Abstract

In this study, the optical characteristics of double triangle wedge prisms are researched and analyzed, and an innovative mathematics model including four apex angle parameters is proposed for the analytical forward and inverse solutions. A mathematical model that defines the four apex angle parameters for DWPS as applied in different configurations has been demonstrated.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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Double-wedge prism scanner for application in thermal imaging systems

Shian-Fu Lai and Cheng-Chung Lee
Appl. Opt. 57(22) 6290-6299 (2018)

References

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  1. S. D. Risley, “A new rotary prism,” Trans. Am. Ophthalmol. Soc. 5, 412–413 (1889).
  2. W. E. Carter and M. S. Carter, “Risley prisms: 125 years of new applications,” Eos 87(28), 273–276 (2006).
    [Crossref]
  3. F. R. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 2001), Sec. 2.
  4. A. Li, , et al., “Radial support analysis for large-aperture rotating wedge prism,” Opt. Laser Technol. 44(6), 1881–1888 (2012).
    [Crossref]
  5. W. C. Warger and Ch. A. Dimarzio, “Dual-wedge scanning confocal reflectance microscope,” Opt. Lett. 32(15), 2140–2142 (2007).
    [Crossref]
  6. G. Garcia-Torales, M. Strojnik, and G. Paez, “Risley prisms to control wave-front tilt and displacement in a vectorial shearing interferometer,” Appl. Opt. 41(7), 1380–1384 (2002).
    [Crossref]
  7. S. -F. Lai and C. -C. Lee, “Double-wedge prism scanner for application in thermal imaging systems,” Appl. Opt. 57(22), 6290–6299 (2018).
    [Crossref]
  8. V. Vuthea and H. Toshiyoshi, “A Design of Risley Scanner for LiDAR Applications,” International Conference on Optical MEMS and Nanophotonics (OMN, 135–136 (2018).
  9. G. Paez, M. Strojnik, and G. G. Torales, “Vectorial shearing interferometer,” Appl. Opt. 39(28), 5172–5178 (2000).
    [Crossref]
  10. Y. Yang, “Analytic solution of free space optical beam steering using Risley prisms,” J. Lightwave Technol. 26(21), 3576–3583 (2008).
    [Crossref]
  11. Y. Li, “Closed form analytical inverse solutions for Risley-prism-based beam steering systems in different configurations,” Appl. Opt. 50(22), 4302–4309 (2011).
    [Crossref]
  12. Q. Peng, X. Wang, G. Ren, H. Chen, L. Cao, and J. Wang, “Analytical direct solutions of the Risley prism systems for tracking and pointing,” Appl. Opt. 53(13), C83–C90 (2014).
    [Crossref]
  13. A. Li, X. Gao, W. Sun, W. Yi, Y. Bian, H. Liu, and L. Liu, “Inverse solutions for a Risley prism scanner with iterative refinement by a forward solution,” Appl. Opt. 54(33), 9981–9989 (2015).
    [Crossref]
  14. Y. Zhao and Y. Yuan, “First-order approximation error analysis of Risley-prism-based beam directing system,” Appl. Opt. 53(34), 8020–8031 (2014).
    [Crossref]
  15. B. Bravo-Medina, M. Strojnik, G. Garcia-Torales, H. Torres-Ortega, R. Estrada-Marmolejo, A. Beltran-Gonzalez, and J. L. Flores, “Error compensation in a pointing system based on Risley prisms,” Appl. Opt. 56(8), 2209–2216 (2017).
    [Crossref]
  16. L. Beiser and R. Barry Johnson, Opticl Instruments, Chapter 19, Scanners.
  17. S. -F. Lai and C. -C. Lee, “Analytic inverse solutions for Risley prisms in four different configurations for positing and tracking systems,” Appl. Opt. 57(35), 10172–10182 (2018).
    [Crossref]
  18. Y. Yang, “Analytic solution of free space optical beam steering using Risley prisms,” J. Lightwave Technol. 26(21), 3576–3583 (2008).
    [Crossref]
  19. M. Born and E. Wolf, Principles of Optics, 7th, ed. (Cambridge University, 1999).
  20. Y. Zhou, S. Fan, Y. Chen, X. Zhou, and G. Liu, “Beam steering limitation of a Risley prism system due to total internal reflection,” Appl. Opt. 56(22), 6079–6086 (2017).
    [Crossref]

2018 (2)

2017 (2)

2015 (1)

2014 (2)

2012 (1)

A. Li, , et al., “Radial support analysis for large-aperture rotating wedge prism,” Opt. Laser Technol. 44(6), 1881–1888 (2012).
[Crossref]

2011 (1)

2008 (2)

2007 (1)

2006 (1)

W. E. Carter and M. S. Carter, “Risley prisms: 125 years of new applications,” Eos 87(28), 273–276 (2006).
[Crossref]

2002 (1)

2000 (1)

1889 (1)

S. D. Risley, “A new rotary prism,” Trans. Am. Ophthalmol. Soc. 5, 412–413 (1889).

Barry Johnson, R.

L. Beiser and R. Barry Johnson, Opticl Instruments, Chapter 19, Scanners.

Beiser, L.

L. Beiser and R. Barry Johnson, Opticl Instruments, Chapter 19, Scanners.

Beltran-Gonzalez, A.

Bian, Y.

Born, M.

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

Bravo-Medina, B.

Cao, L.

Carter, M. S.

W. E. Carter and M. S. Carter, “Risley prisms: 125 years of new applications,” Eos 87(28), 273–276 (2006).
[Crossref]

Carter, W. E.

W. E. Carter and M. S. Carter, “Risley prisms: 125 years of new applications,” Eos 87(28), 273–276 (2006).
[Crossref]

Chen, H.

Chen, Y.

Dimarzio, Ch. A.

Estrada-Marmolejo, R.

Fan, S.

Flores, J. L.

Gao, X.

Garcia-Torales, G.

Jenkins, F. R.

F. R. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 2001), Sec. 2.

Lai, S. -F.

Lee, C. -C.

Li, A.

Li, Y.

Liu, G.

Liu, H.

Liu, L.

Paez, G.

Peng, Q.

Ren, G.

Risley, S. D.

S. D. Risley, “A new rotary prism,” Trans. Am. Ophthalmol. Soc. 5, 412–413 (1889).

Strojnik, M.

Sun, W.

Torales, G. G.

Torres-Ortega, H.

Toshiyoshi, H.

V. Vuthea and H. Toshiyoshi, “A Design of Risley Scanner for LiDAR Applications,” International Conference on Optical MEMS and Nanophotonics (OMN, 135–136 (2018).

Vuthea, V.

V. Vuthea and H. Toshiyoshi, “A Design of Risley Scanner for LiDAR Applications,” International Conference on Optical MEMS and Nanophotonics (OMN, 135–136 (2018).

Wang, J.

Wang, X.

Warger, W. C.

White, H. E.

F. R. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 2001), Sec. 2.

Wolf, E.

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

Yang, Y.

Yi, W.

Yuan, Y.

Zhao, Y.

Zhou, X.

Zhou, Y.

Appl. Opt. (10)

G. Paez, M. Strojnik, and G. G. Torales, “Vectorial shearing interferometer,” Appl. Opt. 39(28), 5172–5178 (2000).
[Crossref]

G. Garcia-Torales, M. Strojnik, and G. Paez, “Risley prisms to control wave-front tilt and displacement in a vectorial shearing interferometer,” Appl. Opt. 41(7), 1380–1384 (2002).
[Crossref]

Y. Li, “Closed form analytical inverse solutions for Risley-prism-based beam steering systems in different configurations,” Appl. Opt. 50(22), 4302–4309 (2011).
[Crossref]

Q. Peng, X. Wang, G. Ren, H. Chen, L. Cao, and J. Wang, “Analytical direct solutions of the Risley prism systems for tracking and pointing,” Appl. Opt. 53(13), C83–C90 (2014).
[Crossref]

Y. Zhao and Y. Yuan, “First-order approximation error analysis of Risley-prism-based beam directing system,” Appl. Opt. 53(34), 8020–8031 (2014).
[Crossref]

A. Li, X. Gao, W. Sun, W. Yi, Y. Bian, H. Liu, and L. Liu, “Inverse solutions for a Risley prism scanner with iterative refinement by a forward solution,” Appl. Opt. 54(33), 9981–9989 (2015).
[Crossref]

B. Bravo-Medina, M. Strojnik, G. Garcia-Torales, H. Torres-Ortega, R. Estrada-Marmolejo, A. Beltran-Gonzalez, and J. L. Flores, “Error compensation in a pointing system based on Risley prisms,” Appl. Opt. 56(8), 2209–2216 (2017).
[Crossref]

Y. Zhou, S. Fan, Y. Chen, X. Zhou, and G. Liu, “Beam steering limitation of a Risley prism system due to total internal reflection,” Appl. Opt. 56(22), 6079–6086 (2017).
[Crossref]

S. -F. Lai and C. -C. Lee, “Double-wedge prism scanner for application in thermal imaging systems,” Appl. Opt. 57(22), 6290–6299 (2018).
[Crossref]

S. -F. Lai and C. -C. Lee, “Analytic inverse solutions for Risley prisms in four different configurations for positing and tracking systems,” Appl. Opt. 57(35), 10172–10182 (2018).
[Crossref]

Eos (1)

W. E. Carter and M. S. Carter, “Risley prisms: 125 years of new applications,” Eos 87(28), 273–276 (2006).
[Crossref]

J. Lightwave Technol. (2)

Opt. Laser Technol. (1)

A. Li, , et al., “Radial support analysis for large-aperture rotating wedge prism,” Opt. Laser Technol. 44(6), 1881–1888 (2012).
[Crossref]

Opt. Lett. (1)

Trans. Am. Ophthalmol. Soc. (1)

S. D. Risley, “A new rotary prism,” Trans. Am. Ophthalmol. Soc. 5, 412–413 (1889).

Other (4)

F. R. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 2001), Sec. 2.

V. Vuthea and H. Toshiyoshi, “A Design of Risley Scanner for LiDAR Applications,” International Conference on Optical MEMS and Nanophotonics (OMN, 135–136 (2018).

L. Beiser and R. Barry Johnson, Opticl Instruments, Chapter 19, Scanners.

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

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Figures (10)

Fig. 1.
Fig. 1. (a) Type 1, (b) Type2, and (c) Type1 and Type2 through arrangement and assembly produce four configurations [11].
Fig. 2.
Fig. 2. From geometry, a triangle is two combined right triangles. Wedge profiles: (a) Type1 at α11 = 0°, (b) Type2 at α12 = 0°, (c) Type3 where apex angle is sum of α11 and α12.
Fig. 3.
Fig. 3. (a) Schematic diagrams illustrating the notation and coordinate system for an incident ray vector in a 3D coordinate system. (b) The incident ray vector from 3D vector form transformed into 2D vector form.
Fig. 4.
Fig. 4. Schematic diagrams illustrating ray propagation paths for a triangle wedge prism. Refractive index: air = n11 and prism = n12, apex angle: α = α1112, rotation angle = θ1. (b), (c), and (d) illustrate incident ray from different directions entering and traveling through wedge prism. For (a)–(d), plane of (eb1b3) corresponds to profile of wedge prism perpendicular to thick-side.
Fig. 5.
Fig. 5. Triangle wedge prism is rotated from 0° to 180°, the incident angle is at rest. For BK-7, the incident angle δ1in = 6°, θ = 0°, n11 = 1, n12 = 1.5, (a) α11 = 5°, α12 = 10°, δ1in > α11; (b) α11 = 15°, α12 = 7°, δ1in < α11. For Ge, the incident angle δ1in = 4° , θ = 0°, n11 = 1, n12 = 4, (c) α11 = 2°, α12 = 5°, δ1in > α11; and (d) α11 = 6°, α12 = 3°, δ1in < α11.
Fig. 6.
Fig. 6. Schematic diagrams illustrating five element types, in Type 1, α11 = 0°; Type 2, α12 = 0°; Type a, α1112; Type b, α11> α12; and Type c, α11 = α12.
Fig. 7.
Fig. 7. (a) Schematic diagram illustrating notation and coordinate system for a configuration of double triangle wedge prisms. Refractive index of S1 = n12, S2 = n22 and air = n11. (b) Two planes (k1, m1, u1) and (k2, m2, u2) correspond to profile perpendicular to base side. In far-field region, plane of (x3, y3, 0) is perpendicular to z-axis. For S2, ray vector emerged from point o2 and parallel shift to point o1.
Fig. 8.
Fig. 8. Schematic diagram illustrating notation and coordinate systems for DWPS in form of 2D vector algebra.
Fig. 9.
Fig. 9. Curves of maximum ray deviation angles are plotted as a function of α11 + α12 (= α21 + α22). (a) and (b) are of BK-7 and n11 = 1, n12 = n22=1.5. (b) Enlargement of partial region of (a); (c) and (d) are of Ge and n11 = 1, n12 = n22 = 4. (d) Enlargement of partial region of (c).
Fig. 10.
Fig. 10. Curve for two identical prisms in two different prism materials, zoomed-in circular region of Fig. 9(a) and (c). For BK-7, n11 = 1, n12 = n22=1.5. (A1.5) α11 = α21 = 0 and α12 = α22; (B1.5) α11 = α22 = 0 and α12 = α21; (C1.5) α12 = α21 = 0 and α11 = α22; (D1.5) α12 = α22 = 0 and α11 = α21; (E1.5) α11 = α21 = α12 = α22 (F1.5) α12=2.3α11, α22=2.3α21; (G1.5) α12=0.3α11; α22=0.3α21. For Ge, n11 = 1, n12 = n22 = 4. (A4) α11 = α21 = 0 and α12 = α22; (B4) α11 = α22 = 0 and α12 = α21; (C4) α12 = α21 = 0 and α11 = α22; (D4) α12 = α22 = 0 and α11 = α21; (E4) α11 = α21 = α12 = α22; (F4) α11=2.5α12, α21=2.5α22; (G4) α11=0.35α12; α21=0.35α22. For each curve, imaginary parts of complex X and/or Y arguments ignored.

Tables (6)

Tables Icon

Table 1. Referring to Fig. 10 for n11 =1, n12 = n22 = 1.5.

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Table 2. Referring to Fig. 10 for n11 =1, n12 = n22 = 4.

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Table 3. n11 =1; n12 = n22 = 1.5; Δθ = θ1 – θ2 > 0.

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Table 4. n11 =1; n12 = n22 = 1.5; Δθ = θ2 – θ1 > 0.

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Table 5. n11 =1; n12 = n22 = 4; Δθ = θ1 – θ2 > 0.

Tables Icon

Table 6. n11 =1; n12 = n22 = 4; Δθ = θ2 – θ1 > 0.

Equations (24)

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δ 1 o u t = α 12 + sin 1 ( sin ( α 12 ) ( n 12 n 11 ) 2 sin 2 ( δ 1 i n cos ( θ 1 θ ) ) + cos ( α 12 ) sin ( δ 1 i n cos ( θ 1 θ ) ) δ 1 i n cos ( θ 1 θ )
δ 1 o u t = sin 1 ( sin ( α 11 ) ( n 12 n 11 ) 2 sin 2 ( δ 1 i n cos ( θ 1 θ ) α 11 ) + cos ( α 11 ) sin ( δ 1 i n cos ( θ 1 θ ) α 11 ) δ 1 i n cos ( θ 1 θ )
δ 1 o u t = ( α 12 ) δ 1 i n cos ( θ 1 θ ) + sin 1 { sin ( α 11 + α 12 ) ( n 12 n 11 ) 2 [ sin ( δ 1 i n cos ( θ 1 θ ) α 11 ) ] 2 + cos ( α 11 + α 12 ) sin ( δ 1 i n cos ( θ 1 θ ) α 11 ) }
δ 1 i n = δ 1 i n cos ( θ ) x ^ + δ 1 i n sin ( θ ) y ^
δ 1 o u t = ( α 12 ) + sin 1 { sin ( α 11 + α 12 ) ( n 12 n 11 ) 2 [ sin ( δ 1 i n cos ( θ 1 θ ) α 11 ) ] 2 + cos ( α 11 + α 12 ) [ sin ( δ 1 i n cos ( θ 1 θ ) α 11 ) ] } δ 1 i n cos ( θ 1 θ )
δ 1 o u t = ( α 12 ) + sin 1 { sin ( α 11 + α 12 ) ( n 12 n 11 ) 2 [ sin ( δ 1 i n cos ( θ 1 θ ) α 11 ) ] 2 cos ( α 11 + α 12 ) [ sin ( α 11 δ 1 i n cos ( θ 1 θ ) ) ] } δ 1 i n cos ( θ 1 θ )
δ 1 o u t = ( α 12 ) + sin 1 { sin ( α 11 + α 12 ) ( n 12 n 11 ) 2 [ sin ( δ 1 i n cos ( θ 1 θ ) + α 11 ) ] 2 cos ( α 11 + α 12 ) [ sin ( δ 1 i n cos ( θ 1 θ ) + α 11 ) ] } + δ 1 i n cos ( θ 1 θ )
δ 1 o u t = ( α 12 ) + sin 1 { sin ( α 11 + α 12 ) ( n 12 n 11 ) 2 [ sin ( δ 1 i n cos ( θ 1 θ ) α 11 ) ] 2 + cos ( α 11 + α 12 ) [ sin ( δ 1 i n cos ( θ 1 θ ) α 11 ) ] } δ 1 i n cos ( θ 1 θ )
δ 1 o u t = δ 1 o u t cos ( θ 1 ) x ^ + δ 1 o u t sin ( θ 1 ) y ^
δ 1 t = δ 1 i n + δ 1 o u t = δ 1 t cos ( β 1 ) x ^ + δ 1 t sin ( β 1 ) y ^
{ β 1 = cos 1 ( ( δ 1 i n cos θ + δ 1 o u t cos θ 1 ) [ ( δ 1 i n ) 2 + ( δ 1 o u t ) 2 + 2 δ 1 i n δ 1 o u t cos ( θ 1 θ ) ] 0.5 ) δ 1 t = [ ( δ 1 i n ) 2 + ( δ 1 o u t ) 2 + 2 δ 1 i n δ 1 o u t cos ( θ 1 θ ) ] 0.5
A 2 = δ 2 i n cos ( Δ θ ) α 21 ; B 2 = δ 2 i n cos ( Δ θ ) + α 22 ; C 2 = sin ( α 21 + α 22 ) ; D 2 = cos ( α 21 + α 22 ) δ 2 o u t = B 2 + sin 1 { C 2 [ ( n 22 n 11 ) 2 ( sin A 2 ) 2 ] 0.5 + D 2 sin ( A 2 ) }
δ 2 o u t = B 2 + sin 1 { C 2 [ ( n 22 n 11 ) 2 ( sin A 2 ) 2 ] 0.5 D 2 sin ( A 2 ) }
δ 2 o u t = B 2 2 α 22 + sin 1 { C 2 [ ( n 22 n 11 ) 2 ( sin A 2 + 2 α 21 ) 2 ] 0.5 D 2 sin ( A 2 + 2 α 21 ) }
δ 2 o u t = B 2 + sin 1 { C 2 [ ( n 22 n 11 ) 2 ( sin A 2 ) 2 ] 0.5 + D 2 sin ( A 2 ) }
δ 2 o u t = δ 2 o u t cos ( θ 2 ) x ^ + δ 2 o u t sin ( θ 2 ) y ^
δ 2 t = δ 1 t + δ 2 o u t
δ 2 t = δ 1 t + δ 2 o u t = δ 2 t cos ( β 2 ) x ^ + δ 2 t sin ( β 2 ) y ^
{ β 2 = cos 1 ( ( δ 1 t cos β 1 + δ 2 o u t cos θ 2 ) [ ( δ 1 t ) 2 + ( δ 2 o u t ) 2 + 2 δ 1 t δ 2 o u t cos ( θ 2 β 1 ) ] 0.5 ) δ 2 t = [ ( δ 1 t ) 2 + ( δ 2 o u t ) 2 + 2 δ 1 t δ 2 o u t cos ( θ 2 β 1 ) ] 0.5
Δ θ = cos 1 ( ( δ h ) 2 ( δ 1 t ) 2 ( δ 2 o u t ) 2 2 δ 1 t δ 2 o u t )
F 21 = cos 1 ( δ 2 o u t + δ 1 t cos Δ θ ) [ ( δ 1 t ) 2 + ( δ 2 o u t ) 2 + 2 δ 1 t δ 2 o u t cos ( Δ θ ) ] 0.5 )
θ 2 = Ω F 21 , θ 1 = Δ θ + θ 2
F 22 = cos 1 ( δ 1 t + δ 2 o u t cos Δ θ ) [ ( δ 1 t ) 2 + ( δ 2 o u t ) 2 + 2 δ 1 t δ 2 o u t cos ( Δ θ ) ] 0.5 )
θ 1 = Ω F 22 , θ 2 = Δ θ + θ 1

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