Abstract

We describe an optical image processing system with regenerative feedback that utilizes a cavity with a phase-conjugate mirror. The system’s characteristics and limitations are discussed, and its application to the implementation of iterative recovery algorithms (e.g., Gerchberg algorithm) is considered and demonstrated experimentally.

© 1992 Optical Society of America

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References

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  1. R. V. Pole, H. Wieder, E. S. Barrekette, “Reactive optical information processing. I: Theory of information recovery and resonator mode structure,” Appl. Opt. 8, 1571–1575 (1967).
    [CrossRef]
  2. M. O. Hagler, “Active synthesis of inverse spatial filters,” Appl. Opt. 10, 2783–2784 (1971).
    [CrossRef] [PubMed]
  3. D. P. Jablonowski, S. H. Lee, “A coherent optical feedback system for optical information processing,” Appl. Phys. 8, 51–58 (1975).
    [CrossRef]
  4. E. Handler, U. Roder, “Suppression of multiplicative disturbance by coherent optical feedback technique,” Opt. Commun. 23, 352–356 (1977).
    [CrossRef]
  5. P. E. Kotljar, E. S. Nezhevenko, B. I. Spektor, V. I. Feldbush, “Optical processing in feedback systems,” in Optical Information Processing, E. S. Barrekette, G. W. Stroke, Y. E. Nesterikhin, W. E. Kock, eds. (Plenum, New York, 1976), Vol. 2, pp. 155–170.
  6. G. Hausler, A. Lohmann, “Hybrid image processing with feedback,” Opt. Commun. 21, 365–368 (1977).
    [CrossRef]
  7. J. N. Cederquist, “Optical feedback processing,” in Optical Signal Processing, J. Horner ed. (Academic, New York, 1987), pp. 525–565.
  8. T. W. Hansch, F. Varsanyi, A. L. Schawlow, “Image amplification with dye lasers,” Appl. Phys. Lett. 18, 108–110 (1971).
    [CrossRef]
  9. R. P. Akins, S. H. Lee, “Coherent optical image amplification by an injection-locked dye amplifier at 632.8 nm,” Appl. Phys. Lett. 35, 660–663 (1979).
    [CrossRef]
  10. F. Laeri, T. Tschudi, J. Albers, “Coherent cw image amplifier and oscillator using two-wave interaction in a BaTiO3 crystal,” Opt. Commun. 47, 387–390 (1983).
    [CrossRef]
  11. P. Pellat-Finet, J. de Bougrenet de la Tocnaye, “Optical generator of spheroidal wave functions, using a BSO crystal,” Opt. Commun. 55, 305–310 (1985).
    [CrossRef]
  12. D. Z. Anderson, “Coherent optical eigenstate memory,” Opt. Lett. 11, 56–58 (1986).
    [CrossRef] [PubMed]
  13. B. H. Soffer, G. J. Dunning, Y. Owechko, E. Marom, “Associative holographic memory with feedback using phase-conjugate mirrors,” Opt. Lett. 11, 118–120 (1986).
    [CrossRef] [PubMed]
  14. H. Rajbenbach, Y. Fainman, S. H. Lee, “Optical implementation of an iterative algorithm for matrix inversion,” Appl. Opt. 26, 1024–1031 (1987).
    [CrossRef] [PubMed]
  15. O. Ikeda, T. Sato, H. Kojima, “Analysis of a high-resolution and large-dynamic-range spatial filter using a pair of facing phase conjugators with gains,” J. Opt. Soc. Am. A 2, 1863–1868 (1985).
    [CrossRef]
  16. O. Ikeda, T. Sato, H. Kojima, “Construction of a Wiener filter using a phase-conjugate filter,” J. Opt. Soc. Am. A 3, 645–650 (1986).
    [CrossRef]
  17. D. Z. Anderson, M. C. Erie, “Resonator memories and optical novelty filter,” Opt. Eng. 26, 434–444 (1987).
  18. M. O. Hagler, S. V. Bell, “Nyquist stability conditions for laser feedback amplifiers with Gaussian or Lorentzian gain profiles,” J. Opt. Soc. Am. A 3, 308–318 (1986).
    [CrossRef]
  19. P. De Santis, F. Gori, G. Guattari, C. Palma, “Optical systems with feedback,” Opt. Acta 23, 505–518 (1976).
    [CrossRef]
  20. M. Cronin-Golumb, B. Fischer, S. K. Kwong, J. O. White, A. Yariv, “Nondegenerate optical oscillation in a resonator formed by two phase-conjugate mirrors,” Opt. Lett. 10, 353–355 (1985).
    [CrossRef]
  21. G. Indebetouw, K. P. Lo, “Real time moire interferometry using a Fabry–Perot cavity with a phase conjugate mirror,” Appl. Opt. 28, 3893–3896 (1989).
    [CrossRef] [PubMed]
  22. R. W. Gerchberg, “Super-resolution through error-energy reduction,” Opt. Acta 21, 709–720 (1974).
    [CrossRef]
  23. A. Papoulis, “A new algorithm in spectral analysis and bandlimited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
    [CrossRef]
  24. D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
    [CrossRef]
  25. C. K. Rushforth, R. L. Frost, “Comparison of some algorithms for reconstructing space-limited images,” J. Opt. Soc. Am. 70, 1539–1544 (1980).
    [CrossRef]
  26. D. Slepian, H. O. Pollak, H. J. Landau, “Prolate spheroidal wave function,” Bell Syst. Tech. J. 40, 43–84 (1961).
  27. R. J. Marks, D. K. Smith, “Iterative coherent processor for bandlimited signal extrapolation,” in International Optical Computing Conference I, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 106–111 (1980).
  28. Y. A. Ananev, A. Y. Bekshaev, “More about the problem of superresolution in optics,” Opt. Spectrosc. (USSR) 64, 139–141 (1988).

1989 (1)

1988 (1)

Y. A. Ananev, A. Y. Bekshaev, “More about the problem of superresolution in optics,” Opt. Spectrosc. (USSR) 64, 139–141 (1988).

1987 (2)

D. Z. Anderson, M. C. Erie, “Resonator memories and optical novelty filter,” Opt. Eng. 26, 434–444 (1987).

H. Rajbenbach, Y. Fainman, S. H. Lee, “Optical implementation of an iterative algorithm for matrix inversion,” Appl. Opt. 26, 1024–1031 (1987).
[CrossRef] [PubMed]

1986 (4)

1985 (3)

1983 (1)

F. Laeri, T. Tschudi, J. Albers, “Coherent cw image amplifier and oscillator using two-wave interaction in a BaTiO3 crystal,” Opt. Commun. 47, 387–390 (1983).
[CrossRef]

1980 (1)

1979 (1)

R. P. Akins, S. H. Lee, “Coherent optical image amplification by an injection-locked dye amplifier at 632.8 nm,” Appl. Phys. Lett. 35, 660–663 (1979).
[CrossRef]

1978 (1)

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

1977 (2)

E. Handler, U. Roder, “Suppression of multiplicative disturbance by coherent optical feedback technique,” Opt. Commun. 23, 352–356 (1977).
[CrossRef]

G. Hausler, A. Lohmann, “Hybrid image processing with feedback,” Opt. Commun. 21, 365–368 (1977).
[CrossRef]

1976 (1)

P. De Santis, F. Gori, G. Guattari, C. Palma, “Optical systems with feedback,” Opt. Acta 23, 505–518 (1976).
[CrossRef]

1975 (2)

D. P. Jablonowski, S. H. Lee, “A coherent optical feedback system for optical information processing,” Appl. Phys. 8, 51–58 (1975).
[CrossRef]

A. Papoulis, “A new algorithm in spectral analysis and bandlimited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

1974 (1)

R. W. Gerchberg, “Super-resolution through error-energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

1971 (2)

T. W. Hansch, F. Varsanyi, A. L. Schawlow, “Image amplification with dye lasers,” Appl. Phys. Lett. 18, 108–110 (1971).
[CrossRef]

M. O. Hagler, “Active synthesis of inverse spatial filters,” Appl. Opt. 10, 2783–2784 (1971).
[CrossRef] [PubMed]

1967 (1)

R. V. Pole, H. Wieder, E. S. Barrekette, “Reactive optical information processing. I: Theory of information recovery and resonator mode structure,” Appl. Opt. 8, 1571–1575 (1967).
[CrossRef]

1961 (1)

D. Slepian, H. O. Pollak, H. J. Landau, “Prolate spheroidal wave function,” Bell Syst. Tech. J. 40, 43–84 (1961).

Akins, R. P.

R. P. Akins, S. H. Lee, “Coherent optical image amplification by an injection-locked dye amplifier at 632.8 nm,” Appl. Phys. Lett. 35, 660–663 (1979).
[CrossRef]

Albers, J.

F. Laeri, T. Tschudi, J. Albers, “Coherent cw image amplifier and oscillator using two-wave interaction in a BaTiO3 crystal,” Opt. Commun. 47, 387–390 (1983).
[CrossRef]

Ananev, Y. A.

Y. A. Ananev, A. Y. Bekshaev, “More about the problem of superresolution in optics,” Opt. Spectrosc. (USSR) 64, 139–141 (1988).

Anderson, D. Z.

D. Z. Anderson, M. C. Erie, “Resonator memories and optical novelty filter,” Opt. Eng. 26, 434–444 (1987).

D. Z. Anderson, “Coherent optical eigenstate memory,” Opt. Lett. 11, 56–58 (1986).
[CrossRef] [PubMed]

Barrekette, E. S.

R. V. Pole, H. Wieder, E. S. Barrekette, “Reactive optical information processing. I: Theory of information recovery and resonator mode structure,” Appl. Opt. 8, 1571–1575 (1967).
[CrossRef]

Bekshaev, A. Y.

Y. A. Ananev, A. Y. Bekshaev, “More about the problem of superresolution in optics,” Opt. Spectrosc. (USSR) 64, 139–141 (1988).

Bell, S. V.

Cederquist, J. N.

J. N. Cederquist, “Optical feedback processing,” in Optical Signal Processing, J. Horner ed. (Academic, New York, 1987), pp. 525–565.

Cronin-Golumb, M.

de Bougrenet de la Tocnaye, J.

P. Pellat-Finet, J. de Bougrenet de la Tocnaye, “Optical generator of spheroidal wave functions, using a BSO crystal,” Opt. Commun. 55, 305–310 (1985).
[CrossRef]

De Santis, P.

P. De Santis, F. Gori, G. Guattari, C. Palma, “Optical systems with feedback,” Opt. Acta 23, 505–518 (1976).
[CrossRef]

Dunning, G. J.

Erie, M. C.

D. Z. Anderson, M. C. Erie, “Resonator memories and optical novelty filter,” Opt. Eng. 26, 434–444 (1987).

Fainman, Y.

Feldbush, V. I.

P. E. Kotljar, E. S. Nezhevenko, B. I. Spektor, V. I. Feldbush, “Optical processing in feedback systems,” in Optical Information Processing, E. S. Barrekette, G. W. Stroke, Y. E. Nesterikhin, W. E. Kock, eds. (Plenum, New York, 1976), Vol. 2, pp. 155–170.

Fischer, B.

Frost, R. L.

Gerchberg, R. W.

R. W. Gerchberg, “Super-resolution through error-energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Gori, F.

P. De Santis, F. Gori, G. Guattari, C. Palma, “Optical systems with feedback,” Opt. Acta 23, 505–518 (1976).
[CrossRef]

Guattari, G.

P. De Santis, F. Gori, G. Guattari, C. Palma, “Optical systems with feedback,” Opt. Acta 23, 505–518 (1976).
[CrossRef]

Hagler, M. O.

Handler, E.

E. Handler, U. Roder, “Suppression of multiplicative disturbance by coherent optical feedback technique,” Opt. Commun. 23, 352–356 (1977).
[CrossRef]

Hansch, T. W.

T. W. Hansch, F. Varsanyi, A. L. Schawlow, “Image amplification with dye lasers,” Appl. Phys. Lett. 18, 108–110 (1971).
[CrossRef]

Hausler, G.

G. Hausler, A. Lohmann, “Hybrid image processing with feedback,” Opt. Commun. 21, 365–368 (1977).
[CrossRef]

Ikeda, O.

Indebetouw, G.

Jablonowski, D. P.

D. P. Jablonowski, S. H. Lee, “A coherent optical feedback system for optical information processing,” Appl. Phys. 8, 51–58 (1975).
[CrossRef]

Kojima, H.

Kotljar, P. E.

P. E. Kotljar, E. S. Nezhevenko, B. I. Spektor, V. I. Feldbush, “Optical processing in feedback systems,” in Optical Information Processing, E. S. Barrekette, G. W. Stroke, Y. E. Nesterikhin, W. E. Kock, eds. (Plenum, New York, 1976), Vol. 2, pp. 155–170.

Kwong, S. K.

Laeri, F.

F. Laeri, T. Tschudi, J. Albers, “Coherent cw image amplifier and oscillator using two-wave interaction in a BaTiO3 crystal,” Opt. Commun. 47, 387–390 (1983).
[CrossRef]

Landau, H. J.

D. Slepian, H. O. Pollak, H. J. Landau, “Prolate spheroidal wave function,” Bell Syst. Tech. J. 40, 43–84 (1961).

Lee, S. H.

H. Rajbenbach, Y. Fainman, S. H. Lee, “Optical implementation of an iterative algorithm for matrix inversion,” Appl. Opt. 26, 1024–1031 (1987).
[CrossRef] [PubMed]

R. P. Akins, S. H. Lee, “Coherent optical image amplification by an injection-locked dye amplifier at 632.8 nm,” Appl. Phys. Lett. 35, 660–663 (1979).
[CrossRef]

D. P. Jablonowski, S. H. Lee, “A coherent optical feedback system for optical information processing,” Appl. Phys. 8, 51–58 (1975).
[CrossRef]

Lo, K. P.

Lohmann, A.

G. Hausler, A. Lohmann, “Hybrid image processing with feedback,” Opt. Commun. 21, 365–368 (1977).
[CrossRef]

Marks, R. J.

R. J. Marks, D. K. Smith, “Iterative coherent processor for bandlimited signal extrapolation,” in International Optical Computing Conference I, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 106–111 (1980).

Marom, E.

Nezhevenko, E. S.

P. E. Kotljar, E. S. Nezhevenko, B. I. Spektor, V. I. Feldbush, “Optical processing in feedback systems,” in Optical Information Processing, E. S. Barrekette, G. W. Stroke, Y. E. Nesterikhin, W. E. Kock, eds. (Plenum, New York, 1976), Vol. 2, pp. 155–170.

Owechko, Y.

Palma, C.

P. De Santis, F. Gori, G. Guattari, C. Palma, “Optical systems with feedback,” Opt. Acta 23, 505–518 (1976).
[CrossRef]

Papoulis, A.

A. Papoulis, “A new algorithm in spectral analysis and bandlimited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

Pellat-Finet, P.

P. Pellat-Finet, J. de Bougrenet de la Tocnaye, “Optical generator of spheroidal wave functions, using a BSO crystal,” Opt. Commun. 55, 305–310 (1985).
[CrossRef]

Pole, R. V.

R. V. Pole, H. Wieder, E. S. Barrekette, “Reactive optical information processing. I: Theory of information recovery and resonator mode structure,” Appl. Opt. 8, 1571–1575 (1967).
[CrossRef]

Pollak, H. O.

D. Slepian, H. O. Pollak, H. J. Landau, “Prolate spheroidal wave function,” Bell Syst. Tech. J. 40, 43–84 (1961).

Rajbenbach, H.

Roder, U.

E. Handler, U. Roder, “Suppression of multiplicative disturbance by coherent optical feedback technique,” Opt. Commun. 23, 352–356 (1977).
[CrossRef]

Rushforth, C. K.

Sato, T.

Schawlow, A. L.

T. W. Hansch, F. Varsanyi, A. L. Schawlow, “Image amplification with dye lasers,” Appl. Phys. Lett. 18, 108–110 (1971).
[CrossRef]

Slepian, D.

D. Slepian, H. O. Pollak, H. J. Landau, “Prolate spheroidal wave function,” Bell Syst. Tech. J. 40, 43–84 (1961).

Smith, D. K.

R. J. Marks, D. K. Smith, “Iterative coherent processor for bandlimited signal extrapolation,” in International Optical Computing Conference I, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 106–111 (1980).

Soffer, B. H.

Spektor, B. I.

P. E. Kotljar, E. S. Nezhevenko, B. I. Spektor, V. I. Feldbush, “Optical processing in feedback systems,” in Optical Information Processing, E. S. Barrekette, G. W. Stroke, Y. E. Nesterikhin, W. E. Kock, eds. (Plenum, New York, 1976), Vol. 2, pp. 155–170.

Tschudi, T.

F. Laeri, T. Tschudi, J. Albers, “Coherent cw image amplifier and oscillator using two-wave interaction in a BaTiO3 crystal,” Opt. Commun. 47, 387–390 (1983).
[CrossRef]

Varsanyi, F.

T. W. Hansch, F. Varsanyi, A. L. Schawlow, “Image amplification with dye lasers,” Appl. Phys. Lett. 18, 108–110 (1971).
[CrossRef]

White, J. O.

Wieder, H.

R. V. Pole, H. Wieder, E. S. Barrekette, “Reactive optical information processing. I: Theory of information recovery and resonator mode structure,” Appl. Opt. 8, 1571–1575 (1967).
[CrossRef]

Yariv, A.

Youla, D. C.

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

Appl. Opt. (4)

Appl. Phys. (1)

D. P. Jablonowski, S. H. Lee, “A coherent optical feedback system for optical information processing,” Appl. Phys. 8, 51–58 (1975).
[CrossRef]

Appl. Phys. Lett. (2)

T. W. Hansch, F. Varsanyi, A. L. Schawlow, “Image amplification with dye lasers,” Appl. Phys. Lett. 18, 108–110 (1971).
[CrossRef]

R. P. Akins, S. H. Lee, “Coherent optical image amplification by an injection-locked dye amplifier at 632.8 nm,” Appl. Phys. Lett. 35, 660–663 (1979).
[CrossRef]

Bell Syst. Tech. J. (1)

D. Slepian, H. O. Pollak, H. J. Landau, “Prolate spheroidal wave function,” Bell Syst. Tech. J. 40, 43–84 (1961).

IEEE Trans. Circuits Syst. (2)

A. Papoulis, “A new algorithm in spectral analysis and bandlimited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Acta (2)

P. De Santis, F. Gori, G. Guattari, C. Palma, “Optical systems with feedback,” Opt. Acta 23, 505–518 (1976).
[CrossRef]

R. W. Gerchberg, “Super-resolution through error-energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Opt. Commun. (4)

F. Laeri, T. Tschudi, J. Albers, “Coherent cw image amplifier and oscillator using two-wave interaction in a BaTiO3 crystal,” Opt. Commun. 47, 387–390 (1983).
[CrossRef]

P. Pellat-Finet, J. de Bougrenet de la Tocnaye, “Optical generator of spheroidal wave functions, using a BSO crystal,” Opt. Commun. 55, 305–310 (1985).
[CrossRef]

E. Handler, U. Roder, “Suppression of multiplicative disturbance by coherent optical feedback technique,” Opt. Commun. 23, 352–356 (1977).
[CrossRef]

G. Hausler, A. Lohmann, “Hybrid image processing with feedback,” Opt. Commun. 21, 365–368 (1977).
[CrossRef]

Opt. Eng. (1)

D. Z. Anderson, M. C. Erie, “Resonator memories and optical novelty filter,” Opt. Eng. 26, 434–444 (1987).

Opt. Lett. (3)

Opt. Spectrosc. (USSR) (1)

Y. A. Ananev, A. Y. Bekshaev, “More about the problem of superresolution in optics,” Opt. Spectrosc. (USSR) 64, 139–141 (1988).

Other (3)

R. J. Marks, D. K. Smith, “Iterative coherent processor for bandlimited signal extrapolation,” in International Optical Computing Conference I, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 106–111 (1980).

J. N. Cederquist, “Optical feedback processing,” in Optical Signal Processing, J. Horner ed. (Academic, New York, 1987), pp. 525–565.

P. E. Kotljar, E. S. Nezhevenko, B. I. Spektor, V. I. Feldbush, “Optical processing in feedback systems,” in Optical Information Processing, E. S. Barrekette, G. W. Stroke, Y. E. Nesterikhin, W. E. Kock, eds. (Plenum, New York, 1976), Vol. 2, pp. 155–170.

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of a cavity with a PCM. The transmittance and reflectance of the mirror are represented by the operators T ^ and R ^, and the reflectivity of the PCM is characterized by the operators μ ^ i. The operators l ^ 1 , 2 ± describe the transfer of the optical field from the input mirror to the PCM and back along the primary and the secondary paths, respectively.

Fig. 2
Fig. 2

Block diagram of the feedback cavity with the PCM shown in Fig. 1.

Fig. 3
Fig. 3

Common path geometry of the Fabry–Perot cavity with a PCM. The output of the cavity is Eo = τ(E1 + E3), and E1 is proportional to the input while E3 is proportional to the phase conjugate of the inverted input.

Fig. 4
Fig. 4

Unfolded cavity with two PCM’s, allowing for the independent control of the reflectivity μ ^ 1 and μ ^ 2 and thus for the control of the phase of the feedback parameter.

Fig. 5
Fig. 5

Diagram of a setup for implementation of the Gerchberg algorithm. The input mirror implements the complementary spatial frequency truncation operator [Î T ^ B]. The primary path contains a spatial truncation operator T ^ D in plane P. The secondary path contains only an identity operator Î.

Fig. 6
Fig. 6

Setup used for the experimental implementation of the Gerchberg algorithm. Phase conjugation is achieved by degenerate four-wave mixing in a single crystal of BaTiO3. The mirror M is slightly tilted to separate the primary and the secondary paths. Output ports 1 and 2 extract the extrapolated spatial frequency information and the enhanced spatial output, respectively.

Fig. 7
Fig. 7

Trace through the intensity distribution of the degraded image of a two-pixel input (dotted curve), and through that of the restored image (solid curve). The restored image shows a 30% improvement in visibility.

Fig. 8
Fig. 8

Output at port 1 showing the extrapolated spatial frequency spectrum. The center portion is the input spectrum. The low intensity at the boundary of the band-limiting slit is due to the soft edges of the mirror, leading to low gain in these regions.

Equations (52)

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

E i = S ^ s E s ,
E o = S ^ f E i + α S ^ f S ^ b E o ,
E j = n e j n ϕ n ,             j = s , i , or o ,
S ^ i ϕ n = λ l n ϕ n ,             l = s , f , or b .
e o n = λ n e i n = λ n λ s n e s n ,
λ n = λ f n 1 - α λ f n λ b n .
λ n ~ ( α λ s n ) - 1 .
λ n = [ 1 - α ( 1 - λ s n ) ] - 1 ~ λ s n - 1 .
μ ^ i = r PCMi p c ^ = r PCMi exp ( i ψ PCMi ) p c ^ ,             i = 1 , 2.
ϕ PCMi = ϕ 1 i + ϕ 2 i .
E 1 = E i + L ^ 1 + R ^ L ^ 2 - E 4 ,
E 2 = μ ^ 1 E 1 ,
E 3 = L ^ 2 + R ^ L ^ 1 - E 2 ,
E 4 = μ ^ 2 E 3 ,
E i = L ^ 1 + T ^ E i .
E 1 = E i + L ^ 1 + R ^ L ^ 2 - μ ^ 2 L ^ 2 + R ^ L ^ 1 - μ ^ 1 E 1 ,
E 3 = L ^ 2 + R ^ L ^ 1 - μ ^ 1 E 1 .
E o = τ E 1 ,
E o = τ E i + L ^ 1 + R ^ L ^ 2 - μ ^ 2 L ^ 2 + R L ^ 1 - μ ^ 1 E o = τ E i + β O ^ E o ,
β = r 2 r PCM 1 * r PCM 2 = r 2 r PCM 1 r PCM 2 exp ( i ψ ) .
E o = τ E i + β L ^ 1 + L ^ 2 - p c ^ L ^ 2 + L ^ 1 - p c ^ E o .
L ^ + ( - ) = exp [ i k L + ( - ) ] F ^ ,
E 1 ( x , y ) = E i ( x , y ) + β E 1 ( x , y ) ,
E 3 ( x , y ) = r r PCM 1 exp ( i k L ) E 1 * ( - x , - y ) ,
E o ( x , y ) = τ [ E 1 ( x , y ) + E 3 ( x , y ) ] = τ 1 - β E i ( x , y ) + τ r r PCM 1 exp ( i k L ) 1 - β * E i * ( - x , - y ) .
L ^ 1 + ( - ) = exp [ i k L i + ( - ) ] F ^ S ^ i F ^ , i = 1 , 2 ,
F ^ p c ^ = p c ^ F ^ - 1 ,
F ^ F ^ f ( x , y ) = f ( - x , - y )             ( coordinate inversion operator ) ,
p c ^ p c ^ = I ^             ( identity operator ) ,
S ^ i p c ^ = p c ^ S ^ i * ,
E o = τ E i + β F ^ S ^ 1 S ^ 2 S ^ 2 * S ^ 1 * F ^ - 1 E o ,
F ^ S ^ i F ^ - 1 ϕ k = λ i k ϕ k ;             i = 1 , 2.
E i ( o ) = k e i ( o ) k ϕ k .
e o k = 1 1 - β λ 1 k 2 λ 2 k 2 τ e i k .
E o = τ E i + n τ λ n E i , ϕ n λ n - β - 1 ϕ n ,
T ^ D f ( x , y ) = { f ( x , y ) , ( x , y ) D 0 , otherwise ,
T ^ B f ˜ ( ξ , η ) = { f ˜ ( ξ , η ) = F ^ f ( x , y ) , ( ξ , η ) B 0 , otherwise .
g ( m ) ( x , y ) = T ^ D { g ( x , y ) + F ^ [ I ^ - T ^ B ] F ^ - 1 T ^ D g ( m - 1 ) ( x , y ) } .
[ I ^ - T ^ B ] g ˜ ( ξ , η ) = { 0 , ( ξ , η ) B g ˜ ( ξ , η ) , ( ξ , η ) B ,
g ( 1 ) ( x , y ) = T ^ D g ( x , y ) = T ^ D F ^ T ^ B F ^ - 1 f ( x , y ) .
O ^ = T ^ D F ^ [ I ^ - T ^ B ] F ^ - 1 T ^ D = [ I ^ - G ] T D ,
T ^ D g ( m ) = e k ( m ) ϕ k
e k ( m ) = 1 - ( 1 - λ k ) m λ k e k ( 1 ) .
e k ( m ) ( m ) 1 λ k e k ( 1 ) ,
F ^ = T ^ B , R ^ = r [ I ^ - T ^ B ] .
E o = τ E i + β T ^ D F ^ [ I ^ - T ^ B ] F ^ p c ^ F ^ [ I ^ - T ^ B ] F ^ T ^ D p c ^ E o ,
E i = T ^ D F ^ T ^ B F ^ g ( x , y ) ,
E o = τ E i + β T ^ D F ^ [ I ^ - T ^ B ] F ^ - 1 T ^ D E o ,
e o k = 1 1 - β ( 1 - λ k ) τ e i k 1 λ k τ e i k             ( β 1 )
E o = τ E i + β ( T ^ D F ^ [ I ^ - T ^ B ] F ^ - 1 ) 2 E o ,
e o k = 1 1 - β ( 1 - λ k ) 2 τ e i k .
e o k 1 λ k ( 2 - λ k ) τ e i k             ( β 1 ) .

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