Modeling and Performance Limits of a Large Aperture High-Resolution Wavefront Control System Based on a Liquid Crystal Spatial Light Modulator

Kent State University Digital Commons @ Kent State University Libraries Chemical Physics Publications Department of Chemical Physics 4-15-2007 Modeling and Performance Limits of a Large Aperture High-Resolution Wavefront Control System Based on a Liquid Crystal Spatial Light Modulator 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 Follow this and additional works at: https://digitalcommons.kent.edu/cpippubs Part of the Physics Commons Recommended Citation Wang, Xinghua; Wang, Bin; and Bos, Philip J. (2007). Modeling and Performance Limits of a Large Aperture High-Resolution Wavefront Control System Based on a Liquid Crystal Spatial Light Modulator. 46(4). doi: 10.1117/1.2730485 Retrieved from https://digitalcommons.kent.edu/cpippubs/130 This Article is brought to you for free and open access by the Department of Chemical Physics at Digital Commons @ Kent State University Libraries. It has been accepted for inclusion in Chemical Physics Publications by an authorized administrator of Digital Commons @ Kent State University Libraries. For more information, please contact digitalcommons@kent.edu.

464, 044001 April 2007 Modeling and performance limits of a large aperture high-resolution wavefront control system based on a liquid crystal spatial light modulator Xinghua Wang Bin Wang Philip J. Bos Kent State University Liquid Crystal Institute Kent, Ohio 44242 E-mail: pbos@kent.edu Paul F. McManamon, FELLOW SPIE Air Force Research Lab Dayton, Ohio 45424-4653 John J. Pouch Felix A. Miranda NASA Glenn Research Center Cleveland, Ohio 44135 Abstract. The aberration introduced by the primary optical element of a lightweight large aperture telescope can be corrected with a diffractive optical element called the liquid crystal spatial light modulator. Such aberration is usually very large, which makes the design and modeling of such a system difficult. A method to analyze the system is introduced, and the performance limitation of the system is studied through extensive modeling. An experimental system is demonstrated to validate the analysis. The connection between the modeling data and the experimental data is given. 2007 Society of Photo-Optical Instrumentation Engineers. DOI: 10.1117/1.2730485 Subject terms: liquid crystal; spatial light modulator; aberration correction; diffractive optical device. Paper 060204RR received Apr. 8, 2006; revised manuscript received Sep. 22, 2006; accepted for publication Oct. 3, 2006; published online Apr. 30, 2007. James E. Anderson Hana Microdisplay Technologies Inc. Twinsburg, Ohio 44087 1 Introduction In a satellite-based laser communication transmitter, a large aperture telescope is required to maintain a small beam divergence for long distance laser communication. The primary optical element needs to be extremely lightweight for space deployment. Ideally, it can be made of a membrane type of material coated with highly reflective coating. However, vibration, thermal expansion, mechanical stress, and manufacturing imperfections can seriously reduce the performance of the primary mirror and introduce large aberrations to the telescope. 1 3 An active wavefront correction system that can compensate for the aberration is important. A schematic of the proposed laser communication transmitter with such an active wavefront control system is shown in Fig. 1. The active correction device used in the system can be a deformable mirror, microelectromechanical systems MEMS, optically addressed spatial light modulator, or a liquid crystal spatial light modulator LC-SLM. 4 13 By having an active device where each element of the device can be programmed to a desired phase value, diffractive wavefront compensation can be achieved for large aberrations. A high-resolution device also enables many new capabilities such as nonmechanical random-access beam pointing, multiple beam control, dynamic focus defocus of laser beam as well as beam steering. A LC-SLM is such 0091-3286/2007/$25.00 2007 SPIE a device that has the advantages of being low cost, highresolution, low profile, lightweight, and capable of handling medium to high laser power. 14 16 Researchers at the Air Force Research Laboratory have pioneered in developing a concept that uses a liquid crystal device to correct for a large aberration in an optical system. For example, McManamon and Watson et al. pioneered a liquid crystal optical phased array device, a onedimensional 1D type of LC-SLM that can be used for laser beam steering and showed the basic efficiency theory related to this type of device. 14 Gruneisen et al. pioneered in high-resolution wavefront control in a large aperture telescope with a diffractive optical element. They studied Fig. 1 A free space laser communication transmitter with LC-SLM as wavefront corrector and beam steerer. 044001-1

the diffraction efficiency, dynamic range of correction, efficiency for correcting large defocus and astigmatism, and the wavelength dependence of such a system using a Fourier optics approach. 1 3 Also, Gruneisen et al. have addressed the efficiency issue associated with the local wavefront slope needed for large correction. In this paper, we give a comprehensive analysis of a wavefront correction system using a LC-SLM, considering the aberration being corrected, the resolution of the correction device, and the LC-SLM efficiency. The paper is organized in the following way. First, a simple model to evaluate the performance of the optical system is proposed. Independent efficiency factors are introduced and discussed. Second, numerical modeling is carried out to study independent efficiency factors. Third, an experimental study of a high-resolution wavefront correction system is shown. Finally, a connection between the modeling data and the experimental data is made. 2 A Model to Describe High-Resolution Wavefront Correction Using LC-SLM To evaluate how well a wavefront correction system works in terms of restoring the wavefront to diffraction limited performance, the Strehl ratio of the system after wavefront correction is a core performance metric. We consider four major factors that contribute to the Strehl ratio of the system, as shown in Eq. 1 total = s ray aber SLM. 2.1 Factor s The first factor, s, is the passive loss associated with the imperfections in the mirror surface, scattering, absorption, front surface reflection, and so on. This factor mainly depends on the material and process that is used to make the optical elements. 2.2 Factor ray The second factor, ray, is the ray aberration term that is related to the ray aberration in the system. It is strongly related to the relative conjugate position of the element being corrected and the corrector. As a simple example, shown in Fig. 2, if there is a substantial aberration in the primary mirror, a ray A can miss the aperture of the correction device. In another case, two rays, B and C, could cross each other and hit the same element on the corrector device. When such things happen, the correction device would not be able to perfectly correct for the wavefront aberrations, even if an infinitely high-resolution correction device is used. The value of this term depends on many factors. First of all, ray depends on the optical system design. In a perfect conjugate system, where the correction device is placed at the conjugate position of the primary optical element, the position of any ray that hits a particular point on the primary mirror will hit a corresponding point on the LC-SLM. There is a one-to-one relationship of the ray interception position on the primary mirror and the correction device, regardless of what is the magnitude and order of the aberration on the primary mirror. In this case, ray =1 always holds true. However, it is hard to design a practical 1 Fig. 2 An illustration of efficiency loss due to ray aberration in a severely aberrated large telescope. high numerical aperture telescope with perfect conjugation. If the wavefront corrector is not in a conjugate position, the wavefront aberrations could cause substantial shift of the ray interception position on the corrector device for different aberrations, and the value of ray will be lower than 1. Second, ray depends strongly on the magnitude and the order of the wavefront aberration for a nonconjugate telescope. As an example, we used CODE V optical design software to trace a uniformly spaced ray bundle through a particular optical system as shown in Table 1. The position of each ray intercepting the LC-SLM correction plane is plotted in Fig. 3. In Fig. 3a, a low order Zernike mode with 60 waves rms of primary coma on the primary mirror of the transmitter is shown. In Fig. 3b, a high order Zernike term, the standard Zernike term Z 33 with 20 waves rms of wavefront aberration is shown. The second case has a much stronger ray aberration than the first case because the shift of the ray interception position on the correction device is much stronger than the first case. When the shift of the ray interception position is much larger than the element size of the corrector element, even if the total rms wavefront aberration is smaller than the low order aberration, a good correction cannot be achieved because the ray is shifted to a wrong correction element. So ray for the second case is much smaller than for the first case. Third, ray also depends on the resolution of the correction device. Although it might be counterintuitive, for the way we have defined this term, its value drops as the resolution of the correction device increases. This is because the significance of the nonconjugate nature of the system is related to the considered area of one corrector element. The larger the resolution, the smaller the considered area of interest, and the more likely the ray will be shifted to a wrong element on the corrector device, which lowers the value of this term. For example, if a particular ray is shifted 100 m due to the nonconjugate nature of the telescope, and assuming the size of the active element is 1000 m, then the shift of the ray is much smaller than the active element, and the effect of the ray aberration on the value of ray is small enough to be neglected. However, if the size of the active element is 50 m, then the shift of the ray is much larger than the size of the active element, and the value of this 044001-2

Table 1 A free space laser communication transmitter telescope design. All units are millimeters. Surface No. Surface Name Surface Type Y Radius Thickness Glass Refract Mode Y Semiaperture Object Sphere Infinity Infinity Refract 500.00 1 Sphere Infinity 400.00 Refract 500.00 Stop Primary mirror Conic 600.00 250.00 Reflect 500.00 3 Sphere Infinity 45.500 Refract 10.00 4 Secondary mirror Conic 9.0000 2.00 Reflect 7.500 5 Sphere Infinity 2.56 Refract 7.500 6 Correction phase plane Sphere Infinity 0.01 BK7 Refract 7.500 7 Sphere Infinity 20.00 Refract 7.500 8 Ideal lens modular Sphere Infinity 50.00 Refract 7.500 Image Sphere Infinity 0.00 Refract 0.200 term is low because the ray does not go through the conjugate corrector element. In this case, the nonconjugate nature of the telescope will have a substantial effect on the value of this term. Not having a simple analytical method to fully quantify the term ray, we can use a numerical method of a simple ray tracing algorithm to describe this term in a physically meaningful way. Consider a certain aberration in a certain optical system, where we can trace a mm bundle of rays through it. The correction device in the system has mm elements across the clear aperture. For each element on the correction device, we can count how many rays intercept its area. When there is no aberration present in the system, the number of rays that intercept each element on the correction device should be one. However, when aberration is present in the system, the number of rays intercepting a certain element is no longer one. We define n 0 to be the number of elements on the corrector device that have no ray intercepting that element, n 1 to be the number of elements on the correction device that have one ray intercepting it, n 2 to be the number of elements on the corrector device that have two rays intercepting it, and n k to be the number of elements that have k rays intercepting it. However, we want to trace more than one ray through a certain element to be more accurate. We use j j rays in the bundle, and mm elements on the correction device jm. There should be j 2 /m 2 rays intercepting one ele- Fig. 3 A ray intersection plot on LC-SLM device plane. a 60 waves rms of primary coma Z 8 on the primary mirror. b 60 waves rms of high order coma Z 33 on the primary mirror. 044001-3

Fig. 4 The ray as function of the resolution of the correction device for different aberrations. ment on the correction device, when there is no aberration in the system. It is convenient to normalize the number of rays intercepting one element on the correction device by the factor of j 2 /m 2 for later calculation. In the case of normalization, if there is an aberration in the system, we can still define n 0 to be the number of elements that has no ray intercepting the element, but n 1 will be the number of elements that has a number of normalized rays intercepting it in the range of 0 1. The same way, n k is the number of elements that has a normalized number of rays in the range of n k 1 n k. Then the percentage of elements that has normalized k rays intercepting the element is k =n k /m 2. For a telescope system shown in Table 1, if the aberration on the primary optical element is a primary spherical aberration Z 13, and the correction device is a 5050 resolution device, then 0 =12.2%, 1 =73.8%, 2 =13.9%, 3 =0.3%, 4 =0.0% can be obtained by tracing 500500 rays through the system. Because the Strehl ratio of the system will be a direct function of how many rays will miss the desired element, which is simply 0, it can be expressed in Eq. 2 ray =1 0. 2 In this case, we can obtain ray =88% for a 5050 correction device. On the other hand, if the correction device has 500500 active elements, then 0 =25.3%, 1 =51.6%, 2 =22.3%, 3 =0.3%, 4 =0.0% can be obtained similarly. In this case, the ray =74.7%. We can also vary the resolution of the correction device and the aberration on the primary telescope to perform the same calculation. The relationship between the ray and the resolution of the correcting device is shown in Fig. 4. We can see a clear trend showing that the Strehl ratio drops as the resolution of the correction device increases, and the trend tends to saturate as the resolution of the correction device becomes very high. 2.3 Factor aber The third factor, aber, is related to the resolution of the corrector. This term becomes larger as the resolution of the device becomes higher. If we assume an ideal piston-phaseplate correction device with finite resolution, the Strehl ratio for correcting random high order aberrations is not 100%. Considering the simplest case of correcting a simple tilt and assuming q is the number of steps for every wave of tilt, the factor aber is the same as the Strehl ratio or diffraction efficiency for a stairlike blazed grating as in Eq. 3. 14 aber = sin/q 2. 3 /q We would like to estimate a factor such as the one in Eq. 3, but which is applicable to the more general case of an arbitrary two-dimensional 2D aberration. Considering again the telescope system shown in Table 1, any aberration on the primary mirror will induce a corresponding aberration on the correction device. The aberration profile on the correcting device can be obtained by diffractive beam propagation of a Gaussian beam from the primary optics to the correction device. More details about the CODE V diffractive beam propagation calculation will be shown in Sec. 3. The whole aperture is divided into small, 11 regions. Within each small region, the aberration profile within region is defined as Px,y x,y, which can be approximated by a simple tilt, where the maximum difference in the phase across the region is = maxpx,y x,y minpx,y x,y. 4 The number of elements along either horizontal or vertical step is S=m/l. Then the number of elements for one wave of tilt q in region is approximately q = S = m l. With such an approximation, the term aber, within region can be estimated for a 2D aberration aber, = sin/q /q 2 = sinl /m l /m 5 2. 6 If we assume the fractional intensity of the beam on region is W, then the total efficiency across the whole aperture is the weighted average of aber, across the whole aperture as in Eq. 7. 17 aber = = W aber, = W sinl /m l /m W sin/q /q 2 2. 7 Following this approach, we can calculate the value of aber in a 2D case. In Fig. 5, the calculated Strehl ratio for different aberrations is shown as a function of the resolution of the correction device. For a particular resolution of the corrector, the Strehl ratio is different for different aber- 044001-4

Fig. 5 The aber as function of the resolution of the correction device for different aberrations. rations. Generally speaking, the larger the magnitude of the aberration and the higher order the aberration, the lower the Strehl ratio. However, it is also clear that if the resolution of the correction device is high enough, this efficiency aber can be very high, approaching aber =1. 2.4 Factor SLM The fourth factor SLM is associated with the nonideal phase profile of the correction device, the LC-SLM. 13,18,19 Consider a simple case of a tip-tilt corrector. To correct for hundreds of waves of aberration, the thickness of the LC- SLM may become impractical as it needs to produce a large optical path difference OPD, and the birefringence of the liquid crystal material is limited. For monochromatic light, this limitation is resolved by considering an approach where the phase profile generated by the LC-SLM is a modulo 2 version of the desired phase profile. In this case, the maximum OPD change required at any point on the LC-SLM is only one wavelength. The phase profile on a LC-SLM is a stairlike blazed grating with phase reset of 2. As has been discussed previously, 14,18 most elements in such a stairlike blazed grating can produce a phase profile very close to the desired phase profile. However, the orientation of the liquid crystal director cannot undergo an abrupt change to produce the phase discontinuity at the phase reset region. There is a fly-back region where the phase slope is in the opposite direction to the desired phase slope. The Strehl ratio of the LC-SLM SLM is related to the ratio of the width of the fly-back region and the grating period as expressed in Eq. 8. 3 A power of 2 in Eq. 8 is to take into account the efficiency for coherent light SLM 1 2 F. 8 Here is the width of each grating segment, and F is the width of the fly-back region. In a 2D LC-SLM discussed in this paper, the exact value of this term depends on many factors such as the cell thickness, the liquid crystal material birefringence, the aperture ratio of the electrode, and so on. In later sections, a detailed study will be carried out to Fig. 6 The optical layout of high-resolution wavefront correction system with large aperture telescope. study the efficiency of a 2D LC-SLM and to relate those results to Eq. 4. 3 Numerical Modeling and Comparison with the Simple Model 3.1 Optical System Modeling for the Ray Aberration Term and the Wavefront Aberration Term Section 2 discussed the factors that contribute to the Strehl ratio of the system. However, the values of the second and third terms are obtained through simple estimation. To obtain the values of the second and third terms in a more rigorous way, CODE V ray tracing software is used to simulate a high-resolution wavefront control system. The optical system is described in the software, and a particular aberration is introduced to the primary mirror. Then the corresponding aberration on the correction device is obtained using diffractive beam propagation. A conjugate phase profile is then introduced to the correction plane and the Strehl ratio of the system before and after correction is obtained using diffractive beam propagation. The Strehl ratio obtained this way should produce the same results as the product of ray and aber in the simple model previously described. As an example, consider the optical layout shown in Fig. 6 and Table 1. A compact telescope design with a parabolic primary and secondary mirror is used. The LC-SLM wavefront corrector is positioned near the conjugate plane of the primary mirror. The clear aperture of the primary mirror is 1 m with a focal length f =300 mm. The secondary mirror has a clear aperture of 20 mm with f =4.5 mm. The LC- SLM device has a clear aperture of 15 mm. The total magnification of the telescope is 66.7. The initial system is aberration free. An ideal lens module is introduced behind the LC-SLM to help analyze the afocal system. The point spread function at the focus of the ideal lens module is evaluated to obtain the Strehl ratio of the telescope system. To simulate an imperfect primary mirror, a surface deformation 10 waves rms, or 16 waves peak-to-valley pv of primary defocus is introduced to the primary mirror, which will introduce a wavefront aberration of 20 waves rms, or 32 waves pv of primary defocus to the optical system. With such a large aberration applied to the primary 044001-5

Fig. 7 The modeled wavefront aberration on the correction device obtained by diffractive beam propagation. Upper graph is the intensity distribution of light at the correction device. Fig. 8 The simulated wavefront of the beam after correction. 044001-6

Fig. 10 The Strehl ratio after correction for different Zernike terms. Assuming all wavefront aberrations are 10 waves rms. Fig. 9 The point spread function at the focus of the ideal lens modular obtained by diffractive beam propagation BPR function of CODE V of a Gaussian beam through the optical system. a Before wavefront compensation. b After wavefront compensation. mirror, ray tracing may not be accurate enough to determine the wavefront aberration in the LC-SLM plane. Instead, the diffractive beam propagation BPR function in CODE V is used to obtain the phase and intensity of light at any given position in the system. The diffractive beam propagation function in CODE V is a combined method of ray tracing and a beam propagation algorithm. Geometrical ray tracing algorithm is used for the rays passing through any nonair surface. Near far field beam propagation algorithm is used for all air paths. In Fig. 7, the wavefront aberration at the LC-SLM plane obtained using diffractive beam propagation is shown. The intensity profile at the LC-SLM plane is illustrated in the upper insert figure of Fig. 7. After the introduction of aberration on the primary mirror, the Strehl ratio of the whole system decreases from 1.0 to 0.0000005. A conjugate phase plate with 500500 active elements over the 1515 mm 2 clear aperture is generated by changing the sign of the aberration and taking the modulo of 2. Such a conjugate phase profile is introduced to the LC- SLM plane as the correction phase plate. Diffractive beam propagation is performed after introducing the correction phase plate. As shown in Fig. 8, the residual wavefront error after correction is near zero within the beam waist, except for some very high spatial frequency phase error at the wavefront resets. The noise at the discontinuity of the wavefront contributes to diffraction loss. The point spread function at the focus of the system can be obtained by using diffractive beam propagation BPR function in CODE V of a Gaussian beam through the system. The point spread function before and after the wavefront compensation is shown in Fig. 9. After wavefront correction, the Strehl ratio of the system improved from 0.0000005 to 0.63. A Strehl ratio improvement factor of =0.63/0.0000005=1.2610 6 is obtained. The reason the Strehl ratio is not 1 is not due to the fact that there is substantial residual phase error after the wavefront correction. In fact, if we ignore the phase error at the wavefront discontinuity, the residual phase error is 0. However, there is loss of energy at the wavefront discontinuity, which is what we would expect for a diffractive wavefront compensation. The complexity of the wavefront and the finite resolution of the correction device are the main factors contributing to this loss. With this technique, the Strehl ratio of the system after correction of different Zernike modes of aberration is studied. First, a fixed magnitude of 10 waves rms of different Zernike mode is introduced to the primary mirror. As shown in Fig. 10, the Strehl ratio after correction is very different. The coma and the spherical aberration can decrease the system Strehl ratio much faster than other terms. Even with a resolution of 500500, the Strehl ratio after correction can sometimes be as low as 0.2 for the 25th standard Zernike term Z 25. For a particular Zernike mode, the Strehl ratio after correction for different magnitudes is shown in Fig. 11. For up to 25 waves rms of low order astigmatism Z 4 and Z 6, the Strehl ratio after correction can be higher than 0.65. On the other hand, the Strehl ratio after correction for primary spherical aberration Z 13 drops very fast as the magnitude of aberration gets larger and larger. If a Strehl ratio of 0.5 is required after correction, the maximum spherical aberration that can be corrected is only about 10 waves rms. The resolution of the LC-SLM also places a limitation on the maximum amount of aberration that can be corrected. In Fig. 12, the correction of primary defocus and spherical aberration with different resolutions is shown. 044001-7

Fig. 11 The Strehl ratio after wavefront correction for different magnitude of primary astigmatism, defocus, coma, and spherical aberration. The Strehl ratio after correction is not a linear function of the resolution of the LC-SLM device, but has a critical value. If the resolution of the LC-SLM is smaller than this critical value, the Strehl ratio after correction would still be low. Such critical resolution is related to the type of the aberration and the magnitude of the aberration. Generally speaking, the larger the magnitude of the aberration, the larger the critical resolution would be. For example, to correct for 10 waves rms of primary spherical aberration Z 13 to a Strehl ratio of 0.3 or more, such a critical resolution is about 150150. However, for 15 waves of primary spherical aberration Z 13 to a Strehl Ratio of 0.3 or more, such a critical resolution is about 300300. The type of aberration also plays an important role. For example, the primary defocus is much easier to correct as compared with spherical aberration, and the critical resolution is lower for the primary defocus. In any case, a high-resolution corrector Fig. 13 The comparison of the Strehl ratio obtained by CODE V and ray aber calculated by Eqs. 2 and 7. device has higher correction efficiency and larger dynamic range of correction. In Fig. 12, it seems that the efficiency becomes saturated for some aberrations, for example at 15 waves rms of Z 13. This is due to the fact that the second efficiency term ray drops as the resolution of LC-SLM increases. The Strehl ratio obtained by CODE V, following the discussion in Sec. 2, can be considered to be the product of ray and aber. In Fig. 13, the plot ray, aber obtained earlier by Eqs. 2 and 7, as well as the Strehl ratio after correction obtained by CODE V simulation are shown. Both data show good agreement with each other. One thing to be noticed is that the overall trend is the efficiency ray, aber increases as the resolution of the LC-SLM increases. Another important factor that could influence the correction efficiency is the pupil apodization the intensity distribution of the beam at the input aperture at the primary optical element. In the system discussed above, the 1/e 2 beam diameter on the LC-SLM plane is 7.5 mm, which is only half the size of the clear aperture of the LC-SLM. The active elements outside of the 1/e 2 beam diameter contribute little to the Strehl ratio because the light intensity is low. More than half of the active elements on the LC-SLM device are not used very efficiently. There is a trade-off between the filling factor of the beam on LC-SLM plane versus the truncation loss of a Gaussian beam. We will not discuss this in full detail because it strongly depends on the system design. Fig. 12 The Strehl ratio after correction as function of the resolution of the correction device. Fig. 14 The two configurations of 1D LC-SLM with line shape electrodes. a Surface alignment direction perpendicular to the electrode direction. b Surface alignment direction parallel to the electrode direction. 044001-8

Fig. 15 The director configuration of the two 1D LC-SLM with eight electrodes. Cell thickness d=2.5 m, element spacing=1.5 m. a Surface alignment direction perpendicular to the electrode direction. No twist structure is present. b Surface alignment direction parallel to the electrode direction. Twist structure is present due to the fringing fields. 3.2 Strehl Ratio Contribution of the SLM SLM We may use the diffraction efficiency DE instead of the Strehl ratio in this part of the discussion using the diffractive nature of the 1D SLM corrector. DE could be defined as a conventional DE equal to the ratio of the peak intensity in the negative first diffraction order to the peak intensity of the nonsteered beam. We assume the diffractive optical element will not introduce any aberration to the system. To obtain the value of the fourth factor in Eq. 1, the loss associated with a 2D liquid crystal device needs to be discussed. The Strehl ratio of a 2D LC-SLM is estimated by considering the Strehl ratio of two 1D LC-SLM cases. These two configurations of 1D LC-SLM have different directions of the surface alignment either perpendicular or parallel with respect to the line shape electrode, as shown in Fig. 14. Light propagation in the two different configurations of the LC-SLM is different. For example, when the surface alignment direction is perpendicular to the line shape electrode of the 1D LC-SLM as shown in Fig. 14a, the LC director configuration is in the xz plane of the LC- SLM, as shown in Fig. 15a. When an incident field Ex polarized along the surface alignment direction propagates through a transmissive LC-SLM, as shown in Fig. 16a, the intensity of the other polarization of the light is always 0. Figure 16 is obtained by a finite difference time domain FDTD simulation as in Refs. 19 and 20. The point spread Fig. 17 The simulated far field diffraction pattern for two different polarizations of light. a The surface alignment perpendicular to the electrode direction. b The surface alignment parallel to the electrode direction. function for the two polarizations of light is shown in Fig. 17a. Because the intensity of the light polarized perpendicular to the incident polarization is always zero in this case, the far field intensity of light for this mode is also 0. The fringing electric field influences for both cases are different. Figure 18a shows the isopotential line in a LC- SLM with 10.5-m element spacing thickness of the liquid crystal cell is 2 m, and Fig. 18b shows the isopotential line in a LC-SLM with 1.5-m element spacing. The voltage profile in Fig. 18a is relatively well defined. However, in Fig. 18b, the isopotential line extends to the element with 0 V, and in parts of the element, the electric field is pointing to directions not perpendicular to the normal direction of the cell surface. Such electric field distribution will produce a phase modulation profile that strongly deviates from the desired phase profile as shown in Fig. 19a. Thus the Strehl ratio of the LC-SLM in this case is only 26.3%, as shown in Fig. 19b. Due to the influence of the fringing electric fields, the diffraction efficiency of the 1D LC-SLM decreases as the element spacing of the device decreases; the efficiency for Fig. 16 The FDTD simulation of light propagation in the two configurations of 1D LC-SLM. a The surface alignment perpendicular to the electrode direction. No polarization is observed in this case. b The surface alignment direction is parallel to the electrode direction. Strong polarization effect is observed, especially at regions close to the phase reset. Fig. 18 The isopotential line in two configurations of 1D LC-SLM. Cell thickness d=2 m. Adjacent isopotential lines have 0.1 V difference. a Electrode width=10 m, gap between electrode =0.5 m. b Electrode width=1 m, gap between electrode =0.5 m. 044001-9

Fig. 19 The simulated phase profile and far field diffraction pattern of a 1D LC-SLM with surface alignment direction perpendicular to the line shape electrode n=0.20. The segmented horizontal lines in a are the desired phase profile of a stairlike blazed grating. The continuous line is the simulated phase profile. Here element spacing=1.5 m, gap between electrode=0.5 m, cell thickness d =6.0 m, and pretilt=3 deg. a blazed grating with eight electrodes per blaze as shown in Fig. 20. In this case, the voltage applied to each element is obtained from a phase versus voltage relationship estimation based on a LC-SLM with infinitely large electrodes as in Ref. 19. However, for a 1D LC-SLM, an optimization of the voltage profile for each individual element can be carried out to improve the diffraction efficiency, 18 which will be described later in this section. It is possible to describe the influence of the fringing electric field on the diffraction efficiency of the LC-SLM in a more quantitative way. Remember there is a term F in Eq. 8 that is the width of the fly-back region. We find F is related to the strength of the fringing electric field, which is again related to the cell thickness of the device. We propose an empirical relationship between the width of the fly-back region and the cell thickness as expressed in Eq. 9 Fig. 20 The diffraction efficiency of 1D LC-SLM as a function of element spacing. Surface alignment direction perpendicular to line shape electrode. Element spacing=1.5 to 19.4 m, gap between electrode=0.5 m, pretilt=3 deg no voltage optimization. Fig. 21 The comparison of diffraction efficiency as function of cell thickness for LC-SLM without voltage optimization and with optimization. Element spacing=1.5 m, gap between electrode=0.5 m, the diffraction angle of the device is 7.4 deg. F = a N d PS. Here F is the width of fly-back region; is the width of one reset; a is a constant related to the strength of the fringing electric field; d is the thickness of the liquid crystal device; PS is the element spacing of the LC-SLM, which is used here as a normalizing factor to make the left side of Eq. 9 dimensionless; N is the average number of elements for every reset. By inserting Eq. 9 into 8, the final expression for the diffraction efficiency becomes Eq. 10, which describes very well the diffraction efficiency data obtained by both director simulation and FDTD simulation =1 F 2 = 1 ad/ps N 9 2. 10 Here N is the average number of elements for every wave of aberration, which is 8 for the eight electrode reset scheme considered in Fig. 20. The fitted line agrees excellently with the FDTD simulation data. The coefficient a is found to be a=1.4, and the value of is found to be 0.7 for all cases. In Fig. 21, the diffraction efficiency before and after voltage optimization is shown as a function of the LC cell thickness. For all cell thicknesses, the voltage optimization improved the diffraction efficiency substantially. We use Eq. 10 to fit the calculated diffraction efficiency as a function of the cell thickness. For cases where no voltage optimization process is carried out, the coefficient a is 1.4. However, for cases where voltage optimization is carried out, the coefficient a is 0.85 and is 0.7. A summary of the diffraction efficiency of a 1D LC- SLM with surface alignment direction perpendicular to the line shape electrode is shown in Fig. 22. If a 1D LC-SLM is built with a commercial off-the-shelf liquid crystal material refractive index anisotropy n=0.2 and no voltage optimization process is performed, such a LC-SLM for a 1550-nm wavelength can deflect a laser beam about 1.8 deg 044001-10

Fig. 22 The comparison of diffraction efficiency of different 1D LC- SLM with surface alignment direction perpendicular to electrodes. Fig. 23 The comparison of diffraction efficiency for 1D LC-SLM with surface alignment direction perpendicular to electrodes and parallel to electrodes no voltage optimization. Fig. 24 The comparison of polarization ratio for 1D LC-SLM with surface alignment direction parallel to electrodes no voltage optimization. with 70.7% efficiency. However, if high birefringence LC material with n=0.35 at a 1550-nm wavelength is utilized, the same 70.7% efficiency can be achieved with a diffraction angle of 4.4 deg. If the voltage optimization process is carried out for the LC-SLM with high birefringence LC material, then the 70.7% efficiency angle could be as high as 7.7 deg. To compare the diffraction efficiency of the 1D LC-SLM with surface alignment direction parallel to the electrode and perpendicular to the electrode, the diffraction efficiency of the two cases of 1D LC-SLM is plotted as function of the element spacing in Fig. 23. The diffraction efficiency for a 1D LC-SLM with parallel surface alignment is about 5 to 15% lower than the case of perpendicular surface alignment. There are two reasons associated with such a drop in the diffraction efficiency. The first reason is that the parallel surface alignment configuration has a strong orthogonal polarization effect associated with the LC-SLM structure. For example, in Fig. 24, the polarization ratio the relative intensity of unwanted light polarization versus the total intensity of light of such a LC-SLM is shown. Another possible effect is that the out-of-plane structure of the LC director can change the phase profile of the LC- SLM from the desired one. As previously discussed, an optimized voltage profile can be obtained by the optimization process for the case where the surface alignment is perpendicular to the electrode. We can use the same optimized voltage profile for the case where the surface alignment is parallel to the electrode. It turned out the optimized voltage profile works well for both cases. The diffraction efficiency for the 1D LC-SLM with both cases improved substantially. The voltage optimization improvement is more substantial for LC-SLM with small element spacing. The results are shown in Fig. 25. The diffraction efficiency of a 2D LC-SLM is a range because it depends on the pattern of the phase profile it generates defined by the two 1D LC-SLM cases in the previous discussion. The high bound value of the diffraction efficiency of a 2D LC-SLM, max corresponds to a 1D LC-SLM with surface alignment perpendicular to the electrode; the low bound value min corresponds to a 1D LC- SLM with surface alignment parallel to the electrode. Such treatment is valid when there is no coupling effect between the phase variation of the two orthogonal axes. The coupling effect between the phase variation in two orthogonal axes can be studied by considering the degree of depolarization of the incident linearly polarized light. Consider one element in the center of a group of nine elements. No volts are applied to the center element and nonzero voltage is applied to others. A strong depolarization effect will happen when the in-plane field gradient is perpendicular to the alignment axis of the device, as shown in Fig. 26. Computer simulation with a LC3D core 21 is used to simulate director configuration and an extended Jones 22 calculation is used to calculate the transmission of the liquid crystal 044001-11

Fig. 25 The comparison of diffraction efficiency for 1D LC-SLM with surface alignment direction perpendicular to electrodes and parallel to electrodes with voltage optimization. The improvement on diffraction efficiency is more apparent for small element spacing. Fig. 27 The polarization effect in a LC-SLM with large element spacing of 16.5 m. device between parallel polarizers. The depolarization effect depends on the voltage difference between neighboring elements and the element spacing of the device. For an element size smaller than 16 m, the depolarization effect is strong, see in Fig. 26, where the simulation was made for element spacing of 8 m. However, for a large element device, as in Fig. 27, where the cell gap is 6 m and the element spacing is 16 m, the de-polarization effect region is pushed toward the edge of the device and the net depolarization effect is small. In this case, the phase variation on the two orthogonal axes can be treated as independent. We can conclude here that for cases when the element spacing is much larger than the cell gap, our efficiency analysis is accurate. The diffraction efficiency of a LC-SLM also depends on whether there will be a trapped wall in the device. The trapped wall occurs when the electric field gradient between the elements causes the LC director to tip with the opposite rotational sense as that defined by the surface Fig. 26 The polarization effect in 2D LC-SLM with small element spacing. a Voltage applied to the center element is 1.2 V; the voltage applied to all other elements is 2.35 V. b Voltage applied to the center element is 0 V; the voltage applied to all other elements is 5V. pretilt direction. Such an effect has been studied previously 23 and can be controlled by increasing the pretilt angle of the surface alignment. 18 3.3 Comparison of the Numerical Modeling Results with the Simple Model With the above discussion, it is possible to relate the Strehl ratio of the system to many design parameters as in Eq. 11 total = s ray aber SLM = s 1 u 0 W sinl 2 /m l /m 2 1 ad/ps. 11 For example, the Strehl ratio is a function of the scattering and reflection loss in the system s, the extent of the ray missing its conjugate element u 0, the resolution of the correction device m, the complexity of the wavefront, the element spacing of the correction device PS, and the cell thickness of the device d. It is possible to estimate the efficiency 1 by using the empirical model described in Eq. 11 or 2 by numerical calculation of the value of the four terms. Let us compare methods 1 and 2 by considering a system that has no passive loss s =1.0. We consider that 32 waves pv 16.6 waves rms of primary defocus is corrected by a 500500 resolution correction device with 9.79.7 mm aperture. The LC-SLM used in the experimental section has an element spacing of 19.4 m and a cell thickness of d=6.0 m. The resolution is 1024768, and the aperture is 2015 mm. Only part of the aperture 9.79.7 mm is used with 500500 resolution. The LC material used has n=0.20. The average wavefront slope of the aberration is about 0.004 rad for 32 waves pv of primary defocus. If we use Eq. 11 to calculate the Strehl ratio, the first factor s =1.0 for an ideal system without passive loss. The second factor ray =82%, as previously calculated in Fig. 4, and the third factor is aber =70%, as N 044001-12

previously calculated in Fig. 5. The fourth factor here and the Strehl ratio of the system calculated by the empirical model is SLM = 1 ad/ps N 2 = 1 1.4 * 6.0/19.4 0.7 /500/2*32 2 = 0.84, total = s ray aber SLM = 1.0 0.82 0.70 0.84 = 0.48. As discussed before, if we use the CODE V numerical modeling method, the Strehl ratio after correcting the 32 waves of aberration is 0.63. The residual aberration is 0 except for the wavefront discontinuities. This is equivalent to the product of the second and third factors in Eq. 11. So ray aber =0.63. The first factor is still s =1.0 for a system with no passive loss. To obtain the value of the fourth term, consider the case in Fig. 23, where the element spacing is 19.4 m, and there are eight elements per blaze, the wavefront slope produced by the LC-SLM is also about 0.004 rad. The Strehl ratio low bound value corresponds to a surface alignment parallel to the electrode case is min =0.815, and the Strehl ratio high bound value corresponds to a surface alignment perpendicular to the electrode case is max =0.858. An average of Strehl ratio for the 2D LC- SLM is SLM =0.837 that is very close to the estimation with Eq. 11 in the last paragraph. The Strehl ratio as calculated by numerical modeling for the considered system is then total = s ray aber SLM =1.00.630.837=0.52. This demonstrates that the concepts of the simple model are consistent with a more detailed numerical calculation. Cell type Table 2 Specification of the LC-SLM correction device. Electric controlled birefringence Active area 2015 mm 2 Fill factor 96% Resolution Bit depth Response speed Stroke length Uniformity Diffraction efficiency 1024768 8 bit, 256 grayscale 50 Hz, visible version 5 Hz, IR version 700 nm at 632.8 nm visible version 2000 nm at 1550 nm IR version 1/10 pv 80% 4 Correction of Aberration in 8-in. Mirror An experimental study was carried out to verify the results of the above-mentioned analysis and modeling, using a liquid crystal on silicon spatial light modulator LCOS SLM. The specification of the LC-SLM wavefront corrector used in the experiment is listed in Table 2. It consists of a thin layer of liquid crystal material sandwiched between the top glass and silicon backplane. Both the top and the bottom substrates are coated with a polyimide alignment layer with a parallel surface alignment. On the silicon backplane, the CMOS gate on each element is hidden behind the segmented aluminum mirror. The size of each mirror is 19 m 2 with a 0.4-m gap between mirrors. The filling factor of the mirror is as high as 96%, and the diffraction loss from the discontinuity of the mirror is small due to the small gap between mirrors. Measured reflectivity is 80% because the aluminum mirror is not coated with a dielectric enhancement layer. The nonuniformity of the liquid crystal layer is less than 1/10 at 632.8 nm. 13 A wavefront measurement and correction system is shown in Fig. 28. The system consists of three parts, an 8-in. telescope, a phase shifting Mach-Zehnder interferometer to measure the wavefront aberration, and a LC-SLM correction device. For the telescope, because the mirror is a diffraction limited mirror with transmitted wavefront aberration less than 1/20 rms, window glass is introduced in front of the mirror to simulate an aberrated primary optical element. Several aberration sources are present in the system, for example, the aberration introduced by the silicon backplane of the LC-SLM device, misalignment of the optics, and the aberration introduced by the window glass. The surface of the silicon backplane is not an optical flat because during the manufacturing process the silicon backplane is glued to a none-flat heat sink, which creates a bowl-shaped surface for the whole chip. Such a bowlshaped surface introduces a wavefront aberration of 18.7 waves pv, as discussed in Ref. 13. The aberration at the exit pupil is measured by interfering a reference beam with the aberrated beam. The standard phase shifting algorithm is used to extract the phase information of the aberrated beam. The phase profile is subsequently unwrapped and filtered with a noise removal filter, as shown in Fig. 29. The measured aberration is decomposed into Zernike components, as listed in Table 3. The most significant Zernike mode is 7.75 waves rms of primary defocus and 2.1 waves rms of astigmatism. The magnitude of the aberration is 34 waves pv which includes the 18.7 waves pv on the LC-SLM device. An optimized conjugate phase profile is introduced to the LC-SLM device to correct for the aberration. The optimization process consists of several steps as follows. First, the wavefront aberration is measured and a conjugate cor- Fig. 28 The experimental setup of aberration measurement and wavefront correction system for 8-in. telescope. 044001-13

Fig. 29 The measured aberration of the telescope before wavefront correction. Fig. 31 The residual wavefront aberration of the telescope after wavefront correction. 044001-14

Table 3 Zernike mode of the transmitted wavefront aberration on LC-SLM clear aperture introduced by surface deformation of silicon backplane. Here we use fringe Zernike polynomials Ref. 24 developed by University of Arizona, which are a subset of the standard Zernike polynomial with the terms of the Zernike mode arranged in a different order. Zernike Term Order Zernike Coefficient waves at 632.8 nm 1 Piston 0 7.781 2 Tilt 1 1.603 3 Tilt 1 0.583 4 Focus 1 7.751 5 Astigmatism 2 2.097 6 Astigmatism 2 0.776 7 Coma 2 0.640 8 Coma 2 0.130 9 Spherical 2 0.0783 10 Astigmatism 3 0.195 11 Astigmatism 3 0.052 12 3 0.295 13 3 0.031 14 Coma 3 0.056 15 Coma 3 0.045 16 Astigmatism 3 0.060 rection phase is put on the LC-SLM. Second, the residual wavefront error is measured and added to the correction phase of the previous step. Then, the residual aberration is measured again, and this procedure is repeated until the residual aberration at the exit pupil at the interferometer is minimal. In Fig. 30, the phase map introduced to the LC- Fig. 30 The correction phase plate introduced to the LC-SLM correction device. Fig. 32 The far field point spread function of the compensated beam a and the noncompensated beam b. SLM device at the final iteration is shown. The residual wavefront map after three loops of the above-mentioned iteration is shown in Fig. 31. The wavefront at the exit pupil of the telescope is well corrected, as the magnitude of the residual wavefront error is less than 1/10 pv or 1/30 rms. Diffraction limited performance is reached after the correction. It has been noticed that the residual wavefront map has a ringlike pattern similar to the reset pattern on the correction phase plate. The wavefront of the front surface reflection is aberrated and could not be corrected because this portion of light does not go through the liquid crystal layer. Thus, in the captured interferogram, such front surface reflection will show up as the abovementioned ringlike patterns. Another possible contributing factor is the diffraction from the discontinuities of the wavefront, although the intensity of such diffracted light is small. For more details regarding this, please see Ref 19. If we focus the laser beam to a charge-coupled device CCD camera, the far field beam diameter of the compensated beam can be obtained to be 136.4 m or 1.31 diffraction limited beam waist. In Fig. 32, the point spread function captured by the CCD camera is shown before wavefront compensation and after wavefront compensation. However, it is hard to determine the Strehl ratio of the system due to the difficulty in measuring the peak intensity of the aberration free system, because several aberration sources are present and there are front surface losses from many optical components. We can only measure the peak intensity of the compensated beam, which is 339 times higher than the noncompensated beam. If we try to approximate the Strehl ratio of the system by the encircled energy of the far field beam, the amount of energy passing through a 120-m pinhole is 51% that of the total laser energy at far field. The detector collects all light at the focus of the lens at 1170 mm. The size of the detector is 55 mm. The intensity of the beam with or without the pinhole is measured. When the pinhole is in the optical path, it is directly mounted on the detector to reduce loss. The reason we choose a 120-m pinhole is that the diffraction limited spot size of the beam is 4/f /D =4*0.6328*1170/8.5/3.1415=98.7 m. For our system, there is no truncation to the Gaussian beam, the point spread function is a Gaussian profile instead of Airy disk. In an ideal situation where there is no aberration in the system, all light should go into the pinhole, and the amount of energy collected by the detector should be 100% of the total laser energy. Then the experimentally measured 044001-15

Strehl ratio of the system is total =51%. This measured Strehl ratio is in line with our analytical estimation results of total =48% and the numerical estimation results of total =52% in Sec. 3.3. 5 Conclusion A model to evaluate the performance of a high-resolution diffractive wavefront compensation system of a severely aberrated optical element is proposed. The idea of the model can be verified by accurate numerical modeling of the high-resolution wavefront compensation, both at the system level and the device level. The aberration correction efficiency for different magnitudes and types of aberration is discussed, as well as the influence of the resolution of a LC-SLM on the correction efficiency. The viability of a large aperture deployable telescope is greatly improved by utilizing the correction power of the LC-SLM to greatly lower the tolerance placed on primary mirror. Up to several tens of waves of aberration can be corrected efficiently with current LC-SLM device, and larger correction with high efficiency can be achieved by increasing the resolution of the device. Experimentally, the correction of an aberrated parabolic reflector is demonstrated and compared to our system analysis. Acknowledgments This work is supported by National Aeronautics and Space Administration NASA space communication project and Defense Advanced Research Projects Agency DARPA TeraHertz Operational Reachback THOR project. The authors thank Michael Fisch for advice and helpful discussions. References 1. M. T. Gruneisen, R. C. Dymale, J. R. Rotge, D. G. Voelz, and M. Deramo, Wavelength-agile telescope system with diffractive wavefront controlled acousto-optic spectral filter, Opt. Eng. 4410, 103202 2005. 2. M. T. Gruneisen, R. C. Dymale, J. R. Rotge, L. F. DeSandre, and D. L. Lubin, Compensated telescope system with programmable diffractive optic, Opt. Eng. 442, 023201 2005. 3. M. T. Gruneisen, R. C. Dymale, J. R. Rotge, L. F. DeSandre, and D. L. Lubin, Wavelength-dependent characteristics of a telescope system with diffractive wavefront compensation, Opt. Eng. 446, 068002 2005. 4. B. Winker, M. Mahajan, and M. Hunwardsen, Liquid crystal beam directors for airborne free-space optical communications, IEEE Aerosp. Conf. Proc., Vol. 3, pp. 1709-1714 2004. 5. M. T. Gruneisen, L. F. DeSandre, J. R. Rotge, R. C. Dymale, and D. L. Lubin, Programmable diffractive optics for wide-dynamic-range wavefront control using liquid crystal spatial light modulators, Opt. Eng. 436, 1387 1393 2004. 6. L. J. Friedman, D. S. Hobbs, S. Lieberman, D. L. Corkum, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and T. A. Dorschner, Spatially resolved phase imaging of a programmable liquid crystal grating, Appl. Opt. 3511, 6236 6240 1996. 7. V. G. Dominic, A. G. Carney, and E. A. Watson, Measurement and modeling of the angular dispersion in liquid crystal broadband beam steering devices, Opt. Eng. 3512, 3371 3799 1996. 8. X. Wang, D. Wilson, R. Muller, P. Maker, and D. Psaltis, Liquidcrystal blazed-grating beam deflector, Appl. Opt. 3935, 6545 6555 2000. 9. D. P. Resler, D. S. Hobbs, R. C. Sharp, L. J. Friedman, and T. A. Dorschner, High efficiency liquid-crystal optical phased-array beam steering, Opt. Lett. 219, 689 693 1996. 10. R. M. Matic, Blazed phase liquid crystal beam steering, Proc. SPIE 2120, 194 205 1994. 11. L. M. Blinov, Electro-optical effects in liquid crystals, Sov. Phys. Usp. 17, 658 672 1975. 12. S. Serati, H. Masterson, and A. Linnenberger, Beam combining using a phased array of phased arrays PAPA, IEEE Aerosp. Conf. Proc., vol. 3, pp. 1729 1735 2004. 13. X. Wang, B. Wang, J. Pouch, F. Miranda, M. Fisch, J. E. Anderson, V. Sergan, and P. J. Bos, Liquid crystal on silicon LCOS wavefront corrector and beam steerer, Proc. SPIE 5162, 139 146 2003. 14. P. F. McManamon, E. A. Watson, T. A. Dorschner, and L. J. Barnes, Applications look at the use of liquid crystal writeable gratings for steering passive radiation, Opt. Eng. 3211, 2657 2664 1993. 15. G. D. Love, Wave-front correction and production of Zernike modes with a liquid crystal spatial light modulator, Appl. Opt. 36, 1517 1520 1997. 16. D. Dayton, S. Restaino, and J. Gonglewski, Novel spatial light modulators for active and adaptive optics, Proc. SPIE 4124, 78 88 2000. 17. J. W. Goodman, Introduction to Fourier Optics, 2nd ed., pp. 210 216, McGraw-Hill Companies Inc., New York 1988. 18. X. Wang, B. Wang, P. J. Bos, P. F. McManamon, J. J. Pouch, F. A. Miranda, and J. E. Anderson, Modeling and design of an optimized liquid-crystal optical phase array, J. Appl. Phys. 987, 073101-1 07310-8 2005. 19. X. Wang, B. Wang, P. J. Bos, J. E. Anderson, J. J. Pouch, and F. A. Miranda, Finite-difference time-domain liquid-crystal optical phased array, J. Opt. Soc. Am. A 222, 364 354 2005. 20. X. Wang, B. Wang, J. Pouch, F. Miranda, J. E. Anderson, and P. J. Bos, Performance evaluation of a liquid-crystal-on-silicon spatial light modulator, Opt. Eng. 4311, 2769 2774 2004. 21. J. E. Anderson, P. E. Watson, and P. J. Bos, Liquid Crystal Display 3-D Director Simulator Software and Technical Guide, Artech House, Norwood, Mass. 2001. 22. A. Lien, Detailed derivation of extended Jones Matrix representation for twisted nematic liquid crystal displays, Liq. Cryst. 222, 171 175 1977. 23. A. Lien and R. A. John, Lateral field effect in twisted nematic cells, IBM J. Res. Dev. 36, 51 58 1992. 24. J. C. Wyant and K. Creath, Basic wavefront aberration theory for optical metrology, in Applied Optics and, vol. 11, pp. 27 39, Academic Press, Inc., New York 1992. Xinghua Wang received his PhD from Liquid Crystal Institute, Kent State University in 2005. His research interest at Kent State is in high-resolution wavefront control and laser beam shaping and manipulation with liquid crystal spatial light modulator, optical phased array, and liquid crystal electrooptical devices. He is currently an optical scientist at ChemImage Corp., leading the development of electro-optical tunable filter for hyperspectral fluorescence, near IR, and Raman imaging. Bin Wang received his BS in physics from Shaanxi Normal University in 1984, MS in solid state physics from Institute of Physics, Chinese Academy of Sciences in 1990, and PhD in chemical physics from Kent State University in 2002. He currently works as a research associate at the Liquid Crystal Institute, Kent State University. His research interests are different applications of liquid crystal materials, computer simulations for multidimensional liquid crystal director fields, and optical calculations. Philip J. Bos received his PhD in physics from Kent State University in 1978. After one year as a research fellow at the Liquid Crystal Institute at Kent State, he joined Tektronix Laboratories in the Display Research Department. In 1994, he returned to the Liquid Crystal Institute, where he is currently an associate director and a professor of chemical physics. He currently has several projects in the area of applications of liquid crystals. He has over 100 publications and 20 patents. 044001-16

Paul F. McManamon is chief scientist for infrared sensors, Sensors Directorate, Air Force Research Laboratory, Air Force Materiel Command, Wright-Patterson Air Force Base, Ohio. He has initiated and technically led in many substantial DARPA technology development efforts. He is responsible for research across a variety of electro-optical sensors technologies, including multifunction laser radar technology, novel electrooptical countermeasure systems, and optical phased-array beam steering technology. He is widely recognized in the electro-optical community. He is the president of SPIE, and sits on the board of directors and the SPIE executive committee. He is also a fellow of the Air Force Research Laboratory AFRL and a fellow of the Military Sensing Symposia MSS. Felix A. Miranda: biography and photograph not available. James E. Anderson earned his PhD in chemical physics from the Liquid Crystal Institute at Kent State University in 2000. He has written numerous papers on the study of liquid crystals. He has been a member of chaired paper selection committees for several conferences. He is currently lead scientist at Hana Microdisplay Technologies. John J. Pouch received his PhD in physics, with a minor degree in mathematics, from Wayne State University, Detroit, Michigan, in 1981. Since 1983, he has been employed at NASA s John H. Glenn Research Center in Cleveland, Ohio. He is a member of the Antenna, Microwave, and Optical Systems Branch. His current activities include managing projects that deal with new optical communications technologies. These efforts are being carried out by academia and industry. He has coauthored over 55 papers in scientific journals and proceedings volumes and over 75 presentations at national and international conferences. He has also coedited 5 books 13 volumes and 2 conference proceedings volumes. 044001-17