Band-limited zone plates for single-sideband holography

Band-limited zone plates for single-sidebandholography

Yasuhiro Takaki* and Yumi TanemotoInstitute of Symbiotic Science and Technology, Tokyo University of Agriculture and Technology,

2–24–16, Naka-cho, Koganei, Tokyo 184–8588, Japan

*Corresponding author: [email protected]

Received 12 June 2009; revised 30 August 2009; accepted 18 September 2009;posted 21 September 2009 (Doc. ID 112687); published 1 October 2009

The single-sideband technique eliminates a conjugate image and zeroth order diffraction light, producingonly a reconstructed image of a hologram. A band-limited cosine zone plate appropriate for use with thesingle-sideband technique is derived. The width of the zone plate is half that of a conventional zone platein one direction. The proper selection of a transmitted spatial frequency band leads to an interlacedband-limited zone plate that has complex amplitudes in odd or even rows. The use of such a zone platereduces calculation time for a hologram to approximately 75%. Experimental verification of this ispresented. © 2009 Optical Society of America

OCIS codes: 090.2870, 070.6120, 090.1760, 090.1970, 100.2000.

1. Introduction

The generation of unwanted conjugate image and zer-oth order diffraction light, in addition to the desiredreconstructed image, is one of the essential problemsof aholographic three-dimensional display technique.This problem is deeply connected to the process of cod-ing the complex-amplitude distribution of an objectwave into the intensity distribution of an interferencepattern, i.e., a hologram. The off-axis holographytechnique proposed by Leith and Upatnieks [1] en-ables the angular separation of a reconstructed imagefrom a conjugate image and zeroth order diffractionlight. The single-sideband technique [2] separatesthese three components spatially in theFourierplane,and a single-sideband filter cuts the two undesiredcomponents. In electronic holography, an object waveis calculated by adding zone plates that generatepoints constituting a three-dimensional image. Theuse of a half zone plate for the object wave calculationis shown to be effective when the single-sidebandtechnique is applied to electronic holography. [3,4]

In this study, we derive zone plates appropriate tothe single-sideband technique by considering theband limitation caused by a single-sideband filterin the Fourier plane, and we show that there are avariety of band-limited zone plates in addition to ahalf zone plate. Moreover, we show that the properselection of the transmitted band leads to aninterlaced band-limited zone plate that reduces thecalculation time of a hologram.

2. Band Limitation by Single-Sideband Technique

Figure 1 shows a 4f optical system used in thesingle-sideband technique. An amplitude spatiallight modulator (SLM) is placed in the object plane,and the reconstructed wave without a conjugatewave and zeroth order diffraction light is obtainedon the image plane. A single-sideband filter is placedin the Fourier plane, which blocks the conjugate im-age and the zeroth order diffraction light. A horizon-tal edge is used as the single-sideband filter in thisstudy. A vertical edge can also be used as a single-sideband filter. The use of the horizontal edge halvesthe vertical viewing zone, and the use of the verticaledge halves the horizontal viewing zone.

The amplitude distribution (real distribution) thatis displayed on the amplitude SLM in the 4f imaging

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system is determined. The Fourier transform of theamplitude distribution is obtained on the Fourierplane of the 4f imaging system and is cut by the sin-gle-sideband filter. From the Fourier-transform rela-tionship, the Fourier transform of a real distributionhas complex-conjugate symmetry. Therefore the dis-tribution in the opaque region of the single-sidebandfilter should be a complex conjugate and should besymmetric with respect to that in the transparent re-gion. A Fourier plane distribution satisfying theabove condition is synthesized as follows: The objectwave is denoted by oðx; yÞ, and its Fourier transformis denoted by Oðνx; νyÞ. When the pixel pitch of theSLM is denoted by p, the spatial frequency band-width of the Fourier transform Oðνx; νyÞ is given byΔν ¼ 1=p. As illustrated in Fig. 2, the spatial fre-quency range is limited to Δν=2 in the νy direction,and the central spatial frequency of the spatial fre-quency range is νc. This band-limited distributionis represented by O0ðνx; νyÞ, the spatial frequencyrange of which is shown by a white box in Fig. 2. Thisband-limited distribution O0ðνx; νyÞ is shifted by theamount Δν=4 − νc in the νy direction to place it inthe transparent region of the single-sideband filteron the Fourier plane. This shifted distribution iswritten as Otðνx; νyÞ ¼ O0ðνx; νy −Δν=4þ νcÞ. Thecomplex conjugate and symmetric distribution of thisdistribution is given by O�

t ð−νx;−νyÞ ¼ O0�ð−νx;−νy

−Δν=4þ νcÞ, where the asterisk represents thecomplex conjugate. This distribution lies in theopaque region of the single-sideband filter. Thereforethe synthesized Fourier plane distribution Sðνx; νyÞ isgiven by

Sðνx; νyÞ ¼ O0ðνx; νy −Δν=4þ νcÞþO0�ð−νx;−νy −Δν=4þ νcÞ: ð1Þ

This synthesized distribution has a complex-conjugate symmetry, i.e., Sðνx; νyÞ ¼ S�ð−νx;−νyÞ.Thus the inverse Fourier transform of Sðνx; νyÞ givesthe real distribution sðx; yÞ, which is displayed on theSLM:

sðx; yÞ ¼ 2Refo0ðx; yÞ exp½i2πðΔν=4 − νcÞy�g þ bðx; yÞ;ð2Þ

where o0ðx; yÞ represents the inverse Fourier trans-form of O0ðνx; νyÞ. As the amplitude SLM can repre-sent a positive real distribution, a bias componentbðx; yÞ, which has a real distribution, is added tomake the distribution nonnegative. Because a con-stant value or a slowly varying distribution can beused as the bias component, the Fourier transformBðνx; νyÞ of the bias component bðx; yÞ generates asharp distribution located around the center onthe Fourier plane. Therefore the bias componentcan be easily removed by slightly shifting the sin-gle-sideband filter in the νy direction to obstructthe sharp distribution.

The spatial shift of the band-limited distributionO0ðνx; νyÞ on the Fourier plane introduces an off-axiscarrier. As shown in Fig. 2, the spatial frequencyrange of the reconstructed image is 0∼Δν=2 inthe y direction, so that the reconstructed image pro-ceeds with an angular range of 0∼ sin−1 λ=2p fromthe optical axis toward the y axis. This means thatthe technique described herein can be applied to com-puter-generated off-axis holography. The relation-ship between the shift process in the frequencydomain and the off-axis holography is explained bythe Fourier analysis in Ref. [1].

3. Band-Limited Zone Plate

We derive a band-limited zone plate that can be usedwith the single-sideband technique by means ofEq. (2). A band-limited spherical wave o0ðx; yÞ iscalculated from a spherical wave oðx; yÞ. The band-limited distribution O0ðνx; νyÞ is extracted from theFourier transform Oðνx; νyÞ. The center spatialfrequency νc of the extracted band can be arbitrarilyselected in the range −Δν=4 ≤ νc ≤ Δν=4.

Figure 3(a) shows an example of a conventionalzone plate. The conventional zone plate is the realpart of a spherical wave oðx; yÞ. A constant bias isadded to make the distribution nonnegative. The cal-culation conditions of the zone plate are as follows:The pixel pitch p of the SLM is 10 μm, and the numberof pixels is 256 × 256. The distance z between a point

Fig. 1. Single-sideband method for elimination of a conjugateimage.

Fig. 2. Band limitation in the Fourier plane.

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and the SLM is 20mm. The wavelength of light isλ ¼ 632:8nm. From the sampling theorem, the radiusof the zone plate is λz=2p ¼ 632:8 μm≃ 63 pixels.Figure 3(b) shows the amplitude distribution of

the Fourier transform Oðνx; νyÞ. Three examples ofthe band-limited distribution O0ðνx; νyÞ are shownin Figs. 3(c)–3(e) for (A) νc ¼ Δν=4, (B) νc ¼ 0, and(C) νc ¼ −Δν=4, respectively. The real-valueddistributions calculated using Eq. (2) are shown inFigs. 3(f)–3(h). These are band-limited zone platesthat can be used with the single-sideband technique.A constant bias value is added to make the distribu-tion nonnegative. The distribution shown in Fig. 3(f)is similar to the half zone plate that was previouslyproposed for the elimination of the conjugate image[3,4]. As shown in these results, there are a variety ofband-limited zone plates depending on the selectionof the transmitted band on the Fourier plane.The reconstructed images of the band-limited zone

plates are calculated. The amplitude distributions ofthe band-limited zone plates are Fourier trans-formed to obtain the distribution on the Fourierplane. The distribution in the lower half of the Four-ier plane, including on the νx axis (νy ≤ 0), is set tozero in order to simulate the single-sideband filterthat eliminates the conjugate image and the zerothorder diffraction light. Then, an inverse Fouriertransform is performed to obtain the distributionon the image plane of the 4f imaging system. The re-constructed images are calculated by Fresnel diffrac-tion at the diffraction distance z ¼ 20mm from theimage plate. Figures 4(a)–4(c) show the recon-structed images of the band-limited zone platesshown in Figs. 3(f)–3(h), respectively. Because a fastFourier transform (FFT) is used to calculate theFresnel diffraction, the sampling pitch of the recon-structed image is 4:96 μm from the sampling theo-rem. The intensity distributions in the center16 × 16 sampling points are shown in the figures.The width of the reconstructed peak distributionin the y direction is twice as large as that in the xdirection, because the bandwidth in the νy directionis half of that in the νx direction. As the beamwidth isinversely proposal to the beam diversity, the verticalviewing zone angle is half that of the horizontal one.Figures 4(d)–4(f) show the conjugate images calcu-lated by Fresnel diffraction at the distance z ¼−20mm from the image plane of the 4f imaging sys-tem. Whole images with 256 × 256 sampling pointsare shown. The intensity distribution of each imagein Fig. 4 has been normalized by each maximum in-tensity. A sharp peak distribution is not observed inthe conjugate images. The ratio of the maximum in-tensities of the conjugate image to the reconstructedimage is (A) 8:96 × 10−4, (B) 6:01 × 10−4, and(C) 1:05 × 10−3. These results show that the conju-gate image is eliminated.

4. Modeled Band-Limited Zone Plate

Because the boundary of the band-limited zoneplates shown in Figs. 3(f)–3(h) is not clear, we model

a zone plate with a finite area. The band-limited zoneplate corresponding to the condition (B) νc ¼ 0 ismodeled as shown in Fig. 5. As the spatial frequencyin the νy direction is limited to −Δν=4 ≤ νy ≤ Δν=4, theminimum sampling pitch in the y direction is 2p.Therefore the spherical wave is sampled with mini-mum sampling pitches of p and 2p in the x and y

Fig. 3. Generation of band-limited zone plates: (a) real part ofspherical wave oðx; yÞ; (b) Fourier transform Oðνx; νyÞ of oðx; yÞ;(c)–(e) band-limited Fourier transforms O0ðνx; νyÞ when(c) νc ¼ Δν=4, (d) νc ¼ 0, and (e) νc ¼ −Δν=4; and (f)–(h) band-limitedzone plates when (f) νc ¼ Δν=4, (g) νc ¼ 0, and (h) νc ¼ −Δν=4.


directions, respectively. From the sampling theorem,the shape of the modeled zone plate should be an el-lipse with a horizontal width of λz=p and a verticalwidth of λz=2p. The multiplication of the phase dis-tribution exp½i2πðΔν=4Þy� in Eq. (2) shifts the centerof the quadric phase distribution upward byΔν=4 ¼ λz=4p. The area of the ellipse is πλ2z2=8p2,which is half the area of a conventional zone plateand is equal to the area of a half zone plate. Onlythe real part of the zone plate needs to be calculated.This zone plate is a band-limited cosine zone plate.Because the real part has negative values as wellas positive values, the real part is called a bipolarintensity [5,6].The reconstructed image and the conjugate image

generated by the band-limited cosine zone plate arecalculated. They are similar to the distributionsshown in Fig. 4, so they are not shown here. The ratioof the maximum intensities of the conjugate image tothe reconstructed image was 6:83 × 10−4. The halfzone plate [3,4] is also one of the band-limited cosinezone plates (νc ¼ Δν=4). We also calculated the recon-structed image and the conjugate image generatedby the half zone plate, and we found that the ratiowas 4:73 × 10−3. These results show that the conju-gate image is effectively eliminated using theband-limited cosine zone plate shown in Fig. 5,and the half zone plate is not the only zone plate thatcan be used with the single-sideband technique.

There are a variety of band-limited zone plates, de-pending on the selection of the transmitting spatialfrequency band.

5. Interlaced Band-Limited Zone Plate

With the band-limited condition (B), the transmittedspatial frequency band is jνyj ≤ Δν=4. Therefore aspherical wave can be calculated with a samplingpitch of 2p in the y direction. The zone plate becomesan interlaced zone plate, i.e., either odd rows or evenrows are calculated. The computation time is re-duced. However, in order to calculate Eq. (2), thecomplex-amplitude distribution of o0ðx; yÞ has to beknown with a sampling pitch of p, because the phasedistribution, exp½i2πðΔν=4Þy� ¼ expðiπy=2pÞ ¼ …; 1;i;−1;−i; 1; i;−1;−i;…, must be multiplied before tak-ing the real part. Therefore interpolation must beperformed in order to obtain the complex-amplitudedistribution o0ðx; yÞ with the sampling pitch of p.

Fourier interpolation is used to interpolate thecomplex-amplitude values. Because Fourier interpo-lation requires performing the Fourier transformtwice, Fourier interpolation is replaced by a convolu-tion function and an approximation is introduced inorder to reduce the computation time. The complex-amplitude distribution of the interlaced band-limitedzone plate with a sampling pitch of 2p is convolvedwith the sinc function sincðy=2pÞ with a samplingpitch of p to obtain complex-amplitude distributionwith a sampling pitch of p in the y direction. The

Fig. 4. Calculated reconstructed peak distributions when (a) νc ¼ Δν=4, (b) νc ¼ 0, and (c) νc ¼ −Δν=4; calculated conjugate images when(d) νc ¼ Δν=4, (e) νc ¼ 0, and (f) νc ¼ −Δν=4.


values of the sinc function with a sampling pitch pare represented by an infinite sequence ð…;−2=7π;0;þ2=5π; 0;−2=3π; 0;þ2=π; 1;þ2=π; 0;−2=3π; 0;þ2=5π; 0;−2=7π;…Þ. This infinite sequence is approxi-mated by finite sequences in order to reduce compu-tation time. We used approximated finite sequenceshaving 3, 7, 11, and 15 elements, the center elementof which is 1.The complex amplitudes of a sphericalwaveare cal-

culated with sampling pitches of p and 2p in the x andy directions in an elliptical area with a horizontalwidth of λz=p and a vertical width of λz=2p to obtainthe interlaced band-limited zone plate. The real partand the imaginary part of the calculated zone plateare shown in Figs. 6(a) and 6(b), respectively. Next,the interpolation was performed. Then the phasedistribution was multiplied to obtain the real partfollowing Eq. (2). The interpolated zone plates areshown in Fig. 7. The reconstructed images and theconjugate images were calculated. The ratios of themaximum intensities of the conjugate image to thereconstructed image were 1:47 × 10−2, 3:24 × 10−3,2:97 × 10−3, and 1:36 × 10−3 for using the sequenceshaving 3, 7, 11, and 15 elements, respectively. The re-constructed peak intensity distributions along the yaxis are shown in Fig. 8(a). The interpolation is eval-uated by comparing the intensity distribution alongthe y direction because the interpolation is performedin the y direction. The intensity distributions for theband-limited zone plate shown in Fig. 3(f), and the

modeled band-limited zone plate shown in Fig. 5,are shown in Fig. 8(b). The use of the finite sequenceswith seven or more elements yields approximatelyequivalent results given by the modeled zone plateshown in Fig. 5.

For comparison, a simple average interpolationwas performed. Uncalculated values were generatedby averaging complex amplitudes in the upper andlower rows. In this case the ratio of the maximum in-tensities of the conjugate image to the reconstructedimage was 2:08 × 10−2. The ratio is higher than thoseobtained using the finite sequences. The intensitydistribution of the reconstructed peak intensityalong the y direction is also shown in Fig. 8(b). Thisintensity distribution is approximately equivalent tothat obtained using the finite sequence with threeelements.

Here, the hologram calculation process is de-scribed. The interlaced complex-amplitude distribu-tions of the band-limited zone plates are calculatedfor all points constituting a three-dimensional image,and they are added to one another to generate an in-terlaced objectwave. In this stage, the complex ampli-tudes in odd rows or even rows are calculated. Then,interpolation is performed to obtain the band-limitedobject wave o0ðx; yÞ. In this stage, the complex ampli-tudes in the uncalculated rows are interpolated fromthose in the precalculated rows. Finally, the hologramdistribution is calculated following Eq. (2).

6. Experiments

Experiments were conducted to verify the effective-ness of using the band-limited cosine zone platesand the interlaced band-limited zone plates. The

Fig. 5. Modeled band-limited cosine zone plate.

Fig. 6. Interlaced band-limited zone plate: (a) real part and(b) imaginary part.

Fig. 7. Interpolated zone plates. The sinc function is approxi-matedby finite sequenceswith (a) 3, (b) 7, (c) 11, and (d)15elements.


amplitude SLM used for the experiment was anLC-R1080 (HoloEye Corporation) The resolutionwas 1920 × 1200, and the pixel pitch p was 8:1 μm.A He–Ne laser (λ ¼ 632:8nm) was used as a lightsource. The focal length of the two Fourier transformlenses constituting the 4f optical system was150mm.First, a hologram was calculated using the band-

limited cosine zone plate shown in Fig. 5. The sam-pling pitch was p in the y direction. The reconstructed

images are shown in Fig. 9. The three-dimensionalimages consisted of (a) 710 points, (b) 1461 points,and (c) 5312 points. The computation time of holo-grams was (a) 22.07 s, (b) 46.28 s, and (c) 165.27 s.The calculation was performed using a personal com-puter with CPU of Intel Core 2 Duo CPU E68503:00GHz. Neither a conjugate image nor zeroth orderdiffraction light appeared. The average vertical view-ing angle was 2:0°. The theoretical vertical viewingangle, which is given by λ=2p, was 2:2°. These resultsshow that the band-limited cosine zone plate effec-tively eliminates the conjugate image and the zerothorder diffraction light. The vertical shift of the hori-zontal slit on theFourier plane to eliminate the zerothorder diffraction light slightly reduces the verticalviewing angle.

Next, the method using interpolated band-limitedzone plates was verified. The sampling pitch was 2pin the y direction. The sinc function was approxi-mated by a finite sequence with seven elements.The reconstructed images are shown in Fig. 10.The computation time was (a) 17.39 s, (b) 36.33 s,and (c) 128.32 s, which are 78.8%, 78.5%, and78.2% of the time required for the method usingband-limited cosine zone plates. Neither a com-plex-conjugate image nor zeroth order diffractionlight appeared. The average vertical viewing anglewas 1:9°, which was approximately equivalent tothat obtained using the band-limited cosine zoneplate. The interpolation method is valid.

7. Discussion

The calculation time is discussed. The number of zoneplates is denoted byM, the average area of band-lim-ited zone plates is denoted by S, and the resolution ofan SLM is Nx ×Ny pixels. A hologram calculationusing band-limited cosine zone plates requires theperformance ofMS square root operations to calculatethe distances between zone plates and points consti-tutinga three-dimensional imageand requiresMS co-sine operations to obtain the real part. A hologramcalculation using interlaced band-limited zone platesrequires the performance ofMS=2 square root opera-tions to calculate the distances, MS=2 cosine opera-tions to obtain the real part, and MS=2 sineoperations to obtain the imaginary part. When the

Fig. 8. Intensity distributions of reconstructed peak distributionsalong the y axis: (a) using zone plates interpolated by sinc function;(b) using zone plates shown in Figs. 3(f) and 5, and zone plate gen-erated by simple average interpolation.

Fig. 9. Reconstructed images from holograms generated using band-limited cosine zone plates: (a) apple, (b) teapot, and (c) wine glass.


sinc function is approximated by a finite sequencewith L elements, the interpolation requires the per-formance of NxNyðLþ 1Þ=2 multiplications andNxNyðLþ 1Þ=2 additions. The square root, cosine,and sine functions are calculated by expansions sothat the calculation time for the interpolation occu-pies a smaller ratio of the total calculation time whenthe number of zone plates increases. Assuming thatthe square root, cosine, and sine functions requirethe same calculation time, the calculation time ofthe method using the interpolated band-limited zoneplates is approximately 75% that of the method usingband-limited cosine zone plates. This considerationwell explains the experimental results.We have so far developed a technique that is used

for the single-sideband technique. The developedtechnique can be also applied to off-axis holography[1]. The use of band-limited cosine zone plates sepa-rates the conjugate image angularly from the recon-structed image. The use of interlaced band-limitedzone plates enables the reduction of calculation time.

8. Conclusions

A band-limited cosine zone plate, whose width ishalf that of a conventional zone plate in one direc-tion, is found to be appropriate for use in the single-

sideband method to eliminate conjugate images. Amethod using the interpolation of interlaced band-limited zone plates reduces the calculation time toapproximately 75% of that using band-limited co-sine zone plates and offers a comparable reconstruc-tion image.

This research is partly supported by the NationalInstitute of Information and Communications Tech-nology, Japan.

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Fig. 10. Reconstructed images from holograms generated using interlaced band-limited zone plates: (a) apple, (b) teapot, and(c) wine glass.


Band-limited zone plates for single-sideband holography

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