nonredundant calculations for creating digital fresnel holograms

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    produce the final composite hologram.2 ~b! Object tion pattern numerically and later added the refer-

    location near the focal plane: the 3-D object issampled in different planes in depth, and each ob-ject plane is multiplied by a quadratic phase factor andFourier transformed, so the superposition forms thesynthesized hologram3,4; however, the problem ofhidden lines is present. ~c! Angular spectrum:The object wave is disassembled in plane waves,each plane wave is multiplied with a phase factor,and the superposition is Fourier transformed.5,6~d! Superpositioned analytic distributions: Acomputer-generated hologram is formed by the su-perposition of wave fronts emanating from the de-composition of the 3-D objects on-line elements.7Most of these methods use the fast Fourier trans-

    ence wave, implementing the FresnelKirchhoffdiffraction equation and simulating an object formedby a two-dimensional array of point sources. Work-ing in this way requires much computer time, as isreported in Refs. 8 and 9. The implementation ofthe discrete diffraction equation is done in such a waythat the registration plane does not include redun-dance in its evaluation. This method can be used toevaluate four points in the registration plane simul-taneously, providing these points have equal kernels.With this technique the required computer time isreduced by a factor of 4 with respect to the direct-calculation method, and the same precision is pre-served.

    Since the Fresnel digital hologram pattern is dis-played on a computer monitor that shows anintensity-fringe pattern modulated by gray levels, aphotoreduction of the monitor is needed to recon-struct the holographic image with a HeNe laser.An optical and a digital reconstruction was done witha plane wave and a convergent wave to observe theholographic image.

    The authors are with the Instituto Nacional de Astrofsica, Op-tica y Electronica, Apartado Postal 51 y 216, Tonantzintla, C.P.72000 Puebla, Mexico.

    Received 20 November 1996; revised manuscript received 5March 1997.

    0003-6935y97y297437-07$10.00y0 1997 Optical Society of AmericaNonredundant calculations focreating digital Fresnel holog

    J. L. Juarez-Perez, A. Olivares-Perez, and L. R. Be

    In this paper we propose an alternuation of a diffraction pattern in acalculations and preserve the precisis calculated at four points in therequired CPU time 4 times with reically and optically reconstructed, aof America

    Key words: Computer-generate

    1. Introduction

    The first communication on making numericalFresnel holograms describes how to employ a Fou-rier technique with a quadratic phase term.1 Dif-ferent techniques for generating three-dimensional~3-D! holograms have been reported: ~a! Holo-graphic stereograms: A sequence of perspectiveviews of an object is calculated, their Fourier trans-form holograms are synthesized by computer, andeach hologram is arranged in the order of viewing torams


    tive technique for producing digital Fresnel holograms. The eval-wide region is implemented in such a way as to avoid redundanton. Because of the symmetry of the kernel, the complex amplituderegistration plane simultaneously. This algorithm decreases thepect to direct calculation. The digital Fresnel hologram is numer-d some qualitative comparisons are made. 1997 Optical Society

    Fresnel holograms, computer diffraction.

    form numerical tool for the calculation of the dif-fraction field.

    More recently some studies have been based on themodel developed by Huygens to determine the inter-ference pattern by means of linear superposition ofspherical wave fronts from point sources in which theobject breaks down and from the reference wave.8,9In contrast to the methods reported in Refs. 8 and 9,we did not derive a mathematical expression of theinterference pattern; we first calculated the diffrac-10 October 1997 y Vol. 36, No. 29 y APPLIED OPTICS 7437

  • 2. Digital Recording Hologram

    The first step in synthesizing a hologram numericallyis to propagate the complex amplitude from the objectplane ~X, Y! to the hologram plane ~x, y! by use of theKirchhoffFresnel diffraction theory. If a waveplane traveling in the z-axis direction is incident onthe object, then the complex amplitude O~x, y! in theholographic plane is10

    O~x, y! 5 *2`




    a~X, Y!exp~ikr!

    rcos~h!dXdY, (1)

    where, according to Fig. ~1!, r is the distance betweeneach point in the object and each point in the holo-graphic plane. The factor cos~h! is called the obliq-uity factor, and the angle h is formed between the zaxis and a straight line connecting points from theobject to points in the hologram planes. In our treat-ment the object is discrete because we use digitalimages; this means that Eq. ~1! takes the discreteform given by

    O~x, y! 5 s (p51




    a~Xp, Yq!exp@ikr~x, y, Xp, Yq!#

    r2~x, y, Xp, Yq!, (2)


    r~x, y, Xp, Yq! 5 @~Xp 2 x!2 1 ~Yq 2 y!

    2 1 s2#1y2 (3)

    is the radius of the spherical wave emerging from theobjects sampling points and detected as the holo-gram sampling points and s is the distance betweenthe hologram and object planes. In Eq. ~2! the obliq-uity factor has been replaced with syr~x, y, Xp, Yq!,and the object is given by a P 3 Q pixel array. FromEq. ~2! it is clearly seen that each object point isrepresented by spherical wave fronts with ampli-tudes given by the values of each point that forms thedigital image a~Xp, Yq!.

    3. Four-Point Property

    We propose a procedure for the calculation of Eq. ~2!in a wide region parallel to the plane of the object.This method is based on coordinate-axis symmetry,and the only restriction is that the object be centeredwith respect to the z axis.

    Fig. 1. Schematic diagram of the reference system.

    7438 APPLIED OPTICS y Vol. 36, No. 29 y 10 October 1997We prove that the amplitudes of the symmetricalpoints ~x0, y0!, ~x0, 2y0!, ~2x0, y0! and ~2x0, 2y0! areobtained by evaluating Eq. ~2! by simply changing thesign in the object function that has the same kernel,that is,

    A~x0, y0! 5 (p51




    a~Xp, Yq!K~x0, y0, Xp, Yq!,

    A~2x0, y0! 5 (p51




    a~2Xp, Yq!K~x0, y0, Xp, Yq!,

    A~x0, 2 y0! 5 (p51




    a~Xp, 2 Yq!K~x0, y0, Xp, Yq!,

    A~2x0, 2 y0! 5 (p51




    a~2Xp, 2 Yq!K~x0, y0, Xp, Yq!,



    K~x0, y0, Xp, Yq! 5exp@ikr~x0, y0, Xp, Yq!#

    r~x0, y0, Xp, Yq!2 . (5)

    This means that the kernel function needs to be eval-uated only one time, and by rotating the image wesimultaneously obtained the complex amplitude offour points in the plane of registration. ~See Appen-dix A for a demostration.!

    4. Interference Pattern

    When the reference is a plane wave with a tilt valuewith respect to the propagation ~z! axis, then thereference wave can be written in discrete form as

    R~xn, ym! 5 A exp~ik z rn,m!, (6)

    where A is the amplitude of the reference wave, thehologram is formed by an N 3 M pixel rectangulararray with the indices n and m running from 0 to Nand M, respectively, k is the propagation vector,which can be written in terms of the direction cosinesas

    k 5 2pyl@cos~ux!i 1 cos~uy!j 1 cos~uz!k#, (7)

    and the vector rn,m can be expressed as

    rn,m 5 xni 1 ymj 1 sk. (8)

    The object- and the reference-wave superpositionhas a complex amplitude at the hologram plane givenby

    H~xn, ym! 5 A exp~ik z rn,m! 1 s (p51




    a~Xp, Yq!

    3exp@ikr~xn, ym, Xp, Yq!#

    r2~xn, ym, Xp, Yq!. (9)

    The intensity distribution field associated with Eq.~9! can be simulated numerically in the computer and

  • stored in computer memory. Therefore, this inten-sive pattern can be obtained by

    I~xn, ym! 5 uH~xn, ym!u2. (10)

    There are two reasons to modify the intensity pat-tern given in Eq. ~10!: ~i! The calculated numericalvalues changed from a very low value to very highvalue. In this case a logarithmic operator is appliedto emphasize the local contrast, so we have

    I9~xn, ym! 5 log@~I~xn, ym! 1 1#, (11)

    where the addition of the constant value 1 is done toprevent singularity values. ~ii! Since there are afinite number of gray levels, we apply normalizationand scaling factors to Ng gray levels while the inter-ference fringes are displayed on the monitor screen.Then the equivalent intensity patterns or the dis-played digital Fresnel hologram is

    I0~xn, ym! 5 NgFI9~xn, ym! 2 Imin9Imax9 2 Imin9 G , (12)where Imin9, Imax9 are the minimum and maximumintensity values, respectively, obtained from I9~xn,ym!.

    Because the holographic image has to be observedwhile it is illuminated with a HeNe laser, the infor-mation contained in Eq. ~12! is recorded on a fine-grain film to make a photoreduction of the monitor.When the film is developed the transparency trans-mittance function is given by

    t~x, y! 5 t0 1 bTI0~x, y!, (13)

    where t0, b, and T are the constant background trans-mittance, the slope, and the film exposure and devel-opment time, respectively. Hence a Fresnelhologram generated by computer is recorded.

    The implementation of Eq. ~13! for making digitalholograms requires the assignment of parameter val-ues. Some of these parameters are restricted by thelimitations of the display device: the hologram max-imum resolution N 3 M and the object resolution P 3Q. Since the holograms physical size depends onthe photoreduction, the sampling points ~xn, ym! haveto be calculated beforehand. Others of these param-eters are freely chosen: the objects physical size orcoordinates ~Xp, Yq! and the distance between theobject and hologram planes. Finally, the amplitudeand tilt of the reference wave ~ux, uy, uz! have to b