Abstract

We present numerical simulations of the complex grating structure that is generated when several longitudinal modes from a laser induce a self-pumped phase conjugator in a photorefractive barium titanate crystal. The results of the numerical analysis clearly show that the detuning curve of the generated grating structure is asymmetric with respect to the center wavelength of the laser that induced it. The asymmetric feedback to the laser, which is generated by diffraction in the gratings of the structure, initiates the frequency scanning of the laser. It is found that the material frequency dispersion of the barium titanate crystal causes the asymmetry and is the origin that initiates the scanning process. The theoretical predictions are in agreement with the reported experimental observations.

© 1999 Optical Society of America

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References

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  1. M. Cronin-Golomb and A. Yariv, “Self-induced frequency scanning and distributed Bragg reflection in semiconductor lasers with phase-conjugate feedback,” Opt. Lett. 11, 455 (1986).
    [CrossRef] [PubMed]
  2. F. C. Jahoda, P. G. Weber, and J. Feinberg, “Optical feedback, wavelength response, and interference effects of a self-pumped phase conjugation in BaTiO3,” Opt. Lett. 9, 362 (1984).
    [CrossRef] [PubMed]
  3. W. B. Whitten and J. M. Ramsey, “Self-scanning of a dye laser due to feedback from a BaTiO3 phase-conjugate reflector,” Opt. Lett. 9, 44 (1984).
    [CrossRef] [PubMed]
  4. J. Feinberg and G. D. Bacher, “Self-scanning of a continuous-wave dye laser having a phase-conjugating resonator cavity,” Opt. Lett. 9, 420 (1984).
    [CrossRef] [PubMed]
  5. J. M. Ramsey and W. B. Whitten, “Phase-conjugate feedback into a continuous-wave ring dye laser,” Opt. Lett. 10, 362 (1985).
    [CrossRef] [PubMed]
  6. M. Lobel, P. M. Petersen, and P. M. Johansen, “Suppressing self-induced frequency scanning of a phase conjugate diode laser array using counterbalance dispersion,” Appl. Phys. Lett. 72, 1263 (1998).
    [CrossRef]
  7. A. Shiratori and M. Obara, “Frequency-stable, narrow linewidth oscillation of red diode laser with phase-conjugate feedback using stimulated photorefractive backscattering,” Appl. Phys. Lett. 69, 1515 (1997).
    [CrossRef]
  8. J. Feinberg, “Self-pumped, continuous-wave phase conjugator using internal reflections,” Opt. Lett. 7, 486 (1982).
    [CrossRef] [PubMed]
  9. M. Cronin-Golomb, K. Y. Lau, and A. Yariv, “Infrared photorefractive passive phase conjugation with BaTiO3: demonstrations with GaAlAs and 1.09-μm Ar+ lasers,” Appl. Phys. Lett. 47, 567 (1985).
    [CrossRef]
  10. A. A. Zozulya, M. Saffman, and D. Anderson, “Double phase-conjugate mirror: convection and diffraction,” J. Opt. Soc. Am. B 12, 255 (1995).
    [CrossRef]
  11. A. A. Zozulya, M. Saffman, and D. Z. Anderson, “Propagation of light beams in photorefractive media: fanning, self-bending, and formation of self-pumped four-wave-mixing phase conjugation geometries,” Phys. Rev. Lett. 73, 818 (1994).
    [CrossRef] [PubMed]
  12. A. A. Zozulya, G. Montemezzani, and D. Z. Anderson, “Analysis of total-internal-reflection phase-conjugate mirror,” Phys. Rev. A 52, 4167 (1995).
    [CrossRef] [PubMed]
  13. P. Xie, J. Dai, P. Wang, and H. Zhang, “Self-pumped phase conjugation in photorefractive crystal: reflectivity and spatial fidelity,” Phys. Rev. A 55, 3092 (1997).
    [CrossRef]
  14. A. Shiratori and M. Obara, “Wavelength-stable, narrow-spectral-width oscillation of an AlGaInP diode laser coupled to a BaTiO3:Co stimulated photorefractive backscattering phase conjugator,” Appl. Phys. B 65, 329 (1997).
    [CrossRef]
  15. P. Lambelet, R. P. Salathe, M. H. Garrett, and D. Rytz, “Characterization of a photorefractive phase conjugator by optical low-coherence reflectometry,” Appl. Phys. Lett. 64, 1079 (1994).
    [CrossRef]
  16. K. R. MacDonald and J. Feinberg, “Theory of a self-pumped phase conjugator with two coupled interaction regions,” J. Opt. Soc. Am. 73, 548 (1983).
    [CrossRef]
  17. M. Delpino, T. Rauch, C. Denz, and M. Carrascosa, “Numerical simulation of the time evolution of photorefractive phase conjugate beams: multigrating operations,” Opt. Mater. 4, 326 (1995).
    [CrossRef]
  18. N. Kukhtarev, “Kinetics of hologram recording and erasure in electrooptic crystals,” Sov. Tech. Phys. Lett. 2, 438 (1976).
  19. R. S. Cudney, R. M. Pierce, G. D. Bacher, D. Mahgerefteh, and J. Feinberg, “Intensity dependence of the photogalvanic effect in barium titanate,” J. Opt. Soc. Am. B 9, 1704 (1992).
    [CrossRef]
  20. G. C. Valley and M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704 (1983).
    [CrossRef]
  21. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes (Cambridge U. Press, London, 1992).
  22. J. Yamanuchi, J. Shibayama, and H. Nakano, “Wide-angle propagating beam analysis based on the generalized Douglas scheme for variable coefficients,” Opt. Lett. 20, 7 (1995).
    [CrossRef]
  23. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909 (1969).
    [CrossRef]
  24. L. Solymar, D. J. Webb, and A. Grunnet-Jepsen, The Physics and Applications of Photorefractive Materials (Clarendon, Oxford, 1996).

1998 (1)

M. Lobel, P. M. Petersen, and P. M. Johansen, “Suppressing self-induced frequency scanning of a phase conjugate diode laser array using counterbalance dispersion,” Appl. Phys. Lett. 72, 1263 (1998).
[CrossRef]

1997 (3)

A. Shiratori and M. Obara, “Frequency-stable, narrow linewidth oscillation of red diode laser with phase-conjugate feedback using stimulated photorefractive backscattering,” Appl. Phys. Lett. 69, 1515 (1997).
[CrossRef]

P. Xie, J. Dai, P. Wang, and H. Zhang, “Self-pumped phase conjugation in photorefractive crystal: reflectivity and spatial fidelity,” Phys. Rev. A 55, 3092 (1997).
[CrossRef]

A. Shiratori and M. Obara, “Wavelength-stable, narrow-spectral-width oscillation of an AlGaInP diode laser coupled to a BaTiO3:Co stimulated photorefractive backscattering phase conjugator,” Appl. Phys. B 65, 329 (1997).
[CrossRef]

1995 (4)

M. Delpino, T. Rauch, C. Denz, and M. Carrascosa, “Numerical simulation of the time evolution of photorefractive phase conjugate beams: multigrating operations,” Opt. Mater. 4, 326 (1995).
[CrossRef]

A. A. Zozulya, G. Montemezzani, and D. Z. Anderson, “Analysis of total-internal-reflection phase-conjugate mirror,” Phys. Rev. A 52, 4167 (1995).
[CrossRef] [PubMed]

A. A. Zozulya, M. Saffman, and D. Anderson, “Double phase-conjugate mirror: convection and diffraction,” J. Opt. Soc. Am. B 12, 255 (1995).
[CrossRef]

J. Yamanuchi, J. Shibayama, and H. Nakano, “Wide-angle propagating beam analysis based on the generalized Douglas scheme for variable coefficients,” Opt. Lett. 20, 7 (1995).
[CrossRef]

1994 (2)

A. A. Zozulya, M. Saffman, and D. Z. Anderson, “Propagation of light beams in photorefractive media: fanning, self-bending, and formation of self-pumped four-wave-mixing phase conjugation geometries,” Phys. Rev. Lett. 73, 818 (1994).
[CrossRef] [PubMed]

P. Lambelet, R. P. Salathe, M. H. Garrett, and D. Rytz, “Characterization of a photorefractive phase conjugator by optical low-coherence reflectometry,” Appl. Phys. Lett. 64, 1079 (1994).
[CrossRef]

1992 (1)

1986 (1)

1985 (2)

J. M. Ramsey and W. B. Whitten, “Phase-conjugate feedback into a continuous-wave ring dye laser,” Opt. Lett. 10, 362 (1985).
[CrossRef] [PubMed]

M. Cronin-Golomb, K. Y. Lau, and A. Yariv, “Infrared photorefractive passive phase conjugation with BaTiO3: demonstrations with GaAlAs and 1.09-μm Ar+ lasers,” Appl. Phys. Lett. 47, 567 (1985).
[CrossRef]

1984 (3)

1983 (2)

G. C. Valley and M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704 (1983).
[CrossRef]

K. R. MacDonald and J. Feinberg, “Theory of a self-pumped phase conjugator with two coupled interaction regions,” J. Opt. Soc. Am. 73, 548 (1983).
[CrossRef]

1982 (1)

1976 (1)

N. Kukhtarev, “Kinetics of hologram recording and erasure in electrooptic crystals,” Sov. Tech. Phys. Lett. 2, 438 (1976).

1969 (1)

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

Anderson, D.

Anderson, D. Z.

A. A. Zozulya, G. Montemezzani, and D. Z. Anderson, “Analysis of total-internal-reflection phase-conjugate mirror,” Phys. Rev. A 52, 4167 (1995).
[CrossRef] [PubMed]

A. A. Zozulya, M. Saffman, and D. Z. Anderson, “Propagation of light beams in photorefractive media: fanning, self-bending, and formation of self-pumped four-wave-mixing phase conjugation geometries,” Phys. Rev. Lett. 73, 818 (1994).
[CrossRef] [PubMed]

Bacher, G. D.

Carrascosa, M.

M. Delpino, T. Rauch, C. Denz, and M. Carrascosa, “Numerical simulation of the time evolution of photorefractive phase conjugate beams: multigrating operations,” Opt. Mater. 4, 326 (1995).
[CrossRef]

Cronin-Golomb, M.

M. Cronin-Golomb and A. Yariv, “Self-induced frequency scanning and distributed Bragg reflection in semiconductor lasers with phase-conjugate feedback,” Opt. Lett. 11, 455 (1986).
[CrossRef] [PubMed]

M. Cronin-Golomb, K. Y. Lau, and A. Yariv, “Infrared photorefractive passive phase conjugation with BaTiO3: demonstrations with GaAlAs and 1.09-μm Ar+ lasers,” Appl. Phys. Lett. 47, 567 (1985).
[CrossRef]

Cudney, R. S.

Dai, J.

P. Xie, J. Dai, P. Wang, and H. Zhang, “Self-pumped phase conjugation in photorefractive crystal: reflectivity and spatial fidelity,” Phys. Rev. A 55, 3092 (1997).
[CrossRef]

Delpino, M.

M. Delpino, T. Rauch, C. Denz, and M. Carrascosa, “Numerical simulation of the time evolution of photorefractive phase conjugate beams: multigrating operations,” Opt. Mater. 4, 326 (1995).
[CrossRef]

Denz, C.

M. Delpino, T. Rauch, C. Denz, and M. Carrascosa, “Numerical simulation of the time evolution of photorefractive phase conjugate beams: multigrating operations,” Opt. Mater. 4, 326 (1995).
[CrossRef]

Feinberg, J.

Garrett, M. H.

P. Lambelet, R. P. Salathe, M. H. Garrett, and D. Rytz, “Characterization of a photorefractive phase conjugator by optical low-coherence reflectometry,” Appl. Phys. Lett. 64, 1079 (1994).
[CrossRef]

Jahoda, F. C.

Johansen, P. M.

M. Lobel, P. M. Petersen, and P. M. Johansen, “Suppressing self-induced frequency scanning of a phase conjugate diode laser array using counterbalance dispersion,” Appl. Phys. Lett. 72, 1263 (1998).
[CrossRef]

Klein, M. B.

G. C. Valley and M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704 (1983).
[CrossRef]

Kogelnik, H.

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

Kukhtarev, N.

N. Kukhtarev, “Kinetics of hologram recording and erasure in electrooptic crystals,” Sov. Tech. Phys. Lett. 2, 438 (1976).

Lambelet, P.

P. Lambelet, R. P. Salathe, M. H. Garrett, and D. Rytz, “Characterization of a photorefractive phase conjugator by optical low-coherence reflectometry,” Appl. Phys. Lett. 64, 1079 (1994).
[CrossRef]

Lau, K. Y.

M. Cronin-Golomb, K. Y. Lau, and A. Yariv, “Infrared photorefractive passive phase conjugation with BaTiO3: demonstrations with GaAlAs and 1.09-μm Ar+ lasers,” Appl. Phys. Lett. 47, 567 (1985).
[CrossRef]

Lobel, M.

M. Lobel, P. M. Petersen, and P. M. Johansen, “Suppressing self-induced frequency scanning of a phase conjugate diode laser array using counterbalance dispersion,” Appl. Phys. Lett. 72, 1263 (1998).
[CrossRef]

MacDonald, K. R.

Mahgerefteh, D.

Montemezzani, G.

A. A. Zozulya, G. Montemezzani, and D. Z. Anderson, “Analysis of total-internal-reflection phase-conjugate mirror,” Phys. Rev. A 52, 4167 (1995).
[CrossRef] [PubMed]

Nakano, H.

Obara, M.

A. Shiratori and M. Obara, “Frequency-stable, narrow linewidth oscillation of red diode laser with phase-conjugate feedback using stimulated photorefractive backscattering,” Appl. Phys. Lett. 69, 1515 (1997).
[CrossRef]

A. Shiratori and M. Obara, “Wavelength-stable, narrow-spectral-width oscillation of an AlGaInP diode laser coupled to a BaTiO3:Co stimulated photorefractive backscattering phase conjugator,” Appl. Phys. B 65, 329 (1997).
[CrossRef]

Petersen, P. M.

M. Lobel, P. M. Petersen, and P. M. Johansen, “Suppressing self-induced frequency scanning of a phase conjugate diode laser array using counterbalance dispersion,” Appl. Phys. Lett. 72, 1263 (1998).
[CrossRef]

Pierce, R. M.

Ramsey, J. M.

Rauch, T.

M. Delpino, T. Rauch, C. Denz, and M. Carrascosa, “Numerical simulation of the time evolution of photorefractive phase conjugate beams: multigrating operations,” Opt. Mater. 4, 326 (1995).
[CrossRef]

Rytz, D.

P. Lambelet, R. P. Salathe, M. H. Garrett, and D. Rytz, “Characterization of a photorefractive phase conjugator by optical low-coherence reflectometry,” Appl. Phys. Lett. 64, 1079 (1994).
[CrossRef]

Saffman, M.

A. A. Zozulya, M. Saffman, and D. Anderson, “Double phase-conjugate mirror: convection and diffraction,” J. Opt. Soc. Am. B 12, 255 (1995).
[CrossRef]

A. A. Zozulya, M. Saffman, and D. Z. Anderson, “Propagation of light beams in photorefractive media: fanning, self-bending, and formation of self-pumped four-wave-mixing phase conjugation geometries,” Phys. Rev. Lett. 73, 818 (1994).
[CrossRef] [PubMed]

Salathe, R. P.

P. Lambelet, R. P. Salathe, M. H. Garrett, and D. Rytz, “Characterization of a photorefractive phase conjugator by optical low-coherence reflectometry,” Appl. Phys. Lett. 64, 1079 (1994).
[CrossRef]

Shibayama, J.

Shiratori, A.

A. Shiratori and M. Obara, “Frequency-stable, narrow linewidth oscillation of red diode laser with phase-conjugate feedback using stimulated photorefractive backscattering,” Appl. Phys. Lett. 69, 1515 (1997).
[CrossRef]

A. Shiratori and M. Obara, “Wavelength-stable, narrow-spectral-width oscillation of an AlGaInP diode laser coupled to a BaTiO3:Co stimulated photorefractive backscattering phase conjugator,” Appl. Phys. B 65, 329 (1997).
[CrossRef]

Valley, G. C.

G. C. Valley and M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704 (1983).
[CrossRef]

Wang, P.

P. Xie, J. Dai, P. Wang, and H. Zhang, “Self-pumped phase conjugation in photorefractive crystal: reflectivity and spatial fidelity,” Phys. Rev. A 55, 3092 (1997).
[CrossRef]

Weber, P. G.

Whitten, W. B.

Xie, P.

P. Xie, J. Dai, P. Wang, and H. Zhang, “Self-pumped phase conjugation in photorefractive crystal: reflectivity and spatial fidelity,” Phys. Rev. A 55, 3092 (1997).
[CrossRef]

Yamanuchi, J.

Yariv, A.

M. Cronin-Golomb and A. Yariv, “Self-induced frequency scanning and distributed Bragg reflection in semiconductor lasers with phase-conjugate feedback,” Opt. Lett. 11, 455 (1986).
[CrossRef] [PubMed]

M. Cronin-Golomb, K. Y. Lau, and A. Yariv, “Infrared photorefractive passive phase conjugation with BaTiO3: demonstrations with GaAlAs and 1.09-μm Ar+ lasers,” Appl. Phys. Lett. 47, 567 (1985).
[CrossRef]

Zhang, H.

P. Xie, J. Dai, P. Wang, and H. Zhang, “Self-pumped phase conjugation in photorefractive crystal: reflectivity and spatial fidelity,” Phys. Rev. A 55, 3092 (1997).
[CrossRef]

Zozulya, A. A.

A. A. Zozulya, M. Saffman, and D. Anderson, “Double phase-conjugate mirror: convection and diffraction,” J. Opt. Soc. Am. B 12, 255 (1995).
[CrossRef]

A. A. Zozulya, G. Montemezzani, and D. Z. Anderson, “Analysis of total-internal-reflection phase-conjugate mirror,” Phys. Rev. A 52, 4167 (1995).
[CrossRef] [PubMed]

A. A. Zozulya, M. Saffman, and D. Z. Anderson, “Propagation of light beams in photorefractive media: fanning, self-bending, and formation of self-pumped four-wave-mixing phase conjugation geometries,” Phys. Rev. Lett. 73, 818 (1994).
[CrossRef] [PubMed]

Appl. Phys. B (1)

A. Shiratori and M. Obara, “Wavelength-stable, narrow-spectral-width oscillation of an AlGaInP diode laser coupled to a BaTiO3:Co stimulated photorefractive backscattering phase conjugator,” Appl. Phys. B 65, 329 (1997).
[CrossRef]

Appl. Phys. Lett. (4)

P. Lambelet, R. P. Salathe, M. H. Garrett, and D. Rytz, “Characterization of a photorefractive phase conjugator by optical low-coherence reflectometry,” Appl. Phys. Lett. 64, 1079 (1994).
[CrossRef]

M. Lobel, P. M. Petersen, and P. M. Johansen, “Suppressing self-induced frequency scanning of a phase conjugate diode laser array using counterbalance dispersion,” Appl. Phys. Lett. 72, 1263 (1998).
[CrossRef]

A. Shiratori and M. Obara, “Frequency-stable, narrow linewidth oscillation of red diode laser with phase-conjugate feedback using stimulated photorefractive backscattering,” Appl. Phys. Lett. 69, 1515 (1997).
[CrossRef]

M. Cronin-Golomb, K. Y. Lau, and A. Yariv, “Infrared photorefractive passive phase conjugation with BaTiO3: demonstrations with GaAlAs and 1.09-μm Ar+ lasers,” Appl. Phys. Lett. 47, 567 (1985).
[CrossRef]

Bell Syst. Tech. J. (1)

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

J. Opt. Soc. Am. (1)

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

Opt. Eng. (1)

G. C. Valley and M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704 (1983).
[CrossRef]

Opt. Lett. (7)

Opt. Mater. (1)

M. Delpino, T. Rauch, C. Denz, and M. Carrascosa, “Numerical simulation of the time evolution of photorefractive phase conjugate beams: multigrating operations,” Opt. Mater. 4, 326 (1995).
[CrossRef]

Phys. Rev. A (2)

A. A. Zozulya, G. Montemezzani, and D. Z. Anderson, “Analysis of total-internal-reflection phase-conjugate mirror,” Phys. Rev. A 52, 4167 (1995).
[CrossRef] [PubMed]

P. Xie, J. Dai, P. Wang, and H. Zhang, “Self-pumped phase conjugation in photorefractive crystal: reflectivity and spatial fidelity,” Phys. Rev. A 55, 3092 (1997).
[CrossRef]

Phys. Rev. Lett. (1)

A. A. Zozulya, M. Saffman, and D. Z. Anderson, “Propagation of light beams in photorefractive media: fanning, self-bending, and formation of self-pumped four-wave-mixing phase conjugation geometries,” Phys. Rev. Lett. 73, 818 (1994).
[CrossRef] [PubMed]

Sov. Tech. Phys. Lett. (1)

N. Kukhtarev, “Kinetics of hologram recording and erasure in electrooptic crystals,” Sov. Tech. Phys. Lett. 2, 438 (1976).

Other (2)

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes (Cambridge U. Press, London, 1992).

L. Solymar, D. J. Webb, and A. Grunnet-Jepsen, The Physics and Applications of Photorefractive Materials (Clarendon, Oxford, 1996).

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of the laser cavity coupled to a self-pumped phase conjugator arranged in a cat geometry. Rend and Rout are the frequency-independent reflectivity of the end mirror and the output mirror, respectively. Rpc is the frequency-dependent phase-conjugate reflectivity of the conjugator. n is the refractive index of the BaTiO3 crystal.

Fig. 2
Fig. 2

Schematic representation of the spectrum of the laser coupled to the conjugator during self-induced frequency scanning: (a) the three-mode spectrum at some moment in time t=t0; (b) the three-mode spectrum at time t=t0+tc, where tc is of the order of a few round trips in the external cavity. The center wavelength is shifted slightly with respect to λ0 owing to asymmetric feedback (detuning curve).

Fig. 3
Fig. 3

Image of the total intensity distribution throughout the crystal. The crystal is 0.25 mm wide and 1.0 mm long. The entrance surface is to the left. The c axis points downward.

Fig. 4
Fig. 4

Cross section of total intensity at the entrance surface (z=0). Both the input beam (solid dots) and the phase-conjugate beam (solid curve) are shown. The profiles are a summation of the intensity profiles corresponding to the three longitudinal modes.

Fig. 5
Fig. 5

Detuning curves (the dispersion coefficient is -1.6×105, and the focal length r is -2.40 mm; the other parameters are given in Table 1): (a) the reflectivity R as a function of the relative detuned wavelength λ-λ0; (b) the phase-conjugate fidelity H as a function of the relative detuned wavelength.

Fig. 6
Fig. 6

Ratios of the phase-conjugate fidelity (H+1/H-1) calculated at the wavelengths corresponding to the two side modes λ+1 and λ-1 as functions of the dispersion coefficient.

Tables (1)

Tables Icon

Table 1 Parameters Used in the Numerical Simulations

Equations (17)

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

E˜s+τ 1I1+1kdxE˜s tE˜s
-1kd22x2E˜s1+1kdxE˜s-1=-1I1kdxI,
E˜s=Es/Eq,Eq=eNA/effkd,
τ=(NAeffγr)/(sNDμe),
kd=(e2NA/κBTeff)1/2.
2E+k2E=0,
n0(λ)=n0(λ0)+(λ-λ0) n0λ,
z-i2k2x2E0f(x, z)=2iγ0E˜sE0f(x, z),
-z-i2k2x2E0b(x, z)=2iγ0E˜sE0b(x, z),
E0f,m(x, z=0)=Am exp-ikm tan(θm)x-2 x-x0w02+ikm (x-x0)22r+Enoisem,
I(x, z)Id+λmIm=Id+λm n02η0[|E0f,m(x, z)|2+|E0b,m(x, z)|2],
E˜s,vu+1+E˜s,vu2+τ 1IvuE˜s,vu+1-E˜s,vuΔt
-12kd2E˜s,v+1u+1-2E˜s,vu+1+E˜s,v-1u+1Δx2
+E˜s,v+1u-2E˜s,vu+E˜s,v-1uΔx2
=-1Ivu+11kdIv+1u+1-Iv-1u+12Δx,
Rm=0lx|E0b,m(z=0)|2dx0lx|E0f,m(z=0)|2dx,
Hm=0lxE0b,m(z=0)E0f,m(z=0)dx20lx|E0b,m(z=0)|2dx0lx|E0f,m(z=0)|2dx.

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