Optimization of the spatial light modulation withtwisted nematic liquid crystals by a genetic algorithm

Joonku Hahn, Hwi Kim, and Byoungho Lee*School of Electrical Engineering, Seoul National University, Gwanak-Gu Sinlim-Dong, Seoul 151-744, South Korea

*Corresponding author: byoungho@snu.ac.kr

Received 12 October 2007; revised 22 January 2008; accepted 22 January 2008;posted 24 January 2008 (Doc. ID 88506); published 17 March 2008

An optimization technique determining the configuration of optical components in a spatial light mod-ulator is proposed. We study a spatial light modulator composed of a twisted nematic liquid crystal, apolarization state generator, and a polarization state detector. To obtain the desired phase and amplitudemodulations, four parameters of the polarization state generator and detector should be optimized.A genetic algorithm is applied in searching the configurations suitable to a given twisted nematic li-quid crystal. By embodying the proposed technique, the evolution of the designed cost functions isproved. © 2008 Optical Society of America

OCIS codes: 050.1970, 090.2890, 160.3710, 230.6120.

1. Introduction

Spatial light modulators (SLMs) are used in a widerange of applications, including telecommunication,displays, optical storage, and optical information pro-cessing, and they are researched intensively. In digi-tal holography, resolution and pixel pitch of the SLMaffect the quality and size of diffraction images. Com-mercial twisted nematic liquid crystals (TNLCs) forprojection display have a lot of pixels, with the size assmall as several tens of micrometers; therefore SLMsbased on TNLC prevail for phase or amplitude modu-lations by the development of the projection displayindustry.The propagation of the light in TNLC is perfectly

described with the extended Jones matrix [1,2].However, practically the optical structure of TNLCis simply modeled to obtain the characteristic para-meters in experimental results. The most simplifiedmodel regards TNLC as a polarization rotator withcascaded wave plates [3]. After that, some ad-vanced models are proposed taking account of edgeeffects of TNLC so that the molecules near both

alignment layers do not tilt independent of an elec-tric field [4–6]. To estimate the modulation througha given TNLC by simulation, the characteristic para-meters of the model of TNLC should be determinedthrough experiments. Even in the most simplifiedmodel, there are ambiguities in the measurementof only three characteristic parameters [7], andtroublesome methods are required to exclude theseambiguities.

Phase and amplitude modulations through TNLCare intuitively understood by the concept of polariza-tion eigenstates based on the most simplified model[8]. With the help of these polarization eigenstates,modulation behaviors of some SLMs are predictedand compared with experimental results [9,10].For convenience, the input and output polariza-tion states are able to be defined as a polarizationstate generator (PSG) and a polarization state detec-tor (PSD) [11]. Recently, to obtain phase-only modu-lations with TNLC, the techniques of analyzing thepolarization states on the Poincaré sphere are ac-tively studied [12,13]. However, since the thicknessof the TNLC decreases on account of smaller pixelsand faster response, the edge effects of TNLC becomemore significant, and the most simplified model thatneglects edge effects is insufficient to predict themodulation behavior.

0003-6935/08/190D87-09$15.00/0© 2008 Optical Society of America

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The thinner TNLC is, the more incorrect the sim-plified model is. In addition, a complicated model re-quires more characteristic parameters, and in themeasurement of the characteristic parameters it istroublesome to exclude the ambiguities. Thereforewe approach the optimization of an SLM, not bymod-eling TNLC, measuring characteristic parameters,and predicting the modulation behavior with a com-putation, but by changing the PSG and PSD config-urations and measuring the modulation behavior. Inthis approach we measure directly the modulationbehavior of an SLM depending on the applied vol-tage; therefore the optimization procedures are sim-plified and free of the errors resulting from modelingand measuring processes.In an optimization problem of several parameters,

a genetic algorithm effectively finds a proper solution[14]. In a digital holographic beam-shaping system,optimization of encoding index and compensationof an aberration is proposed using a genetic feedbackloop [15]. In this paper, we embedded the genetic al-gorithm in the SLM system, simultaneously measur-ing the phase and amplitudemodulation according tothe applied voltage when controlling the rotation an-gles of the optics with motorized stages. Many tech-niques measuring the phase modulation of an SLMhave been studied [16–18]. In this paper, a wavefrontinterferometer using double slits is applied.This paper is organized as follows. In Section 2, the

configuration of the SLM using TNLC is described onthe Poincaré sphere. An ideal TNLC is expressed aspolarization eigenstates, and the edge effects are dis-cussed. The configuration parameters of the PSG andPSD appended to the TNLC are defined. In Section 3,the proposed genetic optimization technique is de-scribed. The chromosome and procedure of thegenetic algorithm are detailed, and the structureof the program is described. In Section 4, the systemembodying the proposed algorithm is described, andthe experimental results are discussed. For the de-signed cost function, the optimization of the SLMis presented. In Section 5, concluding remarks andperspectives on the devised system are provided.

2. Spatial Light Modulation Using a Twisted NematicLiquid Crystal Device

In this section, optical components in a general con-figuration of the SLM using TNLC are representedon the Poincaré sphere, and the configuration para-meters for optimizing the SLM are defined. In theideal TNLC, the eigenstates are elliptic polarizationstates, and the eigenvalues are phase delays of thelight going through TNLC. To obtain a large phasemodulation, the eigenstate with a large change inits eigenvalue according to the applied voltage hasto be chosen. The edge effects are comprehensibleto shift the eigenstates around director axes on thePoincaré sphere, and the shifted eigenstates do nothave the same inclination angle of the polarizationellipse as the applied voltage of TNLC changes.Therefore, these variations in the inclination angles

require a proper PSG and PSD appended to TNLC tochoose the eigenstate with the more desirable phasemodulation in the change of the applied voltage.

The general configuration of the SLM using TNLCis shown in Fig. 1, where a polarizer and a wave platebefore TNLC construct a PSG, and a wave plate anda polarizer after TNLC construct a PSD. The x̂ axis isparallel to the director axis on the input plane ofTNLC, and the ẑ axis is the direction of light propa-gation. The angles of polarizers, θP1 and θP2, are de-fined as the rotations of polarization axes, and theangles of wave plates, θW1 and θW2, are defined asthe rotations of fast axes from the x̂ axis.

An elliptic polarization state can be expressed bythe azimuth angle of the oscillation direction ψ and arelative phase δ, and this is related to Jones vectorrepresentation by

ðδ;ψÞ≡�

cos ψeiδ sin ψ

�: ð1Þ

Figure 2 shows the elliptic polarization state in xyplane and its position on the Poincaré sphere, wherethe inclination angle of the polarization ellipse ϕ andthe ellipticity angle ε is defined by

sin 2ε≡ sin 2ψ sin δ; ð2aÞ

tan 2ϕ≡ tan 2ψ cos δ: ð2bÞIn this paper we represent an ideal TNLC with the

edge effects on the Poincaré sphere, since there arenot many studies about the shifts of eigenstatesowing to the edge effects. Figure 3 shows twist anglesand birefringences as a function of TNLC depth inideal case models. In Figs. 3(a) and 3(b), the simplestmodel is shown that was proposed by Lu and Saleh[3], and in Figs. 3(c) and 3(d), the more advanced

Fig. 1. Schematic of the SLM composed of TNLC, a polarizationstate generator, and a polarization state detector: θP1 and θP2are rotation angles of input and output polarizers; θW1 and θW2are rotation angles of input and output wave plates.

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model with the edge effect is shown that was pro-posed by Coy et al. [4]. In Figs. 3(c) and 3(d), exceptfor the edge of the TNLCwith the thickness t, on bothsides the twist angle increases linearly according tothe TNLC depth, and the birefringence is uniformregardless of the TNLC depth. These bulk propertiesof the TNLC are the same as the Lu and Saleh model[3]. Therefore, the model proposed by Coy et al. [4]is equivalent to the summation of the bulk TNLCmodel and the wave plates on both edges.The Jones matrixMTNLC for TNLCwith edge effect

can be represented as

MTNLC

¼ Rð−αÞWð2πtΔn=λÞRðαÞMbulkTNLCWð2πtΔn=λÞ¼ expð−iβÞRð−αÞWð2πtΔn=λÞMðα; βÞWð2πtΔn=λÞ;

ð3Þ

where MbulkTNLC for the bulk TNLC is given by

MbulkTNLC ¼ expð−iβÞRð−αÞMðα; βÞ: ð4Þ

Here the rotation matrix RðθÞ is defined by

RðθÞ ¼�

cos θ sin θ− sin θ cos θ

�; ð5Þ

and the phase shift matrix WðφÞ is defined by

WðφÞ ¼�1 00 expðiφÞ

�: ð6Þ

The matrix Mðα; βÞ is given by

Mðα; βÞ ¼�cos γ − iβ sin γ=γ α sin γ=γ

−α sin γ=γ cos γ þ iβ sin γ=γ�:

ð7Þ

Here β is the birefringence, α is the twist angle, andγ2 ¼ α2 þ β2. The birefringence β is given by

β ¼ πðd − 2tÞΔneff=λ; ð8Þ

where (d − 2t) is the thickness of a bulk TNLC, λis the incident wavelength, and Δneff is the differ-ence between the ordinary and the effective extraor-dinary indices of refraction perpendicular to thepropagation.

Davis et al. derived the polarization eigenstates ofthe bulk TNLC model in Ref. [8] for the theoreticalapproach to phase modulation. The rotated eigen-states for the matrix Mðα; βÞ are given by

EλðþÞ ¼�

α=ð2γ2 þ 2βγÞ1=2iðβ þ γÞ=ð2γ2 þ 2βγÞ1=2

�; ð9aÞ

Eλð−Þ ¼�ðβ þ γÞ=ð2γ2 þ 2βγÞ1=2−iα=ð2γ2 þ 2βγÞ1=2

�: ð9bÞ

Fig. 2. Elliptic polarization state (a) in the xy plane and (b) itsposition on the Poincaré sphere.

Fig. 3. Twist angles and birefringences as a function of TNLCdepth: (a), (b) Lu and Saleh model; (c), (d) Coy et al. model.

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Here their eigenvalues are given by

λð�Þ ¼ expð−iβÞ expð�iγÞ: ð10ÞTherefore, the negative eigenstate Eλð−Þ has largerphase delay than the positive eigenstate EλðþÞ. Inmost cases, the SLM is optimized to maximize thephase modulation up to 2π, and the negative eigen-state Eλð−Þ is important for designing the PSG andPSD configurations.Figure 4(a) shows the input and output negative

eigenstates Eλð−Þ on the Poincaré sphere and theirtraces according to the applied voltage. In Eq. (9b),the relative phase and azimuth angle of the inputnegative eigenstates are given by

δ ¼ −π=2; ð11aÞ

ψ ¼ tan−1½α=ðβ þ γÞ�: ð11bÞThe output negative eigenstates are rotated by 2αaround the S3 axis on the Poincaré sphere. In

Eq. (8) the birefringence β is a function of the appliedvoltage V defined by

Δneff ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n2en2on2esin2θtiltðVÞ þ n2ocos2θtiltðVÞ

s− no: ð12Þ

Here, θtiltðVÞ is the tilt angle of liquid crystal mole-cules in the bulk TNLC, and no and ne represent theordinary and extraordinary indices in the molecule,respectively. This tilt angle increases monotonouslyaccording to the applied voltage, which is repre-sented as

θtiltðViÞ ≥ θtiltðVjÞ if Vi > Vj: ð13ÞTherefore, the birefringence decreases as the appliedvoltage increases, and negative eigenstates traceto the right handed circle (RHC) point (δ ¼ −π=2,ψ ¼ π=4) with the same inclination angles of the po-larization ellipse ϕ. Figure 4(b) shows the shifts ofinput and output negative eigenstates owing to theedge effect. The input negative eigenstate Eλð−Þshifts by the negative phase −2πtΔn=λ, and the resul-tant state is at (−π=2þ 2πtΔn=λ, tan−1½α=ðβ þ γÞ�). Onthe contrary, the output negative eigenstate Eλð−Þshifts by the positive phase 2πtΔn=λ. Therefore,the edge effects result in the rotation of input andoutput negative eigenstates around the centralpoints ð0; 0Þ and ð0; αÞ, respectively. That is, these el-liptic polarization states on the shifted traces do nothave the same inclination angle of the polarizationellipse. Even though we discussed the ideal TNLCproposed by Coy et al. [4], it is obvious for most non-reflective TNLC models that there are elliptic polar-ization eigenstates whose inclination angles varyaccording to the applied voltage.

In the hypothetical case that the negative eigen-states are placed near the points ð0; 0Þ and ð0; αÞ re-gardless of the applied voltage on the Poincarésphere, the PSG and PSD can be simplified as linearpolarizers to choose the demanded negative eigen-states. However, in a practical TNLC the eigenstatesbecome close to the circular polarization state as theapplied voltage increases, and the edge effects shiftthese eigenstates changing the inclination angle ofthe polarization ellipse. Therefore, proper configura-tions of PSG and PSD are necessary to attain thedesirable phase modulation using TNLC.

Figure 5 shows PSG and PSD constructed withquarter wave plates on the Poincaré sphere. In Fig. 5(a), the bold curve means the conversion from ð0; θP1Þto ðδ;ψÞ by the PSG. The input beam of an SLMshould be circularly polarized for the uniform trans-mittance through the PSG since the first componentof the PSG is a linear polarizer. Any elliptic polariza-tion state can be generated with two variable anglesθP1 and θW1 satisfying

sin 2ðθW1 − θP1Þ ¼ sin 2ψ sin δ; ð14aÞ

tan 2θW1 ¼ tan 2ψ cos δ: ð14bÞ

Fig. 4. Input and output negative eigenstates Eλð−Þ on thePoincaré sphere, and their traces according to the applied voltage(a) without and (b) with the edge effect.

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Similarly, the relative phase and azimuth angle ofthe elliptic polarization state in the PSD satisfy

sin 2ðθP2 − θW2Þ ¼ sin 2ψ sin δ; ð15aÞ

tan 2θW2 ¼ tan 2ψ cos δ: ð15bÞIn Fig. 5(b), the bold curves mean the polarizationstates with equal transmittances chosen by this PSD.The PSG and PSD have the fixed points on the

Poincaré sphere even though the negative eigenstateof TNLC moves according to the applied voltage.So it is impossible to select only the negative eigen-state over the range of the applied voltage. Since thepositive eigenstate is orthogonal to the negativeeigenstate in spite of the phase shift owing to theedge effect, the contributions of two eigenstates de-fined by PSG and PSD are trigonometrical functions.Therefore, the phase modulations resulting from thecombination of two eigenstates are generated and de-tected over the range of the applied voltage, and thefittest configuration of PSG and PSD is searched forthe desired SLM.

3. Genetic Optimization Algorithm

In this section, the proposed genetic optimizationtechnique is described. The chromosome and proce-dure of the genetic algorithm are detailed, and thestructure of the program is described.

Figure 6 shows the schematic of the system opti-mizing the SLM with a genetic algorithm. First,the collimated beam is expanded and circularly po-larized through the quarter wave plate. Next, thecircularly polarized beam enters a PSG and is con-verted to a designed elliptic polarization state. Next,this beam passes the mask with double slits and anaperture before the TNLC. The double slits functionas a wave-front interferometer to measure phasemodulations, and a larger aperture exists to measureamplitude modulations. Next, the beam enters theTNLC for a light modulation. As previously dis-cussed, the elliptic polarization state generated bythe PSG travels on the Poincaré sphere accordingto applied voltage. Next, the modulated beam passesthrough the PSD and is converted to a linear polari-zation state. Last, beams through the double slitsand the aperture on the mask are collected by thelenses at the front of a charge-coupled device (CCD)and an optical power meter (OPM), respectively. Theloop of genetic algorithm is constructed betweenmeasurement and modulation through a computer.The cost value of a genetic algorithm is evaluatedfrom data of the CCD and the OPMunder a full rangeof applied voltage. The rotation stages are controlledand the encoding index of the TNLC is changed bythe computer.

In general, a genetic algorithm has two operationsto search local optima. The crossover operation iseffective in convex optimization problem, since it ismathematically analogous to the linear combina-tion of two chromosomes in a floating point codingscheme. The mutation operation is effective in a gen-eral optimization problem since it makes the smallvariations of the present solution and searches theoptimum solution randomly. Therefore the mutationoperation is more essential in the genetic algorithm,and we employ only this operation in our application.

In this genetic algorithm, a chromosome is com-posed of four configuration parameters of the SLM.

Fig. 5. Elliptic polarization states of (a) PSG and (b) PSD on thePoincaré sphere.

Fig. 6. Schematic of the system optimizing the SLM with geneticalgorithm.

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The angles of polarizer θP1 and the angles of waveplate θW1 define the PSG, and the angles of polarizerθP2 and the angles of wave plate θW2 define the PSD.These angles have the freedom from 0 to π. Hence,the cost function F is parameterized by the chromo-some, and the optimization problem is defined as

max F½θP1; θW1; θP2; θW2� for 0 ≤ θP1; θW1; θP2;and θW2 ≤ π: ð16Þ

Here the formula of F is completely described in thenext section.Figure 7 shows the flow chart of the genetic algo-

rithm implemented with Labview and Matlab. Inthis flow, Labview controls the motorized stageschanging the angles of optics and measures the am-plitude and phase modulations, andMatlab managesthe genetic algorithm and computes the cost func-tion. These two programs are able to be connectedwith the Math script in Labview, and Matlab isinserted in Labview as the object. Even though themain stream of the genetic algorithm is pro-grammed with Matlab, the flow is started and endedby Labview. The system parameters and measureddata are transferred to each other as variables.Therefore, we programmed that Matlab memorizesthe whole data set for generations and that Labviewcalls demanded current variables.In Fig. 7, the flow follows a general genetic algo-

rithm with the mutation operator. First, the initialpopulation P0 with a population size m is selected.In this paper, we set the population size to be five,

and this population is composed of chromosome xihaving four elements in accordance with the rotationangles. Next, the cost values FðPk¼0Þ of chromosomesin the initial population are calculated, and the fit-test chromosome having the largest cost value, i.e.,the elite �x, is determined. Next, the mutation opera-tion is applied to the initial population Pk¼0, andthen the modified population �P0k¼0 is obtained. Inthe mutation operator for floating point coding, themutation probability and the system parameter de-termining the degree of nonuniformity are tuned toabout 0.5 and 1.5. Next, the cost values Fð�P0k¼0Þ ofchromosomes in the mutated population are calcu-lated, and the fittest chromosome x̂ is determined.Next, the new fittest chromosome ~x is selected from�x and x̂. Last, the population is reorganized, and thenew fittest chromosome copies itself by clone size htimes. These sequences are repeated until the kmaxthgeneration.

The main virtual instrument (VI) has three sub-VIs, and each sub-VI calls the functions in dataacquisition (DAQ) and image acquisition (IMAQ) li-braries. The first sub-VI measures the amplitudemodulations of an SLM at 17 encoding indices, thevalues of which are spaced at an interval of 24.The number of encoding indices is determined toobtain a curve of amplitude modulations efficientlyand to avoid the failure to notice singular points inthe curve. This sub-VI is composed of IMAQ func-tions to encode the SLM and a general purpose inter-face bus (GPIB) function to communicate with theOPM. The second sub-VI measures the phase modu-lations of an SLM at 9 encoding indices, the valuesof which are spaced at an interval of 25. The trans-mission is determined by polarizations, and theamplitude modulation could decrease to zero. Onthe other hand, the phase modulation is determinedby the refractive index of liquid crystal molecules.Since the liquid crystal molecule rotates to the posi-tion where the elastic distortion force is equivalent tothe electric force, generally there is no abrupt changein the liquid crystal molecule according to the ap-plied voltage. Therefore, the phasemodulation variesslowly according to an encoding index, and, to obtaina curve of phasemodulations, a relatively small num-ber of measurements is required. This sub-VIs arecomposed of IMAQ functions to encode the SLMand communicate with a CCD through an IMAQboard. To raise the communication speed with theCCD, the size of the region of interest is reducedto 200 × 5 pixels. The third sub-VI drives the motor-ized rotation stages with a full-step resolution. TheDAQ board is used to control the step motor driver.In the event, the whole process of a single generationtakes about 25 seconds.

4. Experimental Embodiment of the OptimizationSystem

In this section, the system embodying the proposedalgorithm is described and the experimental resultsare discussed. The technique for measuring the

Fig. 7. Flow chart of the genetic algorithm implemented withLabview and Matlab.

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phasemodulation is detailed, and a devised economicmotorized stage is explained. The cost function is for-mulated for maximizing the phase modulation, andthe evolution of the SLM is discussed.In this paper, the wave-front interferometer is

used for measuring the phase modulation. The inter-ferometer splits the incident wave front with doubleslits and observes the interference resulting fromdifferent path lengths in the same optical arm.The beam through one slit functions as a referenceand is modulated by the SLM with a fixed encodingindex. The beam through the other slit functions as asignal and is modulated by the SLM with designedencoding index. In this interferogram, the fringemoves according to the phase shift of the signalbeam. When the incident beam with a wavelengthλ passes through the double slits with a gap Δ andis collected by a lens with a focal length f , an inter-ference term of this interferogram on the CCD withpixel pitch pCCD can be represented as

IintðpjÞ ¼ Iint;0 cosð2πpj=ΛþMPhaseÞ: ð17Þ

Here,Λ is the period of a fringe, and pj is the positionon the CCD. The phase modulation MPhase of the sig-nal beam appears in the argument of a cosine func-tion. In the paraxial approximation, Λ is given by

Λ ¼ f λ=Δ; ð18Þ

and pj is defined by

pj ¼ pCCD × ðj − 1Þ: ð19Þ

The phase shift can be simply calculated from theamount of a fringe shift over the period. However,we calculate the phase shift by a discrete Fouriertransform and increase the accuracy. The discreteFourier transform of IintðpjÞ is defined by

XðkÞ ¼XNj¼1

Iint;0 cosð2πpj=ΛþMPhaseÞωðj−1Þðk−1ÞN ; ð20Þ

whereN is the side length of the region of interest onthe CCD over the pixel pitch, and ωN is anNth root ofunity given by

ωN ¼ expð−2πi=NÞ: ð21Þ

The phase modulation MPhase is calculated from

MPhase ¼ unwrapfangle½XðpCCDN=Λþ 1Þ�g: ð22Þ

In this interferometer, the gapΔ of the double slits isdesigned to satisfy

Δ ¼ 4f λ=pCCDN: ð23Þ

The calculation using a discrete Fourier transformfilters a signal in the frequency domain and averagesthe phase modulation by 4 times.

Figure 8 shows the experimental setup. In the sys-tem embodying the proposed algorithm, the opticalparts are realized with the second harmonic Nd:YAGlaser (Coherent Verdi V-5), the CCD (KODAK Mega-Plus ES1.0/MV) with 9 μm pixel pitch, and the OPM(Newport 1835-C). The mechanical parts are actua-lized with motorized rotation stages and motor dri-vers. This motorized stage is driven by a two-phasestep motor with a 1=5 inch pitch timing belt. Theteeth of a timing pulley in an optical mount are twicethose in a step motor, and two revolutions of themotor shaft are equal to one revolution of the optics.

In this paper, the cost function is designed to attainthe maximum phase modulation of the SLM. In thegenetic algorithm, chromosomes evolve to let the costvalue down, and the cost function is formulated as

FðxiÞ ¼ minðMPhase;iÞ −maxðMPhase;iÞ: ð24Þ

Practically we have to exclude chromosomes havingtoo low amplitude transmission MAmpð16j − 1Þ with(16j − 1)th encoding index and set two exclusion con-ditions given by

X16j¼0

½MAmpð16j − 1Þ�2 ≥ 10−5 ðwattÞ; ð25aÞ

4P

16j¼0½MAmpð16j − 1Þ�2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP

16j¼0f½MAmpð16j − 1Þ�2 −MAmp2g2

q > 20: ð25bÞ

Here,MAmp2 is the average of transmittances definedby

MAmp2 ¼X16j¼0

½MAmpð16j − 1Þ�2=17: ð26Þ

Fig. 8. Picture of the experimental setup.

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By Eq. (25b), chromosomes with singular points inthe amplitude transmission curve cannot survive.We use TNLC made by SONY whose model num-

ber is LCX016AL-6. In this TNLC the contrast andbrightness are set at 198 and 102, respectively,with the program Holoeye LC2002. Figure 9 showscharacteristics of SLM during evolution and atstagnated state. Figures 9(a) and 9(b) show the evo-lution of phase modulation and amplitude trans-mission curves, respectively, for 100 generations.At stagnated state, the phase modulation and ampli-tude modulation are shown in Figs. 9(c) and 9(d),and the optimized configuration parameters of theSLM are listed in Table 1. In Fig. 9(a) the values ofthe phase modulation for generations are set to bezero at zeroth encoding index and the phase modula-tion increases monotonically. Therefore the phasemodulations at the 255th encoding index are themaximum values for generations, and it is noticeablefor the range of phase modulation to grow duringthe evolution. At stagnated state, consequently therange of phase modulations in the SLM reaches6:05 ðradÞ. As expected, no significant singular pointsare shown in the amplitude transmission curve atstagnated state, and the numerical value of the sec-ond exclusion condition Eq. (25b) is 41.7. With theproposed genetic system the SLM using TNLC is op-timized to attain the desirable phase and amplitudemodulations.Figure 10 shows the diffraction images using the

SLM. The digital hologram encoded on the SLMis generated by the iterative Fourier transformalgorithm considering the functional relationshipbetween phase and amplitude modulations [19].In Fig. 10(a), the diffraction image is captured with

the configurations at the first generation. The initialconfiguration uses classical linear polarized inputand output states that are aligned parallel to eachother. The SLM with the initial configuration hasonly 2:99 ðradÞ phase modulation, and the twin im-age appears definitely. In Fig. 10(b), the diffractionimage is captured with the configurations at the

Fig. 9. (Color online) Characteristics of SLM during evolutionand at stagnated state: (a) phase modulation and (b) amplitudetransmission during evolution, and (c) phase modulation and(d) amplitude transmission at stagnated state.

Fig. 10. Diffraction images using the SLM (a) at the first genera-tion and (b) at the stagnated state after the genetic optimization.

Table 1. Optimized Configuration Parametersof the Spatial Light Modulator

Configuration ParametersRotation Angle

(degree)

Input polarizer (θP1) 14.4Input wave plate (θW1) 76.5Output wave plate (θW2) 30.6Output polarizer (θP2) 119.7

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stagnated state, and twin image is hard to distin-guish. The existence of a twin image could be evalu-ated by the ratio of twin signal to original signal. Theregions of twin and original signals are defined at thegeneration process by the iterative Fourier transformalgorithm, and the ratio is calculated from the aver-aged intensities in each region. By this calculationwe evaluate the ratios are 0.3271 and 0.0575 beforeand after the optimization, respectively, and it re-flects that the troublesome twin image is reducedeffectively after the genetic optimization.

5. Conclusion

In conclusion, it is shown that the genetic optimiza-tion technique is efficient to determine the configura-tion of an SLM. We represent the optical componentsof an SLM on the Poincaré sphere, and we show thateigenstates of TNLC do not have the same inclina-tion angle according to the applied voltage sincethe edge effect of TNLC shifts eigenstates on thePoincaré sphere. Therefore we show that four para-meters of the polarization state generator and detec-tor are needed to define the characteristics of theSLM. In this genetic optimization system the phaseand amplitude modulation are simultaneously mea-sured according to applied voltages, and the phaseshift is calculated using a discrete Fourier transform.It is expected that the proposed technique can be use-ful in optimizing the SLMusing various kinds of com-mercial TNLCs to attain the desirable phase andamplitude modulation.

This work was supported by Samsung SDI, Ltd.

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