Abstract

Optical spatial phase rectification by limiting quadratic processing in photorefractive two-beam coupling is proposed and demonstrated. We use this limiting quadratic processor to reduce complex multiplicative noise. The principle of compansion (compression and expansion) from serial communication theory is introduced for the first time to our knowledge in parallel optical signal processing. In addition the effect of two-beam coupling compression on the efficiency of beam cleanup is illustrated experimentally and optimized through computer simulation.

© 1994 Optical Society of America

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References

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  1. H. Stark, F. B. Tuteur, and J. B. Anderson, Modern Electrical Communication, Analog, Digital and Optical Systems, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1988), Chap. 4, pp. 175–178; Chap. 6, pp. 247–248.
  2. A. E. Chiou and P. Yeh, "Beam cleanup using photorefractive two-wave mixing," Opt. Lett. 10, 621–623 (1985).
    [CrossRef] [PubMed]
  3. S. Sternklar, S. Weiss, M. Segev, and B. Fischer, "Mach-Zender interferometer with multimode fiber using double-phase conjugate mirror," Appl. Opt. 25, 4518–4519 (1986).
    [CrossRef] [PubMed]
  4. S. MacCormack and R. Eason, "Efficient amplification of single-mode laser diode by photorefractive combination using an injection-locked diode laser array pump," Opt. Lett. 15, 1212–1214 (1990).
    [CrossRef] [PubMed]
  5. G. L. Sicuranza, "Quadratic filter for signal processing," Proc. IEEE, 80, 1263–1285 (1992).
    [CrossRef]
  6. R. D. Martin and C. P. McGath, "Robust detection of stochastic signals," IEEE Trans. Info. Theory IT-20, 537–541 (1974).
    [CrossRef]
  7. M. Schwartz, W. Bennett, and S. Stein, Communication Systems and Techniques (McGraw-Hill, New York, 1966), Chap. 4, pp. 213–216; Chap. 6, p. 247.
  8. J. Khoury, C. L. Woods, and M. Cronin-Golomb, "Noise reduction using adaptive spatial filtering in photorefractive two-beam coupling," Opt. Lett. 16, 747–749 (1991).
    [CrossRef] [PubMed]
  9. J. Khoury, M. Cronin-Golomb, and C. Woods, "Photorefractive deamplification for additive signal dependent noise reduction," Opt. Eng. 32, 2877–2833 (1993).
    [CrossRef]
  10. W. K Pratt, Digital Image Processing, 2nd. ed. (Wiley-Interscience, New York, 1987), Chap. 10, p. 289.
  11. P. Yeh, "Photorefractive phase conjugators," IEEE J. Quantum Electron. 80, 436–450 (1992).
  12. D. W. Vahey, "A nonlinear coupled wave theory of holographic storage in ferroelectric materials," J. Appl. Phys. 46, 3510–3515 (1975).
    [CrossRef]
  13. M. Cronin-Golomb, "Large nonlinearities in four-wave mixing in photorefractive crystals and application in passive optical phase conjugation," Ph.D. dissertation (California Institute of Technology, Pasadena, Calif., 1983).
  14. P. Yeh, "Photorefractive two-beam coupling in cubic crystals," J. Opt. Soc. Am. B 4, 1382–1386 (1987).
    [CrossRef]
  15. L.-J. Cheng and P. Yeh, "Cross-polarization beam coupling in photorefractive GaAs crystal," Opt. Lett. 13, 50–52 (1988).
    [CrossRef] [PubMed]
  16. L.-J. Cheng, G. Gheen, T.-H. Chao, H.-K. Liu, A. Partovi, J. Katz, and E. M. Garmire, "Spatial light modulation by beam coupling in GaAs crystals," Opt. Lett. 12, 705–707 (1987).
    [CrossRef] [PubMed]
  17. J. Khoury, A. M. Biernacki, C. L. Woods, and M. Cronin-Golomb, "Photorefractive quadratic processor for converting multiplicative noise-to-additive noise," Opt. Eng. 32, 2872–2876 (1993).
    [CrossRef]
  18. F. Vachss and P. Yeh, "Image degradation mechanism in photorefractive amplifier," J. Opt. Soc. Am. B 6, 1834–1844 (1989).
    [CrossRef]
  19. M. Cronin-Golomb, "Whole beam method for photorefractive nonlinear optics," Opt. Commun. 89, 276–282 (1992).
    [CrossRef]
  20. H. Kato and J. W. Goodman, "Nonlinear filtering in coherent optical system through halftone screen processing," Appl. Opt. 14,1813–1824 (1975).
    [CrossRef] [PubMed]
  21. J.-P. Huignard, J. P. Herriau, L. Pichon, and A. Marrakehi, "Speckle-free imaging in four-wave mixing experiment with BSO crystals," Opt. Lett. 5, 436–437 (1980).
    [CrossRef] [PubMed]
  22. J. W. Goodman "Statistical properties of laser speckle pattern," in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed., Vol. 9 of Springer Series on Topics in Applied Physics (Springer-Verlag, Berlin, 1984), p. 40.
  23. Y. Fainman, E. Klanick, and S. H. Lee, "Optimal coherent image amplification by two-beam coupling in photorefractive BaTiO3," Opt. Eng. 25, 228–234 (1986).
    [CrossRef]
  24. N. A. Vainos and M. C. Gower, "High-fidelity image amplification and phase conjugation in photorefractive Bi12SiO20 crystals," Opt. Lett. 16, 363–365 (1991).
    [CrossRef] [PubMed]
  25. J. A. Khoury, G. Hussain, and R. W. Eason, "Contrast manipulation and controllable spatial filtering via photore-fractive two-beam coupling," Opt. Commun. 70, 272–276 (1989).
    [CrossRef]
  26. J. Khoury, M. Cronin-Golomb, A. M. Biernacki, and C. L. Woods, "Photorefractive phase-conjugate techniques for measuring surface granularity," Appl. Opt. (to be published).
  27. W. B. Davenport, Jr., and W. L. Root, An Introduction to the Theory to Random Signals and Noise (McGraw-Hill, New York, 1958), Chap. 13, pp. 277–311.

1993 (2)

J. Khoury, M. Cronin-Golomb, and C. Woods, "Photorefractive deamplification for additive signal dependent noise reduction," Opt. Eng. 32, 2877–2833 (1993).
[CrossRef]

J. Khoury, A. M. Biernacki, C. L. Woods, and M. Cronin-Golomb, "Photorefractive quadratic processor for converting multiplicative noise-to-additive noise," Opt. Eng. 32, 2872–2876 (1993).
[CrossRef]

1992 (2)

M. Cronin-Golomb, "Whole beam method for photorefractive nonlinear optics," Opt. Commun. 89, 276–282 (1992).
[CrossRef]

P. Yeh, "Photorefractive phase conjugators," IEEE J. Quantum Electron. 80, 436–450 (1992).

1991 (2)

1990 (1)

1989 (2)

J. A. Khoury, G. Hussain, and R. W. Eason, "Contrast manipulation and controllable spatial filtering via photore-fractive two-beam coupling," Opt. Commun. 70, 272–276 (1989).
[CrossRef]

F. Vachss and P. Yeh, "Image degradation mechanism in photorefractive amplifier," J. Opt. Soc. Am. B 6, 1834–1844 (1989).
[CrossRef]

1988 (1)

1987 (2)

1986 (2)

Y. Fainman, E. Klanick, and S. H. Lee, "Optimal coherent image amplification by two-beam coupling in photorefractive BaTiO3," Opt. Eng. 25, 228–234 (1986).
[CrossRef]

S. Sternklar, S. Weiss, M. Segev, and B. Fischer, "Mach-Zender interferometer with multimode fiber using double-phase conjugate mirror," Appl. Opt. 25, 4518–4519 (1986).
[CrossRef] [PubMed]

1985 (1)

1980 (1)

1975 (2)

D. W. Vahey, "A nonlinear coupled wave theory of holographic storage in ferroelectric materials," J. Appl. Phys. 46, 3510–3515 (1975).
[CrossRef]

H. Kato and J. W. Goodman, "Nonlinear filtering in coherent optical system through halftone screen processing," Appl. Opt. 14,1813–1824 (1975).
[CrossRef] [PubMed]

1974 (1)

R. D. Martin and C. P. McGath, "Robust detection of stochastic signals," IEEE Trans. Info. Theory IT-20, 537–541 (1974).
[CrossRef]

Anderson, J. B.

H. Stark, F. B. Tuteur, and J. B. Anderson, Modern Electrical Communication, Analog, Digital and Optical Systems, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1988), Chap. 4, pp. 175–178; Chap. 6, pp. 247–248.

Bennett, W.

M. Schwartz, W. Bennett, and S. Stein, Communication Systems and Techniques (McGraw-Hill, New York, 1966), Chap. 4, pp. 213–216; Chap. 6, p. 247.

Biernacki, A. M.

J. Khoury, A. M. Biernacki, C. L. Woods, and M. Cronin-Golomb, "Photorefractive quadratic processor for converting multiplicative noise-to-additive noise," Opt. Eng. 32, 2872–2876 (1993).
[CrossRef]

J. Khoury, M. Cronin-Golomb, A. M. Biernacki, and C. L. Woods, "Photorefractive phase-conjugate techniques for measuring surface granularity," Appl. Opt. (to be published).

Chao, T.-H.

Cheng, L.-J.

Chiou, A. E.

Cronin-Golomb, M.

J. Khoury, M. Cronin-Golomb, and C. Woods, "Photorefractive deamplification for additive signal dependent noise reduction," Opt. Eng. 32, 2877–2833 (1993).
[CrossRef]

J. Khoury, A. M. Biernacki, C. L. Woods, and M. Cronin-Golomb, "Photorefractive quadratic processor for converting multiplicative noise-to-additive noise," Opt. Eng. 32, 2872–2876 (1993).
[CrossRef]

M. Cronin-Golomb, "Whole beam method for photorefractive nonlinear optics," Opt. Commun. 89, 276–282 (1992).
[CrossRef]

J. Khoury, C. L. Woods, and M. Cronin-Golomb, "Noise reduction using adaptive spatial filtering in photorefractive two-beam coupling," Opt. Lett. 16, 747–749 (1991).
[CrossRef] [PubMed]

J. Khoury, M. Cronin-Golomb, A. M. Biernacki, and C. L. Woods, "Photorefractive phase-conjugate techniques for measuring surface granularity," Appl. Opt. (to be published).

M. Cronin-Golomb, "Large nonlinearities in four-wave mixing in photorefractive crystals and application in passive optical phase conjugation," Ph.D. dissertation (California Institute of Technology, Pasadena, Calif., 1983).

Davenport, Jr., W. B.

W. B. Davenport, Jr., and W. L. Root, An Introduction to the Theory to Random Signals and Noise (McGraw-Hill, New York, 1958), Chap. 13, pp. 277–311.

Eason, R.

Eason, R. W.

J. A. Khoury, G. Hussain, and R. W. Eason, "Contrast manipulation and controllable spatial filtering via photore-fractive two-beam coupling," Opt. Commun. 70, 272–276 (1989).
[CrossRef]

Fainman, Y.

Y. Fainman, E. Klanick, and S. H. Lee, "Optimal coherent image amplification by two-beam coupling in photorefractive BaTiO3," Opt. Eng. 25, 228–234 (1986).
[CrossRef]

Fischer, B.

Garmire, E. M.

Gheen, G.

Goodman, J. W.

H. Kato and J. W. Goodman, "Nonlinear filtering in coherent optical system through halftone screen processing," Appl. Opt. 14,1813–1824 (1975).
[CrossRef] [PubMed]

J. W. Goodman "Statistical properties of laser speckle pattern," in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed., Vol. 9 of Springer Series on Topics in Applied Physics (Springer-Verlag, Berlin, 1984), p. 40.

Gower, M. C.

Herriau, J. P.

Huignard, J.-P.

Hussain, G.

J. A. Khoury, G. Hussain, and R. W. Eason, "Contrast manipulation and controllable spatial filtering via photore-fractive two-beam coupling," Opt. Commun. 70, 272–276 (1989).
[CrossRef]

Kato, H.

Katz, J.

Khoury, J.

J. Khoury, M. Cronin-Golomb, and C. Woods, "Photorefractive deamplification for additive signal dependent noise reduction," Opt. Eng. 32, 2877–2833 (1993).
[CrossRef]

J. Khoury, A. M. Biernacki, C. L. Woods, and M. Cronin-Golomb, "Photorefractive quadratic processor for converting multiplicative noise-to-additive noise," Opt. Eng. 32, 2872–2876 (1993).
[CrossRef]

J. Khoury, C. L. Woods, and M. Cronin-Golomb, "Noise reduction using adaptive spatial filtering in photorefractive two-beam coupling," Opt. Lett. 16, 747–749 (1991).
[CrossRef] [PubMed]

J. Khoury, M. Cronin-Golomb, A. M. Biernacki, and C. L. Woods, "Photorefractive phase-conjugate techniques for measuring surface granularity," Appl. Opt. (to be published).

Khoury, J. A.

J. A. Khoury, G. Hussain, and R. W. Eason, "Contrast manipulation and controllable spatial filtering via photore-fractive two-beam coupling," Opt. Commun. 70, 272–276 (1989).
[CrossRef]

Klanick, E.

Y. Fainman, E. Klanick, and S. H. Lee, "Optimal coherent image amplification by two-beam coupling in photorefractive BaTiO3," Opt. Eng. 25, 228–234 (1986).
[CrossRef]

Lee, S. H.

Y. Fainman, E. Klanick, and S. H. Lee, "Optimal coherent image amplification by two-beam coupling in photorefractive BaTiO3," Opt. Eng. 25, 228–234 (1986).
[CrossRef]

Liu, H.-K.

MacCormack, S.

Marrakehi, A.

Martin, R. D.

R. D. Martin and C. P. McGath, "Robust detection of stochastic signals," IEEE Trans. Info. Theory IT-20, 537–541 (1974).
[CrossRef]

McGath, C. P.

R. D. Martin and C. P. McGath, "Robust detection of stochastic signals," IEEE Trans. Info. Theory IT-20, 537–541 (1974).
[CrossRef]

Partovi, A.

Pichon, L.

Pratt, W. K

W. K Pratt, Digital Image Processing, 2nd. ed. (Wiley-Interscience, New York, 1987), Chap. 10, p. 289.

Root, W. L.

W. B. Davenport, Jr., and W. L. Root, An Introduction to the Theory to Random Signals and Noise (McGraw-Hill, New York, 1958), Chap. 13, pp. 277–311.

Schwartz, M.

M. Schwartz, W. Bennett, and S. Stein, Communication Systems and Techniques (McGraw-Hill, New York, 1966), Chap. 4, pp. 213–216; Chap. 6, p. 247.

Segev, M.

Sicuranza, G. L.

G. L. Sicuranza, "Quadratic filter for signal processing," Proc. IEEE, 80, 1263–1285 (1992).
[CrossRef]

Stark, H.

H. Stark, F. B. Tuteur, and J. B. Anderson, Modern Electrical Communication, Analog, Digital and Optical Systems, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1988), Chap. 4, pp. 175–178; Chap. 6, pp. 247–248.

Stein, S.

M. Schwartz, W. Bennett, and S. Stein, Communication Systems and Techniques (McGraw-Hill, New York, 1966), Chap. 4, pp. 213–216; Chap. 6, p. 247.

Sternklar, S.

Tuteur, F. B.

H. Stark, F. B. Tuteur, and J. B. Anderson, Modern Electrical Communication, Analog, Digital and Optical Systems, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1988), Chap. 4, pp. 175–178; Chap. 6, pp. 247–248.

Vachss, F.

Vahey, D. W.

D. W. Vahey, "A nonlinear coupled wave theory of holographic storage in ferroelectric materials," J. Appl. Phys. 46, 3510–3515 (1975).
[CrossRef]

Vainos, N. A.

Weiss, S.

Woods, C.

J. Khoury, M. Cronin-Golomb, and C. Woods, "Photorefractive deamplification for additive signal dependent noise reduction," Opt. Eng. 32, 2877–2833 (1993).
[CrossRef]

Woods, C. L.

J. Khoury, A. M. Biernacki, C. L. Woods, and M. Cronin-Golomb, "Photorefractive quadratic processor for converting multiplicative noise-to-additive noise," Opt. Eng. 32, 2872–2876 (1993).
[CrossRef]

J. Khoury, C. L. Woods, and M. Cronin-Golomb, "Noise reduction using adaptive spatial filtering in photorefractive two-beam coupling," Opt. Lett. 16, 747–749 (1991).
[CrossRef] [PubMed]

J. Khoury, M. Cronin-Golomb, A. M. Biernacki, and C. L. Woods, "Photorefractive phase-conjugate techniques for measuring surface granularity," Appl. Opt. (to be published).

Yeh, P.

Appl. Opt. (2)

IEEE J. Quantum Electron. (1)

P. Yeh, "Photorefractive phase conjugators," IEEE J. Quantum Electron. 80, 436–450 (1992).

IEEE Trans. Info. Theory (1)

R. D. Martin and C. P. McGath, "Robust detection of stochastic signals," IEEE Trans. Info. Theory IT-20, 537–541 (1974).
[CrossRef]

J. Appl. Phys. (1)

D. W. Vahey, "A nonlinear coupled wave theory of holographic storage in ferroelectric materials," J. Appl. Phys. 46, 3510–3515 (1975).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Commun. (2)

J. A. Khoury, G. Hussain, and R. W. Eason, "Contrast manipulation and controllable spatial filtering via photore-fractive two-beam coupling," Opt. Commun. 70, 272–276 (1989).
[CrossRef]

M. Cronin-Golomb, "Whole beam method for photorefractive nonlinear optics," Opt. Commun. 89, 276–282 (1992).
[CrossRef]

Opt. Eng. (3)

J. Khoury, A. M. Biernacki, C. L. Woods, and M. Cronin-Golomb, "Photorefractive quadratic processor for converting multiplicative noise-to-additive noise," Opt. Eng. 32, 2872–2876 (1993).
[CrossRef]

Y. Fainman, E. Klanick, and S. H. Lee, "Optimal coherent image amplification by two-beam coupling in photorefractive BaTiO3," Opt. Eng. 25, 228–234 (1986).
[CrossRef]

J. Khoury, M. Cronin-Golomb, and C. Woods, "Photorefractive deamplification for additive signal dependent noise reduction," Opt. Eng. 32, 2877–2833 (1993).
[CrossRef]

Opt. Lett. (7)

Other (8)

W. K Pratt, Digital Image Processing, 2nd. ed. (Wiley-Interscience, New York, 1987), Chap. 10, p. 289.

M. Schwartz, W. Bennett, and S. Stein, Communication Systems and Techniques (McGraw-Hill, New York, 1966), Chap. 4, pp. 213–216; Chap. 6, p. 247.

M. Cronin-Golomb, "Large nonlinearities in four-wave mixing in photorefractive crystals and application in passive optical phase conjugation," Ph.D. dissertation (California Institute of Technology, Pasadena, Calif., 1983).

H. Stark, F. B. Tuteur, and J. B. Anderson, Modern Electrical Communication, Analog, Digital and Optical Systems, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1988), Chap. 4, pp. 175–178; Chap. 6, pp. 247–248.

G. L. Sicuranza, "Quadratic filter for signal processing," Proc. IEEE, 80, 1263–1285 (1992).
[CrossRef]

J. W. Goodman "Statistical properties of laser speckle pattern," in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed., Vol. 9 of Springer Series on Topics in Applied Physics (Springer-Verlag, Berlin, 1984), p. 40.

J. Khoury, M. Cronin-Golomb, A. M. Biernacki, and C. L. Woods, "Photorefractive phase-conjugate techniques for measuring surface granularity," Appl. Opt. (to be published).

W. B. Davenport, Jr., and W. L. Root, An Introduction to the Theory to Random Signals and Noise (McGraw-Hill, New York, 1958), Chap. 13, pp. 277–311.

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Figures (10)

Fig. 1
Fig. 1

Rectification of a sinusoidal input A cos(x). The dashed and the solid curves correspond to values of m = 103 and m = 104, respectively. The intensity of the signal beam A2 is normalized to unity. In these simulations we also used Γl = 5.

Fig. 2
Fig. 2

Results of a beam-cleanup experiment displayed in the Fourier-transform plane. The Gaussian input beam was corrupted by random multiplicative speckle noise.

Fig. 3
Fig. 3

Plot of rectification cleanup efficiency as a function of signal-to-reference intensity ratio m for four different values of Γl = g: (a) linear plot, (b) logarithmic plot.

Fig. 4
Fig. 4

Schematic diagram of the experimental arrangement used for noise conversion. BS, beam splitter; D, diffusing plate; O, object; L1 − L3, lenses.

Fig. 5
Fig. 5

Experimental results for A, clean input information; B, the Fourier transform of A; C, the noisy information; D, the Fourier transform of C; E, the noisy information after optical rectification; and F, the Fourier transformation of E.

Fig. 6
Fig. 6

(a) Plot of the absolute values of amplitude of the three noisy bars; (b) the Fourier transform of (a); (c) the amplitudes of the noisy bars after quadratic operator is imposed on the input data; (d) the absolute values of the squared amplitudes of the three noisy bars; (e) the noisy bars after they are operated on by the rectification nonlinearity of photorefractive two-beam coupling; and (f) the absolute value of the Fourier transform of (e).

Fig. 7
Fig. 7

Sample of the data of Fig. 6, showing the cross sections in the image plane and the diagonal sections in the Fourier plane.

Fig. 8
Fig. 8

Linear (top) and log–log (bottom) plots of the SNR as a function of m (the beam intensity ratio) for two different values of coupling coefficient Γ.

Fig. 9
Fig. 9

Block diagram outlining the steps in the general technique for reducing multiplicative noise.

Fig. 10
Fig. 10

Plot of F1(ν) as a function of ν, the order nonlinearity.

Equations (38)

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

d A 1 d z = - γ I 0 A 1 A 2 2 - α 2 A 1 ,
d A 2 * d z = γ I 0 A 2 * A 1 2 - α 2 A 2 * ,
A 1 ( x , z ) = A 1 ( x , 0 ) [ 1 + m ( x ) - 1 1 + m ( x ) - 1 exp ( Γ z ) ] 1 / 2 exp ( - α 2 z ) ,
A 2 ( x , z ) = A 2 ( x , 0 ) [ 1 + m ( x ) 1 + m ( x ) exp ( - Γ z ) ] 1 / 2 exp ( - α 2 z ) ,
m = | A 1 ( 0 ) A 2 ( 0 ) | 2 .
A 2 ( z ) = A 2 ( 0 ) f ( s ) exp ( - α 2 z ) ,
f ( s ) = ( 1 + m s 2 1 + m b s 2 ) 1 / 2 ,
f ( s ) = 1 + ( - ½ m b + ½ m ) s 2 = 1 + ( m γ l s 2 ) .
A 2 ( z ) = A 1 ( 0 ) sin [ u ( m , Γ z ) ] ,
u ( m , Γ z ) = { 2 m [ 1 + cosh ( m - 1 ) ] ( 1 + m ) 2 } 1 / 2 × [ tan - 1 exp ( Γ z 2 + 1 2 m ) - tan - 1 exp ( 1 2 m ) ] .
η t ( z ) = A 2 ( z ) 2 A 1 ( 0 ) 2 = sin 2 [ u ( m , Γ z ) ] .
η c = A 2 ( u ) 2 ¯ m n 2 ¯ = Mean of deflected output Mean of the input ,
f ( s ) = i = 0 a i s 2 i .
A out = c i = 0 a i s n 2 i = c i = 0 a i s 2 i n 2 i = c i = 0 a i s 2 i ( b 2 i + N 2 i ) = c [ a 0 + i = 1 a i i ! ( 2 σ s 2 ) i + i = 0 a i s 2 i N 2 i ] ,
η s = ( a i i 2 σ s i ) 2 m ( n s ) 2 .
s 2 > [ 2 + 3 exp ( Γ l ) ] 1 / 2 - 1 m .
s 2 < [ 2 + 3 exp ( Γ l ) ] 1 / 2 - 1 m .
( S N ) 0 F 1 ( ν ) ( S N ) i 2 ,
F 1 ( ν ) = 1 Γ m 2 ( ν 2 ) k = 2 1 ( k 2 ) ! Γ m 2 ( 1 - k - ν 2 ) ,
( S N ) 0 2 ν 2 ( S N ) i .
d A 1 d z = - C A 2 ,
d A 2 * d z = C A 1 ,
d A 1 d u = - A 2 ,
d A 2 * d u = A 1 .
A 1 ( u ) = A 1 ( 0 ) cos ( u ) - A 2 ( 0 ) sin ( u ) ,
A 2 ( u ) = A 2 ( 0 ) cos ( u ) + A 1 ( 0 ) sin ( u ) .
A 2 ( u ) = A 1 ( 0 ) sin ( u ) .
A 2 ( u ) = A 1 ( 0 ) sin ( u ) .
d C d z = C γ ( I 1 - I 2 ) I 0 .
C ( z ) = C ( z = 0 ) [ 1 + cosh Ψ 1 + cosh ( 2 γ z + Ψ ) ] 1 / 2 ,
C ( z = 0 ) = γ [ I 1 ( 0 ) I 2 ( 0 ) ] 1 / 2 I 0 ,
u = [ 2 I 1 I 2 ( 1 + cosh Ψ ) ] 1 / 2 I 0 × { tan - 1 [ exp ( γ z + Ψ 2 ) ] - tan - 1 [ exp ( Ψ 2 ) ] } .
u ( m , Γ z ) = [ 2 m ( 1 + m ) 2 ( 1 + cosh ( m - 1 ) ] 1 / 2 × { tan - 1 [ exp ( Γ z 2 + 1 2 m ) ] - tan - 1 [ exp ( 1 2 m ) ] } ,
η t = A 2 [ u ( 1 ) ] 2 A 1 ( 0 ) 2 = sin [ u ( 1 ) ] 2 .
n 2 i = i ! ( 2 σ 2 ) i = i ! n 2 i ,
F ( n 2 ) = b 2 δ ( 0 , 0 ) + N 2 ( ν x , ν y )
n 2 = b 2 + N 2 ,
N 2 i = b 2 i + N 2 i ,

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