Motion control of the wedge prisms in Risley-prism-based beam steering system for precise target tracking

Transcription

1 Motion control of the wedge prisms in Risley-prism-based beam steering system for precise target tracking Yuan Zhou,,2, * Yafei Lu, 2 Mo Hei, 2 Guangcan Liu, and Dapeng Fan 2 Department of Electronic and Communication Engineering, Changsha University, Changsha 40003, China 2 College of Engine-electronic and Automation, National University of Defense Technology, Changsha 40073, China *Corresponding author: zhouyuan304@63.com Received 3 January 203; revised 22 March 203; accepted 23 March 203; posted 25 March 203 (Doc. ID 8359); published 8 April 203 Two exact inverse solutions of Risley prisms have been given by previous authors, based on which we calculate the gradients of the scan field that open a way to investigate the nonlinear relationship between the slewing rate of the beam and the required angular velocities of the two wedge prisms in the Risleyprism-based beam steering system for target tracking. The limited regions and singularity point at the center and the edge of the field of regard are discussed. It is found that the maximum required rotational velocities of the two prisms for target tracking are nearly the same and are dependent on the altitude angle. The central limited region is almost independent of the prism parameters. The control singularity at the crossing center path can be avoided by switching the two solutions. 203 Optical Society of America OCIS codes: ( ) Prisms; ( ) Geometric optics; ( ) Mathematical methods (general); ( ) Optomechanics; (280.00) Aerosol detection; ( ) Geometric optical design. Introduction Beam steering devices are critical for optical systems in which beam alignment and target tracking are required, such as free-space optical communications, countermeasure, laser weapons, and fiber-optic switches []. Risley prisms are attractive because they offer numerous advantages over competitive beam steering systems, including compact size, conformal aperture, low moment of inertia, wide field of regard (FOR), and relative immunity to vibration [2,3]. However, Risley prisms still have to face several problems as a beam steering system. Some problems are associated with optical design, such as chromatic dispersion [4,5], Fresnel reflection losses [6], beam compression [7,8], and wavefront X/3/ $5.00/0 203 Optical Society of America quality, which have been studied in a number of publications. Some more difficult problems in the steering processes are related to the motion control of prisms, which is actually more important for some applications such as target tracking. To perform target tracking, the prisms must be controlled to rotate to a certain orientation in a short time so as to track the dynamic targets rapidly, which is a challenge for Risley-prism-based beam steering systems. The prevalent challenges include nonlinearity and singularity problems [3,6,9,0]. Unlike the conventional gimbal-based beam steering system, the relation between the pointing position and the corresponding prisms orientations is nonlinear in the Risley-prism-based systems, which leads to nonlinearity of the system control. Furthermore, while the emerging beam traces the target through the center of the FOR along a continuous path, control singularities can occur that require infinite rotational velocities of the prisms. 20 April 203 / Vol. 52, No. 2 / APPLIED OPTICS 2849

2 In order to eliminate the singularities, some companies propose the use of a third prism, which allows the system to perform smooth tracking of arbitrary target trajectories across the FOR [3,9]. However, this also gives rise to numerous problems, such as Fresnel reflection losses and backreflections. Additionally, using a third prism increases the overall system complexity and may limit the maneuverability of the beam steering system []. In many specific applications, having two prisms still seems like the logical solution, so it is desirable to discuss and evaluate the nonlinearity and singularity problems in the Risley-prism-based beam steering system for tracking applications. Detailed analysis of these problems is very rare in the literature, and the previous studies generally provide only an introduction to the topic. The outline of this study is as follows. In Section 2, the two-step method [2] is employed to investigate the relation between any given pointing position and the corresponding prisms orientations based on the nonparaxial ray-tracing method through the prism systems. Two exact inverse solutions are obtained in a closed analytical form. In Section 3, by calculating the two-dimensional numerical gradients of the solutions, the nonlinearity relation between the required angular velocities of the prisms and the slew rate of the beam is studied. The angular velocity requirements of the two prisms are tested in the optical tracking application. The limited regions and nonlinearity point that occur at the center and the edge of the FOR are discussed in Section 4. Conclusions are drawn at the end of this study. 2. Relation Between Pointing Position and Prisms Orientations Figure is a schematic diagram illustrating the configuration of Risley prisms. The two prisms Π and Π 2, whose principal cross sections are right triangles, are arranged with their flat sides together and in the position perpendicular to the z axis, which is also the axis of rotation. The indices and opening angles of Π and Π 2 are expressed as n, n 2 and α, α 2. The pointing position of the output ray is depicted with altitude angle Φ and azimuth angle Θ in a polar coordinate space. The nonparaxial ray-tracing method, based on the vector form of Snell s law, can be performed to give the exact solution of the pointing position of the emerging ray. When the incident ray is collinear with the z axis, the direction cosines K; L; M of the ray vector for the emerging ray are in the form [3] K a cos θ a 3 sin α 2 cos θ 2 ; L a sin θ a 3 sin α 2 sin θ 2 ; (a) (b) Fig.. Schematic diagram illustrating the notation and coordinate systems for Risley prisms. The incident ray is collinear with the z axis, which is also the axis of rotation for the two prisms Π and Π 2. The rotational angles θ and θ 2 are measured from the x axis. The direction of the emerging ray is described as altitude angle Φ and azimuth angle Θ in a polar coordinate space. a n 2 sin α n cos α n 2 sin2 α 2 ; a 2 n 2 n 2 n sin2 α 2 cos α sin 2 α ; a 3 a sin α 2 cos Δθ a 2 cos α 2 n 2 2 a sin α 2 cos Δθ a 2 cos α ; Δθ θ 2 θ : The azimuth Θ of the beam emerging from the system is given as Θ L K; For K 0 L K π; For K<0 : (2) The altitude angle Φ is given as Φ arccos M : (3) For a specific system, the refractive indices and opening angles of the two prisms are invariable, so that the altitude angle Φ only depends on jδθj. For a given altitude angle Φ, jδθj can be calculated from Eqs. (a) (3) in the form jδθj arccos a a tan α a 2 cos Φ 2 : (4) n 2 2 a2 cos Φ cos α 2 where M a 2 a 3 cos α 2 ; (c) To obtain the rotational angles θ and θ 2 of the two prisms to steer the beam to the direction specified by Φ; Θ, the two-step method can be employed. The 2850 APPLIED OPTICS / Vol. 52, No. 2 / 20 April 203

3 first step is to keep one of the prisms stationary at 0 rotational angle and rotate the other prism until the desired altitude Φ is achieved. The angle of azimuth rotation between Π and Π 2 will be jδθj, given by Eq. (4). The second step is a simultaneous rotation of Π and Π 2 until the desired azimuth Θ is reached. There are two sets of solutions for the rotational angles of the two prisms. The first one, expressed as θ 0 ; θ0 2, is related to jδθj θ 2 θ, while the other one, corresponding to jδθj θ θ 2, is expressed as θ 00 ; θ00 2. To obtain the first set of solutions θ 0 ; θ0 2 with the two-step method, the prism Π is kept stationary at 0 orientation and we rotate the prism Π 2 along the forward direction (counterclockwise) up to jδθj, given by Eq. (4). The beam will point to a new direction specified by the angles Φ; Θ 0 0. The azimuth displacement Θ 0 0 introduced in the first step is given as L Θ 0 0 arctan ; K θ 0 For K 0; (5a) θ 2 jδθj L Θ 0 0 arctan K θ 0 π; For K<0: (5b) θ 2 jδθj θ 0 2 Θ0 jδθj Θ Θ0 0 jδθj: (7b) Similarly, the second set of solutions θ 00 ; θ00 2 to the rotational angles of the two prisms can be obtained with the two-step method. The prism Π is kept stationary at 0 orientation and we rotate the prism Π 2 in a clockwise direction up to jδθj, which introduces azimuth displacement Θ 00 0, given by L Θ 00 0 arctan ; For K 0; (8a) K θ 0 θ 2 jδθj L Θ 00 0 arctan K θ 0 π; For K<0: (8b) θ 2 jδθj A close examination of Eqs. (5) and(8) reveals the relation Θ 0 0 Θ00 0 : (9) The corotational angle Θ 00 is given as Θ 00 Θ Θ00 0 Θ Θ0 0 : (0) The final rotation angles of Π and Π 2 can be given as Then the two prisms are rotated simultaneously until the desired azimuth Θ is reached. The corotational angle Θ 0 is represented as θ 00 Θ00 Θ Θ0 0 ; θ 00 2 Θ00 jδθj Θ Θ0 0 jδθj: (a) (b) Θ 0 Θ Θ0 0 : (6) The final rotation angles of Π and Π 2 can be expressed, respectively, as θ 0 Θ0 Θ Θ0 0 ; (7a) In Fig. 2, the color maps describe the first set of solutions θ 0 ; θ0 2 from Eqs. () (7) in the various pointing positions of the emerging beam for systems including two identical prisms with opening angle α α 2 0 and refractive index n n 2.5 (glass prisms). The horizontal deviation Φ x and Fig. 2. First set of solutions θ 0 ; θ0 2 for the various pointing positions of the output beam. 20 April 203 / Vol. 52, No. 2 / APPLIED OPTICS 285

4 the vertical deviation Φ y in the Cartesian coordinate can be defined as Φ x Φ cos Θ; Φ y Φ sin Θ: (2) It can be seen that the rotational angle θ 0 and θ0 2 of the two prisms is nonlinear with the pointing position of the output beam. Analogous results can also be obtained for the second set of solutions θ 00 ; θ Nonlinearity Analysis Since the prisms are rotated about the optical axis, the consequent beam motion is naturally described in a polar coordinate space. The radial slew rate ω r and the tangential slew rate ω t of beam steering are expressed, respectively, as the time derivative of altitude and azimuth angle, in the form ω r dφ dt; ω t dθ dt: (3) In order to test the angular velocity requirements of the two prisms in the optical tracking application, one can take the time derivative of the rotational angles solved with the two-step method. From Eqs. (7) and (), the required angular velocity of Π and Π 2 can be shown in the form ω 0 dθ0 dt ω t dθ0 0 dt ; ω 00 dθ00 dt ω t dθ0 0 dt ω 0 2 dθ0 2 dt ω t dθ0 0 dt d Δθ ; dt ω 00 2 dθ00 2 dt ω t dθ0 0 dt d Δθ dt For prism Π ; (4a) For prism Π 2 : (4b) Upon substituting from Eqs. () (8), the angular velocities can be calculated in detail by a long derivation, but the specific analytical expressions are complicated and omitted here. Since the indices and opening angles of the prisms remain invariable for a given system, a close examination of the equations reveals that Θ 0 0 and Δθ are only determined by the target altitude angle Φ and are independent of the azimuth Θ. Therefore, for a track target with tangential motion i.e., the target moves only around the rotation axis; the azimuth angle Θ changes, while the altitude angle Φ remains stable the angular velocities of the two prisms from Eq. (4) are expressed as ω 0 ω00 ω0 2 ω00 2 ω t: (5) It is interesting to mention that, for tangential motion, the equation is independent of the pointing position of the output beam. Contrarily, if the target only moves radially, ω t 0, the angular velocities of the two prisms would vary with Φ and be independent of Θ. The relations can readily derived from Eq. (4) as dθ 00 dφ dθ0 dφ and dθ00 2 dφ dθ0 2 dφ: (6) For a random radial path, the rotation angles of the two prisms can be derived from Eqs. (a) () asa function of the altitude Φ, so that the ratios of the rotational velocities of the prisms to the slew rate of the radial beam steering, expressed by dθ 0 dφ ω0 ω r, dθ 00 dφ ω00 ω r, dθ 0 2 dφ ω0 2 ω r, and dθ 00 2 dφ ω 00 2 ω r, can be obtained by the numerical derivation versus Φ. These ratios are plotted in Fig. 3 under the condition of two identical prisms. The solid curves in Fig. 3 show the ratio dθ 0 dφ, while the dotted curves show the ratio dθ 0 2 dφ, as a function of the altitude Φ. Specifically, Fig. 3(a) is for the system with glass prisms of n.5 and α 0, with the maximum deviation Φ m It can be found that the ratio dθ 0 dφ increases with increasing altitude Φ. Especially, as Φ approaches Φ m, dθ 0 dφ increases dramatically and tends to infinity, which means that the ability of the prism pair to move the optical path in altitude is reduced. A similar situation can also be found in dθ 0 2 dφ, which has a negative value, just implying a reversed rotation direction. Figure 3(b) Fig. 3. Ratios of the rotational velocities of the prisms to the slew rates of the radial beam steering for the systems with (a) n n 2.5, α α 2 0, Φ m and (b) n n 2 4, α α 2 8, Φ m APPLIED OPTICS / Vol. 52, No. 2 / 20 April 203

5 is for the system with silicon prisms of n 4 and α 8, in which the maximum deviation Φ m increases up to It is notable that the ratios are much smaller than those of the glass prisms system. Similarly, the ratios also tend to infinity as the system reaches the edge of the FOR. The angular velocities required for the two prisms depend on the motion path of the target while tracking arbitrary target trajectories. To study the nonlinearity relation between the required angular velocities of prisms and the slew rate of the emerging beam for various tracking paths, the Cartesian coordinate is used to express the pointing position of the emerging beam instead of the polar coordinate. From Fig. 2, the rotational angles of the two prisms can be obtained for all beam pointing positions within the full FOR. The two-dimensional numerical gradients for the rotational angles can be computed in Φ x (horizontal) and Φ y (vertical) directions. In Fig. 4, the gradient vector fields of θ 0 and θ0 2 are displayed with solid lines and the level curves are drawn with dotted lines for the glass prisms system. One can deduce from the gradient vector fields that the moving direction of the target, requiring the two prisms to rotate most rapidly, tends to the tangential direction as the target moves near the rotation axis, while it tends to the radial direction as the target reaches the edge of the FOR. A reverse trend can also be realized from the level curves. Given the direction of target motion, the gradient vector can be used to calculate the direction derivative and estimate the ratios of the rotational velocities of the prisms to the slew rates of the beam steering for arbitrary target trajectories. In the gradient direction, the prisms are required to rotate more quickly and the magnitude of the gradient can be used to represent the maximum ratio of the rotational velocities of the prisms to the slew rates of the beam steering. To trace a target continuously along arbitrary path, the prisms need to provide sufficient rotational velocities for arbitrary direction of target motion. So the magnitude of the gradient can be used to express the strictest speed requirement of prism rotation for smooth tracking. The level curves in Figs. 5(a) and 5(b) are plotted to show the magnitudes of the gradients of θ 0 and θ0 2 for the glass prisms system. It can be seen that the distributions of the magnitudes over the pointing positions show rotational symmetry, which implies that the magnitudes only depend on altitude Φ and have nothing to do with azimuth Θ. Figure 5(c) shows the magnitudes plotted as a function of altitude Φ. The solid curve describing the magnitude of the gradient for θ 0 almost overlaps with the dotted curve describing that for θ 0 2, which means that the maximum rotational velocities required of the two prisms are nearly the same. As altitude Φ approaches zero or the maximum value Φ m, the magnitudes tend to infinity. The analogous curves are also plotted in Fig. 6 for the silicon prisms system, and similar trends can be seen. Compared with glass prisms, silicon prisms result in smaller magnitudes for most regions of the FOR. Specifically, the minimum magnitudes for silicon prisms are only about.6, while those of glass prisms are more than. For a given maximum deviation Φ m, the magnitudes of gradients are calculated for the prisms with varied indices and opening angles, and the results (omitted here) are almost the same. So one can safely draw the conclusion that the distribution of the magnitude of the gradient mainly depends on the maximum ray deviation angle Φ m in the prism system. It is worth mentioning that the large ratio of the rotational velocity of the prisms to the slew rate of the beam steering, which means that a large-angle rotation is required to perform a small beam deviation, is a mixed blessing; i.e., the steering Fig. 4. Gradient vector field and equipotential contour of (a) θ 0 and (b) θ0 2 for the glass prisms system. 20 April 203 / Vol. 52, No. 2 / APPLIED OPTICS 2853

6 Fig. 5. Magnitudes of the gradients of (a) θ 0 and (b) θ0 2 for the glass prisms system. (c) The magnitudes are plotted as a function of altitude Φ. resolution is enhanced, whereas the response performance is degraded. 4. Singularity Problem As shown in Figs. 5 and 6, the magnitudes of gradients increase dramatically while altitude Φ approaches zero or maximum deviation Φ m, which means that the motors are required to be capable of strong accelerations. Since the rotational velocities of the prisms are finite for the given motors, there are some regions in which smooth tracking cannot be performed at the center or the edge of the FOR, which may limit the maneuverability of the target tracking. The ranges of altitude Φ for the limited regions depend on the maximum rotational velocities of the motors as well as the required slew rates of beam steering for target tracking. Figures 7(a) and 7(b) show the magnitudes of gradients of θ 0 and θ0 2 at the center and edge of the FOR for the glass and silicon prism systems, respectively. In all cases, nearly the same results are obtained for θ 0 and θ0 2, which means that the requirements of rotational velocities for the two prisms are almost the same. At the center of the FOR, as shown in Fig. 7(a), the silicon and glass prism systems have approximately the same magnitudes of gradients. So the range of altitude Φ for the limited region at the center of the FOR is almost independent of the prism parameters, specifically, the indices of refraction and the opening angles. The reason is that the gradient vector tends to the tangential direction near the center and the maximum rotational velocities required of the two prisms are mostly determined by the tangential angular velocity ω t of beam steering [seen from Eq. (5)] and are almost independent of the prism parameters. Based on the curves in Fig. 7(a), the range of the limited region at the center can be estimated in terms of the maximum rotational velocities of the motors and the slew rates of beam steering required for target tracking. For a given constant slew rate ω s, the maximum magnitude of gradient θ m, which can be achieved with the system, is determined by the maximum rotational velocities ω m of the motors, and can be used to estimate the angular radius ΔΦ c of the limited region, which is shown as a cone in Fig. 7(c). The curves in Fig. 8 show the angular radius ΔΦ c as a function of the maximum rotational velocities ω m with the slew rate ω s as a parameter. This shows that ΔΦ c decreases with increasing ω m, whereas it increases with the required ω s. For example, we assume a laser beam is required to track a target along arbitrary path at a 40 s slew rate. It can be seen from Fig. 8 that the angular radii of the limited region at the center are 2.29,.5, and 0.76 with maximum rotational velocities ω m 000 s, 2000 s, and 3000 s, respectively. In addition, for a system with two prisms of maximum velocities ω m 2000 s, continuous target tracking can be safely performed only outside a 0.57,.72, or 2.87 cone at the center of the FOR for slew rates ω s 20 s, 60 s, and 00 s, respectively APPLIED OPTICS / Vol. 52, No. 2 / 20 April 203

7 Fig. 6. Magnitudes of the gradients of (a) θ 0 and (b) θ0 2 for the silicon prisms system. (c) The magnitudes are plotted as a function of altitude Φ. Fig. 7. Magnitudes of gradients of θ 0 and θ0 2 at (a) center and (b) edge of the FOR for the glass and silicon prism systems. (c) Schematic diagram of the limited regions at the center and edge of the FOR. There is another limited region at the edge of the FOR, as shown in Fig. 7(b). The reason is that the gradient vector tends to the radial direction at the edge and the maximum ratio of the rotational velocity of the prisms to the slew rate of the beam steering tends to infinity as Φ approaches Φ m (seen from Fig. 3). The limited region looks like a hollow cone in Fig. 7(c), and the angular width ΔΦ e can be estimated with the same method. Figure 9 shows the angular width ΔΦ e for the glass and silicon prism systems, respectively. Similar to the angular Fig. 8. Angular radius ΔΦ c of the limited region at the center of the FOR. 20 April 203 / Vol. 52, No. 2 / APPLIED OPTICS 2855

8 Fig. 9. Angular width ΔΦ e of the limited region at the edge of the FOR. Fig. 0. Sudden changes in the solutions to the rotational angles of (a) prism Π and (b) prism Π 2 for the glass prisms system tracking through the center of the FOR along the x axis coordinate. radius ΔΦ c of the central limited region, ΔΦ e also decreases with increasing ω m and increases with the required ω s, whereas it is much smaller under the same condition. Compared with the glass prisms, the silicon prisms offer smaller angular width for the given ω m and ω s. For example, if the maximum rotational velocity is ω m 000 s and a slew rate of 40 s is required, the angular width for the silicon prisms is only 0.05, while that for the glass prisms is up to Attention is now turned to the control singularity point at the center of the FOR, which is also the coordinate origin where altitude Φ 0 and azimuth Θ is undefined. For a system with two identical prisms, i.e., n n 2 and α α 2, we can obtain from Eq. (4) that the relative azimuthal rotation angle Δθ between Π and Π 2 is 80 and the azimuth displacements Θ 0 0 and Θ00 0 are undefined, so that the final rotation angles of the prisms cannot be obtained from Eqs. (7) and (). However, in practice, the center can be pointed as long as one prism is rotated by 80 with respect to the other and there are countless possible prism orientations in the center. In the tracking applications, it is interesting to investigate the prism position change when the target is crossing the center of the FOR. As mentioned above, the angular velocities of the two prisms are independent of azimuth Θ for a radial motion, so one only needs to focus on an arbitrary path crossing the center. Figure 0 illustrates the changes of the solutions to the rotational angles of the two glass prisms along the x axis coordinate. Since the crossing center motion gives rise to a sudden 80 azimuth change, the two solutions θ 0 ; θ0 2 and θ00 ; θ00 2 also have a sudden 80 change when the system moves the beam through the center of the FOR. However, it should be noted that the two solutions coincide with each other at the zero position, which means that the control singularity can be avoided by switching the two solutions. 5. Conclusions In conclusion, two closed-form analytical inverse solutions for Risley-prism-based beam steering systems are investigated in the two-step method on the basis of the nonparaxial ray-tracing method. The twodimensional numerical gradients for the solutions are calculated to study the nonlinearity relation between the required angular velocities of the prisms and the slew rate of the beam. It is found that the moving direction of the target, requiring the two prisms to rotate most rapidly, tends to the tangential direction as the target moves near the rotation axis, while it tends to the radial direction as the target reaches the edge of the FOR. The maximum rotational velocities required for the two prisms are nearly the same and are dependent on altitude rather than azimuth. The angular velocity requirements of the prisms are alleviated by using large-deviation prisms for most regions of the FOR. The limited regions at the center and edge of the FOR are discussed. It is shown that the central 2856 APPLIED OPTICS / Vol. 52, No. 2 / 20 April 203

9 limited region is almost independent of the prism parameters and is determined by the maximum rotational velocities of the prisms and the required slew rate of the beam. For example, in the case of a 2000 s rotational velocity, continuous target tracking can be safely performed only outside a 0.57,.72, or 2.87 cone at the center for the slew rates 20 s, 60 s, and 00 s, respectively. Compared with the central limited region, the limited region at the edge is much smaller under the same condition. Finally, the singularity point at the center is also analyzed, and it is found that the control singularity at the crossing center path can be avoided by switching the two solutions. This research is supported by the National Natural Science Foundation of China (Grant No ), the Natural Science Foundation of Hunan Province (Grant No. 3JJ322), and the Scientific Research Fund of Hunan Provincial Education Department (Grant No. 2C0473). References. C. Oh, J. Kim, J. Muth, S. Serati, and M. J. Escuti, Highthroughput continuous beam steering using rotating polarization gratings, IEEE Photon. Technol. Lett. 22, (200). 2. E. Schundler, D. Carlson, R. Vaillancourt, J. R. Dupuis, and C. Schwarze, Compact, wide field DRS explosive detector, Proc. SPIE 808, 808O (20). 3. M. Sanchez and D. Gutow, Control laws for a three-element Risley prism optical beam pointer, Proc. SPIE 6304, (2006). 4. C. Florea, J. Sanghera, and I. Aggarwal, Broadband beam steering using chalcogenide-based Risley prisms, Opt. Eng. 50, (20). 5. B. D. Duncan, P. J. Bos, and V. Sergan, Wide-angle achromatic prism beam steering for infrared countermeasure applications, Opt. Eng. 42, (2003). 6. M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, Risley prism beam pointer, Proc. SPIE 6304, (2006). 7. J. Sun, L. Liu, M. Yun, L. Wan, and M. Zhang, Distortion of beam shape by a rotating double-prism wide-angle laser beam scanner, Opt. Eng. 45, (2006). 8. J. L. Gibson, B. D. Duncan, P. Bos, and V. Sergen, Wide angle beam steering for infrared countermeasures applications, Proc. SPIE 4723, 00 (2002). 9. P. J. Bos, H. Garcia, and V. Sergan, Wide-angle achromatic prism beam steering for infrared countermeasures and imaging applications: solving the singularity problem in the two-prism design, Opt. Eng. 46, 300 (2007). 0. J. Lacoursiere, M. Doucet, E. O. Curatu, M. Savard, S. Verreault, S. Thibault, P. C. Chevrette, and B. Ricard, Large-deviation achromatic Risley prisms pointing systems, Proc. SPIE 4773, 23 3 (2002).. R. Winsor and M. Braunstein, Conformal beam steering apparatus for simultaneous manipulation of optical and radio frequency signals, Proc. SPIE 625, 6250G (2006). 2. C. T. Amirault and C. A. Dimarzio, Precision pointing using a dual-wedge scanner, Appl. Opt. 24, (985). 3. Y. Li, Closed form analytical inverse solutions for Risleyprism-based beam steering systems in different configurations, Appl. Opt. 50, (20). 20 April 203 / Vol. 52, No. 2 / APPLIED OPTICS 2857