Abstract

We study modulational instability and beam propagation through a photorefractive polymer in the presence of absorption losses. The one-dimensional beam propagation through the nonlinear medium is studied using variational and numerical methods. Stable soliton propagation is observed both analytically and numerically.

© 2008 Optical Society of America

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References

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  1. R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479-482 (1964).
    [CrossRef]
  2. M. Segev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923-926 (1992).
    [CrossRef] [PubMed]
  3. 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]
  4. D. N. Christodoulides and M. J. Carvalho, “Bright, dark, and gray spatial soliton states in photorefractive media,” J. Opt. Soc. Am. B 12, 1628-1633 (1995).
    [CrossRef]
  5. 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]
  6. M.-F. Shih and F.-W. Sheu, “Photorefractive polymeric optical spatial solitons,” Opt. Lett. 24, 1853-1855 (1999).
    [CrossRef]
  7. Z. Chen, M. Asaro, O. Ostroverkhova, W. E. Moerner, M. He, and R. J. Twieg, “Self-trapping of light in an organic photorefractive glass,” Opt. Lett. 28, 2509-2511 (2003).
    [CrossRef] [PubMed]
  8. S. Ducharme, J. C. Scott, R. J. Twieg, and W. E. Moerner, “Observation of the photorefractive effect in a polymer,” Phys. Rev. Lett. 66, 1846-1849 (1991).
    [CrossRef] [PubMed]
  9. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “The physics and applications of photorefractive materials,” Ferroelectrics 22, 949-960 (1979).
    [CrossRef]
  10. P. J. Melz, “Photogeneration in trinitrofluorenone-poly(n-vinylcarbazole),” J. Chem. Phys. 57, 1694-1699 (1972).
    [CrossRef]
  11. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 2002).
  12. J. M. Bilbaut, P. Marquite, and B. Michaux, “Modulational instability of two counterpropagating waves in an experimental transmission line,” Phys. Rev. E 51, 817-820 (1995).
    [CrossRef]
  13. A. Hasegawa, Plasma Instabilities and Nonlinear Effects (Springer, 1975).
    [CrossRef]
  14. H. He, A. Arraf, C. M. de Sterke, P. D. Drummond, and B. A. Malomed, “Theory of modulational instability in Bragg gratings with quadratic nonlinearity,” Phys. Rev. E 59, 6064-6078 (1999).
    [CrossRef]
  15. J. F. Corney and O. Bang, “Complete modulational-instability gain spectrum of nonlinear quasi-phase-matching gratings,” J. Opt. Soc. Am. B 21, 617-621 (2004).
    [CrossRef]
  16. A. Mohamadou and T. C. Kofané, “Modulational instability and pattern formation in discrete dissipative systems,” Phys. Rev. E 73, 046607 (2006).
    [CrossRef]
  17. M. Saffman, D. Montgomery, A. A. Zozulya, K. Kuroda, and D. Z. Anderson, “Transverse instability of counterpropagating waves in photorefractive media,” Phys. Rev. A 48, 3209-3215 (1993).
    [CrossRef] [PubMed]
  18. M. I. Carvalho, S. R. Singh, and D. Christodoulides, “Modulational instability of quasi-plane-wave optical beams biased in photorefractive crystals,” Opt. Commun. 126, 167-174 (1996).
    [CrossRef]
  19. R. Boyd, Nonlinear Optics, 3rd, ed. (Wiley, 1998).
  20. N. Zhu, R. Guo, S. Liu, Z. Liu, and T. Song, “Spatial modulation instability in self-defocusing photorefractive crystal LiNbO3:Fe,” J. Opt. A, Pure Appl. Opt. 8, 149154 (2006).
  21. D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135-3145 (1983).
    [CrossRef]
  22. C. P. Jisha, V. C. Kuriakose, and K. Porsezian, “Variational approach to spatial optical solitons in bulk cubic-quintic media stabilized by self-induced multiphoton ionization,” Phys. Rev. E 71, 056615 (2005).
    [CrossRef]
  23. W. E. Moerner, A. Grunnet-Jepsen, and C. L. Thompson, “Photorefractive polymers,” Annu. Rev. Mater. Sci. 27, 585-623 (1997).
    [CrossRef]

2006 (2)

A. Mohamadou and T. C. Kofané, “Modulational instability and pattern formation in discrete dissipative systems,” Phys. Rev. E 73, 046607 (2006).
[CrossRef]

N. Zhu, R. Guo, S. Liu, Z. Liu, and T. Song, “Spatial modulation instability in self-defocusing photorefractive crystal LiNbO3:Fe,” J. Opt. A, Pure Appl. Opt. 8, 149154 (2006).

2005 (1)

C. P. Jisha, V. C. Kuriakose, and K. Porsezian, “Variational approach to spatial optical solitons in bulk cubic-quintic media stabilized by self-induced multiphoton ionization,” Phys. Rev. E 71, 056615 (2005).
[CrossRef]

2004 (1)

2003 (1)

1999 (2)

M.-F. Shih and F.-W. Sheu, “Photorefractive polymeric optical spatial solitons,” Opt. Lett. 24, 1853-1855 (1999).
[CrossRef]

H. He, A. Arraf, C. M. de Sterke, P. D. Drummond, and B. A. Malomed, “Theory of modulational instability in Bragg gratings with quadratic nonlinearity,” Phys. Rev. E 59, 6064-6078 (1999).
[CrossRef]

1997 (1)

W. E. Moerner, A. Grunnet-Jepsen, and C. L. Thompson, “Photorefractive polymers,” Annu. Rev. Mater. Sci. 27, 585-623 (1997).
[CrossRef]

1996 (1)

M. I. Carvalho, S. R. Singh, and D. Christodoulides, “Modulational instability of quasi-plane-wave optical beams biased in photorefractive crystals,” Opt. Commun. 126, 167-174 (1996).
[CrossRef]

1995 (2)

J. M. Bilbaut, P. Marquite, and B. Michaux, “Modulational instability of two counterpropagating waves in an experimental transmission line,” Phys. Rev. E 51, 817-820 (1995).
[CrossRef]

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

1993 (1)

M. Saffman, D. Montgomery, A. A. Zozulya, K. Kuroda, and D. Z. Anderson, “Transverse instability of counterpropagating waves in photorefractive media,” Phys. Rev. A 48, 3209-3215 (1993).
[CrossRef] [PubMed]

1992 (1)

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923-926 (1992).
[CrossRef] [PubMed]

1991 (1)

S. Ducharme, J. C. Scott, R. J. Twieg, and W. E. Moerner, “Observation of the photorefractive effect in a polymer,” Phys. Rev. Lett. 66, 1846-1849 (1991).
[CrossRef] [PubMed]

1983 (1)

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135-3145 (1983).
[CrossRef]

1979 (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “The physics and applications of photorefractive materials,” Ferroelectrics 22, 949-960 (1979).
[CrossRef]

1972 (1)

P. J. Melz, “Photogeneration in trinitrofluorenone-poly(n-vinylcarbazole),” J. Chem. Phys. 57, 1694-1699 (1972).
[CrossRef]

1964 (1)

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479-482 (1964).
[CrossRef]

Annu. Rev. Mater. Sci. (1)

W. E. Moerner, A. Grunnet-Jepsen, and C. L. Thompson, “Photorefractive polymers,” Annu. Rev. Mater. Sci. 27, 585-623 (1997).
[CrossRef]

Ferroelectrics (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “The physics and applications of photorefractive materials,” Ferroelectrics 22, 949-960 (1979).
[CrossRef]

J. Chem. Phys. (1)

P. J. Melz, “Photogeneration in trinitrofluorenone-poly(n-vinylcarbazole),” J. Chem. Phys. 57, 1694-1699 (1972).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

N. Zhu, R. Guo, S. Liu, Z. Liu, and T. Song, “Spatial modulation instability in self-defocusing photorefractive crystal LiNbO3:Fe,” J. Opt. A, Pure Appl. Opt. 8, 149154 (2006).

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

Opt. Commun. (1)

M. I. Carvalho, S. R. Singh, and D. Christodoulides, “Modulational instability of quasi-plane-wave optical beams biased in photorefractive crystals,” Opt. Commun. 126, 167-174 (1996).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (3)

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. Saffman, D. Montgomery, A. A. Zozulya, K. Kuroda, and D. Z. Anderson, “Transverse instability of counterpropagating waves in photorefractive media,” Phys. Rev. A 48, 3209-3215 (1993).
[CrossRef] [PubMed]

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135-3145 (1983).
[CrossRef]

Phys. Rev. E (4)

C. P. Jisha, V. C. Kuriakose, and K. Porsezian, “Variational approach to spatial optical solitons in bulk cubic-quintic media stabilized by self-induced multiphoton ionization,” Phys. Rev. E 71, 056615 (2005).
[CrossRef]

H. He, A. Arraf, C. M. de Sterke, P. D. Drummond, and B. A. Malomed, “Theory of modulational instability in Bragg gratings with quadratic nonlinearity,” Phys. Rev. E 59, 6064-6078 (1999).
[CrossRef]

A. Mohamadou and T. C. Kofané, “Modulational instability and pattern formation in discrete dissipative systems,” Phys. Rev. E 73, 046607 (2006).
[CrossRef]

J. M. Bilbaut, P. Marquite, and B. Michaux, “Modulational instability of two counterpropagating waves in an experimental transmission line,” Phys. Rev. E 51, 817-820 (1995).
[CrossRef]

Phys. Rev. Lett. (4)

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479-482 (1964).
[CrossRef]

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923-926 (1992).
[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]

S. Ducharme, J. C. Scott, R. J. Twieg, and W. E. Moerner, “Observation of the photorefractive effect in a polymer,” Phys. Rev. Lett. 66, 1846-1849 (1991).
[CrossRef] [PubMed]

Other (3)

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 2002).

A. Hasegawa, Plasma Instabilities and Nonlinear Effects (Springer, 1975).
[CrossRef]

R. Boyd, Nonlinear Optics, 3rd, ed. (Wiley, 1998).

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Figures (8)

Fig. 1
Fig. 1

Normalized growth rate as a function of normalized spatial frequency κ for PR crystal (curve), PR polymer with no loss (dotted curve) and PR polymer with loss (thick curve).

Fig. 2
Fig. 2

Normalized gain G versus normalized spatial frequency κ and normalized input power P for PR polymer with loss.

Fig. 3
Fig. 3

Normalized gain G versus normalized spatial frequency κ and normalized input power P for PR polymer with no loss.

Fig. 4
Fig. 4

Normalized gain G versus normalized spatial frequency κ and normalized input power P for PR crystal.

Fig. 5
Fig. 5

Qualitative plot of the normalized potential function versus normalized width when β is positive. Dotted curve gives the linear case.

Fig. 6
Fig. 6

Qualitative plot of the normalized potential function versus normalized width when β is negative (intermediately high nonlinearity). Dotted curve gives the linear case.

Fig. 7
Fig. 7

Qualitative plot of the normalized potential function versus normalized width showing trapping of the beam. Dotted curve gives the linear case.

Fig. 8
Fig. 8

Soliton propagation through the PR polymer 1. The soliton propagates 3 cm through the material, which corresponds to 10 diffraction length.

Tables (1)

Tables Icon

Table 1 Absorption Coefficient of Some Photorefractive Polymers

Equations (28)

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i ϕ z + 1 2 k ϕ x x + k 0 2 2 k C x , y E 0 2 ( 1 + γ 1 + ϕ 2 ) 2 ( m + 1 ) ϕ + i 2 α 0 ϕ = 0 ,
C x = 0.54 N c h Δ α ( μ d k b T a ) 2 ,
C y = C x 2 ,
i ϕ ζ + ϕ ξ ξ + β ( 1 + γ 1 + ϕ 2 ) 2 ( m + 1 ) ϕ + i α ϕ = 0 ,
i Q ζ + Q ξ ξ + β ( 1 + γ 1 + Q 2 e 2 α ζ ) 2 ( m + 1 ) Q = 0 .
Q ( ζ , ξ ) = P exp [ i Φ ( ζ ) ] ,
H [ 2 m + 1 , 2 m + 1 , 3 + m m + 1 , e 2 α ζ P ]
Q ( ζ , ξ ) = ( P + a ( ζ , ξ ) ) exp [ i Φ ( ζ ) ] .
i a ζ + a ξ ξ β ( 1 + γ ) 2 m + 1 ( 1 ( 1 + P e 2 α ζ ) 2 m + 1 1 ( 1 + P e 2 α ζ ) 4 m + 1 ) a β ( 1 + γ ) 2 m + 1 2 1 + m P e 2 α ζ ( 1 + P e 2 α ζ ) 4 m + 1 a * = 0 .
a ( ζ , ξ ) = U ( ζ ) exp [ i κ ξ ] + V ( ζ ) exp [ i κ ξ ] .
U ζ = i ( κ 2 + A ) U i B V * ,
V ζ * = i B U + i ( κ 2 + A ) V * .
M = ( a 11 a 22 a 33 a 44 ) ,
A = β ( 1 + γ ) 2 m + 1 ( 1 ( 1 + P e 2 α ζ ) 2 m + 1 1 ( 1 + P e 2 α ζ ) 4 m + 1 ) ,
B = β ( 1 + γ ) 2 m + 1 2 1 + m ( P e 2 α ζ ( 1 + P e 2 α ζ ) 4 m + 1 ) .
Λ ± = ± ( B 2 A 2 2 A κ 2 κ 4 ) 1 2 .
L = ( i 2 ( ϕ ϕ ζ * ϕ * ϕ ζ ) + ϕ ξ 2 + β ( 1 + γ ) 1 + m 1 m ( 1 + γ 1 + ϕ 2 ) ( 1 m ) ( 1 + m ) ) e 2 α ζ .
ϕ ( ζ , ξ ) = A ( ζ ) exp [ ξ 2 2 ( a ) ζ 2 + i b ( ζ ) ξ 2 ] ,
L = e 2 α ζ { i ( A A ζ * A * A ζ ) a π 4 + 2 b ζ A 2 a 3 π 4 + ( 1 a 4 + 4 b 2 ) A 2 a 3 π 4 ( 1 + γ ) 2 ( 1 + m ) β A 2 a π 2 + ( 1 + γ ) 2 ( 1 + m ) β A 4 a 2 ( 1 + m ) π 4 + C } ,
z ( L q z ) L q = 0 ,
A L A = e 2 α ζ { i A A ζ * a π 4 + 2 b ζ A 2 a 3 π 4 + A 2 ( 1 a 4 + 4 b 2 ) a 3 π 4 ( 1 + γ ) 2 ( 1 + m ) β A 2 a π 2 + ( 1 + γ ) 2 ( 1 + m ) β A 4 a π 2 ( m + 1 ) } ,
A * L A * = e 2 α ζ { i A * A ζ a π 4 + 2 b ζ A 2 a 3 π 4 + A 2 ( 1 a 4 + 4 b 2 ) a 3 π 4 ( 1 + γ ) 2 ( 1 + m ) β A 2 a π 2 + ( 1 + γ ) 2 ( 1 + m ) β A 4 a π 2 ( m + 1 ) } ,
a ( ζ ) A ( ζ ) 2 e 2 α ζ = a 0 A 0 2 = E 0 ,
b = 1 2 a d a d ζ ,
d 2 a d ζ 2 = 3 a 3 β ( 1 + γ ) 2 ( 1 + m ) 2 a + 3 2 E 0 β ( 1 + γ ) 2 ( 1 + m ) 1 a 2 e 2 α ζ .
1 2 ( d y d ζ ) 2 + Π ( y ) = 0 .
Π ( y ) = μ y 2 + ν log [ y ] η y + K ,
Π ( y ) = μ y 2 μ .

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