Abstract

We present an improved two-in-one polarization-interlaced liquid-crystal-on-silicon (LCoS) stereoscopic projection prototype employing a novel prism-array configuration and a specially designed illumination freeform lens group. The parallel prism configuration is designed based on the balance analysis between stereoscopic channels. For further system simplification, the illumination lens group, which consists of three prepositive aspherical surfaces and a single postpositive freeform one, is synthetically obtained from the Monge–Ampère method and feedback optimization. Design results show that the proposed prototype can well solve the problem of stereo-channel separation and integration, and provide both better performance and lower volume. It is proven to have potentiality replacing existing stereoscopic projectors.

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

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OSA Recommended Articles
Three-dimensional display technologies

Jason Geng
Adv. Opt. Photon. 5(4) 456-535 (2013)

References

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2018 (3)

S. R. Soomro and H. Urey, “Integrated 3D display and imaging using dual purpose passive screen and head-mounted projectors and camera,” Opt. Express 26(2), 1161–1173 (2018).
[Crossref] [PubMed]

Y. Huang, E. Liao, R. Chen, and S.-T. Wu, “Liquid-crystal-on-silicon for augmented reality displays,” Appl. Sci. (Basel) 8(12), 2366 (2018).
[Crossref]

R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, J. C. Miñano, and F. Duerr, “Design of freeform illumination optics,” Laser Photonics Rev. 12(7), 1700310 (2018).
[Crossref]

2017 (2)

X. Zeng, X. Zhou, T. Guo, L. Yang, E. Chen, and Y. Zhang, “Crosstalk reduction in large-scale autostereoscopic 3D-LED display based on black-stripe occupation ratio,” Opt. Commun. 389, 159–164 (2017).
[Crossref]

V. Oliker, “Controlling light with freeform multifocal lens designed with supporting quadric method(SQM),” Opt. Express 25(4), A58–A72 (2017).
[Crossref] [PubMed]

2016 (4)

2014 (1)

2013 (5)

2012 (4)

Z. L. Zhenjie Liu, P. L. Peng Liu, and F. Y. Feihong Yu, “Parametric optimization method for the design of high-efficiency free-form illumination system with a LED source,” Chin. Opt. Lett. 10(11), 112201 (2012).
[Crossref]

E. Chen and F. Yu, “Design of an elliptic spot illumination system in LED-based color filter-liquid-crystal-on-silicon pico projectors for mobile embedded projection,” Appl. Opt. 51(16), 3162–3170 (2012).
[Crossref] [PubMed]

E. Chen, P. Liu, and F. Yu, “Synchronized parameter optimization of the double freeform lenses illumination system used for the CF-LCoS pico-projectors,” Opt. Laser Technol. 44(7), 2080–2087 (2012).
[Crossref]

E. Chen, P. Liu, and F. Yu, “Optical design of single plano-convex freeform lens-based illumination system for color filter liquid crystal on silicon pico projectors,” Opt. Eng. 51(3), 033002 (2012).
[Crossref]

2011 (2)

2010 (2)

L. Bogaert, Y. Meuret, J. Vanderheijden, H. D. Smet, and H. Thienpont, “Stereoscopic projector for polarized viewing with extended color gamut,” Displays 31(2), 73–81 (2010).
[Crossref]

Y. Luo, Z. Feng, Y. Han, and H. Li, “Design of compact and smooth free-form optical system with uniform illuminance for LED source,” Opt. Express 18(9), 9055–9063 (2010).
[Crossref] [PubMed]

2008 (2)

2006 (3)

S. Kim and E. Kim, “A novel configuration of LCD projectors for efficient orthogonal polarization of two projected views,” Opt. Commun. 266(1), 55–66 (2006).
[Crossref]

K. Takizawa, “Three-dimensional large screen display using polymer-dispersed liquid-crystal light valves and a schlieren optical system: proposal and basic experiments,” Opt. Rev. 13(1), 1–7 (2006).
[Crossref]

S. C. Kim and E. S. Kim, “A novel configuration of LCD projectors for efficient orthogonal polarization of two projected views,” Opt. Commun. 266(1), 55–66 (2006).
[Crossref]

2005 (2)

M. Brown, A. Majumder, and R. Yang, “Camera-based calibration techniques for seamless multiprojector displays,” IEEE Trans. Vis. Comput. Graph. 11(2), 193–206 (2005).
[Crossref] [PubMed]

S. Shen, J. She, and T. Tao, “Optimal design of achromatic true zero-order waveplates using twisted nematic liquid crystal,” J. Opt. Soc. Am. A 22(5), 961–965 (2005).
[Crossref] [PubMed]

2002 (1)

1999 (1)

I. Sexton and P. Surman, “Stereoscopic and autostereoscopic display systems,” IEEE Signal Process. Mag. 16(3), 85–99 (1999).
[Crossref]

1998 (1)

1997 (2)

B. Jacobson, R. Gengelbach, and J. Ferri, “Beam shape transforming devices in high-efficiency projection systems,” Proc. SPIE 3139, 141–150 (1997).
[Crossref]

H. Kikuta, Y. Ohira, and K. Iwata, “Achromatic quarter-wave plates using the dispersion of form birefringence,” Appl. Opt. 36(7), 1566–1572 (1997).
[Crossref] [PubMed]

1996 (1)

S.-T. Wu and C.-S. Wu, “Mixed-mode twisted nematic liquid crystal cells for reflective displays,” Appl. Phys. Lett. 68(11), 1455–1457 (1996).
[Crossref]

Amano, T.

T. Amano, “Projection center calibration for a co-located projector camera system,” In Proc. of the IEEE Conference on Computer Vision and Pattern Recognition Workshops, 449 (2014).
[Crossref]

Bäuerle, A.

Benítez, P.

R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, J. C. Miñano, and F. Duerr, “Design of freeform illumination optics,” Laser Photonics Rev. 12(7), 1700310 (2018).
[Crossref]

Berens, M.

Bogaert, L.

Brown, M.

M. Brown, A. Majumder, and R. Yang, “Camera-based calibration techniques for seamless multiprojector displays,” IEEE Trans. Vis. Comput. Graph. 11(2), 193–206 (2005).
[Crossref] [PubMed]

Bruneton, A.

Chen, E.

X. Zeng, X. Zhou, T. Guo, L. Yang, E. Chen, and Y. Zhang, “Crosstalk reduction in large-scale autostereoscopic 3D-LED display based on black-stripe occupation ratio,” Opt. Commun. 389, 159–164 (2017).
[Crossref]

E. Chen, P. Liu, and F. Yu, “Synchronized parameter optimization of the double freeform lenses illumination system used for the CF-LCoS pico-projectors,” Opt. Laser Technol. 44(7), 2080–2087 (2012).
[Crossref]

E. Chen, P. Liu, and F. Yu, “Optical design of single plano-convex freeform lens-based illumination system for color filter liquid crystal on silicon pico projectors,” Opt. Eng. 51(3), 033002 (2012).
[Crossref]

E. Chen and F. Yu, “Design of an elliptic spot illumination system in LED-based color filter-liquid-crystal-on-silicon pico projectors for mobile embedded projection,” Appl. Opt. 51(16), 3162–3170 (2012).
[Crossref] [PubMed]

Chen, N.

Chen, R.

Y. Huang, E. Liao, R. Chen, and S.-T. Wu, “Liquid-crystal-on-silicon for augmented reality displays,” Appl. Sci. (Basel) 8(12), 2366 (2018).
[Crossref]

Chen, Y.

Y. Chen, D. W. Clark, A. Finkelstein, T. C. Housel, and K. Li, “Automatic alignment of high-resolution multi-projector display using an un-calibrated camera,” In Proc. of the IEEE conference on Visualization, 125–130 (2000).

Cheng, D.

Y. Gao, D. Cheng, C. Xu, and Y. Wang, “Design of an ultra-short throw catadioptric projection lens with a freeform mirror,” Proc. SPIE 10154, 101540 (2016).
[Crossref]

Choi, H. J.

Clark, D. W.

Y. Chen, D. W. Clark, A. Finkelstein, T. C. Housel, and K. Li, “Automatic alignment of high-resolution multi-projector display using an un-calibrated camera,” In Proc. of the IEEE conference on Visualization, 125–130 (2000).

Deng, H.

Dietz, P. H.

J. C. Lee, P. H. Dietz, D. Maynes-Aminzade, R. Raskar, and S. E. Hudson, “Automatic projector calibration with embedded light sensors,” In Proc. of the 17th annual ACM symposium on User interface software and technology, 123–126, (2004).
[Crossref]

Duerr, F.

R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, J. C. Miñano, and F. Duerr, “Design of freeform illumination optics,” Laser Photonics Rev. 12(7), 1700310 (2018).
[Crossref]

Escuti, M. J.

Feihong Yu, F. Y.

Feng, Z.

Ferri, J.

B. Jacobson, R. Gengelbach, and J. Ferri, “Beam shape transforming devices in high-efficiency projection systems,” Proc. SPIE 3139, 141–150 (1997).
[Crossref]

Finkelstein, A.

Y. Chen, D. W. Clark, A. Finkelstein, T. C. Housel, and K. Li, “Automatic alignment of high-resolution multi-projector display using an un-calibrated camera,” In Proc. of the IEEE conference on Visualization, 125–130 (2000).

Froese, B. D.

Fujii, T.

Gao, Y.

Y. Gao, D. Cheng, C. Xu, and Y. Wang, “Design of an ultra-short throw catadioptric projection lens with a freeform mirror,” Proc. SPIE 10154, 101540 (2016).
[Crossref]

Geng, J.

J. Geng, “Three-dimensional display technologies,” Adv. Opt. Photonics 5(4), 456–535 (2013).
[Crossref] [PubMed]

Gengelbach, R.

B. Jacobson, R. Gengelbach, and J. Ferri, “Beam shape transforming devices in high-efficiency projection systems,” Proc. SPIE 3139, 141–150 (1997).
[Crossref]

Giel, B. V.

Guo, T.

X. Zeng, X. Zhou, T. Guo, L. Yang, E. Chen, and Y. Zhang, “Crosstalk reduction in large-scale autostereoscopic 3D-LED display based on black-stripe occupation ratio,” Opt. Commun. 389, 159–164 (2017).
[Crossref]

Hahn, J.

Han, Y.

Hong, J.

Hong, J. Y.

Housel, T. C.

Y. Chen, D. W. Clark, A. Finkelstein, T. C. Housel, and K. Li, “Automatic alignment of high-resolution multi-projector display using an un-calibrated camera,” In Proc. of the IEEE conference on Visualization, 125–130 (2000).

Huang, Y.

Y. Huang, E. Liao, R. Chen, and S.-T. Wu, “Liquid-crystal-on-silicon for augmented reality displays,” Appl. Sci. (Basel) 8(12), 2366 (2018).
[Crossref]

Hudson, S. E.

J. C. Lee, P. H. Dietz, D. Maynes-Aminzade, R. Raskar, and S. E. Hudson, “Automatic projector calibration with embedded light sensors,” In Proc. of the 17th annual ACM symposium on User interface software and technology, 123–126, (2004).
[Crossref]

Iwata, K.

Jacobson, B.

B. Jacobson, R. Gengelbach, and J. Ferri, “Beam shape transforming devices in high-efficiency projection systems,” Proc. SPIE 3139, 141–150 (1997).
[Crossref]

Ji, C. C.

Kikuta, H.

Kim, E.

S. Kim and E. Kim, “A novel configuration of LCD projectors for efficient orthogonal polarization of two projected views,” Opt. Commun. 266(1), 55–66 (2006).
[Crossref]

Kim, E. S.

S. C. Kim and E. S. Kim, “A novel configuration of LCD projectors for efficient orthogonal polarization of two projected views,” Opt. Commun. 266(1), 55–66 (2006).
[Crossref]

Kim, H.

Kim, S.

S. Kim and E. Kim, “A novel configuration of LCD projectors for efficient orthogonal polarization of two projected views,” Opt. Commun. 266(1), 55–66 (2006).
[Crossref]

Kim, S. C.

S. C. Kim and E. S. Kim, “A novel configuration of LCD projectors for efficient orthogonal polarization of two projected views,” Opt. Commun. 266(1), 55–66 (2006).
[Crossref]

Kim, S. J.

Kim, Y.

Kishi, K.

Komanduri, R. K.

Lawler, K. F.

Lee, B.

Lee, C. K.

Lee, J. C.

J. C. Lee, P. H. Dietz, D. Maynes-Aminzade, R. Raskar, and S. E. Hudson, “Automatic projector calibration with embedded light sensors,” In Proc. of the 17th annual ACM symposium on User interface software and technology, 123–126, (2004).
[Crossref]

Li, H.

Li, K.

Y. Chen, D. W. Clark, A. Finkelstein, T. C. Housel, and K. Li, “Automatic alignment of high-resolution multi-projector display using an un-calibrated camera,” In Proc. of the IEEE conference on Visualization, 125–130 (2000).

Li, L.

Liang, R.

R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, J. C. Miñano, and F. Duerr, “Design of freeform illumination optics,” Laser Photonics Rev. 12(7), 1700310 (2018).
[Crossref]

Z. Feng, B. D. Froese, and R. Liang, “Freeform illumination optics construction following an optimal transport map,” Appl. Opt. 55(16), 4301–4306 (2016).
[Crossref] [PubMed]

Liao, E.

Y. Huang, E. Liao, R. Chen, and S.-T. Wu, “Liquid-crystal-on-silicon for augmented reality displays,” Appl. Sci. (Basel) 8(12), 2366 (2018).
[Crossref]

Liu, P.

R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation,” Opt. Lett. 38(2), 229–231 (2013).
[Crossref] [PubMed]

R. Wu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “A mathematical model of the single freeform surface design for collimated beam shaping,” Opt. Express 21(18), 20974–20989 (2013).
[Crossref] [PubMed]

E. Chen, P. Liu, and F. Yu, “Optical design of single plano-convex freeform lens-based illumination system for color filter liquid crystal on silicon pico projectors,” Opt. Eng. 51(3), 033002 (2012).
[Crossref]

E. Chen, P. Liu, and F. Yu, “Synchronized parameter optimization of the double freeform lenses illumination system used for the CF-LCoS pico-projectors,” Opt. Laser Technol. 44(7), 2080–2087 (2012).
[Crossref]

Liu, X.

Loosen, P.

Luo, Y.

Majumder, A.

M. Brown, A. Majumder, and R. Yang, “Camera-based calibration techniques for seamless multiprojector displays,” IEEE Trans. Vis. Comput. Graph. 11(2), 193–206 (2005).
[Crossref] [PubMed]

Maynes-Aminzade, D.

J. C. Lee, P. H. Dietz, D. Maynes-Aminzade, R. Raskar, and S. E. Hudson, “Automatic projector calibration with embedded light sensors,” In Proc. of the 17th annual ACM symposium on User interface software and technology, 123–126, (2004).
[Crossref]

Meuret, Y.

Min, S. W.

Miñano, J. C.

R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, J. C. Miñano, and F. Duerr, “Design of freeform illumination optics,” Laser Photonics Rev. 12(7), 1700310 (2018).
[Crossref]

Moon, S.

Moreno, I.

I. Moreno and C. C. Sun, “LED array: where does far-field begin?” Proc. SPIE 7058, 70580R (2008).
[Crossref]

Müller, G.

Murat, H.

Muschaweck, J.

Ohira, Y.

Oliker, V.

Park, J. H.

Park, S. G.

Peng Liu, P. L.

Raskar, R.

J. C. Lee, P. H. Dietz, D. Maynes-Aminzade, R. Raskar, and S. E. Hudson, “Automatic projector calibration with embedded light sensors,” In Proc. of the 17th annual ACM symposium on User interface software and technology, 123–126, (2004).
[Crossref]

Ries, H.

Sexton, I.

I. Sexton and P. Surman, “Stereoscopic and autostereoscopic display systems,” IEEE Signal Process. Mag. 16(3), 85–99 (1999).
[Crossref]

She, J.

Shen, S.

Smet, H. D.

Soomro, S. R.

Stollenwerk, J.

Sun, C. C.

I. Moreno and C. C. Sun, “LED array: where does far-field begin?” Proc. SPIE 7058, 70580R (2008).
[Crossref]

Sunaga, T.

Surman, P.

I. Sexton and P. Surman, “Stereoscopic and autostereoscopic display systems,” IEEE Signal Process. Mag. 16(3), 85–99 (1999).
[Crossref]

Takizawa, K.

K. Takizawa, “Three-dimensional large screen display using polymer-dispersed liquid-crystal light valves and a schlieren optical system: proposal and basic experiments,” Opt. Rev. 13(1), 1–7 (2006).
[Crossref]

K. Takizawa, T. Fujii, T. Sunaga, and K. Kishi, “Three-dimensional large-screen display with reflection-mode spatial light modulators and a single-projection optical system: analysis of a retardation-modulation method,” Appl. Opt. 37(26), 6182–6195 (1998).
[Crossref] [PubMed]

Tao, T.

Thienpont, H.

Urey, H.

Vanderheijden, J.

L. Bogaert, Y. Meuret, J. Vanderheijden, H. D. Smet, and H. Thienpont, “Stereoscopic projector for polarized viewing with extended color gamut,” Displays 31(2), 73–81 (2010).
[Crossref]

Völl, A.

Wang, Q. H.

Wang, Y.

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X. Zeng, X. Zhou, T. Guo, L. Yang, E. Chen, and Y. Zhang, “Crosstalk reduction in large-scale autostereoscopic 3D-LED display based on black-stripe occupation ratio,” Opt. Commun. 389, 159–164 (2017).
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X. Zeng, X. Zhou, T. Guo, L. Yang, E. Chen, and Y. Zhang, “Crosstalk reduction in large-scale autostereoscopic 3D-LED display based on black-stripe occupation ratio,” Opt. Commun. 389, 159–164 (2017).
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Zhenjie Liu, Z. L.

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X. Zeng, X. Zhou, T. Guo, L. Yang, E. Chen, and Y. Zhang, “Crosstalk reduction in large-scale autostereoscopic 3D-LED display based on black-stripe occupation ratio,” Opt. Commun. 389, 159–164 (2017).
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Appl. Opt. (6)

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Y. Huang, E. Liao, R. Chen, and S.-T. Wu, “Liquid-crystal-on-silicon for augmented reality displays,” Appl. Sci. (Basel) 8(12), 2366 (2018).
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R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, J. C. Miñano, and F. Duerr, “Design of freeform illumination optics,” Laser Photonics Rev. 12(7), 1700310 (2018).
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S. Kim and E. Kim, “A novel configuration of LCD projectors for efficient orthogonal polarization of two projected views,” Opt. Commun. 266(1), 55–66 (2006).
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V. Oliker, “Controlling light with freeform multifocal lens designed with supporting quadric method(SQM),” Opt. Express 25(4), A58–A72 (2017).
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Opt. Laser Technol. (1)

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Figures (11)

Fig. 1
Fig. 1 Light propagation process of the 1-PBSP configuration. (a) Left-eye channel modulated by the L-LCoS panel; (b) right-eye channel modulated by the R-LCoS panel. In this figure, p stands for p-polarized light and s for s-polarized light.
Fig. 2
Fig. 2 Light propagation process of the improved 2-PBSP configuration. (a) Left-eye channel modulated by the L-LCoS panel; (b) right-eye channel modulated by the R-LCoS panel.
Fig. 3
Fig. 3 Collimated beam generation by the prepositive aspherical lens group.
Fig. 4
Fig. 4 (a) Beam collimation by the first three aspherical surfaces and shaping by the freeform exit surface; (b) design principle of the freeform exit surface.
Fig. 5
Fig. 5 (a) Surface profile curves and (b) the spatial intensity distribution of the designed aspherical lens group with the planar exit surface. 3D model and the corresponding illuminance distribution of (c) the initial and (d) the optimized freeform surface.
Fig. 6
Fig. 6 (a) Reflectivity of the PPSF varying with the DAOI and the wavelength. Ray-tracing results of the light separation and integration of (b) R-LCoS and (c) L-LCoS channels of the 2-PBSP system.
Fig. 7
Fig. 7 The optical paths of projection imaging lenses within the prisms for (a) the left-eye and (b)the right-eye stereoscopic channels; (c) MTF curves and 2D cross-sectional plot; (d) optical aberration curves of the designed stereoscopic projection lens group.
Fig. 8
Fig. 8 Ray tracing of (a) the complete projection prototype, (b) the left-eye and the right-eye stereoscopic channels. Results of the designed prototype, including (c) degree of polarization, (d) actual color mixing, and (e) 3D illuminance distribution.
Fig. 9
Fig. 9 Light propagation process of the 4-PBSP configuration. (a) Left-eye channel modulated by the L-LCoS panel; (b) right-eye channel by from the R-LCoS panel.
Fig. 10
Fig. 10 Normalized transmission efficiencies varying with the DAOI among 1-PBSP, 2-PBSP, and 4-PBSP configurations.
Fig. 11
Fig. 11 Comparison of the overall transmission efficiencies among the 1-PBSP, 2-PBSP, and 4-PBSP configurations.

Tables (5)

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Table 1 Parameters of the LCoS Panel

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Table 2 Lens Parameters of the Advanced Illumination Engine

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Table 3 Comparisons of the Parameters between the Original and Optimized Projection Lens Groups

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Table 4 The Lens Data of the Optimized Projection Imaging Lens Group

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Table 5 Comparisons of the Projection Prototypes Equipped with Different PBSP Configurations

Equations (19)

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

Φ L 1 = 1 2 Φ LED τ s σ P τ LCoS Φ R 1 = 1 2 Φ LED σ P τ s τ LCoS ,
Φ L 1 = 1 2 Φ LED τ s σ S τ LCoS .
Λ L 1 = Φ L 1 Φ L 1 = τ s σ P τ s σ S = σ P σ S .
Λ R 1 = Φ R 1 Φ R 1 = σ P τ s σ P τ P = τ s τ P .
Φ L 2 = 1 2 Φ LED τ S 2 σ P 2 τ LCoS .
Φ R 2 = 1 2 Φ LED τ S 2 σ P 2 τ LCoS ,
Φ L 2 = 1 2 Φ LED τ S σ P τ P σ S τ LCoS .. Φ R 2 = 1 2 Φ LED σ P τ S σ S τ P σ LCoS ,
Λ L 2 = τ s σ P τ P σ S ...... Λ R 2 = σ P τ S σ S τ P .
Κ L = Κ R = τ P 2 τ S 2 σ P 2 τ S 2 + σ P σ S ,
y 1 = z 1 tanθ,
sin( α 1 +θ )=nsin( β 1 + α 1 ),
d y 1 d z 1 =cot α 1 = ncos θ 2 cosθ sinθnsin θ 2 ,
d z 1 dθ = d z 1 d y 1 × d y 1 dθ = z 1 cos θ 2 × sinθnsin θ 2 ncos θ 2 cosθ 1 sinθnsin θ 2 ncos θ 2 cosθ ×tanθ .
d z 2 dθ = d z 2 d y 2 × d y 2 dθ = nsin θ 2 sin θ 3 ncos θ 2 cos θ 3 ×( d z 1 dθ ×tanθ+ z 1 cos θ 2 d z 1 dθ ×tan θ 2 + z 2 z 1 2cos θ 2 2 ) 1+ nsin θ 2 sin θ 3 ncos θ 2 cos θ 3 ×tan θ 2 ,
d z 3 dθ = sin θ 3 ncos θ 3 ×( d z 1 dθ ×tanθ+ z 1 cos θ 2 +( d z 2 dθ d z 1 dθ )×tan θ 2 + z 2 z 1 2cos θ 2 2 d z 2 dθ ×tan θ 3 + z 3 z 2 3cos θ 3 2 ) 1 sin θ 3 ncos θ 3 ×tan θ 3 .
{ E[ t x ( x,y ), t y (x,y) ]| J( T ) |=I( x,y ) BC:{ t x = t x ( x,y,z, z x , z y ) t y = t y ( x,y,z, z x , z y ) : S 1 S 2 ,
E i+1 ( x,y )= F i ( x,y )× E i ( x,y ),
F i ( x,y )= E 0 ( x,y ) γ E 0 ( x,y )+( 1-γ )× E i ( x,y ) ,
RSD= 1 M1 × j=1 M ( E j E ¯ 1 ) 2 ,

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