Abstract

Based on a multiwave mixing model, we explain the symmetrical and broad light-induced scattering effect in LiNbO3:Fe crystal sheets. Analytical results are given in the undepleted-pumps approximation. The competition between forward small-angle scattering and climbing lights and its relation to the ratio of the size of the incident-light spot to the crystal thickness are discussed. The intensity angle distribution of the forward small-angle light-induced scattering is studied theoretically and experimentally. We suggest that the self-defocusing effect of the LiNbO3:Fe crystal plays an important role in the formation of the fanning intensity angle distribution.

© 1997 Optical Society of America

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  1. E. Parshall, M. Cronin-Golomb, and R. Barakat, “Model of amplified scattering in photorefractive media: comparison of numerical results and experiment,” Opt. Lett. 20, 432–434 (1995).
    [Crossref] [PubMed]
  2. M. Segev, D. Engin, and A. Yariv, “Temporal evolution of fanning in photorefractive materials,” Opt. Lett. 18, 956–958 (1993).
    [Crossref] [PubMed]
  3. Y.-H. Hong, P. Xie, J.-H. Dai, Y. Zhu, H.-G. Yang, and H.-J. Zhang, “Fanning effects in photorefractive crystals,” Opt. Lett. 18, 772–774 (1993).
    [Crossref] [PubMed]
  4. P. P. Banerjee and R. M. Misra, “Dependence of photorefractive beam fanning on beam parameters,” Opt. Commun. 100, 166–172 (1993).
    [Crossref]
  5. W. P. Brown and G. C. Valley, “Kinky beam paths inside photorefractive crystals,” J. Opt. Soc. Am. B 101901–1906 (1993).
    [Crossref]
  6. P. Yeh, Introduction to Photorefractive Nonlinear Optics, Wiley Series in Pure and Applied Optics (Wiley, New York, 1993), Chap. 4, pp. 118–134, 150–177.
  7. I. C. Khoo and T. H. Liu, “Theory and experiments on multiwave-mixing-mediated probe-beam amplification,” Phys. Rev. A 39, 4036–4044 (1989).
    [Crossref] [PubMed]
  8. F. Sanchez, “Two-wave mixing in thin nonlinear local-response media: a simple theoretical model,” J. Opt. Soc. Am. B 9, 2196–2205 (1992).
    [Crossref]
  9. L. B. Au and L. Solymar, “Amplification in photorefractive materials via a higher order wave,” Appl. Phys. B 45, 125–128 (1988).
    [Crossref]
  10. F. Jermann and K. Buse, “Light-induced thermal gratings in LiNbO3:Fe,” Appl. Phys. B 59, 437–443 (1994).
    [Crossref]
  11. Liu Simin, Xu Jingjun, Zhang Guangyin, and Wu Yuanqing, “Light-climbing effect in LiNbO3:Fe crystal,” Appl. Opt. 33, 997–999 (1994).
    [Crossref] [PubMed]
  12. G. Zhang, Y. Wu, S. Liu, and J. Wang, “Light-climbing effect in thin LiNbO3:Fe wafers,” Chin. Phys. Lasers 14, 606–609 (1987).
  13. M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
    [Crossref]
  14. Q. Wang Song, C.-p. Zhang, and P. J. Talbot, “Self-defocusing, self-focusing, and speckle in LiNbO3 and LiNbO3:Fe crystals,” Appl. Opt. 32, 7266–7271 (1993).
    [Crossref]
  15. A. A. Zozulya and D. Z. Anderson, “Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field,” Phys. Rev. A 51, 1520–1531 (1995).
    [Crossref] [PubMed]
  16. P. A. Augustov and K. K. Shvarts, “Surface recombination and photorefraction in LiNbO3-Fe crystals,” Appl. Phys. 18, 399–401 (1979).
    [Crossref]
  17. K. K. Shvarts, P. A. Augustov, A. O. Ozols, and A. K. Popelis, “Photorefraction kinetics in LiNbO3 crystals under irradiation and heating,” Ferroelectrics 22, 655–657 (1978).
    [Crossref]
  18. T. R. Volk, N. V. Razumovski, A. V. Mamaev, and N. M. Rubinina, “Hologram recording in Zn-doped LiNbO3 crystals,” J. Opt. Soc. Am. B 13, 1457–1460 (1996).
    [Crossref]
  19. Guangyin Zhang, Jingjun Xu, Simin Liu, Qian Sun, Guoquan Zhang, Qiyin Fang, and Chaoli Ma, “Study of resistance against photorefractive light-induced scattering in LiNbO3:Fe, Mg crystals,” in Photorefractive Fiber and Crystal Devices: Materials, Optical Properties, and Applications, F. T. S. Yu, ed., Proc. SPIE, 2529, 14–17 (1995).
    [Crossref]

1996 (1)

1995 (2)

A. A. Zozulya and D. Z. Anderson, “Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field,” Phys. Rev. A 51, 1520–1531 (1995).
[Crossref] [PubMed]

E. Parshall, M. Cronin-Golomb, and R. Barakat, “Model of amplified scattering in photorefractive media: comparison of numerical results and experiment,” Opt. Lett. 20, 432–434 (1995).
[Crossref] [PubMed]

1994 (2)

F. Jermann and K. Buse, “Light-induced thermal gratings in LiNbO3:Fe,” Appl. Phys. B 59, 437–443 (1994).
[Crossref]

Liu Simin, Xu Jingjun, Zhang Guangyin, and Wu Yuanqing, “Light-climbing effect in LiNbO3:Fe crystal,” Appl. Opt. 33, 997–999 (1994).
[Crossref] [PubMed]

1993 (5)

1992 (1)

1989 (1)

I. C. Khoo and T. H. Liu, “Theory and experiments on multiwave-mixing-mediated probe-beam amplification,” Phys. Rev. A 39, 4036–4044 (1989).
[Crossref] [PubMed]

1988 (1)

L. B. Au and L. Solymar, “Amplification in photorefractive materials via a higher order wave,” Appl. Phys. B 45, 125–128 (1988).
[Crossref]

1987 (1)

G. Zhang, Y. Wu, S. Liu, and J. Wang, “Light-climbing effect in thin LiNbO3:Fe wafers,” Chin. Phys. Lasers 14, 606–609 (1987).

1984 (1)

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[Crossref]

1979 (1)

P. A. Augustov and K. K. Shvarts, “Surface recombination and photorefraction in LiNbO3-Fe crystals,” Appl. Phys. 18, 399–401 (1979).
[Crossref]

1978 (1)

K. K. Shvarts, P. A. Augustov, A. O. Ozols, and A. K. Popelis, “Photorefraction kinetics in LiNbO3 crystals under irradiation and heating,” Ferroelectrics 22, 655–657 (1978).
[Crossref]

Anderson, D. Z.

A. A. Zozulya and D. Z. Anderson, “Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field,” Phys. Rev. A 51, 1520–1531 (1995).
[Crossref] [PubMed]

Au, L. B.

L. B. Au and L. Solymar, “Amplification in photorefractive materials via a higher order wave,” Appl. Phys. B 45, 125–128 (1988).
[Crossref]

Augustov, P. A.

P. A. Augustov and K. K. Shvarts, “Surface recombination and photorefraction in LiNbO3-Fe crystals,” Appl. Phys. 18, 399–401 (1979).
[Crossref]

K. K. Shvarts, P. A. Augustov, A. O. Ozols, and A. K. Popelis, “Photorefraction kinetics in LiNbO3 crystals under irradiation and heating,” Ferroelectrics 22, 655–657 (1978).
[Crossref]

Banerjee, P. P.

P. P. Banerjee and R. M. Misra, “Dependence of photorefractive beam fanning on beam parameters,” Opt. Commun. 100, 166–172 (1993).
[Crossref]

Barakat, R.

Brown, W. P.

Buse, K.

F. Jermann and K. Buse, “Light-induced thermal gratings in LiNbO3:Fe,” Appl. Phys. B 59, 437–443 (1994).
[Crossref]

Cronin-Golomb, M.

E. Parshall, M. Cronin-Golomb, and R. Barakat, “Model of amplified scattering in photorefractive media: comparison of numerical results and experiment,” Opt. Lett. 20, 432–434 (1995).
[Crossref] [PubMed]

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[Crossref]

Dai, J.-H.

Engin, D.

Fang, Qiyin

Guangyin Zhang, Jingjun Xu, Simin Liu, Qian Sun, Guoquan Zhang, Qiyin Fang, and Chaoli Ma, “Study of resistance against photorefractive light-induced scattering in LiNbO3:Fe, Mg crystals,” in Photorefractive Fiber and Crystal Devices: Materials, Optical Properties, and Applications, F. T. S. Yu, ed., Proc. SPIE, 2529, 14–17 (1995).
[Crossref]

Fischer, B.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[Crossref]

Guangyin, Zhang

Hong, Y.-H.

Jermann, F.

F. Jermann and K. Buse, “Light-induced thermal gratings in LiNbO3:Fe,” Appl. Phys. B 59, 437–443 (1994).
[Crossref]

Jingjun, Xu

Khoo, I. C.

I. C. Khoo and T. H. Liu, “Theory and experiments on multiwave-mixing-mediated probe-beam amplification,” Phys. Rev. A 39, 4036–4044 (1989).
[Crossref] [PubMed]

Liu, S.

G. Zhang, Y. Wu, S. Liu, and J. Wang, “Light-climbing effect in thin LiNbO3:Fe wafers,” Chin. Phys. Lasers 14, 606–609 (1987).

Liu, Simin

Guangyin Zhang, Jingjun Xu, Simin Liu, Qian Sun, Guoquan Zhang, Qiyin Fang, and Chaoli Ma, “Study of resistance against photorefractive light-induced scattering in LiNbO3:Fe, Mg crystals,” in Photorefractive Fiber and Crystal Devices: Materials, Optical Properties, and Applications, F. T. S. Yu, ed., Proc. SPIE, 2529, 14–17 (1995).
[Crossref]

Liu, T. H.

I. C. Khoo and T. H. Liu, “Theory and experiments on multiwave-mixing-mediated probe-beam amplification,” Phys. Rev. A 39, 4036–4044 (1989).
[Crossref] [PubMed]

Ma, Chaoli

Guangyin Zhang, Jingjun Xu, Simin Liu, Qian Sun, Guoquan Zhang, Qiyin Fang, and Chaoli Ma, “Study of resistance against photorefractive light-induced scattering in LiNbO3:Fe, Mg crystals,” in Photorefractive Fiber and Crystal Devices: Materials, Optical Properties, and Applications, F. T. S. Yu, ed., Proc. SPIE, 2529, 14–17 (1995).
[Crossref]

Mamaev, A. V.

Misra, R. M.

P. P. Banerjee and R. M. Misra, “Dependence of photorefractive beam fanning on beam parameters,” Opt. Commun. 100, 166–172 (1993).
[Crossref]

Ozols, A. O.

K. K. Shvarts, P. A. Augustov, A. O. Ozols, and A. K. Popelis, “Photorefraction kinetics in LiNbO3 crystals under irradiation and heating,” Ferroelectrics 22, 655–657 (1978).
[Crossref]

Parshall, E.

Popelis, A. K.

K. K. Shvarts, P. A. Augustov, A. O. Ozols, and A. K. Popelis, “Photorefraction kinetics in LiNbO3 crystals under irradiation and heating,” Ferroelectrics 22, 655–657 (1978).
[Crossref]

Razumovski, N. V.

Rubinina, N. M.

Sanchez, F.

Segev, M.

Shvarts, K. K.

P. A. Augustov and K. K. Shvarts, “Surface recombination and photorefraction in LiNbO3-Fe crystals,” Appl. Phys. 18, 399–401 (1979).
[Crossref]

K. K. Shvarts, P. A. Augustov, A. O. Ozols, and A. K. Popelis, “Photorefraction kinetics in LiNbO3 crystals under irradiation and heating,” Ferroelectrics 22, 655–657 (1978).
[Crossref]

Simin, Liu

Solymar, L.

L. B. Au and L. Solymar, “Amplification in photorefractive materials via a higher order wave,” Appl. Phys. B 45, 125–128 (1988).
[Crossref]

Sun, Qian

Guangyin Zhang, Jingjun Xu, Simin Liu, Qian Sun, Guoquan Zhang, Qiyin Fang, and Chaoli Ma, “Study of resistance against photorefractive light-induced scattering in LiNbO3:Fe, Mg crystals,” in Photorefractive Fiber and Crystal Devices: Materials, Optical Properties, and Applications, F. T. S. Yu, ed., Proc. SPIE, 2529, 14–17 (1995).
[Crossref]

Talbot, P. J.

Valley, G. C.

Volk, T. R.

Wang, J.

G. Zhang, Y. Wu, S. Liu, and J. Wang, “Light-climbing effect in thin LiNbO3:Fe wafers,” Chin. Phys. Lasers 14, 606–609 (1987).

Wang Song, Q.

White, J. O.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[Crossref]

Wu, Y.

G. Zhang, Y. Wu, S. Liu, and J. Wang, “Light-climbing effect in thin LiNbO3:Fe wafers,” Chin. Phys. Lasers 14, 606–609 (1987).

Xie, P.

Xu, Jingjun

Guangyin Zhang, Jingjun Xu, Simin Liu, Qian Sun, Guoquan Zhang, Qiyin Fang, and Chaoli Ma, “Study of resistance against photorefractive light-induced scattering in LiNbO3:Fe, Mg crystals,” in Photorefractive Fiber and Crystal Devices: Materials, Optical Properties, and Applications, F. T. S. Yu, ed., Proc. SPIE, 2529, 14–17 (1995).
[Crossref]

Yang, H.-G.

Yariv, A.

M. Segev, D. Engin, and A. Yariv, “Temporal evolution of fanning in photorefractive materials,” Opt. Lett. 18, 956–958 (1993).
[Crossref] [PubMed]

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[Crossref]

Yeh, P.

P. Yeh, Introduction to Photorefractive Nonlinear Optics, Wiley Series in Pure and Applied Optics (Wiley, New York, 1993), Chap. 4, pp. 118–134, 150–177.

Yuanqing, Wu

Zhang, C.-p.

Zhang, G.

G. Zhang, Y. Wu, S. Liu, and J. Wang, “Light-climbing effect in thin LiNbO3:Fe wafers,” Chin. Phys. Lasers 14, 606–609 (1987).

Zhang, Guangyin

Guangyin Zhang, Jingjun Xu, Simin Liu, Qian Sun, Guoquan Zhang, Qiyin Fang, and Chaoli Ma, “Study of resistance against photorefractive light-induced scattering in LiNbO3:Fe, Mg crystals,” in Photorefractive Fiber and Crystal Devices: Materials, Optical Properties, and Applications, F. T. S. Yu, ed., Proc. SPIE, 2529, 14–17 (1995).
[Crossref]

Zhang, Guoquan

Guangyin Zhang, Jingjun Xu, Simin Liu, Qian Sun, Guoquan Zhang, Qiyin Fang, and Chaoli Ma, “Study of resistance against photorefractive light-induced scattering in LiNbO3:Fe, Mg crystals,” in Photorefractive Fiber and Crystal Devices: Materials, Optical Properties, and Applications, F. T. S. Yu, ed., Proc. SPIE, 2529, 14–17 (1995).
[Crossref]

Zhang, H.-J.

Zhu, Y.

Zozulya, A. A.

A. A. Zozulya and D. Z. Anderson, “Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field,” Phys. Rev. A 51, 1520–1531 (1995).
[Crossref] [PubMed]

Appl. Opt. (2)

Appl. Phys. (1)

P. A. Augustov and K. K. Shvarts, “Surface recombination and photorefraction in LiNbO3-Fe crystals,” Appl. Phys. 18, 399–401 (1979).
[Crossref]

Appl. Phys. B (2)

L. B. Au and L. Solymar, “Amplification in photorefractive materials via a higher order wave,” Appl. Phys. B 45, 125–128 (1988).
[Crossref]

F. Jermann and K. Buse, “Light-induced thermal gratings in LiNbO3:Fe,” Appl. Phys. B 59, 437–443 (1994).
[Crossref]

Chin. Phys. Lasers (1)

G. Zhang, Y. Wu, S. Liu, and J. Wang, “Light-climbing effect in thin LiNbO3:Fe wafers,” Chin. Phys. Lasers 14, 606–609 (1987).

Ferroelectrics (1)

K. K. Shvarts, P. A. Augustov, A. O. Ozols, and A. K. Popelis, “Photorefraction kinetics in LiNbO3 crystals under irradiation and heating,” Ferroelectrics 22, 655–657 (1978).
[Crossref]

IEEE J. Quantum Electron. (1)

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[Crossref]

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

Opt. Commun. (1)

P. P. Banerjee and R. M. Misra, “Dependence of photorefractive beam fanning on beam parameters,” Opt. Commun. 100, 166–172 (1993).
[Crossref]

Opt. Lett. (3)

Phys. Rev. A (2)

I. C. Khoo and T. H. Liu, “Theory and experiments on multiwave-mixing-mediated probe-beam amplification,” Phys. Rev. A 39, 4036–4044 (1989).
[Crossref] [PubMed]

A. A. Zozulya and D. Z. Anderson, “Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field,” Phys. Rev. A 51, 1520–1531 (1995).
[Crossref] [PubMed]

Other (2)

Guangyin Zhang, Jingjun Xu, Simin Liu, Qian Sun, Guoquan Zhang, Qiyin Fang, and Chaoli Ma, “Study of resistance against photorefractive light-induced scattering in LiNbO3:Fe, Mg crystals,” in Photorefractive Fiber and Crystal Devices: Materials, Optical Properties, and Applications, F. T. S. Yu, ed., Proc. SPIE, 2529, 14–17 (1995).
[Crossref]

P. Yeh, Introduction to Photorefractive Nonlinear Optics, Wiley Series in Pure and Applied Optics (Wiley, New York, 1993), Chap. 4, pp. 118–134, 150–177.

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Figures (4)

Fig. 1
Fig. 1

Wave-vector diagram of the model that we have suggested. k1 and k2 are the wave vectors of the incident light and the reflected light from the second interface of the crystal, respectively. k3, k5, k4, and k6 are the wave vectors of light-induced symmetrical scattering. θ is the angle in the crystal between pump beam Ip and its scattering.

Fig. 2
Fig. 2

Schematic of the experimental setup used in our experiments: S, beam splitter; M, mirror; B, block; D, detector. The dashed circle shows the places where the detector measures the intensity distribution of scatterings.

Fig. 3
Fig. 3

Intensity angle distribution of the forward scattering of a LiNbO3:Fe (0.07 wt. % Fe) crystal with 1-mm thickness. The power of the incident light is 15 mW, the size of the incident pump beam is 1 mm, and the pump light is incident normally onto the LiNbO3:Fe crystal surface and perpendicularly to the crystalline c axis. The measured intensity is in arbitrary units.

Fig. 4
Fig. 4

Intensity angle distribution of the forward scattering I3(l) [or I5(l)]: (a) The dashed curve shows the results calculated from Eqs. (34) and (35), the solid curve shows the results calculated through three-wave interaction model by Eqs. (A4)–(A7), and the squares denote the measured results. (b) Results calculated from Eqs. (34) and (35) when the value of the photovoltaic field is 1.0×106 V/m and the other parameters are unchanged. The angles in(a) are in air, and the angles in (b) are in crystal. In all calculations the ratio dI/d is assumed to be ∞, and the calculated intensity is in arbitrary units. The parameters used in these calculations are listed in Table 1.

Tables (1)

Tables Icon

Table 1 Parameters Used to Calculate the Intensity Angle Distribution of Forward Scatteringa

Equations (53)

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

Ej(r, t)=Aj(r)exp[i(kj·r-ωt)]+c.c.
n=n0+nI2exp(-iϕI)(A1A3*+A4A2*)I0×exp[i(k1-k3)·r]+c.c.+nII2exp(-iϕII)(A1A5*+A6A2*)I0×exp[i(k1-k5)·r]+c.c.,
I0=|Aj|2
2E+ω2c2n2E=0,
2ikA1z+ω2c2n0nI×exp(-iϕI)(A1A3*+A4A2*)I0A3+ω2c2n0nI×exp(iϕI)(A1*A3+A4*A2)I0A5 exp(-iδkz)+ω2c2n0nII exp(-iϕII)(A1A5*+A6A2*)I0A5+ω2c2n0nII exp(iϕII)(A1*A5+A6*A2)I0A3
×exp(-iδkz)=0,
-2ikA2z+ω2c2n0nI×exp(-iϕI)(A1A3*+A4A2*)I0A6×exp(iδkz)+ω2c2n0nI×exp(iϕI)(A1*A3+A4*A2)I0A4+ω2c2n0nII×exp(-iϕII)(A1A5*+A6A2*)I0A4×exp(iδkz)+ω2c2n0nII
×exp(iϕII)(A1*A5+A6*A2)I0A6=0,
A3z=-γ1*(A1*A3+A4*A2)I0A1+γ2(A1A5*+A6A2*)I0A1 exp(iδkz),
A4*z=-γ1*(A1*A3+A4*A2)I0A2*+γ2(A1A5*+A6A2*)I0A2* exp(iδkz),
A5*z=γ1*(A1*A3+A4*A2)I0A1* exp(-iδkz)-γ2(A1A5*+A6A2*)I0A1*,
A6z=γ1*(A1*A3+A4*A2)I0A2 exp(-iδkz)-γ2(A1A5*+A6A2*)I0A2,
γ1=iω2n0nI2kc2 cos θexp(-iϕI),
γ2=iω2n0nII2kc2 cos θexp(-iϕII).
A1A5*z+A1* exp(-iδkz)A3z=0,
A2*A6z+A2 exp(-iδkz)A4*z=0,
A2*A3z-A1A4*z=0,
A2A5*z-A1*A6z=0,
A1A6z+A2 exp(-iδkz)A3z=0,
A2*A5*z+A1* exp(-iδkz)A4*z=0.
2A3z2+(γ1*+γ2-iδk)A3z
-iδkγ1*A1*A3+A2A4*I0A1=0,
2A4*z2+(γ1*+γ2-iδk)A4*z
-iδkγ1*A1*A3+A2A4*I0A2*=0,
2A5*z2+(γ1*+γ2+iδk)A5*z
+iδkγ2A1A5*+A2*A6I0A1*=0,
2A6z2+(γ1*+γ2+iδk)A6z
+iδkγ2A1A5*+A2*A6I0A2=0.
A3z=iδkγ1*γ1*+γ2-iδkA1*A3+A2A4*I0A1-γ1*A1*A3+A2A4*I0A1,
A4*z=iδkγ1*γ1*+γ2-iδkA1*A3+A2A4*I0A2*-γ1*A1*A3+A2A4*I0A2*,
A5*z=-iδkγ2γ1*+γ2+iδkA1A5*+A2*A6I0A1*-γ2A1A5*+A2*A6I0A1*,
A6z=-iδkγ2γ1*+γ2+iδkA1A5*+A2*A6I0A2-γ2A1A5*+A2*A6I0A2.
2A3z2+(γ1*+γ2-iδk)A3z-iδkγ1*A3=0,
2A5*z2+(γ1*+γ2+iδk)A5*z+iδkγ2A5*=0,
2A4*z2+(γ1*+γ2-iδk)A4*z-iδkγ1*A2*A1A3=0,
2A6z2+(γ1*+γ2+iδk)A6z+iδkγ2A2A1*A5*=0.
A3(z)=[a exp(βz/2)+b exp(-βz/2)]×exp[-(γ1*+γ2-iδk)z/2],
A5*(z)=[c exp(βz/2)+d exp(-βz/2)]×exp[-(γ1*+γ2+iδk)z/2],
a=A1*A3(0)[(γ1*+γ2)2-(β-iδk)2]-A1A5*(0)[β2-(γ1*+γ2+iδk)2]4iβδkA1*,
b=A1*A3(0)[(β+iδk)2-(γ1*+γ2)2]+A1A5*(0)[β2-(γ1*+γ2+iδk)2]4iβδkA1*,
c=A1A5*(0)[(β+iδk)2-(γ1*+γ2)2]+A1*A3(0)[β2-(γ1*+γ2-iδk)2]4iβδkA1,
d=A1A5*(0)[(γ1*+γ2)2-(β-iδk)2]-A1*A3(0)[β2-(γ1*+γ2-iδk)2]4iβδkA1,
β=[(γ1*+γ2)2-δk2+2iδk(γ1*-γ2)]1/2.
γ1=γ2=χ+iσ,
I3(z)=I5(z)=I3(0)(β2-4χ2)(δk2+4χ2-β2)2β2δk2+(4χ2-β2)2+δk2(β+2χ)24β2δk2×exp(βz)exp(-2χz)+I3(0)(4χ2-β2)2+δk2(β-2χ)24β2δk2×exp(-βz)exp(-2χz)
I3(z)=I5(z)=I3(0)×8χ2δk2+2|β|2δk2+2(4χ2+|β|2)24|β|2δk2+2χ|β|sin(|β|z)-(4χ2+|β|2)2+δk2(4χ2-|β|2)4|β|2δk2×cos(|β|z)exp(-2χz).
ni=-n032reffEsc,
reff=r33ne4 cos θ cos(θ/2)+n02ne2r51 sin θ sin(θ/2)n03ne
Esc=EqEd+EqEph2+Ed2
Φx=2σLI0(2I3 cos θ-I1)(1+cos Φ)+δkL,
I3x=-2σLI1I3I0sin Φ,
I1x=4σL cos θI1I3I0sin Φ,
I0=I1+2I3.

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