Abstract

Two different analytical methods, the first-order paraxial approximation method and the nonparaxial ray tracing method, are applied to determine the steering mechanism of the Risley prism system, including the pointing prediction and the complete and exact inverse orientation solutions. The analytical results obtained with the two different methods are investigated in detail about the pointing prediction and the two groups of inverse orientation solutions, respectively. Risley prism equipment for wide angular range beam scanning is assembled and the experimental setup is built to test the steering mechanism of the Risley prism system. Experimental results validate the availability of the nonparaxial ray tracing method to discuss the beam steering mechanism for the Risley prism system.

© 2013 Optical Society of America

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  1. 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]
  2. 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]
  3. T. G. Garcia, M. Stronjnik, and G. Paez, “Risley prisms to control wave-front tilt and displacement in a vectorial shearing interferometer,” Appl. Opt. 41, 1380–1384 (2002).
    [CrossRef]
  4. D. C. Weber, J. D. Trolinger, R. G. Nichols, and A. K. Lal, “Diffractively corrected Risley prism for infrared imaging,” Proc. SPIE 4025, 79–86 (2000).
    [CrossRef]
  5. C. Chu, “Double Risley prism pairs for optical beam steering and alignment,” U.S. patent 20040057656A1 (25March2004).
  6. C. Schwarze, R. Vaillancourt, D. Carlson, E. Schundler, T. Evans, and J. Engel, “Risley-prism based compact laser beam steering for IRCM, laser communications, and laser radar,” Critical Technology 9, 1–9 (2005).
  7. C. Schwarze, “A new look at Risley prism,” Photon. Spectra 40, 67–70 (2005).
  8. O. Miroslaw, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 1–10 (2006).
    [CrossRef]
  9. Y. Li, “Closed form analytical inverse solutions for Risley-prism-based beam steering systems in different configurations,” Appl. Opt. 50, 4302–4309 (2011).
    [CrossRef]
  10. G. C. Boisset, B. Robertson, and H. S. Hinton, “Design and construction of an active alignment demonstrator for a free-space optical interconnect,” IEEE Photon. Technol. Lett. 7, 676–679 (1995).
    [CrossRef]
  11. J. J. Degnan, “Ray matrix approach for the real time control of SLR2000 optical elements,” presented at Proceedings of the 14th International Workshop on Laser Ranging, San Fernando, Spain, 2004.
  12. Y. Yang, “Analytic solution of free space optical beam steering using Risley prisms,” J. Lightwave Technol. 26, 3576–3583 (2008).
    [CrossRef]
  13. J. S. Horng and Y. J. Li, “Error source and their impact on the performance of dual-wedge beam steering system,” Appl. Opt. 51, 4168–4175 (2012).
    [CrossRef]
  14. Y. Li, “Third-order theory of the Risley-prism-based beam steering system,” Appl. Opt. 50, 679–686 (2011).
    [CrossRef]

2012 (1)

2011 (2)

2008 (1)

2006 (1)

O. Miroslaw, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 1–10 (2006).
[CrossRef]

2005 (2)

C. Schwarze, R. Vaillancourt, D. Carlson, E. Schundler, T. Evans, and J. Engel, “Risley-prism based compact laser beam steering for IRCM, laser communications, and laser radar,” Critical Technology 9, 1–9 (2005).

C. Schwarze, “A new look at Risley prism,” Photon. Spectra 40, 67–70 (2005).

2003 (1)

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 (2)

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]

T. G. Garcia, M. Stronjnik, and G. Paez, “Risley prisms to control wave-front tilt and displacement in a vectorial shearing interferometer,” Appl. Opt. 41, 1380–1384 (2002).
[CrossRef]

2000 (1)

D. C. Weber, J. D. Trolinger, R. G. Nichols, and A. K. Lal, “Diffractively corrected Risley prism for infrared imaging,” Proc. SPIE 4025, 79–86 (2000).
[CrossRef]

1995 (1)

G. C. Boisset, B. Robertson, and H. S. Hinton, “Design and construction of an active alignment demonstrator for a free-space optical interconnect,” IEEE Photon. Technol. Lett. 7, 676–679 (1995).
[CrossRef]

Boisset, G. C.

G. C. Boisset, B. Robertson, and H. S. Hinton, “Design and construction of an active alignment demonstrator for a free-space optical interconnect,” IEEE Photon. Technol. Lett. 7, 676–679 (1995).
[CrossRef]

Bos, P. J.

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]

Carlson, D.

C. Schwarze, R. Vaillancourt, D. Carlson, E. Schundler, T. Evans, and J. Engel, “Risley-prism based compact laser beam steering for IRCM, laser communications, and laser radar,” Critical Technology 9, 1–9 (2005).

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]

Chu, C.

C. Chu, “Double Risley prism pairs for optical beam steering and alignment,” U.S. patent 20040057656A1 (25March2004).

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]

Degnan, J. J.

J. J. Degnan, “Ray matrix approach for the real time control of SLR2000 optical elements,” presented at Proceedings of the 14th International Workshop on Laser Ranging, San Fernando, Spain, 2004.

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.

O. Miroslaw, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 1–10 (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]

Engel, J.

C. Schwarze, R. Vaillancourt, D. Carlson, E. Schundler, T. Evans, and J. Engel, “Risley-prism based compact laser beam steering for IRCM, laser communications, and laser radar,” Critical Technology 9, 1–9 (2005).

Evans, T.

C. Schwarze, R. Vaillancourt, D. Carlson, E. Schundler, T. Evans, and J. Engel, “Risley-prism based compact laser beam steering for IRCM, laser communications, and laser radar,” Critical Technology 9, 1–9 (2005).

Garcia, T. G.

Gutow, D.

O. Miroslaw, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 1–10 (2006).
[CrossRef]

Harford, S.

O. Miroslaw, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 1–10 (2006).
[CrossRef]

Hinton, H. S.

G. C. Boisset, B. Robertson, and H. S. Hinton, “Design and construction of an active alignment demonstrator for a free-space optical interconnect,” IEEE Photon. Technol. Lett. 7, 676–679 (1995).
[CrossRef]

Hoffman, C.

O. Miroslaw, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 1–10 (2006).
[CrossRef]

Horng, J. S.

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]

Lal, A. K.

D. C. Weber, J. D. Trolinger, R. G. Nichols, and A. K. Lal, “Diffractively corrected Risley prism for infrared imaging,” Proc. SPIE 4025, 79–86 (2000).
[CrossRef]

Li, Y.

Li, Y. J.

Miroslaw, O.

O. Miroslaw, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 1–10 (2006).
[CrossRef]

Nichols, R. G.

D. C. Weber, J. D. Trolinger, R. G. Nichols, and A. K. Lal, “Diffractively corrected Risley prism for infrared imaging,” Proc. SPIE 4025, 79–86 (2000).
[CrossRef]

Paez, G.

Pierce, R.

O. Miroslaw, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 1–10 (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]

Robertson, B.

G. C. Boisset, B. Robertson, and H. S. Hinton, “Design and construction of an active alignment demonstrator for a free-space optical interconnect,” IEEE Photon. Technol. Lett. 7, 676–679 (1995).
[CrossRef]

Sanchez, M.

O. Miroslaw, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 1–10 (2006).
[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.

C. Schwarze, R. Vaillancourt, D. Carlson, E. Schundler, T. Evans, and J. Engel, “Risley-prism based compact laser beam steering for IRCM, laser communications, and laser radar,” Critical Technology 9, 1–9 (2005).

Schwarze, C.

C. Schwarze, R. Vaillancourt, D. Carlson, E. Schundler, T. Evans, and J. Engel, “Risley-prism based compact laser beam steering for IRCM, laser communications, and laser radar,” Critical Technology 9, 1–9 (2005).

C. Schwarze, “A new look at Risley prism,” Photon. Spectra 40, 67–70 (2005).

Sergan, V.

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]

Stronjnik, M.

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]

Trolinger, J. D.

D. C. Weber, J. D. Trolinger, R. G. Nichols, and A. K. Lal, “Diffractively corrected Risley prism for infrared imaging,” Proc. SPIE 4025, 79–86 (2000).
[CrossRef]

Vaillancourt, R.

C. Schwarze, R. Vaillancourt, D. Carlson, E. Schundler, T. Evans, and J. Engel, “Risley-prism based compact laser beam steering for IRCM, laser communications, and laser radar,” Critical Technology 9, 1–9 (2005).

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]

Weber, D. C.

D. C. Weber, J. D. Trolinger, R. G. Nichols, and A. K. Lal, “Diffractively corrected Risley prism for infrared imaging,” Proc. SPIE 4025, 79–86 (2000).
[CrossRef]

Yang, Y.

Appl. Opt. (4)

Critical Technology (1)

C. Schwarze, R. Vaillancourt, D. Carlson, E. Schundler, T. Evans, and J. Engel, “Risley-prism based compact laser beam steering for IRCM, laser communications, and laser radar,” Critical Technology 9, 1–9 (2005).

IEEE Photon. Technol. Lett. (1)

G. C. Boisset, B. Robertson, and H. S. Hinton, “Design and construction of an active alignment demonstrator for a free-space optical interconnect,” IEEE Photon. Technol. Lett. 7, 676–679 (1995).
[CrossRef]

J. Lightwave Technol. (1)

Opt. Eng. (1)

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]

Photon. Spectra (1)

C. Schwarze, “A new look at Risley prism,” Photon. Spectra 40, 67–70 (2005).

Proc. SPIE (3)

O. Miroslaw, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 1–10 (2006).
[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]

D. C. Weber, J. D. Trolinger, R. G. Nichols, and A. K. Lal, “Diffractively corrected Risley prism for infrared imaging,” Proc. SPIE 4025, 79–86 (2000).
[CrossRef]

Other (2)

C. Chu, “Double Risley prism pairs for optical beam steering and alignment,” U.S. patent 20040057656A1 (25March2004).

J. J. Degnan, “Ray matrix approach for the real time control of SLR2000 optical elements,” presented at Proceedings of the 14th International Workshop on Laser Ranging, San Fernando, Spain, 2004.

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Figures (11)

Fig. 1.
Fig. 1.

Notation and coordinate systems for Risley prism system. The unit vector s^1i for the incident ray is collinear and negative with the Z axis, which is also the optical axis of the system. The zero orientation of both prisms is described with the opening angle pointing up and the positive rotational angle follows the right-hand rule around axis Z.

Fig. 2.
Fig. 2.

Steering mechanism study with the first-order paraxial approximation method (a) pointing position prediction. The total ray deviation of the prisms can be obtained with the vector sum of δ^1 and δ^2 and (b) orientations inverse solution. Two triangle diagrams can be graphed for a pointing position. Therefore, there are two groups of inverse oriental solutions.

Fig. 3.
Fig. 3.

Schematic diagram of the nonparaxial ray tracing method. The refractive direction of the incident ray is deviated at the left surface of prism Π1 and the right surface of prism Π2. Angle deviation between the left surface of prism Π1 and the right surface of prism Π2 can be neglected, because the two sides of the air gap are parallel.

Fig. 4.
Fig. 4.

Difference between the two groups of prisms’ orientations solutions with the nonparaxial ray tracing method: Δ=θ1N1θ2N2=θ2N1θ1N2. The difference is rotationally symmetrical around the center axis, i.e., it only relies on the altitude angle and is independent on the azimuth angle.

Fig. 5.
Fig. 5.

Comparison of pointing position prediction with the first-order paraxial approximation method and the nonparaxial ray tracing method. (a) The difference of altitude angles obtained by two different methods when α1=α2=10° and n1=n2=1.5195, (b) the difference of azimuth angles when α1=α2=10° and n1=n2=1.5195, (c) the difference of altitude angles obtained by two different methods when α1=α2=7° and n1=n2=3.478, (d) the difference of azimuth angles when α1=α2=7° and n1=n2=3.478, (e) the absolute difference of altitude angle varied with the deflection angles, and (f) the absolute difference of azimuth angle varied with the deflection angles.

Fig. 6.
Fig. 6.

Comparison of the reverse solutions obtained with the two methods. (a) Angle difference of the first group of inverse orientation solution of prism Π1, Δ11=θ1F1θ1N1, (b) angle difference of the first group of inverse orientation solution of prism Π2Δ12=θ1F2θ1N2, (c) angle difference of the second group of inverse orientation solution of prism Π1Δ21=θ2F1θ2N1, and (d) angle difference of the second group of inverse orientation solution of prism Π2Δ22=θ2F2θ2N2.

Fig. 7.
Fig. 7.

Experiment principle. Two torque actuators are controlled to rotate independently, and the angle position of the two prisms is measured with two high-resolution encoders.

Fig. 8.
Fig. 8.

Experimental setup to investigate the steering mechanism of Risley prism. A green laser source is mounted in front of the Risley prisms system. The pointing position is surveyed by measuring the position of the laser pot on the grid paper.

Fig. 9.
Fig. 9.

Comparison of the experimental results and the theoretical results obtained with the first-order approximate method and the nonparaxial ray tracing method. (a) ΔΦ1=ΦFΦE, (b) ΔΦ2=ΦNΦE, (c) ΔΘ1=ΘFΘE, and (d) ΔΘ2=ΘNΘE.

Fig. 10.
Fig. 10.

Experimental values of the altitude angles with the first group of solutions by the two different methods. (a) The expected altitude angle is 1°, (b) the expected altitude angle is 7°, and (c) the expected altitude angle is 10°.

Fig. 11.
Fig. 11.

Difference between the experimental values and the expected values of the azimuth angles with the first group of solutions by the two different methods. (a) The expected altitude angle is 1°, (b) the expected altitude angle is 7°, and (c) the expected altitude angle is 10°.

Equations (28)

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δ1=α1(n11),
δ2=α2(n21).
δX=δ1cosθ1+δ2cosθ2,
δY=δ1sinθ1+δ2sinθ2.
Φ=δX2+δY2=δ12+δ22+2δ1δ2cos(θ1θ2),
Θ=arctanδYδX=arctanδ1sinθ1+δ2sinθ2δ1cosθ1+δ2cosθ2.
α=arccos(δ12+Φ2δ222δ1Φ),
β=arccos(δ22+Φ2δ122δ2Φ).
{θ1F1=Θα=Θarccos(δ12+Φ2δ222δ1Φ)θ2F1=Θ+β=Θ+arccos(δ22+Φ2δ122δ2Φ)
{θ1F2=Θ+α=Θ+arccos(δ12+Φ2δ222δ1Φ)θ2F2=Θβ=Θarccos(δ22+Φ2δ122δ2Φ).
n^1=(sinα1cosθ1,sinα1sinθ1,cosα1).
s^1r=1n1[s^1i(s^1i·n^1)n^1]n^111n12+1n12(s^1i·n^1)2.
s^2i=s^1r.
n^2=(sinα2cosθ2,sinα2sinθ2,cosα2).
s^2r=n2[s^2i(s^2i·n^2)n^2]n^21n22+n22(s^2i·n^2)2.
{K=a1cosθ1+a3sinα2cosθ2L=a1sinθ1+a3sinα2sinθ2M=a2a3cosα2,
{a1=n2n1sinα1(cosα1n12sin2α1);a2=n2n1(n12sin2α1cosα1+sin2α1);a3=(a1sinα2cosΔθa2cosα2)+1n22+(a1sinα2cosΔθa2cosα2)2
Φ=arccos(M),
Θ={arctan(LK);whenK0andL0arctan(LK)+2π;whenK0andL<0arctan(LK)+π;whenK<0.
cosΦ=M=a2+a3cosα2|Δθ|=arccos(1a1tanα2(a2+12(a2+cosΦ)1n22(a2+cosΦcosα2)2)).
ψ01={arctan(LK)θ1=0θ2=|Δθ|,whenK0andL0arctan(LK)θ1=0θ2=|Δθ|+2π,whenK0andL<0arctan(LK)θ1=0θ2=|Δθ|+π,whenK<0.
θs1=Θψ01.
{θ1N1=θs1=Θψ01θ2N1=θs1+|Δθ|=Θψ01+|Δθ|.
ψ02={arctan(LK)θ1=|Δθ|θ2=0,whenK0andL0arctan(LK)θ1=|Δθ|θ2=0+2π,whenK0andL<0arctan(LK)θ1=|Δθ|θ2=0+π,whenK<0.
θs2=Θψ02.
{θ1N2=θs2+|Δθ|=Θψ02+|Δθ|θ2N2=θs2=Θψ02.
ΦE=arctan(XP2+YP2/L),
ΘE={arccos(YPXP),whenYP0arccos(YPXP)+π,whenYP<0,

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