Abstract

Two exact inverse solutions of Risley prisms have been given by previous authors, based on which we calculate the gradients of the scan field that open a way to investigate the nonlinear relationship between the slewing rate of the beam and the required angular velocities of the two wedge prisms in the Risley-prism-based beam steering system for target tracking. The limited regions and singularity point at the center and the edge of the field of regard are discussed. It is found that the maximum required rotational velocities of the two prisms for target tracking are nearly the same and are dependent on the altitude angle. The central limited region is almost independent of the prism parameters. The control singularity at the crossing center path can be avoided by switching the two solutions.

© 2013 Optical Society of America

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References

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  1. C. Oh, J. Kim, J. Muth, S. Serati, and M. J. Escuti, “High-throughput continuous beam steering using rotating polarization gratings,” IEEE Photon. Technol. Lett. 22, 200–202 (2010).
    [CrossRef]
  2. E. Schundler, D. Carlson, R. Vaillancourt, J. R. Dupuis, and C. Schwarze, “Compact, wide field DRS explosive detector,” Proc. SPIE 8018, 80181O (2011).
    [CrossRef]
  3. M. Sanchez and D. Gutow, “Control laws for a three-element Risley prism optical beam pointer,” Proc. SPIE 6304, 630403 (2006).
  4. C. Florea, J. Sanghera, and I. Aggarwal, “Broadband beam steering using chalcogenide-based Risley prisms,” Opt. Eng. 50, 033001 (2011).
    [CrossRef]
  5. B. D. Duncan, P. J. Bos, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasure applications,” Opt. Eng. 42, 1038–1047 (2003).
    [CrossRef]
  6. M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006).
    [CrossRef]
  7. J. Sun, L. Liu, M. Yun, L. Wan, and M. Zhang, “Distortion of beam shape by a rotating double-prism wide-angle laser beam scanner,” Opt. Eng. 45, 43001–43004 (2006).
    [CrossRef]
  8. J. L. Gibson, B. D. Duncan, P. Bos, and V. Sergen, “Wide angle beam steering for infrared countermeasures applications,” Proc. SPIE 4723, 100–111 (2002).
    [CrossRef]
  9. P. J. Bos, H. Garcia, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasures and imaging applications: solving the singularity problem in the two-prism design,” Opt. Eng. 46, 113001 (2007).
    [CrossRef]
  10. J. Lacoursiere, M. Doucet, E. O. Curatu, M. Savard, S. Verreault, S. Thibault, P. C. Chevrette, and B. Ricard, “Large-deviation achromatic Risley prisms pointing systems,” Proc. SPIE 4773, 123–131 (2002).
    [CrossRef]
  11. R. Winsor and M. Braunstein, “Conformal beam steering apparatus for simultaneous manipulation of optical and radio frequency signals,” Proc. SPIE 6215, 62150G (2006).
    [CrossRef]
  12. C. T. Amirault and C. A. Dimarzio, “Precision pointing using a dual-wedge scanner,” Appl. Opt. 24, 1302–1308 (1985).
    [CrossRef]
  13. Y. Li, “Closed form analytical inverse solutions for Risley-prism-based beam steering systems in different configurations,” Appl. Opt. 50, 4302–4309 (2011).
    [CrossRef]

2011

E. Schundler, D. Carlson, R. Vaillancourt, J. R. Dupuis, and C. Schwarze, “Compact, wide field DRS explosive detector,” Proc. SPIE 8018, 80181O (2011).
[CrossRef]

C. Florea, J. Sanghera, and I. Aggarwal, “Broadband beam steering using chalcogenide-based Risley prisms,” Opt. Eng. 50, 033001 (2011).
[CrossRef]

Y. Li, “Closed form analytical inverse solutions for Risley-prism-based beam steering systems in different configurations,” Appl. Opt. 50, 4302–4309 (2011).
[CrossRef]

2010

C. Oh, J. Kim, J. Muth, S. Serati, and M. J. Escuti, “High-throughput continuous beam steering using rotating polarization gratings,” IEEE Photon. Technol. Lett. 22, 200–202 (2010).
[CrossRef]

2007

P. J. Bos, H. Garcia, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasures and imaging applications: solving the singularity problem in the two-prism design,” Opt. Eng. 46, 113001 (2007).
[CrossRef]

2006

M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006).
[CrossRef]

J. Sun, L. Liu, M. Yun, L. Wan, and M. Zhang, “Distortion of beam shape by a rotating double-prism wide-angle laser beam scanner,” Opt. Eng. 45, 43001–43004 (2006).
[CrossRef]

M. Sanchez and D. Gutow, “Control laws for a three-element Risley prism optical beam pointer,” Proc. SPIE 6304, 630403 (2006).

R. Winsor and M. Braunstein, “Conformal beam steering apparatus for simultaneous manipulation of optical and radio frequency signals,” Proc. SPIE 6215, 62150G (2006).
[CrossRef]

2003

B. D. Duncan, P. J. Bos, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasure applications,” Opt. Eng. 42, 1038–1047 (2003).
[CrossRef]

2002

J. L. Gibson, B. D. Duncan, P. Bos, and V. Sergen, “Wide angle beam steering for infrared countermeasures applications,” Proc. SPIE 4723, 100–111 (2002).
[CrossRef]

J. Lacoursiere, M. Doucet, E. O. Curatu, M. Savard, S. Verreault, S. Thibault, P. C. Chevrette, and B. Ricard, “Large-deviation achromatic Risley prisms pointing systems,” Proc. SPIE 4773, 123–131 (2002).
[CrossRef]

1985

Aggarwal, I.

C. Florea, J. Sanghera, and I. Aggarwal, “Broadband beam steering using chalcogenide-based Risley prisms,” Opt. Eng. 50, 033001 (2011).
[CrossRef]

Amirault, C. T.

Bos, P.

J. L. Gibson, B. D. Duncan, P. Bos, and V. Sergen, “Wide angle beam steering for infrared countermeasures applications,” Proc. SPIE 4723, 100–111 (2002).
[CrossRef]

Bos, P. J.

P. J. Bos, H. Garcia, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasures and imaging applications: solving the singularity problem in the two-prism design,” Opt. Eng. 46, 113001 (2007).
[CrossRef]

B. D. Duncan, P. J. Bos, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasure applications,” Opt. Eng. 42, 1038–1047 (2003).
[CrossRef]

Braunstein, M.

R. Winsor and M. Braunstein, “Conformal beam steering apparatus for simultaneous manipulation of optical and radio frequency signals,” Proc. SPIE 6215, 62150G (2006).
[CrossRef]

Carlson, D.

E. Schundler, D. Carlson, R. Vaillancourt, J. R. Dupuis, and C. Schwarze, “Compact, wide field DRS explosive detector,” Proc. SPIE 8018, 80181O (2011).
[CrossRef]

Chevrette, P. C.

J. Lacoursiere, M. Doucet, E. O. Curatu, M. Savard, S. Verreault, S. Thibault, P. C. Chevrette, and B. Ricard, “Large-deviation achromatic Risley prisms pointing systems,” Proc. SPIE 4773, 123–131 (2002).
[CrossRef]

Curatu, E. O.

J. Lacoursiere, M. Doucet, E. O. Curatu, M. Savard, S. Verreault, S. Thibault, P. C. Chevrette, and B. Ricard, “Large-deviation achromatic Risley prisms pointing systems,” Proc. SPIE 4773, 123–131 (2002).
[CrossRef]

Dimarzio, C. A.

Doucet, M.

J. Lacoursiere, M. Doucet, E. O. Curatu, M. Savard, S. Verreault, S. Thibault, P. C. Chevrette, and B. Ricard, “Large-deviation achromatic Risley prisms pointing systems,” Proc. SPIE 4773, 123–131 (2002).
[CrossRef]

Doughty, N.

M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006).
[CrossRef]

Duncan, B. D.

B. D. Duncan, P. J. Bos, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasure applications,” Opt. Eng. 42, 1038–1047 (2003).
[CrossRef]

J. L. Gibson, B. D. Duncan, P. Bos, and V. Sergen, “Wide angle beam steering for infrared countermeasures applications,” Proc. SPIE 4723, 100–111 (2002).
[CrossRef]

Dupuis, J. R.

E. Schundler, D. Carlson, R. Vaillancourt, J. R. Dupuis, and C. Schwarze, “Compact, wide field DRS explosive detector,” Proc. SPIE 8018, 80181O (2011).
[CrossRef]

Escuti, M. J.

C. Oh, J. Kim, J. Muth, S. Serati, and M. J. Escuti, “High-throughput continuous beam steering using rotating polarization gratings,” IEEE Photon. Technol. Lett. 22, 200–202 (2010).
[CrossRef]

Florea, C.

C. Florea, J. Sanghera, and I. Aggarwal, “Broadband beam steering using chalcogenide-based Risley prisms,” Opt. Eng. 50, 033001 (2011).
[CrossRef]

Garcia, H.

P. J. Bos, H. Garcia, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasures and imaging applications: solving the singularity problem in the two-prism design,” Opt. Eng. 46, 113001 (2007).
[CrossRef]

Gibson, J. L.

J. L. Gibson, B. D. Duncan, P. Bos, and V. Sergen, “Wide angle beam steering for infrared countermeasures applications,” Proc. SPIE 4723, 100–111 (2002).
[CrossRef]

Gutow, D.

M. Sanchez and D. Gutow, “Control laws for a three-element Risley prism optical beam pointer,” Proc. SPIE 6304, 630403 (2006).

M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006).
[CrossRef]

Harford, S.

M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006).
[CrossRef]

Hoffman, C.

M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006).
[CrossRef]

Kim, J.

C. Oh, J. Kim, J. Muth, S. Serati, and M. J. Escuti, “High-throughput continuous beam steering using rotating polarization gratings,” IEEE Photon. Technol. Lett. 22, 200–202 (2010).
[CrossRef]

Lacoursiere, J.

J. Lacoursiere, M. Doucet, E. O. Curatu, M. Savard, S. Verreault, S. Thibault, P. C. Chevrette, and B. Ricard, “Large-deviation achromatic Risley prisms pointing systems,” Proc. SPIE 4773, 123–131 (2002).
[CrossRef]

Li, Y.

Liu, L.

J. Sun, L. Liu, M. Yun, L. Wan, and M. Zhang, “Distortion of beam shape by a rotating double-prism wide-angle laser beam scanner,” Opt. Eng. 45, 43001–43004 (2006).
[CrossRef]

Muth, J.

C. Oh, J. Kim, J. Muth, S. Serati, and M. J. Escuti, “High-throughput continuous beam steering using rotating polarization gratings,” IEEE Photon. Technol. Lett. 22, 200–202 (2010).
[CrossRef]

Oh, C.

C. Oh, J. Kim, J. Muth, S. Serati, and M. J. Escuti, “High-throughput continuous beam steering using rotating polarization gratings,” IEEE Photon. Technol. Lett. 22, 200–202 (2010).
[CrossRef]

Ostaszewski, M.

M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006).
[CrossRef]

Pierce, R.

M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006).
[CrossRef]

Ricard, B.

J. Lacoursiere, M. Doucet, E. O. Curatu, M. Savard, S. Verreault, S. Thibault, P. C. Chevrette, and B. Ricard, “Large-deviation achromatic Risley prisms pointing systems,” Proc. SPIE 4773, 123–131 (2002).
[CrossRef]

Sanchez, M.

M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006).
[CrossRef]

M. Sanchez and D. Gutow, “Control laws for a three-element Risley prism optical beam pointer,” Proc. SPIE 6304, 630403 (2006).

Sanghera, J.

C. Florea, J. Sanghera, and I. Aggarwal, “Broadband beam steering using chalcogenide-based Risley prisms,” Opt. Eng. 50, 033001 (2011).
[CrossRef]

Savard, M.

J. Lacoursiere, M. Doucet, E. O. Curatu, M. Savard, S. Verreault, S. Thibault, P. C. Chevrette, and B. Ricard, “Large-deviation achromatic Risley prisms pointing systems,” Proc. SPIE 4773, 123–131 (2002).
[CrossRef]

Schundler, E.

E. Schundler, D. Carlson, R. Vaillancourt, J. R. Dupuis, and C. Schwarze, “Compact, wide field DRS explosive detector,” Proc. SPIE 8018, 80181O (2011).
[CrossRef]

Schwarze, C.

E. Schundler, D. Carlson, R. Vaillancourt, J. R. Dupuis, and C. Schwarze, “Compact, wide field DRS explosive detector,” Proc. SPIE 8018, 80181O (2011).
[CrossRef]

Serati, S.

C. Oh, J. Kim, J. Muth, S. Serati, and M. J. Escuti, “High-throughput continuous beam steering using rotating polarization gratings,” IEEE Photon. Technol. Lett. 22, 200–202 (2010).
[CrossRef]

Sergan, V.

P. J. Bos, H. Garcia, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasures and imaging applications: solving the singularity problem in the two-prism design,” Opt. Eng. 46, 113001 (2007).
[CrossRef]

B. D. Duncan, P. J. Bos, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasure applications,” Opt. Eng. 42, 1038–1047 (2003).
[CrossRef]

Sergen, V.

J. L. Gibson, B. D. Duncan, P. Bos, and V. Sergen, “Wide angle beam steering for infrared countermeasures applications,” Proc. SPIE 4723, 100–111 (2002).
[CrossRef]

Sun, J.

J. Sun, L. Liu, M. Yun, L. Wan, and M. Zhang, “Distortion of beam shape by a rotating double-prism wide-angle laser beam scanner,” Opt. Eng. 45, 43001–43004 (2006).
[CrossRef]

Thibault, S.

J. Lacoursiere, M. Doucet, E. O. Curatu, M. Savard, S. Verreault, S. Thibault, P. C. Chevrette, and B. Ricard, “Large-deviation achromatic Risley prisms pointing systems,” Proc. SPIE 4773, 123–131 (2002).
[CrossRef]

Vaillancourt, R.

E. Schundler, D. Carlson, R. Vaillancourt, J. R. Dupuis, and C. Schwarze, “Compact, wide field DRS explosive detector,” Proc. SPIE 8018, 80181O (2011).
[CrossRef]

Verreault, S.

J. Lacoursiere, M. Doucet, E. O. Curatu, M. Savard, S. Verreault, S. Thibault, P. C. Chevrette, and B. Ricard, “Large-deviation achromatic Risley prisms pointing systems,” Proc. SPIE 4773, 123–131 (2002).
[CrossRef]

Wan, L.

J. Sun, L. Liu, M. Yun, L. Wan, and M. Zhang, “Distortion of beam shape by a rotating double-prism wide-angle laser beam scanner,” Opt. Eng. 45, 43001–43004 (2006).
[CrossRef]

Winsor, R.

R. Winsor and M. Braunstein, “Conformal beam steering apparatus for simultaneous manipulation of optical and radio frequency signals,” Proc. SPIE 6215, 62150G (2006).
[CrossRef]

Yun, M.

J. Sun, L. Liu, M. Yun, L. Wan, and M. Zhang, “Distortion of beam shape by a rotating double-prism wide-angle laser beam scanner,” Opt. Eng. 45, 43001–43004 (2006).
[CrossRef]

Zhang, M.

J. Sun, L. Liu, M. Yun, L. Wan, and M. Zhang, “Distortion of beam shape by a rotating double-prism wide-angle laser beam scanner,” Opt. Eng. 45, 43001–43004 (2006).
[CrossRef]

Appl. Opt.

IEEE Photon. Technol. Lett.

C. Oh, J. Kim, J. Muth, S. Serati, and M. J. Escuti, “High-throughput continuous beam steering using rotating polarization gratings,” IEEE Photon. Technol. Lett. 22, 200–202 (2010).
[CrossRef]

Opt. Eng.

C. Florea, J. Sanghera, and I. Aggarwal, “Broadband beam steering using chalcogenide-based Risley prisms,” Opt. Eng. 50, 033001 (2011).
[CrossRef]

B. D. Duncan, P. J. Bos, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasure applications,” Opt. Eng. 42, 1038–1047 (2003).
[CrossRef]

J. Sun, L. Liu, M. Yun, L. Wan, and M. Zhang, “Distortion of beam shape by a rotating double-prism wide-angle laser beam scanner,” Opt. Eng. 45, 43001–43004 (2006).
[CrossRef]

P. J. Bos, H. Garcia, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasures and imaging applications: solving the singularity problem in the two-prism design,” Opt. Eng. 46, 113001 (2007).
[CrossRef]

Proc. SPIE

J. Lacoursiere, M. Doucet, E. O. Curatu, M. Savard, S. Verreault, S. Thibault, P. C. Chevrette, and B. Ricard, “Large-deviation achromatic Risley prisms pointing systems,” Proc. SPIE 4773, 123–131 (2002).
[CrossRef]

R. Winsor and M. Braunstein, “Conformal beam steering apparatus for simultaneous manipulation of optical and radio frequency signals,” Proc. SPIE 6215, 62150G (2006).
[CrossRef]

J. L. Gibson, B. D. Duncan, P. Bos, and V. Sergen, “Wide angle beam steering for infrared countermeasures applications,” Proc. SPIE 4723, 100–111 (2002).
[CrossRef]

M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006).
[CrossRef]

E. Schundler, D. Carlson, R. Vaillancourt, J. R. Dupuis, and C. Schwarze, “Compact, wide field DRS explosive detector,” Proc. SPIE 8018, 80181O (2011).
[CrossRef]

M. Sanchez and D. Gutow, “Control laws for a three-element Risley prism optical beam pointer,” Proc. SPIE 6304, 630403 (2006).

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Figures (10)

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 rotational angles θ1 and θ2 are measured from the x axis. The direction of the emerging ray is described as altitude angle Φ and azimuth angle Θ in a polar coordinate space.

Fig. 2.
Fig. 2.

First set of solutions (θ1,θ2) for the various pointing positions of the output beam.

Fig. 3.
Fig. 3.

Ratios of the rotational velocities of the prisms to the slew rates of the radial beam steering for the systems with (a) n1=n2=1.5, α1=α2=10°, Φm=10.268° and (b) n1=n2=4, α1=α2=8°, Φm=67.489°.

Fig. 4.
Fig. 4.

Gradient vector field and equipotential contour of (a) θ1 and (b) θ2 for the glass prisms system.

Fig. 5.
Fig. 5.

Magnitudes of the gradients of (a) θ1 and (b) θ2 for the glass prisms system. (c) The magnitudes are plotted as a function of altitude Φ.

Fig. 6.
Fig. 6.

Magnitudes of the gradients of (a) θ1 and (b) θ2 for the silicon prisms system. (c) The magnitudes are plotted as a function of altitude Φ.

Fig. 7.
Fig. 7.

Magnitudes of gradients of θ1 and θ2 at (a) center and (b) edge of the FOR for the glass and silicon prism systems. (c) Schematic diagram of the limited regions at the center and edge of the FOR.

Fig. 8.
Fig. 8.

Angular radius (ΔΦ)c of the limited region at the center of the FOR.

Fig. 9.
Fig. 9.

Angular width (ΔΦ)e of the limited region at the edge of the FOR.

Fig. 10.
Fig. 10.

Sudden changes in the solutions to the rotational angles of (a) prism Π1 and (b) prism Π2 for the glass prisms system tracking through the center of the FOR along the x axis coordinate.

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

K=a1cosθ1+a3sinα2cosθ2,
L=a1sinθ1+a3sinα2sinθ2,
M=a2a3cosα2,
a1=n2n1sinα1[cosα1(n12sin2α1)1/2],a2=n2n1[(n12sin2α1)1/2cosα1+sin2α1],a3=(a1sinα2cosΔθa2cosα2)+[1n22+(a1sinα2cosΔθa2cosα2)2]1/2,(Δθ=θ2θ1).
Θ={L/K,ForK0L/K+π,ForK<0.
Φ=arccos(M).
|Δθ|=arccos(1a1tanα2{a2+12(a2+cosΦ)[1n22(a2+cosΦcosα2)2]}).
Θ0=arctan(LK)θ1=0θ2=|Δθ|,ForK0,
Θ0=arctan(LK)θ1=0θ2=|Δθ|+π,ForK<0.
Θ1=ΘΘ0.
θ1=Θ1=ΘΘ0,
θ2=Θ1+|Δθ|=ΘΘ0+|Δθ|.
Θ0=arctan(LK)θ1=0θ2=|Δθ|,ForK0,
Θ0=arctan(LK)θ1=0θ2=|Δθ|+π,ForK<0.
Θ0=Θ0.
Θ1=ΘΘ0=Θ+Θ0.
θ1=Θ1=Θ+Θ0,
θ2=Θ1|Δθ|=Θ+Θ0|Δθ|.
Φx=ΦcosΘ,Φy=ΦsinΘ.
ωr=dΦ/dt,ωt=dΘ/dt.
ω1=dθ1dt=ωtdΘ0dt,ω1=dθ1dt=ωt+dΘ0dt(For prismΠ1),
ω2=dθ2dt=ωtdΘ0dt+d(Δθ)dt,ω2=dθ2dt=ωt+dΘ0dtd(Δθ)dt(For prismΠ2).
ω1=ω1=ω2=ω2=ωt.
dθ1/dΦ=dθ1/dΦanddθ2/dΦ=dθ2/dΦ.

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