Abstract

Based on the vector form Snell’s law, ray tracing is performed to quantify the pointing errors of Risley-prism-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

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References

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    [CrossRef]
  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).
    [CrossRef]
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    [CrossRef]
  8. Y. Li, “Third-order theory of the Risley-prism-based beam steering system,” Appl. Opt. 50, 679–686 (2011).
    [CrossRef]
  9. B. J. Frey, D. B. Leviton, and T. J. Madison, “Temperature-dependent refractive index of silicon and germanium,” Proc. SPIE 6273, 62732J (2006).
    [CrossRef]
  10. J. E. Mebius, “Derivation of the Euler–Rodrigues formula for three-dimensional rotations from the general formula for four-dimensional 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.
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2012 (2)

2011 (3)

2008 (1)

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).
[CrossRef]

2006 (1)

B. J. Frey, D. B. Leviton, and T. J. Madison, “Temperature-dependent refractive index of silicon and germanium,” Proc. SPIE 6273, 62732J (2006).
[CrossRef]

1985 (1)

1960 (1)

Aggarwal, I.

C. Florea, J. Sanghera, and I. Aggarwal, “Broadband beam steering using chalcogenide-based Risley prisms,” Opt. Eng. 50, 33001 (2011).
[CrossRef]

Amirault, C. T.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Christenson, T. R.

W. C. Sweatt and T. R. Christenson, “Optical switch using Risley prisms,” U.S. patent6,549,700 (15April2003).

Degnan, J.

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).
[CrossRef]

DiMarzio, C. A.

Field, C.

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).
[CrossRef]

Florea, C.

C. Florea, J. Sanghera, and I. Aggarwal, “Broadband beam steering using chalcogenide-based Risley prisms,” Opt. Eng. 50, 33001 (2011).
[CrossRef]

Frey, B. J.

B. J. Frey, D. B. Leviton, and T. J. Madison, “Temperature-dependent refractive index of silicon and germanium,” Proc. SPIE 6273, 62732J (2006).
[CrossRef]

Horng, J.

Jiang, X.

Jodor, G.

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).
[CrossRef]

Koks, D.

D. Koks, “A roundabout route to geometric algebra,” in Explorations in Mathematical Physics: The Concepts behind an Elegant Language (Springer, 2006), p. 147.

Lawrence, D.

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).
[CrossRef]

Leventhal, E.

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).
[CrossRef]

Leviton, D. B.

B. J. Frey, D. B. Leviton, and T. J. Madison, “Temperature-dependent refractive index of silicon and germanium,” Proc. SPIE 6273, 62732J (2006).
[CrossRef]

Li, A.

Li, Y.

Li, Z.

Liu, L.

Machan, R.

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).
[CrossRef]

Madison, T. J.

B. J. Frey, D. B. Leviton, and T. J. Madison, “Temperature-dependent refractive index of silicon and germanium,” Proc. SPIE 6273, 62732J (2006).
[CrossRef]

Mebius, J. E.

J. E. Mebius, “Derivation of the Euler–Rodrigues formula for three-dimensional rotations from the general formula for four-dimensional rotations,” arXiv:0701759 (2007).

Rosel, F. A.

Sanghera, J.

C. Florea, J. Sanghera, and I. Aggarwal, “Broadband beam steering using chalcogenide-based Risley prisms,” Opt. Eng. 50, 33001 (2011).
[CrossRef]

Sun, J.

Sweatt, W. C.

W. C. Sweatt and T. R. Christenson, “Optical switch using Risley prisms,” U.S. patent6,549,700 (15April2003).

Wang, L.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Appl. Opt. (5)

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

C. Florea, J. Sanghera, and I. Aggarwal, “Broadband beam steering using chalcogenide-based Risley prisms,” Opt. Eng. 50, 33001 (2011).
[CrossRef]

Proc. SPIE (2)

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).
[CrossRef]

B. J. Frey, D. B. Leviton, and T. J. Madison, “Temperature-dependent refractive index of silicon and germanium,” Proc. SPIE 6273, 62732J (2006).
[CrossRef]

Other (4)

J. E. Mebius, “Derivation of the Euler–Rodrigues formula for three-dimensional rotations from the general formula for four-dimensional rotations,” arXiv:0701759 (2007).

D. Koks, “A roundabout route to geometric algebra,” in Explorations in Mathematical Physics: The Concepts behind an Elegant Language (Springer, 2006), p. 147.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

W. C. Sweatt and T. R. Christenson, “Optical switch using Risley prisms,” U.S. patent6,549,700 (15April2003).

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Figures (9)

Fig. 1.
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.

Fig. 2.
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 |Δn| is 0.001. (c) The maximum pointing error Δm versus the index error |Δn| 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.

Fig. 3.
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.

Fig. 4.
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.

Fig. 5.
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.

Fig. 6.
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.

Fig. 7.
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.

Fig. 8.
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.

Fig. 9.
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.

Tables (1)

Tables Icon

Table 1. Error Allowances for 100 μrad Pointing Accuracy

Equations (25)

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(Δt)m=0.000090.0004=0.225°C.
MpT=Ap+cosδ·(IAp)+sinδ·Bp,
Ap=[ux2uxuyuxuzuyuxuy2uyuzuzuxuzuyuz2]=[sin2θ0sinθ0cosθ00sinθ0cosθ0cos2θ00000],
Bp=[0uzuyuz0uxuyux0]=[00cosθ000sinθ0cosθ0sinθ00].
n^110=(sinα1,0,cosα1)·Mp,n^120=(0,0,1)·Mp.
n^11=n^110·[cosθ1sinθ10sinθ1cosθ10001],
n^12=n^120·[cosθ1sinθ10sinθ1cosθ10001].
n^21=(0,0,1),
n^22=(sinα2cosθ2,sinα2sinθ2,cosα2).
s^11r=1n1[s^1i(s^1i·n^11)n^11]n^1111n12+1n12(s^1i·n^11)2,
s^12r=n1[s^11r(s^11r·n^12)n^12]n^121n12+n12(s^11r·n^12)2,
s^21r=1n2[s^12r(s^12r·n^21)n^21]n^2111n22+1n22(s^12r·n^21)2,
s^22r=n2[s^21r(s^21r·n^22)n^22]n^221n22+n22(s^21r·n^22)2.
Δ=arccos(s^r·s^22r).
n^11=(sinα1cosθ1,sinα1sinθ1,cosα1),
n^12=(0,0,1).
MbT=Ab+cosθ·(IAb)+sinθ·Bb,
Ab=[uxuxuxuyuxuzuyuxuyuyuyuzuzuxuzuyuzuz]=(sin2δcos2θ0sin2δsinθ0cosθ0sinδcosδcosθ0sin2δsinθ0cosθ0sin2δsin2θ0sinδcosδsinθ0sinδcosδcosθ0sinδcosδsinθ0cos2δ)
Bb=[0uzuyuz0uxuyux0]=(0cosδsinδsinθ0cosδ0sinδcosθ0sinδsinθ0sinδcosθ00).
n^11=n^110·Mb,
n^12=n^120·Mb.
n^210=(0,0,1)·Mp,
n^220=(sinα2,0,cosα2)·Mp.
n^21=n^210·Mb,
n^22=n^220·Mb.

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