spatial amplitude and phase modulation using commercial...

Spatial amplitude and phase modulation usingcommercial twisted nematic LCDs

E. G. van Putten,* I. M. Vellekoop, and A. P. MoskComplex Photonic Systems, Faculty of Science and Technology and MESAþ Institute for Nanotechnology,

University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

*Corresponding author: [email protected]

Received 27 November 2007; revised 27 February 2008; accepted 19 March 2008;posted 20 March 2008 (Doc. ID 90209); published 16 April 2008

We present a method for full spatial phase and amplitude control of a laser beam using a twisted nematicLCD combined with a spatial filter. By spatial filtering we combine four neighboring pixels into onesuperpixel. At each superpixel we are able to independently modulate the phase and the amplitudeof light. We experimentally demonstrate the independent phase and amplitude modulation using thisnovel technique. Our technique does not impose special requirements on the spatial light modulator andallows precise control of fields even with imperfect modulators. © 2008 Optical Society of America

OCIS codes: 090.1995, 120.5060, 230.3720, 230.6120.

1. Introduction

Several excitingnewapplications of optics rely onspa-tial phase and amplitude control of light. In adaptiveoptics, light ismodulated tocorrect foraberrations inavariety of optical systems suchas thehuman eye [1] ortheatmosphere [2].Digitalholographyisanotherareaof optics that relies on the spatial modulation of light.Holographic data storage [3], three-dimensional dis-play technology [4], and diffractive optical elements[5] are just a few examples of themany exciting topicsin digital holography that require a very precise mod-ulation of light. Very recently it was shown that lightcan be focussed through and inside opaque stronglyscattering materials by sending in spatially shapedwavefronts [6,7].Liquid crystal (LC) spatial lightmodulators (SLMs)

are used in many cases where modulation of light isrequired because of their high optical efficiency, highnumber of degrees of freedom, and wide availability.An LC SLM can modulate the phase and the ampli-tude of light at typical refresh rates of ∼60Hz andat resolutions in the order of 106 pixels. Aside fromtheir advantages, commercially available SLMs have

some limitations. Inmost LC SLMs themodulation ofphase is coupled to a modulation of the polarization.This makes it hard to independently modulate thephase or the amplitude.

Several techniques have been proposed to achieveindependent spatial phase and amplitude controlusing an SLM. Examples are setups where two SLMsare used to balance out the cross-modulation [8,9],doublepassconfigurationswhere the lightpropagatestwice through the sameSLM [10,11], and double pixelsetups thatdivide the encoding of complex values overtwo neighboring pixels [12–14]. Each of these techni-ques has its specific limitations. The use of two SLMsin one setup [8,9] requires a sensitive alignment andvaluable space. Double pass configurations [10,11]double the phase modulation capacities of the SLM,usually delivering a full 2π phase shift. Unfortunatelythough, to keep the amplitude constant, the SLM hastowork in amostly phase condition that requires lightof a specific elliptical polarization.Doublepixel setups[12–14] also require special modulation propertiessuch as amplitude-only modulation [14], phase-onlymodulation [12], or a phase modulation range of 2π[13].Thesepropertiesareusuallyachievedbysendingin light under certain specific polarizations. Small de-viations from these polarizations already modify the0003-6935/08/122076-06$15.00/0

© 2008 Optical Society of America

2076 APPLIED OPTICS / Vol. 47, No. 12 / 20 April 2008

modulation properties, leading to drift in the achievedamplitudes and phases.We propose a novel modulation technique to

achieve amplitude and phase modulation of lightusing an SLM together with a spatial filter. Thistechnique does not pose special requirements onthe SLM or the incoming light, making our techniquemore widely applicable than previous techniques. Inour experiments we used a HoloEye LC-R 2500twisted nematic LCD (TN-LCD) that has 1024 ×768 pixels (19 μm square in size), and can modulatewith 8 bit accuracy (256 voltage levels). This SLMhas a strong cross-modulation between the phaseand the polarization of the light. We demonstratethe use of this SLM to independently modulate phaseand amplitude.

2. Cross-Modulation in Twisted Nematic LCDs

Twisted nematic LCDs owe their name to the helicalstructure formed by the molecules inside the LCDcells, in absence of an electric field. When a voltageis applied over the edges of the LCD cell, the mole-cules are aligned to the electric field and the twistedstructure disappears. By changing the voltage overthe cells, the internal structure is changed continu-ously from a twisted to a straight alignment of themolecules, affecting both the phase and the polariza-tion of the reflected light.To determine the relation between the applied vol-

tage and the modulation of the light (the modulationcurve of our SLM), we used a diffractive technique,where we placed a binary grating with a 50% dutycycle on the display, comparable with the techniquedescribed in [15]. Using a polarizing beam splitter,we illuminated the SLM with vertically polarizedlight and detected the horizontally polarized compo-nent of the modulated light. We measured the inten-sity in the zeroth order of the far-field diffractionpattern for different contrast levels of the grating.The contrast difference is achieved by keeping thenotches of the grating steady at a constant valuewhile we cycled the voltage values of the rules ofthe grating over the complete voltage range in 256steps. We determined the real and imaginary partsof the field as a function of the voltage level encodedon the pixels by comparing the measured zeroth or-der intensity for different pixel values on both thenotches and the rules of the grating. The results ofthese measurements are plotted in Fig. 1. Whenno voltage is applied, there is a 0:75π phase shift,and the amplitude is close to zero. With increasingvoltage the phase is modulated from 0:75π back to0, where the amplitude is at a maximum. If the vol-tage is increased further to the maximal voltage, thephase is modulated up to −0:62π while the amplitudedecreases slightly. The deviation of the modulationcurve from a circle centered around the origin is aclear indication of cross-modulation.

3. Decoupling of Phase and Amplitude by CombiningMultiple Pixels

By combining four neighboring pixels into one super-pixel, we decouple the modulation of the phase andthe amplitude, thereby achieving full independentphase and amplitude control of light. Any arbitrarycomplex amplitude can be synthesized even thougheach individual pixel on the SLM has a limited com-plex response (see Fig. 1). This technique is based onthe work done by Birch et al. [14,16], where they com-bined two neighboring pixels of an amplitude-onlySLM to achieve full independent phase and ampli-tude modulation. By using four pixels instead oftwo, our technique no longer depends on a specialmodulation curve. There are only two requirementson the modulation curve. First, the real part of themodulation curve should be an increasing functionof the applied voltage. Second, for each imaginary va-lue on the modulation curve where the derivative isnonzero, there should exist at least two different so-lutions. For every known SLM the modulation curve,or a suitable subset, follows these two requirements.

The setup used to decouple the phase and ampli-tude modulation is shown in Fig. 2. A monochromaticbeam of light at a wavelength of 532nm is incidentnormal to the SLM surface. The modulated light isreflected from the SLM. We choose an observationplane at an angle at which the contribution of eachneighboring pixel is π=2 out of phase, as seen in theinset. The fields from the individual pixels are super-posed using a spatial filter. By choosing the correctcombination of pixel voltages, any complex valuewithin a square (−1::1), (−i::i) can be synthesized.

Fig. 1. Experimental modulation plot for the SLM illuminatedwith vertical polarized light. After modulation the light with a hor-izontal polarization is detected. This plot shows the amplitude ver-sus the phase modulation. The arbitrary phase reference is chosensuch that the maximum amplitude coincides with the a zero phasemodulation. The modulation voltage increases in the direction ofthe gray arrows.

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To construct a complex value f ¼ gþ ih, we use thefirst and third pixel of a superpixel to construct g,and the second and fourth pixels to construct h.The voltage for the first and third pixels, V1 andV3, are chosen such that the imaginary parts ofthe modulated fields are equal and the difference be-tween the real parts of the modulated fields is equalto g (a detailed discussion about how to find the vol-tages can be found in Appendix A). The fields modu-lated by the two pixels are then

E1 ¼ E1r þ iΔ; ð1Þ

E3 ¼ E3r þ iΔ; ð2Þ

E1r − E3r ¼ g; ð3Þ

where E1r and E3r are the real parts of the fieldsmodulated by pixels 1 and 3, and Δ is the imaginaryterm of the modulated fields. In the plane of recon-struction, the two pixels are π out of phase. Thereforethe imaginary parts of the modulated fields canceleach other while the real parts are subtracted andproduce a nonzero real value equal to g. A geometri-cal representation of this construction is given inFig. 3. We construct h in a similar way using the sec-ond and the fourth pixels. The fields modulated bythe these two pixels are then chosen to be

E2 ¼ E2r þ iϵ; ð4Þ

E4 ¼ E4r þ iϵ; ð5Þ

E2r − E3r ¼ h: ð6Þ

The π=2 phase shift between all the neighboring pix-els in a superpixel causes the reconstructed values gand h to be π=2 out of phase. The total reconstructedfield is therefore gþ ih, which is exactly the complexvalue f we wanted to modulate.

Instead of a single value f , we use the SLM to con-struct a two-dimensional function f ðx; yÞ ¼ gðx; yÞþihðx; yÞ. We will calculate the reconstructed fieldin analogy to the analysis given by Birch et al.[16]. When we modulate the function f ðx; yÞ, the fieldat the SLM surface is described by

f sðx; yÞ ¼X∞

m;n¼−∞

δðy −ma=4Þ½δðx − naÞE1ðx; yÞ

þ δðx − 1=4a − naÞE2ðx − 1=4a; yÞþ δðx − 1=2a − naÞE3ðx − 1=2a; yÞþ δðx − 3=4a − naÞE4ðx − 1=4a; yÞ�; ð7Þ

where the shape of the pixels is simplified to a deltafunction response. The functions E1, E2, E3, and E4are constructed such that for each superpixel coordi-nate (na, ma=4) they agree with Eqs. (2)–(6). A lensL1 positioned behind our SLM Fourier transforms

Fig. 2. (Color online) Experimental setup to decouple phase andamplitude modulation. Four neighboring pixels (pixels 1, 2, 3, and4) are combined to one superpixel that can modulate one complexvalue. The modulated light is focused by a lens, L1, onto a dia-phragm placed in the focal plane. The focal length of L1 is200mm. The diaphragm is positioned such that it only transmitslight under a certain angle. This angle is chosen so that, in thehorizontal dimension, two neighboring pixels are exactly π=2out of phase with each other as can be seen from the inset. Behindthe diaphragm the light is collimated by a lens, L2.

Fig. 3. (Color online) The electric field of a superpixel is super-posed out of four distinct neighboring pixels whose phases areall shifted over π=2 with respect to each other. In this figure weshow a limited modulation range of pixels 1, 2, 3, and 4. The num-bers outside the plot relate the curves to the pixel numbers. Bychoosing the right pixel values for the different pixels, every com-plex value can be synthesized by the superpixel. In the figure, pix-els 1 and 3 are used to modulate a value on the real axis. Theelectric fieldsE1 andE3 are chosen such that their imaginary partscancel andE1 − E3 lies on the real axis. Pixels 2 and 4 are used in asimilar way to modulate any value on the imaginary axis (notshown).


the field. To find the field in the focal plane, wecalculate the Fourier transform of Eq. (7) to the spa-tial frequencies ωx and ωy and recenter the coordi-nate system to the center of the first diffractionmode by defining Ωx ≡ ωx − ð2π=aÞ,

FsðΩx;ωyÞ ¼12π

X∞k;l¼−∞

n~E1rðΩxk;ωylÞ

− ~E3rðΩxk;ωylÞe−iaΩx2 þ i

h~E2rðΩxk;ωylÞ

− ~E4rðΩxk;ωylÞe−iaΩx2

ie−i

aΩx4 þ i

h~ΔðΩxk;ωylÞ

þ ~ϵðΩxk;ωylÞe−iaΩx4

ih1 − e−i

aΩx2

io; ð8Þ

with Ωxk ≡ Ωx − 2πk=a and ωyl ≡ ωy − 8πl=a. Fromthis equation we see that at Ωx ¼ 0, the ~Δ and ~ϵterms completely vanish, and only the terms ~E1r −~E3r þ ið~E2r −

~E4rÞ ¼ ~gþ i~h survive, which is exactlythe Fourier transform of the original function writtento the SLM. At nonzero spatial frequencies the ~Δ and~ϵ terms appear and introduce an error that increaseswith spatial frequency. To reduce these errors andcut out unwanted diffraction orders, we place a dia-phragm of width ðλfwÞ=a around ðΩx;ωyÞ ¼ ð0; 0Þwith λ the wavelength of the light, f the focal lengthof lens L1, andw the relative width of the diaphragm.The relative width can be chosen w ¼ 1 for maximalresolution, but can also be chosen smaller. We notethat in the original analysis by Birch et al. [16], allterms inside the summation, except one, were ne-glected. However, at the SLM pixels, light may dif-fract into different orders. Therefore we do notmake this approximation, and include the summa-tion over all orders into our calculation.A second lens L2 transforms the filtered field at the

diaphragm into the reconstructed field. We find thisreconstructed field by taking the inverse Fouriertransform of Eq. (8),

f rðx; yÞ ¼X∞

m;n¼−∞

δðy −ma=4Þ�δðx − naÞE1ðx; yÞ

− δðx − 1=2a − naÞE3ðx − 1=2a; yÞþ iδðx − 1=4a − naÞE2ðx − 1=4a; yÞ− iδðx − 3=4a − naÞE4ðx − 1=4a; yÞ�

⊗2w

a2 sinc�xwa

;4ya

�: ð9Þ

In the reconstructed field the pixels of a superpixelare spatially separated in the x direction, a limitationthat is present in every multipixel technique. Be-cause of the spatial filter the fields modulated by dif-ferent subpixels are spatially broadened, whicheffectively averages them. In Fig. 4 the reconstructedfield in the x direction is shown for the case where the128th superpixel is set to 1þ i while the other super-pixels are set to 0. The real and imaginary part of the

reconstructed superpixel are spatially separated bya=4. Decreasing the size of the spatial filter, w, re-sults in an improved quality of the reconstructedfield by cutting out the ~Δ and ~ϵ terms in Eq. (8). Thisquality improvement happens at the cost of spatialresolution. As for small w, the sinc-function inEq. (9) is broadened.

4. Experimental Demonstration of ModulationTechnique

We tested the modulation technique with the setupdepicted in Fig. 2. We placed an extra lens and a CCDcamera behind the spatial filter to detect the modu-lated light in the far-field.

First, we demonstrate the decoupling of the phaseand amplitude modulation. We encoded the sameamplitudes and phases on all superpixels of theSLM forming a plane light wave. The plane wavewas then focused onto the CCD camera. We mea-sured the intensity in the focus while changing thephase and amplitude of the plane wave. In Fig. 5the relative amplitudes from the intensity measure-ments are plotted against the phase of the planewave. For different amplitude modulations the phaseis cycled from 0 to 2π. It is seen from the results thatwe modulate the phase while keeping the amplitudeof the light constant within 2.5%.

Next, we show that the set phase modulation is inagreement with the measured phase modulation. Weplaced a binary grating with a 50% duty cycle on theSLM. We modified the contrast by keeping one partof the grating constant at a relative amplitude, A, of0.5 while changing the relative amplitude of theother part, B, between 0 and 1. For each value ofB, we changed the phase difference between thetwo parts of the grating, Δθ, from 0 to 2π. Becauseof interference the recorded intensity, I, at theCCD camera changes as a function of A, B, and Δθin the following way:

Fig. 4. Calculation of the reconstructed field in the x direction forthe case where the 128th superpixel is set to 1þ i while the othersuperpixels are set to 0. The vertical lines represent the borders ofthe superpixels. The reconstructed real and imaginary parts arespatially separated by a=4.


I ¼ A2 þ B2 þ 2AB cosðΔθÞ: ð10Þ

In Fig. 6 the results are presented for three differentvalues of B. Both the amplitude and the phase of themeasured oscillations match the theory well, demon-strating an independent phase modulation in agree-

ment with the set phase difference. With the newmodulation technique, we are able to modulate thephase of the light from 0 to 2π, even though each in-dividual pixel is not able to modulate a full 2π phaseshift (see Fig. 1).

Finally, we demonstrate complex spatial modula-tion of light. We encoded the Fourier transform ofan image onto the SLM. The original image is recon-structed in the far-field and recorded by the CCDcamera. In Fig. 7 the reconstructed intensity isshown on a logarithmic scale. The letters in the im-age stand out with high contrast. Around the letters,small areas containing noise are seen. The relativeintensity of the noise is < 10%.

5. Conclusions

We have presented a novel approach to achieve fullindependent spatial phase and amplitude controlusing a TN-LCD together with a spatial filter. Ournovel modulation scheme combines four neighboringpixels into one superpixel. Each superpixel is able toindependently generate any complex amplitude va-lue. We demonstrated that we were able to freelymodulate the phase over a range of 2π while keepingthe amplitude of the light constant within 2.5%.Furthermore we showed that we could achieve a spa-tially complex modulation of the light with a high re-solution.

The presented modulation technique can be usedwith any SLM. No special modulation response isneeded, therefore the illumination light can have lin-ear polarization. This polarization makes a relativesimple calibration possible. Using linearly polarizedlight, the setup can easily be expanded to control twoindependent polarizations with two SLMs.

Fig. 5. (Color online) Results of the measurements to demon-strate the independent phase and amplitude modulation. Usingthe superpixels we synthesized a plane wave of which we variedthe amplitude and the phase. In this plot see the relative ampli-tude A=

ffiffiffiffiffiI0

pfrom intensity measurements as a function of the pro-

gramed phase. The intensity I0 ¼ 19:7 · 103 counts/seconds. Therelative amplitudes are set to 0.25, 0.5, and 0.75, correspondingto the red, green, and blue lines, respectively.

Fig. 6. (Color online) Results of the measurements demonstratethat the set phase difference corresponds with the actual phasedifference. In this plot we show the relative measured intensityof a grating interference experiment. The intensities are plottedas a function of the set phase difference Δθ, and are referenceto I0 ¼ 4:56 · 103 counts/seconds. The experiment was done formultiple amplitude differences between the rules and the notchesof the grating. The solid lines represent the expected intensities.

Fig. 7. (Color online) Demonstration of complexmodulation usingthe novelmodulation technique. In this experimentwe encoded theFourier transform of an image containing the two words “Phase”and “Amplitude” onto the SLM and reconstructed the image inthe far-field. This plot shows themeasured intensity in the far-fieldon a logarithmic scale. The intensities are in counts/seconds.


Appendix A: How to Find the Proper Pixel Voltages

In this appendix we prove that for every modulationcurvethat fulfills therequirementsstatedinSection3,we can find pixel voltages fulfilling Eqs. (2)–(6). Thereal and imaginary parts of the field modulated bya pixel, Er and Ei, are functions of the voltage onthe pixel. We only use the voltage range for whichErðVÞ is a rising function of the applied voltage. Bymultiplying themodulation curvewithaproperphasefactor, this range can be maximized. The functionErðVÞ can be inverted while EiðVÞ has two inverses.We divide the voltage range into two parts, V < Vcand V > Vc, where Vc is the voltage for which thederivative of EiðVÞ equals 0. On both ranges we caninvert the function EiðVÞ.To modulate a positive real value gðV1Þ, we use

pixels 1 and 3, which are π out of phase. On the firstpixel we place V1 > Vc, and on the third pixel weplace V3 ¼ VLðEiðV1ÞÞ, where VL is the inverse ofEiðVÞ for V < Vc. The total field modulated by thesetwo pixels is now

gðV1Þ ¼ ErðV1Þ − ErðVLðEiðV1ÞÞÞ: ðA1ÞIt can be shown that gðV1Þ is an increasing functionof V1, and that gðVcÞ ¼ 0. We find the function V1ðgÞby inverting gðV1Þ. To construct a negative real va-lue, the voltage V1 should be chosen V1 < Vc. Thevoltage V3 is then V3 ¼ VHðEiðV1ÞÞ, where VH isthe inverse of EiðVÞ for V > Vc. Similar calculationscan be done to find the functions V2ðhÞ and V4ðhÞ tomodulate a positive and negative value h usingpixels 2 and 4.As a technical note, we reconfigured the gamma

lookup curve of the modulation electronics afterthe calibration measurements to perform the calcu-lations discussed above. The conversion of amplitudeand phase to real and imaginary values is performedin real time in the computer’s video hardware.

The authors thank Ad Lagendijk and Willem Vosfor support and valuable discussions. This work ispart of the research program of the Stichting voorFundamenteel Onderzoek der Materie (FOM), whichis financially supported by the Nederlandse Organi-satie voor Wetenschappelijk Onderzoek (NWO). A. P.Mosk is supported by a VIDI grant from the NWO.

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