Abstract

A general approach to photorefractive kinetics, based on the band-transport model, has been developed for nonphotovoltaic materials in the diffusion regime. For small grating periods (ΛΛd, where Λd is the Debye screening length) analytical solutions for the space-charge field have been found and compared with accurate numerical solutions in Bi12SiO20. The quantitative agreement is excellent when the solution includes all harmonics and greatly improves on the linear approach, even if a few (two or three) harmonics are taken into account. The analytical solutions show the most relevant features of the reported experimental results. Moreover, they include as particular cases other previous theoretical results.

© 1996 Optical Society of America

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References

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  1. N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic-crystals I. Steady state,” Ferroelectrics 22, 949–961 (1979).
    [Crossref]
  2. M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Holographic grating formation in photorefractive crystals with arbitrary electron transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
    [Crossref]
  3. E. Ochoa, F. Vachss, and L. Hesselink, “Higher-order analysis of the photorefractive effect for large modulation depths,” J. Opt. Soc. Am. A 3, 181–187 (1986).
    [Crossref]
  4. Ph. Réfrégier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive B12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
    [Crossref]
  5. R. Saxena and T. Y. Chang, “Perturbative analyses of higher-order photorefractive gratings,” J. Opt. Soc. Am. B 9, 1467–1472 (1992).
    [Crossref]
  6. L. Boutsikaris and F. Davidson, “Perturbative analysis of higher-order photorefractive gratings in InP: Fe,” Opt. Commun. 105, 411–420 (1994).
    [Crossref]
  7. D. A. Temple and C. Warde, “High-order anisotropic diffraction in photorefractive crystals,” J. Opt. Soc. Am. B 5, 1800–1806 (1988).
    [Crossref]
  8. E. Serrano, V. López, M. Carrascosa, and F. Agulló-López, “Recording and erasure kinetics in photorefractive materials at high modulation depths,” J. Opt. Soc. Am. B 11, 670–675 (1994).
    [Crossref]
  9. L. B. Au and L. Solymar, “Higher harmonic gratings in photorefractive materials at large modulation with moving fringes,” J. Opt. Soc. Am. A 7, 1554–1562 (1990).
    [Crossref]
  10. Y. H. Lee and R. W. Hellwarth, “Spatial harmonics of photorefractive gratings in a barium titanate crystal,” J. Appl. Phys. 71, 916–923 (1992).
    [Crossref]
  11. G. A. Brost, “Numerical analysis of photorefractive grating formation dynamics at large modulation in BSO,” Opt. Commun. 96, 113–116 (1993).
    [Crossref]
  12. M. Horowitz, R. Daisy, and B. Fischer, “Signal-to-pump ratio dependence of buildup and decay rates in photorefractive nonlinear two-beam coupling,” J. Opt. Soc. Am. B 9, 1685–1688 (1992).
    [Crossref]
  13. P. P. Banerjee and J. M. Jarem, “Transient wave mixing and recording kinetics in photorefractive barium titanate: a nonlinear coupled mode approach,” Opt. Eng. 34, 2254–2260 (1995).
    [Crossref]
  14. M. Carrascosa and F. Agulló-López, “Kinetics for optical erasure of sinusoidal holographic gratings in photorefractive materials,” IEEE J. Quantum Electron. QE-22, 1369–1375 (1986).
    [Crossref]
  15. J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980).
    [Crossref]
  16. J. V. Alvarez-Bravo, M. Carrascosa, and L. Arizmendi, “Experimental effects of light intensity modulation on the recording and erasure of holographic gratings in BSO crystals,” Opt. Commun. 103, 22–28 (1993).
    [Crossref]
  17. F. V. Atkinson, Discrete and Continuous Boundary Problems (Academic, New York, 1964), Chap. 6.
  18. E. Serrano, “Análisis teórico del efecto fotorrefractivo en el régimen no lineal,” Ph.D. dissertation (Universidad Autónoma de Madrid, Madrid, Spain, 1994).
  19. E. Serrano, M. Carrascosa, and F. Agulló-López, “Nonperturbative analytical solution for steady-state photorefractive recording,” Opt. Lett. 20, 1910–1912 (1995).
    [Crossref] [PubMed]
  20. E. Serrano, V. López, M. Carrascosa, and F. Agulló-López, “Steady state photorefractive gratings in LiNbO3 for strong light modulation depths,” IEEE J. Quantum Electron. 30, 875–880 (1994).
    [Crossref]

1995 (2)

P. P. Banerjee and J. M. Jarem, “Transient wave mixing and recording kinetics in photorefractive barium titanate: a nonlinear coupled mode approach,” Opt. Eng. 34, 2254–2260 (1995).
[Crossref]

E. Serrano, M. Carrascosa, and F. Agulló-López, “Nonperturbative analytical solution for steady-state photorefractive recording,” Opt. Lett. 20, 1910–1912 (1995).
[Crossref] [PubMed]

1994 (3)

L. Boutsikaris and F. Davidson, “Perturbative analysis of higher-order photorefractive gratings in InP: Fe,” Opt. Commun. 105, 411–420 (1994).
[Crossref]

E. Serrano, V. López, M. Carrascosa, and F. Agulló-López, “Steady state photorefractive gratings in LiNbO3 for strong light modulation depths,” IEEE J. Quantum Electron. 30, 875–880 (1994).
[Crossref]

E. Serrano, V. López, M. Carrascosa, and F. Agulló-López, “Recording and erasure kinetics in photorefractive materials at high modulation depths,” J. Opt. Soc. Am. B 11, 670–675 (1994).
[Crossref]

1993 (2)

J. V. Alvarez-Bravo, M. Carrascosa, and L. Arizmendi, “Experimental effects of light intensity modulation on the recording and erasure of holographic gratings in BSO crystals,” Opt. Commun. 103, 22–28 (1993).
[Crossref]

G. A. Brost, “Numerical analysis of photorefractive grating formation dynamics at large modulation in BSO,” Opt. Commun. 96, 113–116 (1993).
[Crossref]

1992 (3)

1990 (1)

1988 (1)

1986 (2)

E. Ochoa, F. Vachss, and L. Hesselink, “Higher-order analysis of the photorefractive effect for large modulation depths,” J. Opt. Soc. Am. A 3, 181–187 (1986).
[Crossref]

M. Carrascosa and F. Agulló-López, “Kinetics for optical erasure of sinusoidal holographic gratings in photorefractive materials,” IEEE J. Quantum Electron. QE-22, 1369–1375 (1986).
[Crossref]

1985 (1)

Ph. Réfrégier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive B12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[Crossref]

1980 (1)

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980).
[Crossref]

1979 (2)

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic-crystals I. Steady state,” Ferroelectrics 22, 949–961 (1979).
[Crossref]

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Holographic grating formation in photorefractive crystals with arbitrary electron transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
[Crossref]

Agulló-López, F.

E. Serrano, M. Carrascosa, and F. Agulló-López, “Nonperturbative analytical solution for steady-state photorefractive recording,” Opt. Lett. 20, 1910–1912 (1995).
[Crossref] [PubMed]

E. Serrano, V. López, M. Carrascosa, and F. Agulló-López, “Recording and erasure kinetics in photorefractive materials at high modulation depths,” J. Opt. Soc. Am. B 11, 670–675 (1994).
[Crossref]

E. Serrano, V. López, M. Carrascosa, and F. Agulló-López, “Steady state photorefractive gratings in LiNbO3 for strong light modulation depths,” IEEE J. Quantum Electron. 30, 875–880 (1994).
[Crossref]

M. Carrascosa and F. Agulló-López, “Kinetics for optical erasure of sinusoidal holographic gratings in photorefractive materials,” IEEE J. Quantum Electron. QE-22, 1369–1375 (1986).
[Crossref]

Alvarez-Bravo, J. V.

J. V. Alvarez-Bravo, M. Carrascosa, and L. Arizmendi, “Experimental effects of light intensity modulation on the recording and erasure of holographic gratings in BSO crystals,” Opt. Commun. 103, 22–28 (1993).
[Crossref]

Arizmendi, L.

J. V. Alvarez-Bravo, M. Carrascosa, and L. Arizmendi, “Experimental effects of light intensity modulation on the recording and erasure of holographic gratings in BSO crystals,” Opt. Commun. 103, 22–28 (1993).
[Crossref]

Atkinson, F. V.

F. V. Atkinson, Discrete and Continuous Boundary Problems (Academic, New York, 1964), Chap. 6.

Au, L. B.

Banerjee, P. P.

P. P. Banerjee and J. M. Jarem, “Transient wave mixing and recording kinetics in photorefractive barium titanate: a nonlinear coupled mode approach,” Opt. Eng. 34, 2254–2260 (1995).
[Crossref]

Boutsikaris, L.

L. Boutsikaris and F. Davidson, “Perturbative analysis of higher-order photorefractive gratings in InP: Fe,” Opt. Commun. 105, 411–420 (1994).
[Crossref]

Brost, G. A.

G. A. Brost, “Numerical analysis of photorefractive grating formation dynamics at large modulation in BSO,” Opt. Commun. 96, 113–116 (1993).
[Crossref]

Carrascosa, M.

E. Serrano, M. Carrascosa, and F. Agulló-López, “Nonperturbative analytical solution for steady-state photorefractive recording,” Opt. Lett. 20, 1910–1912 (1995).
[Crossref] [PubMed]

E. Serrano, V. López, M. Carrascosa, and F. Agulló-López, “Recording and erasure kinetics in photorefractive materials at high modulation depths,” J. Opt. Soc. Am. B 11, 670–675 (1994).
[Crossref]

E. Serrano, V. López, M. Carrascosa, and F. Agulló-López, “Steady state photorefractive gratings in LiNbO3 for strong light modulation depths,” IEEE J. Quantum Electron. 30, 875–880 (1994).
[Crossref]

J. V. Alvarez-Bravo, M. Carrascosa, and L. Arizmendi, “Experimental effects of light intensity modulation on the recording and erasure of holographic gratings in BSO crystals,” Opt. Commun. 103, 22–28 (1993).
[Crossref]

M. Carrascosa and F. Agulló-López, “Kinetics for optical erasure of sinusoidal holographic gratings in photorefractive materials,” IEEE J. Quantum Electron. QE-22, 1369–1375 (1986).
[Crossref]

Chang, T. Y.

Daisy, R.

Davidson, F.

L. Boutsikaris and F. Davidson, “Perturbative analysis of higher-order photorefractive gratings in InP: Fe,” Opt. Commun. 105, 411–420 (1994).
[Crossref]

Feinberg, J.

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980).
[Crossref]

Fischer, B.

Gaylord, T. K.

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Holographic grating formation in photorefractive crystals with arbitrary electron transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
[Crossref]

Heiman, D.

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980).
[Crossref]

Hellwarth, R. W.

Y. H. Lee and R. W. Hellwarth, “Spatial harmonics of photorefractive gratings in a barium titanate crystal,” J. Appl. Phys. 71, 916–923 (1992).
[Crossref]

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980).
[Crossref]

Hesselink, L.

Horowitz, M.

Huignard, J.-P.

Ph. Réfrégier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive B12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[Crossref]

Jarem, J. M.

P. P. Banerjee and J. M. Jarem, “Transient wave mixing and recording kinetics in photorefractive barium titanate: a nonlinear coupled mode approach,” Opt. Eng. 34, 2254–2260 (1995).
[Crossref]

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic-crystals I. Steady state,” Ferroelectrics 22, 949–961 (1979).
[Crossref]

Lee, Y. H.

Y. H. Lee and R. W. Hellwarth, “Spatial harmonics of photorefractive gratings in a barium titanate crystal,” J. Appl. Phys. 71, 916–923 (1992).
[Crossref]

López, V.

E. Serrano, V. López, M. Carrascosa, and F. Agulló-López, “Steady state photorefractive gratings in LiNbO3 for strong light modulation depths,” IEEE J. Quantum Electron. 30, 875–880 (1994).
[Crossref]

E. Serrano, V. López, M. Carrascosa, and F. Agulló-López, “Recording and erasure kinetics in photorefractive materials at high modulation depths,” J. Opt. Soc. Am. B 11, 670–675 (1994).
[Crossref]

Magnusson, R.

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Holographic grating formation in photorefractive crystals with arbitrary electron transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
[Crossref]

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic-crystals I. Steady state,” Ferroelectrics 22, 949–961 (1979).
[Crossref]

Moharam, M. G.

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Holographic grating formation in photorefractive crystals with arbitrary electron transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
[Crossref]

Ochoa, E.

Odoulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic-crystals I. Steady state,” Ferroelectrics 22, 949–961 (1979).
[Crossref]

Rajbenbach, H.

Ph. Réfrégier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive B12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[Crossref]

Réfrégier, Ph.

Ph. Réfrégier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive B12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[Crossref]

Saxena, R.

Serrano, E.

E. Serrano, M. Carrascosa, and F. Agulló-López, “Nonperturbative analytical solution for steady-state photorefractive recording,” Opt. Lett. 20, 1910–1912 (1995).
[Crossref] [PubMed]

E. Serrano, V. López, M. Carrascosa, and F. Agulló-López, “Recording and erasure kinetics in photorefractive materials at high modulation depths,” J. Opt. Soc. Am. B 11, 670–675 (1994).
[Crossref]

E. Serrano, V. López, M. Carrascosa, and F. Agulló-López, “Steady state photorefractive gratings in LiNbO3 for strong light modulation depths,” IEEE J. Quantum Electron. 30, 875–880 (1994).
[Crossref]

E. Serrano, “Análisis teórico del efecto fotorrefractivo en el régimen no lineal,” Ph.D. dissertation (Universidad Autónoma de Madrid, Madrid, Spain, 1994).

Solymar, L.

L. B. Au and L. Solymar, “Higher harmonic gratings in photorefractive materials at large modulation with moving fringes,” J. Opt. Soc. Am. A 7, 1554–1562 (1990).
[Crossref]

Ph. Réfrégier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive B12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[Crossref]

Soskin, M.

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic-crystals I. Steady state,” Ferroelectrics 22, 949–961 (1979).
[Crossref]

Tanguay, A. R.

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980).
[Crossref]

Temple, D. A.

Vachss, F.

Vinetskii, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic-crystals I. Steady state,” Ferroelectrics 22, 949–961 (1979).
[Crossref]

Warde, C.

Ferroelectrics (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic-crystals I. Steady state,” Ferroelectrics 22, 949–961 (1979).
[Crossref]

IEEE J. Quantum Electron. (2)

M. Carrascosa and F. Agulló-López, “Kinetics for optical erasure of sinusoidal holographic gratings in photorefractive materials,” IEEE J. Quantum Electron. QE-22, 1369–1375 (1986).
[Crossref]

E. Serrano, V. López, M. Carrascosa, and F. Agulló-López, “Steady state photorefractive gratings in LiNbO3 for strong light modulation depths,” IEEE J. Quantum Electron. 30, 875–880 (1994).
[Crossref]

J. Appl. Phys. (4)

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980).
[Crossref]

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Holographic grating formation in photorefractive crystals with arbitrary electron transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
[Crossref]

Ph. Réfrégier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive B12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[Crossref]

Y. H. Lee and R. W. Hellwarth, “Spatial harmonics of photorefractive gratings in a barium titanate crystal,” J. Appl. Phys. 71, 916–923 (1992).
[Crossref]

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

J. Opt. Soc. Am. B (4)

Opt. Commun. (3)

G. A. Brost, “Numerical analysis of photorefractive grating formation dynamics at large modulation in BSO,” Opt. Commun. 96, 113–116 (1993).
[Crossref]

L. Boutsikaris and F. Davidson, “Perturbative analysis of higher-order photorefractive gratings in InP: Fe,” Opt. Commun. 105, 411–420 (1994).
[Crossref]

J. V. Alvarez-Bravo, M. Carrascosa, and L. Arizmendi, “Experimental effects of light intensity modulation on the recording and erasure of holographic gratings in BSO crystals,” Opt. Commun. 103, 22–28 (1993).
[Crossref]

Opt. Eng. (1)

P. P. Banerjee and J. M. Jarem, “Transient wave mixing and recording kinetics in photorefractive barium titanate: a nonlinear coupled mode approach,” Opt. Eng. 34, 2254–2260 (1995).
[Crossref]

Opt. Lett. (1)

Other (2)

F. V. Atkinson, Discrete and Continuous Boundary Problems (Academic, New York, 1964), Chap. 6.

E. Serrano, “Análisis teórico del efecto fotorrefractivo en el régimen no lineal,” Ph.D. dissertation (Universidad Autónoma de Madrid, Madrid, Spain, 1994).

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Figures (5)

Fig. 1
Fig. 1

Time evolution for the first harmonic N1 of the modulation carrier density: numerical solution (solid curves), linear solution (dashed curves). The time has been normalized to the time constant g1 defined in Eq. (12). The modulation depth is m=0.9.

Fig. 2
Fig. 2

Time evolution of the first three Fourier space-charge gratings from nonlinear field Eqs. (5b) and (8) (solid curves) and simplified Eq. (11) (dashed curves). A basis of N=15 harmonics has been used in both cases. The data for the linear solution (fundamental grating) are shown as a dotted curve. The modulation depth is m=0.9.

Fig. 3
Fig. 3

Time evolution of the first three Fourier space-charge gratings from nonlinear field Eqs. (5b) and (8) (solid curves) and simplified Eq. (14) (dashed curves). A basis of N=15 harmonics has been used in both cases. (a) Λ=10 µm, (b) Λ=20 µm. The data for the linear solution (fundamental grating) are shown as dotted curves. The modulation depth is m=0.9.

Fig. 4
Fig. 4

Comparison of the time evolution of the three first harmonics of the space-charge field given by the numerical solution of Eqs. (5b) and (8), with N=15 (solid curves); analytical solution (16) (dashed curves); Eqs. (17) (dotted–dashed curves); and the linear solution (dotted curve). The modulation depth is m=0.9.

Fig. 5
Fig. 5

Dependence of the analytical steady-state values N=1 (dotted curve), N=2 (dotted–dashed curves), and N=3 (dashed curves) for 1, 2, and 3 on the modulation depth m. The values obtained from the full numerical calculations of Eqs. (5b) and (8) at steady state for N=15 are also shown for comparison (solid curves).

Tables (1)

Tables Icon

Table 1 Material Parameters for BSO

Equations (59)

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

ND+t=sI(ND0-ND+)-γNND+,
Nt=ND+t-1eJx,
J=eμN-μkbTNx,
x(0)=e(N+NA-ND+)
-Jx=et(N+NA-ND+).
Nt-etx=sI(ND0-ND+)-γN×NA-ex,
-tx=eμNx+eμNx-μkbT2Nx2.
I(x)=I0[1+(m/2)exp(iKx)+c.c.],
(x,t)0+j=1Nj(t)exp(ijKx)+c.c.,
N(x,t)N0+j=1NNj(t)exp(ijKx)+c.c.,
dnjdt-ijdEjdt=sI0NA2γ(ND0-ND+)δj0+m2δj1-nj+ip=j-NNpEpn(j-p),
dEjdt=-EQEMp=j-NNEpn(j-p)+iEDEMjnj.
Ej=jEQ, nj=NjNA, t=tτe,
EQ=eNAK, EM=γNAμK, ED=kbTKe,
τe=1γNA.
nj=sI0ND0NA2γδj0+m2δj1+ip=j-NNpEpn(-jp)+ijdEjdt.
1+j2EDEMdEjdt=-jEDEM[jEjn0+(j-1)Ej-1n1+(j+1)Ej+1n1]-EQEM(Ejn0+n1Ej-1+n1Ej+1)+isI0ND0NA2γm2EDEMδj1,
n0=sI0ND0γNA2,
n1(t)=n01+ED/EMm2+E1(t)EQEM-1,
n1(0)=m2n01+EDEM,
n1()m2n01+EDEM1-ED(EQ+ED)EQEM-1,
dE1dtdENdt=d1/c1b1/c100a2/c2d2/c2b2/c20000aN/cNdN/cNAˆ×E1EN+in0mED2EM100,
aj=EQEM+j2EDEM-jEDEMn1,
bj=EQEM+j2EDEM+jEDEMn1,
cj=1+j2EDEM,
dj=EQEM+j2EDEMn0.
gj=τd-11+j2EDEQ1+j2EDEM,
τd=0eμN0.
d1dtdNdt=1τd1m/200m/21m/2000m/21Aˆ 1N+imED2τd100,
1(t)=i(m/2)ED[1-exp(-t/τd)].
1(t)=i2m4-m2ED1-m+24exp[-(2-m)t/2τd]-2-m4exp[-(2+m)t/2τd],
2(t)=-im24-m2ED1-m+22mexp[-(2-m)t/τd]-m-22mexp[-(2+m)t/2τd]
1(t)=im4m2-8ED[m2-4-(m2-2)exp(-t/τd)+λ1 exp(-λ3t/τd)+λ3 exp(-λ1t/τd)],
2(t)=imED4m2-8[2m-(m+2)exp(-λ1t/τd)+(2-m)exp(-λ3t/τd)],
3(t)=imED4m2-8[-m2+(m2-2)exp(-t/τd)+λ1 exp(-λ3t/τd)+λ3 exp(-λ1t/τd)],
λ1=1-(m2/2),
λ2=1,
λ3=1+(m2/2),
1()=i4m-m34m2-8EDim2ED×1+m22+2m24+Om7,
t=tτd, M=m2, j=-ijMED
d1dtdNdt=-1M0 M1 0M0 0M 011N+100.
ΔN=βM0 Mβ 0M0 0M 0β=0,
ΔN=βΔN-1-M2ΔN-2,
Δ1=β, Δ2=β2-M2,
Δ0=1.
ΔN=aMN exp(iNΘ)+bMN exp(-iNΘ)=MN[g cos(NΘ)+ih sin(NΘ)],
ΔN=MNsin[(N+1)Θ]sin(Θ).
Θj=jπN+1, j=1,,N,
λj=1-2M cosjπN+1.
λ1=1-(m/2), λ2=1+(m/2),
λ1=1-m(2/2), λ2=1, λ3=1+m(2/2),
vsj=(-M)s-1ΔN-sΔN-1, s=2,,N,
s(t)=j=1NCjvsj exp(-λjt)+us,
1M0 M1 0M0 0M 01u1uN=100.
us=(-M)s-1ΔN-sΔN.
u1=1-1-4M22M2,
us=(-M)s-1(u1)s.
1()=iED(1-1-m2)m,
s()=iED(-1)s-11-1-m2ms.

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