1228 OPTICS LETTERS / Vol. 19, No. 16 / August 15, 1994

High-speed analog complex-amplitudeliquid-crystal light modulator

Gary D. Sharp and Kristina M. Johnson

Department of Electrical and Computer Engineering, Optoelectronic Computing Systems Center,University of Colorado at Boulder, Boulder, Colorado 80309-0425

Received March 3, 1994

A homogeneously aligned smectic A* liquid-crystal quarter-wave retarder is used to produce analog complex-amplitude modulation of light through the electroclinic effect. The device consists of a reflection-mode polariza-tion rotative phase shifter enhanced by an asymmetric resonator. With an entrance mirror reflectivity of 70%,an analog modulation of 65% of a fringe was observed.measured to be 25 Aus at 25'C.

High-speed analog phase-only and phase-amplitudespatial light modulators are currently required forcoherent optical processing, adaptive optics, pro-grammable diffractive optical elements, and agilebeam directors. Nematic liquid crystals have beenthe materials of choice for many of these applica-tions because of the requirement for high modulatordensity, large effective change in refractive indexwith a small interaction length, low voltage, and low-power dissipation. Homogeneously aligned nematicdevices exhibit analog variable retardation as a resultof rotation of anisotropic molecules in a plane con-taining the device normal. Phase-only modulationis achieved by extraordinary polarization of the inputlight. A current limitation of nematic liquid-crystaldevices is the slow response time [1-10 ms (Ref. 1)].

The microsecond switching speed of chiral smec-tic liquid crystals is a potential solution to thecurrent material-limited device reconfigurationrate. Unlike nematic liquid-crystal modulators,chiral smectic liquid-crystal devices are usuallyaligned such that an electro-optic rotation of themolecular directors occurs about the device nor-mal. Furthermore, the common surface-stabilizedferroelectric liquid-crystal device, fabricated withsmectic C* mixtures, exhibits bistable switching.These characteristics relegate the surface-stabilizedferroelectric liquid-crystal device to the two-levelbinary phase.

The electroclinic effect in the smectic A* phase ischaracterized by a molecular tilt in linear proportionto the electric-field amplitude. The response of ahomogeneously aligned smectic A* device has beenmeasured to be 300 ns with a tilt angle of 100,2 withtilt angles exceeding 22' with microsecond responsetimes. To achieve analog variable retardation with arotative element, one must use a compound retarder.

Compound rotative variable retarders were used asphase shifters in the first active phased-array radar3

and have been discussed by several researchers foruse at optical frequenciesk4 However, the realiza-tion that a rotatable half-wave retarder bounded byparallel quarter-wave retarders is identical to a vari-able retarder was reportedly known to Fresnel.7

The 10-90% response time of the liquid crystal was

A phase shifter is formed when the input quarter-wave retarder functions as a circular polarizer. Thecentral half-wave retarder changes the instanta-neous state of polarization (and handedness), produc-ing an orientation-dependent absolute phase. Thesecond quarter-wave plate restores the linear po-larization state. The transformation, often calledthe Pancharatnam phase, is represented by a closedcircuit on the Poincar6 sphere in which the phasechange is twice the rotation angle of the half-waveplate.8 Previously, a two-level binary phase modu-lator of this type was demonstrated with a surface-stabilized smectic C* device.9

In this Letter we present preliminary results foran analog phase shifter, using a homogeneouslyaligned smectic A* device. The compound liquid-crystal orientational variable retarder (CLOVAR) isimplemented in reflection and uses only a zero-orderliquid-crystal quarter-wave retarder preceded by acircular polarizer. Because the induced phase istwice the reorientation angle of the half-wave re-tarder, the modulation is substantially less than the2v required for many applications (a maximum phasechange of vr/2 is currently feasible). An increase inphase shift can be achieved by cascading half-waveretarders with opposing tilts3 or by multiple interac-tions with the same modulator. In this research wetake the latter approach for reasons of compactnessand for the benefits associated with a single activelayer. An asymmetric Gires-Turnois resonator10

centered at resonance is utilized to achieve a phaseenhancement.

We implement the resonated CLOVAR device,shown in Fig. 1, by placing a partial reflector at theentrance of a reflection-mode phase shifter. Sincea modest phase enhancement is required, the in-put partial reflector need not be highly reflective(40-70%). The following Jones calculus analysisgives the complex reflection coefficient of the res-onated CLOVAR device as a function of the designparameters.

Consider a monochromatic plane wave incidentnormally upon the structure shown in Fig. 1, con-sisting of plane-parallel mirrors bounding a sequence

0146-9592/94/161228-03$6.00/0 © 1994 Optical Society of America

August 15, 1994 / Vol. 19, No. 16 / OPTICS LETTERS 1229

Jones matrixfields,

relating the input and output optical

exp(ixy) E0°

where

Smectic A' Quarter-WaveRetarder

Quarter-WaveRetarder

Front Mirror

Fig. 1. Diagram of the resonated CLOVAR device.

of dielectric slabs. A sum of partial waves providesthe Jones vector for the reflected field amplitude, Er,in terms of Jones vector for the incident field ampli-tude, Eo:

Er = [r' + t-Q-r+Q'[I - rLQ-1r'Q]-t+]Eo, (1)

where the complex field reflection and transmissioncoefficients for the partial reflector are ri and ti,respectively, the back mirror reflection coefficient isr2 , and I is the 2 X 2 identity matrix. The positive(negative) superscript corresponds to a Jones matrixfor light propagating along the positive (negative) zaxis of Fig. 1. The Jones matrix, Q, is the productof Jones matrices describing the sequence of slabscontained between the 6talon mirrors.

For mirrors consisting of lossless high- or low-indexdielectric quarter-wave layers at the wavelength ofincident light,

01

1 ], r2 - [7

XX = 2 tan'1[G tan(y - a)],Xy = 2 tan-1[G tan(y + a)], (5)

and y is the single-pass absolute phase of the cav-ity. The effect of the resonator is contained in theenhancement factor G = (1 + >/X_)/(1 - R-i). Theoutput phase is therefore a nonlinear function ofthe cavity phase, determined by the molecular tiltand the single-pass absolute phase of the two re-tarders. Note that in the absence of intracavitylosses, or a partially transmissive back mirror, thematrix describes a phase-only modulation. Coupledphase-amplitude modulation would be observed fora device swept through resonance in the presence ofsignificant round-trip optical losses.

With the operating point of the modulator se-lected to be on resonance, y = rm (with m an in-teger), phase modulation enhancement occurs thatincreases with entrance mirror reflectivity. Con-sider a i-polarized monochromatic plane wave in-cident normally upon a resonated CLOVAR devicewith a resonant operating point. Figure 2 showsthe relationship between molecular orientation andinduced phase for various values of the partial mirrorreflectivity. For an ideal stable and lossless modu-lator, the selection of the front mirror reflectivity isarbitrary, based on a balance between linearity andmodulation depth for an available range in molecu-lar tilt. In practice, stability of the operating point,parallelism, mirror defects, and absorption dictatean acceptable reflectivity based on usable apertureand efficiency.

A resonated CLOVAR phase shifter was fabricatedat the University of Colorado at Boulder to demon-strate the principle. Optical flats were coated with

where the front partial reflector has a reflectivity Rand the back mirror is an ideal high reflector.

For the present modulator, the cavity contains apassive quarter-wave retarder and an electronicallyrotatable quarter-wave retarder. If we take the opticaxes of the retarders to be crossed at the zero field, sothat the structure appears isotropic at zero tilt, theJones matrices for the forward and reverse passesare given by

Q+ = W(ir/2, -7r/4 + a)W(ir/2, 7r/4),Q- = W(wr/2, -7r/4)W(vr/ 2 , 7r/4 - a), (3)

where a is the electronically controlled tilt of themolecular director with respect to the smectic layernormal and W(r, A) represents the Jones matrix for alinear retarder with retardation F and orientation 0.Substituting Eqs. (2) and (3) into Eq. (1) gives the

,r/80.0a

Fig. 2. CLOVAR device output phase (X) as a functionof the active quarter-wave retarder orientation (a) forvarious entrance mirror reflectivities (R).

*^y

/ x

Back Mirror(R=I)

Er [-exp(iXx) (4)

r+ = -r- = -,IR- -11 1 0

t+t- = (1 - R) 1 01 1 0 i ,

1230 OPTICS LETTERS / Vol. 19, No. 16 / August 15, 1994

Fig. 3. Experimentally measured fringe shift with theresonated CLOVAR device used as an arm of a Michelsoninterferometer. The upper and lower fringe sets corre-spond to bias voltages of +30 and -30 V, respectively,with a relative shift of approximately 65%.

dielectric partial reflectors (70%) and high reflec-tors (99.9%). A quartz second-order quarter-waveretarder, which is indium tin oxide coated on the in-ner surface, was optically contacted to the partial re-flector. A transparent electrode was integrated intothe high reflector in the interest of light efficiency.A rubbed polyvinyl alcohol layer was used to alignthe liquid-crystal molecules. Spacers mixed with ITVepoxy were deposited on the perimeter of one part toprovide a liquid-crystal layer thickness with a retar-dance of one quarter wave at 633 nm. The devicewas filled with the British Drug House 764E smecticA* mixture.

We observed phase modulation by using theCLOVAR device as one arm of a Michelson inter-ferometer.8 By ramping the voltage to the device,we observed an analog fringe shift with the magni-tude dependent on position relative to resonance.With the beam positioned for modulation aboutresonance, a large fringe shift of 65% (234') wasobserved. A single-pass CLOVAR device with a sim-ilar tilt would provide a phase modulation of only35', indicating the large phase enhancement. Thediameter of device aperture that provides straightfringes is roughly 1 mm, limited by mirror paral-

lelism. Figure 3 shows the output for extreme op-posite voltages (±30 V) applied to the cell, showingthe spatial shift of the fringes. With the presentdevice the response of the liquid crystal permits a25-us modulation of the fringes at room tempera-ture. For maximum modulation the material wasoperated near the C*-A* transition, with a net tiltangle of approximately 90. This low tilt was theconsequence of the voltage drop across the multi-layer reflector, which limited the field amplitudethat could be applied to the material.

Complex-amplitude light modulation based on areflection-mode CLOVAR phase shifter, resonantlyenhanced with an asymmetric Gires-Turnois cav-ity, is theoretically and experimentally verified. Asmall-aperture device (1 mm) that provided a phasemodulation of 65% of a fringe (234') was fabricated.Future work involves fabrication of large-area res-onantly enhanced devices for use as spatial lightmodulators and beam-steering devices. Attempts todecrease the cavity thickness, currently determinedby the passive retarder, are now being made. Forexample, a polymer cholesteric liquid-crystal film"is an ideal replacement for the multilayer partialreflector/quarter-wave retarder structure. A thinfilm of polymer material on a cell substrate wouldact as a partial reflector for circular polarizationwhile preserving handedness on reflection.

The authors gratefully acknowledge cell fabricationby Dave Doroski and the support of this researchby the U.S. Air Force and the NASA Johnson SpaceCenter.

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