Typically, binary phase holograms, corresponding totwo-dimensional pixel arrays that consist of onlytwo phase values, diffract not only into the desiredfirst diffraction order, but with equal efficiency alsointo the disturbing minus first order [1,2]. The ap-pearance of a conjugate diffraction order is particu-larly disturbing for on-axis Fourier hologramswhere the reconstructed image field is superposedwith a second, inverted (“pseudoscopic”) image ofthe same intensity. Depending on the size of the phasejump between the two available phase levels, alsohigher order components appear, which show en-larged copies of the image with amagnification factorcorresponding to the diffraction order. As an example,a comparison between reconstructed continuous andbinary phase holograms, displayed at the same spa-tial light modulator, is shown in Fig. 1. Whereas con-tinuous phase holograms [Figs. 1(a) and 1(c)]reconstruct just the desired images, in the case of bin-
ary phase holograms [Figs. 1(b) and 1(d)], these aresuperposed by their corresponding inverted images.
These effects limit the usefulness of ferroelectricspatial light modulators (SLMs) as display or projec-tion devices for computer-generated holograms(CGHs), since this type of SLM can display only bin-ary phase structures. In other respects, ferroelectricSLMs would be highly suitable as holographic dis-plays, since they allow a very fast switching ratein the 10kHz regime (or even faster) [3–7]. Thiscan be useful for displaying time-multiplexed holo-grams [8] for, e.g., reproducing color holograms byswitching the displayed phase patterns in synchroni-zation with light sources of different colors, opticalmanipulation of mesoscopic particles in microscopy[9–15], or the trapping of atoms [16,17].
There are well-known methods that separate thedesired image field from the undesired diffraction or-ders for binary holograms.One approach is to produceoff-axis holograms by multiplying the transmissionfunction of the original on-axis DOE with an inclinedplane term∝ expði~kp ·~rÞ, where~kp is the grating vec-tor that defines thedirection and themagnitude of thetilt. The undesired negative diffraction orders are
thus spatially separated from the positive ones andcan be blocked [18]. However, this operation discardsthe half planes in k space that carry unwanted infor-mation and, thus, reduces the available bandwidth ofthe DOE and the utilizable field of view of the pro-jected image by one-half. Another method is to useFresnel-type holograms, instead of Fourier holo-grams,whichmeans that the original on-axis transferfunction is multiplied by a lens term ∝ expð−iπj~rj2=λf Þ. In this case, the desired image is not re-constructed at infinite distance (as for Fourier holo-grams), but at a certain distance f behind thehologram,whereas the undesired negative diffractionorders are simultaneously defocused and produce adiffuse background [15,19]. However, this methodalso reduces the available bandwidth and it doesnot allow the use of certain advantageous featuresof Fourier holograms, such as their maximal depthof sharpness or the shift invariance of the outputimage with respect to the transverse position of thehologram.In this paper we demonstrate an alternative meth-
od to eliminate theundesired diffraction orders of bin-aryCGHsby illuminating themwith apseudorandomlight field with a known phase distribution. Thismight be achieved with a lithographically producedphase mask that is directly attached to the SLM dis-play and that shifts the phase of each SLM pixel by arandomly chosen, but known, value in ½0; 2π�. Notethat for reflective displays the double pass throughthe mask has to be considered. Alternatively, a givenrandom phase mask, for instance, a slab of groundglass, could be used after measuring its phase distri-bution interferometrically. This known “offset” phasedistribution is stored in the computer and taken intoaccount when calculating the binary CGH of any de-sired image field, as explained later. First attempts inthis direction have been reported in [20].As wewill explain, in the ideal case of two available
phase levels that differ by π, the maximal first-orderdiffraction efficiency that can be achieved is 40.5% ofthe incoming light, which is identical to the maximalefficiency of a standard binary phase hologram and isthe same efficiency that can be reachedwith the otherabove-mentioned methods (off-axis or Fresnel holo-grams) to suppress the undesired diffraction orders.However, the dispersion of the diffuse backgroundwave is maximized by the random phase plate, i.e.,the background noise is smaller than in the Fresnel
approach. The contrast between the programmed im-age and the diffuse background depends on the imagestructure, i.e., the 40.5% “usable” light is concen-trated in the programmed bright image areas,whereas the remaining 59.5% is diffusely distributedover the entire image plane. Therefore, the contrast isparticularly high for images with only a few brightpixels, or one-dimensional line structures, like thoseused for optical trapping patterns.
To demonstrate the principle, we simulated thepseudorandom phase mask by a phase pattern thatwas displayed at a part of the same SLM that wasused for displaying the binary CGH. In our case,we could use a nematic SLM, which could displaycontinuous phase patterns (with 256 programmablephase levels in the interval between 0 and 2π), aswell as simulating a binary ferroelectric SLM by dis-playing only the two phase values 0 and π. Section 2describes the basic principle of the method, followedby its experimental demonstration.
2. Underlying Concept
Thediffractionefficiency of an ideal phaseCGH(with-out absorption andwith a 100% fill factor) depends onthe number n of phase values in the interval between0 and 2π that are actually used to construct the under-lying grating structure. If we consider a blazed grat-ing (“sawtooth grating”) as an elementary example, nwould be the number of different phase levels that areaddressed within one grating period. In this case, thediffraction efficiency η becomes [21]
η ¼�nπ sin
πn
�2: ð1Þ
A one-dimensional sawtooth grating with a gratingperiod of 14pixels (with a linear phase spacing inthe interval ½0; 2π�) is plotted in Fig. 2(a).
According to Eq. (1), the corresponding first-orderdiffraction efficiency of such a blazed grating is98.3%. The corresponding diffraction pattern is dis-played in the inset of Fig. 2(a), where m denotes thediffraction order and η the corresponding diffractionefficiency. The diffraction efficiency η is calculated bysimply computing the squared absolute value of theFourier transform of the corresponding complextransmission function. In this case, there is a domi-nant peak at m ¼ 1 with η ≈ 0:98.
The next part of the figure [Fig. 2(b)] shows a phasepattern corresponding to a binary grating (with theoptimal phase step of π). In this case, Eq. (1) predictsan efficiency of 40.5% for the first-order efficiency.However, the same energy is also scattered to theminus first order (and the residual energy to higher,odd-numbered diffraction orders), such that the dif-fraction pattern looks symmetrical [see inset ofFig. 2(b)].
Next, Fig. 2(c) shows the diffraction pattern of aone-dimensional random phase mask with phase va-lues that are uniformly distributed in the interval
Fig. 1. Reconstructed computer-generated on-axis Fourier holo-grams using a spatial light modulator as a phase display. (a)and (c) were displayed using 256 phase levels in the interval be-tween 0 and 2π, whereas (b) and (d) are the corresponding binaryholograms using only the phase levels 0 and π.
½0; 2π�. In this case, the scattered light is randomlydistributed as a speckle pattern [inset of Fig. 2(c)].In Fig. 2(d), the random phase pattern of Fig. 2(c)
has been corrected with a binary phasemask, by add-ing either 0 or π to the phase of each pixel, such thatthe best possible approximation to the ideal sawtoothgrating of Fig. 2(a) is obtained. This one-dimensionalsituation can straightforwardly be generalized to ourtwo-dimensional diffuse illumination method, wherea first pseudorandom phase distribution (c) is cor-rected by a binary phase mask to get the closestpossible approximation to a desired ideal (i.e., contin-uous) phase hologram. The corresponding diffractionefficiency of the approximated blazed grating inFig. 2(d) is displayed again in the inset. The first-order diffraction efficiency is close to 40.5%, whereasthe residual intensity is distributed over the wholeimage plane as a fine speckle pattern.An intuitive explanation for this is given in the re-
maining part of the figure: the approximated blazedgrating of Fig. 2(d) can be numerically decomposedinto the sum of two other phase patterns, namely,an ideal sawtooth grating [Fig. 2(e)] and a randomphase mask [Fig. 2(f)]. However, now the modulationamplitude of the random phase pattern in Fig. 2(f) isconfined to the interval ½0; π�, rather than ½0; 2π� asbefore [Fig. 2(c)]. The reason for this confinementis the “intelligent” choice to add 0 or π to the randomfield in Fig. 2(c) in order to approximate the desiredphase pattern in Fig. 2(e). Therefore, the differenceof the created phase pattern from its ideal shape[Fig. 2(e)] cannot exceed π.The overall scattering property of Fig. 2(d) can now
be obtained by first diffracting the incident wave atthe sawtooth grating [Fig. 2(e)], followed by scatter-ing at the random phase mask [Fig. 2(f)]. The firstdiffraction process at the “perfect” blazed gratinghas an efficiency of 98.3% [as in Fig. 2(a)] for the de-sired field distribution (see inset), which is then scat-tered by the second random phase mask. However,its reduced phase distribution amplitude of only π re-duces its scattering strength. It can be shown [22]that such a phase mask diffusely scatters only59.5% of the incoming light, whereas the remaining40.5% is still confined in the zero order [see inset ofFig. 2(f)], which means that this percentage of the
incoming light field is transmitted without distor-tion. Thus, the overall efficiency of the desired imageis the product of the first-order efficiency of the firstpattern and the zero-order efficiency of the secondone, yielding about 40.5%. This efficiency is also ob-tainable by an “ordinary” binary phase CGH, withthe advantage that now the energy of the undesireddiffraction orders is distributed over the whole imagefield, since the nonzero-order light behind the maskin Fig. 2(f) is homogeneously distributed [see inset ofFig. 2(f)] such that no conjugate image is formed.
3. Experimental Realization
The basic outline of the experiment is sketched inFig. 3. The light source for the following experimentswas a continuous wave 5mWhelium–neon laser witha wavelength of 633nm, emitting a linearly polarizedpure TEM00 mode. It illuminated a reflective ne-matic SLM (Holoeye, HEO 1080P) with a resolutionof 1920pixels × 1080pixels, each with a size of8 μm × 8 μm, and a fill factor (corresponding to theusable phase modulating area) of 87%. For the cor-rectly adjusted incident light polarization the SLMacted as a pure phase modulator with a resolutionof 256 linearly adjustable phase levels in the intervalbetween 0 and 2π.
For the proof-of-principle of the diffuse illumina-tion method, the SLM was used both for generatingthe pseudorandom diffuse illumination wave and fordisplaying the binary phase CGH. For this purpose,the SLM displayed two different phase masks side byside, each with a size of 960pixels × 960pixels. Thelaser beamwas expanded to a plane wave that homo-geneously illuminated the first phase pattern (H1),acting as the pseudorandom diffuser. The reflected
Fig. 2. Explanation for the diffraction properties of different grat-ing types. Details are explained in the text.
Fig. 3. (Color online) Experimental setup. The expanded beam ofa He–Ne laser illuminates one-half of a high-resolution SLM witha size of 1920pixels × 1080pixels, acting as a diffuser by display-ing a continuous pseudorandom phase pattern. From there, thelight is reflected and sharply imaged (by a set of two lenses) atthe other half of the SLM display. There a binary CGH is dis-played, which takes the pseudorandom illumination into account.The hologram is reconstructed in its Fourier plane, using a Four-ier-transforming lens in front of a CCD camera. In another config-uration (not displayed), the Fourier transform of the light reflectedfrom the first SLM pattern is projected at the second one, usingonly one Fourier-transforming lens arranged in the middle ofthe optical path between the two SLM displays.
light was then guided by two mirrors and a (variable)lens arrangement to the second phase mask (H2),which displayed the CGH (corrected by the illumina-tion phase). Finally, the light reflected from H2 wasprojected through a Fourier-transforming lens to aCCD camera, where the reconstructed hologramwas recorded.This setup was used to simulate different situa-
tions; Fig. 4 shows the results. The lower diagramssketch the realized experimental configurations.First, a standard binary Fourier CGH reconstruc-
tion was performed in Fig. 4(a). In this case, H1 wasunmodulated, just acting as a mirror. Thus, the inci-dent laser beam was reflected without being changedconsiderably and it illuminated H2 as a plane wave.Displaying a hologram pattern composed of the twophase levels 0 and π at H2, this corresponded to a“normal” binary Fourier hologram readout. As ex-pected, the reconstructed image in Fig. 4(a) showsthe superposition of the desired image (i.e., the “X”and “O” characters) in the upper part of the figurewith their conjugates in the lower part. The brightspot in the center results from light reflected fromthe nonactive area of the SLM, due to its limited fillfactor, and makes up 13% of the total reflected light.Second, the setup could be used to simulate the si-
tuation where a pseudorandom phase mask is “at-tached” to the CGH. Since it was not possible tophysically attach such a diffuser to the SLM, this si-tuation was simulated by sharply imaging the pseu-dorandom phase mask H1 with a size-preservingtelescopic lens arrangement into the plane of H2[as indicated below Fig. 4(b)]. For the simulation ofthe diffuser,H1 was chosen as a pseudorandom arraywith a uniform continuous phase distribution ϕðx; yÞin the interval between 0 and 2π. To calculate a CGHthat can be displayed in the plane ofH2, and that cor-rects for the illuminating phase distribution, first the“normal” CGH Ψðx; yÞ for the desired image was cal-culated with a standard Gerchberg–Saxton optimiza-tion algorithm [23], just assuming the availability ofcontinuous (i.e., nonbinary) phase levels. This pattern
was then corrected by subtracting theknownpseudor-andom offset phase distribution ϕðx; yÞ. The resultingphase distribution was “wrapped” to the interval½0; 2π� by taking it modulo-2π and, finally, “binarized”by setting all phase values ofmod2πfΨðx; yÞ − ϕðx; yÞgin ½0; π� to 0 and in ½π; 2π� to π.
The resulting reconstructed image is shown inFig. 4(b). As expected, there is no conjugate imageanymore. Instead, its energy is dispersed as a diffusespeckle pattern, which is sufficiently diluted to dis-play the desired image with high contrast. Becauseof the first pseudorandom mask, the zero-order spot(i.e., the just specularly reflected spot) in the centerof the image [Fig. 4(a)] has now vanished.
A third alternative simulates the situation wherethe CGH is illuminated by a diffuse wave that resultsfrom a first scattering process at a random phasemask situated in its Fourier plane (or in its far field).For this purpose, the optics between H1 and H2 aredesigned such that the Fourier transform of H1 is re-constructed at H2. The pseudorandom phase patternH1 thus produces a diffuse illumination wave in theplane of H2. It consists of a diffuse speckle patternwith a phase distribution that can be calculated bynumerically computing the two-dimensional Fouriertransform of H1 (i.e., of Ffexp½iϕðx; yÞ�g), and consid-ering the correct scaling factor that depends on thefocal length of the Fourier-transforming lens. Thecorrection of the binary CGH displayed at H2 canthen be performed with this calculated phase offsetas described before in Fig. 4(b). The position of thehologram H2 must be adjusted properly, such thatthe actually generated diffuse wave field in the planeH2 matches the one expected from the numericalcomputation of the Fourier transform of H1.
The experimentally reconstructed CGH in Fig. 4(c)shows that this method works with approximatelythe same performance as in Fig. 4(b). Particularly,there is again a suppression of other diffraction or-ders and of the central bright spot. The reconstruc-tion efficiency in the cases of Figs. 4(b) and 4(c) isapproximately equal. This demonstrates that it ispossible to consider the phase distribution of an illu-mination wave scattered by a distant diffuser in thecalculation of a corrected CGH. In the case where thepseudorandom scattering mask H1 is located exactlyin the Fourier plane of H2, a lateral shift of H1 justresults in a corresponding lateral shift of the recon-structed image in the camera plane [24].
4. Conclusion
A method was demonstrated to reconstruct on-axisbinary CGHs without superposition of the unwantedconjugated order. A practical motivation for this ap-proach is the possibility to extend the use of binaryphase displays, such as ferroelectric SLMs, for holo-graphic displays. To date these have the advantage ofbeing the fastest available high-resolution SLMs,with switching rates of almost 100kHz. In the future,they could be used to reconstruct color hologramsby time-multiplexed displaying of different CGHs
Fig. 4. (Color online) Reconstructed holograms (upper line) in dif-ferent reconstruction geometries (below). (a) shows a classic binaryCGH with plane wave illumination. In (b) the pseudorandomphase mask H1 is sharply imaged onto the corrected binaryCGHH2. In (c) the pseudorandom phase maskH1 acts as a diffuseillumination source by locating it in the Fourier plane of the binaryCGH H2.
illuminated with synchronously switched diodes ofdifferent colors. They could also be used in opticaltweezers or optical atom traps, or as fast lithographytools to, for example, write sophisticated hologrampatterns into photopolymers.Themost convenient implementation of themethod
would probably be to attach a specially fabricatedpseudorandom phase mask with a known phase pro-file close to the liquid crystal layer by using, e.g., amicrostructured coverglass or a diffusely reflectingrear mirror. Alternatively, the coverglass of a SLMcould be replaced by a ground-glass slide acting asa random phase mask. Its offset spatial phase distri-bution could afterward be measured interferometri-cally so that it could be used as a pseudorandomphasemask. For a randomphasemask that is directlyattached to the SLM, the offset phase of each pixel ismuch better controlled than in our actual experimentand, therefore, we expect also a better signal-to-noiseratio. Nevertheless, as we have shown, it is also pos-sible to illuminate the SLM with a diffuse light fieldscattered by a distant diffuser with a known phaseprofile. In any case, provided that the offset phasedistribution in the CGH plane is known, it can beconsidered in all further hologram calculations.The theoretically achievable diffraction efficiency
of the method is 40.5%, and equals the efficiency ofa standard binary CGH. The remaining 59.5% ofthe diffracted light is dispersed as a uniform back-ground over thewhole image plane and creates a non-avoidable uniform background. However, the methodmaximizes the dispersion of the unused light suchthat, in this respect, it is in any case better thanthe Fresnel method explained above, which has al-ready been demonstrated to be practicable for certainapplications, such as optical tweezers [15] or imagefiltering [19]. Therefore, themethod allows producingCGHs for on-axis reconstruction (i.e.,without a super-posed grating), which has the advantages that the fullbandwidth of the display is used to encode the imageinformation and that the whole field of view in theimage plane is accessible.
This work was supported by the Austrian ScienceFund (FWF) project P19582-N20.
References and Notes1. B. Kress and P. Meyrueis, Digital Diffractive Optics (Wiley,
2000).2. G. O. Reynolds, J. B. Develis, and B. J. Thompson, The New
Physical Optics Notebook: Tutorials in Fourier Optics (SPIEPress, 1989).
3. T. H. Barnes, T. Eiju, K. Matsuda, H. Ichikawa, M. R.Taghizadeh, and J. Turunen, “Reconfigurable free-spaceoptical interconnections with a phase-only liquid-crystalspatial light modulator,” Appl. Opt. 31, 5527–5535 (1992).
4. J. Gourlay, S. Samus, P. McOwan, D. G. Vass, I. Underwood,and M. Worboys, “Real-time binary phase holograms on a re-flective ferroelectric liquid-crystal spatial light modulator,”Appl. Opt. 33, 8251–8254 (1994).
5. W. Hossack, E. Theofanidou, J. Crain, K. Heggarty, and M.Birch, “High-speed holographic optical tweezers using a ferro-
electric liquid crystal microdisplay,” Opt. Express 17, 2053–2059 (2003).
6. K. M. Johnson, M. A. Handschy, and L. A. Pagano-Stauffer,“Optical computing and image processing with ferroelectricliquid crystals,” Opt. Eng. 26, 385–391 (1987).
7. C. J. Henderson, D. G. Leyva, and T. D. Wilkinson, “Free spaceadaptive optical interconnect at 1:25Gb=s, with beam steeringusing a ferroelectric liquid-crystal SLM,” J. LightwaveTechnol. 24, 1989–1997 (2006).
8. A. Lafong, W. J. Hossack, J. Arlt, T. J. Nowakowski, and N. D.Read, “Time-multiplexed Laguerre–Gaussian holographicoptical tweezers for biological applications,” Opt. Express14, 3065–3072 (2006).
9. J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, “Multi-functional optical tweezers using computer-generated holo-grams,” Opt. Commun. 185, 77–82 (2000).
10. E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, andD. G. Grier, “Computer-generated holographic optical tweezerarrays,” Rev. Sci. Instrum. 72, 1810–1816 (2001).
11. K. Dholakia, G. Spalding, and M. MacDonald, “Optical twee-zers: the next generation,” Phys. World 15, 31–35 (2002).
12. V. Bingelyte, J. Leach, J. Courtial, and M. J. Padgett, “Opti-cally controlled three-dimensional rotation of microscopicobjects,” Appl. Phys. Lett. 82, 829–831 (2003).
13. G. Gibson, L. Barron, F. Beck, G. Whyte, and M. Padgett,“Optically controlled grippers for manipulating micron-sizedparticles,” New J. Phys. 9, 14 (2007), doi: 10.1088/1367-2630/9/1/014.
14. H.Melville,G. F.Milne,G.C. Spalding,W.Sibbett,K.Dholakia,and D. McGloin, “Optical trapping of three-dimensional struc-tures using dynamic hologram,” Opt. Express 11, 3562–3567 (2003).
15. A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte,“Diffractive optical tweezers in the Fresnel regime,” Opt.Express 12, 2243–2250 (2004).
16. S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimen-tal observation of optically trapped atoms,” Phys. Rev. Lett.57, 314–317 (1986).
17. D. McGloin, G. Spalding, H. Melville, W. Sibbett, and K.Dholakia, “Applications of spatial light modulators in atomoptics,” Opt. Express 11, 158–166 (2003).
18. E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filteringfor zero-order and twin-image elimination in digital off-axisholography,” Appl. Opt. 39, 4070–4075 (2000).
19. J. A. Davis, D. M. Cottrell, J. E. Davis, and R. A. Lilly, “Fresnellens-encoded binary phase-only filters for optical pattern re-cognition,” Opt. Lett. 14, 659–661 (1989).
20. M. A. A. Neil and E. G. S. Paige, “Breaking of inversion sym-metry in 2-level binary, Fourier holograms,” in Fourth Inter-national Conference on Holographic Systems, Componentsand Applications (IEEE, 1993), pp. 85–90.
21. V. Arrizón and M. Testorf, “Efficiency limit of spatially quan-tized Fourier array illuminators,” Opt. Lett. 15, 197–199(1997).
22. The undistorted zero-order efficiency of a wave field trans-mitted through a phase mask is given by η0 ¼ jðx2 − x1Þ−1Rx2x1
TðxÞdxj2, where TðxÞ is the complex transmission function.In the case of a random, uniformly distributed phase grating(in an interval between 0 and π), this becomesη0 ¼ jπ−1 R π
0 expðiϕÞdϕj2 ¼ 0:405.23. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for
the determination of phase from image and diffraction planepictures,” Optik (Stuttgart) 35, 237–246 (1972).
24. A. Jesacher, C. Maurer, A. Schwaighofer, S. Bernet, and M.Ritsch-Marte, “Near-perfect hologram reconstruction with aspatial light modulator,” Opt. Express 16, 2597–2603 (2008).
