Abstract

Nonparaxial ray tracing is performed to investigate the field scanned out by a single beam through two rotatable thick prisms with different parameters, and a general solution is obtained and then expanded into a power series to establish the third-order theory for Risley prisms that paves the way to investigate topics of interest such as optical distortions in the scan pattern and an analytical solution of the inverse problem of a Risley-prism-based laser beam steering system; i.e., the problem is concerned with how to direct a laser beam to any specified direction within the angular range of the system.

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References

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  1. F. A. Rosell, “Prism scanners,” J. Opt. Soc. Am. 50, 521–526 (1960).
    [CrossRef]
  2. C. T. Amirault and C. A. DiMarzio, “Precision pointing using a dual-wedge scanner,” Appl. Opt. 24, 1302–1308(1985).
    [CrossRef] [PubMed]
  3. 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]
  4. W. L. Wolfe, “Optical-mechanical scanning techniques and devices,” in The Infrared Handbook, W.L.Wolfe and G.J.Zissis, eds. (Environmental Research Institute of Michigan, 1989), Chap. 10.
  5. Y. Yang, “Analytic solution of free space optical beam steering using Risley prisms,” J. Lightwave Technol. 26, 3576–3583(2008).
    [CrossRef]
  6. F. R. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 2001), Sec. 2.7.
  7. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), Sec. 3.2.2.
  8. Y. Li and J. Katz, “Laser beam scanning by rotary mirrors. I. modeling mirror scanning devices,” Appl. Opt. 34, 6403–6416 (1995).
    [CrossRef] [PubMed]
  9. H. Erfle, “Über die durch ein Drehkeilpaar erzeugte Ablenkung und über eine als Kennzeichen für die Beibehaltung des ‘Hauptschnittes’ dienende Sinusbedingung,” Zeitschrift für Physik 1, 57–81 (1920).
    [CrossRef]
  10. http://vova1001.narod.ru/00009295.htm.

2008

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]

1995

1985

1960

1920

H. Erfle, “Über die durch ein Drehkeilpaar erzeugte Ablenkung und über eine als Kennzeichen für die Beibehaltung des ‘Hauptschnittes’ dienende Sinusbedingung,” Zeitschrift für Physik 1, 57–81 (1920).
[CrossRef]

Amirault, C. T.

Born, M.

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

DiMarzio, C. A.

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]

Erfle, H.

H. Erfle, “Über die durch ein Drehkeilpaar erzeugte Ablenkung und über eine als Kennzeichen für die Beibehaltung des ‘Hauptschnittes’ dienende Sinusbedingung,” Zeitschrift für Physik 1, 57–81 (1920).
[CrossRef]

Gutow, D.

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]

Jenkins, F. R.

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

Katz, J.

Li, Y.

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]

Rosell, F. A.

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]

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 University, 1999), Sec. 3.2.2.

Wolfe, W. L.

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

Yang, Y.

Appl. Opt.

J. Lightwave Technol.

J. Opt. Soc. Am.

Proc. SPIE

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]

Zeitschrift für Physik

H. Erfle, “Über die durch ein Drehkeilpaar erzeugte Ablenkung und über eine als Kennzeichen für die Beibehaltung des ‘Hauptschnittes’ dienende Sinusbedingung,” Zeitschrift für Physik 1, 57–81 (1920).
[CrossRef]

Other

http://vova1001.narod.ru/00009295.htm.

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

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

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

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram illustrating the notation and coordinate systems for ray tracing through Risley prisms. The incident beam is collinear with the z axis, which is also the axis of rotation of the two prisms Π 1 and Π 2 of indices n 1 and n 2 and apex angles α 1 and α 2 , respectively.

Fig. 2
Fig. 2

Curves plotted from Eqs. (16, 17, 18, 19, 20, 22) under the condition of θ 1 = θ 2 = 0 for a comparison of the percentage difference Δ ( % ) in the calculations of the patterns scanned out by a single ray through Risley prisms having two identical prisms of index n = 1.5 and the apex angle of α = 0 ° 30 ° on a plane in the far-field region ( D / P 0 ) and planes at different distances D / P = 0.15 , 0.3, and 0.45 in the near-field region.

Fig. 3
Fig. 3

Distortions of line scan patterns produced by two prisms counter-rotated at equal angular velocities. The prisms are identical with index n = 1.5 and apex angle α = 15 ° : (a) near-field and (b) far-field region scan patterns.

Fig. 4
Fig. 4

Far-field region scan patterns predicted by the third-order theory to show the scanning spot along the three-petal-rose curve produced by a pair of identical prisms of n = 1.5 , apex angle α = 15 ° and counter-rotated with the angular frequency ratio m = 2 . Incident beam with (a) circular profile of radius 0.015 P and (b) elliptical profile with the major and minor axes ( 0.02 P , 0.01 P ).

Fig. 5
Fig. 5

Curves plotted under the condition of two identical prisms of index n = 1.5 and different apex angle α ranging from 2.5 ° to 10 ° to show the third-order solution of the inverse problem, i.e., to direct a laser beam to the desired altitude Φ and azimuth Θ. (a) Curves, plotted from Eq. (26), showing the dependence of the azimuth rotation angle ( Δ θ ) 0 between the two prisms on the desired altitude Φ, (b) Curves, plotted from Eq. (30), showing the azimuth ψ 0 of the beam after the desired altitude is achieved.

Tables (1)

Tables Icon

Table 1 Percentage Error of the Approximate Formulas Eqs. (18, 20, 22) of Different Orders as Compared with the Prediction of the Exact Solution in Eq. (16) under the Condition of θ 1 = θ 2 = 0

Equations (37)

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δ q = ( n q 1 ) α q ( q = 1 , 2 ) .
θ q = ω q t + θ 0 q ( q = 1 , 2 ) ,
D = d 1 + d air + d 2 , D n = d 1 n 1 + d air + d 2 n 2 ,
m = ω 2 / ω 1 .
ξ = x 0 + ξ 0 cos τ and η = y 0 + η 0 sin τ .
( X Y ) = P [ δ 1 ( cos θ 1 sin θ 1 ) + δ 2 ( cos θ 2 sin θ 2 ) ] .
R 1 F = X + i Y = P [ δ 1 exp ( i θ 1 ) + δ 2 exp ( i θ 2 ) ] ,
r 10 = ( ξ , η , 0 ) .
s ^ 11 ( i ) × ( r r 10 ) = 0 ,
n ^ 11 = ( sin α 1 cos θ 1 , sin α 1 sin θ 1 , cos α 1 ) ,
n ^ 11 · r = 0 .
s ^ 11 ( r ) = 1 n 1 [ s ^ 11 ( i ) ( s ^ 11 ( i ) · n ^ 11 ) n ^ 11 ] ( n ^ 11 ) 1 ( 1 n 1 ) 2 + ( 1 n 1 ) 2 ( s ^ 11 ( i ) · n ^ 11 ) 2 .
s ^ 12 ( r ) = n 1 [ s ^ 11 ( r ) ( s ^ 11 ( r ) · n ^ 12 ) n ^ 12 ] ( n ^ 12 ) 1 n 1 2 + n 1 2 ( s ^ 11 ( r ) · n ^ 12 ) 2 .
K 22 ( r ) = sin α 1 ( cos α 1 n 1 2 sin 2 α 1 ) cos θ 1 + A sin α 2 cos θ 2 , L 22 ( r ) = sin α 1 ( cos α 1 n 1 2 sin 2 α 1 ) sin θ 1 + A sin α 2 sin θ 2 , M 22 ( r ) = n 2 2 n 1 2 + [ sin 2 α 1 + cos α 1 n 1 2 sin 2 α 1 ] 2 A cos α 2 } ,
A = 1 n 2 2 + [ cos α 2 n 2 2 n 1 2 + [ sin 2 α 1 + cos α 1 n 1 2 sin 2 α 1 ] 2 + sin α 1 sin α 2 ( cos α 1 n 1 2 sin 2 α 1 ) cos ( θ 2 θ 1 ) ] 2 cos α 2 n 2 2 n 1 2 + [ sin 2 α 1 + cos α 1 n 1 2 sin 2 α 1 ] 2 sin α 1 sin α 2 ( cos α 1 n 1 2 sin 2 α 1 ) cos ( θ 2 θ 1 ) .
R = r 22 ( P + D ) + k ^ · r 22 k ^ · s ^ 22 ( r ) s ^ 22 ( r ) .
( X Y ) = P sin α 1 ( cos α 1 n 1 2 sin 2 α 1 ) ( cos θ 1 sin θ 1 ) + A sin α 2 ( cos θ 2 sin θ 2 ) n 2 2 n 1 2 + [ sin 2 α 1 + cos α 1 n 1 2 sin 2 α 1 ] 2 + K 22 cos α 2 ,
R = X + i Y = P sin α 1 ( cos α 1 n 1 2 sin 2 α 1 ) exp ( i θ 1 ) + A sin α 2 exp ( i θ 2 ) n 2 2 n 1 2 + [ sin 2 α 1 + cos α 1 n 1 2 sin 2 α 1 ] 2 + K 22 cos α 2 .
R 3 = R 1 F + P { [ 3 n 1 3 6 n 1 2 + 2 n 1 + 3 6 n 1 ( n 1 1 ) 2 δ 1 3 + 2 n 2 1 + n 2 exp [ 2 i ( θ 2 θ 1 ) ] 2 ( n 2 1 ) δ 1 δ 2 2 ] exp ( i θ 1 ) + [ 2 n 2 + 1 + n 2 exp [ 2 i ( θ 1 θ 2 ) ] 2 n 2 δ 1 2 δ 2 + 3 n 2 2 3 n 2 + 2 6 ( n 2 1 ) 2 δ 2 3 ] exp ( i θ 2 ) } + R 3 N ,
R 3 N = ρ 10 δ 1 2 n 1 ( n 1 1 ) g ( θ 1 ) exp ( i θ 1 ) + g ( θ 2 ) [ δ 1 n 2 exp ( i θ 1 ) + δ 2 n 2 1 exp ( i θ 2 ) ] δ 2 + D n [ ( 1 + 2 n 1 2 3 n 1 + 3 6 n 1 2 ( n 1 1 ) 2 δ 1 2 + n 2 + 1 n 2 ( n 2 1 ) δ 1 δ 2 cos ( θ 1 θ 2 ) ) δ 1 exp ( i θ 1 ) + δ 1 δ 2 2 cos ( θ 1 θ 2 ) n 2 1 exp ( i θ 2 ) ] ,
R 2 = R 1 F ρ 10 + δ 1 [ D n δ 1 n 1 ( n 1 1 ) g ( θ 1 ) ] exp ( i θ 1 ) + δ 2 [ δ 1 n 2 exp ( i θ 1 ) + δ 2 n 2 1 exp ( i θ 2 ) ] g ( θ 2 ) .
R 1 = P [ δ 1 exp ( i θ 1 ) + δ 2 exp ( i θ 2 ) ] ρ 10 + D n δ 1 exp ( i θ 1 ) .
R 1 = P [ δ 1 ( 1 + D n P ) exp ( i θ 1 ) + δ 2 exp ( i θ 2 ) ] .
K 22 ( r ) = ( δ 1 cos θ 1 ) κ 1 ( δ 2 cos θ 2 ) κ 2 ,
L 22 ( r ) = ( δ 1 sin θ 1 ) κ 1 ( δ 2 sin θ 2 ) κ 2 ,
M 22 ( r ) 1 + δ 1 2 2 + δ 2 2 2 + δ 1 δ 2 cos ( θ 2 θ 1 ) ,
κ 1 = 1 + 3 n 1 6 n 1 ( n 1 1 ) 2 δ 1 2 ,
κ 2 = 1 + δ 1 2 2 n 2 + 3 n 2 1 6 ( n 2 1 ) 2 δ 2 2 + δ 1 δ 2 cos ( θ 2 θ 1 ) n 2 1 .
cos Φ = M 22 ( r ) = 1 δ 1 2 2 δ 2 2 2 δ 1 δ 2 cos ( Δ θ ) 0 ,
( Δ θ ) 0 = arccos [ 2 ( 1 cos Φ ) ( δ 1 2 + δ 2 2 ) 2 δ 1 δ 2 ] .
ψ 0 = arctan ( L 22 ( r ) K 22 ( r ) ) θ 1 = 0 , θ 2 = ( Δ θ ) 0 .
( L 22 ( r ) K 22 ( r ) ) θ 1 = 0 , θ 2 = ( Δ θ ) 0 = κ 20 4 δ 1 2 δ 2 2 [ 2 ( 1 cos Φ ) ( δ 1 2 + δ 2 2 ) ] 2 2 κ 1 δ 1 2 + κ 20 [ 2 ( 1 cos Φ ) ( δ 1 2 + δ 2 2 ) ] ,
κ 20 = κ 2 | θ 1 = 0 , θ 2 = ( Δ θ ) 0 = 1 + δ 1 2 2 n 2 + 3 n 2 1 6 ( n 2 1 ) 2 δ 2 2 + 2 ( 1 cos Φ ) ( δ 1 2 + δ 2 2 ) 2 ( n 2 1 ) .
ψ 0 = arctan { ( n 2 cos Φ ) 4 δ 1 2 δ 2 2 [ 2 ( 1 cos Φ ) ( δ 1 2 + δ 2 2 ) ] 2 ( n 2 1 ) δ 1 2 + ( n 2 cos Φ ) [ 2 ( 1 cos Φ ) ( δ 1 2 + δ 2 2 ) ] } .
θ 1 = Θ ψ 0 and θ 2 = θ 1 + ( Δ θ ) 0 .
θ 1 = 120 ° 25.958 ° = 94.042 ° ,
θ 2 = 120 ° 25.958 ° + 51.745 ° = 145.787 ° .

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