Modeling and Design of an Optimized Liquid- Crystal Optical Phased Array

Transcription

1 Kent State University Digital Kent State University Libraries Chemical Physics Publications Department of Chemical Physics Modeling and Design of an Optimized Liquid- Crystal Optical Phased Array Xinghua Wang Kent State University - Kent Campus Bin Wang Kent State University - Kent Campus Philip J. Bos Kent State University - Kent Campus, pbos@kent.edu Paul F. McManamon John J. Pouch See next page for additional authors Follow this and additional works at: Part of the Physics Commons Recommended Citation Wang, Xinghua; Wang, Bin; Bos, Philip J.; McManamon, Paul F.; Pouch, John J.; Miranda, Felix A.; and Anderson, James E. (2005). Modeling and Design of an Optimized Liquid-Crystal Optical Phased Array. Journal of Applied Physics 98(7). doi: / Retrieved from This Article is brought to you for free and open access by the Department of Chemical Physics at Digital Kent State University Libraries. It has been accepted for inclusion in Chemical Physics Publications by an authorized administrator of Digital Kent State University Libraries. For more information, please contact digitalcommons@kent.edu.

2 Authors Xinghua Wang, Bin Wang, Philip J. Bos, Paul F. McManamon, John J. Pouch, Felix A. Miranda, and James E. Anderson This article is available at Digital Kent State University Libraries:

3 Modeling and design of an optimized liquid-crystal optical phased array Xinghua Wang, Bin Wang, and Philip J. Bos a Liquid Crystal Institute, Kent State University, Kent, Ohio Paul F. McManamon Air Force Research Lab, Dayton, Ohio JOURNAL OF APPLIED PHYSICS 98, John J. Pouch and Felix A. Miranda National Aeronautics and Space Administration (NASA) Glenn Research Center, Cleveland, Ohio James E. Anderson Hana Microdisplay Technologies, Inc., Twinsburg, Ohio Received 13 December 2004; accepted 25 August 2005; published online 10 October 2005 In this paper, the physics that determines the performance limits of a diffractive optical element based on a liquid-crystal LC optical phased array OPA is investigated by numerical modeling. The influence of the fringing electric fields, the LC material properties, and the voltage optimization process is discussed. General design issues related to the LC OPA configuration, the diffraction angle, and the diffraction efficiency are discussed. A design for a wide-angle LC OPA is proposed for high-efficiency laser beam steering. This work provides fundamental understanding for a light beam deflected by a diffractive liquid-crystal device American Institute of Physics. DOI: / I. INTRODUCTION A liquid-crystal optical phased array LC OPA is a liquid-crystal device proposed by McManamon et al. 1,2 for laser beam steering. The device has the characteristics of being simple to fabricate, nonmechanical, having low power consumption, and having a low size-weight-and-profile factor. It has many important applications such as free space laser communications, missile countermeasures, and laser radar It enables capabilities such as random-access laser beam pointing, multibeam control, and phased arrays. 12,13 As shown in Fig. 1, a LC OPA consists of two glass cover substrates. One of the substrates is coated with patterned transparent electrodes. The other substrate is coated with a uniform transparent conductor for transmissive LC OPA or a metal reflector for reflective LC OPA, which serves as the common electrode for driving the LC OPA. On top of the conductive layer of both substrates, a very thin layer of rubbed polymer is used to induce surface alignment of the liquid-crystal material. The two plates are spaced 1 20 m apart to create a cavity of uniform thickness, which is then vacuum filled with a liquid-crystal material which is aligned along the rubbing direction by the polymer layer. By applying different voltages to different electrodes, the orientation of the LC director at each electrode can be controlled. The incident light polarized along the rubbing direction of the LC OPA then experiences a different effective refractive index at each electrode across the clear aperture of the LC OPA. If the phase delay induced by the LC OPA is a linear function of position, light will be deflected to a nonzero angle. The maximum deflection angle of the LC OPA is determined by the maximum phase delay across a particular clear aperture. If we assume a continuous change in the effective extraordinary refractive index across a LC OPA, the maximum deflection angle is given by sin = nd/w, where n is the change in the refractive index across the aperture w and d is the thickness of the liquid-crystal material. However, since the birefringence of the liquid-crystal materials is limited, the thickness of the LC OPA may become impractically large if large-angle steering is required. For monochromatic light, this limitation is resolved by considering an approach where the phase profile generated by the LC OPA is a modulo 2 version of the desired phase profile. In this case, the maximum optical path difference required at any point on the array is only being the wavelength of light. The modulo 2 version of a linear phase ramp will then be a blazed grating that has m segments, and L =w/m is the a Author to whom correspondence should be addressed; pbos@lci.kent.edu FIG. 1. A liquid-crystal optical phased array /2005/98 7 /073101/8/$ , American Institute of Physics

4 Wang et al. J. Appl. Phys. 98, FIG. 2. A LC OPA with ideal director configuration. grating period of the LC OPA. The maximum steering angle of the device is now determined by sin =m /w. In this case, the total optical path difference at the phase reset nd is one wavelength. This type of blazed grating is called a first-order blazed grating. II. A LC OPA WITH AN IDEAL REFRACTIVE INDEX PROFILE To study the maximum deflection angle that can be achieved with a LC OPA, we first consider a LC OPA with an ideal LC director configuration, where the pixelation effect and the interelectrode coupling effect is neglected. From geometrical optics considerations, the ideal phase profile that gives the highest diffraction efficiency is one in which the phase delay is a linear function with respect to position, as in Eq. 1. The ideal LC director configuration that corresponds to a linear phase profile is given by Eq. 2 and is shown in Fig. 2. The coordinate system is defined such that the thickness direction for the LC OPA is along the Z axis and rubbing direction is along X axis. The angle is the tilt angle of the LC director, which is defined as the angle between the surface normal of the LC cell and the LC director. n eff x = n x L + n o, n eff x = n o n e no 2 sin 2 + n e 2 cos 2. With the finite difference time domain FDTD simulation method described previously, 14,15 the diffraction efficiency of such an ideal LC OPA can be calculated. We find that a LC OPA with first-order resets for 1550 nm wavelength has an efficiency of 70.7% for a steering angle of The diffraction angle is 40.5 for an efficiency of 50%, as shown in Table I. From such results, it is clear that an ideal LC OPA can have excellent efficiency at large diffraction angles. III. INTRODUCTION TO THE EFFECT OF FRINGING ELECTRIC FIELDS IN A LC OPA DEVICE In a real LC OPA with pixelated electrodes, the diffraction efficiency strongly depends on the phase profile of the 1 2 FIG. 3. Isopotential line in an ECB LC OPA for cell thickness d=2 m. dv=0.1 V for adjacent isopotential lines. a Electrode width=10 m and gap between electrode=0.5 m. b Electrode width=1 m and gap between electrode=0.5 m. grating, especially when the periodicity of the diffractive grating is close to the wavelength of light. The main loss is related to the nonideal phase profile at the wave-front discontinities or phase resets. The sharpness of the phase resets and the accuracy of the phase profile become the most critical issue that determines the diffraction efficiency of a real LC OPA. To produce the optimal director configuration, the electric field in a LC OPA device must be normal to the surface of the LC cell, as in the case when the electrode of the LC OPA is an infinitly large one. However, this is not possible for a wide-angle LC OPA, where the electrode width is small compared with the thickness of the LC cell. In this case, the electric field at the edge of the electrode is not perpendicular to the surface but has some tangential component. Such tangential components of the electric field are generally referred to as fringing electric fields. If we define a width/height ratio E, which is the ratio between the electrode width and the cell thickness, it can be an indicator of how strong the fringing field effect will be. E = W d. Here E is the width/height ratio, and W is the width of one electrode and d is the LC cell thickness. A comparison of the electric-field distribution between a high-resolution LC OPA small W value and a low-resolution LC OPA large W value is shown in Fig. 3. The isopotential line in a LC OPA with 10.5 m electrode spacing is shown in case a, and isopotential line in a LC OPA with 1.5 m electrode spacing is shown in case b. The voltage profile in case a is relatively well defined. However, in case b, the isopotential lines extend to the electrode with 0 V, and in a substantial portion of the region, the electric field is pointing to directions not perpendicular to the normal direction of the cell surface. Such an electric-field distribution will produce phase 3 TABLE I. A LC OPA with ideal director configuration. Serial no. Wavelength nm n Cell thickness d m Reset order 50% efficiency steering angle 70.7% efficiency steering angle

5 Wang et al. J. Appl. Phys. 98, FIG. 4. Basic configuration of an ECB LC OPA with 1D line-shape electrodes. a Rubbing direction perpendicular to the electrode direction. b Rubbing direction parallel to the electrode direction. modulation profile that strongly deviates from the desired phase profile and will decrease the diffraction efficiency of the LC OPA. 16 It is very hard to quantify the fringing field effect analytically because in a LC device, the electric-field distribution is coupled to the LC director configuration. If the orientation of the LC director changes, the field distribution in a LC OPA changes correspondingly. In order to study the physics behind a high-resolution LC OPA, we use computer modeling with a modified LC3D Ref. 17 core routine to simulate the director configuration considering fringing electric fields. The FDTD Ref. 15 simulation is performed to model the light propagation in the vicinity of the phase reset region. The accuracy of this simulation is very high, as can be seen in the comparison between the results from the theory, the FDTD simulation, and a test LC OPA. 15 IV. THE EFFECT OF THE SURFACE ALIGNMENT CONDITION ON THE DESIRED PHASE PROFILE There are two possible configurations for a LC OPA with line-shape electrodes, as shown in Fig. 4. The first one has the surface alignment direction parallel to the electrodes and the second one has the surface alignment direction perpendicular to the electrodes. The two structures are not equivalent in terms of optical performance. For the second configuration, the fringing fields are not in the same plane as the LC director at the cell surface. The field gradient may reorient the LC director to have an outof-the-rubbing-plane component as shown in Fig. 5 b. This leads to a depolarization of the incident light. FIG. 5. The director configuration of an ECB LC OPA in an eight-level stairlike blazed grating configuration. k 11 = N, k 22 = N, k 33 = N, =12.1, =4.1, n e =1.85, and n o =1.50. Cell thickness d=2.5 m, electrode spacing=1.5 m, gap between electrode=0.5 m, and pretilt=3. a Rubbing direction perpendicular to the electrode direction. No twist structure is present. b Rubbing direction parallel to the electrode direction. Twist structure is present due to the fringing fields. FIG. 6. The out-of-xz-plane twist director configuration of an ECB LC OPA in an eight electrode blazed grating configuration, with initial perturbation of out-of-plane twist azimuthal angle of 1. k 11 = N, k 22 = N, k 33 = N, =12.1, =4.1, n e =1.85, and n o =1.50. Cell thickness d=2.5 m, electrode spacing=1.5 m, gap between electrode=0.5 m, rubbing direction perpendicular to the electrode direction, and pretilt=0. a Equilibrium director configuration in the XZ plane. b ny component of such out-of-plane twist. For the first configuration, the fringing electric fields are in the same plane as the surface alignment direction and the cell normal the X-Z plane in Fig. 5, but it is still possible for the director to rotate out from that plane. The mechanism for this can be explained by considering Fig. 6. The director configuration, if no voltage was applied to any of the electrodes, would be nearly horizontally aligned except for a slight rotation in a clockwise direction that is due to the surface pretilt of the director. When the voltages that provide the shown director configuration are applied, the strongest fringing fields are on either side of the first full electrode on the right side because it has a much higher voltage applied to it than those on either side of it see Fig. 5. These fringing fields cause the director to rotate with a counter clockwise sense on the right side of the electrode and clockwise along the left side, preventing the director over the top of the electrode from tipping upwards in a uniform manner and trapping a tilt wall 18 over the electrode. In liquid-crystal materials, the elastic constant for a twist deformation is much lower than that for a bend configuration, so the elastic energy in this tilt wall can be reduced by the director twisting out of the plane of the figure as shown. This twisting of the director field causes a depolarization of light and lowers the device efficiency. However, if the pretilt angle of the surface alignment is high enough, it can overwhelm the effect of the fringing field on the right side of the high-voltage electrode to rotate in the counterclockwise direction, and therefore prevent the formation of the tilt wall and its possible change to a twisted structure. In order to estimate the minimum pretilt angle required to prevent an out-of-plane component of the director field, we modeled a system with the condition specified in Fig. 6. However, in the computer modeling method we use, metastable LC director configurations can cause the modeling software to give a false result for the equilibrium director configuration. To prevent this, the initial director configuration in the LC bulk is set to have an initial out-of-the- XZ-plane component that we call the perturbation angle. This is the angle between the projection of the director on the XY plane and the X axis. We vary the perturbation angle from 0 to 3 uniformly across the thickness direction of the cell, except at the boundary where the director was fixed along

6 Wang et al. J. Appl. Phys. 98, TABLE II. The stability of ECB structure in high-resolution LC OPA as a function of pretilt angle. N: The final equilibrium director configuration has no significant out-of-xz-plane twist component. Y: The final equilibrium director configuration has significant out-of-xz-plane twist component. Initial out of XZ plane azimuthal angle perturbation angle Pretilt angle the X direction. If the in plane director configuration of the LC OPA has a lower energy than the out-of-plane twist structure, the director in the LC bulk will rotate back into the XZ plane, otherwise, the director will twist out of the XZ plane to form a twist structure. In Table II, we show the out-ofplane twist structure at the equilibrium state for the example device considered in Fig. 6. When the pretilt angle of the LC OPA is 0 and the initial perturbation is 1, a twist structure is present. The director configuration in this case is shown in Fig. 6 a, and the out-of-plane director component ny is shown in Fig. 6 b. It is clear that for a pretilt angle less than 1, the out-of-plane twist structure is more likely to form, and for a pretilt higher than 3, the twist structure can be prevented. One thing to be noted is that the stability condition of 3 pretilt is a function of the electrode spacing, the cell gap, and the voltage applied to the LC OPA. When the electrode spacing of the LC OPA is low, a 1 pretilt may be enough to prevent the out-of-plane twist; but for a highspatial-resolution LC OPA, or when a large voltage difference is applied to the neighboring electrodes, a higher pretilt is required to prevent out-of-plane twist structure. V. CONTROL OF THE INDEX PROFILE AT THE RESET 0 N N N Y Y Y Y 1 N N N N N N Y 3 N N N N N N N FIG. 7. The simulated phase profile and far-field diffraction pattern of a reflective LC OPA. The segmented horizontal lines in a is the desired phase profile of a stairlike blazed grating. The continuous line is the simulated phase profile. Here k 11 = N, k 22 = N, k 33 = N, =12.1, =4.1, n e =1.7100, n o =1.5076, electrode spacing =1.5 m, gap between electrode=0.5 m, cell thickness d=6.0 m, and pretilt=3. a Phase profile of the LC OPA. b Far-field diffraction peaks. With the above consideration, a LC OPA with the first configuration in Fig. 4 is chosen for further discussion. As a standard design considered in this article unless otherwise specified, the LC OPA parameters are as follows: a pretilt angle of 3, a driving voltage of 0 5 V which is enough to produce over 90% of the retardation change as determined by the material birefringence and the cell thickness, a cell thickness of d=6 m, and eight electrodes per reset as discussed by Mcmanamon et al. 1. For this design, the operational wavelength of 1550 nm is considered and the electrode spacing is 1.5 m. The steering angle of the LC OPA in this case is The first step toward optimization of such a LC OPA is to determine the driving voltage for each electrode of the LC OPA by using a one-dimensional 1D director configuration simulation this method neglects the fringing field effect and the interelectrode coupling that can determine the retardation or optical path difference, OPD of the LC OPA as a function of the applied voltage. The two-dimensional 2D director simulation is then carried out. 10,15 to produce the phase profile of the eight-level stairlike blazed grating, as shown in Fig. 7 a. The phase profile of this device strongly deviates from the desired phase profile. The far-field diffraction pattern is obtained with a FDTD simulation as shown in Fig. 7 b. If we define the diffraction efficiency DE to be the peak intensity of the 1 diffraction order versus the peak intensity of the nonsteered beam, then the DE is only 26.3% for the LC OPA with design parameter considered in the previous paragraph. For this configuration, although the electrode spacing is small enough to achieve wide-angle beam steering, the wideangle performance of such a LC OPA is still very poor. The 70.7% efficiency angle is limited to 1.85 and the 50% efficiency angle to In order to determine how we can improve the diffraction efficiency at large diffraction angles, we will discuss the parameters that affect it, such as the cell thickness, the birefringence of the LC material, the electrode spacing, the voltage profile, the gap between electrodes, the pretilt angle, the elastic constants, and the surface alignment direction. The DE depends strongly on the spatial resolution of the LC OPA as shown in Fig. 8. As the electrode spacing becomes smaller, the corresponding steering angle increases, but the DE decreases as shown in Table III. However, there is a critical value of the electrode spacing where the DE decreases dramatically. For different cell thicknesses, the critical value is different. For example, in the case where the FIG. 8. The diffraction efficiency of LC OPA as a function of spatial resolution. Here k 11 = N, k 22 = N, k 33 = N, =12.1, =4.1, n o =1.50, n e =n o + n, electrode spacing= m, gap between electrode=0.5 m, and pretilt=3.

7 Wang et al. J. Appl. Phys. 98, TABLE III. The diffraction efficiency as a function of electrode spacing of the LC OPA. Here eight-electrode LC OPA is considered. k 11 = N, k 22 = N, k 33 = N, =12.1, =4.1, n o =1.50, n e =n o + n, electrode spacing= m, gap between electrode=0.5 m, cell thickness d=2.5, 3.0, and 6.0 m, and pretilt=3. Electrode spacing m Steering angle MLC6080 n= d=6 m n=0.35 d=3 m 1 order 26.3% 49.4% 63.7% 74.8% 79.9% 86.9% 0 order 49.4% 16.9% 3.3% 0.9% 0.5% 0.2% +1 order 18.7% 20.4% 8.7% 2.8% 1.2% 0.2% 1 order 47.6% 60.6% 70.8% 80.1% 83.5% 88.3% 0 order 11.2% 3.8% 3.3% 0.9% 0.7% 0.2% +1 order 17.1% 7.4% 6.9% 0.7% 0.4% 0.5% n=0.35 d=2.5 m 1 order 55.0% 69.7% 80.0% 84.0% 86.9% 91.0% 0 order 11.3% 3.9% 1.2% 0.2% 0.1% 0.1% +1 order 15.5% 6.5% 1.6% 0.4% 0.2% 0.1% cell thickness d is 6 m, the critical value of the electrode spacing is around 8 10 m. However, in the case where the cell thickness is 2.5 m, the critical value of electrode spacing is around 4 m. The critical value of electrode spacing is related to the fringing field effect and corresponds to a width/ height ratio E of between 0.5 and 2. Since the phase modulation depth in Fig. 7 a is reduced by the fringing field effect, it is natural to think that by increasing the cell thickness thus increasing the nd/ the phase modulation depth will increase. However, we find this is only true for a LC OPA with E not much smaller than 1. For cases when E 1, the phase modulation depth decreases as the cell thickness becomes larger due to the increase of the effect of fringing fields. A reflective LC OPA has substantial advantage in both reducing the fringing field effect and improving the switching speed for the wide-angle LC OPA. The cell thickness of a LC OPA is determined by the operational wavelength of the LC OPA and the birefringence of the LC material available. Generally, the double-pass retardation value, since we are now talking about reflective devices, 2 nd/ is desired to be larger than that necessary to achieve a modulo 2 version of the desired phase profile. Different combinations of cell thickness and n of the LC material that gives 2 nd/ value ranging from 0.86 to 2.61 are compared in Table IV. Surprisingly, the highest DE corresponds to when the cell thickness is 2.1 m, which gives a 2 nd/ value of The reason for this is that the total modulation depth only needs to be 1 1/8=0.875 waves for an eight-level blazed gratings. However, if we choose such a cell thickness, the DE will be lower for small steering angles because the number of steps in the grating will increase. A trade off has to be made between high efficiency at large and small steering angles. In any case, it is very important to reduce the cell thickness and improve the diffraction angle of a LC OPA. Using high birefringence LC material and a very thin reflective cell can improve the performance of the LC OPA dramatically. For example, using n=0.35 material can allow a very thin LC OPA with cell thickness of 2.5 m to be able to operate at 1550 nm wavelength. The diffraction efficiency of such a LC OPA is much higher than the LC OPA with lower birefringence. As shown in Fig. 8, to achieve a 70.7% diffraction efficiency for a LC OPA with a n=0.35 LC material, the electrode spacing needs to be 2.6 m, which corresponds to a diffraction angle of 4.2. To achieve 50% diffraction efficiency, the electrode spacing needs to be 1.5 m, which corresponds to a diffraction angle of 8.7. This means, a high birefringence LC material if very critical for high-efficiency wide-angle LC OPA. Further optimization of the DE is possible by optimizing the voltages applied to the electrodes. Consider Fig. 9, where the voltages on the electrodes have been set without taking into account the fringing electric field or interelectrode coupling. The DE of such a LC OPA is not optimized, as the phase at electrode numbers 1 and 2 is much smaller than the desired value. Also the phase slope of electrode numbers 2 8 is larger than desired. One way to alleviate this phase error is to optimize the voltage profile on each of the eight pixels. Since no analytical solution is available, we use an iterative process to achieve such optimization. If the phase at the center of a particular electrode is lower than the desired phase TABLE IV. Diffraction efficiency as a function of cell thickness. Here k 11 = N, k 22 = N, k 33 = N, =12.1, =4.1, n o =1.50, n e =n o + n, electrode spacing=1.5 m, gap between electrode =0.5 m, and pretilt=3. n d m 2 nd/ DE % % % % % % % % %

8 Wang et al. J. Appl. Phys. 98, FIG. 9. The phase profile of LC OPA before voltage optimization. Here k 11 = N, k 22 = N, k 33 = N, =12.1, =4.1, n o =1.50, n e =1.85, d=2.5 m, electrode spacing=1.5 m, gap between electrode=0.5 m, and pretilt=3. profile, the voltage applied to that electrode is adjusted lower; if the phase at the center of the electrode is higher than the desired one, the voltage is adjusted higher. Within several iterations, the phase value on each electrode is close to the desired phase profile. The voltage introduced on each of the electrodes for five iterations of our process is listed in Table V. We call the difference between the optimized voltage on each electrode after iteration 5 and the original voltage profile on iteration 1 as the bias voltage. The bias voltage for each of the electrodes is shown in Fig. 10, where the horizontal axis is the normalized position of each pixel the first electrode is defined to be in normalized position 0, and the last electrode is in normalized position 1. We can see that, in order to optimize the voltage profile, the first electrode has to have a large negative bias voltage, the second electrode has to have a medium positive bias, and the last electrode has to have a large positive bias. The eight discrete bias voltages is fitted with a seventh-order polynomial. The appropriate bias voltages can be precalibrated to obtain a high-efficiency LC OPA for different steering angles. Such an optimization has been carried out experimentally by Harris 19 in a slightly different way, and a substantial improvement to the DE was also reported. The phase profile of the wide-angle LC OPA after optimization and the corresponding far-field diffraction pattern is shown in Fig. 11. The DE of the LC OPA is 72.7% for 7.4 of steering after voltage optimization. In Fig. 12, the DE as a function of the cell thickness before and after voltage optimization is shown. For a LC OPA with a large cell thickness, the DE improved by a factor of 2. For a LC OPA with a small cell thickness, because the fringing field effect is less strong, FIG. 10. The bias voltage to correct for fringing-field-induced phase error in high-resolution LC OPA. Discrete data points are from the optimization process from simulation. The continuous line is the seventh-order polynomial fit of the data points. The coefficients for the seventh-order polynomial are: p1=1192.8, p2= , p3=6700.6, p4= , p5=2236.8, p6= , p7=49.621, and p8= the DE improvement is less apparent. For example, when d=2.5 m, the DE improved by 17.7% after voltage optimization. A comparison of wide-angle performance of different wide-angle LC OPAs is shown in Fig. 13, as well as in Table VI. For the optimized design, the 70.7% efficiency angle is 7.7 and the 50% efficiency angle is VI. OTHER CONSIDERATIONS Another factor related to the De of a LC OPA is the influence of the gap between electrodes. For the LC OPA considered in this article, a continuous mirror as a common electrode is assumed to avoid possible diffraction loss from a segmented mirror. The patterned electrode is a transparent conductor, for example, indium tin oxide ITO. To describe the efficiency factor associated with the patterned electrodes, we define the aperture ratio of the electrodes to be the ratio between the size of the electrode and the electrode spacing. The DE as function of the aperture ratio of a LC OPA is shown in Table VII no voltage optimization is carried out for these cases. We can see that for an electrode gap less than one-half of the cell thickness, the DE loss is almost independent of its exact value. However, for electrode gap larger than one-half of the cell thickness, the DE loss increases. In this consideration, we use a 0.5 m gap between electrodes and the aperture ratio larger than 66.7% in our TABLE V. Voltage optimization process on each electrode of the LC OPA. Voltage on each electrode v Electrode number DE Iteration number % % % % %

9 Wang et al. J. Appl. Phys. 98, FIG. 11. The optimized phase profile and far-field diffraction pattern of a wide-angle LC OPA. Simulation parameters used here are k 11 = N, k 22 = N, k 33 = N, =12.1, =4.1, n e =1.85, and n o =1.50. Cell thickness d=2.3 m, electrode width W=1 m, and gap between electrode=0.5 m. a Optimized phase profile by adjusting voltage profile on the eight electrodes. b Corresponding far-field diffraction pattern. FIG. 13. Comparison of diffraction efficiency as a function of diffraction angle for different cases: k 11 = N, k 22 = N, k 33 = N, =12.1, =4.1, n e =1.85, n o =1.50, electrode spacing =1.5 m, gap between electrode=0.5 m, and pretilt=3. later consideration. In an actual device, there is a dielectric layer between the electrodes and the LC material as an alignment layer. This layer could have an effect on the fringing fields, but typically, this layer is very thin, so we will not discuss its influence. In our modeling, we assumed the alignment layer to be a 50-nm-thick nonconductive dielectric layer with a dielectric constant of 4.5. The influence of the elastic constants of the liquidcrystal material on the DE also needs to be considered. The larger the splay, twist, and bend elastic constants k 11, k 22, and k 33, the larger the elastic energy associated with a nonuniform director configuration. It is reasonable to think that if the elastic constants become smaller, it will be easier to deform the LC director to achieve the desired phase profile. However, through our study, we find this is not a significant factor related to the DE because the dominating factor is the fringing-electric-field effect. As an example, the DE without voltage optimization improved only to 59.3% from 58.9% as shown in Table VIII, when the splay and bend elastic constants are reduced to N. The main effect of the surface pretilt of the director is mainly related to the control of a fringing-electric-fieldinduced tilt wall. Excluding this effect, the effect of the pretilt angle on the DE is small as shown in Table IX. Another effect on the DE of the OPA is related to the symmetry of the device. If the LC OPA is programmed to steer to the 1 or the +1 diffraction order, the peak intensity or DE is different in these two cases. Two effects could contribute to the lack of symmetry in a LC OPA. The first one has to do with the interaction of the surface director tilt orientation with the desired phase gradient direction. Due to this effect, the phase profile of the LC OPA corresponding to steering to 1 and +1 diffraction orders is not exactly a mirror image of each other and the DE for these two cases may be different. The other effect that contributes to this nonsymmetrical behavior is related to the tilting of the optic axis in a LC OPA. For example, if we change the sign of the tilt angle everywhere in the cell, the phase gradient will stay the same, but the diffraction efficiency is not the same. This is caused by the change in the direction of energy flow of light Poynting vector in the LC OPA associated with the change in the tilt angle of the optical axis. This effect has been discussed by Titus et al. 20 for a transmissive LC OPA, where the difference between the DE in the +1 and 1 diffraction orders can be as high as 5% 10%. In the reflective LC OPA discussed in this article, these effects are small, as shown in Table X. VII. CONCLUSION The design of an optimized wide-angle LC OPA is systematically discussed, and the performance limits of a liquidcrystal-based optical phased array is studied. The influence factors of the diffraction efficiency for a real LC OPA is TABLE VI. Comparison of wide-angle properties of different LC OPAs. FIG. 12. Comparison of diffraction efficiency as function of cell thickness for a LC OPA without voltage optimization and with optimization. Simulation parameters used here are k 11 = N, k 22 = N, k 33 = N, =12.1, =4.1, n e =1.85, n o =1.50, electrode spacing =1.5 m, gap between electrode=0.5 m, and pretilt= % efficiency angle Off the shelf LC material High n LC High n LC+ voltage optimization Ideal LC OPA

10 Wang et al. J. Appl. Phys. 98, TABLE VII. The diffraction efficiency of a LC OPA for different electrode configurations. Here k 11 = N, k 22 = N, k 33 = N, =12.1, =4.1, n o =1.50, n e =1.85, cell thickness d=2.5 m, and pretilt=3. Electrode spacing m Size of electrode m Size of gap between electrodes m Aperture ratio % 56.0% % 55.0% % 54.6% % 52.5% % 43.4% DE determined by accurate modeling of the LC director configuration considering fringing field effects and the optical properties of the OPA. ACKNOWLEDGMENTS This work is supported by the DARPA THOR and STAB projects, as well as the NASA space communication project. The authors wish to thank Michael Fisch for very helpful discussions. TABLE VIII. The diffraction efficiency of a LC OPA with different elastic constants. Here =12.1, =4.1, n o =1.50, n e =1.85, cell thickness d =2.3 m, electrode spacing=1.5 m, gap between electrode=0.5 m, and pretilt=3. k N k N k N DE % % % TABLE IX. Diffraction efficiency of a d=2.5 m LC OPA for different pretilt angles. Note that pretilt smaller than 3 may not be a stable structure. Pretilt angle DE 53.4% 55.0% 55.6% 56.2% TABLE X. The diffraction efficiency for steering to 1 and +1 diffraction orders. Forward voltage ramp to steer to 1 order Reverse voltage ramp to steer to +1 order d=2.1 m d=2.3 m d=2.5 m 60.1% 58.9% 55.0% 60.0% 61.0% 57.5% 1 P. F. Mcmanamon, E. A. Watson, T. A. Dorschner et al., Opt. Eng. 32, P. F. McManamon et al., Proc. IEEE 84, D. P. Resler, D. S. Hobbs, R. C. Sharp, L. J. Friedman, and T. A. Dorschner, Opt. Lett. 21, R. M. Matic, Proc. SPIE 2120, L. J. Friedman, D. S. Hobbs, S. Lieberman, D. L. Corkum, H. W. Nguyen, R. C. Sharp, and T. A. Dorschner, Appl. Opt. 35, V. G. Dominic and E. A. Watson, Opt. Eng. 35, X. Wang, D. Wilson, R. Muller, P. Maker, and D. Psaltis, Appl. Opt. 39, M. T. Gruneisen and J. M. Wilkes, Proceeding of TOPS, 1997, edited by G. Burdge and S. C. Esener unpublished, Vol. 14, p G. D. Love, Appl. Opt. 36, X. Wang, B. Wang, J. Pouch et al., Opt. Eng. 43, M. T. Gruneisen, T. Martinez, D. V. Wick, J. M. Wilkes, J. T. Baker, and I. Percheron, Proc. SPIE 3760, P. F. Mcmanamon, Great Lakes Photonics Symposium, 2004 unpublished. 13 S. Serati, H. Masterson, and A. Linnenberger, 2004 IEEE Aerospace Conference Proceedings, 2004 unpublished, No C. M. Titus, P. J. Bos, J. R. Kelly, and E. C. Gartland, Jpn. J. Appl. Phys., Part 1 38, X. Wang, B. Wang, P. J. Bos et al., J. Opt. Soc. Am. A 22, B. Apter, U. Efron, and E. Bahat-Treidel, Appl. Opt. 43, J. Anderson, P. Watson, and P. Bos, Liquid Crystal Display 3-D Director Simulator Software and Technical Guide Artech House, Norwood, MA, A. Lien and R. A. John, IBM J. Res. Dev. 36, S. Harris, Proc. SPIE 5162, C. M. Titus, J. R. Kelly, E. C. Gartland, S. V. Shiyanovskii, J. A. Anderson, and P. J. Bos, Opt. Lett. 26,