLICATIONS IN COMPUTER-GENERATED HOLOGRAMS

position of the fringes because of quantization error creates a wavefront with phase variation given by (PYX,

Y) = d x , Y ) + n,(x, Y ) .

(3.14)

The root mean square phase error is

RMSE= JE[n,(x, y ) ' ] = .rr/(,Np3t).

(3.15)

The result in eq. (3.15) indicates that with Np=30 the root mean square phase error in the reconstructed wavefront is less than 1/100 of a wavelength. Although the off -axis reference beam hologram has only amplitude quantization, the phase reconstructed from the hologram is also affected by the quantization noise. Suppose that the off-axis reference beam hologram is just a sinusoidal grating. The amplitude transmittance of the grating after quantization is

t'(x, y) =$[1+sin 27rax]+ n(x, y).

(3.16)

In the absence of quantization noise, the transmittance of the grating is equal to 1 / 2 at x = n / 2 a . With the quantization noise, the location where the transmittance is 1 / 2 occurs at sin 2 ~ a =x- 2 n ( x , y )

or

2rax

= m.rr+sin-'

2n(x, y ) .

(3.17)

For small n(x, y ) the phase error is approximately given by 2 n ( x , y). The RMSE phase noise in this case is equal to l/(&N,). The phase error in an off-axis reference beam hologram is less affected by the amplitude quantization. However, note that this result is obtained by assuming that the sampled data are recorded without jitter. Jitter in the sampling position in the off-axis reference beam hologram will produce phase error whose RMS fluctuation will be determined by an equation similar to eq. (3.15).

P

4. Applications of Computer-Generated Holograms

4.1. 3-D IMAGE DISPLAY W M COMPUTER-GENERATED HOLOGRAMS

The feasibility of a particular technique for making computergenerated holograms is often demonstrated experimentally by using the

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technique to make a Fourier transform hologram of a simple object. This, indirectly, demonstrates the usefulness of computer-generated holograms as a medium for displaying images that may not physically exist. In § 2, we have discussed the many techniques for recording wavefronts in computer-generated holograms. These methods can also record the wavefronts from a three-dimensional solid object in computer-generated holograms. However, because the exact calculation of the wavefront of a 3-D object at the hologram plane is rather complicated, if not impossible, this causes some difficulties in realization of such a computer-generated hologram. This rather difficult problem has been solved by two methods which we will discuss here. One way to simplify the computation of the wavefront of a 3-D object is to assume that the object consists of many independent scatterers (WATERS[1968], LESEM,HIRSCHand JORDAN[1969], BROWNand LOHMANN [1969]). Each scatterer is considered as a point source with a parabolic wavefront at the hologram plane. Figure 23 illustrates the optical system used for the calculation of the wavefront at the hologram plane. The 3-D object is located at the front focal plane of lens L. The hologram is assumed to be at the back focal plane. For an object point at (x, y, 2,) the wavefront at the hologram plane with the usual parabolic approximation is equal to

where S(x, y, 2 , ) is the amplitude of the point source and f is the focal length of lens L. When there is more than one point at a distance z, from the lens, the collective wavefronts at the hologram from these points is given by

(4.2)

'-2

--

f -

Fig. 23. Optical system used in computing the wavefront of a 3-D object.

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175

The summation in eq. (4.2) is the discrete two-dimensional Fourier transform of the points at a distance z,, from the lens. The total wavefront at the hologram for the entire 3-D object is the superposition of the functions w(u, TI, zn): (4.3) The summation in eq. (4.3) is carried out for all the planes that intersect the object. W(u,u) is a paraxial approximation of the wavefront of the object at the hologram plane. For If- z,,I << f the amount of phase variation in the quadratic phase term is small and can easily be incorporated into the computer-generated hologram. Figure 24 shows the images recorded at three different image planes from such a hologram. The separations between the different image planes can be noted by the varying sizes of the undifiacted light shown on the right-hand side of the reconstructed image. Because of the small size of the hologram, the reconstructed image shows very little parallax effect. A more practical approach to this 3-D image display problem was [1970]. Starting with a description of suggested by KING,NOLLand BERRY a 3-D object, the computer calculates the two-dimensional perspective view of the objects as it would appear on a projection screen at a particular viewing angle. Many of these perspective projection images are recorded in sequence on film. The film is then used in the optical system in Fig. 25 to produce a composite hologram. Each frame in the film is sequentially imaged onto a diffuse screen. A narrow hologram about 3mm in width is recorded at a distance from the screen. After each exposure, the input film is advanced to the next frame and the hologram is incremented to the next position. In reconstruction because each eye observes a different perspective of the original object, this produces a stereoscopic image. As the composite hologram is moved before the viewer, he sees a 3-D object rotate in front of him. Recently, YATAGAI [1974, 19761 has adopted this method to make computer-generated holograms to displaying 3-D images. Instead of recording the perspective projection images on film, he converts the projection images into Fourier transform holograms which are then plotted on a large scale on a plotter. These holograms are then arranged in the same order as the projection images are obtained. The holograms along the horizontal direction correspond to the different perspective projection images. To increase the viewing aperture of the hologram,

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Fig. 24. Images obtained from a computer-generated hologram containing depth information. (Courtesy of Brown and Lohmann.)

Film

Fig. 25. Optical setup for making a composite hologram.

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each of the holograms is duplicated six times along the vertical direction. Therefore, the hologram will not have the parallax effect in the vertical direction. If vertical parallax is needed, the holograms in the vertical direction have to be obtained from their corresponding perspective projection images. In viewing the 3-D object recorded in the hologram, the hologram is illuminated by a point source. The viewer looks through the hologram. Since the holograms are Fourier transform holograms, each eye of the viewer will see a different image of the object occurring at the same location. This produces a stereoscopic image. By using holograms, the use of special viewing aids for fusing the two images to form the stereoscopic image is not needed. Some of the perspective projection images from Yatagai’s holograms are shown in Fig. 26. The quality of the images displayed by computer-generated holograms have not yet reached the level of conventional holograms because of the limitations of the graphic devices. In this regard the method developed by King, No11 and Berry for displaying 3-D images obtained from computer calculations is far more effective than the computer-generated holograms approach used by Yatagai. However, it is worth pointing out that with proper display devices, the Yatagai approach can be used to display computer-generated 3-D images in real time.

Fig. 26. Perspective projections of a 3-D object reconstructed from the mosaic hologram. (Courtesy of Yatagai.)

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4.2. OPTICAL DATA PROCESSING

A coherent imaging system such as that shown in Fig. 27 can be considered a linear system (GOODMAN [1968]). The spatial frequencies of the images at plane Pi can be modified by a mask (or spatial frequency filter) at plane F to produce a filtered image at the Po. The light waves at planes Pi and Po are related by the equation

So(u,v) = Si(u, v)F(u,u ) ,

(4.4)

where Si(u, v ) and So(u, v) are the Fourier transformations of the light waves at the two planes indicated. F(u, u ) is the transmittance function of the spatial filter at plane F and is generally a complex function. In the 1950’s when the spatial frequency filtering experiments were demonstrated, the complex filter function F( u, u ) was synthesized by combining an amplitude mask with a corresponding phase plate (TSUJIUCHI [1963]). With the development of holography, it has become possible to make LUGT[1964]). The early certain types of complex spatial filters (VANDER works in computer-generated holograms were motivated by the need for a general method of synthesizing spatial filters for use in optical data and LOHMANN [1966], BROWN, LOHMANN and processing systems (BROWN PARIS[1966], Loand PARIS[1968]). With the different techniques discussed in § 2 for making computer-generated holograms, it is now possible to make rather complicated spatial filters for processing twodimensional images. In this section, we choose three examples to illustrate this potential application of computer-generated holograms.

Fig. 27. The input plane of the coherent optical processor is located at P, and the output plane is at Po.F is the spatial filter.

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4.2,l. Edge enhancement The edges in a low-contrast picture can be enhanced by differentiating the amplitude transmittance function of the picture. Suppose that we wish to enhance the horizontal edges of the picture by differentiating it in the y-direction. The Fourier transform of the differentiated picture is given by

So( u, v ) = Si(U, v)v.

(4.5)

A comparision of eq. (4.4) with eq (4.9, shows that the variation of the differentiation filter needed in the frequency plane F is equal to F(u, v) = 2).

(4.6)

The amplitude transmittance of this filter is 1 ~ 1 .The phase of the filter is equal to 0 on the upper half-plane and +rron the lower half-plane. This simple filter can be made in a number of ways. One realization of this spatial filter is shown in Fig. 28(a). The carrier frequency used in making the filter is along the u-direction. The width of the fringes in the filter is proportional to ( v ( . The maximum fringe width is equal to half of the grating period. The phase information is recorded by shifting the fringe pattern in the lower half-plane of the filter by a half-grating period. When this filter is used in the optical system in Fig. 27, the output picture at plane Po will be shown in Fig. 28(b). The picture at the input of the optical system is the letter H. Four diffracted outputs from the optical system can be seen in Fig. 28(b). The first-order output shows the enhanced horizontal edges of the letter H. Because the phase information in the second diffracted order from the filter is doubled, the phase on the lower half-plane of the filter is increased to 27r and gets in phase with the upper-half plane. No differentiation occurs at the second-order diffracted output. The third-order diffracted output again shows the differentiation of the letter H. The increase in phase in the third-order output makes the enhanced edges wider than those in the first-order output. There is no differentiation in the fourth-order output because the two half-planes in the filter are once more in phase. Another spatial filter that can produce a differentiated output picture is shown in Fig. 29(a). The spatial filter is made by combining the method suggested by Kirk and Jones (cf. 0 2.3.3) for modifying the kinoform and the binary interferogram. The grating equation for generating the spatial

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Fig. 28. The hologram in (a) can be used as a spatial filter for edge enhancement. (b) shows the output from the optical processor. The horizontal lines in the letter H at the oddnumbered diffracted orders of the spatial filter are enhanced.

filter in Fig. 29(a) is au

+ a(v)cos 27rpv = n.

(4.7)

As explained in 9 2.4, the first-order diffracted wave from the spatial filter made using the grating equation given by eq. (4.7) is

exp { j 2 m ( u ) cos 27rpv) = where bo = 1 and b,

=2

for n # 0.

C bn~,,(2.rra(v))cos 2 ~ n p v , n

(4.8)

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Fig. 29. (a) shows another realization of the differentiating filter. The differentiated "H" occurs at the +1 diffracted orders along the vertical direction.

If a ( v ) is such that or

J,[2%-a(v)]= u,

a ( u )= (1/2Tf)JT'(v), (4.9) the n = 1 term in eq. (4.8) will have the proper characteristics for

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differentiating the input picture. For small values of u, the function a ( u ) is approximately equal to u. This approximation is used in making the filter in Fig. 29(a). The outputs at Po when this filter is used in the optical system are shown in Fig. 29(b). The *1 diffracted orders along the y-direction clearly show the enhanced edges of the letter H. Because the filter is recorded with a carrier frequency along the u-direction, the phase of the higher diffracted orders along x will be multiplied by an integer corresponding to their diffracted order. The phase multiplication in the diffracted orders along x changes only the widths of the edges in the enhanced images. This can be seen in Fig. 29(b). By using the filters in Figs. 28(a) and 29(a) only the edges along the x-direction are enhanced. To enhance the edges in any direction, a spatial filter with the filter function, (4.10) F(u, v ) = u +ju, can be used. The spatial filter in Fig. 30(a) is a binary off-axis reference beam hologram of the function F(u, u ) in eq. (4.10). Lee's encoding method is used in making the filter. In eq. (4.10), the variation along u is 90" out of phase with the variation along u. By using Lee's method, the filter has the appearance of two interlaced filters. The data samples in one of the filters have been shifted by 1/4 of the sampling period. An enhanced picture obtained by using this filter is shown in Fig. 30(b). The edges of the bar target in both directions are enhanced. The multiple, filtered images in Fig. 30(b) are the result of the uniform spacing between the data samples along the u-direction in the filter. The circles enclosing the letter H shown in Figs. 28 to 30 are 3 mm in diameter. The input picture processed by these filters is small mainly because of the low carrier frequency of the spatial filter. If the spatial filter is copied by using the optical system in Fig. 31, the interferometrically copied spatial filter with its higher carrier frequency can be used to process a much larger picture (LOWENTHAL and CHAVEL [1974]). The copying system images one of the diffracted waves from the computergenerated spatial filter to the film plane. This reconstructed wave from the original spatial filter is then recorded on film as a hologram by using reference beam R. 4.2.2. Image deblurring Image deblurring is another problem that can be solved by using a coherent optical system with the proper computer-generated hologram.

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Fig. 30. The spatial filter in (a) is made as a modified off -axis hologram according to Lee’s method. This filter can enhance the edges in an image in any direction as can be seen in (b).

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Fig. 31. Optical system for copying a computer-generated hologram to increase the carrier frequency of the hologram.

So long as the process that degrades the image can be modelled as a linear operation on the image, the degraded image can be improved by spatial filtering. The improvement in the filtered image is limited by the noise in the degraded picture. The spatial filter that minimizes the mean squared error in a degraded image is given by HELSTROM [1967]:

F(u, v ) = B * ( U , v) S ( u ,u)/{(B(u,v)(ZS(u,v ) + N ( u ,v)}.

(4.1I)

B ( u , u ) is the Fourier transform of the impulse response of the degrading process. S(u, u ) and N(u, v ) are, respectively, the power spectra of the image and the noise in the picture. For lack of information about the image or the noise process a useful filter function would be

F(u, v) = B*(u, v>l{lB(u,v>l”+N),

(4.12)

for u and z, less than the cutoff frequency of the image. The constant N represents the mean squared value of the noise in the degraded image. For example, the pictures in Figs. 32(a) and (d) are degraded by convoluting the original picture with a small aperture. The orginal picture has 512 X 512 elements and the blurring aperture has 16 x 16 elements. For this aperture, the function B(u, v ) in eq. (4.12) is equal to sin rrwx sin rrwy (4.13) B(U, v) = ___. rrwx rrwy The parameter, w, is the width of the aperture. A realization of the deblurring filter is shown in Fig. 33. The filter function recorded is equal ~

to

(4.14) F,(u,u ) = /F(u, u)(iei+,u), where q(u, u ) is the phase function of F(u, v ) in eq. (4.12). The square

root of the amplitude is used instead of the amplitude in the filter to reduce the dynamic range required for the deblurring filter (ANDREW, TESCHER and KRUGER [1972]). The optically processed pictures are shown in Figs. 32(c) and (f). Note that the phase reversal of the radial bar target

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Fig. 32. The pictures in (a) and (d) have been “blurred” by convoluting the original image with a square aperture. (b) and (e) are the output pictures restored by using the digital technique. ( c ) and (f) are pictures restored by using a coherent optical system and the spatial filter in Fig. 33. (Courtesy of Campbell, Wecksung and Mansfield.)

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Fig. 33. Spatial filter for deblumng images. (Courtesy of Campbell, Wecksung and Mansfield.)

has been corrected by the filter. Some of the details around the mouth of the girl in the blurred picture become more visible after deblurring. Figures 32(b) and (e) are the same pictures after they are processed digitally by the computer with the same filter function Fl(u, u ) . Without such things as coherent noise in the optical system, the digital computer produces a better restoration of the orginal picture than the optical system. 4.2.3. Matched filters optical processor One of the most important applications of a coherent optical system is in processing data f l m recorded by a synthetic aperture radar system. The coherent optical system is basically used as a multichannel correlator (CUTRONA, LEITH,PORCELLO and VIVIAN[1966], LEITHand INGALLS [1968], LEITH[1976]). The returned radar signals, after being recorded o n photographic film, can be compressed by cross correlating them with a set

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of reference signals to form a high resolution terrain map. The reference signals used in the early optical processors are provided by the phase variation of a conical lens. With the process of data collection by the synthetic aperture radar system better understood, KOZMA,LEITHand MASSEY[1972] developed a new optical processor for obtaining a terrain map from radar film. The new processor did not use a conical lens in the optical system, using, instead, tilted input and output planes. The tilt angle of the input plane is related to the angle of elevation of the antenna used in -collecting the radar data. This tilted-plane processor has the advantage that the magnification in the output terrain image is independent of the distance of the terrain from the antenna. Thus, there is a oneto-one correspondence between the movements of the input data film and the output data. The synchronism in movement between the input and output planes (also called tracking) increases the exposure time for recording the output image and averages out the coherent noise from the dust particles in the optical system. Tracking in the tilted-plane processor can also be achieved by using multichannel spatial matched filters. The matched filters needed for processing the synthetic aperture radar film, unlike a conical lens, cannot be fabricated easily by conventional optical techniques. In this section, we discuss how to use computer-generated holograms as multichannel matched filters. An optical processor using mutichannel matched filters does not require tilted planes in the optical system.

A. Properties of the radar signal film In collecting data for making a terrain map, the synthetic aperture radar illuminates the terrain with a broad beam of monochromatic radiation. The returned signals from the terrain at different slant ranges are detected and recorded on film. Suppose that there is a point target at a slant range R from the aircraft carrying the radar. The photographic film that records the returned signals from this target has amplitude transmittance given by s ( x , R ) = t , + a c o s ( 2 ~ a x - ~ x ~ p ~ / h , for R ) IxlsW2

(4.15)

The corrdinate x in eq. (4.15) is along the flight path of the aircraft. A, is the wavelength of the signal radiated from the radar, tb is a bias constant, CT is proportional to the reflectivity of the target, p is a scaling factor for compressing the azimuth data on film and a! is a carrier

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frequency for recording the radar data. The length L of the radar data is determined by the diffraction pattern of the aperture of the antenna. For an antenna with aperture D (in the azimuth dimension) and uniform illumination, the far-field pattern of the antenna is of the form sin xlx. The beam width at slant range R is equal to &RID. Therefore, the length of the azimuth data recorded on film is equal to

L

(4.16)

= h,R/pD.

The transmittance function of the radar film can be rewritten as s(x,

I “x2p211

R ) = t , + f a exp j 27rax -h,R

1

+ $ a e x p -j 27rax---7 r x 2 p 2 ] ) h,R

(4.17)

Suppose that this film record is illuminated by a collimated laser beam with wavelength A. The quadratic phase variation in the second term will focus the laser beam along x at a distance f ( R )= h,R/(2hp2) behind the data film. The width of the focused spot along x (the azimuth) direction is p = h f ( R ) / L= D/2p.

(4.18)

The actual azimuth resolution on the terrain is equal to p p = D/2 (half of the aperture of the broad beam antenna). This resolution is independent of the slant range R. The third term in eq. (4.13) produces a divergent beam which appears to originate from a point at a distance f ( R ) in front of the data film.

B. Optical processor Because the focusing power of the azimuth data is linearly dependent on R, the image of the terrain obtained directly from the data film is formed on a tilted plane. Moreover, since the signal film has no focal power along the range direction, the range focal plane does not coincide with the azimuth focal plane. The optical processor for the radar signal film must be able to made erect the tilted azimuth focal plane and also bring the range focal plane into coincidence with the azimuth focal plane. In most synthetic aperture radar systems, the magnifications in the azimuth and range directions are different. The optical processor must also correct for the aspect ratio recorded on the radar film. An optical system that performs these various tasks is shown in Fig. 34. The radar film is located at the front focal plane, P,, of lens L,, and is illuminated by a collimated laser beam. The azimuth data s(x, R )as given

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Fig. 34. Matched filter optical processor.

in eq. (4.17) have two focal points (one real and one virtual) corresponding to the two quadratic phase variations in eq. (4.17). Figure 34(a) shows that virtual focal point at a distance f ( R ) in front of the film. The spherical lens L, images the virtual focal point 0 to 0’.The distance of 0’from the back focal plane of lens L, is determined by the Newtonian equation f ( R M R ) = F2,

(4.19)

where F is the focal length of lens L,. The lateral magnification of image 0’ is equal to

MB= y ( R ) / F .

(4.20)

A computer-generated matched filter is placed at the back focal plane of L1. This filter has properties similar to a cylindrical lens with focal power -1/y(R)- l/f. The R dependent focal power in the matched filter forms an image of 0 at infinity. The additional focal power in the computer-generated filter brings the image at infinity back to the plane P,. As a result, the tilted azimuth focal plane of the radar film is now made erect at plane P2. The magnification produced by the computergenerated filter is

M: = f/y(Rj.

(4.2 1j

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The total magnification along the azimuth direction is then equal to

Ma = MLM: = f/F.

(4.22)

Note that the magnification, Ma, is independent of R. Thus, the images from the different slant ranges will move at the same velocity as the data film moves across the input plane of the optical processor. In the range meridian of the optical system, the telescopic system with lenses L, and C, images the range focal (film) plane to the back focal plane of lens L, with magnification

Mi = fJF.

(4.23)

fc, is the focal length of cylindrical lens C1. Cylindrical lens C , creates an image of the range focal plane at P, to coincide with the erect azimuth focal plane. The separation, z, between cylindrical lens C , and the computer-generated filter is determined by the equation: l / z - 1/cf+2) = l/fc,.

(4.24)

The magnification produced by cylindrical lens C2 is

M:' = (f + z)/z,

(4.25)

with the distance, z, found from eq. (4.24). The total magnification of the image of the range focal plane at P, is

M, = M:M: = (f + z)f,,/zF.

(4.26)

The unequal magnification in the original data film along the azimuth and along the range directions can be corrected by choosing the proper focal length, f, in the matched filter. Finally, spherical lens L, in the optical system projects the processed terrain image from plane, P,, to the output plane, P,, for viewing or for recording on film.

C. Multichannel matched filter The filter in the optical system shown in Fig. 34 is in the frequency plane of the azimuth data. It is a matched filter, and its variation is given by the Fourier transform of the second term or the third term in eq. (4.17). The structure of the signal, s ( x , R ) , is similar to that of a cylindrical lens. To form an image of 0' at infinity, the matched filter must also act as a cylindrical lens with focal length (4.27)

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Recall that the focal length, f(R), is equal to X,R/(2Xp2). In radar film, the range is recorded along the y-direction. For convenience in the remaining discussion, the range variable R is now replaced by y. The functional form of the matched filter as a function of x and y can be written as m(x, Y)=expIj.rrx2(y/y,+fm/f)lhf,). (4.28) The focal power of the matched filter as given in eq. (4.38) has two components. The one proportional to y is used to correct the tilted azimuth focal plane. The focal length, fm, is the focal length of the range channel R = yo and is determined by the radar parameters and eq. (4.28). The constant focal length, f, is used to correct the aspect ratio in the radar film and to bring the range focal plane and the azimuth focal plane into coincidence. The width of the matched filter is fixed by the bandwidth of the radar signal, s(x, R ) . In the optical system in Fig. 34, the width of the Fourier transform of the radar signal is equal to U = h L ( R ) l f ( R ) . Since L ( R ) and f(R) are both linearly proportional to R, the width of the matched filter, U, is independent of R. An example of the matched filter in eq. (4.28) is shown in Fig. 35(a). The focal length, f, is infinite for the matched filter shown. Since the matched filter has only phase variation, it is made as a computergenerated interferogram. The carrier frequency of the filter is along the x-direction. The two-dimensional Fourier transform of the filter shown in Fig. 35(b) has spatial frequencies mainly along the x-direction. Figure 35(c) is a one-dimensional Fourier transform o f the same matched filter. The y-dimension of the filter is imaged onto the frequency plane by a cylindrical lens. This one-dimensional transform of the matched filter shows the impulse response of the matched filter at the different range channels. The width of the transform in Fig. 33c) varies linearly as y. This shows that the impulse response of the matched filter has the same properties as the signal recorded in the radar film. The narrow spatial frequency bandwidth of the matched filter along the y-direction suggests that the carrier frequency of the filter should be set in the y-direction to make better use of the bandwidth of the plotter. Figure 36(a) is a matched filter recorded with a carrier frequency along the y-direction. The magnitudes of the carrier frequencies in Figs. 35(a) and 36(a) are the same. But, by arranging the carrier frequency to be normal to the phase variation of the filter, more phase variation is recorded in Fig. 36(a). This can be seen in the larger bandwidth in the Fourier transform of the matched filter in Figs. 36(b) and 36(c).

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Fig. 35. The hologram (a) for tilted plane correction has the carrier frequency in the same direction as the major phase variation. The two dimensional Fourier transform (b) of the hologram shows only small frequency variation in the y direction. (c) is the one dimensional Fourier transform of the hologram in (a). The y dimension of the hologram is imaged onto the frequency plane. This illustrates the relationship between the bandwidth of the filter function along the x-direction and its y coordinates.

The focal power due to l/f can be included directly in the making of a computer-generated hologram. However, the bandwidth required to make a filter with a 100 mm focal length easily exceeds the bandwidth of most display devices. Alternatively, this constant focal power can be added to the computer-generated filter by copying it interferometrically in the optical system in Fig. 31. The reconstructed wave from the computer-generated filter interferes with a diverging reference beam coming from a cylindrical lens. The reference beam provides the cylindrical focal power, l/f,needed in the matched filter. By copying the matched filter, we increase the carrier frequency of the matched filter. This permits

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Fig. 36. This hologram (a) for tilted plane correction has the carrier frequency along the y direction. This allows more phase variations to be recorded in the hologram for the same number fringes. This can be seen in the two spectra shown in (b) and (c).

processing and tracking the data film over most of the input aperture of the optical processor. A radar map of Detroit metropolitan airport obtained by the matched filter processor is shown in Fig. 37. The information needed to make the matched filter was obtained by analyzing the Fourier transform and the focusing properties of the original signal film. The focal length, f(R), of the radar film is measured by illuminating the film with a collimated laser beam and observing where each of the range channels comes into focus. The bandwidth of the signal film is measured directly at plane P, in the processor in Fig. 34. These measured data are then used by the computer to generate the matched filter without the focal power, l/f. Additonal focal power is added to the matched filter by the copying procedure discussed previously.

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Fig. 37. A radar map of the Detroit metropolitan airport obtained by the matched filter optical processor.

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4.3 INTERFEROMETRY

4.3.1. Shaping a reference wave in optical testing The phase variation of a wavefront can be made visible by using an interferometer such as the one shown in Fig. 38. A laser beam is split into two beams. The phase object under test is put in the path of one beam. The wavefront from the second beam serves as a reference wave to produce an interference pattern with the object wave at plane I. The intensity variation of the interference pattern is proportional to

(po(x, y) and qr(x, y) are, respectively, the phase variation of the object wavefront and the reference wavefront. The reference wavefronts commonly used in interferometers are limited to plane waves with or without a small tilt. Spherical wavefronts are sometimes used. By putting a computer-generated hologram in the reference branch of the interferometer, it is possible to create any type of reference wavefront for displaying the phase variation of the object wavefront. The special wavefront from the computer-generated holograms can reduce the complexities in the interferograms. For example, in testing aspheric mirrors or lenses, the residual aberrations of the optical components are so large that it is difficult to detect the fabrication errors in the optical elements. A special reference wavefront from a computer-generated hologram can remove the residual phase variations from the mirrors and produce an interferogram that displays only the fabrication errors (MACGOVERN and WYANT[1971], WYANT[1974], WYANTand BENNETT [1972]). An interesting application of computer-generated holograms to interferometry was demonstrated by BRYNGDAHL [1973]. He used special

0

L2

L,

I

Fig. 38. The reference wave in this Mach-Zender interferometer is provided by a computer-generated hologram.

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wavefronts from computer-generated holograms to display the phase variation of a wavefront in interferograms with circular or radial fringes. To produce a circular fringe system in the interferogram, the phase variation of the reference wave is equal to

(PAX, y ) = 27i-(x2+ y2F/ro,

(4.30)

where r,, is a constant. This reference wave is just the conical wave discussed in 3 2.4.3. A computer-generated hologram with this phase variation is shown in Fig. 17(a). The interferogram obtained by using this reference wave has a circular fringe pattern (see Fig. 17(c)). The interferograms in Fig. 39 show how the circular fringe system can enhance display of the information of the phase object under test. Figure 39(a) is

Fig. 39. Two phase objects are displayed in interferograms with parallel fringes [(a) and (c)] and with circular fringes [(b) and (d)].

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an interferogram of a phase object obtained by using a plane reference wave. An interferogram recorded using a conical reference wave is shown in Fig. 39(b). It shows clearly that the phase object consists of five blades extending in the radial direction. Figures 39(c) and (d) shows the interferograms of another phase object which consists of five radial bars. Because the circular fringes are normal to the phase variation of the object in both cases, the deformation of the fringes from their circular geometry directly displays the phase variation o f the object. The phase variation along the radial direction can be displayed with the same result by using radial fringes. The reference wave in this case is the helical wavefront described by d X , Y) = 2.rrO/Oo,

(4.31)

where 8 =tan-' ylx, and O,, is a constant. The computer-generated hologram of this wavefront is shown in Fig. 19(a). The interferograms in Fig. 40 demonstrate how this reference wave can help to display the phase information along the radial direction. The phase objects used in Fig. 40 are a phase disk and a phase ring. Figures 40(a) and 40(c) are the interferograms of these phase objects when a plane reference wave is used. Figures 40(b) and 40(d) show the corresponding interferograms with the radial fringes.

4.3.2. Shearing interferometry in polar coordinates

Another method for observing the phase variation of a wavefront is with a shearing interferometer. In a shearing interferometer, the wavefront from the test object is split into two wavefronts which are later recombined with a small displacement (shear) between the wavefronts. The interference pattern thus produced shows the gradients of the wavefront along the direction of the shear. One advantage of the shearing interferometer over the previous type is that the two wavefronts of the test object can be: made to travel along the same path in the interferometer. This makes the adjustment and stability requirements of the interferometer less critical. An example of a common-path-shearing interferometer is shown in Fig. 41. The phase object is shown to be in contact with a grating at H,. The phase object can also be imaged onto the grating plane by a telecentric lens system. Grating, H I , serves as a beam splitter; it also

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Fig. 40. Two phase objects are displayed in interferograms with parallel fringes [(a) and (c)] and with radial fringes [(b) and (d)].

introduces the desired type of shear to the wavefront. Mask, F,, passes only the +1 order diffracted waves from H, through the optical system. The second grating, H2, is located at a small distance from the image plane of the first grating. When the two diffracted beams from the first grating are recombined by the second grating, the small displacement of H, from the image plane of HI produces a small shear between the two

+/ --=I q I

F1

H2

F2

- .

----

I

Fig. 41. Shearing interferometer using gratings as beam splitter and recombiner. A reference wavefront for displaying the shear interferogram can also be included in one or both of the gratings.

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wavefronts. The amount of shear can be adjusted by changing the position of H2. Mask, F2, passes the two diffracted beams that propagate along the optical axis to form the interference pattern at plane I. The shearing interferometer just described consists of a series of imaging optics with spatial frequency planes for selecting the different diffracted waves from gratings HI and H,. Since the lenses are mainly used for imaging, the light source for the interferometer can be broadband. But, because spatial filterings are performed in the interferometer. the light source has to be spatially coherent. The interferometer in Fig. 41 uses two regular gratings for illustrating the basic function of a shearing interferometer. The regular gratings produce a lateral displacement of the two wavefronts of the object. The interferogram obtained at plane I is then a lateral shearing interferogram. Shearing interferograms with shear along the radial or azimuthal direction have been obtained with computer-generated holograms (BRYNGDAHL and LEE [1974]). To display the gradient of a wavefront with a radial shearing interferogram, the grating, HI, has circular fringes. The constant frequency of the grating along the radial direction can produce a displacement along the radial direction in the same way that the regular grating produces lateral displacement on the wavefronts. The grating used at H, has spiral fringes similar to the one shown in Fig. 42. This grating is actually a computer-generated hologram of a helical shape wavefront using a conical shape wavefront as the reference wave. The grating equation for the hologram in Fig. 42 is r/rO+ 0/0, = n.

(4.32)

The period, r,, of this circular carrier hologram (LEE[1975]) is matched to the period of the circular grating at H,. When the two diffracted waves are recombined by the second grating, the phase variations of the two wavefronts that interfere at plane I are

and

cpz(r, 0) = q o ( r - h r , 0) - 2rrtj/0,,.

(4.33)

The interferogram formed by these two wavefronts has variation given by

I ( r , 0) = 2 + 2 cos [ d r + Ar, 0) - cpo(r - Ar, 0) + 4 ~ 00,]. /

(4.34)

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COMPUTER-GENERATED HOLOGRAMS TECHNIQUES AND APPLICATION5

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Fig. 42. Circular carrier hologram for producing constant radial shear interferograms with radial fringe system.

po(r +Ar, 0) - cpo(r-Ar, 0) is proportional to &po(r, 0 ) l d r for small Ar. The helical wavefront in the second grating provides a reference wave for displaying the gradient of the phase variation with radial fringes. Figure 43(b) is the Fraunhofer diffraction pattern of the circular carrier hologram in Fig. 42. The conical reference wave produces many diffracted orders that are symmetric about the origin. Unlike the plane reference wave hologram, the diffraction patterns of the conjugate waves in the circular carrier holograms overlap in the Fraunhofer plane and cannot be separated. However, since the two conjugate waves are used simultaneously in the shearing interferometer, they need not be separated. The *1 order diffracted waves from the circular carrier hologram can be isolated with a mask that has an annulus opening at plane F,. The interferogram in Fig. 43(a) is recorded in the shearing interferometer with the phase object removed. The radial shear interferogram of a phase disk and a phase ring are shown in Fig. 44. The phase disk has an abrupt transition along the radius. Its gradient along the radial direction is a narrow pulse which is

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20 1

Fig. 43. The interferogram in (a) shows the fringe system in the constant radial shear interferogram. The Fraunhofer diffraction pattern of the hologram in Fig. 42 is shown in (b).

Fig. 44. Two phase objects are shown in regular interferograms [(a) and (c)] and in constant radial shear interferograms [(b) and (d)].

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COMPUTER-GENERATED HOLOGRAMS TECHNIQUES A V D APPLICATIOUS

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clearly shown by the radial fringes in Fig. 44(b). On the other hand, the phase variation of the phase ring is a narrow pulse. Its gradient is a doublet which is positive at one edge of the ring and negative at the other (see Fig. 44(d)). The gradient of the wavefront along the azimuthal direction can be shown in an interferogram with shear along the same direction. The grating that can produce shearing in this direction is a circular carrier hologram made using the following grating equation r/rl + r8/(r000)= n.

(4.35)

The reference wave for the hologram is a conical wave with period r , . The object wave of the hologram has phase variation proportional to re. This wavefront with a constant slope along the azimuthal direction can displace the wavefront of the test object by a constant amount along the azimuthal direction. The hologram shown in Fig. 45 is made in ten sectors according to eq. (4.35).By limiting the angle 8 to a smaller angular range, we can increase the slope in the azimuthal direction without using a

Fig. 45. Circular carrier hologram for producing constant azimuthal shear interferograrns.

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203

Fig. 46. The interferogram in (a) shows one of the phase variations in the circular carrier hologram in Fig. 45. The constant slope of this phase variation along the azimuthal direction will produce constant azimuthal displacements on the wavefront, independent of the radius. (b) is the Fraunhofer diffraction pattern of the hologram in Fig. 45.

higher carrier frequency along the radial direction. Figure 46 shows the diffraction pattern of the hologram in Fig. 45 and the interferogram of the phase variation, re. The fringes inside each sector in Fig. 46(a) are parallel. If a circle is drawn inside the interferogram, the circumference of the circle will be divided by the fringes into small arcs with the same length. The arc length is independent of the radius of the circle drawn. Figure 47(a) illustrates the azimuthal displacement that can be obtained with this hologram. The displacement shown in Fig. 47(a) is not the same as the displacement obtained by a simple rotation on one of the wavefronts. A rotation of the wavefront will result in a larger displacement

,

I

I

/

I

(a)

(b)

Fig. 47. (a) illustrates the displacement As produced by the hologram in Fig. 45. In (b), the displacement As is obtained by rotating the wavefront. The rotation produces a displacement As which is dependent on the radius r.

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along the azimuthal direction at the outer radius and a smaller displacement at the inner radius (see Fig. 47(b)). The hologram at H2 is similar to the one used in HI except that the period of the hologram is equal to r,: r/r2+ r@/(r,,@o) = n.

(4.36)

The two diffracted waves, after being recombined by the second hologram H2, have phase variations cpl(r, @)=cpo(r, @ + A s / r ) + 2 7 r r ( l / r l - l / r , )

and

(p2(r,O ) = cpo(r, @ - A s / r ) - 2 m ( l / r l - l / r 2 ) .

(4.37)

Fig. 48. Two phase objects are shown in regular interferograms [(a) and (c)] and constant azimuthal shear interferograms [(b) and (d)].

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APPLICATIONS IN COMPUTER-GENERATED HOLOGRAMS

The resulting shearing interferogram has variation

r( 3

I(r,O)=2+2cos cpo r,O+-

-cpo

(r.O---.3+47rr

31

---

. (438)

The first argument inside the cosine function is proportional to r-'acp,(r, O ) / M The gradient information of the wavefront is displayed with circular fringes produced by the second term in the cosine function. Figure 48 shows the shearing interferograms of two phase objects. One has five blades and the other has five bars. The gradients of these two phase variations are shown by the deformations on the circular fringes. The circular carrier holograms used in the shearing interferometer are made by using a conical reference wave. Although conical wavefronts can be obtained from axicons, it is difficult, if not impossible, to make holograms of this type by conventional techniques. The use of a special reference wave to make computer-generated hologram, has been discussed by ENGEL and HERZIGER [1973] and ICHIOKAand LOHMANN [1972]. Instead of using a conical wave, they use a spherical wave as the reference wave in making computer-generated holograms. Because of the on-axis nature of the reference wavefront, this type of computer-generated hologram can reduce the alignment difficulty in interferometers used for optical testing. Additional references to the use of spherical reference wave holograms in interferometers can be found in the review paper by SCHULZ and SCHWIDER [1976].

4.4. OPTICAL DATA STORAGE AND RANDOM PHASE CODING

An optical memory has been built to store digital data in the form of one-dimensional computer-generated holograms (KOZMA[19731). This is so far the only commercial application of the computer-generated holograms. This optical data storage system is shown in block diagram form in Fig. 49. The digital data from the data source are segmented into sequences of N bits in the buffer. Each bit in the sequence is then multiplied by +1 or -1 selected from a random table stored in the computer. The cosine transform of this N-bit sequence is then calculated: C(X)=

LI,, n

cos [27r(cu + n A a ) . x+ cp,,].

(4.39)

The coefficients { a , } in eq. (4.39) are either 0 or 1. The phase {cp,}

is

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COMPUTER-GENERATED HOLOGRAMS TECHNIQUES AND APPLICATIONS

’ ’’ ’

PHASE

[III, 8 4

COSINE XFORM

Fig. 49. Block diagram of a data storage system using computer-generated holograms

either 0 or T,selected independent of the values of a,. In eq. (4.39), each data bit is encoded by a sinusoid with frequency a! + n A a , where a is the carrier frequency and Aa! is the difference frequency between the sinusoidal functions of two adjacent bits. The transmittance of the hologram is related to the transform C(x) by t ( x ) = +++qC(x).

(4.40)

q is a normalization constant. For all practical purposes, the transmit-

tance function, t ( x ) , is identical to that of a Burch hologram of a sequence of point sources. The value of t ( x ) calculated by digital electronics is converted into an analog signal by a D / A converter to control the electrooptical modulator in the laser recorder to record the holograms o n photographic film. By using an electronic system to calculate the amplitude transmittance of the holograms for the digital data, the page composer used in other types of holographic optical memory is no longer needed. However, digital calculation of the cosine transform of the digital data eventually limits the input data rate of such a data storage system. The optical data storage system described by KOZMA[1973] has an input data rate of 2.5 lo5 bits/sec. Another characteristic of this type of data storage system is that the packing density is not limited by the recording material itself but by the spot diameter of the laser beam in the laser recorder. The constant, q, in eq. (4.40) is equal to the reciprocal of the maximum value in the cosine transform. Its function is to normalize the value of C(x) so that the amplitude transmittance t ( x ) in eq. (4.40) is between 0 and 1. The intensity of the light of a “1” bit in the reconstruction is proportional to q2/16.The larger the peak value in the cosine transform, the smaller is the light efficiency of the hologram. Therefore, it is important to use the random phase, qn,in eq. (4.39) to minimize the peak value in the cosine transform C(x).

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207

The effectiveness of phase coding in reducing the dynamic range in Fourier transforms of images or digital data has been studied by AKAHORI [1973], DALLAS[1973a] and GALLAGHER and LIU [1973]. In terms of digital data storage, the phase codes can be roughly divided into four categories: (1) deterministic binary phase code, (2) random binary phase code, (3) deterministic polyphase code, (4) random polyphase code. The random binary phase code was used in the data storage system described earlier. The 0 and T phase are selected from a random number table. The probabilities of selecting either phase are equal. This phase code not only can reduce the dynamic range of the cosine transform, but it also makes the cosine transform in eq. (4.39) an even function so that only half of the transform has to be calculated in generating the onedimensional holograms. The deterministic binary code has not been used for reducing the dynamic range in computer-generated holograms. In studying the properties of Fourier coefficients, RUDIN[1959] derived an upper bound on the amplitude of the Fourier transform of an all 1 sequence using a deterministic binary phase code. His result is important because the upper bound on the Fourier coefficients can determine the normalization constant q in eq. (4.40). His derivation of the upper bound as applied to the cosine transform, ANDERSON [19611, will be discussed later in this section. Two deterministic polyphase codes (Frank’s code and Schroeder’s code) have been studied by AKAHOKI [1973] for reducing the dynamic range in Fourier transform holograms. The phase in Schroeder’s code is (4.41) (P, = T n ‘1 N. This phase code, when applied to an image, has the same effect as putting a very long focal length thin lens in contact with the image transparency. The hologram made by using this phase code with the original image is a near Fourier transform hologram, because the equivalent interferometric hologram is recorded near the Fourier transform plane of the image. On the other hand, the random polyphase code is the digital analog to the ground glass diffuser in optical holography. Random polyphase code is the most commonly used code for making computergenerated Fourier transform holograms. Recently, GALLACHER and LIU [1973] used an iterative method to further improve the polyphase code. The effectiveness of a phase code in reducing the dynamic range in

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COMPUTER-GENERATED HOLOGRAMS TECHNIQUES A N C ) APPLICATION5

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Fourier transform holograms can be determined by the maximum amplitude in the Fourier coefficients, or by their RMS value. The RMS of the cosine transform C(x) is defined as (4.42) where T is the period of the cosine transform C ( x ) .If we substitute C ( x ) in eq. (4.39) into eq. (4.42), we obtain (4.43) The constant K is the Hamming weight (total number of nonzero bits) of the data sequence. Therefore, the RMS value of C(x) is independent of the phase code and is dependent only on the Hamming weight of the data sequence. To use the maximum value of C(x) as a criterion for determining the effectiveness of a phase code we have to obtain an upper bound and a lower bound on the maximum value of IC(x)l for each of the phase codes under consideration. The lower bound on the maximum value of C(x),as will be shown, is equal to (+ and is independent of the choice of {cp,,}. A tight upper bound on the maximum value of IC(x)l for an abitrary {cp,,} is generally difficult to obtain. However, an upper bound for a particular deterministic binary phase code has been derived by RUDIN [1959] and will be discussed here. The lower bound on the maximum value of the cosine transform can be derived from the following inequality:

(1/T)

IT 0

C’(x) dx 5 Max C’(x) 0 S X < T

5[

Max ICCX)~]’.

(4.44)

osxs?’

The average value of C”(x)given on the left-hand side of cq. (4.34) is equal to c2.Therefore, the inequality in eq. (4.44) can be written as (4.45) This lower bound is a universal one, because it is independent of the choice of {cp,}. In eq. (4.40), the constant q is selected independent of the Hamming weight of the data sequence. To be certain that the value of t(x) in eq. (4.40) is within the range [O, 11, the value of q is generally selected for

111, 8 41

209

APPLICATIONS IN COMPUTER-GENERATED HOLOGRAMS

the worst case where K = N. If occasional clippings of the values of t(x) at 0 and 1 are allowed, then the constant 7 could be selected on the basis of the most probable Hamming weight, K. In the following, an upper bound on the maximum value of IC(x)l for an all 1's sequence will be derived. Specifically, it will be shown that (4.46) The integer N must be equal to 2k for some integer k in the inequality in eq. (4.46). I , is either +1 or -1. The inequality in eq. (4.46) can be proved with the help of the following two polynomials: ~ O ( X >=

Q o ( x ) = X,

p k + i ( x ) =Pk(X)fX2'Qk(.X) Qk+i(X)

(4.47)

= p k (X) - X2"Qk (.XI.

Examples of the polynomials pk(x)and in Table 2 .

Qk(x)

for k

= 0,

1 , 2 , 3 are shown

TABLE2 Examples of the polynomials P k ( x ) and Q k ( x ) for k = 0 , 1 , 2 , 3

k

-

o 1 2 3

Qk(x)

Pkb)

x x+x2 x+xz+x3-x4 x + X2+X'-X4+XS+x6-n'+x8

X

x-x2 x +xz-x3 x +x2+x7

+X4 --

x4-x5-

Xh+

x7-xx

Table 2 shows the degree, N, of the polynomial Pk(x) to be 2k(N=2 k ) , and the coefficient of p k ( x ) is either -t-1 or --I. Therefore, Pk(eJne)is identical to the left-hand side of the inequality in eq. (4.46). From eq. (4.47), it can be shown that for 1x1 = 1,

1p k

4

1

(x l2 + I Qk

/* I

+1 (x) = p k (x

+ x2' Qk (x)I

= 2{Ipk(x)12+

-t1 p k (x)- x 2 k o k (x) I

10k(x>l'}-

Since for

IPo(x>I2+IQd~)I2=2

1x1 = 1,

(4.48)

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we can show by deduction that

for 1x1 = 1. Hence,

IPk(X)lS(2N)f.

(4.50)

The inequality in eq. (4.46) is obtained by substituting eJnefor x in eq. (4.50). From this inequality we can further show that I

I n

5 (2N);.

(4.51)

The left-hand side of eq. (4.51) is just the cosine transform of an all 1 sequence. Therefore, the inequality in eq. (4.51) gives us the upper bound o n the amplitude of the cosine transform. With this upper bound, it is clear that the value of t ( x ) will not exceed the range [0,1] if the value of can be set equal to 1/(2N)f. In the optical memory application, the number of bits in the data sequence is always equal to 2k where the bound in eq. (4.51) is applicable. For other values of N,Rudin showed that the upper bound is equal to (5N)f. To compare Rudin's phase sequence with other phase coding schemes, graphs of the cosine transform of the four phase coding methods are calculated and shown plotted in Fig. 50. The number N in the cosine transform is equal to 8. The phase codes used in obtaining the curves are (in order): (a) deterministic binary code: (b) random binary code: (c) Schroeder code:

cpn/.rr = 0, 0, 0, 1 , 0 , 0 , 1 , 0

(d) random polyphase code:

cp,/.rr = .96, .72, .26, .32, 1.14,1.84,1.12,

cp"l.rr

= 0, 1 , 1 , 1 , 0 , 1, 1 , o

cpn/.rr = 0, .125, .5,1.125,2,3.125,4.5,

6.125 0.6.

As can be seen in all the cases except for the random polyphase code, the maximum values of the cosine transforms are bounded by 4 which is in agreement with the bound derived by Rudin. The binary phase code (deterministic or random) produces a cosine transform that has even symmetry about the origin. This permits a 50% reduction in computation time for the cosine transform. Polyphase codes do not have this advantage.

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APPLICATIONS IN COMPUTER-GENERATED HOLOGRAMS

21 1

4

2 0

-2 -4

t

Fig. 50. The cosine transforms of an all 1 sequence with four different phase codings: (a) deterministic binary phase code, (b) random binary phase code, (c) deterministic polyphase code, (d) random polyphase code.

Although the selection of the phase code has centered on the cosine transform for making one-dimensional holograms, the fluctuation in the cosine transform is related to the amplitude of the Fourier transform of the same digital data. Therefore, the results derived in this section can also be applied to other types of Fourier transform holograms.

4.5. LASER BEAM SCANNING

Laser beam scanning devices such as the galvanometer mirror scanner, acoustooptical beam deflectors or polygonal mirror scanners have now been widely used in electronic systems for facsimile reproduction, document reading, display, pattern generation for IC circuitry or computer

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[ 111,

4

output printers. Besides these well developed laser beam scanning devices, a new type of mechanical scanner using a holographically recorded grating to deflect laser beams has been investigated (see BRYNGDAHL and LEE [19761 and their references therein). Holographic grating scanners have properties similar to both the polygonal mirror scanner and the acoustooptical beam deflectors. Like the polygonal mirror scanner, the grating scanner requires mechanical motions to change the deflection of the laser beam. However, it deflects the laser beam by means of diffraction rather than reflection. The holographic grating scanner is capable of scanning over large angles and with high resolution. In comparison with other types of mechanical scanners, holographic scanners have these advantages: (1) The holographic grating scanner can be used in transmission. This has the advantage that the grating scanner can, therefore, be used in a prescan mode. This permits the focusing lens following the scanner to produce a flat scan line at the back focal plane of the lens. (2) The holographic grating scanner can be made in such a way that the wobbling of the scanner has a very small effect on the scan line. (3) The scan angle in a holographic grating scanner is independent of the number of hologram facets on the scanner. This permits construction of a multifaceted scanner that can scan over a larger cone angle than an equivalent polygonal mirror scanner of the same size. (4) Holographic gratings can be recorded on the circumference of a disc to make a high speed rotating disc scanner. (5) The holographic grating scanner can be used without any focusing lens in the optical system. In this section two methods for making holographic grating scanners will be discussed. In both of these methods the computer-generated hologram plays an important part in construction of the laser scanner. 4.5.1. Computer-generated holograms for laser beam scanning

There are two ways to use a holographic grating to scan a laser beam. One way is to scan the laser beam by rotating a constant frequency grating (see Fig. 51). As the grating rotates, the focused spot at the back focal plane of the lens generates a circular scan line. The radius of the circular scan is proportional to the spatial frequency of the grating and the focal length of the lens. An arc from this circular scan is used as the

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213

A

Fig. 51. A laser beam is scanned by rotating a constant frequency grating, G

scan line. To obtain a straight scan line, the document or recording surface must conform to the curvature of the circular scan. The circular scan can be corrected by additional optical components (BRAMLEY [1973], WYANTr197.51). Because the spatial frequency of the grating is constant across the grating, partially illuminating the grating will have no effect on the location of the scan spot. Therefore, the scan spot is invariant to the location in the grating illuminated by the laser beam. A different method of scanning the laser beam is to use a grating which has linear spatial frequency variation across the grating. When this grating is partially illuminated by a laser beam, the frequency variation in the different parts of the grating can change the direction of the diffracted laser beam. This is the major distinction between the constant-frequency grating scanner and the variable-frequency grating scanner. An example of a variable-frequencv scanner is shown in Fig. 52. The grating is X

Fig. 52. A drum scanner with holographic grating wrapped around its circumference.

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mounted on the circumference of a drum. The rotation of the drum moves the different parts of the grating across the laser beam. In practice, there is usually more than one grating on the circumference of the drum. The scan line from this drum scanner does not have the curvature problem of the constant-frequency grating scanner. The variable-frequency grating for scanning the laser beam can be made as a computer-generated interferogram with the grating equation: 27$y

+ q ( x , y) = 2 m .

(4.52)

The carrier frequency in the computer-generated interferogram is along the y-direction. The phase q ( x , y) is chosen so that the laser beam will be scanned along the x-direction. The spatial frequency of the grating made according to eq. (4.52) will be %(X,

Y)

= (1/2n)Mx, y)ldx.

(4.53)

In raster scanning, the spatial frequency of the grating must be a linear function in x. Hence, the phase function, q ( x , y), in eq. (4.53) is the solution to the following differential equation: a q ( ~ ,ypax = ~ ~ X I W A X .

(4.54)

The parameters w and Ax represent the hologram width and the incremental displacement of the grating needed to deflect the laser beam to the next resolvable position along the scan line. A function satisfying eq. (4.54) is d x , Y ) = (.rrx2/wAx)+d y ) .

(4.55)

g(y) is the constant of integration when eq. (4.54) is integrated to obtain the solution in eq. (4.55). Two different forms of g(y) can be used in making the grating for scanning the laser beam: (4.56)

The computer-generated holograms of q ( x , y) with two different forms of g(y) is shown in Fig. 53. When gl(y) is used, the spacing between the fringes along the y-direction is constant as shown in Fig. 53(b). This hologram is an off-axis cylindrical zone plate. When g,(y) is used to make the hologram in Fig. 53(a), the resulting hologram is just an off-axis Fresnel zone plate. In either hologram, the spatial frequency along x is a linear function of x.

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215

Fig. 53. The holograms in (a) and (b) can produce a linear scan line by moving across a laser beam. (a) is obtained from an off-axis section of a Fresnel zone plate. (b) is a cylindrical zone plate with carrier frequency normal to the phase variation.

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Suppose that the hologram in Fig. 53(a) is illuminated by a collimated beam at x = x f . The phase variation of one of the diffracted beams is equal to

cp(x - X I ,

y) = - T { ( x -x’)’+ y2}/(wAx)-27Tpy T(X2 --

+ y’)

w Ax

- 2mpy

2TXXI TXf2 +-w Ax w Ax’

(4.57)

The first term in the phase function produces a spherical wavefront that converges to a point at a distance wAx/A from the hologram. Since this quadratic phase function is independent of X I , it can be compensated for by using a diverging beam originated from a point source at a distance wAx/A from the hologram. The second term in eq. (4.57) causes a tilt of the diffracted beam along the y-direction. The third term which is linear is x, and x’ is responsible for the continuous deflection of the laser beam when the grating is moved across it. The deflection angle as a function of x’ is

Since the diameter of the laser beam can be equal to w, the angular resolution of the diffracted beam is A 6 = A / w . From eq. (4.58) if the grating is moved from x f to x f + Ax, the angle of the diffracted beam will be changed by he. This affirms that the parameter A x in the phase function q ( x , y) is the incremental distance that the grating must travel to address the next resolution position. For a grating with length L, the number of resolvable spots along a scan line is given by N

=L/Ax.

(4.59)

Some experimental results demonstrating the scanning capability of the linear frequency grating scanner are shown in Figs. 54-56. The computergenerated hologram used in the experiment is similar to the hologram in Fig. 53(b). The hologram was plotted on a Calcomp plotter and was 120cm in length and 20cm in width. The parameter Ax is equal to 3/8 cm. There are a total of 640 fringes in the hologram. From eq. (4.59) the number of resolution elements that can be addressed by this grating is 320. Because this grating is made using gl(y) in eq. (4.59), the grating has focusing power only in the x-direction. When this grating is used in the drum scanner in Fig. 52, the laser beam along the y-direction will focus at

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APPLICATIONS IN COMPUTER-GEKERATED HOLOGRAMS

217

the back focal plane of the lens. But, the laser beam along x will focus at * A ~ / W A Xfrom the back focal plane of the lens shown. F is the focal length of the lens in Fig. 52. Without any cylindrical component added to the scanner, a line rather than a well-focused spot will be scanned in the frequency plane. Because the grating has a constant frequency along y-direction, any movement of the grating along y will have no effect on the scan line. This is the advantage of this kind of astigmatic grating. Figure 54 illustrates the focusing properties of the grating in Fig. 53(b). When the area marked a in Fig. 53(b) is illuminated, the diffraction pattern at the back focal plane of the lens is as shown in Fig. 54(a). The focused spot in the center in Fig. 54(a) is from the undiffracted beam of the grating. The lines on either side of the center spot are from the diffracted waves of the hologram. These waves are focused along the y-direction at the back focal plane of the lens. The length of the line along the x-direction is caused by the quadratic phase variation in the hologram. One of the diffracted waves from the hologram will focus at a

Fig. 54. Recording made with the rotating drum scanner using the hologram in Fig. S3(b). (a) is the Fraunhofer diffraction pattern of the hologram when it is illuminated in the area marked a in Fig. S3(b). (b) is taken at a short distance from the back focal plane of the lens. It shows the astigmatic focus of the diffracted wave from the hologram. The center disk is due to the undiffracted laser beam. (c) shows the correction of the astigmatism in the hologram with a cylindrical lens inserted just behind the hologram.

21x

COMPUTER-GENERATED

HOLOGRAMS TECHNIQUES A N D APPLICA now

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small distance from the back focal plane as shown in Fig. 54(b). However, at that plane the diffracted wave along the y-direction gets out of focus. Figure 54(c) shows correction of the astigmatism by a cylindrical lens. The dependence of the scan spot on the different parts of the grating is demonstrated in Fig. 55. Figures 55(a) through 55(c) show the positions of the scan spot at the back focal plane when the holograms are illuminated at areas marked “a”, “b” and “c”. When the drum rotates, the scan line produced by the grating is shown in Fig. 55(d). To test the resolution of the scanner, the laser beam is passed through a modulator which turns the intensity of the laser beam on and off at selected rates. The modulated scan line is then recorded on film. Figure 56 shows the resolution of the scanner at 100 spots/scan, 200 spots/scan and 300 spots/scan. The capacity of the scanner is as predicted by eq. (4.59).

Fig. 5 5 . (a) to (c) show the positions of the scan spots when the areas marked a. b, and c o f the hologram in Fig. 53(b) are illuminated; (d) shows the complete scan line when the drum rotates.

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Fig. 56. Shows resolution of a feasibility scanner. Recording made with the rotating drum scanner using the hologram in Fig. 53(b): (a) 100 spots/scan; (b) 200 spotslscan; (c) 300 spots/scan,

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4.5.2. Interferometric grating scanner with aberration correction The number of resolution elements that can be scanned by a computergenerated hologram scanner is limited by the recording system used to make the hologram. For instance, the Calcomp plotter in the experiment described in the last section was a 28 mm (11in.) plotter with positional accuracy of 0.25 mm (0.01 in.). The period of the highest frequency in the experimental grating is about 1.25 mm which is only five pen positions of the plotter. It is difficult to increase the spatial frequency of the computer-generated hologram further without causing other type errors. The limitation of the Calcomp plotter can be circumvented by going to a laser recorder which uses a focused laser beam to write the computergenerated hologram directly on film. Even then, to make a computergenerated hologram scan over a cone angle exceeding +30", the laser recorder will require a precision similar to that of a ruling engine for making diffraction gratings. In this section, an alternative method for making holograms for scanning laser beams will be discussed. This method also illustrates a potentially important application of computergenerated holograms in correcting aberration in holographic optical elements. We have shown that the phase variation needed to produce a linear (4.60)

where f = wAlA. As written in eq. (4.60), f is the focal length of the spherical wavefront. The phase q ( x , y) in eq. (4.60) is equal to 2 m at radius r,, = ( 2 ~ n f ) d . (4.61) The radii {r"} are equal to the radii of the zones in a Fresnel zone plate. Therefore, the problem of making a computer-generated hologram to scan over a large cone angle is equivalent to that of making an off-axis zone plate with a large number of zones. Besides using plotters or laser recorders to make zone plates, Fresnel zone plates can be made by recording the interference pattern between a divergent wavefront from a point source and a collimated reference wave (HORMAN and CHAU[1967], CHAMPAGNE [1968], CHAU[1969]). The divergent wave recorded in the hologram has a phase variation of (4.62)

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The constant phase term, 27rflA, is included in eq. (4.62) so that (p2(x, y) = 0 at x = y = 0. The radius of the constant phase of (p2(x,y ) is equal to

(4.63)

rl,= (n2h2+2nAf):.

The radii {rln) of the interferometric zone plate are different from those of the Fresnel zone plate. Only when f > >nh/4 and the difference between r,, and r; is small can the point source hologram approximate a Fresnel zone plate. When the interferometric zone plate is not used in an optical system similar to its construction geometry, the diffracted wave from an off-axis part of the grating will have aberrations. This is the case when the interferometric zone plate is used in the drum scanner in Fig. 52. The presence of off-axis aberration in the interferometric zone plate can be shown by expanding the phase function (p2(x, y) about an off-axis point (xo, Yo): V2(X - xo7 Y - Yo) = (27r/h){(1/2f)[(x-x,)’+(~

- Y ~ ) ’ I - ( ~ / ~ ~ ~ ) I I ( X - X ~ ) ~ + ( Y.-.I~ ~ ) ~ I ’ + .

= (p2(x, Y)+2.rrl~){-(1/f)(xox

+YYd

+ ( ~ / S ~ ’ > [ - ~ ( X ~ + Y ~Y) (YX~X) + ~~ + ( . K X ” + ~ ~., .J}.~ ] +(4.64) .

In eq. (4.64) the first term is again the spherical wavefront. The second term which is linear in x and y will change the direction of the diffracted beam. The remaining terms in the equation are the third-order aberration of the interferometric zone plate. By letting x = p cos 4 and y = p sin d ~ , the third-order phase error in the interferometric zone plate can be written as @’ = (27r/h){- (p3/2f3)(xOcos 4 + yo sin 4) + (p’/2f3)(xi cos2 4 + y i sin’ 4 + 2x,y,

cos 4 sin 4)).

(4.65)

The two terms in @3 are the coma and the astigmatism. These aberrations occur because the playback geometry of the hologram is not the same as its constructional geometry. Although the interferometric zone plate is recorded by using a large cone from a divergent wavefront and thus has a small f-number, the actual f-number of the zone plate used in scanning the laser beam is typically about 10. A t this f-number, the coma has less than one wavelength variation within the illuminated part of the zone plate. Hence, they have no noticable effect on the scan spot. The

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most serious aberration in the diffracted wave from the interferometric zone plate is caused by the astigmatism which makes the wavefront in the x-direction to focus at a distance

(4.66) from the zone plate. Whereas, with the laser beam centered about yo = 0, the wavefront along y will always focus at f, =f. As a result, there are two focal planes for the diffracted wave emerging from an off-axis part of the interferometric zone plate. The separation between the two focal planes is a function of the scan angle 8 because sin 8 = x,/f. The following experiment will demonstrate how this off -axis astigmatism affects the focused spot of the scanning laser beam. An interferometric zone plate is recorded in the setup shown in Fig. 57.

Fig. 57. Side view of an interferometer for making an IZP. The mirror, M, and the beam splitter, B, are used to combine the collimated reference beam with the divergent wavefront from the point source 0 for recording the IZP. The IZP recorded in this setup is narrow and has a width, W, along the y-direction. (b) Top view of the interferometer. The length of the IZP along x is equal to L.

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The optical system is basically an interferometer. The mirror, M, and the beam splitter, B, combine a collimated beam with the divergent wave from a point source, 0.The distance between the film plane and the point source is f. The mirror and beam splitter arrangement in the interferometer provides a reference beam angle, $, that can be as small as a few degrees. The interferometric zone plate recorded has width w. The top view of the interferometer is shown in Fig. 57(b). It shows that the cone angle from the point source is 8 and the length of the zone plate is L. The cone angle that can be scanned by this grating is equal to the cone angle 8 of the divergent beam. For the experiment, a zone plate was recorded using.f=37.5 mm, L = 18 mm and w = 3 mm. The cone angle 8 of the divergent beam from the point source was about 24 degrees. The reference beam angle $ was about 5 degrees. The interferometric zone plate was recorded on Kodak 649F plate. In reconstruction, the interferometric zone plate was illuminated by a divergent beam whose diameter on the zone plate was about 3mm. The distance of the reconstruction point source was such that one of the diffracted waves from the zone plate focused at a distance of 40cm from the zone plate. Photographs of the magnified spots at different scan angles are shown in Fig. %(a). It can be seen that the spot becomes a line at large scan angles because of the off-axis astigmatism. The angular range in which the scan spots are diffraction limited is small. For comparison the diameter of the on-axis spot in Fig. %(a) is about 100 pm. The off-axis aberration in the interferometric zone plate can be corrected by a corrector plate. Since the interferometric zone plate is recorded in an interferometer, the correction o f the aberration can be incorporated in the zone plate during the recording process. The phase variation needed to correct the aberration can be obtained from a computer-generated hologram. In the following we will discuss some of the considerations in using computer-generated holograms to correct aberrations in the interferometric zone plate. As pointed out in eq (4.60), the ideal phase variation for scanning a laser beam is cpl(x, y). This phase function has the property that the focusing power is independent of the part of the grating illuminated. Therefore, the phase variation needed to correct the aberration in an interferometric zone plate is given by the difference between (p2(x, y ) and PO,(X, Y):

Fig. 58. The magnified spots in (a) illustrate the off-axis astigmatism in the drum scanner. The focal power along the x-direction is strongly dependent on the scan angle, especially at large angles. As a reference, the diameter of the on-axis spot in (a) and (b) is about 100 pm.The scan angle hetween the two adjacent scan spots shown is about 0.15". (b) shows the spots from a corrected IZP. Most of the astigmatism in (a) has been removed by the correction recorded in the IZP.

e

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Equation (4.67) is derived by expanding cp2(x, y ) in a Taylor series about the origin. The terms 6(x, y) are the third- and higher-order terms in the expansion of (p2(x,y). In comparison to the first term in eq. (4.67), 6(x, y ) is generally small and can be neglected in making the corrector plate. The maximum phase deviation in A(p,(x, y) occurs at the boundary of the function. Its value depends on the focal length, f, and the f-number o f the zone plate. In the experiment just described, the maximum phase deviation is about 25h. The phase variation, Acp,(x, y), is only one of the many phase variations that can be used to correct the aberrations in the zone plate scanner. Another one that can correct the aberrations and at the same time minimize the phase deviation in the corrector plate is

A’(P~(x, y ) = - (d16Af)(L/f)’(x2 + y’)

+ (2r/8Af3)(x2+ Y’)~. (4.68)

The first term in A(p2(x,y) will change the focal length of the corrected interferometric zone plate by a small amount. ‘The addition of that phase term makes the phase function A(p2(x,y) vanish at x = y = 0 and x = L/2 and y = 0. The maximum value of hq2(x, y ) occurs at x = L/& and is 1/4 of that in Acpl(x, y). The aberration in the interferometric zone plate is corrected by the second term in A(p2(x,y). A computer-generated hologram of Acp2(x, y) is shown in Fig. 59(a). The carrier frequency of the hologram is along the y-direction, because most of the phase variation of A(pz(x,y) is along the x-direction. The interferogram of the wavefront reconstructed from this hologram is shown in Fig. 59(b). The interferometer in Fig. 60 is modified from the

(b) Fig. 59. Computer-generated hologram used as corrector plate. For clearness, the hologram contains fewer fringes than the one used in the experiment. The interferograrn (b) is obtained by combining an inline collimated reference beam with the wavefront reconstructed from the hologram in (a). The fringes show the phase variation added to the IZP.

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Fig. 60. Side view of the modified interferometer for making 1ZP's with aberration correction. The phase variation from the hologram corrector plate is imaged onto the film plane by the telecentric system. Because the computer-generated hologram is a diffracted optical element, the mask in the back focal plane of the first lens is used to select one of the diffracted waves from the hologram.

interferometer in Fig. 57 to include the computer-generated corrector plate in the making of the interferometric zone plate. The diffracted wave from the computer-generated hologram replaces the collimated beam used in the previous interferometer. As shown in Fig. 60, the computergenerated hologram is illuminated by a collimated beam and is imaged one-to-one by the telecentric lens system to the film plane. A mask in the back focal plane of lens L1 selects only one of the diffracted waves from the hologram to form the interference pattern on the film plane. The intensity variation of the interference pattern on the film plane is proportional to

The first term in eq. (4.69) is from the computer-generated hologram; the second term comes from the point source. The reference angle is 4.When Aq2(x, y) in eq. (4.68) is substituted into eq. (4.69), the function I(x, y ) becomes

(4.70) This interference pattern will produce an interferometric zone plate identical to a Fresnel zone plate.

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The photographs in Fig. 58(b) show the magnified scan spots from an aberration corrected, interferometric zone plate having the same parameters as the uncorrected zone plate discussed previously. A significant improvement in the focused spots at large scan angles can be seen in Fig. 58(b). This method of using a computer-generated hologram as a corrector plate is not only useful in making gratings for scanning laser beams, but is also useful for correcting the aberrations in other types of holographic optical elements. Up to now, the aberrations inherent in the holographic elements could only be minimized by the recording geometry or by using multiple elements. Using computer-generated holograms as corrector plates in recording holographical optical elements can help to eliminate certain aberrations in the holographic optical elements.

0 5. SummaryandComments We have discussed in some detail the many methods for making computer-generated holograms. Since each of the methods discussed is unique, it is difficult to specify the best method for making computergenerated holograms. When a laser recorder is available, Burch’s method of making off-axis reference beam holograms is the one to use. His method takes advantage of the uniform sampling in a raster scanning system and permits maximum use of the spatial bandwidth of the recording system. On the other hand, if a binary hologram is desired, the computer-generated interferogram or the improved detour phase method should be used. Kinoforms with their various extensions offer better light efficiency than any other methods. Kinoforms do not require a carrier frequency to record the complex wavefront; however, it is better to add a linear phase term to the kinoform so that if the kinoform has phase mismatch, the linear phase term can separate the different diffracted waves from the kinoform. In recent years there has been considerable interest in the effect of quantization on computer-generated holograms. In 8 3, we looked at two types of quantization. Quantization in assigning the sampled value of a function into one of a finite number of levels, introduces irrecoverable errors to the function. These errors are often random in nature. On the other hand, the term quantization has also been used to mean the conversion of a continuous function into a discrete-valued function. In

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this case, it has been shown that under certain conditions the orginal function can be recovered intact from the discrete-valued functions. Five different applications of computer-generated holograms have been discussed. These are by no means the only applications of computergenerated holograms. For example, we have not discussed the use of computer-generated holograms in long optical wavelengths for laser machining (ENGELand HERZIGER [19731, SWEENEY, STEVENSON, CAMPBELL and SHAFFER[1976]), because more work is expected from this area. In all the applications discussed, except for the 3-D display and data storage applications, computer-generated holograms can be considered as optical elements. The only difference between computer-generated holograms and other optical components such as lenses or mirrors etc., is that computer-generated holograms are diffractive optical elements. Their usefulness is limited to narrow band and spatially coherent light sources. The computer-generated holograms are most useful in supplementing other types of optical elements rather than replacing them. This is especially evident in the laser-beam scanning application. The computergenerated hologram can be made to scan a laser beam. But, the combination of an interferometric hologram and computer-generated hologram is a more practical approach to making laser scanners for scanning over large cone angles. We want to emphasize again the importance of copying computer-generated holograms in an interferometer. By doing so, we make an optical hologram which has the same wavefront as the original computer-generated hologram, possibly with additional phase variation. Development of the computer-generated hologram is strongly motivated by the need to synthesize spatial filters for coherent optical data processing. However, the filtering done by the coherent optical systems is mainly limited to linear filtering. Recently, it has been shown that with computer-generated holograms the optical system can be converted into a space-variant system (BRYNGDAHL [19741, CASASENI and KRAUS[ 19761, CASASENTand SZCZUTKOWSKI [1976]). However, more work must be done to improve the quality of this kind of nonlinear optical transformation. Image processing for many applications will invariably be dominated by digital methods. The computer-generated holograms have their greatest potential in the area of interferometry. They have been shown to be useful in supplementing existing methods of optical testing. Laser beam scanning is another promising area for computer-generated holograms. Holographic grating scanners are in many ways better than other mechanical mirror

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scanners. Complicated scan patterns, in addition to the raster scan, can be obtained by using holographic gratings. With other types of scanners these patterns will require the use of two different beam deflectors. Laser scanners have other optical components in them, some of which can be combined into the holographic grating to produce a more compact scanning system. In this chapter we have not discussed recording materials or the diffraction efficiency of computer-generated holograms because the considerations in selecting recording materials for an optically recorded hologram could be similarly applied to computer-generated holograms. A good review on the recording materials for holography can be found in the book by SMITH [1976].

References* AKAHORI, H.. 1973, Appl. Opt. 12, 2336. ALLEBACH, J. P. and B. LIU, 1976, Appl. Opt. 14, 3062. ANDERSON, G. B. and T. S. HUANG,1969, Spring Joint Computer Conf., AFIPS Conf. Proc., Vol. 34, pp. 173-185. ANDERSON, D. R., 1961, Proc. IRE 49, 357. H.C., A. G. TESCHERand R. P. KRUGER,1972, IEEE Spectrum, Vol. 9, no. 7. p. ANDREW, 20. BECKER,H. and W. J. DALLAS,1975, Opt Comm. 15, 50. BESTE,D. C. and E. N. LEITH,1966, IEEE Trans. Aerosp. Electron. Syst. AES-2, 376. BIRCH,K. G. and F. J. GREEN,1972, J. Phys. D: Apply. I’hys. 5, 1982. BRAMLEY,A,, 1973, U. S. Patent 3,721, 486. BRAUNECKER, B. and A. W. LOHMANN, 1974, Opt. Comm. 11, 141. BROWN,B. R. and A. W. LOHMANN, 1966, Appl. Opt. 5, 967. BROWN, B. R., A. W. LOHMANN and D. P. PARIS,1966, Opt. Acta 13, 377. BROWN,B. R. and A. W. LOHMANN, 1969, IBM J. Res. Develop. 13, 160. 0.and A. W. LOHMANN, 1968, J. Opt. SOC.Am. 58, 141. BRYNGDAHL, O., 1973, J. Opt. Soc. Am. 63, 1098. BRYNGDAHL, O.,1974a, Opt. Comm. 10, 164. BRYNGDAHL, O.,1974b, J. Opt. Soc. Am. 64, 1092. BRYNGDAHL, 0. and W-H. LEE, 1974, J. Opt. SOC.Am. 64, 1606. BRYNGDAHL, 0.and W-H. LEE, 1975, J. Opt. SOC.Am. 65, 1124(a). BRYNGDAHI, 0. 1975, Opt. Comm. 15, 237. BRYNGDAHL, BRYNGDAHL, O.,1975, Opt. Eng. 14, 426. 0.and W-H. LEE, 1976, Appl. Opt. 15, 183. BRYNGDAHL, BURCH,J. J., 1967, Proc. IEEE 55, 599. BURCKHARDT, C. B., 1970, Appl. Opt. 9, 1949. and C. R. MANSFIELD, 1974a Opt. Eng. 13, 175. CMBEIL, K., G. W. WECKSUNG and C. R. MANSFIEID,1974b, Proc. S.P.I.E. 48, 69. CAMPBELL, K., G. W. WECKSLJNG

* Some of the references are not quoted in the main text. They are included here to provide a reasonably complete bibliography on the subject.

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HELSTROM,C. W., 1967, J. Opt. SOC.Am. 57, 297. HORMAN, M. H. and H. M. CHAU,1967, Appl. Opt. 6, 317. HUANG,T.S. and B. PRASADA,1966, MIT/RLE Quar. Prog. Rep. 81, 199. HUANG,T.S., 1971, Proc. IEEE 59, 1335. HUGONIN J. P. and P. CHAVEL,1975, Opt. Comm. 16, 342. Y.and A. W. LOHMANN, 1972, Appl. Opt. 11, 2597. ICHIOKA, ICHIOKA, Y., M. IZUMIand T. SUZUKI,1969, Appl. Opt. 8, 2461. ICHIOKA,Y.,M. IZUMIand T. SUZUKI,1971, Appl. Opt. 10, 403. KATYL,R. H., 1972, Appl. Opt. 11, 198. KEETON,S. C., 1968, Proc. IEEE 56, 325. KETTERER, G., 1972, Optik 36, 565. KING.M. C., A. M. NOLLand D. H. BERRY,1970, Appl. Opt. 9, 471. KIRMISCH, D., 1970, J. Opt. SOC.Am. 60, 15. KIRK,J. P. and A. L. JONES,1971, J. Opt. SOC.Am. 61, 1023. KOZMA,A. and D. L. KELLY,1965, Appl. Opt. 4, 387. KOZMA,A,, W-H. LEEand P. J. PETERS, 1971, IEEE/OSA Conf. on Laser Engineering and Aplications. KOZMA,A,, E. N. LEITH and N. G. ~ ~ A S S E1972, Y , Appl. Opt. 11, 1766. KOZMA,A., 1973, in: Topical Meeting on Optical Storage of Digital Data, Digest of Technical Papers (Optical Society of America, March 1973). LEE, W-H., 1970a, Appl. Opt. 9, 639. LEE, W-H., 1970b, Pattern Recog., 2, 127. LEE, W-H. and M. 0. GREER,1974, Appl. Opt. 13, 925. LEE, W-H., 1974, Appl. Opt. 13, 1677. 1974, Opt. Comm. 12, 382. LEE, W-H. and 0. BRYNGDAHL, LEE, W-H., 1974, Proc. S.P.I.E. 48,77. LEE, W-H., 1975a, J. Opt. Soc. Am. 65, 518. LEE, W-H., 1975b, Appl. Opt. 14, 2217. LEE, W-H., 1975c, Appl. Opt. 14, 2447. LEE, W-H., 1977, Appl. Opt. 16, 1392. LEITH,E. N. and J. UPATNIEKS, 1962, J. Opt. Soc. Am. S2, 1123. 1968, Appl. Opt. 7, 539. LEITH,E. N. and A. L. INGALLS, LEITH,E. N., 1968, IEEE Trans. Aerosp. Electron. Syst. AES-4, 879. LEITH,E. N., 1971, Proc. IEEE 59, 1305. LEITH,E. N., 1976, in: Advances in Holography, Vol. 2, ed. N. H. Farhat (Marcel Dekker Inc.). LESEM,L. B., P. M. HIR~CH and J. A. JORDAN Jr., 1967a, Roc. Symp. Modern Optics (New York, Polytechnic Institute of Brooklyn). LESEM,L. B., P. M. HIRSCHand J. A. JORDAN Jr., 1967b, Proc. 1967 Fall Joint Comp. Conf., Vol. 31 (Washington D.C., Thompson Books) pp. 41-47. LESEM,L. B., P. M. HIRSCHand J. A. JORDAN Jr., 1968, Commun. ACM 11, 661. and J. A. JORDAN Jr., 1969, IBM J. Res Develop. 13, 150. LESEM,L. B., P. M. HIRSCH LESEM, L. B., P. M. HIRSCHand J. A. JORDAN Jr., 1970, Opt. Spectra 4, 18. LINFOOT,E. H., 1958, Recent Advances in Optics (Oxford) p. 176. LIU, B. and N. C. GALLEGHERJr., 1974, Appl. Opt. 13, 2470. A.W., D. P. PARISand H. W. WERLICH,1967, Appl. Opt. 6, 1139. LOHMANN, LQHMANN, A. W. and D. P. PARIS, 1967, Appl. Opt. 6, 1567, 1739. LOHMANN, A. W. and D. P. PARIS,1968, Appl. Opt. 7, 651. LOHMANN, A. W., 1973, NEREM Record, Part 2, p. 148. S. and P. CHAVEL, 1974, Appl. Opt. 13, 718. LQWENTHAL,

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COMPUTER-GENERATF.D HOLOGRAMS: TECHNIQUES A N D AI'PI lCAl ION\

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