Applied Optics Letters to the Editor

digital generation and

C. A. Irby, M. O. Hagler, and T. F. Krile When this work was done all authors were with Texas Tech University, Department of Electrical Engineering, Lubbock, Texas 79409; C. A. Irby is now with BDM Cor­poration, Albuquerque, New Mexico 87101. Received 13 September 1981. 0003-6935/82/020169-03$01.00/0. © 1982 Optical Society of America.

A new technique for multiplexed hologram storage and retrieval is proposed in this Letter in which a multiplexed hologram and the appropriate functions for retrieving the different multiplexed functions stored on the hologram are generated by computer. Previous experiments have proved the feasibility of using multiplex holograms for coherent space-variant optical system representation.1,2 With this method a complicated space-variant system can be replaced with a simpler system that uses multiplexed holograms which are referenced independently but simultaneously to simulate the output of the system to be represented.

In the work referenced here each hologram was generated experimentally and multiplexed using ground glass diffusers onto the film by means of multiple sequential exposures. The necessity of experimentally generating each function to be stored holographically severely restricts the class of functions that can be multiplexed with this approach. Because each multiplexed function in these applications represents a sample of the point spread function of a space-variant system, this restriction limits the class of space-variant systems that can be represented by multiplexed holograms.

Computer generation of the holograms would clearly broaden the class of functions that can be multiplexed holo­graphically. The major difficulty is in devising a set of ref­erence beam diffusers that have (1) analytical descriptions simple enough for computer calculation; (2) physical reali­zations simple enough to be fabricated experimentally; and (3) above all, suitable statistical properties. This Letter de­scribes how separable 2-D pseudorandom binary codes such as Gold codes can be used to construct a set of diffusers that allows the simultaneous computer generation of a multiplexed hologram and a set of masks that can be used to make a set of diffusers for individually retrieving the multiplexed holo­grams. Other potential advantages of this new technique over the previously reported methods3 of storing and retrieving multiplexed holograms include: less noise in the output plane, less bias buildup on the recording film, a greater dy­namic range in the multiplexed hologram, and an optical re­trieval system whose space-bandwidth product is indepen­dent of the number of functions stored. The disadvantages of this approach to multiplex holography include the quan­tization noise inherent in the digital generation of the holo­gram, limited resolution, and the matched-filter range of ac­curacy required in the alignment of the optical retrieval system.4

The steps in this technique—called computer-generated multiplex holography—are as follows5: First, the computer generates and plots dn a pseudorandom reference function for each function tn to be multiplexed. Using the method proposed by Burckhardt6 to represent both magnitude and phase information the computer next generates and plots the function

in which D*n is the complex conjugate of the Fourier transform Dn of the diffuser reference function dn; Tn is the Fourier transform of the desired output tn; and M is the number of functions to be multiplexed. In this way each function to be retrieved is associated with a different diffuser.

The holographic plot and the reference plots are photore-duced, and the resulting masks are implemented in the optical system of Fig. 1. Plane P1 contains one (or possibly more) of the reference function masks and is located a focal length in front of lens L1. Plane P2 which contains the multiplexed hologram mask is located in the back focal plane of L1 which is also the front focal plane of L2. The output plane P3 is in the back focal plane of L2.

Each reference function that can be placed in the input plane P1 of the retrieval system is the same as a reference function used in the generation of the hologram. It is thus associated with one of the desired outputs. For example, if the reference dk is inserted in P1, the function tk should ap­pear as the output in P3. This is accomplished as follows. A coherent source illuminates dk in the input plane. The first lens displays the Fourier transform Dk of the diffuser incident on the hologram mask. This Fourier transform incident on P2 is multiplied by the hologram H in P2 to form

immediately to the right of P2. Lens L 2 displays the inverse transform in P3 as:

Fig. 1. Optical retrieval system.

15 January 1982 / Vol. 21, No. 2 / APPLIED OPTICS 169

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Fig. 2. Stored input functions 64 × 64 arrays.

Fig. 3. Smoothed cross sections of: (a) diffuser autocorrelations and (b) cross correlations.

Fig. 4. Experimental results: (a) M = 1; (b) and (c) M = 2.

in which ó stands for correlation and Ý stands for convolu­tion. If the autocorrelation dk ó dk of each pseudorandom diffuser is a Dirac delta function, and the cross correlations of each diffuser with every other diffuser are zero, tk the de­sired output appears in the output plane. Hence, the ideal statistical characteristics for each reference function are

A set of Gold codes can be used to construct diffusers that approximate these characteristics, and, because Gold codes are binary in nature and can be described analytically, com­puter calculations with them and fabrication of the physical diffusers from them are straightforward.7,8

A preliminary experiment has been constructed to deter­mine the practical difficulties involved in the implementation of this technique particularly in photoreduction or scaling and alignment. For this purpose Gold code binary phase masks were passed over in favor of random checkerboard binary amplitude masks which considerably simplify construction

of the diffusers. Although the correlation properties of these diffusers are relatively poor, they were satisfactory for the initial studies of difficulties in scaling and alignment.

The steps outlined in this Letter were followed in the gen­eration and photoreduction of the masks. The fast Fourier transform (FFT) subroutine HARM was used for the genera­tion of the multiplexed hologram. The diffuser and hologram masks were photoreduced simultaneously so that the array element or cell size of the hologram mask would be the same as the cell size of any of the diffuser masks. This approach ensures an equal amount of reduction in the two film masks. Although it is not strictly necessary that the two masks be the same size, making them so decreases quality control problems in fabricating the masks separately. Since the Fourier transform of a diffuser must fit exactly on the holographic mask, the length of a side T of a diffuser is given by5

in which N is the number of cells per side, λ is the wavelength of coherent illumination, and F1 is the focal length of lens L1.

The experiment was performed first for simplicity by storing and retrieving only one function (M = 1) and then by storing two different functions on a single multiplexed holo­gram (M = 2) and retrieving each separately. The function stored in the first case is shown in Fig. 2(a). The two functions stored in the second case are shown in Figs. 2(b) and 2(c). In both cases the array size was sixty-four elements squared (N = 64).

The autocorrelation and cross correlation for the binary amplitude random checkerboard masks used are shown in Fig. 3. Note that the autocorrelation as desired exhibits a spike at the origin but also includes large undesirable tails on each side. The cross correlations were also far from being ideal when measured by Eq. (1) because of similar nonzero tail structure. These tails resulted in considerable smearing of the desired functions for both M = 1 and M = 2 in the exper­imental results of Fig. 4 but particularly for the M = 2 case in which the nonideal cross-correlation properties of the random checkerboard masks, in addition to the nonideal autocorre­lation properties important in the M = 1 case, came into play. Nevertheless, the desired functions can be identified in the output [indicated by arrows for the M = 2 case shown in Figs. 4(b) and (c)].

More important, however, is that in obtaining these results we found that scaling and alignment though difficult are certainly not impossible. Scaling accuracy to within ~0.5% (about a quarter of a cell width since there were sixty-four cells) proved adequate to get discernible patterns with proper alignment. In practice, alignment of the mask in P2 of Fig. 1 proved to be the most critical, comparable in difficulty to aligning most interferometric devices.5

The research described in this Letter was funded by the Air Force Office of Scientific Research under grants AFOSR 75-2855 and 79-0076.

References 1. M. I. Jones, "Multiplex Holography for Space-Variant Optical

Processing," M.S.E.E. Thesis, Texas Tech U., Lubbock, Aug. 1979.

2. T. F. Krile, R. J. Marks II, J. F. Walkup, and M. O. Hagler, Appl. Opt. 16, 3131 (1977).

3. J. F. Walkup, Opt.Eng. 19, 339 (1980). 4. C. A. Irby, M. O. Hagler, and T. F. Krile, J. Opt. Soc. Am. 69, 1409

(1979).

170 APPLIED OPTICS / Vol. 21, No. 2 / 15 January 1982

5. C. A. Irby "Computer Generated Multiplex Holography," MSEE Thesis, Texas Tech U., May, 1980.

6. C. B. Burckhart, Appl. Opt. 9, 1949 (1970); 9, 2813 (1970). 7. T. F. Krile, M. O. Hagler, W. D. Redus, and J. F. Walkup, Appl.

Opt. 18, 52 (1979). 8. E. L. Kral, "Correlation Properties of Diffusers for Multiplex

Holography," MSEE Thesis, Texas Tech U., Aug. 1979.

15 January 1982 / Vol. 21, No. 2 / APPLIED OPTICS 171