Implementation of phase-only modulation utilizing a twisted nematic liquid crystal spatial light

modulator

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2008 J. Opt. A: Pure Appl. Opt. 10 085007

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IOP PUBLISHING JOURNAL OF OPTICS A: PURE AND APPLIED OPTICS

J. Opt. A: Pure Appl. Opt. 10 (2008) 085007 (7pp) doi:10.1088/1464-4258/10/8/085007

Implementation of phase-only modulationutilizing a twisted nematic liquid crystalspatial light modulatorMohammad Sultan Mahmud1,3, Izabela Naydenova1 andVincent Toal1,2

1 Centre for Industrial and Engineering Optics, Dublin Institute of Technology,Dublin 8, Republic of Ireland2 School of Physics, Faculty of Science, Dublin Institute of Technology, Dublin 8,Republic of Ireland

E-mail: mohammad.mahmud@dit.ie

Received 27 June 2008, accepted for publication 17 July 2008Published 5 August 2008Online at stacks.iop.org/JOptA/10/085007

AbstractTwisted nematic liquid crystal spatial light modulators (SLM) are widely used for amplitudemodulation of light. The technique for measuring phase and amplitude modulation introducedby a twisted nematic cell is not new but the novelty here is that, after characterization of aHoloeye LC2002 TNLC device, we were able to identify specific conditions that allow for itsuse as a pure 180◦ or more phase modulator with little or no amplitude modulation at 514.5 nmlaser wavelength. The phase shift introduced by the SLM was measured using a nearcommon-path Mach–Zehnder interferometer. The experimentally measured and thetheoretically predicted intensity transmission and phase shift as functions of gray levels arecompared. Finally, the effects of errors in the angular settings of the polarizers and quarterwaveplates on the intensity transmission and phase shift are studied.

Keywords: spatial light modulator, phase shifts due to bulk and edges birefringences,amplitude modulation, phase modulation, transmission and phase shift error

1. Introduction

The pure phase mode of operation of an SLM can be achievedby careful study of its physical properties—twist angle (α),input director orientation, maximum (off-state) phase shift andboth voltage-dependent phase shifts due to bulk and edgebirefringences (β and δ) [1–6]. The intensity transmittance (T )for a polarizer–SLM–analyzer system when the input directoris parallel to one of the coordinate axes is expressed as [5, 6]

T = [X1cos(ϕ1−ϕ2+α)+ Z sin(ϕ1−ϕ2+α)]2

+ [Y1cos(ϕ1+ϕ2−α)]2 (1)

where

X1 = cos γ cos 2δ − β

γsin γ sin 2δ;

3 Author to whom any correspondence should be addressed.

Y1 = cos γ cos 2δ + β

γsin γ cos 2δ;

Z = α

γsin γ, γ =

√α2+β2.

Here, ϕ1 and ϕ2 are the polarizer and analyzer angles,respectively, measured with respect to the input directorof the SLM. Here, all angles are taken as positive whenmeasured clockwise looking from the detector side and thetransmissions are normalized in order to eliminate the intensityfluctuations [7]. By using the Solver fitting utility of Excel atthree experimental configurations (ϕ1 = 0, ϕ2 = α + 90◦),(ϕ1 = 45◦, ϕ2 = α− 45◦), (ϕ1 = 22.5◦, ϕ2 = α+ 112.5◦) andat four wavelengths the values of β(V ) and δ(V ) as functionsof the gray level [5, 6] were obtained. It was then possibleto search for any desired modulation behavior by introducingexternal polarization components. The most general way toobtain any desired amplitude or phase modulation behavior is

1464-4258/08/085007+07$30.00 © 2008 IOP Publishing Ltd Printed in the UK1

J. Opt. A: Pure Appl. Opt. 10 (2008) 085007 M S Mahmud et al

to use a polarizer–QWP–SLM–QWP–analyzer set-up [6, 8].This set-up was used to generate a desired elliptically polarizedeigenvector whose ellipticity changes as a function of thebirefringence of the LC which in turn is proportional to theapplied voltage [9]. The intensity transmittance and total phaseshift for such a system are given by [6]

T = A2RE + A2

IM (2)

ψ = −β − 2δ + tan−1

(AIM

ARE

)(3)

where ARE and AIM are the real and imaginary parts of thewhole system with

ARE = X1{cos2 �cos(ϕ1 − ϕ2 + α)− sin2�

× cos[2(η1 − η2)− ϕ1 + ϕ2 + α]}− Y1cos�sin�{cos[2η2 + ϕ1 − ϕ2 − α]+ cos[2η1 − ϕ1 + ϕ2 − α]}+ Z1{cos2�sin(ϕ1 − ϕ2 + α] − sin2 �

× sin[2(η1 − η2)− ϕ1 + ϕ2 + α]}. (4A)

and

AIM = −X1cos�sin�{cos(2η2 − ϕ1 − ϕ2 − α)

+ cos[2η1 − ϕ1 − ϕ2 + α]}+ Y1{− cos2�cos(ϕ1 + ϕ2 − α)+ sin2 �

× cos[2(η1 + η2)− ϕ1 − ϕ2 − α]}+ Z1cos�sin�{sin[2η2 − ϕ1 − ϕ2 − α]− sin[2η1 − ϕ1 − ϕ2 + α]}. (4B)

Here η1 and η2 are the orientations of the slow axes of thefirst and second QWPs. The value of � is 45◦, fixed by theQWPs at the desired 514.5 nm wavelength.

In order to evaluate the influence of the misalignmentof the angular positions of the polarizing components onthe output light properties (transmitted intensity and phase)equations (2) and (3) must be analyzed by differentiation.Transmission and phase errors (T,ψ) due to small angularmisalignments of polarizer (ϕ1), analyzer (ϕ2) and first andsecond QWPs (η1,η1) are given by:

Tϕ1 = ∂T

∂ϕ1ϕ1; Tϕ2 = ∂T

∂ϕ2ϕ2;

Tη1 = ∂T

∂η1η1; Tη2 = ∂T

∂η2η2

(5A)

and

ψϕ1 = ∂ψ

∂ϕ1ϕ1; ψϕ2 = ∂ψ

∂ϕ2ϕ2;

ψη1 = ∂ψ

∂η1η1; ψη2 = ∂ψ

∂η2η2

(5B)

where

∂T

∂ϕ1= 2

[ARE

∂ARE

∂ϕ1+ AIM

∂AIM

∂ϕ1

];

∂T

∂ϕ2= 2

[ARE

∂ARE

∂ϕ2+ AIM

∂AIM

∂ϕ2

]

∂T

∂η1= 2

[ARE

∂ARE

∂η1+ AIM

∂AIM

∂η1

];

∂T

∂η2= 2

[ARE

∂ARE

∂η2+ AIM

∂AIM

∂η2

]

∂ψ

∂ϕ1=

(ARE

∂ AIM∂ϕ1

− AIM∂ ARE∂ϕ1

)

A2RE + A2

IM

; (5C)

∂ψ

∂ϕ2=

(ARE

∂ AIM∂ϕ2

− AIM∂ ARE∂ϕ2

)

A2RE + A2

IM

∂ψ

∂η1=

(ARE

∂ AIM∂η1

− AIM∂ ARE∂η1

)

A2RE + A2

IM

; and

∂T

∂η2=

(ARE

∂ AIM∂η2

− AIM∂ ARE∂η2

)

A2RE + A2

IM

.

The theoretical values for T and ψ can be calculatedby calculation of the partial differentials (5A) and (5B) forspecific values of the misalignments in the angular positions ofeach element—ϕ1,ϕ2,η1 andη2. For practical reasonsthis deviation should not exceed ±2◦.

The total transmission and phase errors (δT, δψ) due toerrors in the positioning of all the elements in the set-up can befound as

δT ={(∣∣∣∣

∂T

∂ϕ1

∣∣∣∣ϕ1

)2

+(∣∣∣∣∂T

∂ϕ2

∣∣∣∣ϕ2

)2

+(∣∣∣∣∂T

∂η1

∣∣∣∣η1

)2

+(∣∣∣∣∂T

∂η2

∣∣∣∣η2

)2}−1/2

(6A)

and

δψ ={(∣∣∣∣

∂ψ

∂ϕ1

∣∣∣∣ϕ1

)2

+(∣∣∣∣∂ψ

∂ϕ2

∣∣∣∣ϕ2

)2

+(∣∣∣∣∂ψ

∂η1

∣∣∣∣η1

)2

+(∣∣∣∣∂ψ

∂η2

∣∣∣∣η2

)2}−1/2

. (6B)

2. Experimental methods for measuring phase shiftand intensity modulation

The Mach–Zehnder interferometer is commonly used formeasuring the phase shift introduced after transmissionthrough an SLM [10–14]. Zhang [15] and McClain [16] useddiffraction techniques for measuring the phase shift, which arenoninterferometric but have the disadvantage of being derivedfrom intensity measurements, as higher measurement accuracydepends upon camera resolution and quantization by the imageboard. Marquez [6] used diffraction gratings for measuring thephase shift experimentally, which required very precise Ronchicomponents, in order to obtain the required equal intensities inthe first orders. We used a near-common-path Mach–Zehnderinterferometer (figure 1) to measure phase shift because thistechnique is less sensitive to temperature, vibration, and issimple but cost-effective as the SLM is effectively split into two

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J. Opt. A: Pure Appl. Opt. 10 (2008) 085007 M S Mahmud et al

Figure 1. Experimental set-up for measuring phase shift. Here, NDF: neutral density filter, SF: spatial filter, P: polarizer, WP: quarter-waveplate, A: analyzer, SLM: spatial light modulator, M: mirror, BS: beamsplitter, L: lens.

Figure 2. Phase shifts due to (a) bulk birefringence, β(V ), and (b) edge birefringences, δ(V ), at four different wavelengths: (+) 514.5, (•)496.5, (�) 488 and ( ) 458 nm.

parts. An ordinary CCD camera or a webcam could be usedto capture changes in the fringe pattern for measuring phaseshift. A laser beam was expanded and collimated to illuminatethe SLM through a double-hole mask in front of the SLM,separating it into two distinct parts; one part is addressed witha constant gray level of 0, whereas the other part is addressedwith gray levels from 0 to 255. The holes had to be identical toobtain the best contrast in the fringe pattern.

It is a usual practice to use a 4f system [17] to enable thebeams to interfere with each other (for example, Holoeye usedthis technique) but in our experiment an ordinary beamsplitterbehind the SLM was used to overlap the two beams (figure 1).A lens (L) was used to expand the interference pattern onto awebcam (Pro 5000, Logitech). The phase shift appeared asa shift in the interference pattern and was evaluated by the‘Phase cam’ software [17]. A program written in LabView wasused to change the gray level of the SLM in order to measurethe intensity modulation. Intensity modulation was measuredby blocking one of the beams and replacing the webcam witha photodetector and the whole display was addressed with ahomogeneous gray level.

3. Results

3.1. Characterization of the basic properties of the SLM

By using the Solver utility of the Excel module at threeexperimental configurations (ϕ1 = 0, ϕ2 = α + 90◦), (ϕ1 =45◦, ϕ2 = α − 45◦), (ϕ1 = 22.5◦, ϕ2 = α + 112.5◦) and atfour wavelengths, the values of β(V ) and δ(V ) as functions of

gray level were found [5, 6] (figure 2). Once β(V ) and δ(V )were obtained, we used the polarizer–QWP1–SLM–QWP2–analyzer set-up to get the desired phase-only or amplitudemodulation.

3.2. Polarizer–QWP–SLM–analyzer set-up

In this study we used a polarizer–QWP1–SLM–analyzer set-up to get the desired phase-only or amplitude modulation. Tosimplify the set-up, three parameters (ϕ1, η1 and ϕ2) were takenas free parameters for Solver searching and a maximum of 182◦phase shift was found with nearly flat intensity modulation overthe entire gray level range. The required orientation angles ofthe polarizer, analyzer and QWP1 were found to be −14.24◦,−33.42◦ and 6.84◦, respectively, with respect to the inputdirector of the SLM. The theoretical intensity modulation wasalmost flat, having a variation of 6% over the entire range ofgray levels.

From figure 3(a) it is seen that the experimental intensitymodulation is slightly different from the theoretical results.To investigate the influences of angular misalignment of theoptical components, the analyzer, polarizer and quarter-waveplate angular positions were deliberately changed by up to±2◦ from their theoretically determined values. Measurementswere carried out when two of the three polarizing elementswere kept constant at their theoretically predicted values usingthe Solver method and only one of them was misaligned by±2◦ in order to analyze misalignment errors. The experimentalresults showed that significant changes in the transmission

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J. Opt. A: Pure Appl. Opt. 10 (2008) 085007 M S Mahmud et al

Figure 3. Theoretical (•) and experimental ( ) transmission (a) and phase shift (b) at 514.5 nm for the P–QWP1–SLM–A set-up.

Figure 4. Experimental intensity transmission due to ±2◦ misalignment of angles of (a) analyzer, (b) polarizer and (c) QWP and (d) theircorresponding phase shifts (PS) for the P–QWP–SLM–A set-up. Here the angles of the polarizer, QWP and analyzer are IM1(−14.24◦, 6.84◦,−33.42◦), IM2(−14.24◦, 6.84◦, −35.42◦), IM3(−14.24◦, 6.84◦, −31.42◦), IM4(−16.24◦, 6.84◦, −33.42◦), IM5(−12.24◦, 6.84◦, −33.42◦),IM6(−14.24◦, 4.84◦, −33.42◦) and IM7(−14.24◦, 8.84◦, −33.42◦), respectively.

occurred even with small changes of these angles (figure 4).The influence of the misalignment in the position of theanalyzer is shown in figure 4(a). It is seen that deviation ofthe analyzer’s angle by ±2◦ from the theoretically predictedvalue of −33.42◦ would lead to more than 10% deviation inthe transmission, especially at low gray levels. Similar resultswere observed for the misalignment in the polarizer’s position(figure 4(b)) and the QWP1 position (figure 4(c)). Therefore,

the slight disagreement between the theoretically predicted andthe experimentally measured values for the SLM transmission(figure 3(a)) could be due to small errors in the misalignmentof the optical components. At the same time, measurements ofthe phase of the light transmitted through the SLM when theangular positions of the polarizing elements are varied within±2◦ from the theoretically predicted values show that the phasevaries 1% from the expected value.

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J. Opt. A: Pure Appl. Opt. 10 (2008) 085007 M S Mahmud et al

Figure 5. Theoretical ( ) and experimental ( ) transmission (a) and phase shift (b) at 514.5 nm for the P–QWP1–SLM–QWP2–A set-up.

Figure 6. Theoretically calculated transmission error,(T ) (a) and phase error (ψ) (b) due to ±2◦ misalignment of each optical componentfor the P–QWP–SLM–QWP–A set-up. Each graph represents the misalignment of one component: analyzer (squares), polarizer (triangles),QWP1 (circles) and QWP2 (crosses) while the positions of the other polarizing components remain unchanged and their angles are equal tothose theoretically predicted by the Solver method which are 144◦ for the analyzer, 3.3◦ for the polarizer, −48.25◦ for QWP1 and −88.78◦ forQWP2.

3.3. Polarizer–QWP1–SLM–QWP2–analyzer set-up

The best match for constant intensity modulation and morethan 180◦ phase shift was found when all parameters (ϕ1,η1, ϕ2 and η2) were taken as free parameters during computersearching and the values of ϕ1, η1ϕ2 and η2 were foundto be 144◦, 3.3◦, −48.25◦ and −88.78◦, respectively, fromthe input director looking from the detector side. Thetheoretical and experimental intensity transmission and phaseshift agree with each other (figure 5) and a phase shift upto 224◦ (figure 5(b)) was achieved at a laser wavelengthof 514.5 nm. One conclusion from this result is that purephase modulation of 180◦ could be achieved at even longerwavelengths using this SLM. For these angular settings ofthe polarizing components the theoretically predicted intensityvaries by 3% over the entire range of gray levels. Theexperimentally measured variation was within 3%, which isin good agreement with the theoretically predicted dependenceon the gray level (figure 5(a)).

3.4. Theoretical study of the transmission and phase shifterrors due to misalignment of the polarizing components

To investigate transmission and phase shift error (T andψ)due to the influences of angular misalignment of polarizersand both QWPs, their angular positions were deliberatelychanged ±2◦ from their theoretical values. These errors wereanalyzed using equation (5C), where ϕ1, ϕ2,η1 andη2 are the misalignment errors. The results showed that,when both QWPs and polarizers were taken as free parametersduring Solver searching, ±2◦ changes in the angle do notaffect transmission significantly. The maximum variation intransmittance is 2.5% (figure 6(a)) and in phase shift themaximum variation is 3◦ (figure 6(b)), whereas when one QWPand polarizer angles were taken as free parameters then themaximum variation of transmittance is 12% (figure 7(a)) andthe phase shift variance is 5◦ (figure 7(b)). Therefore the P–QWP1–SLM–QWP2–A set-up is more stable against smallerrors in the positions by the polarizing components.

In order to estimate the total transmission and phaseerror (δT, δψ) due to simultaneous misalignment of 2◦ of

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J. Opt. A: Pure Appl. Opt. 10 (2008) 085007 M S Mahmud et al

Figure 7. Theoretically calculated transmission error (T ) (a) and phase error (ψ) (b) due to ±2◦ misalignment of each optical componentfor the P–QWP1–SLM–A set-up. Each graph represents the misalignment of one component: analyzer (squares), polarizer (triangles) andQWP1 (circles), while the positions of the other polarizing components remain unchanged and their angles are equal to those theoreticallypredicted by the Solver method, which are −14.24◦ for the analyzer, −33.42◦, for the polarizer and 6.84◦ for QWP1.

Figure 8. Theoretically calculated total error in transmittance (a) and in phase (b) due to 2◦ misalignment of all polarizing components for theP–QWP1–SLM–QWP2–A system ( ) and the P–QWP1–SLM–A system ( ).

all polarizing components, equations (6A) and (6B) wereused for the P–QWP1–SLM–A and P–QWP1–SLM–QWP2–Asystems. The results for the total error in transmission (δT ) andphase (δψ) are shown in figures 8(a) and (b), respectively. It isseen from these two figures that the P–QWP1–SLM–QWP2–Aset-up is characterized by a smaller error in transmission andphase than the P–QWP1–SLM–A set-up.

4. Conclusion

It is found that the LC2002 SLM can be used for implementing180◦ phase-only modulation for phase coded reference beamholographic optical data storage. A near common-path Mach–Zehnder interferometer was used to measure the phase shift.It is shown experimentally that the transmissivity is moresensitive to the precise orientations of the optical componentswhen the optical system contains one QWP than in the casewhen two QWPs are utilized in the system. Variation of±2◦ in the positions of the optical components in the systemcontaining one QWP lead to a maximum variation at 12% in

transmission and 6◦ in phase shift variation, whereas the sameangular errors in the system containing two QWPs leads to amaximum of 2.5% variation in transmission and 3◦ in phase.

Acknowledgments

The Dublin Institute of Technology (DIT) provided financialsupport for this project. The authors thank Professor I Moreno(Universidad Miguel Hemandez), Professor A Marquez(Universidad de Alicante), Dr R Jallapuram (DIT), S Osten(Holoeye) and Dr T Babeva (DIT).

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