Abstract

The expression for an induced space-charge electric field, including higher-order space-charge field terms, in biased photovoltaic–photorefractive crystals is obtained. There are two higher-order odd perturbation terms in the expression, and their signs depend on the polarity of the bias and the photovoltaic fields, respectively. The effects of the higher-order terms on self-deflection of screening photovoltaic bright solitons in the crystals are investigated by both numerical and perturbation methods under steady-state conditions. When the bias field is positive and the photovoltaic field is negative, the screening photovoltaic bright solitons always bend in the direction opposite the crystal’s c axis, and the absolute value of the spatial shift that is due to the first-order diffusion term alone is always smaller than that which is due to the first-order diffusion term and the higher-order terms together. When the bias field is negative and the photovoltaic field is positive or the two fields are both positive, bending in both the same direction as and in the direction opposite the crystal’s c axis is possible. Whether the direction is in the same or in the opposite direction will depend on the polarity and strength of the two fields and on the incident optical intensity; it is possible that no self-deflection will occur if the strength and polarity of the two fields and the incident intensity are appropriately selected. Relevant examples are provided.

© 2002 Optical Society of America

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References

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  1. M.-F. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. Di Porto, “Observation of two-dimensional steady-state photorefractive screening solitons,” Electron. Lett. 31, 826 (1995).
    [CrossRef]
  2. Z. Chen, M. Mitchell, M.-F. Shih, M. Segev, M. H. Garrett, and G. C. Valley, “Steady-state dark photorefractive screening solitons,” Opt. Lett. 21, 629–631 (1996).
    [CrossRef] [PubMed]
  3. C. Montemezzani and P. Gunter, “Profile of photorefractive one-dimensional bright spatial solitons,” Opt. Lett. 22, 451–453 (1997).
    [CrossRef] [PubMed]
  4. M. Segev, G. C. Valley, B. Crosignani, P. Diporto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73, 3211–3214 (1994).
    [CrossRef] [PubMed]
  5. D. N. Christodoulides and M. I. Carvalho, “Bright, dark, and gray spatial soliton states in photorefractive media,” J. Opt. Soc. Am. B 12, 1628–1633 (1995).
    [CrossRef]
  6. S. R. Singh and D. N. Christodoulides, “Evolution of spatial optical solitons in biased photorefractive media under steady state conditions,” Opt. Commun. 118, 569–576 (1995).
    [CrossRef]
  7. M. Segev, George C. Valley, M. C. Bashaw, M. Taya, and M. M. Fejer, “Photovoltaic spatial solitons,” J. Opt. Soc. Am. B 14, 1772–1781 (1997).
    [CrossRef]
  8. M. Taya, M. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, “Observation of dark photovoltaic spatial solitons,” Phys. Rev. A 52, 3095–3100 (1995).
    [CrossRef] [PubMed]
  9. G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, and M. C. Bashaw, “Dark and bright photovoltaic spatial solitons,” Phys. Rev. A 50, R4457–R4460 (1994).
    [CrossRef] [PubMed]
  10. J. Liu and K. Lu, “Screening-photovoltaic spatial solitons in biased photovoltaicphotorefractive crystals and their self-deflection,” J. Opt. Soc. Am. B 16, 550–555 (1999).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  13. J. Petter, C. Weilbau, C. Denz, A. Stepken, and F. Kaiser, “Self-bending of photorefractive solitons,” Opt. Commun. 170, 291–297 (1999).
    [CrossRef]
  14. M. I. Carvalho, S. R. Singh, and D. N. Christodoulides, “Self-deflection of steady-state bright spatial solitons in biased photorefractive crystals,” Opt. Commun. 120, 311–315 (1995).
    [CrossRef]
  15. S. R. Singh, M. I. Carvalho, and D. N. Christodoulides, “Higher-order space charge field effects on the evolution of spatial solitons in biased photorefractive crystals,” Opt. Commun. 130, 288–294 (1996).
    [CrossRef]
  16. B. I. Sturman and V. M. Fridkin, The Photovoltaic and Photorefractive Effect in Non-centrosymmetric Materials (Gordon & Breach, Philadelphia, Pa., 1992).
  17. Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23, 510–524 (1987).
    [CrossRef]
  18. K. J. Blow, N. J. Doran, and D. Wood, “Generation and stabilization of short soliton pulses in the amplified nonlinear Schrödinger equation,” J. Opt. Soc. Am. B 5, 1301–1304 (1988).
    [CrossRef]

2000 (1)

J. Liu, D. Zhang, and C. Liang, “Stability of bright screening photovoltaic spatial solitions,” Chin. Phys. 9, 667–671 (2000).
[CrossRef]

1999 (2)

J. Petter, C. Weilbau, C. Denz, A. Stepken, and F. Kaiser, “Self-bending of photorefractive solitons,” Opt. Commun. 170, 291–297 (1999).
[CrossRef]

J. Liu and K. Lu, “Screening-photovoltaic spatial solitons in biased photovoltaicphotorefractive crystals and their self-deflection,” J. Opt. Soc. Am. B 16, 550–555 (1999).
[CrossRef]

1997 (2)

1996 (2)

Z. Chen, M. Mitchell, M.-F. Shih, M. Segev, M. H. Garrett, and G. C. Valley, “Steady-state dark photorefractive screening solitons,” Opt. Lett. 21, 629–631 (1996).
[CrossRef] [PubMed]

S. R. Singh, M. I. Carvalho, and D. N. Christodoulides, “Higher-order space charge field effects on the evolution of spatial solitons in biased photorefractive crystals,” Opt. Commun. 130, 288–294 (1996).
[CrossRef]

1995 (6)

M. I. Carvalho, S. R. Singh, and D. N. Christodoulides, “Self-deflection of steady-state bright spatial solitons in biased photorefractive crystals,” Opt. Commun. 120, 311–315 (1995).
[CrossRef]

M.-F. Shih, P. Leach, M. Segev, M. H. Garret, G. J. Salamo, and G. C. Valley, “Two-dimensional steady-state photorefractive screening solitons,” Opt. Lett. 21, 324–326 (1995).
[CrossRef]

M.-F. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. Di Porto, “Observation of two-dimensional steady-state photorefractive screening solitons,” Electron. Lett. 31, 826 (1995).
[CrossRef]

M. Taya, M. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, “Observation of dark photovoltaic spatial solitons,” Phys. Rev. A 52, 3095–3100 (1995).
[CrossRef] [PubMed]

D. N. Christodoulides and M. I. Carvalho, “Bright, dark, and gray spatial soliton states in photorefractive media,” J. Opt. Soc. Am. B 12, 1628–1633 (1995).
[CrossRef]

S. R. Singh and D. N. Christodoulides, “Evolution of spatial optical solitons in biased photorefractive media under steady state conditions,” Opt. Commun. 118, 569–576 (1995).
[CrossRef]

1994 (2)

G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, and M. C. Bashaw, “Dark and bright photovoltaic spatial solitons,” Phys. Rev. A 50, R4457–R4460 (1994).
[CrossRef] [PubMed]

M. Segev, G. C. Valley, B. Crosignani, P. Diporto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73, 3211–3214 (1994).
[CrossRef] [PubMed]

1988 (1)

1987 (1)

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23, 510–524 (1987).
[CrossRef]

Bashaw, M.

M. Taya, M. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, “Observation of dark photovoltaic spatial solitons,” Phys. Rev. A 52, 3095–3100 (1995).
[CrossRef] [PubMed]

Bashaw, M. C.

M. Segev, George C. Valley, M. C. Bashaw, M. Taya, and M. M. Fejer, “Photovoltaic spatial solitons,” J. Opt. Soc. Am. B 14, 1772–1781 (1997).
[CrossRef]

G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, and M. C. Bashaw, “Dark and bright photovoltaic spatial solitons,” Phys. Rev. A 50, R4457–R4460 (1994).
[CrossRef] [PubMed]

Blow, K. J.

Carvalho, M. I.

S. R. Singh, M. I. Carvalho, and D. N. Christodoulides, “Higher-order space charge field effects on the evolution of spatial solitons in biased photorefractive crystals,” Opt. Commun. 130, 288–294 (1996).
[CrossRef]

M. I. Carvalho, S. R. Singh, and D. N. Christodoulides, “Self-deflection of steady-state bright spatial solitons in biased photorefractive crystals,” Opt. Commun. 120, 311–315 (1995).
[CrossRef]

D. N. Christodoulides and M. I. Carvalho, “Bright, dark, and gray spatial soliton states in photorefractive media,” J. Opt. Soc. Am. B 12, 1628–1633 (1995).
[CrossRef]

Chen, Z.

Christodoulides, D. N.

S. R. Singh, M. I. Carvalho, and D. N. Christodoulides, “Higher-order space charge field effects on the evolution of spatial solitons in biased photorefractive crystals,” Opt. Commun. 130, 288–294 (1996).
[CrossRef]

M. I. Carvalho, S. R. Singh, and D. N. Christodoulides, “Self-deflection of steady-state bright spatial solitons in biased photorefractive crystals,” Opt. Commun. 120, 311–315 (1995).
[CrossRef]

S. R. Singh and D. N. Christodoulides, “Evolution of spatial optical solitons in biased photorefractive media under steady state conditions,” Opt. Commun. 118, 569–576 (1995).
[CrossRef]

D. N. Christodoulides and M. I. Carvalho, “Bright, dark, and gray spatial soliton states in photorefractive media,” J. Opt. Soc. Am. B 12, 1628–1633 (1995).
[CrossRef]

Crosignani, B.

M.-F. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. Di Porto, “Observation of two-dimensional steady-state photorefractive screening solitons,” Electron. Lett. 31, 826 (1995).
[CrossRef]

M. Segev, G. C. Valley, B. Crosignani, P. Diporto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73, 3211–3214 (1994).
[CrossRef] [PubMed]

G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, and M. C. Bashaw, “Dark and bright photovoltaic spatial solitons,” Phys. Rev. A 50, R4457–R4460 (1994).
[CrossRef] [PubMed]

Denz, C.

J. Petter, C. Weilbau, C. Denz, A. Stepken, and F. Kaiser, “Self-bending of photorefractive solitons,” Opt. Commun. 170, 291–297 (1999).
[CrossRef]

Di Porto, P.

M.-F. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. Di Porto, “Observation of two-dimensional steady-state photorefractive screening solitons,” Electron. Lett. 31, 826 (1995).
[CrossRef]

Diporto, P.

M. Segev, G. C. Valley, B. Crosignani, P. Diporto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73, 3211–3214 (1994).
[CrossRef] [PubMed]

Doran, N. J.

Fejer, M. M.

M. Segev, George C. Valley, M. C. Bashaw, M. Taya, and M. M. Fejer, “Photovoltaic spatial solitons,” J. Opt. Soc. Am. B 14, 1772–1781 (1997).
[CrossRef]

M. Taya, M. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, “Observation of dark photovoltaic spatial solitons,” Phys. Rev. A 52, 3095–3100 (1995).
[CrossRef] [PubMed]

G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, and M. C. Bashaw, “Dark and bright photovoltaic spatial solitons,” Phys. Rev. A 50, R4457–R4460 (1994).
[CrossRef] [PubMed]

Garret, M. H.

Garrett, M. H.

Gunter, P.

Hasegawa, A.

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23, 510–524 (1987).
[CrossRef]

Kaiser, F.

J. Petter, C. Weilbau, C. Denz, A. Stepken, and F. Kaiser, “Self-bending of photorefractive solitons,” Opt. Commun. 170, 291–297 (1999).
[CrossRef]

Kodama, Y.

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23, 510–524 (1987).
[CrossRef]

Leach, P.

Liang, C.

J. Liu, D. Zhang, and C. Liang, “Stability of bright screening photovoltaic spatial solitions,” Chin. Phys. 9, 667–671 (2000).
[CrossRef]

Liu, J.

Lu, K.

Mitchell, M.

Montemezzani, C.

Petter, J.

J. Petter, C. Weilbau, C. Denz, A. Stepken, and F. Kaiser, “Self-bending of photorefractive solitons,” Opt. Commun. 170, 291–297 (1999).
[CrossRef]

Salamo, G.

M.-F. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. Di Porto, “Observation of two-dimensional steady-state photorefractive screening solitons,” Electron. Lett. 31, 826 (1995).
[CrossRef]

Salamo, G. J.

Segev, M.

M. Segev, George C. Valley, M. C. Bashaw, M. Taya, and M. M. Fejer, “Photovoltaic spatial solitons,” J. Opt. Soc. Am. B 14, 1772–1781 (1997).
[CrossRef]

Z. Chen, M. Mitchell, M.-F. Shih, M. Segev, M. H. Garrett, and G. C. Valley, “Steady-state dark photorefractive screening solitons,” Opt. Lett. 21, 629–631 (1996).
[CrossRef] [PubMed]

M.-F. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. Di Porto, “Observation of two-dimensional steady-state photorefractive screening solitons,” Electron. Lett. 31, 826 (1995).
[CrossRef]

M. Taya, M. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, “Observation of dark photovoltaic spatial solitons,” Phys. Rev. A 52, 3095–3100 (1995).
[CrossRef] [PubMed]

M.-F. Shih, P. Leach, M. Segev, M. H. Garret, G. J. Salamo, and G. C. Valley, “Two-dimensional steady-state photorefractive screening solitons,” Opt. Lett. 21, 324–326 (1995).
[CrossRef]

G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, and M. C. Bashaw, “Dark and bright photovoltaic spatial solitons,” Phys. Rev. A 50, R4457–R4460 (1994).
[CrossRef] [PubMed]

M. Segev, G. C. Valley, B. Crosignani, P. Diporto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73, 3211–3214 (1994).
[CrossRef] [PubMed]

Shih, M.-F.

Singh, S. R.

S. R. Singh, M. I. Carvalho, and D. N. Christodoulides, “Higher-order space charge field effects on the evolution of spatial solitons in biased photorefractive crystals,” Opt. Commun. 130, 288–294 (1996).
[CrossRef]

M. I. Carvalho, S. R. Singh, and D. N. Christodoulides, “Self-deflection of steady-state bright spatial solitons in biased photorefractive crystals,” Opt. Commun. 120, 311–315 (1995).
[CrossRef]

S. R. Singh and D. N. Christodoulides, “Evolution of spatial optical solitons in biased photorefractive media under steady state conditions,” Opt. Commun. 118, 569–576 (1995).
[CrossRef]

Stepken, A.

J. Petter, C. Weilbau, C. Denz, A. Stepken, and F. Kaiser, “Self-bending of photorefractive solitons,” Opt. Commun. 170, 291–297 (1999).
[CrossRef]

Taya, M.

M. Segev, George C. Valley, M. C. Bashaw, M. Taya, and M. M. Fejer, “Photovoltaic spatial solitons,” J. Opt. Soc. Am. B 14, 1772–1781 (1997).
[CrossRef]

M. Taya, M. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, “Observation of dark photovoltaic spatial solitons,” Phys. Rev. A 52, 3095–3100 (1995).
[CrossRef] [PubMed]

Valley, G. C.

Z. Chen, M. Mitchell, M.-F. Shih, M. Segev, M. H. Garrett, and G. C. Valley, “Steady-state dark photorefractive screening solitons,” Opt. Lett. 21, 629–631 (1996).
[CrossRef] [PubMed]

M.-F. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. Di Porto, “Observation of two-dimensional steady-state photorefractive screening solitons,” Electron. Lett. 31, 826 (1995).
[CrossRef]

M. Taya, M. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, “Observation of dark photovoltaic spatial solitons,” Phys. Rev. A 52, 3095–3100 (1995).
[CrossRef] [PubMed]

M.-F. Shih, P. Leach, M. Segev, M. H. Garret, G. J. Salamo, and G. C. Valley, “Two-dimensional steady-state photorefractive screening solitons,” Opt. Lett. 21, 324–326 (1995).
[CrossRef]

G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, and M. C. Bashaw, “Dark and bright photovoltaic spatial solitons,” Phys. Rev. A 50, R4457–R4460 (1994).
[CrossRef] [PubMed]

M. Segev, G. C. Valley, B. Crosignani, P. Diporto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73, 3211–3214 (1994).
[CrossRef] [PubMed]

Valley, George C.

Weilbau, C.

J. Petter, C. Weilbau, C. Denz, A. Stepken, and F. Kaiser, “Self-bending of photorefractive solitons,” Opt. Commun. 170, 291–297 (1999).
[CrossRef]

Wood, D.

Yariv, A.

G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, and M. C. Bashaw, “Dark and bright photovoltaic spatial solitons,” Phys. Rev. A 50, R4457–R4460 (1994).
[CrossRef] [PubMed]

M. Segev, G. C. Valley, B. Crosignani, P. Diporto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73, 3211–3214 (1994).
[CrossRef] [PubMed]

Zhang, D.

J. Liu, D. Zhang, and C. Liang, “Stability of bright screening photovoltaic spatial solitions,” Chin. Phys. 9, 667–671 (2000).
[CrossRef]

Chin. Phys. (1)

J. Liu, D. Zhang, and C. Liang, “Stability of bright screening photovoltaic spatial solitions,” Chin. Phys. 9, 667–671 (2000).
[CrossRef]

Electron. Lett. (1)

M.-F. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. Di Porto, “Observation of two-dimensional steady-state photorefractive screening solitons,” Electron. Lett. 31, 826 (1995).
[CrossRef]

IEEE J. Quantum Electron. (1)

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23, 510–524 (1987).
[CrossRef]

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

Opt. Commun. (4)

S. R. Singh and D. N. Christodoulides, “Evolution of spatial optical solitons in biased photorefractive media under steady state conditions,” Opt. Commun. 118, 569–576 (1995).
[CrossRef]

J. Petter, C. Weilbau, C. Denz, A. Stepken, and F. Kaiser, “Self-bending of photorefractive solitons,” Opt. Commun. 170, 291–297 (1999).
[CrossRef]

M. I. Carvalho, S. R. Singh, and D. N. Christodoulides, “Self-deflection of steady-state bright spatial solitons in biased photorefractive crystals,” Opt. Commun. 120, 311–315 (1995).
[CrossRef]

S. R. Singh, M. I. Carvalho, and D. N. Christodoulides, “Higher-order space charge field effects on the evolution of spatial solitons in biased photorefractive crystals,” Opt. Commun. 130, 288–294 (1996).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (2)

M. Taya, M. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, “Observation of dark photovoltaic spatial solitons,” Phys. Rev. A 52, 3095–3100 (1995).
[CrossRef] [PubMed]

G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, and M. C. Bashaw, “Dark and bright photovoltaic spatial solitons,” Phys. Rev. A 50, R4457–R4460 (1994).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

M. Segev, G. C. Valley, B. Crosignani, P. Diporto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73, 3211–3214 (1994).
[CrossRef] [PubMed]

Other (1)

B. I. Sturman and V. M. Fridkin, The Photovoltaic and Photorefractive Effect in Non-centrosymmetric Materials (Gordon & Breach, Philadelphia, Pa., 1992).

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Figures (9)

Fig. 1
Fig. 1

(a) Intensity profile evolution of a SP bright soliton when the diffusion effect and higher-order terms are taken into account in a biased photovoltaic–photorefractive crystal with r=10, Ep=105 V/m, E0=5×105 V/m, α=31.6, β=158, γ=0.204, γ1=0.177, γ2=0.035, γ3=4.57×10-4, and γ4=2.28×10-4. The input fundamental SP bright soliton was obtained from Eq. (11) with r=10, α=158, and β=158. (b) Comparison of diffusion term and all γi term contributions. The two curves were obtained for the γ term alone (dashed curve) and for all the γ terms together (solid curve).

Fig. 2
Fig. 2

Evolution of spatial shift Δs when E0>0, Ep<0, and E0>|Ep|. In each case the two curves were obtained for the γ term alone (dashed curve) and for all the γ terms together (solid curve). (a) The input SP bright soliton was obtained from Eq. (11) with r=10. Comparisons of values of (1) E0=2×105 V/m and Ep=-105 V/m, (2) E0=5×105 V/m, and Ep=-4×105 V/m, and (3) E0=2×106 V/m and Ep=-106 V/m. (b) Comparisons of values of (1) r=1, (2) r=10, and (3) r=100 with E0=5×105 V/m and Ep=-4×105 V/m.

Fig. 3
Fig. 3

Evolution of spatial shift Δs when E0<0, Ep>0, and Ep>|E0|. (a) Comparisons of values of (1) E0=-105 V/m and Ep=2×105 V/m and (2) E0=-106 V/m and Ep=2×106 V/m. The input SP bright soliton was obtained from Eq. (11) with r=10. (b) Comparison of values of (1) r=1, (2) r=10, and (3) r=100 with E0=-106 V/m and Ep=2×106 V/m.

Fig. 4
Fig. 4

Evolution of spatial shift Δs when E0>0 and Ep>0. (a) Comparison of values of (1) E0=5×105 V/m and Ep=5×105 V/m and (2) E0=5×105 V/m and Ep=2×106 V/m. The input SP bright soliton was obtained from Eq. (11) with r=10. (b) Comparison of values of (1) r=0.5, (2) r=10, and (3) r=100 with E0=5×105 V/m and Ep=2×106 V/m.

Fig. 5
Fig. 5

Evolution of spatial shift of SP screening and photovoltaic bright solitons. (a) Ep=106 V/m: (1) E0=5×105 V/m, (2) E0=0, and (3) E0=-5×105 V/m; (b) E0=106 V/m: (1) Ep=5×105 V/m, (2) Ep=0, and (3) Ep=-5×105 V/m.

Fig. 6
Fig. 6

(a) Dependence of K(r), K1(r), and K2(r) on r. (b), (c) Expanded views of K1(r) and K2(r), respectively. The values of α and β are 300 and 200, respectively.

Fig. 7
Fig. 7

Dependence of (a) γK(r), γ1K1(r), and γ2K2(r) on r and (b) γK(r)+γ1K1(r)-γ2K2(r) on r, with γ=0.204, E0=-106 V/m, and Ep=2×106 V/m. The corresponding values of α, β, γ1, and γ2 are α=632, β=-316, γ1=-0.589, and γ2=1.18.

Fig. 8
Fig. 8

Evolution of spatial shift Δs obtained from the numerical solution (solid curves) and from the analytical model (dashed curves). Comparison of values of r=1, 14.1, 50. SP bright soliton solutions were obtained at E0=-106 V/m and Ep=2×106 V/m. The corresponding values of α, β, γ1, and γ2 are α=632, β=-316, γ1=-0.589, γ2=1.18, γ3=7.61×10-4, and γ4=3.81×10-4. First-order diffusion term γ equals 0.204.

Fig. 9
Fig. 9

Dependence of (a) γK(r), γ1K1(r), and γ2K2(r) on r and (b) γK(r)+γ1K1(r)-γ2K2(r) on r with γ=0.204, E0=5×105 V/m, and Ep=2×106 V/m. The corresponding values of α, β, γ1, and γ2 are α=632, β=158, γ1=0.74, and γ2=2.95.

Equations (20)

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Esc=Esc01+ε0εreNAEscx-KBTe1I+IdIx+KBTeε0εreNA2Escx2,
Esc0=E0IdI+Id-EpII+Id,
Esc=Esc0+Eγ+Eγ1+Eγ2+Eγ3+Eγ4,
Eγ=-KBTe1I+IdIx,
Eγ1=-ε0εreNAE0(E0+EP)Id2(I+Id)3Ix,
Eγ2=ε0εreNAEp(E0+Ep)IId(I+Id)3Ix,
Eγ3=2KBTeε0εreNA(E0+Ep)Id(I+Id)3Ix2,
Eγ4=-KBTeε0εreNA(E0+Ep)Id(I+Id)22Ix2.
iUξ+½ Uss-βU1+|U|2+α|U|2U1+|U|2+γ(|U|2)sU1+|U|2+γ1(|U|2)sU(1+|U|2)3-γ2|U|2(|U|2)sU(1+|U|2)3
-γ3[(|U|2)s]2U(1+|U|2)3+γ4(|U|2)ssU(1+|U|2)2=0,
iUξ+12Uss-βU1+|U|2+α|U|2U1+|U|2=0.
[2(β+α)]1/2s=±y1r1/2dyˆ[ln(1+ryˆ2)-yˆ2 ln(1+r)]1/2,
U=r1/2y[s+u(ξ)]exp{i[νξ+ω(ξ)(s+u(ξ))+σ(ξ)]},
K(r)=-+ds2y2(s)1+ry2(s){y2(s)ln(1+r)-ln[1+ry2(s)]}-+dsy2(s)-1,
K1(r)=-+ds2y2(s)[1+ry2(s)]3{y2(s)ln(1+r)-ln[1+ry2(s)]}-+dsy2(s)-1,
K2(r)=-+ ds2ry4(s)[1+ry2(s)]3{y2(s)ln(1+r)-ln[1+ry2(s)]}-+dsy2(s)-1.
ω(ξ)=4(β+α)[γK(r)+γ1K1(r)-γ2K2(r)]ξ,
u(ξ)=-2(β+α)[γK(r)+γ1K1(r)-γ2K2(r)]ξ2,
σ(ξ)=8{(β+α)[γK(r)+γ1K1(r)-γ2K2(r)]}2ξ3/3.
Δs(ξ)=2(β+α)[γK(r)+γ1K1(r)-γ2K2(r)]ξ2.

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