Abstract

We present analytical expressions for time-dependent space charge fields and two wave-mixing gains under the external applied fields accompanying with the grating translation. We analyzed the variations of complex space charge fields in a complex plane, and also obtained the explicit expressions for the resonance and optimum frequencies (or moving grating velocities), which maximize the magnitude and imaginary part of space charge fields. We also conducted two wave-mixing experiments with the grating translation technique without an external applied field in a BaTiO3 crystal. The transient behaviors of measured gains look like damped harmonic oscillations, showing excellent agreement with the theory for the entire time range.

©2008 Optical Society of America

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References

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  1. M. Z. Zha, P. Amrhein, and P. Günter, “Measurement of phase shift of photorefractive gratings by a novel method,” IEEE Quantum Electron. 26, 788–792 (1990).
    [Crossref]
  2. K. Sutter and P. Günter, “Photorefractive gratings in the organic crystal 2-cyclooctylamino-5-nitropyridine doped with 7,7,8,8-tetracyanoquinodimethane,” J. Opt. Soc. Am. B 7, 2274–2278 (1990).
    [Crossref]
  3. R. S. Cudney, G. D. Bacher, R. M. Pierce, and J. Feinberg, “Measurement of the photorefractive phase shift,” Opt. Lett. 17, 67–69 (1992).
    [Crossref] [PubMed]
  4. R. Hofmeister, A. Yariv, A. Kewitsch, and S. Yagi, “Simple methods of measuring the net photorefractive phase shift and coupling constant,” Opt. Lett. 18, 488–490 (1993).
    [Crossref]
  5. D. G. Gray, M. G. Moharam, and T. M. Ayres, “Heterodyne technique for the direct measurement of the amplitude and phase of photorefractive space-charge field,” J. Opt. Soc. Am. B 11, 470–475 (1994).
    [Crossref]
  6. C. H. Kwak and S. J. Lee, “Approximate analytic solution of photochromic and photorefractive gratings in photorefractive materials,” Opt. Commun. 183, 547–554 (2000).
    [Crossref]
  7. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics22, 949–960 (1979), and idem, ibid22, 961–964 (1979).
    [Crossref]
  8. Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
    [Crossref]
  9. S. I. Stepanov and M. P. Petrov, in Photorefractive materials and their applications I, P. Günter and J. P. Huignard, eds., (Springer, Berlin, 1988) Chap. 9.
  10. P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
    [Crossref]
  11. C. H. Kwak, S. Y. Park, and E. H. Lee, “Intensity dependent two-wave mixing at large modulation depth in photorefractive BaTiO3 crystal,” Opt. Commun. 115, 315–322 (1995).
    [Crossref]
  12. G. C. Valley and M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704–711 (1983).
  13. C. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E. H. Lee, “An analytical solution for large modulation effects in photorefractive two-wave couplings,” Opt. Commun. 105, 353–358 (1994).
    [Crossref]
  14. J. P. Huignard and A. Marrakchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi12SiO20 crystals,” Opt. Commun. 38, 249–254 (1981).
    [Crossref]
  15. I. McMichael and P. Yeh, “Phase shift of photorefractive gratings and phase-conjugate waves,” Opt. Lett. 12, 48–50 (1987).
    [Crossref] [PubMed]
  16. K. H. Kim, E. J. Kim, S. J. Lee, J. H. Lee, C. H. Kwak, and J. E. Kim, “Effects of applied electric field on orientational photorefraction in porphyrin:Zn-doped nematic liquid crystals,” Appl. Phys. Lett. 85, 366–368 (2004).
    [Crossref]
  17. E. J. Kim, H. R. Yang, S. J. Lee, G. Y. Kim, and C. H. Kwak, “Orientational photorefractive holograms in porphyrin:Zn-doped nematic liquid crystals,” Opt. Express 16, 17329–17341 (2008).
    [Crossref] [PubMed]

2008 (1)

2004 (1)

K. H. Kim, E. J. Kim, S. J. Lee, J. H. Lee, C. H. Kwak, and J. E. Kim, “Effects of applied electric field on orientational photorefraction in porphyrin:Zn-doped nematic liquid crystals,” Appl. Phys. Lett. 85, 366–368 (2004).
[Crossref]

2000 (1)

C. H. Kwak and S. J. Lee, “Approximate analytic solution of photochromic and photorefractive gratings in photorefractive materials,” Opt. Commun. 183, 547–554 (2000).
[Crossref]

1995 (1)

C. H. Kwak, S. Y. Park, and E. H. Lee, “Intensity dependent two-wave mixing at large modulation depth in photorefractive BaTiO3 crystal,” Opt. Commun. 115, 315–322 (1995).
[Crossref]

1994 (2)

C. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E. H. Lee, “An analytical solution for large modulation effects in photorefractive two-wave couplings,” Opt. Commun. 105, 353–358 (1994).
[Crossref]

D. G. Gray, M. G. Moharam, and T. M. Ayres, “Heterodyne technique for the direct measurement of the amplitude and phase of photorefractive space-charge field,” J. Opt. Soc. Am. B 11, 470–475 (1994).
[Crossref]

1993 (1)

1992 (1)

1990 (2)

M. Z. Zha, P. Amrhein, and P. Günter, “Measurement of phase shift of photorefractive gratings by a novel method,” IEEE Quantum Electron. 26, 788–792 (1990).
[Crossref]

K. Sutter and P. Günter, “Photorefractive gratings in the organic crystal 2-cyclooctylamino-5-nitropyridine doped with 7,7,8,8-tetracyanoquinodimethane,” J. Opt. Soc. Am. B 7, 2274–2278 (1990).
[Crossref]

1989 (1)

P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[Crossref]

1987 (1)

1985 (1)

Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[Crossref]

1983 (1)

G. C. Valley and M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704–711 (1983).

1981 (1)

J. P. Huignard and A. Marrakchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi12SiO20 crystals,” Opt. Commun. 38, 249–254 (1981).
[Crossref]

Amrhein, P.

M. Z. Zha, P. Amrhein, and P. Günter, “Measurement of phase shift of photorefractive gratings by a novel method,” IEEE Quantum Electron. 26, 788–792 (1990).
[Crossref]

Ayres, T. M.

Bacher, G. D.

Cudney, R. S.

Feinberg, J.

Gray, D. G.

Günter, P.

K. Sutter and P. Günter, “Photorefractive gratings in the organic crystal 2-cyclooctylamino-5-nitropyridine doped with 7,7,8,8-tetracyanoquinodimethane,” J. Opt. Soc. Am. B 7, 2274–2278 (1990).
[Crossref]

M. Z. Zha, P. Amrhein, and P. Günter, “Measurement of phase shift of photorefractive gratings by a novel method,” IEEE Quantum Electron. 26, 788–792 (1990).
[Crossref]

Hofmeister, R.

Huignard, J. P.

Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[Crossref]

J. P. Huignard and A. Marrakchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi12SiO20 crystals,” Opt. Commun. 38, 249–254 (1981).
[Crossref]

Jeong, J. S.

C. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E. H. Lee, “An analytical solution for large modulation effects in photorefractive two-wave couplings,” Opt. Commun. 105, 353–358 (1994).
[Crossref]

Kewitsch, A.

Kim, E. J.

E. J. Kim, H. R. Yang, S. J. Lee, G. Y. Kim, and C. H. Kwak, “Orientational photorefractive holograms in porphyrin:Zn-doped nematic liquid crystals,” Opt. Express 16, 17329–17341 (2008).
[Crossref] [PubMed]

K. H. Kim, E. J. Kim, S. J. Lee, J. H. Lee, C. H. Kwak, and J. E. Kim, “Effects of applied electric field on orientational photorefraction in porphyrin:Zn-doped nematic liquid crystals,” Appl. Phys. Lett. 85, 366–368 (2004).
[Crossref]

Kim, G. Y.

Kim, J. E.

K. H. Kim, E. J. Kim, S. J. Lee, J. H. Lee, C. H. Kwak, and J. E. Kim, “Effects of applied electric field on orientational photorefraction in porphyrin:Zn-doped nematic liquid crystals,” Appl. Phys. Lett. 85, 366–368 (2004).
[Crossref]

Kim, K. H.

K. H. Kim, E. J. Kim, S. J. Lee, J. H. Lee, C. H. Kwak, and J. E. Kim, “Effects of applied electric field on orientational photorefraction in porphyrin:Zn-doped nematic liquid crystals,” Appl. Phys. Lett. 85, 366–368 (2004).
[Crossref]

Klein, M. B.

G. C. Valley and M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704–711 (1983).

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics22, 949–960 (1979), and idem, ibid22, 961–964 (1979).
[Crossref]

Kwak, C. H.

E. J. Kim, H. R. Yang, S. J. Lee, G. Y. Kim, and C. H. Kwak, “Orientational photorefractive holograms in porphyrin:Zn-doped nematic liquid crystals,” Opt. Express 16, 17329–17341 (2008).
[Crossref] [PubMed]

K. H. Kim, E. J. Kim, S. J. Lee, J. H. Lee, C. H. Kwak, and J. E. Kim, “Effects of applied electric field on orientational photorefraction in porphyrin:Zn-doped nematic liquid crystals,” Appl. Phys. Lett. 85, 366–368 (2004).
[Crossref]

C. H. Kwak and S. J. Lee, “Approximate analytic solution of photochromic and photorefractive gratings in photorefractive materials,” Opt. Commun. 183, 547–554 (2000).
[Crossref]

C. H. Kwak, S. Y. Park, and E. H. Lee, “Intensity dependent two-wave mixing at large modulation depth in photorefractive BaTiO3 crystal,” Opt. Commun. 115, 315–322 (1995).
[Crossref]

C. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E. H. Lee, “An analytical solution for large modulation effects in photorefractive two-wave couplings,” Opt. Commun. 105, 353–358 (1994).
[Crossref]

Lee, E. H.

C. H. Kwak, S. Y. Park, and E. H. Lee, “Intensity dependent two-wave mixing at large modulation depth in photorefractive BaTiO3 crystal,” Opt. Commun. 115, 315–322 (1995).
[Crossref]

C. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E. H. Lee, “An analytical solution for large modulation effects in photorefractive two-wave couplings,” Opt. Commun. 105, 353–358 (1994).
[Crossref]

Lee, J. H.

K. H. Kim, E. J. Kim, S. J. Lee, J. H. Lee, C. H. Kwak, and J. E. Kim, “Effects of applied electric field on orientational photorefraction in porphyrin:Zn-doped nematic liquid crystals,” Appl. Phys. Lett. 85, 366–368 (2004).
[Crossref]

Lee, S. J.

E. J. Kim, H. R. Yang, S. J. Lee, G. Y. Kim, and C. H. Kwak, “Orientational photorefractive holograms in porphyrin:Zn-doped nematic liquid crystals,” Opt. Express 16, 17329–17341 (2008).
[Crossref] [PubMed]

K. H. Kim, E. J. Kim, S. J. Lee, J. H. Lee, C. H. Kwak, and J. E. Kim, “Effects of applied electric field on orientational photorefraction in porphyrin:Zn-doped nematic liquid crystals,” Appl. Phys. Lett. 85, 366–368 (2004).
[Crossref]

C. H. Kwak and S. J. Lee, “Approximate analytic solution of photochromic and photorefractive gratings in photorefractive materials,” Opt. Commun. 183, 547–554 (2000).
[Crossref]

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics22, 949–960 (1979), and idem, ibid22, 961–964 (1979).
[Crossref]

Marrakchi, A.

J. P. Huignard and A. Marrakchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi12SiO20 crystals,” Opt. Commun. 38, 249–254 (1981).
[Crossref]

McMichael, I.

Moharam, M. G.

Odulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics22, 949–960 (1979), and idem, ibid22, 961–964 (1979).
[Crossref]

Park, S. Y.

C. H. Kwak, S. Y. Park, and E. H. Lee, “Intensity dependent two-wave mixing at large modulation depth in photorefractive BaTiO3 crystal,” Opt. Commun. 115, 315–322 (1995).
[Crossref]

C. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E. H. Lee, “An analytical solution for large modulation effects in photorefractive two-wave couplings,” Opt. Commun. 105, 353–358 (1994).
[Crossref]

Petrov, M. P.

S. I. Stepanov and M. P. Petrov, in Photorefractive materials and their applications I, P. Günter and J. P. Huignard, eds., (Springer, Berlin, 1988) Chap. 9.

Pierce, R. M.

Rajbenbach, H.

Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[Crossref]

Refregier, Ph.

Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[Crossref]

Solymar, L.

Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[Crossref]

Soskin, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics22, 949–960 (1979), and idem, ibid22, 961–964 (1979).
[Crossref]

Stepanov, S. I.

S. I. Stepanov and M. P. Petrov, in Photorefractive materials and their applications I, P. Günter and J. P. Huignard, eds., (Springer, Berlin, 1988) Chap. 9.

Suh, H. H.

C. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E. H. Lee, “An analytical solution for large modulation effects in photorefractive two-wave couplings,” Opt. Commun. 105, 353–358 (1994).
[Crossref]

Sutter, K.

Valley, G. C.

G. C. Valley and M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704–711 (1983).

Vinetskii, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics22, 949–960 (1979), and idem, ibid22, 961–964 (1979).
[Crossref]

Yagi, S.

Yang, H. R.

Yariv, A.

Yeh, P.

Zha, M. Z.

M. Z. Zha, P. Amrhein, and P. Günter, “Measurement of phase shift of photorefractive gratings by a novel method,” IEEE Quantum Electron. 26, 788–792 (1990).
[Crossref]

Appl. Phys. Lett. (1)

K. H. Kim, E. J. Kim, S. J. Lee, J. H. Lee, C. H. Kwak, and J. E. Kim, “Effects of applied electric field on orientational photorefraction in porphyrin:Zn-doped nematic liquid crystals,” Appl. Phys. Lett. 85, 366–368 (2004).
[Crossref]

IEEE J. Quantum Electron. (1)

P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[Crossref]

IEEE Quantum Electron. (1)

M. Z. Zha, P. Amrhein, and P. Günter, “Measurement of phase shift of photorefractive gratings by a novel method,” IEEE Quantum Electron. 26, 788–792 (1990).
[Crossref]

J. Appl. Phys. (1)

Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[Crossref]

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

Opt. Commun. (4)

C. H. Kwak and S. J. Lee, “Approximate analytic solution of photochromic and photorefractive gratings in photorefractive materials,” Opt. Commun. 183, 547–554 (2000).
[Crossref]

C. H. Kwak, S. Y. Park, and E. H. Lee, “Intensity dependent two-wave mixing at large modulation depth in photorefractive BaTiO3 crystal,” Opt. Commun. 115, 315–322 (1995).
[Crossref]

C. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E. H. Lee, “An analytical solution for large modulation effects in photorefractive two-wave couplings,” Opt. Commun. 105, 353–358 (1994).
[Crossref]

J. P. Huignard and A. Marrakchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi12SiO20 crystals,” Opt. Commun. 38, 249–254 (1981).
[Crossref]

Opt. Eng. (1)

G. C. Valley and M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704–711 (1983).

Opt. Express (1)

Opt. Lett. (3)

Other (2)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics22, 949–960 (1979), and idem, ibid22, 961–964 (1979).
[Crossref]

S. I. Stepanov and M. P. Petrov, in Photorefractive materials and their applications I, P. Günter and J. P. Huignard, eds., (Springer, Berlin, 1988) Chap. 9.

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Figures (7)

Fig. 1.
Fig. 1. Representations of the complex space charge field during the grating translation when an external DC field applies (i.e., E 0=1 kV cm). (a) Complex representation of the space charge field. The circle exhibits the steady state representation and the spirals depict the transient behaviors for each moving frequency, (b) the imaginary part of the space charge field Y(t,Ω), (c) the amplitude of the space charge field |Z(t,Ω)and (d) the phase shift Ψ(t,Ω) with an initial grating phase ϕ g against dimensionless time t/τg for various r=Ω/ΩOPT, where the moving frequency Ω is divided by the optimum frequency ΩOPT, where ΩOPT=4 Hz and τg=4.3×10-2 s are used for calculations.

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Fig. 2.
Fig. 2. Representations of the complex space charge field during the grating translation when no external field applies (i.e., E 0=0kV/cm). (a) Complex representation of the space charge field. The circle exhibits the steady state representation and the spirals depict the transient behaviors for each dimensionless moving frequency b=Ωτg , (b) the imaginary part of the space charge field Y(t,Ω), (c) the amplitude of the space charge field |Z(t,Ω) and (d) the phase shift Ψ(t,Ω) with an initial phase ϕ g against dimensionless time t/τg for various b values. In the case of no external field, the optimum frequency ΩOPT becomes zero.

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Fig. 3.
Fig. 3. Representations of the complex space charge field for a fast grating translation (b=100) and no external field applies (i.e., E 0=0kV/cm). (a) Complex representation of the space charge field. The circle exhibits the steady state representation and the spirals behaves like a squirrel cage with gradually decreasing the radii, (b) the transient behavior of Y(t,Ω)represent nearly sinusoidal oscillation, (c) |Z(t,Ω)|decays very slowly with time and (d) the phase shift Ψ(t,Ω) is linearly proportional to the grating translation time.

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Fig. 4.
Fig. 4. Experimental setup for two wave mixing with grating translational method.

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Fig. 5.
Fig. 5. Typical experimental data for the transient behaviors of the pump beam and signal beam powers during the grating formation and the grating translation periods. No external electric field was applied to a BaTiO3 photorefractive crystal.

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Fig. 6.
Fig. 6. Transient behaviors of two wave mixing gains for various dimensionless moving frequencies, b=Ωτg . Theoretical gain curves are from Eq. (18b) with Eq. (19).

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Fig. 7.
Fig. 7. Normalized gain coefficient as a function of dimensionless moving frequency b. The lines are theoretical curves of Eq.(20).

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Equations (41)

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

I ( x , t ) = I 0 + 1 2 I 1 [ j ( K g x − Ω t ) ] + c . c . ,
n ( t ) = n o + 1 2 n 1 ( t ) e j ( ϕ g + Ψ ( t ) ) I P I S I 0 exp [ j ( K g x − Ω t ) ] + c . c .
n 1 ( t ) = n b 3 r eff ∣ E 1 ( t ) ∣ ⁄ m ,
I P ( z , t ) = I P ( z = 0 ) exp [ − α z ⁄ cos ⁡ θ ] 1 + β − 1 exp [ Γ ( t ) z ] + β − 1 ,
I S ( z , t ) = I S ( z = 0 ) exp [ − α z ⁄ cos ⁡ θ ] 1 + β 1 + β exp [ − Γ ( t ) z ] ,
Γ ( t ) = 2 π n 1 ( t ) λ cos ⁡ θ sin ⁡ ( ϕ g + Ψ ( t ) ) = 2 π n b 3 r eff λ cos ⁡ θ Im [ E 1 ( t ) ] m .
∂ N D + ∂ t = ( N D − N D + ) s I − γ R N N D + ,
∂ N ∂ t = ∂ N D + ∂ t + 1 e ∂ J ∂ x ,
∂ E ∂ x = e ε ε 0 ( N D + − N A − N ) ,
J = e μ NE + k B T μ ∂ N ∂ x ,
F ( x , t ) = F 0 ( t ) + 1 2 F 1 ( t ) exp [ j ( K g x − Ω t ) ] + c . c . ,
d E 1 dt + g E 1 = mh ,
g = 1 D τ d ( − τ ∂ E 0 ⁄ ∂ t E 0 + i E D + 1 + E D E q − j E 0 E q − j Ω τ d D ) ,
h = 1 D τ d ( E 0 + j E D ) ,
D = − τ ∂ E 0 ⁄ ∂ t E 0 + j E T + 1 + E D E M − j E 0 E M ,
1 τ g ≡ Re [ g o ] = 1 τ d ( 1 + E D E q ) ( 1 + E D E M ) + E 0 2 ⁄ ( E q E M ) ( 1 + E D ⁄ E D ) 2 + ( E 0 ⁄ E M ) 2 ,
ω o ≡ Im [ g o ] = 1 τ d ( 1 ⁄ E M − 1 ⁄ E q ) E 0 ( 1 + E D ⁄ E M ) 2 + ( E 0 ⁄ E M ) 2 .
E 1 ( t ) = m E 10 [ 1 − exp ( − g o t ) ] ,
E 10 = E 0 + j E D 1 + E D ⁄ E q − j E 0 ⁄ E q = ∣ E 10 ∣ exp [ j ϕ g ] .
∣ E 10 ∣ = E q [ E 0 2 + E D 2 E 0 2 + ( E D + E q ) 2 ] 1 2 ,
tan ⁡ ϕ g = E D E 0 ( 1 + E D E q + E 0 2 E D E q ) ,
Γ ( t ) = Im [ E 1 ( t ) ] m = ∣ E 10 ∣ [ sin ⁡ ϕ g − exp ( − t τ g ) sin ⁡ ( ϕ g − ω o t ) ] .
E 1 ( t ) = m E 10 [ ∣ g 0 ∣ 2 − 2 Ω Im [ g 0 exp ( − g * t ) ] + Ω 2 exp ( − 2 t ⁄ τ g ) ∣ g 0 ∣ 2 − 2 Ω Im [ g 0 ] + Ω 2 ] 1 2 exp [ j Ψ ( t ) ] ,
tan ⁡ Ψ ( t ) = Ω Re [ g 0 − g exp ( − g * t ) ] ∣ g 0 ∣ 2 − Ω Im [ g 0 + g exp ( − g * t ) ] ,
E 1 ( t = ∞ ) = m E 10 1 + j ω o τ g 1 + j ( ω o − Ω ) τ g = m h g ,
E 1 ( t = ∞ ) = m ( E 0 + j E D ) 1 + E D ⁄ E q − b E 0 ⁄ E M − j [ E 0 ⁄ E q + ( 1 + E D ⁄ E D ) b ] ,
( X ( ∞ , Ω ) − 1 2 X RES ) 2 + ( Y ( ∞ , Ω ) − 1 2 Y RES ) 2 = ( 1 2 ∣ Z RES ∣ ) 2 ,
X ( ∞ , Ω ) = Re [ E 1 ( t = ∞ , Ω ) m ∣ E 10 ∣ ] , Y ( ∞ , Ω ) = Im [ E 1 ( t = ∞ , Ω ) m ∣ E 10 ∣ ] ,
X RES = X ( ∞ , Ω RES ) , Y RES = Y ( ∞ , Ω RES ) , Z RES = X RES + j Y RES ,
Ω RES = ω o = Im [ g o ] = 1 τ d ( 1 ⁄ E M − 1 ⁄ E q ) E 0 ( 1 + E D ⁄ E D ) 2 + ( E 0 ⁄ E M ) 2 .
Ω OPT = 1 τ g · ( 1 + ( ω 0 τ g ) 2 ) tan ⁡ ϕ g 1 − ( ω 0 τ g ) tan ⁡ ϕ g [ 1 + 1 − ( ω 0 τ g ) 2 tan 2 ϕ g ( 1 + ( ω 0 τ g ) 2 ) tan 2 ϕ g − 1 ] ,
E 1 ( t ) = m ∣ E 10 ∣ exp [ − t ⁄ τ g ] exp [ j ( ϕ g + ( Ω − ω o ) t ) ] ,
tan ⁡ Ψ ( t ) = tan ⁡ ( Ω − ω o ) t ⇒ Ψ ( t ) = ( Ω − ω o ) t ,
E 1 ( t ) = m E 10 [ 1 − exp ( − t ⁄ τ g ) ] .
E 1 ( t ) = m E 10 1 + b 2 [ 1 + 2 b e − t ⁄ τ g sin ⁡ Ω t + b 2 e − 2 t ⁄ τ g ] 1 2 e j Ψ ( t ) ,
tan ⁡ Ψ ( t ) = b 1 − e − t τ g [ cos ⁡ Ω t − b sin ⁡ Ω t ] 1 + b e − t τ g [ sin ⁡ Ω t + b cos ⁡ Ω t ] , with b = Ω τ g .
G ( t ) = I S ( L , t ) with pump beam I S ( L ) without pump beam ,
G ( t ) = 1 + β 1 + β exp [ − Γ ( t ) L ] ,
Γ ( t ) = [ ln ( β G ) − ln ( β + 1 − G ) ] ⁄ L .
Γ ( t ) = Γ ( 0 ) 1 + b 2 [ 1 + 2 b e − t τ g sin ⁡ Ω t + b 2 e − 2 t ⁄ τ g ] 1 2 sin ⁡ ( ϕ g + Ψ ( t ) ) ,
Γ ( ∞ ) = Γ ( 0 ) 1 + b 2 sin ⁡ ( ϕ g + tan − 1 b ) ,

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