Abstract

We propose a new method for the optical implementation of the Hopfield neural network with a universal tool. The tool is a matrix grating constituted with a group of element gratings. The algorithms for designing a matrix grating are proposed, and a matrix grating is created to execute recognition experiments by use of the Hopfield neural network. The experimental results demonstrate that the proposed method performs well. The stability of the light efficiencies of different optical components used in optical networks is also considered.

© 2004 Optical Society of America

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References

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    [CrossRef]
  2. D. Armitage, “Micromirror spatial light modulator,” U.S. patent4,698,602 (6October1987).
  3. S. Wu, T. Lu, X. Xu, F. T. S. Yu, “An adaptive optical neural network using a high resolution video monitor,” Microwave Opt. Technol. Lett. 2, 252–257 (1989).
    [CrossRef]
  4. D. Psaltis, D. Brady, X. G. Gu, S. Lin, “Holography in artificial neural networks,” Nature 343, 325–330 (1990).
    [CrossRef] [PubMed]
  5. T. Galstyan, G. Pauliat, A. Villing, G. Roosen, “Adaptive photorefractive neurons for self-organizing networks,” Opt. Commun. 109, 35–42 (1994).
    [CrossRef]
  6. Y. Frauel, G. Pauliat, A. Villing, G. Roosen, “High-capacity photorefractive neural network implementing a Kohonen topological map,” Appl. Opt. 40, 5162–5169 (2001).
    [CrossRef]
  7. F. Ito, K. Kitayama, “Optical implementation of the Hopfield neural network using multiple fiber nets,” Appl. Opt. 28, 4176–4181 (1989).
    [CrossRef] [PubMed]
  8. D. M. Chiarulli, S. P. Levitan, P. Derr, R. Hofmann, B. Greiner, M. Robinson, “Demonstration of a multichannel optical interconnection by use of imaging fiber bundles butt coupled to optoelectronic circuits,” Appl. Opt. 39, 698–703 (2000).
    [CrossRef]
  9. H. Mills, D. R. Burton, M. J. Lalor, “Applying backpropagation neural networks to fringe analysis,” Opt. Lasers Eng. 23, 331–341 (1995).
    [CrossRef]
  10. F. J. Cuevas, M. Servin, R. Rodriguez-Vera, “Depth object recovery using radial basis functions,” Opt. Commun. 163, 270–277 (1999).
    [CrossRef]
  11. F. J. Cuevas, M. Servin, O. N. Stavroudis, R. Rodriguez-Vera, “Multi-layer neural network applied to phase and depth recovery from fringe patterns,” Opt. Commun. 181, 239–259 (2000).
    [CrossRef]
  12. N. H. Farhat, D. Psaltis, A. Prata, E. Paek, “Optical implementation of the Hopfield model,” Appl. Opt. 24, 1469–1475 (1985).
    [CrossRef] [PubMed]
  13. M. Kranzdorf, B. J. Bigner, L. Zhang, K. M. Johnson, “Optical connectionist machine with polarization-based bipolar weight values,” Opt. Eng. 28, 844–848 (1989).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  19. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  20. For more information, see http://www.ahead.com.tw .
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    [CrossRef]
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    [CrossRef]
  26. M. G. Moharam, D. A. Pommet, E. B. Grann, T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
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  28. G. R. Fowles, Introduction to Modern Optics, 2nd ed. (Holt, Rinehart & Winston, New York, 1975).
  29. For more information, see http://www.townetech.com/holoplat.htm .
  30. M. J. Beesley, J. G. Castledine, “The use of photoresist as a holographic recording medium,” Appl. Opt. 9, 2720–2724 (1970).
    [CrossRef] [PubMed]

2003

S. L. Yeh, Y. K. Shen, “Optical matrix structures for optical interconnection between two groups of points,” Opt. Eng. 42, 2068–2074 (2003).
[CrossRef]

2002

Y. K. Shen, S. L. Yeh, S. H. Chen, “Three-dimensional non-Newtonian computations of micro-injection molding with the finite element method,” Int. Commun. Heat Mass Transf. 29, 643–652 (2002).
[CrossRef]

2001

2000

D. M. Chiarulli, S. P. Levitan, P. Derr, R. Hofmann, B. Greiner, M. Robinson, “Demonstration of a multichannel optical interconnection by use of imaging fiber bundles butt coupled to optoelectronic circuits,” Appl. Opt. 39, 698–703 (2000).
[CrossRef]

F. J. Cuevas, M. Servin, O. N. Stavroudis, R. Rodriguez-Vera, “Multi-layer neural network applied to phase and depth recovery from fringe patterns,” Opt. Commun. 181, 239–259 (2000).
[CrossRef]

1999

F. J. Cuevas, M. Servin, R. Rodriguez-Vera, “Depth object recovery using radial basis functions,” Opt. Commun. 163, 270–277 (1999).
[CrossRef]

1996

S. Jutamulia, F. T. S. Yu, “Overview of hybrid optical neural networks,” Opt. Laser Technol. 28, 59–72 (1996).
[CrossRef]

1995

1994

T. Galstyan, G. Pauliat, A. Villing, G. Roosen, “Adaptive photorefractive neurons for self-organizing networks,” Opt. Commun. 109, 35–42 (1994).
[CrossRef]

1993

1991

1990

D. Psaltis, D. Brady, X. G. Gu, S. Lin, “Holography in artificial neural networks,” Nature 343, 325–330 (1990).
[CrossRef] [PubMed]

1989

S. Wu, T. Lu, X. Xu, F. T. S. Yu, “An adaptive optical neural network using a high resolution video monitor,” Microwave Opt. Technol. Lett. 2, 252–257 (1989).
[CrossRef]

F. Ito, K. Kitayama, “Optical implementation of the Hopfield neural network using multiple fiber nets,” Appl. Opt. 28, 4176–4181 (1989).
[CrossRef] [PubMed]

M. Kranzdorf, B. J. Bigner, L. Zhang, K. M. Johnson, “Optical connectionist machine with polarization-based bipolar weight values,” Opt. Eng. 28, 844–848 (1989).
[CrossRef]

1985

1970

1969

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

1964

G. Oster, M. Wasserman, C. Zwerling, “Theoretical interpretation of moiré patterns,” J. Opt. Soc. Am. 54, 169–175 (1964).
[CrossRef]

A. V. Lugt, “Signal detection by complex filtering,” IEEE Trans. Inf. Theory IT-10, 139–146 (1964).
[CrossRef]

Armitage, D.

D. Armitage, “Micromirror spatial light modulator,” U.S. patent4,698,602 (6October1987).

Beesley, M. J.

Bigner, B. J.

M. Kranzdorf, B. J. Bigner, L. Zhang, K. M. Johnson, “Optical connectionist machine with polarization-based bipolar weight values,” Opt. Eng. 28, 844–848 (1989).
[CrossRef]

Brady, D.

D. Psaltis, D. Brady, X. G. Gu, S. Lin, “Holography in artificial neural networks,” Nature 343, 325–330 (1990).
[CrossRef] [PubMed]

Burton, D. R.

H. Mills, D. R. Burton, M. J. Lalor, “Applying backpropagation neural networks to fringe analysis,” Opt. Lasers Eng. 23, 331–341 (1995).
[CrossRef]

Castledine, J. G.

Chen, S. H.

Y. K. Shen, S. L. Yeh, S. H. Chen, “Three-dimensional non-Newtonian computations of micro-injection molding with the finite element method,” Int. Commun. Heat Mass Transf. 29, 643–652 (2002).
[CrossRef]

Chiarulli, D. M.

Cuevas, F. J.

F. J. Cuevas, M. Servin, O. N. Stavroudis, R. Rodriguez-Vera, “Multi-layer neural network applied to phase and depth recovery from fringe patterns,” Opt. Commun. 181, 239–259 (2000).
[CrossRef]

F. J. Cuevas, M. Servin, R. Rodriguez-Vera, “Depth object recovery using radial basis functions,” Opt. Commun. 163, 270–277 (1999).
[CrossRef]

Derr, P.

Farhat, N. H.

Fowles, G. R.

G. R. Fowles, Introduction to Modern Optics, 2nd ed. (Holt, Rinehart & Winston, New York, 1975).

Frauel, Y.

Galstyan, T.

T. Galstyan, G. Pauliat, A. Villing, G. Roosen, “Adaptive photorefractive neurons for self-organizing networks,” Opt. Commun. 109, 35–42 (1994).
[CrossRef]

Gaylord, T. K.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Grann, E. B.

Greiner, B.

Gu, X. G.

D. Psaltis, D. Brady, X. G. Gu, S. Lin, “Holography in artificial neural networks,” Nature 343, 325–330 (1990).
[CrossRef] [PubMed]

Hofmann, R.

Ito, F.

Johnson, K. M.

M. Kranzdorf, B. J. Bigner, L. Zhang, K. M. Johnson, “Optical connectionist machine with polarization-based bipolar weight values,” Opt. Eng. 28, 844–848 (1989).
[CrossRef]

Jutamulia, S.

S. Jutamulia, F. T. S. Yu, “Overview of hybrid optical neural networks,” Opt. Laser Technol. 28, 59–72 (1996).
[CrossRef]

Kitayama, K.

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Kranzdorf, M.

M. Kranzdorf, B. J. Bigner, L. Zhang, K. M. Johnson, “Optical connectionist machine with polarization-based bipolar weight values,” Opt. Eng. 28, 844–848 (1989).
[CrossRef]

Lalor, M. J.

H. Mills, D. R. Burton, M. J. Lalor, “Applying backpropagation neural networks to fringe analysis,” Opt. Lasers Eng. 23, 331–341 (1995).
[CrossRef]

Levitan, S. P.

Li, L.

Lin, S.

D. Psaltis, D. Brady, X. G. Gu, S. Lin, “Holography in artificial neural networks,” Nature 343, 325–330 (1990).
[CrossRef] [PubMed]

Lo, R. C.

S. L. Yeh, R. C. Lo, C. Y. Shi, “Implement of Hopfield neural network using optical grating approach and its application,” presented at the 14th Conference on Computer Vision, Graphics and Image Processing, Taiwan, 19–21 Aug. 2001.

Lu, T.

F. T. S. Yu, X. Yang, T. Lu, “Space-time-sharing optical neural networks,” Opt. Lett. 16, 247–249 (1991).
[CrossRef] [PubMed]

S. Wu, T. Lu, X. Xu, F. T. S. Yu, “An adaptive optical neural network using a high resolution video monitor,” Microwave Opt. Technol. Lett. 2, 252–257 (1989).
[CrossRef]

Lugt, A. V.

A. V. Lugt, “Signal detection by complex filtering,” IEEE Trans. Inf. Theory IT-10, 139–146 (1964).
[CrossRef]

Mills, H.

H. Mills, D. R. Burton, M. J. Lalor, “Applying backpropagation neural networks to fringe analysis,” Opt. Lasers Eng. 23, 331–341 (1995).
[CrossRef]

Moharam, M. G.

Oster, G.

Paek, E.

Pauliat, G.

Y. Frauel, G. Pauliat, A. Villing, G. Roosen, “High-capacity photorefractive neural network implementing a Kohonen topological map,” Appl. Opt. 40, 5162–5169 (2001).
[CrossRef]

T. Galstyan, G. Pauliat, A. Villing, G. Roosen, “Adaptive photorefractive neurons for self-organizing networks,” Opt. Commun. 109, 35–42 (1994).
[CrossRef]

Pommet, D. A.

Prata, A.

Psaltis, D.

D. Psaltis, D. Brady, X. G. Gu, S. Lin, “Holography in artificial neural networks,” Nature 343, 325–330 (1990).
[CrossRef] [PubMed]

N. H. Farhat, D. Psaltis, A. Prata, E. Paek, “Optical implementation of the Hopfield model,” Appl. Opt. 24, 1469–1475 (1985).
[CrossRef] [PubMed]

Robinson, M.

Rodriguez-Vera, R.

F. J. Cuevas, M. Servin, O. N. Stavroudis, R. Rodriguez-Vera, “Multi-layer neural network applied to phase and depth recovery from fringe patterns,” Opt. Commun. 181, 239–259 (2000).
[CrossRef]

F. J. Cuevas, M. Servin, R. Rodriguez-Vera, “Depth object recovery using radial basis functions,” Opt. Commun. 163, 270–277 (1999).
[CrossRef]

Roosen, G.

Y. Frauel, G. Pauliat, A. Villing, G. Roosen, “High-capacity photorefractive neural network implementing a Kohonen topological map,” Appl. Opt. 40, 5162–5169 (2001).
[CrossRef]

T. Galstyan, G. Pauliat, A. Villing, G. Roosen, “Adaptive photorefractive neurons for self-organizing networks,” Opt. Commun. 109, 35–42 (1994).
[CrossRef]

Servin, M.

F. J. Cuevas, M. Servin, O. N. Stavroudis, R. Rodriguez-Vera, “Multi-layer neural network applied to phase and depth recovery from fringe patterns,” Opt. Commun. 181, 239–259 (2000).
[CrossRef]

F. J. Cuevas, M. Servin, R. Rodriguez-Vera, “Depth object recovery using radial basis functions,” Opt. Commun. 163, 270–277 (1999).
[CrossRef]

Shen, Y. K.

S. L. Yeh, Y. K. Shen, “Optical matrix structures for optical interconnection between two groups of points,” Opt. Eng. 42, 2068–2074 (2003).
[CrossRef]

Y. K. Shen, S. L. Yeh, S. H. Chen, “Three-dimensional non-Newtonian computations of micro-injection molding with the finite element method,” Int. Commun. Heat Mass Transf. 29, 643–652 (2002).
[CrossRef]

Shi, C. Y.

S. L. Yeh, R. C. Lo, C. Y. Shi, “Implement of Hopfield neural network using optical grating approach and its application,” presented at the 14th Conference on Computer Vision, Graphics and Image Processing, Taiwan, 19–21 Aug. 2001.

Stavroudis, O. N.

F. J. Cuevas, M. Servin, O. N. Stavroudis, R. Rodriguez-Vera, “Multi-layer neural network applied to phase and depth recovery from fringe patterns,” Opt. Commun. 181, 239–259 (2000).
[CrossRef]

Vest, C. M.

C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).

Villing, A.

Y. Frauel, G. Pauliat, A. Villing, G. Roosen, “High-capacity photorefractive neural network implementing a Kohonen topological map,” Appl. Opt. 40, 5162–5169 (2001).
[CrossRef]

T. Galstyan, G. Pauliat, A. Villing, G. Roosen, “Adaptive photorefractive neurons for self-organizing networks,” Opt. Commun. 109, 35–42 (1994).
[CrossRef]

Wasserman, M.

Wu, S.

S. Wu, T. Lu, X. Xu, F. T. S. Yu, “An adaptive optical neural network using a high resolution video monitor,” Microwave Opt. Technol. Lett. 2, 252–257 (1989).
[CrossRef]

Xu, X.

S. Wu, T. Lu, X. Xu, F. T. S. Yu, “An adaptive optical neural network using a high resolution video monitor,” Microwave Opt. Technol. Lett. 2, 252–257 (1989).
[CrossRef]

Yang, X.

Yeh, S. L.

S. L. Yeh, Y. K. Shen, “Optical matrix structures for optical interconnection between two groups of points,” Opt. Eng. 42, 2068–2074 (2003).
[CrossRef]

Y. K. Shen, S. L. Yeh, S. H. Chen, “Three-dimensional non-Newtonian computations of micro-injection molding with the finite element method,” Int. Commun. Heat Mass Transf. 29, 643–652 (2002).
[CrossRef]

S. L. Yeh, R. C. Lo, C. Y. Shi, “Implement of Hopfield neural network using optical grating approach and its application,” presented at the 14th Conference on Computer Vision, Graphics and Image Processing, Taiwan, 19–21 Aug. 2001.

Yu, F. T. S.

S. Jutamulia, F. T. S. Yu, “Overview of hybrid optical neural networks,” Opt. Laser Technol. 28, 59–72 (1996).
[CrossRef]

F. T. S. Yu, X. Yang, T. Lu, “Space-time-sharing optical neural networks,” Opt. Lett. 16, 247–249 (1991).
[CrossRef] [PubMed]

S. Wu, T. Lu, X. Xu, F. T. S. Yu, “An adaptive optical neural network using a high resolution video monitor,” Microwave Opt. Technol. Lett. 2, 252–257 (1989).
[CrossRef]

Zhang, L.

M. Kranzdorf, B. J. Bigner, L. Zhang, K. M. Johnson, “Optical connectionist machine with polarization-based bipolar weight values,” Opt. Eng. 28, 844–848 (1989).
[CrossRef]

Zwerling, C.

Appl. Opt.

Bell Syst. Tech. J.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

IEEE Trans. Inf. Theory

A. V. Lugt, “Signal detection by complex filtering,” IEEE Trans. Inf. Theory IT-10, 139–146 (1964).
[CrossRef]

Int. Commun. Heat Mass Transf.

Y. K. Shen, S. L. Yeh, S. H. Chen, “Three-dimensional non-Newtonian computations of micro-injection molding with the finite element method,” Int. Commun. Heat Mass Transf. 29, 643–652 (2002).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Microwave Opt. Technol. Lett.

S. Wu, T. Lu, X. Xu, F. T. S. Yu, “An adaptive optical neural network using a high resolution video monitor,” Microwave Opt. Technol. Lett. 2, 252–257 (1989).
[CrossRef]

Nature

D. Psaltis, D. Brady, X. G. Gu, S. Lin, “Holography in artificial neural networks,” Nature 343, 325–330 (1990).
[CrossRef] [PubMed]

Opt. Commun.

T. Galstyan, G. Pauliat, A. Villing, G. Roosen, “Adaptive photorefractive neurons for self-organizing networks,” Opt. Commun. 109, 35–42 (1994).
[CrossRef]

F. J. Cuevas, M. Servin, R. Rodriguez-Vera, “Depth object recovery using radial basis functions,” Opt. Commun. 163, 270–277 (1999).
[CrossRef]

F. J. Cuevas, M. Servin, O. N. Stavroudis, R. Rodriguez-Vera, “Multi-layer neural network applied to phase and depth recovery from fringe patterns,” Opt. Commun. 181, 239–259 (2000).
[CrossRef]

Opt. Eng.

S. L. Yeh, Y. K. Shen, “Optical matrix structures for optical interconnection between two groups of points,” Opt. Eng. 42, 2068–2074 (2003).
[CrossRef]

M. Kranzdorf, B. J. Bigner, L. Zhang, K. M. Johnson, “Optical connectionist machine with polarization-based bipolar weight values,” Opt. Eng. 28, 844–848 (1989).
[CrossRef]

Opt. Laser Technol.

S. Jutamulia, F. T. S. Yu, “Overview of hybrid optical neural networks,” Opt. Laser Technol. 28, 59–72 (1996).
[CrossRef]

Opt. Lasers Eng.

H. Mills, D. R. Burton, M. J. Lalor, “Applying backpropagation neural networks to fringe analysis,” Opt. Lasers Eng. 23, 331–341 (1995).
[CrossRef]

Opt. Lett.

Other

C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).

D. Armitage, “Micromirror spatial light modulator,” U.S. patent4,698,602 (6October1987).

For more information, see http://www.taileng.com.tw/en/index.htm .

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

For more information, see http://www.ahead.com.tw .

S. L. Yeh, R. C. Lo, C. Y. Shi, “Implement of Hopfield neural network using optical grating approach and its application,” presented at the 14th Conference on Computer Vision, Graphics and Image Processing, Taiwan, 19–21 Aug. 2001.

G. R. Fowles, Introduction to Modern Optics, 2nd ed. (Holt, Rinehart & Winston, New York, 1975).

For more information, see http://www.townetech.com/holoplat.htm .

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Figures (13)

Fig. 1
Fig. 1

Grating for diffraction.

Fig. 2
Fig. 2

4f-transform device.

Fig. 3
Fig. 3

Setup for implementing the Hopfield neural network.

Fig. 4
Fig. 4

Matrix grating on a photoresist plate.

Fig. 5
Fig. 5

8 × 8 points diffracted by 64 × 64 gratings. (a) λ = 633 nm; (b) λ = 477 nm.

Fig. 6
Fig. 6

Target patterns for recognition.

Fig. 7
Fig. 7

Values of w jilk with f = (j - 1) × 8 + l and g = (i - 1) × 8 + k.

Fig. 8
Fig. 8

Distorted patterns for recognition.

Fig. 9
Fig. 9

Recognition processes for a distorted pattern D. (a) First input distorted pattern, (b) first output brightness distribution, (c) second input pattern got produced by (b), (d) second output brightness distribution, and (e) final correct pattern produced by (d).

Fig. 10
Fig. 10

Seriously distorted patterns for recognition.

Fig. 11
Fig. 11

Diffraction efficiencies η1 of the first-order diffraction of surface-relief sine-profile gratings with n i = 1 and n t = 1.5. (a) d/λ = 0.3, (b) d/λ = 0.4.

Fig. 12
Fig. 12

Diffraction efficiencies η B of phase-volume transmission holograms for the Bragg condition being obeyed. (a) ϕ = 60°; (b) ϕ = 70°.

Fig. 13
Fig. 13

Refraction efficiencies η r for an interface. (a) n 1 = 1 and n 2 = 1.5, (b) n 1= 1.5 and n 2 = 1.

Equations (29)

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

xjlnew=fvjl,
vjl= ik wjilkxikold.
xjlnew=1, if vjl>T,
xjlnew=xjlold, if vjl=T,
xjlnew=-1, if vjl<T,
For ji and lk, wjilk=p=1M xjlpxikp,
For j=i or l=k, wjilk=0,
wjilkxik= wjilkxik+- wjilkxik-.
p= nλ2 sincos-1cos αd cos αi+cos βd cos βi+cos γd cos γi21-a12+a22+a321/2,
a1=cos αd-cos αicos αvcos αd-cos αi2+cos βd-cos βi2+cos γd-cos γi21/2,
a2=cos βd-cos βicos βvcos αd-cos αi2+cos βd-cos βi2+cos γd-cos γi21/2,
a3=cos γd-cos γicos γvcos αd-cos αi2+cos βd-cos βi2+cos γd-cos γi21/2,
cos αθ=b1 cos γθ,
cos βθ=b2 cos γθ,
cos γθ=±1b12+b22+11/2,
cos αp=s1+b1 cos γp,
cos βp=s2+b2 cos γp,
cos γp= -s1b1+s2b2±s1b1+s2b22-b12+b22+1s12+s22-11/2b12+b22+1,
b1=cos βd-cos βicos γv-cos γd-cos γicos βvcos αd-cos αicos βv-cos βd-cos βicos αv,
b2=cos αd-cos αicos γv-cos γd-cos γicos αvcos βd-cos βicos αv-cos αd-cos αicos βv,
s1= nλpcos βvcos αd-cos αicos βv-cos βd-cos βicos αv,
s2= nλpcos αvcos βd-cos βicos αv-cos αd-cos αicos βv.
cos αd=t1+c1 cos γd,
cos βd=t2-c2 cos γd,
cos γd= -t1c1+t2c2+t1c1+t2c22-c12+c22+1t12+t22-11/2c12+c22+1,
t1=cos αi+ nλ cos βθpcos αp cos βθ-cos αθ cos βp-c1 cos γi,
t2=cos βi- nλ cos αθpcos αp cos βθ-cos αθ cos βp+c2 cos γi,
c1=cos γθ cos βp-cos γp cos βθcos αp cos βθ-cos αθ cos βp,
c2=cos γp cos βθ-cos γθ cos βpcos αp cos βθ-cos αθ cos βp.

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