Abstract

Nonparaxial ray tracing through Risley prisms of four different configurations is performed to give the exact solution of the inverse problem arisen from applications of Risley prisms to free space commu nications. Predictions of the exact solution and the third-order theory [Appl. Opt. 50, 679 (2011)] are compared and results are shown by curves for systems using prisms of different materials. The exact solution for the problem of precision pointing is generalized to investigate the synthesis of the scan pattern, i.e., to create a desirable scan pattern on some plane perpendicular to the optical axis of the system by controlling the circular motion of the two prisms.

© 2011 Optical Society of America

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References

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  1. F. R. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 2001), Sec. 2.7.
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    [CrossRef]
  3. W. L. Wolfe, “Optical-mechanical scanning techniques and devices,” in W.L.Wolfe and G.J.Zissis (eds.), The Infrared Handbook (Environmental Research Institute of Michigan, 1989), Chapt. 10.
  4. Y. Li, “Third-order theory of Risley-prism based beam steering system,” Appl. Opt. 50, 679–686 (2011).
    [CrossRef] [PubMed]
  5. C. T. Amirault and C. A. DiMarzio, “Precision pointing using a dual-wedge scanner,” Appl. Opt. 24, 1302–1308(1985).
    [CrossRef] [PubMed]
  6. 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, 676–678(1995).
    [CrossRef]
  7. J. J. Degnan, “Ray matrix approach for the real time control of SLR2000 optical elements,” in 14th International Workshop on Laser Ranging (2004).
  8. Y. Yang, “Analytic solution of free space optical beam steering using Risley prisms,” J. Lightwave Technol. 26, 3576–3583(2008).
    [CrossRef]
  9. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Sec. 3.2.2.
  10. Y. Li, “Single-mirror beam steering system: analysis and synthesis of high-order conic-section scan patterns,” Appl. Opt. 47, 386–397 (2008).
    [CrossRef] [PubMed]
  11. G. Garcia-Torales, M. Strojnik, and G. Paez, “Risley prisms to control wave-front tilt and displacement in a vectorial interferometer,” Appl. Opt. 41, 1380–1384 (2002).
    [CrossRef] [PubMed]
  12. W. C. Warger II and C. A. DiMarzio, “Dual-wedge scanning confocal reflectance microscope,” Opt. Lett. 32, 2140–2142(2007).
    [CrossRef] [PubMed]

2011

2008

2007

2004

J. J. Degnan, “Ray matrix approach for the real time control of SLR2000 optical elements,” in 14th International Workshop on Laser Ranging (2004).

2002

2001

F. R. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 2001), Sec. 2.7.

1999

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Sec. 3.2.2.

1995

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, 676–678(1995).
[CrossRef]

1989

W. L. Wolfe, “Optical-mechanical scanning techniques and devices,” in W.L.Wolfe and G.J.Zissis (eds.), The Infrared Handbook (Environmental Research Institute of Michigan, 1989), Chapt. 10.

1960

Amirault, C. T.

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, 676–678(1995).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Sec. 3.2.2.

Degnan, J. J.

J. J. Degnan, “Ray matrix approach for the real time control of SLR2000 optical elements,” in 14th International Workshop on Laser Ranging (2004).

DiMarzio, C. A.

Garcia-Torales, G.

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, 676–678(1995).
[CrossRef]

Jenkins, F. R.

F. R. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 2001), Sec. 2.7.

Li, Y.

Paez, G.

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, 676–678(1995).
[CrossRef]

Rosell, F. A.

Strojnik, M.

Warger, W. C.

White, H. E.

F. R. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 2001), Sec. 2.7.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Sec. 3.2.2.

Wolfe, W. L.

W. L. Wolfe, “Optical-mechanical scanning techniques and devices,” in W.L.Wolfe and G.J.Zissis (eds.), The Infrared Handbook (Environmental Research Institute of Michigan, 1989), Chapt. 10.

Yang, Y.

Appl. Opt.

IEEE Photon. Technol. Lett.

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, 676–678(1995).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am.

Opt. Lett.

Other

F. R. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 2001), Sec. 2.7.

W. L. Wolfe, “Optical-mechanical scanning techniques and devices,” in W.L.Wolfe and G.J.Zissis (eds.), The Infrared Handbook (Environmental Research Institute of Michigan, 1989), Chapt. 10.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Sec. 3.2.2.

J. J. Degnan, “Ray matrix approach for the real time control of SLR2000 optical elements,” in 14th International Workshop on Laser Ranging (2004).

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram illustrating the notation and coordinate systems for Risley prisms. The unit vector s ^ 1 ( i ) for the incident ray is collinear with the z axis, which is also the axis of rotation for the two prisms Π 1 and Π 2 , which may not be identical and having indexes n 1 and n 2 and opening angles α 1 and α 2 , respectively. The rotational angles θ 1 and θ 2 are measured from the x axis and di agram shows the prisms at θ 1 = 90 ° , θ 2 = + 90 ° .

Fig. 2
Fig. 2

Diagram illustrating the Risley prisms in four different configurations. The symbol 1 represents the plane face of the wedge prism perpendicular to the axis of rotation and two for the face inclined to the axis. The Risley prisms in groups A and B share the same form expressions for the ray emergent from the prism systems.

Fig. 3
Fig. 3

Comparison of the maximum ray deviation angle Φ m for systems in the groups A and B under the conditions of two identical wedge prisms with different opening angle α and refractive index (a)  n = 1.5 and (b)  n = 4.0 .

Fig. 4
Fig. 4

Position error introduced by the third-order theory as the theory is applied to the Risley prisms of the 21-12 configuration having two identical wedge prisms of (a)  n = 1.5 and (b)  n = 4.0 .

Fig. 5
Fig. 5

Synthesis of an elliptical scan pattern by using the Risley prisms of the 21-12 configuration containing two identical wedge prisms of refractive index n = 1.5 and opening angle α = 5 ° . (a) The elliptical pattern; (b) The azimuthal rotation angle between the two prisms as a function of the time-dependent parameter τ; (c) Rotation angles θ 1 and θ 2 for the two prisms.

Tables (2)

Tables Icon

Table 1 Coefficients ( a 1 , a 2 , a 3 ) in Eqs. (2.5a, 2.5b, 2.5c) for the Direction Cosines of the Ray Emergent from in the Risley Prisms in the Group A Configurations

Tables Icon

Table 2 Coefficients ( b 1 , b 2 , b 3 ) in Eqs. (2.6a, 2.6b, 2.6c) for the Direction Cosines of the Ray Emergent from in the Risley Prisms in the Group B Configurations

Equations (48)

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n ^ 1 = ( sin α 1 cos θ 1 , sin α 1 sin θ 1 , cos α 1 ) .
s ^ 1 ( r ) = 1 n 1 [ s ^ 1 ( i ) ( s ^ 1 ( i ) · n ^ 1 ) n ^ 1 ] ( n ^ 1 ) 1 ( 1 n 1 ) 2 + ( 1 n 1 ) 2 ( s ^ 1 ( i ) · n ^ 1 ) 2 .
n ^ 2 = ( sin α 2 cos θ 2 , sin α 2 sin θ 2 , cos α 2 ) .
s ^ 2 ( r ) = n 2 [ s ^ 1 ( r ) ( s ^ 1 ( r ) · n ^ 2 ) n ^ 2 ] ( n ^ 2 ) 1 n 2 2 + n 2 2 ( s ^ 1 ( r ) · n ^ 2 ) 2 .
K A = a 1 cos θ 1 + a 3 sin α 2 cos θ 2 ,
L A = a 1 sin θ 1 + a 3 sin α 2 sin θ 2
M A = a 2 a 3 cos α 2 .
K B = b 1 cos θ 1 b 3 sin α 2 cos θ 2 ,
L B = b 1 sin θ 1 b 3 sin α 2 sin θ 2
M B = 1 n 2 2 + ( b 2 + b 3 cos α 2 ) 2 .
cos Φ = M A = ( a 2 a 3 cos α 2 ) ( For system in the Group   A ) ,
cos Φ = M B = 1 n 2 2 + ( b 2 + b 3 cos α 2 ) 2 ( For system in the Group   B ) .
( Δ θ ) 0 = arccos ( 1 a 1 tan α 2 { a 2 + 1 2 ( a 2 + cos Φ ) [ 1 n 2 2 ( a 2 + cos Φ cos α 2 ) 2 ] } ) .
( Δ θ ) 0 = arccos ( 1 b 1 tan α 2 { b 2 + 1 b 2 ± cos 2 Φ 1 + n 2 2 [ 1 n 2 2 ( b 2 ± cos 2 Φ 1 + n 2 2 cos α 2 ) 2 ] } ) .
ψ 0 = arctan ( L A K A ) θ 1 = 0 , θ 2 = ( Δ θ ) 0 = arctan ( tan α 2 ( a 2 + cos Φ ) sin ( Δ θ ) 0 a 1 + tan α 2 ( a 2 + cos Φ ) cos ( Δ θ ) 0 ) .
ψ 0 = arctan ( L B K B ) θ 1 = 0 , θ 2 = ( Δ θ ) 0 = arctan ( tan α 2 ( b 2 cos 2 Φ 1 + n 2 2 ) sin ( Δ θ ) 0 b 1 + tan α 2 ( b 2 cos 2 Φ 1 + n 2 2 ) cos ( Δ θ ) 0 ) .
θ 1 = Θ ψ 0 and θ 2 = θ 1 + ( Δ θ ) 0 = Θ ψ 0 + ( Δ θ ) 0 .
δ 1 = ( θ 1 ) Exact ( θ 1 ) 3 rd _ order and δ 2 = ( θ 2 ) Exact ( θ 2 ) 3 rd _ order ,
δ 1 = ( ψ 0 ) Exact + ( ψ 0 ) 3 rd _ order and δ 2 = Δ θ 1 + [ ( Δ θ ) 0 ] Exact [ ( Δ θ ) 0 ] 3 rd _ order ,
x 1 = P tan Φ and y 1 = 0 .
d = ( x 2 x 1 ) 2 + ( y 2 y 1 ) 2 .
x 2 K A = y 2 L A = P M A ,
d P = [ ( K A M A ) + tan Φ ] 2 + ( L A M A ) | θ 1 = ( θ 1 ) 3 rd _ order , θ 2 = ( θ 2 ) 3 rd _ order .
x = x ( τ ) and y = y ( τ ) ,
ρ ( τ ) = x 2 ( τ ) + y 2 ( τ )
φ ( τ ) = arctan ( y ( τ ) x ( τ ) ) .
x ( τ ) = K A M A P and y ( τ ) = L A M A P ,
[ ρ ( τ ) P ] 2 = x 2 ( τ ) + y 2 ( τ ) = K A 2 + L A 2 M A 2 = K A 2 + L A 2 1 ( K A 2 + L A 2 ) .
a 1 2 + a 3 2 sin 2 α 2 + 2 a 1 a 3 sin α 2 cos ( Δ θ ) = ρ 2 ( τ ) P 2 + ρ 2 ( τ ) ,
cos Δ θ = 1 a 1 sin α 2 ( 1 n 2 2 a 3 2 2 a 3 + a 2 cos α 2 ) .
A a 3 2 + B a 3 + C = 0 ,
a 3 = B ± B 2 4 A C 2 B ,
Δ θ = arccos [ 1 a 1 sin α 2 ( 1 n 2 2 a 3 2 2 a 3 + a 2 cos α 2 ) ] .
tan φ ( τ ) = y ( τ ) x ( τ ) = L A K A = a 1 sin θ 1 + a 3 sin α 2 sin θ 2 a 1 cos θ 1 + a 3 sin α 2 cos θ 2 .
tan φ ( τ ) = a 1 sin θ 1 + a 3 sin α 2 sin ( θ 1 + Δ θ ) a 1 cos θ 1 + a 3 sin α 2 cos ( θ 1 + Δ θ ) = sin ( θ 1 + θ 10 ) cos ( θ 1 + θ 10 ) = tan ( θ 1 + θ 10 ) ,
θ 1 = φ ( τ ) θ 10 and θ 2 = φ ( τ ) θ 10 + Δ θ .
x = w x cos τ and y = w y sin τ ,
cos ( Δ θ ) 1 / 2 cos ( Δ θ ) 2 / 2 = cos ( 24.6 ° ) cos ( 62.8 ° ) = 0.9092 0.4571 2
sin α 1 ( cos α 1 n 1 2 sin 2 α 1 )
sin α 1 ( n 1 cos α 1 + 1 n 1 2 sin 2 α 1 )
n 2 2 n 1 2 + [ sin 2 α 1 + cos α 1 n 1 2 sin 2 α 1 ] 2
n 2 2 1 + [ n 1 sin 2 α 1 + cos α 1 1 n 1 2 sin 2 α 1 ] 2
( a 1 sin α 2 cos Δ θ a 2 cos α 2 ) + 1 n 2 2 + ( a 1 sin α 2 cos Δ θ a 2 cos α 2 ) 2 ( Δ θ = θ 2 θ 1 )
sin α 1 ( cos α 1 n 1 2 sin 2 α 1 )
sin α 1 ( n 1 cos α 1 + 1 n 1 2 sin 2 α 1 )
1 n 1 2 + ( sin 2 α 1 + cos α 1 n 1 2 sin 2 α 1 ) 2
n 1 sin 2 α 1 + cos α 1 1 n 1 2 sin 2 α 1
b 1 sin α 2 cos Δ θ b 2 cos α 2 + n 2 2 1 + ( b 1 sin α 2 cos Δ θ b 2 cos α 2 ) 2 ( Δ θ = θ 2 θ 1 )

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