Closed form analytical inverse solutions for Risley-prism-based beam steering systems in different configurations

Closed form analytical inverse solutions for Risley-prism-based beam steering systems in different configurations Yajun Li P.O. Box 975, Great River, New York 11739, USA ([email protected]) Received 20 April 2011; accepted 30 May 2011; posted 16 June 2011 (Doc. ID 146180); published 21 July 2011 Nonparaxial ray tracing through Risley prisms of four different configurations is performed to give the exact solution of the inverse problem arisen from applications of Risley prisms to free space commu- nications. Predictions of the exact solution and the third-order theory [Appl. Opt. 50, 679 (2011)] are compared and results are shown by curves for systems using prisms of different materials. The exact solution for the problem of precision pointing is generalized to investigate the synthesis of the scan pat- tern, i.e., to create a desirable scan pattern on some plane perpendicular to the optical axis of the system by controlling the circular motion of the two prisms. © 2011 Optical Society of America OCIS codes: 080.0080, 080.2720, 120.5800. 1. Introduction Risley prisms are pairs of rotatable prisms that can be used to continuously scan a laser beam over a wide angular range with a high resolution. There are two basic problems encountered in the applications of Risley-prism-based systems to laser beam steering. First, given the two prisms’ orientations, what is the pointing position of the laser beam emergent from the system? The second problem may be regarded as the inverse of the first problem, i.e., given the required pointing position, what will be the orienta- tions of two prisms? The answer to the first problem can be found in a number of publications [1–4]. Inves- tigations of the second problem, which is known as the inverse problem or the problem of precision pointing [4–8], have been stimulated by the applica- tions of Risley prisms to optical tracking and point- ing of targets in free space. Solutions for the inverse problem, which can be regarded as exact, are very rare in literature. Historically, in the 1985 paper of Amirault and DiMarzio [5], they found difficulties in inverting the vector equations for refraction at the surfaces of the prisms for an exact solution to the problem of precise pointing using Risley prisms. However, they proposed a realistic approach to ob- tain an inverse solution through a two-step process. Ten years later in 1995, Boisset et al. [6] proposed a paraxial method and developed an iterative algo- rithm to solve the inverse problem. Furthermore, Degnan [7] developed in 2002 a first-order approxi- mation method that does not need an iterative algo- rithm to find prisms’ orientations for a given pointing position. Recently, Li proposed a third-order solution for the inverse problem [4]. The first exact solution was given in 2008 by Yang [8], who investigated the case of Risley prisms having two identical dispersive prisms, i.e., the prisms having the isosceles triangu- lar cross sections, whereas this study presents exact inverse solutions for systems containing two wedge prisms on account of the wedge prisms being ideal for laser beam steering and are now commonly used in the real devices. Figure 1 is a schematic diagram illustrating the configuration of Risley prisms, in which the two wedge prisms Π1 and Π2 may have different indexes n1 and n2 and different opening angles α1 and α2. The principal cross sections of Π1 and Π2 are right 0003-6935/11/224302-08$15.00/0 © 2011 Optical Society of America 4302 APPLIED OPTICS / Vol. 50, No. 22 / 1 August 2011 triangles. If the opening angle of the prism is taken as a reference to define the three sides of a right tri- angle and if symbol 1 is used to represent the side adjacent to the opening angle and symbol 2 for the hypotenuse, the configuration shown in Fig. 1 can therefore be described by a combination of the sym- bols as 21-12. Making use of this symbology, we may have a system in the 12-12 configuration if the first prismΠ1 in Fig. 1 is turned 180° around the y axis. In general, there are four different configurations of the two wedge-prism systems as shown in Figs. 2(a)–2(d) described by 21-12, 12-12, 21-21 and 12-21, respec- tively. Different configurations may have different power in ray deviation as we will show in Section 2 of this study. The outline of this study is as follows. In Section 2, nonparaxial ray tracing through the prism systems in different configurations is performed under the basis of the more exact thick prism theory. An exact solution to the problem of precision pointing is developed in Section 3. Predictions of the exact solu- tion and the third-order theory are compared in Section 4. Generalization of the exact result for the problem of precision pointing to scan pattern synth- esis is investigated in Section 5. Conclusions are drawn at the end of this study. This study is conducted under the error-free assumptions: no prism tilt and no beam and mechan- ical axes misalignment will be considered through- out this study. 2. Nonparaxial Ray Tracing Having illustrated the two prisms system in different configurations in Figs. 1 and 2, we can shine a ray through the systems shown in Fig. 2 to compare their powers of ray deviation. The two prisms shown in Fig. 1 are rotatable about the z axis, counted from the xz plane. Their angular positions are specified by their respective rotation angles θ1 and θ2. The incident ray propagates in the direction specified by the ray vector s^ðiÞ1 ¼ ð0; 0;−1Þ and hits the center of the first surface of prism Π1. In case of the 21-12 configuration and when Π1 is in the position of θ1, the first surface of Π1 is now in the position specified by the unit normal vector n^1 ¼ ðsin α1 cos θ1; sin α1 sin θ1; cos α1Þ: ð2:1Þ Apply the vector form Snell’s law [9] to Π1; the in- cident ray is refracted at the first surface of Π1, separating air and glass, into the direction specified by the ray vector s^ðrÞ1 ¼ 1 n1 ½^sðiÞ1 − ðs^ðiÞ1 · n^1Þn^1� − ðn^1Þ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 − � 1 n1 � 2 þ � 1 n1 � 2 ðs^ðiÞ1 · n^1Þ2 s : ð2:2Þ Because the two plane surfaces on the two sides of the air gap in the 21-12 configuration are parallel, they do not affect the direction of the ray propagating over the air gap (see, e.g., Section 14.10 in [1]); we may therefore neglect their contributions to the re- fraction calculations. After doing so, we may extend the formulation shown in Eqs. (2.1) and (2.2) for the first surface of Π1 to the second surface of Π2 that is now in the position specified by the unit normal vector n^2 ¼ ð− sin α2 cos θ2;− sin α2 sin θ2; cos α2Þ: ð2:3Þ Again, apply the vector form Snell’s law to refrac- tion calculations for the ray emergent from the system and we obtain the ray vector s^ðrÞ2 ¼ n2 ½^sðrÞ1 − ðs^ðrÞ1 · n^2Þn^2� − ðn^2Þ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 − n22 þ n22ðs^ðrÞ1 · n^2Þ2 q : ð2:4Þ After a long mathematical development, omitted here, we can write expressions for the direction co- sines ðKA;LA;MAÞ of the ray vector s^ðrÞ2 for the ray emergent from the system in the 21-12 configuration in the form KA ¼ a1 cos θ1 þ a3 sin α2 cos θ2; ð2:5aÞ Fig. 1. Schematic diagram illustrating the notation and coordi- nate systems for Risley prisms. The unit vector s^ðiÞ1 for the incident ray is collinear with the z axis, which is also the axis of rotation for the two prisms Π1 and Π2, which may not be identical and having indexes n1 and n2 and opening angles α1 and α2, respectively. The rotational angles θ1 and θ2 are measured from the x axis and di- agram shows the prisms at θ1 ¼ −90°, θ2 ¼ þ90°. Fig. 2. Diagram illustrating the Risley prisms in four different configurations. The symbol 1 represents the plane face of the wedge prism perpendicular to the axis of rotation and two for the face inclined to the axis. The Risley prisms in groups A and B share the same form expressions for the ray emergent from the prism systems. 1 August 2011 / Vol. 50, No. 22 / APPLIED OPTICS 4303 LA ¼ a1 sin θ1 þ a3 sin α2 sin θ2 ð2:5bÞ and MA ¼ a2 − a3 cos α2: ð2:5cÞ Expressions for the coefficients ða1;a2; a3Þ are shown in Eqs. (T1.1) to (T1.5) in Table 1. The next step is to perform the nonparaxial ray tracing for the Risley prisms in the 12-12 configura- tion as shown in Fig. 2(b). Interestingly, we found that the direction cosines of the ray vector for the ray emergent from the system in the 12-12 con- figuration can be expressed in the same forms as Eqs. (2.5a)–(2.5c) for the system in the 21-12 config- uration. This finding can be explained as follows. The first prism Π1 in the two configurations is in different arrangement, i.e., the 21 and 12 arrange- ment, and, according to the more exact thick prisms theory, these two prisms may have different power of ray deviation, say, δ21 and δ12, and δ21≠δ12. As Π1 is rotated about the z axis, the ray deviated by Π1 will trace out a circular cone symmetrical to the z axis. The half-vertex angles of the circular cones are δ21 and δ12 for the two arrangements, respectively. How- ever, they are all homocentric pencils and symmetri- cally distributed around the optical axes of the systems. The point of great importance is that the second prism Π2 in the two systems is in the same 12 arrangement and the incident rays to Π2 are uni- formly distributed on circular cones scanned out by the first prism Π1. The difference of the half-vertex angles δ21 and δ12 of the circular cones may exert some quantitative influence on the numerical out- comes of the scan field but may not be able to change the mathematical formulism of the scan field. This is why systems in the 21-12 and 12-12 configurations share the same expressions as shown in Eqs. (2.5a)– (2.5c). Therefore, we may consider the two systems in the 21-12 and 12-21 configurations as two members in one group, as shown by the group A in Fig. 2. Similarly, the prisms systems in the 21-21 and 12-21 configurations as shown in Figs. 2(c) and 2(d) can be considered as the two members in the group B (see Fig. 2). Because the direction cosines ðKB;LB; MBÞ of the rays emerging from these two systems can be expressed in the same forms as KB ¼ b1 cos θ1 − b3 sin α2 cos θ2; ð2:6aÞ LB ¼ b1 sin θ1 − b3 sin α2 sin θ2 ð2:6bÞ and MB ¼ − ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 − n22 þ ðb2 þ b3 cos α2Þ2 q : ð2:6cÞ Expressions for the coefficients ðb1; b2; b3Þ are shown in Eqs. (T2.1) to (T2.5) in Table 2. The next step is a comparison of the power of ray deviation of systems in different configurations. It is known that a pair of rotatable prisms is equivalent to a single prism of variable power and the maximum ray deviation angle Φm is obtained when the prism apexes are aligned [1,2]. Variations ofΦm are plotted from Eqs. (2.5) and (2.6) as a function of prism open- ing angle. Results are shown by curves in Fig. 3(a) for systems having two identical prisms of opening angle α1 ¼ α2 ¼ α and indices n1 ¼ n2 ¼ n ¼ 1:5 (glass prisms) and n1 ¼ n2 ¼ n ¼ 4:0 (silicon prisms) by the curves in Fig. 3(b). All the curves in Figs. 3(a) and 3(b) have a linear portion when the prism open- ing angle is small. More specifically, in the case shown in Fig. 3(a), the difference between the predic- tions of the exact solution and the third-order theory [4] is the linear variation range shown in Figs. 3(a) and 3(b). 3. Inverse Solutions for Risley Prisms of Different Configurations Attention is now turned to the exact solutions of the problem of precision pointing, i.e., the problem of steering a laser beam to any specific altitude Φ and azimuth Θ within the angular range of the sys- tem [2,3]. HereΦ is the angle of the beam relative to the z axis and Θ is the angle around the z axis, counted from the x-z plane (see Fig. 1). To steer the beam to the direction specified by ðΦ;ΘÞ, we follow the two-step method proposed by Amirault and DiMarzio [5]. The first step is to keep the prism Π1 stationary at θ1 ¼ 0 and rotate the sec- ond prism Π2 until the desired altitude Φ is achieved, i.e., a rotation of Π2 relative to Π1 until the following condition is satisfied: cosΦ ¼ −MA ¼ −ða2 − a3 cos α2Þ ðFor system in the GroupAÞ; ð3:1aÞ cosΦ ¼ −MB ¼ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 − n22 þ ðb2 þ b3 cos α2Þ2 q ðFor system in the GroupBÞ: ð3:1bÞ Upon substituting from Eqs. (T1.3) and (T1.5) in Table 1 for the coefficient a2 and a3 into Eq. (3.1a), we may express the angle of azimuthal rotation between Π1 and Π2 as ðΔθÞ0 ¼ arccos � 1 a1 tan α2 � a2 þ 1 2ða2 þ cosΦÞ × � 1 − n22 − � a2 þ cosΦ cos α2 � 2 �� : ð3:2aÞ Similarly, for the prism systems in the group B, the angle ðΔθÞ0 can be obtained by substituting from Eqs. (T2.3) and (T2.5) in Table 2 for b2 and b3 into Eq. (7b). The result is Table 2. Coefficients �b1; b2; b3� in Eqs. (2.6a)–(2.6c) for the Direction Cosines of the Ray Emergent from in the Risley Prisms in the Group B Configurations 21-21 Configuration [Fig. 2(c)] 12-21 Configuration [Fig. 2(d)] b1 ¼ sin α1ðcos α1 − ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ n21 − sin 2 α1 q Þ ðT2:1Þ EQ-TARGET;temp:intralink-;dT2.2;376;686 sin α1ð−n1 cos α1 þ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 − n21 sin 2 α1 q Þ ðT2:2Þ b2 ¼ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 − n21 þ ðsin2 α1 þ cosα1 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ n21 − sin 2 α1 q Þ2 r ðT2:3Þ n1 sin2 α1 þ cos α1 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 − n21 sin 2 α1 q ðT2:4Þ b3 ¼ b1 sin α2 cosΔθ − b2 cos α2 þ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ n22 − 1þ ðb1 sin α2 cosΔθ − b2 cos α2Þ2 q ðΔθ ¼ θ2 − θ1Þ ðT2:5Þ Fig. 3. Comparison of the maximum ray deviation angle Φm for systems in the groups A and B under the conditions of two iden- tical wedge prisms with different opening angle α and refractive index (a) n ¼ 1:5 and (b) n ¼ 4:0. 1 August 2011 / Vol. 50, No. 22 / APPLIED OPTICS 4305 ðΔθÞ0¼arccos � 1 b1tanα2 × � b2þ 1 −b2� ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ cos2Φ−1þn22 q × � 1−n22− � −b2� ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ cos2Φ−1þn22 q cosα2 � 2 �� : ð3:2bÞ After the desired altitude is achieved, the beam is pointing to a new direction specified by the angles ðΦ;ψ0Þ. For systems in the group A, the azimuth dis- placement ψ0 introduced in the first step is given by ψ0 ¼ arctan � LA KA � θ1 ¼ 0; θ2 ¼ ðΔθÞ0 ¼ arctan � tan α2ða2 þ cosΦÞ sinðΔθÞ0 a1 þ tan α2ða2 þ cosΦÞ cosðΔθÞ0 � : ð3:3aÞ Similarly, for prism systems in group B, we have ψ0¼arctan � LB KB � θ1¼0; θ2¼ðΔθÞ0 ¼arctan 0 @ tanα2ðb2∓ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ cos2Φ−1þn22 q ÞsinðΔθÞ0 b1þtanα2ðb2∓ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ cos2Φ−1þn22 q ÞcosðΔθÞ0 1 A: ð3:3bÞ The second step in the two-step process is a simul- taneous rotation of the prisms Π1 and Π2 about the axis of the system until the desired azimuth Θ is reached. The final rotation angles of Π1 and Π2 are given, respectively, by θ1 ¼ Θ − ψ0 and θ2 ¼ θ1 þ ðΔθÞ0 ¼ Θ − ψ0 þ ðΔθÞ0: ð3:4Þ 4. Comparison of the Predictions of Different Theories This section is devoted to a comparison of the exact solution with the predictions of the third-order the- ory [4]. Results in Sections 4 and 5 will be obtained for the system in the 21-12 configuration. However, they can be easily obtained for systems in other con- figurations if the appropriate formulas in Tables 1 and 2 are used to replace Eqs. (T1.1), (T1.3), and (T1.5) in the calculations of direction cosines of the ray emergent from the system. Assume δ1 and δ2 represent the difference between prism rotation angles θ1 and θ2 predicted by the exact solution and the third-order theory for the problem of precision pointing of a beam to the direction specified by ðΦ;ΘÞ. Hence we can write δ1 ¼ ðθ1ÞExact − ðθ1Þ3rdorder and δ2 ¼ ðθ2ÞExact − ðθ2Þ3rdorder; ð4:1Þ Upon substitution of Eq. (3.4) in this study and Eq. (3.9) in [4] into Eq. (4.1), we can further express Eq. (4.1) in the form δ1 ¼ −ðψ0ÞExact þ ðψ0Þ3rdorder and δ2 ¼ Δθ1 þ ½ðΔθÞ0�Exact − ½ðΔθÞ0�3rdorder; ð4:2Þ where the expression for ðψ0ÞExact is given by Eq. (3.3a), whereas the expression for ðψ0Þ3rdorder can be found in [4]. A close examination of these two expressions for ðψ0ÞExact and ðψ0Þ3rdorder reveals that δ1 is a function of the altitudeΦ and has nothing to do with the azimuth Θ. The same conclusion can be drawn for δ2. Taking advantage of this finding, we may assume the azimuthal angle Θ ¼ 0 for simpli- city and our attention is concentrated to the evalua- tion of ðδ1; δ2Þ in the range of ð0;ΦmÞ. Under the condition ofΘ ¼ 0, assume the target under tracking is located at the point ðx1; y1;−PÞ in the Cartesian co- ordinates system shown in Fig. 1, and then we have x1 ¼ P tanΦ and y1 ¼ 0: ð4:3Þ To evaluate the error introduced by the third-order theory, we first substitute the target location speci- fied by ðΦ;Θ ¼ 0Þ into the third-order equation to determine prisms’ rotation angles ðθ1Þ3rdorder and ðθ2Þ3rdorder and then substitute from ðθ1Þ3rdorder and ðθ2Þ3rdorder into the expressions for exact solution pre- sented in Section 3 of this study and then we obtain the point ðx2; y2;−PÞ where the laser beam hits the target plane located at z ¼ −P if the two prisms are orientated according to the predictions of the third-order theory. The distance d between the two points ðx1; y1;−PÞ and ðx2; y2;−PÞ represents the po- sition finding error in precision pointing the target that can be evaluated by using the formula d ¼ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðx2 − x1Þ2 þ ðy2 − y1Þ2 q : ð4:4Þ Because ðx2; y2;PÞ is proportional to direction co- sines ½KA;LA;MA� of the ray emergent from the sys- tem, we may obtain ðx2; y2Þ from the relationship x2 KA ¼ y2 LA ¼ −P MA ; ð4:5Þ where the direction cosines should be evaluated at prism rotation angles θ1 ¼ ðθ1Þ3rdorder and θ1 ¼ ðθ1Þ3rdorder. Upon substituting from Eqs. (4.3) and (4.4), we arrived at 4306 APPLIED OPTICS / Vol. 50, No. 22 / 1 August 2011 d P ¼ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ�� KA MA � þ tanΦ � 2 þ � LA MA �s �� θ1 ¼ ðθ1Þ3rdorder; θ2 ¼ ðθ2Þ3rdorder : ð4:6Þ This equation is plotted in Fig. 4 under the condi- tion of two identical prisms. The curves in Fig. 4 show the normalized position finding error d=P as a func- tion of the normalized altitudeΦ=Φm with the prism opening angle α as a parameter. Specifically, Fig. 4(a) is for systems using glass prisms of n ¼ 1:5 and when the opening angle of the two prisms α ¼ 1°; 2°;…10°. It is seen from Fig. 4(a) if the position finding error d=P < 10−6 is required, then the third-order theory provides results with sufficient high accuracy only when the prism opening angle α < 2° and the target should be located in the paraxial region of the scan field where Φ ≤ 0:2Φm. Similar situations can be seen in Fig. 4(b), which is plotted under the condi- tions of systems using silicon prisms of n ¼ 4:0 with opening angles of α ¼ 1°; 2°;…5°. Finally, let us see a case example: assuming that a laser beam is required to be steered into the direction specified by the altitude Φ ¼ 4:5° and azimuth Θ ¼ 120°. The prism system has two identical wedge prisms of index n ¼ 4:0 and opening angle α ¼ 5°. If the two wedges are in the 21-12 arrangement, the first step is to obtain from Eq. (3.2a) the relative azi- muthal rotation angle ðΔθÞ0 ¼ 162:768° and then from Eq. (3.3a) we obtain the angular displacement ψ0 ¼ 81:578° introduced by the azimuthal rotation of Π2 relative to Π1. The second step is a co-rotation of Π1 and Π2 to reach the required azimuth Θ ¼ 120° and, finally, the total rotation angle for prism Π1 is θ1 ¼ Θ − ψ0 ¼ 120° − 81:578° ¼ 38:422° and for prism Π2 is θ2 ¼ θ1 þΔθ0 ¼ 38:422° þ 162:768° ¼ 201:190°. Compared to the predictions of the third- order theory, we obtain θ1 ¼ 38:431° and θ2 ¼ 201:182° from corresponding equations in [4]. The angular difference between the predictions of differ- ent theories is as small as 0.025%. This is not a sur- prising outcome because under the condition of α ≤ 5° the third-order theory predicts results with high ac- curacy as discussed in Section 3. The surprising issue is that a small angular error may cause a huge posi- tion finding error if the target is located several hun- dred miles away, e.g., at 500km the position error will be ∼125m, which may be much larger than the maximum linear dimension of the target. From this case example we may conclude that δ1 and δ2 in Eq. (4.2) relating to the differences between the predictions of different theories are small quan- tities. Therefore, the graphs in Fig. 5 of [4] showing the prism rotation angles ðθ1; θ2Þ as functions of beam steering angles ðΦ;ΘÞ will be good approxima- tions for the prediction of the exact solution. 5. Motion Equations of the Risley Prisms to Create a Desirable Scan Pattern Create a desirable scan pattern on some plane per- pendicular to the optical axis of the system is also known as synthesis of the scan pattern [10–12]. In this section, scan pattern synthesis is considered as a generalization of the inverse problem, i.e., from steering a laser beam to address a single point to cre- ate a desirable pattern on the plane of observation. Assume the desirable scan pattern is described by the parametrization x ¼ xðτÞ and y ¼ yðτÞ; ð5:1Þ where τ is a time-dependent parameter. The polar form of Eq. (5.1) is given by ρðτÞ ¼ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ x2ðτÞ þ y2ðτÞ q ð5:2aÞ and φðτÞ ¼ arctan � yðτÞ xðτÞ � : ð5:2bÞ Similar to Eq. (4.5) in Section 4, the Cartesian co- ordinates ½xðτÞ; yðτÞ� for the scan pattern can be ex- pressed in terms of the direction cosines of the ray emergent from the system: xðτÞ ¼ − KA MA P and yðτÞ ¼ − LA MA P; ð5:3Þ where P is the distance from the Risley prisms to the plane of observation (see Fig. 1). Upon substituting from Eq. (5.3) into Eq. (5.2) and then making use of the fact that the square sum of direction cosines is 1, we arrive at �ρðτÞ P � 2 ¼ x2ðτÞ þ y2ðτÞ ¼ K 2 A þ L2A M2A ¼ K 2 A þ L2A 1 − ðK2A þ L2AÞ : ð5:4Þ Fig. 4. Position error introduced by the third-order theory as the theory is applied to the Risley prisms of the 21-12 configuration having two identical wedge prisms of (a) n ¼ 1:5 and (b) n ¼ 4:0. 1 August 2011 / Vol. 50, No. 22 / APPLIED OPTICS 4307 Again, upon substituting from Eqs. (T1.1), (T1.3), and (T1.5) into Eqs. (2.5a) and (2.5b) for KA and LA and then into Eq. (5.4), we obtain, after some re- arranging, the expression a21 þ a23 sin2 α2 þ 2a1a3 sin α2 cosðΔθÞ ¼ ρ2ðτÞ P2 þ ρ2ðτÞ ; ð5:5Þ where Δθ ¼ θ2 − θ1 represents the difference of prisms’ orientations and Δθ is a function of the time- dependent parameter τ. After a close examination of the expressions for the coefficients ða1; a2;a3Þ in Table 1, we found a1 and a2 are independent from Δθ but a3 is a function of cosΔθ and their functional relationship can be ex- pressed as cosΔθ ¼ 1 a1 sin α2 � 1 − n22 − a 2 3 2a3 þ a2 cos α2 � : ð5:6Þ Upon substituting from Eq. (5.6) into Eq. (5.5), we obtain the following second-order equation: Aa23 þ Ba3 þ C ¼ 0; ð5:7Þ this can be solved by using the formula a3 ¼ −B� ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ B2 − 4AC p 2B ; ð5:8Þ where A ¼ cos2 α2, B ¼ −2a2 cos α2 and C ¼ −a21 − 1þ n22 þ ρ2ðτÞ P2 þ ρ2ðτÞ : Again, upon substituting from a3 in Eq. (5.8) into Eq. (5.6), we obtain Δθ ¼ arccos � 1 a1 sin α2 � 1 − n22 − a 2 3 2a3 þ a2 cos α2 �� : ð5:9Þ To determine the rotation angles θ1ðτÞ and θ2ðτÞ for the two prisms, we return to Eqs. (5.2b) and (5.3) and re-express Eq. (5.2b) in the form tanφðτÞ ¼ yðτÞ xðτÞ ¼ LA KA ¼ a1 sin θ1 þ a3 sin α2 sin θ2 a1 cos θ1 þ a3 sin α2 cos θ2 : ð5:10Þ On account of Δθ ¼ θ2 − θ1, Eq. (5.10) can be re- arranged in the form tanφðτÞ ¼ a1 sin θ1 þ a3 sin α2 sinðθ1 þΔθÞ a1 cos θ1 þ a3 sin α2 cosðθ1 þΔθÞ ¼ sinðθ1 þ θ10Þ cosðθ1 þ θ10Þ ¼ tanðθ1 þ θ10Þ; ð5:11Þ where θ10 ¼ arctan � a3 sin α2 sinΔθ a1 þ a3 sin α2 cosΔθ � : To this stage, we have sufficient parameters to ex- press the rotation angles θ1 ¼ θ1ðτÞ and θ2 ¼ θ2ðτÞ of the two prisms in the form θ1 ¼ φðτÞ − θ10 and θ2 ¼ φðτÞ − θ10 þΔθ: ð5:12Þ Finally, let us see a case example regarding the creation of an elliptical scan pattern [see Fig. 5(a)] on the plane of observation located at a distance P from the Risley prisms system using two identical wedges of n1 ¼ n2 ¼ n ¼ 1:5 and α1 ¼ α2 ¼ α ¼ 5°. The pattern to be created can be expressed param- etrically in the form x ¼ wx cos τ and y ¼ wy sin τ; ð5:13Þ where wx and wy are the major and minor axes coinciding with the x and y axes and τ is the time-dependent parameter running from 0 to 2π. Fig. 5. Synthesis of an elliptical scan pattern by using the Risley prisms of the 21-12 configuration containing two identical wedge prisms of refractive index n ¼ 1:5 and opening angle α ¼ 5°. (a) The elliptical pattern; (b) The azimuthal rotation angle between the two prisms as a function of the time-dependent parameter τ; (c) Rotation angles θ1 and θ2 for the two prisms. 4308 APPLIED OPTICS / Vol. 50, No. 22 / 1 August 2011 Under the condition of wx ¼ 0:08P and wy ¼ 0:04P, the azimuthal rotation angle Δθ between Π1 and Π2 is plotted from Eq. (5.9) and shown by the curve in Fig. 5(b), from which we found two extreme values ðΔθÞ1 ¼ 49:1° and ðΔθÞ2 ¼ 129:6°. The two curves in Fig. 5(c) are plotted from Eq. (5.11) to show the de- pendence of prism rotation angles θ1 and θ2 on the time-dependent parameter τ. The spacing between the two curves in Fig. 5(c) represents the variations of Δθ when the parameter τ is running from 0 to 2π. The spacing attains the minimum value of ðΔθÞ1 ¼ 49:1° when τ ¼ 0 and π…, where the two prism apexes are aligned the prism system attains the maximum power of ray deviation for generation of the major axis of the ellipse, whereas another extreme value ðΔθÞ2 ¼ 129:6° is reached when τ ¼ 0:5π and 1:5π… that implies the power of ray devia- tion of the system attains minima for generation of the minor axis of the ellipse. Correctness of the above discussion can be checked by using the first-order theory for rotating prisms [1,3,4], which states that the resultant deviation of the light ray is the vector sum of the deviations contributed by the two prisms as shown in Fig. 1. For example, when τ ¼ 0 laser spot is at the point (x ¼ wx, y ¼ 0), the orientations of the prisms Π1 and Π2 are at θ1 ¼ −ðΔθÞ1=2 ¼ −24:6° and θ2 ¼ þðΔθÞ1=2 ¼ 24:6°, respectively. The resultant devia- tion at this point is proportional to 2 cosðΔθÞ1=2. Si- milarly, the resultant deviation is proportional to 2 cosðΔθÞ2=2 at the point of (x ¼ 0, y ¼ wy). If the dis- cussion is correct, then the ratio of the two resultant deviations cosðΔθÞ1=2 cosðΔθÞ2=2 ¼ cosð24:6 °Þ cosð62:8°Þ ¼ 0:9092 0:4571 ≈ 2 should take the value close to the ellipticity of the elliptical scan pattern, i.e., ε ¼ wx=wy ¼ 2. In this section, we have shown the generation of an elliptical scan pattern by a pair of identical and co- rotational prisms when their motions are under the control of the predictions of Eq. (5.12). However, generation of an elliptical scan pattern has been con- sidered in existing literature [3] as a special case of optical generation of the hypotrochoid by using a pair of counter-rotational prisms with different power in ray deviation. 6. Concluding Remarks In conclusion, nonparaxial ray tracing through the Risley-prism-based systems in four different config- urations has been performed under the conditions of the two prisms having different refractive index and opening angle. It is found that the four different con- figurations can be divided into two groups and a uni- fied approach is developed that allows the use of two mathematical models to describe the four systems in different configurations. Closed form analytic solu- tions of the inverse problem, including both precision pointing and scan pattern synthesis, were developed and results obtained are compared with the predic- tions of the third-order theory and we found that even if the angular difference between the predic- tions of different theories are very small, the corre- sponding position error in tracking the target may not be small and could be much larger than the max- imum linear dimension of the target. References 1. F. R. Jenkins andH. E.White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 2001), Sec. 2.7. 2. F. A. Rosell, “Prism scanners,” J. Opt. Soc. Am. 50, 521–526 (1960). 3. W. L. Wolfe, “Optical-mechanical scanning techniques and de- vices,” inW. L.Wolfe andG. J. Zissis (eds.),The InfraredHand- book (Environmental Research Institute of Michigan, 1989), Chapt. 10. 4. Y. Li, “Third-order theory of Risley-prism based beam steering system,” Appl. Opt. 50, 679–686 (2011). 5. C. T. Amirault and C. A. DiMarzio, “Precision pointing using a dual-wedge scanner,” Appl. Opt. 24, 1302–1308(1985). 6. G. C. Boisset, B. Robertson, andH. S. Hinton, “Design and con- struction of an active alignment demonstrator for a free-space optical interconnect,” IEEE Photon. Technol. Lett. 7, 676–678 (1995). 7. J. J. Degnan, “Ray matrix approach for the real time control of SLR2000 optical elements,” in 14th International Workshop on Laser Ranging (2004). 8. Y. Yang, “Analytic solution of free space optical beam steering using Risley prisms,” J. Lightwave Technol. 26, 3576–3583 (2008). 9. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Sec. 3.2.2. 10. Y. Li, “Single-mirror beam steering system: analysis and synthesis of high-order conic-section scan patterns,” Appl. Opt. 47, 386–397 (2008). 11. G. Garcia-Torales, M. Strojnik, and G. Paez, “Risley prisms to control wave-front tilt and displacement in a vectorial inter- ferometer,” Appl. Opt. 41, 1380–1384 (2002). 12. W. C. Warger II and C. A. DiMarzio, “Dual-wedge scanning confocal reflectance microscope,” Opt. Lett. 32, 2140–2142 (2007). 1 August 2011 / Vol. 50, No. 22 / APPLIED OPTICS 4309

Closed form analytical inverse solutions for Risley-prism-based beam steering systems in different configurations

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