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Nonredundant calculations forcreating digital Fresnel holograms

J. L. Juarez-Perez, A. Olivares-Perez, and L. R. Berriel-Valdos

In this paper we propose an alternative technique for producing digital Fresnel holograms. The eval-uation of a diffraction pattern in a wide region is implemented in such a way as to avoid redundantcalculations and preserve the precision. Because of the symmetry of the kernel, the complex amplitudeis calculated at four points in the registration plane simultaneously. This algorithm decreases therequired CPU time 4 times with respect to direct calculation. The digital Fresnel hologram is numer-ically and optically reconstructed, and some qualitative comparisons are made. © 1997 Optical Societyof America

Key words: Computer-generated Fresnel holograms, computer diffraction.

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 toproduce the final composite hologram.2 ~b! Objectlocation 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 object’s on-line elements.7Most of these methods use the fast Fourier trans-

The authors are with the Instituto Nacional de Astrofısica, 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 America

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.8,9

In 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-tion pattern numerically and later added the refer-ence wave, implementing the Fresnel–Kirchhoffdiffraction 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 He–Ne laser.An optical and a digital reconstruction was done witha plane wave and a convergent wave to observe theholographic image.

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 theKirchhoff–Fresnel 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`

`

*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

P

(q51

Q

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

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

where

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 theobject’s 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 1997

We 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

P

(q51

Q

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

A~2x0, y0! 5 (p51

P

(q51

Q

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

A~x0, 2 y0! 5 (p51

P

(q51

Q

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

A~2x0, 2 y0! 5 (p51

P

(q51

Q

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

(4)

where

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

P

(q51

Q

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 Imin9

Imax9 2 Imin9G , (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 He–Ne 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 hologram’s physical size depends onthe photoreduction, the sampling points ~xn, ym! haveto be calculated beforehand. Others of these param-eters are freely chosen: the object’s 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 beknown.

According to the sampling theorem, the hologramwith the highest spatial frequency is fmax 5 ~2d!21,where d is the sampled period. Because the gratingequation implies that waves of wavelength l inter-fering at an angle u determine the spatial frequency,this frequency is given by

f 51l

sinSu

2D . (14)

Since d is equal to ~xN21 2 x0!y~N 2 1! and f 5 fmax,we can therefore find the maximum allowed anglebetween the object and reference waves.

To produce an adequate holographic image in thereconstruction step, the interference pattern formedby the object and reference waves should have a highcontrast value.11 A high contrast value occurs if theobject and reference waves have equal amplitude, soone possible way to choose the amplitude A of thereference wave is

A 5 U(n51

N

(m51

M

O~xn, ym!U. (15)

5. Experimental Verification

A. Digital Hologram Construction

Two digital holograms have been obtained: one forthe Instituto Nacional de Astrofısica, Optica y Elec-tronic’s ~INAOE’s! logo @Fig. 2~a!#, which is a binarydigital image with ~P 5 330! 3 ~Q 5 380! pixels andthe other, which is a gray-level digital image with~P 5 100! 3 ~Q 5 100! pixels, for a girl’s face @Fig.2~b!#. To obtain either of these holograms, threesteps have to be followed: ~i! We calculate the dif-fracting field using Eq. ~2! in direct form and with theoptimization given in Section 3; ~ii! we obtain the

Fig. 3. Computer time employed to make the digital Fresnel ho-lograms.

Fig. 2. Object planes used in this study: ~a! INAOE’s logo ~bi-nary image!. ~b! Girl’s face ~gray-level image!.


Fig. 4. Setup employed in the reconstruction of the digital Fresnel–Kirchhoff hologram.

reference-wave amplitude with Eq. ~15! and incorpo-rate it with the maximum angle that is allowed by Eq.~14!; and ~iii! by using Eq. ~10! we calculate the in-tensity pattern and exhibit it on a monitor using Eq.~11!. For implementation of all these steps the re-quired parameters are the monitor’s squared maxi-

mum resolution, N 5 M 5 768 pixels; the distancebetween the object plane and the hologram plane,2 m; the final reduced hologram size, 2 cm2; the max-imum angle between the object and the referencewaves, umax 5 1°159; and the recording wavelength, l5 632.8 nm ~He–Ne laser!. One possible choice of

Fig. 5. Optical reconstruction of the INAOE logo in the three diffraction planes: ~a! pseudoscopic plane, ~b! focal plane, and ~c!orthoscopic plane.


Fig. 6. Optical and numerical reconstruction without the lens of the hologram of girl’s face. Both kinds of reconstruction show threediffraction orders: the primary image ~reconstructed object!, the zero order ~square form!, and the conjugate image ~diffuse form!.

direction-cosine angles is ~90° 2 umax, 90°, umax!.Figure 3 shows the time used in the generation ofthat hologram with both methods when a 486y66 PCis used.

Since each hologram point is independent of theothers it is possible to calculate the total area inindependent regions, and therefore parallel program-ming is feasible. We implemented parallel comput-ing in seven PC’s, and the time required to make adigital hologram was reduced proportionally.

B. Digital Hologram Recording

The digital holograms were displayed with Ng 5 64gray levels on the computer monitor. The interfer-ence pattern on the screen was deamplified 10 timeswith a photoreduction technique by use of a NikonFM2 camera with SO-253 holographic film. The ex-posure time was 3 s for a fully opened diaphragm.The distance between the screen and the camera was80 cm; at this distance the lens’s camera can resolveall the pixels and reduce the digital hologram to 2cm2. The last step in the digital hologram-recordingstep is film development, which in our case was donewith the conventional technique used for holographicdevelopment.

C. Optical Hologram Reconstruction

To observe the holographic image we illuminated theobtained digital hologram slide with a collimatedbeam emerging from a He–Ne laser. We carried outoptical reconstruction with the lens to determinethat, in the Fraunhofer plane, no holographic imageswere formed and that the pseudoscopic and ortho-scopic real images were located in the focal plane.11

We also achieved lensless optical reconstructionshowing only the pseudoscopic real image. This lat-ter technique proves the feasibility of our method.

The setup that uses a lens ~Fig. 4! carries out op-tical reconstruction. The Fraunhofer patterns of thelogo hologram @Fig. 5~b!# are seen on a screen at afocal distance of 1.07 m. In Fraunhofer plane, thepseudoscopic real images @Fig. 5~a!# can be observedseparated by approximately 30 cm, and the ortho-scopic real image @Fig. 5~c!# can also be seen, sepa-

rated by approximately 50 cm, rotated 180°, andhaving different amplification values. The recon-struction of any hologram generates the appearanceof four luminous forms; this fact has been provedmathematically11; however, only three of the termsare of interest since two of them overlap. We canobserve the diffraction patterns in both planes, andthey show three different orders: the zero order ~asquare form!, a primary image ~synthesized object!,and a conjugate image ~diffuse order!.

With lensless optical reconstruction to a distance of2 m from the digital hologram slide, a reconstructionsimilar to that shown in Fig. 5~a! is observed. Theadvantages of this reconstruction method are that ~i!we do not need to use an optical system during thehologram-reconstruction step, and ~ii! the object isrecovered with identical dimensions, as is shown inFig. 6~a!. In this way we can observe qualitativelythat, with a digital Fresnel hologram, it is possible toobtain good hologram images.

D. Digital Hologram Reconstruction

A holographic process ends with the reconstructionstep, so we carried out a numerical simulation of thisstep to have an alternative form for comparison withthe results of the analogous optical reconstruction.Digital reconstruction was realized by the simulationof a lensless optical reconstruction and implementedby use of Eq. ~2!, for which we substituted I0~xn, ym!@Eq. ~12!# with a~X, Y!. Therefore the values ~P, Q!and ~N, M! changed to ~768 3 768! pixels and ~300 3100! pixels, respectively.

To show the result of reconstruction of the image ofthe girl’s face in the monitor @Fig. 6~b!#, we applied alogarithmic transformation @Eq. ~11!# to attenuatethe zero order. As a result of that logarithmic trans-formation, we also can see the Fresnel diffractionfringes of a square aperture.

6. Conclusions

Digital Fresnel holograms were obtained when thediscrete Fresnel–Kirchhoff diffraction equation wasused. Because we do not use an approximation of thekernel, the diffraction pattern of the object is suffi-


cient to yield excellent digital holograms. Recon-struction of the diffraction field of the object is verytime consuming with conventional methods; howeverour technique employed in this study reduced by 4times the required calculation time, plus permittingthe use of parallel programming, with seven PC’s forcalculation. The usefulness of the proposed methodcan be confirmed through the optical-reconstructionstep, with which we can observe a holographic imagewith a high signal-to-noise ratio at the holographicplane.

Appendix A: Demonstration of the Four-Point Property

Sufficient demonstration of the definitions in Eqs. ~4!can be realized with the unidimensional case. Be-fore carrying out the demonstration let us observe aproperty of Eq. ~3!. It is easy demonstrate the fol-lowing relation of symmetry:

r~x, y, 2 X, Y! 5 $@~2X! 2 x#2 1 ~Y 2 y!2%1y2

5 @~X 1 x!2 1 ~Y 2 y!2#1y2

5 $@X 2 ~2x!#2 1 ~Y 2 y!2%1y2

5 r~2x, y, X, Y!, (A1)

where both coordinates X and Y are

r~x, y, X, 2Y! 5 r~x, 2y, X, Y!,

r~x, y, 2X, 2Y! 5 r~2x, 2y, X, Y!. (A2)

These relations extend to Eq. ~5!:

K~x, y, 2X, Y! 5 K~2x, y, X, Y!,

K~x, y, X, 2Y! 5 K~x, 2y, X, Y!,

K~x, y, 2X, 2Y! 5 K~2x, 2y, X, Y!. (A3)

Because of restrictions on the object, sampling inthe object plane take values in the range of ~2c, c!:

S 5 HXp 52cP

yp 5 1, . . . , PJ. (A4)

Separating the coordinate values into the positivesand negatives yields

S1 5 HXiyi 5 1, . . . ,P2J,

S2 5 HXjyj 5P2

1 1, . . . , PJ, (A5)

given the following coordinate symmetry:

X1 5 2XP

X2 5 2XP21

······

XPy221 5 2XPy212

XPy2 5 2XPy211. (A6)


From the first two equalities of Eq. ~A6! is derived thefollowing recurrence equation:

Xk 5 2XP2k11, (A7)

and from the last two equalities of Eq. ~A6! we have

XPy22k11 5 2XPy21k, k 5 1, . . . ,P2

. (A8)

The complex amplitude at the point 2x0 is

A~2x0! 5 (p51

P

a~Xp!K~2x0, Xp!. (A9)

Applying one of the relations given in Eqs. ~A3! yields

A~2x0! 5 (p51

P

a~Xp!K~x0, 2Xp!. (A10)

Separating the sum in the point subsets S1 and S2

@Eq. ~A5!# yields

A~2x0! 5 (k51

Py2

a~Xk!K~x0, 2Xk!

1 (k5Py211

P

a~Xk!K~x0, 2Xk!, (A11)

where the second term on the right-hand side couldbe represented with the indices starting from thesecond half, that is,

A~2x0! 5 (k51

Py2

a~Xk!K~x0, 2Xk!

1 (k51

Py2

a~XPy21k!K~x0, 2XPy21k!. (A12)

Applying the recurrence equations ~A7! and ~A8!, re-spectively, leads to

A~2x0! 5 (i51

Py2

a~XP2k11!K~x0, 2XP2k11!

1 (k51

Py2

a~2XPy22k11!K~x0, XPy22k11!. (A13)

Carrying out a change of index in each term respec-tively yields

l 5 P 2 k 1 1, k 5 17 l 5 P, k 5P27 l 5

P2

1 1,

l 5P2

2 k 1 1 k 5 17 l 5P2

k 5P27 l 5 1,

(A14)

and we have

A~2x0! 5 (l5P

Py211

a~2Xl!K~x0, Xl! 1 (l5Py2

1

a~2Xl!K~x0, Xl!.

(A15)

Carrying out the sum, we have

A~2x0! 5 (l51

P

a~2Xl!K~x0, Xl!. (A16)

Since the indices do not overlap, the property of thefour points has been demonstrated.

References1. J. W. Goodman, An Introduction to Fourier Optics ~McGraw-

Hill, New York, 1968!.2. Toyohiko Yatagai, “Three-dimensional display using

computer-generated holograms,” Opt. Commun. 12, 43–45~1974!.

3. J. P. Waters, “Three-dimensional Fourier-transform methodfor synthesizing binary holograms,” J. Opt. Soc. Am. 58, 1284–1288 ~1968!.

4. D. Leseberg, “Sizable Fresnel-type hologram generated by

computer,” J. Opt. Soc. Am. A 6, 229–233 ~1989!.5. D. Leseberg and C. Frere, “Computer-generated holograms of

3-D objects composed of tilted planes segments,” Appl. Phys.27, 3020–3024 ~1988!.

6. D. Leseberg, “Computer-generated three-dimensional imageholograms,” Appl. Phys. 31, 223–229 ~1992!.

7. D. Leseberg, “Computer generated holograms: cylindrical,conical and helical waves,” Appl. Phys. 26, 4385–4390~1987!.

8. A. D. Stein, Z. Wang, and J. S. Leigh, Jr., “Computer-generatedhologram: a simplified ray-tracing approach,” Comput. Phys.6, 389–392 ~1992!.

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11. P. Harihara, Optical Holography, Principles, Techniques andApplications ~Cambridge U. Press, Cambridge, UK, 1984!.


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