Pointing error analysis of Risley-prism-based beam steering system Yuan Zhou,1,2 Yafei Lu,3 Mo Hei,1 Guangcan Liu,2 and Dapeng Fan1,* 1
College of Mechanical Engineering and Automation, National University of Defense Technology, Changsha 410073, China 2
Department of Electronic and Communication Engineering, Changsha University, Changsha 410003, China
3
College of Aerospace and Material Engineering, National University of Defense Technology, Changsha 410073, China *Corresponding author: [email protected] Received 6 May 2014; revised 28 June 2014; accepted 30 June 2014; posted 30 July 2014 (Doc. ID 211471); published 28 August 2014
Based on the vector form Snell’s law, ray tracing is performed to quantify the pointing errors of Risleyprism-based beam steering systems, induced by component errors, prism orientation errors, and assembly errors. Case examples are given to elucidate the pointing error distributions in the field of regard and evaluate the allowances of the error sources for a given pointing accuracy. It is found that the assembly errors of the second prism will result in more remarkable pointing errors in contrast with the first one. The pointing errors induced by prism tilt depend on the tilt direction. The allowances of bearing tilt and prism tilt are almost identical if the same pointing accuracy is planned. All conclusions can provide a theoretical foundation for practical works. © 2014 Optical Society of America OCIS codes: (230.5480) Prisms; (080.0080) Geometric optics; (080.2720) Mathematical methods (general); (120.4880) Optomechanics; (280.1100) Aerosol detection; (220.2740) Geometric optical design. http://dx.doi.org/10.1364/AO.53.005775
1. Introduction
Risley prisms, composed of two rotatable prisms with a small apex angle [1], are widely used for beam steering in optical systems in which beam alignment and scanning are required, such as freespace optical communications, countermeasure, fiber-optic switches, and laser radar [2–5]. Highaccuracy beam pointing control is the key factor in determining the performance of a beam steering system. Therefore, there is a need to investigate the beam pointing accuracy for Risley-prism-based beam steering systems. Numerous error sources have an impact on the beam pointing accuracy and the relevant analysis is very rare in the literature. Horng 1559-128X/14/255775-09$15.00/0 © 2014 Optical Society of America
and Li investigated the distortions in the scan patterns produced by wedge elements with slightly different parameters and the impact of assembly errors [6], whereas this study focuses on the pointing accuracy and its sensitivity on various errors. Figure 1 is a schematic diagram illustrating the configuration of Risley prisms, in which the two prisms Π1 and Π2 may have different indices n1 and n2 and different apex angles α1 and α2 . The error sources that impact the beam pointing accuracy mainly include (1) component errors, that is, component parameters that deviate from the nominal or theoretical values, which may happen when systems work in environments of variable climatic conditions; (2) errors in prism orientation, that is, rotation angles of prisms that deviate from the nominal or theoretical values because of the performance limitation of the prism rotation and control components or the 1 September 2014 / Vol. 53, No. 25 / APPLIED OPTICS
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Fig. 1. 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 Π1 and Π2 . The rotation angles θ1 and θ2 are measured from the x axis.
system with index errors can be calculated by employing Eqs. (2.5) in [7]. First, numerous pointing positions in the field of regard (FOR) are chosen to evaluate the distribution of the pointing errors. Equations (3.1)–(3.4) in [7] are adopted to calculate the inverse solutions θ1 ; θ2 for the ideal system with nominal indices. The relevant rotational angles of the prisms are used to calculate the actual pointing positions of emerging beams for the system with index errors. Then, the pointing errors Δ can be readily obtained by comparing the actual and nominal beam pointing positions. Two systems are considered as case examples. The first system has the glass prisms of opening angle α 10° and refractive index n 1.50 and the other one is made of germanium prisms with α 8° and n 4.00. Figure 2(a) shows the distribution of the pointing errors arising from the index error in the field of view (FOV) for the glass system where the index error jΔnj is 0.001. It is seen that the
calibration errors of the angular positions of the prisms; and (3) assembly errors, such as prism tilt and bearing tilt. For the system mounted perfectly, the relation between any given pointing position and the corresponding prisms’ orientations was described efficiently by the analytic formulas in Li’s papers (see, e.g., [7,8]). It is therefore convenient to employ the formulas to estimate the pointing errors Δ induced by the component errors and the errors in prism orientation, described in Section 2 and Section 3, respectively. As to the assembly errors, described in Section 4, ray tracing can be performed to calculate the actual pointing position based on the thick prism theory, so that the pointing error can be obtained by comparing the ideal and actual pointing position. For all the error sources, the tolerance limits can be estimated for any planned pointing accuracy. Conclusions are drawn at the end of this study. 2. Pointing Errors Induced by Component Errors
The component errors are common because of the inevitable machining error and the changeable working environments. The two main component errors, that is, errors in refractive index and opening angles, will be investigated. A.
Pointing Errors Induced by Errors in Refractive Index
The refractive index of the prism material, such as optical glass and infrared material, changes with temperature. In many applications such as space optical communications, the beam steering system works in a wide temperature range. Therefore, it is desirable to investigate the pointing errors induced by errors in refractive index. In most applications, the two prisms are made from same materials and work in the same environments, so one can assume that the two prisms have identical indices errors, denoted as jΔnj. The resultant deviation of the ray emerging from the 5776
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Fig. 2. Pointing errors arising from the index errors. (a) The distribution of the pointing errors in the FOV, and (b) the pointing error versus the altitude Φ for the glass system where jΔnj is 0.001. (c) The maximum pointing error Δm versus the index error jΔnj for the glass and germanium system. (d) The allowances of index error for systems using prisms of opening angles ranging from 4° to 10° and refractive indices n 1.5; 2.0; …; 4.0 for 100 μrad pointing accuracy.
pointing error Δ is independent of the azimuth Θ but depends on the altitude Φ of the emerging beam. Variation of Δ is plotted as a function of Φ and the result is shown by the curve in Fig. 2(b). The pointing error Δ increases with the altitude Φ and approaches the maximum value Δm when the maximum ray deviation angle Φm is obtained, which means that the pointing accuracy is degraded gradually when the system steers a beam from the center to the edge of the FOR. Similar trends can also be obtained for the germanium system. The maximum pointing errors Δm in the two systems are plotted as a function of the index errors jΔnj in Fig. 2(c). It is shown that Δm varies almost linearly with jΔnj. If the pointing accuracy is planned, the upper limit of the index error can therefore be estimated. The allowance of index error is 0.00027 and 0.00009 for the glass and germanium system for 100 μrad pointing accuracy. In Fig. 2(d), the allowances of index error are shown for systems using prisms of opening angles ranging from 4° to 10° and refractive indices n 1.5; 2.0; …; 4.0 for 100 μrad pointing accuracy. It is obvious that the allowance decreases rapidly as the refractive index or opening angle increases. The results are helpful to guide the design of the prisms for the wide range of temperature applications. For example, the thermo-optic coefficient of germanium is about 4.0 × 10−4 ∕°C at temperatures ranging from 250 to 295 K at wavelengths from 3.5 to 5.5 μm [9]. If 100 μrad pointing accuracy is planned, then we obtain the allowance of temperature deviation 0.00009 0.225°C: Δtm 0.0004 B.
Pointing Errors Induced by Errors in Opening Angles
Due to imperfect manufacture, the opening angles of the two prisms may deviate from the nominal values, which will deteriorate the pointing errors. To achieve a given pointing accuracy, the errors in opening angles need to be limited to a level well below the corresponding tolerance, so that a suitable manufacturing requirement can be proposed. We assume one of the two opening angles deviates slightly from the nominal value, while the other one keeps invariable at the nominal value. Similarly, Eqs. (2.5) in [7] are adopted to calculate the pointing errors arising from the errors in opening angle for either of the two prisms. Figure 3(a) shows the distribution of the pointing errors Δ for the glass system where the error of opening angle for prism Π1 is 0.01°. Δ is also independent of azimuth Θ but changes with the altitude Φ, which is also the case for the prism Π2 . The two curves in Fig. 3(b) plot the pointing error as a function of the altitude Φ for Π1 and Π2 with 0.01° opening angle error. Δ approaches the maximum value Δm when the maximum ray deviation angle Φm is obtained. The above results are also suitable for the germanium system. Figure 3(c) compares Δm for Π1 and Π2 in the two
Fig. 3. Pointing errors arising from the errors in opening angle. (a) The distribution of the pointing errors in the FOV for the glass system where the error of opening angle for prism Π1 is 0.01°, and (b) the pointing error versus the altitude Φ for Π1 and Π2 with 0.01° opening angle error. (c) The maximum pointing error Δm versus the opening angle error of Π1 and Π2 for the glass and germanium system. (d) The allowances of opening angle error for systems using prisms of opening angles ranging from 0° to 10° and refractive indices n 1.5; 2.0; …; 4.0 for 100 μrad pointing accuracy.
systems, for different values of the errors in opening angle. The maximum pointing error, induced by the opening angle error of Π2, is little larger than that of Π1 . If 100 μrad pointing accuracy is planned, the allowance jΔαjm of opening angle error for Π1 and Π2 is 0.0108° and 0.0103° for the glass system, while that for the germanium system is 0.0005° and 0.0004°, respectively. Variations of jΔαjm are plotted in Fig. 3(d) as a function of opening angle for different refractive indices. These ranges of parameters can cover almost all the possible applications today. One can readily confirm that the large refractive index and opening angle pose significant challenges for prism manufacturing. 3. Pointing Errors Induced by Prism Orientation Errors
The errors in prism orientations, which are inevitable because of the performance limitations of the 1 September 2014 / Vol. 53, No. 25 / APPLIED OPTICS
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rotation control system of the prism and the errors in the calibration procedure, will also have an impact on beam pointing accuracy. We will investigate the pointing errors produced by systems in which one of the two rotation angles deviates slightly from the nominal value, while the other one stays at its nominal value. Similarly, the inverse solutions θ1 ; θ2 for the ideal system are firstly calculated for numerous target pointing positions in FOR. Then, a small deviation Δθ1 or Δθ2 is appended to the theoretical rotation angles θ1 or θ2 and the actual rotation angles, written as θ1 Δθ1 or θ2 Δθ2, are used to deduce the actual beam pointing positions. Finally, the pointing errors Δ can be obtained by comparing the direction cosines of the actual and nominal beam pointing positions. The distribution of the pointing errors Δ is shown in Fig. 4(a) for the glass system where the error of rotation angle for prism Π1 is 0.01°. Δ has nothing
to do with azimuth Θ but depends on altitude Φ, which is also shown by the curves in Fig. 4(b) for Π1 and Π2 with 0.01° rotation angle error. Then, the maximum pointing error Δm can be readily obtained for different values of the rotation angle error, and the results are shown in Fig. 4(c) for Π1 and Π2 in the glass and germanium systems. Similarly, the maximum pointing error arising from the rotation angle error of Π2 is little larger than that of Π1. The allowance jΔθjm of rotation angle error for Π1 and Π2 is 0.064° and 0.063° for the glass system, while that for the germanium system is 0.0103° and 0.0099° for 100 μrad pointing accuracy. The curves in Fig. 4(d) show the variations of jΔθjm with opening angle for different refractive indices. It can be seen that jΔθjm decreases dramatically as the opening angle or refractive index increases. It is interesting to mention that the prism orientation errors as well as the opening angle errors in prism Π1 and Π2 produce different pointing errors, which result from the difference of prism position in the propagated path of the beam. Rays impinged onto the two prisms have different incident directions, which have an impact on the angular deviations of prisms and their dependence on the two error sources. For a thin prism system, the impact of incident direction is negligible. With the increase of opening angle, the angular deviation of the prism depends gradually on the incident direction. Therefore, it can be readily seen from Figs. 3(d) and 4(d) that different error allowances are obtained for the two prisms with large opening angle. 4. Pointing Errors Induced by Assembly Errors
For an ideal Risley-prism-based beam steering system, the bearing rotational axes of the two prisms are collinear with the optical axis of the system and the two inner flat surfaces of the prism pair are parallel to each other and perpendicular to the optical axis. Due to imperfect mounting, the two prism elements may be tilted and the bearing axes of the two prisms may be misaligned with the optical axis. The two assembly errors are, respectively, shown in Figs. 5(a) and 5(b), where the two inner flat
Fig. 4. Pointing errors arising from the errors in rotation angle. (a) The distribution of the pointing errors in the FOV for the glass system where the error of rotation angle for prism Π1 is 0.01°, and (b) the pointing error versus the altitude Φ for Π1 and Π2 with 0.01° rotation angle error. (c) The maximum pointing error Δm versus the rotation angle error of Π1 and Π2 for the glass and germanium system. (d) The allowances of rotation angle error for systems using prisms of opening angles ranging from 0° to 10° and refractive indices n 1.5; 2.0; …; 4.0 for 100 μrad pointing accuracy. 5778
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Fig. 5. Diagrams illustrating the assembly errors. (a) Prism tilt: the surface 12 or 21 not perpendicular to the optical axis of the system. (b) Bearing tilt: a misalignment of a bearing axis with respect to the optical axis of the system.
surfaces, that is, the second surface of the prism Π1 and the first surface of the prism Π2 , are described by symbol 12 and symbol 21. In the prism tilt case [see Fig. 5(a)], both the prisms Π1 and Π2 rotate about the optical axis (i.e., the z axis) and either of them is tilted by a small angle δ. In the bearing tilt case [see Fig. 5(b)], the bearing rotational axis of the prism Π1 or Π2 is tilted and the system changes from one axis of rotation for the two elements into a system of two axes of rotation, one for Π1 and one for Π2. It is worth mentioning that not only the tilt angle but also the tilt direction should be considered for discussing the pointing errors induced by assembly errors, as Risley prisms are nonrotational symmetry systems. A.
Tilted Prism Element
The prism tilt is described in Fig. 5(a). The tilt angle and the tilt direction are, respectively, expressed by δ and θ0 when the rotation angle of the tilted prism is 0°. That is, the whole prism is rotated by δ about the axis specified by the unit vector ux ; uy ; uz − sin θ0 ; cos θ0 ; 0. Based on the well-known Rodrigues’ rotation formula [10,11], which is an efficient matrix for rotating an object around an arbitrary axis, the transpose of the rotation matrix M p can be written as M Tp Ap cos δ · I − Ap sin δ · Bp ;
(1)
where I is a unit vector and the expression for AP and Bp is given by Eqs. (2) and (3), respectively: 2
u2x
ux uy
ux uz
3
6 7 u2y uy uz 7 Ap 6 4 uy ux 5 uz ux uz uy u2z 2 − sin θ0 cos θ0 sin2 θ0 6 6 4 − sin θ0 cos θ0 cos2 θ0 0 0 2
0 6 Bp 6 4 uz −uy 2 0 6 6 0 4 − cos
−uz 0 ux
uy
3
7 07 5; (2) 0
7 −ux 7 5 0
0 − sin θ0
nˆ 120 0; 0; 1 · M p :
cos θ0
3
7 sin θ0 7 5: 0
(3)
For the first case, we consider the configuration where the prism Π1 is tilted but Π2 is mounted perfectly. When the rotation angle of Π1 is 0°, the unit normal vector of the first and second surface, described by symbol 11 and symbol 12, respectively, can be written as
(4)
When Π1 is in the position of θ1, the two surfaces are now in the position specified by the vectors 2
nˆ 11
cos θ1 nˆ 110 · 4 − sin θ1 0 2
nˆ 12
cos θ1 nˆ 120 · 4 − sin θ1 0
sin θ1 cos θ1 0
3 0 0 5; 1
(5)
sin θ1 cos θ1 0
3 0 0 5: 1
(6)
When Π2 is in the position of θ2, the unit normal vectors of its two surfaces, described by symbol 21 and symbol 22, respectively, can be written as nˆ 21 0; 0; 1; nˆ 22 − sin α2 cos θ2 ; − sin α2 sin θ2 ; cos α2 :
(7) (8)
The incident ray propagates in the direction specified by the ray vector sˆ i1 0; 0; −1 and hits the center of the surface 11. The refractive rays at the surface 11, 12, 21, and 22, calculated by applying the vector form Snell’s law [12,13], can be expressed by the vectors sˆr11
1 i ˆs − ˆsi1 · nˆ 11 nˆ 11 n1 1 s 1 1 − nˆ 11 1 − 2 2 ˆsi1 · nˆ 11 2 ; n1 n1
sˆ r12 n1 ˆsr11 − ˆsr11 · nˆ 12 nˆ 12 q − nˆ 12 1 − n21 n21 ˆsr11 · nˆ 12 2 ; sˆr21
3
0 θ0
0
nˆ 110 sin α1 ; 0; cos α1 · M p ;
1 r ˆs − ˆsr12 · nˆ 21 nˆ 21 n2 12 s 1 1 − nˆ 21 1 − 2 2 ˆsr12 · nˆ 21 2 ; n2 n2
sˆr22 n2 ˆsr21 − ˆsr21 · nˆ 22 nˆ 22 q − nˆ 22 1 − n22 n22 ˆsr21 · nˆ 22 2 :
(9)
(10)
(11)
(12)
First, numerous pointing positions in the FOR are chosen and the direction vector sˆr is calculated for each position. The rotation angles of Π1 and Π2 , calculated by using Eqs. (3.1)–(3.4) in [7], are then substituted into Eqs. (5)–(12) to calculate the actual ray vector sˆ r22. Then, the pointing errors caused by the prism tilt can be expressed in the form 1 September 2014 / Vol. 53, No. 25 / APPLIED OPTICS
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Δ arc cosˆsr · sˆ r22 .
(13)
Figure 6(a) shows the color map of the pointing error Δ in the FOV for the glass system, in which prism Π1 is tilted by δ 0.01° and θ0 30°. It can be seen that the distributions of the pointing error in the FOV show rotational symmetry, which implies that the pointing errors caused by prism tilt have nothing to do with azimuth Θ. Variations of Δ are plotted as a function of altitude Φ and the results are shown by curves in Fig. 6(b) for different θ0 . It can be found that the pointing error Δ and its dependency on altitude Φ depend on the tilt direction, described by θ0. The curve in Fig. 6(c) represents the variations of the maximum pointing errors ΔM in the FOV when the parameter θ0 is running from 0° to 360°. Interestingly, ΔM attains the maximum value Δm when θ0 0° or 180°, where the prism Π1 is tilted in the direction of the thinnest or thickest part of the prism, so that the allowance of prism tilt can be estimated in the two cases. Variations of Δm are plotted in Fig. 6(d) as a function of δ. If 100 μrad pointing accuracy is planned, the tilt allowance δm of prism Π1 is 0.7523° and 0.0191° for the glass system and the germanium system, respectively. Attention is now turned to the configuration where the prism Π2 is tilted but Π1 is mounted perfectly. The unit normal vectors of 11 and 12 surfaces can be written as
nˆ 11 sin α1 cos θ1 ; sin α1 sin θ1 ; cos α1 ; nˆ 12 0; 0; 1:
(14)
(15)
The unit normal vectors of 21 and 22 surfaces, described by nˆ 21 and nˆ 22 , can be deduced in an analogous form as Eqs. (4)–(6). Again, the ray tracing through the four interfaces 11, 12, 21, and 22 of the system is performed and the pointing errors Δ caused by the tilt of Π2 are calculated in the same method as that of Π1. The results are analogous to that of Π1, that is, the distributions of Δ show rotational symmetry and the pointing error attains the maximum value Δm when Π2 is tilted in the direction of the thinnest or the thickest part of the prism. The two curves in Fig. 6(e) plot the maximum pointing errors Δm as a function of δ for the glass system and the germanium system. The relevant tilt allowance δm of prism Π2 is 0.1653° and 0.0023° for 100 μrad pointing accuracy. B. Tilted Bearing Axis
The bearing tilt is described in Fig. 4(b). The direction of the tilted bearing axis can be expressed by the unit vector u0x ; u0y ; u0z sin δ cos θ0 ; sin δ sin θ0 ; cos δ. Again, by applying the Rodrigues’ rotation formula [10,11], we obtain the transpose of the matrix M b for the rotation about the bearing axis in the form
Fig. 6. Pointing errors arising from the prism tilt. (a) The distribution of the pointing errors in the FOV for the glass system, in which prism Π1 is tilted by δ 0.01° and θ0 30°. (b) The pointing error Δ versus the altitude Φ for different θ0 . (c) The maximum pointing error ΔM in the FOV versus θ0 . (d) The maximum value Δm of ΔM versus the tilt angle δ of prism Π1 for the glass and germanium system. (e) The maximum value Δm of ΔM versus the tilt angle δ of prism Π2 for the glass and germanium system. 5780
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the normal unit vector of the two surfaces of Π1 in the case of the bearing tilt of Π2. When the bearing axis of Π1 is tilted, the normal unit vector of the two surfaces is nˆ 110 and nˆ 120 , specified by Eq. (4) for 0° rotation angle. Rotating Π1 about the tilted bearing axis, specified by the unit vector sin δ cos θ0 ; sin δ sin θ0 ; cos δ, will steer the normal vectors. By using the rotation matrix defined by Eqs. (16)–(18), the resultant normal vectors can be written in the form nˆ 11 nˆ 110 · M b ;
(19)
nˆ 12 nˆ 120 · M b :
(20)
When the bearing axis of Π2 is tilted and the rotation angle is 0°, the normal unit vector of its two surfaces can be given by Fig. 7. Pointing error Δ in the FOV for the glass system, in which the bearing axis of Π1 or Π2 is tilted by δ 0.01° and the tilt directions are expressed by θ0 30° and 120°. (a) θ0 30° for Π1 ; (b) θ0 120° for Π1 ; (c) θ0 30° for Π2 ; and (d) θ0 120° for Π2.
M Tb Ab cos θ · I − Ab sin θ · Bb ;
(16)
u0x u0x
6 0 0 Ab 6 4 uy ux u0z u0x
2
0
6 0 Bb 6 4 uz −u0y 0 B B @
−u0z 0 u0x
u0y
u0x u0y
u0x u0z
3
0
u0z u0y
u0z u0z
sin2 δ cos2 θ0
sin δ cos δ cos θ0
3
nˆ 220 − sin α2 ; 0; cos α2 · M p :
(22)
nˆ 21 nˆ 210 · M b ;
7 B 2 B u0y u0z 7 5 @ sin δ sin θ0 cos θ0
u0y u0y
(21)
By using the rotation matrix defined by Eqs. (16)–(18), the normal unit vectors for rotation angle θ2 can be given by
where θ is the rotation angle of Π1 or Π2, I is the unit matrix, and the following definitions are used: 2
nˆ 210 0; 0; 1 · M p ;
sin2 δ sin θ0 cos θ0 sin2 δ sin2 θ0 sin δ cos δ sin θ0
sin δ cos δ cos θ0
(23) 1
C sin δ cos δ sin θ0 C A cos2 δ
nˆ 22 nˆ 220 · M b :
7 −u0x 7 5 0
0
− cos δ
cos δ
0
− sin δ sin θ0
sin δ cos θ0
sin δ sin θ0
1
C − sin δ cos θ0 C A: 0 (18)
Due to imperfect mounting, bearing tilt is probable for either of the two prisms. To explore the impact of bearing tilt for each prism separately and compare the results, we assume that only one bearing axis is tilted, while the other one is mounted perfectly. Therefore, Eqs. (7) and (8) can still be employed to express the normal unit vector of the two surfaces of Π2 in the case of the bearing tilt of Π1, while Eqs. (14) and (15) can also be employed to express
(17)
(24)
Similarly, ray tracing through the four interfaces of the system can be performed based on the four normal unit vectors. For the bearing tilt of Π1, the emerging ray vector sˆ r22 can be derived by substituting from Eqs. (19), (20), (7), and (8) for the vectors nˆ 11, nˆ 12 , nˆ 21 , and nˆ 22 into Eqs. (9)–(12). For the bearing tilt of Π2, the emerging ray vector sˆ r22 can be derived by substituting from Eqs. (14), (15), (23), and (24) for the vectors nˆ 11 , nˆ 12 , nˆ 21 , and nˆ 22 into Eqs. (9)–(12). Then, the pointing errors Δ caused by the bearing tilt of Π1 and Π2 can be investigated in accordance with the analogous methodologies mentioned above. The color maps in Fig. 7 show the pointing error Δ in the FOV for the glass system, in which the bearing axis of Π1 or Π2 is tilted by δ 0.01° and the tilt directions are expressed by θ0 30° and 120°, respectively. The maps look different for the two prisms. Additionally, the pointing error induced by the 1 September 2014 / Vol. 53, No. 25 / APPLIED OPTICS
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Fig. 8. Maximum pointing error Δm versus the tilt angle δ. (a) Δm for bearing tilt of Π1 ; and (b) Δm for bearing tilt of Π2 .
the allowance δm of bearing tilt and prism tilt are identical, and the results are shown in Fig. 9 for different refractive indices and opening angles. Similarly, a larger refractive index and opening angle result in a smaller allowance of tilt. Compared with that of prism Π1 , the tilt allowance of prism Π2 is smaller for the same pointing accuracy. Also, Horng and Li investigate the impact of assembly errors on the pointing accuracy in [6]. Compared with their work, this study considers the tilt direction of the prism element as well as the bearing axes. Additionally, in [6], they discuss the impacts of prism tilt and bearing tilt on the circular and line scan pattern, while we characterize the pointing error distribution in the FOV for all the cases. 5. Conclusion
In this paper, the pointing errors of Risley-prismbased beam steering systems, induced by component errors, prism orientation errors, and assembly errors, are investigated by ray tracing based on the vector form Snell’s law. Case examples were given to characterize the pointing error distributions in the FOV and evaluate the allowances of the above error sources, summarized in Table 1. From the above analyses, it can be concluded that: Fig. 9. Allowances δm of bearing tilt for systems using prisms of opening angles ranging from 0° to 10° and refractive indices n 1.5; 2.0; …; 4.0 for 100 μrad pointing accuracy for (a) Π1 , and (b) Π2 .
bearing tilt of Π2 is much more notable than that of Π1 . The differences between the results for the two prisms come from their position difference in the propagated path, which results in different directions of incident ray. The analogical distributions of Δ can be seen in the FOV for the two tilt directions, and the color map in Figs. 7(b) and 7(d) can actually be obtained by rotating, respectively, the color map in Figs. 7(a) and 7(c) about the center of the FOV. The maximum pointing error Δm can be evaluated by calculating and comparing the resultant Δ for different pointing positions within the FOV, and its variations are plotted in Fig. 8 as a function of δ for the glass system and the germanium system. It should be noted that the curves in Figs. 8(a) and 8(b) are almost identical to those in Figs. 6(d) and 6(e), which implies that the variations of Δm with δ in the bearing tilt case are the same as those in the prism tilt case. Therefore, for the same pointing accuracy,
Table 1.
Case Example
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This research is supported by the National Natural Science Foundation of China (Grant No. 51135009), the Natural Science Foundation of Hunan Province (Grant No. 13JJ3122), and China Postdoctoral Science Foundation (Grant No. 2013M532117).
Error Allowances for 100 μrad Pointing Accuracy
Opening Angle Error
Glass Ge
(1) The pointing error is independent of azimuth Θ but depends on altitude Φ of emerging beam for component errors, prism orientation errors, and prism tilts. (2) For a given pointing accuracy, all the error allowances for the germanium system are much smaller than those for the glass system. Larger refractive index results in a smaller error allowance. (3) The pointing errors induced by assembly errors of the second prism, including prism tilt and bearing tilt, are more notable in contrast with the first one. (4) The allowance of bearing tilt and prism tilt are almost identical if the same pointing accuracy is planned. (5) The pointing errors induced by prism tilt depend on the tilt direction and attain the maximum value when the prism is tilted in the direction of the thinnest or thickest part.
Prism Orientation Error
Prism Tilt or Bearing Tilt
Index Error
Π1
Π2
Π1
Π2
Π1
Π2
0.00027 0.00009
0.0108° 0.0005°
0.0103° 0.0004°
0.0640° 0.0103°
0.0630° 0.0099°
0.7523° 0.0191°
0.1653° 0.0023°
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References 1. F. A. Rosel, “Prism scanners,” J. Opt. Soc. Am. 50, 521–526 (1960). 2. A. Li, X. Jiang, J. Sun, L. Wang, Z. Li, and L. Liu, “Laser coarse-fine coupling scanning method by steering double prisms,” Appl. Opt. 51, 356–364 (2012). 3. C. Florea, J. Sanghera, and I. Aggarwal, “Broadband beam steering using chalcogenide-based Risley prisms,” Opt. Eng. 50, 33001 (2011). 4. J. Degnan, R. Machan, E. Leventhal, D. Lawrence, G. Jodor, and C. Field, “Inflight performance of a second generation, photon counting, 3D imaging lidar,” Proc. SPIE 6950, 695001 (2008). 5. W. C. Sweatt and T. R. Christenson, “Optical switch using Risley prisms,” U.S. patent 6,549,700 (15 April 2003). 6. J. Horng and Y. Li, “Error sources and their impact on the performance of dual-wedge beam steering systems,” Appl. Opt. 51, 4168–4175 (2012).
7. Y. Li, “Closed form analytical inverse solutions for Risleyprism-based beam steering systems in different configurations,” Appl. Opt. 50, 4302–4309 (2011). 8. Y. Li, “Third-order theory of the Risley-prism-based beam steering system,” Appl. Opt. 50, 679–686 (2011). 9. B. J. Frey, D. B. Leviton, and T. J. Madison, “Temperature-dependent refractive index of silicon and germanium,” Proc. SPIE 6273, 62732J (2006). 10. J. E. Mebius, “Derivation of the Euler–Rodrigues formula for three-dimensional rotations from the general formula for fourdimensional rotations,” arXiv:0701759 (2007). 11. D. Koks, “A roundabout route to geometric algebra,” in Explorations in Mathematical Physics: The Concepts behind an Elegant Language (Springer, 2006), p. 147. 12. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999). 13. C. T. Amirault and C. A. DiMarzio, “Precision pointing using a dual-wedge scanner,” Appl. Opt. 24, 1302–1308 (1985).
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College of Mechanical Engineering and Automation, National University of Defense Technology, Changsha 410073, China 2
Department of Electronic and Communication Engineering, Changsha University, Changsha 410003, China
3
College of Aerospace and Material Engineering, National University of Defense Technology, Changsha 410073, China *Corresponding author: [email protected] Received 6 May 2014; revised 28 June 2014; accepted 30 June 2014; posted 30 July 2014 (Doc. ID 211471); published 28 August 2014
Based on the vector form Snell’s law, ray tracing is performed to quantify the pointing errors of Risleyprism-based beam steering systems, induced by component errors, prism orientation errors, and assembly errors. Case examples are given to elucidate the pointing error distributions in the field of regard and evaluate the allowances of the error sources for a given pointing accuracy. It is found that the assembly errors of the second prism will result in more remarkable pointing errors in contrast with the first one. The pointing errors induced by prism tilt depend on the tilt direction. The allowances of bearing tilt and prism tilt are almost identical if the same pointing accuracy is planned. All conclusions can provide a theoretical foundation for practical works. © 2014 Optical Society of America OCIS codes: (230.5480) Prisms; (080.0080) Geometric optics; (080.2720) Mathematical methods (general); (120.4880) Optomechanics; (280.1100) Aerosol detection; (220.2740) Geometric optical design. http://dx.doi.org/10.1364/AO.53.005775
1. Introduction
Risley prisms, composed of two rotatable prisms with a small apex angle [1], are widely used for beam steering in optical systems in which beam alignment and scanning are required, such as freespace optical communications, countermeasure, fiber-optic switches, and laser radar [2–5]. Highaccuracy beam pointing control is the key factor in determining the performance of a beam steering system. Therefore, there is a need to investigate the beam pointing accuracy for Risley-prism-based beam steering systems. Numerous error sources have an impact on the beam pointing accuracy and the relevant analysis is very rare in the literature. Horng 1559-128X/14/255775-09$15.00/0 © 2014 Optical Society of America
and Li investigated the distortions in the scan patterns produced by wedge elements with slightly different parameters and the impact of assembly errors [6], whereas this study focuses on the pointing accuracy and its sensitivity on various errors. Figure 1 is a schematic diagram illustrating the configuration of Risley prisms, in which the two prisms Π1 and Π2 may have different indices n1 and n2 and different apex angles α1 and α2 . The error sources that impact the beam pointing accuracy mainly include (1) component errors, that is, component parameters that deviate from the nominal or theoretical values, which may happen when systems work in environments of variable climatic conditions; (2) errors in prism orientation, that is, rotation angles of prisms that deviate from the nominal or theoretical values because of the performance limitation of the prism rotation and control components or the 1 September 2014 / Vol. 53, No. 25 / APPLIED OPTICS
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Fig. 1. 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 Π1 and Π2 . The rotation angles θ1 and θ2 are measured from the x axis.
system with index errors can be calculated by employing Eqs. (2.5) in [7]. First, numerous pointing positions in the field of regard (FOR) are chosen to evaluate the distribution of the pointing errors. Equations (3.1)–(3.4) in [7] are adopted to calculate the inverse solutions θ1 ; θ2 for the ideal system with nominal indices. The relevant rotational angles of the prisms are used to calculate the actual pointing positions of emerging beams for the system with index errors. Then, the pointing errors Δ can be readily obtained by comparing the actual and nominal beam pointing positions. Two systems are considered as case examples. The first system has the glass prisms of opening angle α 10° and refractive index n 1.50 and the other one is made of germanium prisms with α 8° and n 4.00. Figure 2(a) shows the distribution of the pointing errors arising from the index error in the field of view (FOV) for the glass system where the index error jΔnj is 0.001. It is seen that the
calibration errors of the angular positions of the prisms; and (3) assembly errors, such as prism tilt and bearing tilt. For the system mounted perfectly, the relation between any given pointing position and the corresponding prisms’ orientations was described efficiently by the analytic formulas in Li’s papers (see, e.g., [7,8]). It is therefore convenient to employ the formulas to estimate the pointing errors Δ induced by the component errors and the errors in prism orientation, described in Section 2 and Section 3, respectively. As to the assembly errors, described in Section 4, ray tracing can be performed to calculate the actual pointing position based on the thick prism theory, so that the pointing error can be obtained by comparing the ideal and actual pointing position. For all the error sources, the tolerance limits can be estimated for any planned pointing accuracy. Conclusions are drawn at the end of this study. 2. Pointing Errors Induced by Component Errors
The component errors are common because of the inevitable machining error and the changeable working environments. The two main component errors, that is, errors in refractive index and opening angles, will be investigated. A.
Pointing Errors Induced by Errors in Refractive Index
The refractive index of the prism material, such as optical glass and infrared material, changes with temperature. In many applications such as space optical communications, the beam steering system works in a wide temperature range. Therefore, it is desirable to investigate the pointing errors induced by errors in refractive index. In most applications, the two prisms are made from same materials and work in the same environments, so one can assume that the two prisms have identical indices errors, denoted as jΔnj. The resultant deviation of the ray emerging from the 5776
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Fig. 2. Pointing errors arising from the index errors. (a) The distribution of the pointing errors in the FOV, and (b) the pointing error versus the altitude Φ for the glass system where jΔnj is 0.001. (c) The maximum pointing error Δm versus the index error jΔnj for the glass and germanium system. (d) The allowances of index error for systems using prisms of opening angles ranging from 4° to 10° and refractive indices n 1.5; 2.0; …; 4.0 for 100 μrad pointing accuracy.
pointing error Δ is independent of the azimuth Θ but depends on the altitude Φ of the emerging beam. Variation of Δ is plotted as a function of Φ and the result is shown by the curve in Fig. 2(b). The pointing error Δ increases with the altitude Φ and approaches the maximum value Δm when the maximum ray deviation angle Φm is obtained, which means that the pointing accuracy is degraded gradually when the system steers a beam from the center to the edge of the FOR. Similar trends can also be obtained for the germanium system. The maximum pointing errors Δm in the two systems are plotted as a function of the index errors jΔnj in Fig. 2(c). It is shown that Δm varies almost linearly with jΔnj. If the pointing accuracy is planned, the upper limit of the index error can therefore be estimated. The allowance of index error is 0.00027 and 0.00009 for the glass and germanium system for 100 μrad pointing accuracy. In Fig. 2(d), the allowances of index error are shown for systems using prisms of opening angles ranging from 4° to 10° and refractive indices n 1.5; 2.0; …; 4.0 for 100 μrad pointing accuracy. It is obvious that the allowance decreases rapidly as the refractive index or opening angle increases. The results are helpful to guide the design of the prisms for the wide range of temperature applications. For example, the thermo-optic coefficient of germanium is about 4.0 × 10−4 ∕°C at temperatures ranging from 250 to 295 K at wavelengths from 3.5 to 5.5 μm [9]. If 100 μrad pointing accuracy is planned, then we obtain the allowance of temperature deviation 0.00009 0.225°C: Δtm 0.0004 B.
Pointing Errors Induced by Errors in Opening Angles
Due to imperfect manufacture, the opening angles of the two prisms may deviate from the nominal values, which will deteriorate the pointing errors. To achieve a given pointing accuracy, the errors in opening angles need to be limited to a level well below the corresponding tolerance, so that a suitable manufacturing requirement can be proposed. We assume one of the two opening angles deviates slightly from the nominal value, while the other one keeps invariable at the nominal value. Similarly, Eqs. (2.5) in [7] are adopted to calculate the pointing errors arising from the errors in opening angle for either of the two prisms. Figure 3(a) shows the distribution of the pointing errors Δ for the glass system where the error of opening angle for prism Π1 is 0.01°. Δ is also independent of azimuth Θ but changes with the altitude Φ, which is also the case for the prism Π2 . The two curves in Fig. 3(b) plot the pointing error as a function of the altitude Φ for Π1 and Π2 with 0.01° opening angle error. Δ approaches the maximum value Δm when the maximum ray deviation angle Φm is obtained. The above results are also suitable for the germanium system. Figure 3(c) compares Δm for Π1 and Π2 in the two
Fig. 3. Pointing errors arising from the errors in opening angle. (a) The distribution of the pointing errors in the FOV for the glass system where the error of opening angle for prism Π1 is 0.01°, and (b) the pointing error versus the altitude Φ for Π1 and Π2 with 0.01° opening angle error. (c) The maximum pointing error Δm versus the opening angle error of Π1 and Π2 for the glass and germanium system. (d) The allowances of opening angle error for systems using prisms of opening angles ranging from 0° to 10° and refractive indices n 1.5; 2.0; …; 4.0 for 100 μrad pointing accuracy.
systems, for different values of the errors in opening angle. The maximum pointing error, induced by the opening angle error of Π2, is little larger than that of Π1 . If 100 μrad pointing accuracy is planned, the allowance jΔαjm of opening angle error for Π1 and Π2 is 0.0108° and 0.0103° for the glass system, while that for the germanium system is 0.0005° and 0.0004°, respectively. Variations of jΔαjm are plotted in Fig. 3(d) as a function of opening angle for different refractive indices. These ranges of parameters can cover almost all the possible applications today. One can readily confirm that the large refractive index and opening angle pose significant challenges for prism manufacturing. 3. Pointing Errors Induced by Prism Orientation Errors
The errors in prism orientations, which are inevitable because of the performance limitations of the 1 September 2014 / Vol. 53, No. 25 / APPLIED OPTICS
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rotation control system of the prism and the errors in the calibration procedure, will also have an impact on beam pointing accuracy. We will investigate the pointing errors produced by systems in which one of the two rotation angles deviates slightly from the nominal value, while the other one stays at its nominal value. Similarly, the inverse solutions θ1 ; θ2 for the ideal system are firstly calculated for numerous target pointing positions in FOR. Then, a small deviation Δθ1 or Δθ2 is appended to the theoretical rotation angles θ1 or θ2 and the actual rotation angles, written as θ1 Δθ1 or θ2 Δθ2, are used to deduce the actual beam pointing positions. Finally, the pointing errors Δ can be obtained by comparing the direction cosines of the actual and nominal beam pointing positions. The distribution of the pointing errors Δ is shown in Fig. 4(a) for the glass system where the error of rotation angle for prism Π1 is 0.01°. Δ has nothing
to do with azimuth Θ but depends on altitude Φ, which is also shown by the curves in Fig. 4(b) for Π1 and Π2 with 0.01° rotation angle error. Then, the maximum pointing error Δm can be readily obtained for different values of the rotation angle error, and the results are shown in Fig. 4(c) for Π1 and Π2 in the glass and germanium systems. Similarly, the maximum pointing error arising from the rotation angle error of Π2 is little larger than that of Π1. The allowance jΔθjm of rotation angle error for Π1 and Π2 is 0.064° and 0.063° for the glass system, while that for the germanium system is 0.0103° and 0.0099° for 100 μrad pointing accuracy. The curves in Fig. 4(d) show the variations of jΔθjm with opening angle for different refractive indices. It can be seen that jΔθjm decreases dramatically as the opening angle or refractive index increases. It is interesting to mention that the prism orientation errors as well as the opening angle errors in prism Π1 and Π2 produce different pointing errors, which result from the difference of prism position in the propagated path of the beam. Rays impinged onto the two prisms have different incident directions, which have an impact on the angular deviations of prisms and their dependence on the two error sources. For a thin prism system, the impact of incident direction is negligible. With the increase of opening angle, the angular deviation of the prism depends gradually on the incident direction. Therefore, it can be readily seen from Figs. 3(d) and 4(d) that different error allowances are obtained for the two prisms with large opening angle. 4. Pointing Errors Induced by Assembly Errors
For an ideal Risley-prism-based beam steering system, the bearing rotational axes of the two prisms are collinear with the optical axis of the system and the two inner flat surfaces of the prism pair are parallel to each other and perpendicular to the optical axis. Due to imperfect mounting, the two prism elements may be tilted and the bearing axes of the two prisms may be misaligned with the optical axis. The two assembly errors are, respectively, shown in Figs. 5(a) and 5(b), where the two inner flat
Fig. 4. Pointing errors arising from the errors in rotation angle. (a) The distribution of the pointing errors in the FOV for the glass system where the error of rotation angle for prism Π1 is 0.01°, and (b) the pointing error versus the altitude Φ for Π1 and Π2 with 0.01° rotation angle error. (c) The maximum pointing error Δm versus the rotation angle error of Π1 and Π2 for the glass and germanium system. (d) The allowances of rotation angle error for systems using prisms of opening angles ranging from 0° to 10° and refractive indices n 1.5; 2.0; …; 4.0 for 100 μrad pointing accuracy. 5778
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Fig. 5. Diagrams illustrating the assembly errors. (a) Prism tilt: the surface 12 or 21 not perpendicular to the optical axis of the system. (b) Bearing tilt: a misalignment of a bearing axis with respect to the optical axis of the system.
surfaces, that is, the second surface of the prism Π1 and the first surface of the prism Π2 , are described by symbol 12 and symbol 21. In the prism tilt case [see Fig. 5(a)], both the prisms Π1 and Π2 rotate about the optical axis (i.e., the z axis) and either of them is tilted by a small angle δ. In the bearing tilt case [see Fig. 5(b)], the bearing rotational axis of the prism Π1 or Π2 is tilted and the system changes from one axis of rotation for the two elements into a system of two axes of rotation, one for Π1 and one for Π2. It is worth mentioning that not only the tilt angle but also the tilt direction should be considered for discussing the pointing errors induced by assembly errors, as Risley prisms are nonrotational symmetry systems. A.
Tilted Prism Element
The prism tilt is described in Fig. 5(a). The tilt angle and the tilt direction are, respectively, expressed by δ and θ0 when the rotation angle of the tilted prism is 0°. That is, the whole prism is rotated by δ about the axis specified by the unit vector ux ; uy ; uz − sin θ0 ; cos θ0 ; 0. Based on the well-known Rodrigues’ rotation formula [10,11], which is an efficient matrix for rotating an object around an arbitrary axis, the transpose of the rotation matrix M p can be written as M Tp Ap cos δ · I − Ap sin δ · Bp ;
(1)
where I is a unit vector and the expression for AP and Bp is given by Eqs. (2) and (3), respectively: 2
u2x
ux uy
ux uz
3
6 7 u2y uy uz 7 Ap 6 4 uy ux 5 uz ux uz uy u2z 2 − sin θ0 cos θ0 sin2 θ0 6 6 4 − sin θ0 cos θ0 cos2 θ0 0 0 2
0 6 Bp 6 4 uz −uy 2 0 6 6 0 4 − cos
−uz 0 ux
uy
3
7 07 5; (2) 0
7 −ux 7 5 0
0 − sin θ0
nˆ 120 0; 0; 1 · M p :
cos θ0
3
7 sin θ0 7 5: 0
(3)
For the first case, we consider the configuration where the prism Π1 is tilted but Π2 is mounted perfectly. When the rotation angle of Π1 is 0°, the unit normal vector of the first and second surface, described by symbol 11 and symbol 12, respectively, can be written as
(4)
When Π1 is in the position of θ1, the two surfaces are now in the position specified by the vectors 2
nˆ 11
cos θ1 nˆ 110 · 4 − sin θ1 0 2
nˆ 12
cos θ1 nˆ 120 · 4 − sin θ1 0
sin θ1 cos θ1 0
3 0 0 5; 1
(5)
sin θ1 cos θ1 0
3 0 0 5: 1
(6)
When Π2 is in the position of θ2, the unit normal vectors of its two surfaces, described by symbol 21 and symbol 22, respectively, can be written as nˆ 21 0; 0; 1; nˆ 22 − sin α2 cos θ2 ; − sin α2 sin θ2 ; cos α2 :
(7) (8)
The incident ray propagates in the direction specified by the ray vector sˆ i1 0; 0; −1 and hits the center of the surface 11. The refractive rays at the surface 11, 12, 21, and 22, calculated by applying the vector form Snell’s law [12,13], can be expressed by the vectors sˆr11
1 i ˆs − ˆsi1 · nˆ 11 nˆ 11 n1 1 s 1 1 − nˆ 11 1 − 2 2 ˆsi1 · nˆ 11 2 ; n1 n1
sˆ r12 n1 ˆsr11 − ˆsr11 · nˆ 12 nˆ 12 q − nˆ 12 1 − n21 n21 ˆsr11 · nˆ 12 2 ; sˆr21
3
0 θ0
0
nˆ 110 sin α1 ; 0; cos α1 · M p ;
1 r ˆs − ˆsr12 · nˆ 21 nˆ 21 n2 12 s 1 1 − nˆ 21 1 − 2 2 ˆsr12 · nˆ 21 2 ; n2 n2
sˆr22 n2 ˆsr21 − ˆsr21 · nˆ 22 nˆ 22 q − nˆ 22 1 − n22 n22 ˆsr21 · nˆ 22 2 :
(9)
(10)
(11)
(12)
First, numerous pointing positions in the FOR are chosen and the direction vector sˆr is calculated for each position. The rotation angles of Π1 and Π2 , calculated by using Eqs. (3.1)–(3.4) in [7], are then substituted into Eqs. (5)–(12) to calculate the actual ray vector sˆ r22. Then, the pointing errors caused by the prism tilt can be expressed in the form 1 September 2014 / Vol. 53, No. 25 / APPLIED OPTICS
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Δ arc cosˆsr · sˆ r22 .
(13)
Figure 6(a) shows the color map of the pointing error Δ in the FOV for the glass system, in which prism Π1 is tilted by δ 0.01° and θ0 30°. It can be seen that the distributions of the pointing error in the FOV show rotational symmetry, which implies that the pointing errors caused by prism tilt have nothing to do with azimuth Θ. Variations of Δ are plotted as a function of altitude Φ and the results are shown by curves in Fig. 6(b) for different θ0 . It can be found that the pointing error Δ and its dependency on altitude Φ depend on the tilt direction, described by θ0. The curve in Fig. 6(c) represents the variations of the maximum pointing errors ΔM in the FOV when the parameter θ0 is running from 0° to 360°. Interestingly, ΔM attains the maximum value Δm when θ0 0° or 180°, where the prism Π1 is tilted in the direction of the thinnest or thickest part of the prism, so that the allowance of prism tilt can be estimated in the two cases. Variations of Δm are plotted in Fig. 6(d) as a function of δ. If 100 μrad pointing accuracy is planned, the tilt allowance δm of prism Π1 is 0.7523° and 0.0191° for the glass system and the germanium system, respectively. Attention is now turned to the configuration where the prism Π2 is tilted but Π1 is mounted perfectly. The unit normal vectors of 11 and 12 surfaces can be written as
nˆ 11 sin α1 cos θ1 ; sin α1 sin θ1 ; cos α1 ; nˆ 12 0; 0; 1:
(14)
(15)
The unit normal vectors of 21 and 22 surfaces, described by nˆ 21 and nˆ 22 , can be deduced in an analogous form as Eqs. (4)–(6). Again, the ray tracing through the four interfaces 11, 12, 21, and 22 of the system is performed and the pointing errors Δ caused by the tilt of Π2 are calculated in the same method as that of Π1. The results are analogous to that of Π1, that is, the distributions of Δ show rotational symmetry and the pointing error attains the maximum value Δm when Π2 is tilted in the direction of the thinnest or the thickest part of the prism. The two curves in Fig. 6(e) plot the maximum pointing errors Δm as a function of δ for the glass system and the germanium system. The relevant tilt allowance δm of prism Π2 is 0.1653° and 0.0023° for 100 μrad pointing accuracy. B. Tilted Bearing Axis
The bearing tilt is described in Fig. 4(b). The direction of the tilted bearing axis can be expressed by the unit vector u0x ; u0y ; u0z sin δ cos θ0 ; sin δ sin θ0 ; cos δ. Again, by applying the Rodrigues’ rotation formula [10,11], we obtain the transpose of the matrix M b for the rotation about the bearing axis in the form
Fig. 6. Pointing errors arising from the prism tilt. (a) The distribution of the pointing errors in the FOV for the glass system, in which prism Π1 is tilted by δ 0.01° and θ0 30°. (b) The pointing error Δ versus the altitude Φ for different θ0 . (c) The maximum pointing error ΔM in the FOV versus θ0 . (d) The maximum value Δm of ΔM versus the tilt angle δ of prism Π1 for the glass and germanium system. (e) The maximum value Δm of ΔM versus the tilt angle δ of prism Π2 for the glass and germanium system. 5780
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the normal unit vector of the two surfaces of Π1 in the case of the bearing tilt of Π2. When the bearing axis of Π1 is tilted, the normal unit vector of the two surfaces is nˆ 110 and nˆ 120 , specified by Eq. (4) for 0° rotation angle. Rotating Π1 about the tilted bearing axis, specified by the unit vector sin δ cos θ0 ; sin δ sin θ0 ; cos δ, will steer the normal vectors. By using the rotation matrix defined by Eqs. (16)–(18), the resultant normal vectors can be written in the form nˆ 11 nˆ 110 · M b ;
(19)
nˆ 12 nˆ 120 · M b :
(20)
When the bearing axis of Π2 is tilted and the rotation angle is 0°, the normal unit vector of its two surfaces can be given by Fig. 7. Pointing error Δ in the FOV for the glass system, in which the bearing axis of Π1 or Π2 is tilted by δ 0.01° and the tilt directions are expressed by θ0 30° and 120°. (a) θ0 30° for Π1 ; (b) θ0 120° for Π1 ; (c) θ0 30° for Π2 ; and (d) θ0 120° for Π2.
M Tb Ab cos θ · I − Ab sin θ · Bb ;
(16)
u0x u0x
6 0 0 Ab 6 4 uy ux u0z u0x
2
0
6 0 Bb 6 4 uz −u0y 0 B B @
−u0z 0 u0x
u0y
u0x u0y
u0x u0z
3
0
u0z u0y
u0z u0z
sin2 δ cos2 θ0
sin δ cos δ cos θ0
3
nˆ 220 − sin α2 ; 0; cos α2 · M p :
(22)
nˆ 21 nˆ 210 · M b ;
7 B 2 B u0y u0z 7 5 @ sin δ sin θ0 cos θ0
u0y u0y
(21)
By using the rotation matrix defined by Eqs. (16)–(18), the normal unit vectors for rotation angle θ2 can be given by
where θ is the rotation angle of Π1 or Π2, I is the unit matrix, and the following definitions are used: 2
nˆ 210 0; 0; 1 · M p ;
sin2 δ sin θ0 cos θ0 sin2 δ sin2 θ0 sin δ cos δ sin θ0
sin δ cos δ cos θ0
(23) 1
C sin δ cos δ sin θ0 C A cos2 δ
nˆ 22 nˆ 220 · M b :
7 −u0x 7 5 0
0
− cos δ
cos δ
0
− sin δ sin θ0
sin δ cos θ0
sin δ sin θ0
1
C − sin δ cos θ0 C A: 0 (18)
Due to imperfect mounting, bearing tilt is probable for either of the two prisms. To explore the impact of bearing tilt for each prism separately and compare the results, we assume that only one bearing axis is tilted, while the other one is mounted perfectly. Therefore, Eqs. (7) and (8) can still be employed to express the normal unit vector of the two surfaces of Π2 in the case of the bearing tilt of Π1, while Eqs. (14) and (15) can also be employed to express
(17)
(24)
Similarly, ray tracing through the four interfaces of the system can be performed based on the four normal unit vectors. For the bearing tilt of Π1, the emerging ray vector sˆ r22 can be derived by substituting from Eqs. (19), (20), (7), and (8) for the vectors nˆ 11, nˆ 12 , nˆ 21 , and nˆ 22 into Eqs. (9)–(12). For the bearing tilt of Π2, the emerging ray vector sˆ r22 can be derived by substituting from Eqs. (14), (15), (23), and (24) for the vectors nˆ 11 , nˆ 12 , nˆ 21 , and nˆ 22 into Eqs. (9)–(12). Then, the pointing errors Δ caused by the bearing tilt of Π1 and Π2 can be investigated in accordance with the analogous methodologies mentioned above. The color maps in Fig. 7 show the pointing error Δ in the FOV for the glass system, in which the bearing axis of Π1 or Π2 is tilted by δ 0.01° and the tilt directions are expressed by θ0 30° and 120°, respectively. The maps look different for the two prisms. Additionally, the pointing error induced by the 1 September 2014 / Vol. 53, No. 25 / APPLIED OPTICS
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Fig. 8. Maximum pointing error Δm versus the tilt angle δ. (a) Δm for bearing tilt of Π1 ; and (b) Δm for bearing tilt of Π2 .
the allowance δm of bearing tilt and prism tilt are identical, and the results are shown in Fig. 9 for different refractive indices and opening angles. Similarly, a larger refractive index and opening angle result in a smaller allowance of tilt. Compared with that of prism Π1 , the tilt allowance of prism Π2 is smaller for the same pointing accuracy. Also, Horng and Li investigate the impact of assembly errors on the pointing accuracy in [6]. Compared with their work, this study considers the tilt direction of the prism element as well as the bearing axes. Additionally, in [6], they discuss the impacts of prism tilt and bearing tilt on the circular and line scan pattern, while we characterize the pointing error distribution in the FOV for all the cases. 5. Conclusion
In this paper, the pointing errors of Risley-prismbased beam steering systems, induced by component errors, prism orientation errors, and assembly errors, are investigated by ray tracing based on the vector form Snell’s law. Case examples were given to characterize the pointing error distributions in the FOV and evaluate the allowances of the above error sources, summarized in Table 1. From the above analyses, it can be concluded that: Fig. 9. Allowances δm of bearing tilt for systems using prisms of opening angles ranging from 0° to 10° and refractive indices n 1.5; 2.0; …; 4.0 for 100 μrad pointing accuracy for (a) Π1 , and (b) Π2 .
bearing tilt of Π2 is much more notable than that of Π1 . The differences between the results for the two prisms come from their position difference in the propagated path, which results in different directions of incident ray. The analogical distributions of Δ can be seen in the FOV for the two tilt directions, and the color map in Figs. 7(b) and 7(d) can actually be obtained by rotating, respectively, the color map in Figs. 7(a) and 7(c) about the center of the FOV. The maximum pointing error Δm can be evaluated by calculating and comparing the resultant Δ for different pointing positions within the FOV, and its variations are plotted in Fig. 8 as a function of δ for the glass system and the germanium system. It should be noted that the curves in Figs. 8(a) and 8(b) are almost identical to those in Figs. 6(d) and 6(e), which implies that the variations of Δm with δ in the bearing tilt case are the same as those in the prism tilt case. Therefore, for the same pointing accuracy,
Table 1.
Case Example
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This research is supported by the National Natural Science Foundation of China (Grant No. 51135009), the Natural Science Foundation of Hunan Province (Grant No. 13JJ3122), and China Postdoctoral Science Foundation (Grant No. 2013M532117).
Error Allowances for 100 μrad Pointing Accuracy
Opening Angle Error
Glass Ge
(1) The pointing error is independent of azimuth Θ but depends on altitude Φ of emerging beam for component errors, prism orientation errors, and prism tilts. (2) For a given pointing accuracy, all the error allowances for the germanium system are much smaller than those for the glass system. Larger refractive index results in a smaller error allowance. (3) The pointing errors induced by assembly errors of the second prism, including prism tilt and bearing tilt, are more notable in contrast with the first one. (4) The allowance of bearing tilt and prism tilt are almost identical if the same pointing accuracy is planned. (5) The pointing errors induced by prism tilt depend on the tilt direction and attain the maximum value when the prism is tilted in the direction of the thinnest or thickest part.
Prism Orientation Error
Prism Tilt or Bearing Tilt
Index Error
Π1
Π2
Π1
Π2
Π1
Π2
0.00027 0.00009
0.0108° 0.0005°
0.0103° 0.0004°
0.0640° 0.0103°
0.0630° 0.0099°
0.7523° 0.0191°
0.1653° 0.0023°
APPLIED OPTICS / Vol. 53, No. 25 / 1 September 2014
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