Abstract

The nonlinear one-dimensional model of controllable photonic crystals is considered. A photonic crystal is formed in a dielectric film on which the electromagnetic pump and signal waves fall. It is shown that the shape of the superlattice is controlled by the variation of the external electrostatic field imposed on the film. Thus it leads to changing boundaries of the allowed and forbidden zones of the signal wave spectrum.

© 2007 Optical Society of America

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References

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  1. Y. S. Kivshar and G. P. Agraval, Optical Solutions: from Fibers to Photonic Crystals (Academic, 2003).
  2. K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, 2001).
  3. I. V. Guryev, O. V. Shulika, I. A. Sukhoivanov, and O. V. Mashoshina, "Improvement of characterization accuracy of the nonlinear photonic crystals using finite elements-iterative method," Appl. Phys. B: Photophys. Laser Chem. 84, 83-87 (2006).
    [CrossRef]
  4. R. Chiao, "Bogolubov dispersion relation for a 'photon fluid': Is this a superfluid?" Opt. Commun. 179, 157-166 (2000).
    [CrossRef]
  5. I. V. Dzedolik, "Polariton Vortices in a Transparent Medium," in Proceedings of International Conference on Laser and Fiber-Optical Networks Modeling (IEEE-LEOS, 2006), pp. 96-99.
  6. M. B. Vinogradova, O. V. Rudenko, and A. P. Sukhorukov, Wave Theory (Nauka, 1990) (in Russian).
  7. L. D. Landau and E. M. Lifshitz, Theoretical Physics: VIII. Electrodynamics of Continuums (Nauka, 1982) (in Russian).
  8. I. V. Dzedolik, "Spontaneous symmetry breaking in an electromagnetic field--insulator system," Tech. Phys. 51, 932-937 (2006).
    [CrossRef]
  9. L. D. Landau and E. M. Lifshitz, Theoretical Physics: IX Statistical Physics. Part II (Nauka, 1978) (in Russian).
  10. N. D. Mermin and H. Wagner, "Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models," Phys. Rev. Lett. 17, 1133-1136 (1966).
    [CrossRef]
  11. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, 1927).
  12. L. D. Landau and E. M. Lifshitz, Theoretical Physics: III. Quantum Mechanics (Non-Relative Theory) (Nauka, 1989) (in Russian).

2006

I. V. Guryev, O. V. Shulika, I. A. Sukhoivanov, and O. V. Mashoshina, "Improvement of characterization accuracy of the nonlinear photonic crystals using finite elements-iterative method," Appl. Phys. B: Photophys. Laser Chem. 84, 83-87 (2006).
[CrossRef]

I. V. Dzedolik, "Spontaneous symmetry breaking in an electromagnetic field--insulator system," Tech. Phys. 51, 932-937 (2006).
[CrossRef]

2000

R. Chiao, "Bogolubov dispersion relation for a 'photon fluid': Is this a superfluid?" Opt. Commun. 179, 157-166 (2000).
[CrossRef]

1966

N. D. Mermin and H. Wagner, "Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models," Phys. Rev. Lett. 17, 1133-1136 (1966).
[CrossRef]

Agraval, G. P.

Y. S. Kivshar and G. P. Agraval, Optical Solutions: from Fibers to Photonic Crystals (Academic, 2003).

Chiao, R.

R. Chiao, "Bogolubov dispersion relation for a 'photon fluid': Is this a superfluid?" Opt. Commun. 179, 157-166 (2000).
[CrossRef]

Dzedolik, I. V.

I. V. Dzedolik, "Spontaneous symmetry breaking in an electromagnetic field--insulator system," Tech. Phys. 51, 932-937 (2006).
[CrossRef]

I. V. Dzedolik, "Polariton Vortices in a Transparent Medium," in Proceedings of International Conference on Laser and Fiber-Optical Networks Modeling (IEEE-LEOS, 2006), pp. 96-99.

Guryev, I. V.

I. V. Guryev, O. V. Shulika, I. A. Sukhoivanov, and O. V. Mashoshina, "Improvement of characterization accuracy of the nonlinear photonic crystals using finite elements-iterative method," Appl. Phys. B: Photophys. Laser Chem. 84, 83-87 (2006).
[CrossRef]

Kivshar, Y. S.

Y. S. Kivshar and G. P. Agraval, Optical Solutions: from Fibers to Photonic Crystals (Academic, 2003).

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Theoretical Physics: IX Statistical Physics. Part II (Nauka, 1978) (in Russian).

L. D. Landau and E. M. Lifshitz, Theoretical Physics: VIII. Electrodynamics of Continuums (Nauka, 1982) (in Russian).

L. D. Landau and E. M. Lifshitz, Theoretical Physics: III. Quantum Mechanics (Non-Relative Theory) (Nauka, 1989) (in Russian).

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Theoretical Physics: III. Quantum Mechanics (Non-Relative Theory) (Nauka, 1989) (in Russian).

L. D. Landau and E. M. Lifshitz, Theoretical Physics: VIII. Electrodynamics of Continuums (Nauka, 1982) (in Russian).

L. D. Landau and E. M. Lifshitz, Theoretical Physics: IX Statistical Physics. Part II (Nauka, 1978) (in Russian).

Mashoshina, O. V.

I. V. Guryev, O. V. Shulika, I. A. Sukhoivanov, and O. V. Mashoshina, "Improvement of characterization accuracy of the nonlinear photonic crystals using finite elements-iterative method," Appl. Phys. B: Photophys. Laser Chem. 84, 83-87 (2006).
[CrossRef]

Mermin, N. D.

N. D. Mermin and H. Wagner, "Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models," Phys. Rev. Lett. 17, 1133-1136 (1966).
[CrossRef]

Rudenko, O. V.

M. B. Vinogradova, O. V. Rudenko, and A. P. Sukhorukov, Wave Theory (Nauka, 1990) (in Russian).

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, 2001).

Shulika, O. V.

I. V. Guryev, O. V. Shulika, I. A. Sukhoivanov, and O. V. Mashoshina, "Improvement of characterization accuracy of the nonlinear photonic crystals using finite elements-iterative method," Appl. Phys. B: Photophys. Laser Chem. 84, 83-87 (2006).
[CrossRef]

Sukhoivanov, I. A.

I. V. Guryev, O. V. Shulika, I. A. Sukhoivanov, and O. V. Mashoshina, "Improvement of characterization accuracy of the nonlinear photonic crystals using finite elements-iterative method," Appl. Phys. B: Photophys. Laser Chem. 84, 83-87 (2006).
[CrossRef]

Sukhorukov, A. P.

M. B. Vinogradova, O. V. Rudenko, and A. P. Sukhorukov, Wave Theory (Nauka, 1990) (in Russian).

Vinogradova, M. B.

M. B. Vinogradova, O. V. Rudenko, and A. P. Sukhorukov, Wave Theory (Nauka, 1990) (in Russian).

Wagner, H.

N. D. Mermin and H. Wagner, "Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models," Phys. Rev. Lett. 17, 1133-1136 (1966).
[CrossRef]

Watson, G. N.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, 1927).

Whittaker, E. T.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, 1927).

Appl. Phys. B: Photophys. Laser Chem.

I. V. Guryev, O. V. Shulika, I. A. Sukhoivanov, and O. V. Mashoshina, "Improvement of characterization accuracy of the nonlinear photonic crystals using finite elements-iterative method," Appl. Phys. B: Photophys. Laser Chem. 84, 83-87 (2006).
[CrossRef]

Opt. Commun.

R. Chiao, "Bogolubov dispersion relation for a 'photon fluid': Is this a superfluid?" Opt. Commun. 179, 157-166 (2000).
[CrossRef]

Phys. Rev. Lett.

N. D. Mermin and H. Wagner, "Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models," Phys. Rev. Lett. 17, 1133-1136 (1966).
[CrossRef]

Tech. Phys.

I. V. Dzedolik, "Spontaneous symmetry breaking in an electromagnetic field--insulator system," Tech. Phys. 51, 932-937 (2006).
[CrossRef]

Other

L. D. Landau and E. M. Lifshitz, Theoretical Physics: IX Statistical Physics. Part II (Nauka, 1978) (in Russian).

Y. S. Kivshar and G. P. Agraval, Optical Solutions: from Fibers to Photonic Crystals (Academic, 2003).

K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, 2001).

I. V. Dzedolik, "Polariton Vortices in a Transparent Medium," in Proceedings of International Conference on Laser and Fiber-Optical Networks Modeling (IEEE-LEOS, 2006), pp. 96-99.

M. B. Vinogradova, O. V. Rudenko, and A. P. Sukhorukov, Wave Theory (Nauka, 1990) (in Russian).

L. D. Landau and E. M. Lifshitz, Theoretical Physics: VIII. Electrodynamics of Continuums (Nauka, 1982) (in Russian).

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, 1927).

L. D. Landau and E. M. Lifshitz, Theoretical Physics: III. Quantum Mechanics (Non-Relative Theory) (Nauka, 1989) (in Russian).

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Figures (3)

Fig. 1
Fig. 1

Fabry–Perot resonator based on the dielectric film.

Fig. 2
Fig. 2

Signal-wave spectrum at 2 b 1 g ̃ = 1 2 .

Fig. 3
Fig. 3

Signal-wave spectrum at 2 b 1 g ̃ = 1 .

Equations (38)

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2 A ( A ) 1 c 2 ( t ϵ t ) A = 0 .
1 c 2 ( t ϵ t ) A = 1 c 2 t ( ϵ 1 A t ) + 4 π χ 3 c 4 ( 2 A * t 2 A t A t + 2 A * t 2 A t 2 A t ) ,
( t ϵ 1 t ) = f ̂ ( ω + Δ ω ) .
f ̂ ( ω + Δ ω ) A { f ̂ ( ω ) A ̃ + i [ d f ̂ ( ω ) ( d ω ) ] A ̃ t } exp ( i ω t ) ,
i ψ j t = α 0 2 ψ j + α 0 j ( j j ψ j ) α 1 ψ j α 2 ( j ψ j * ψ j ) ψ j ,
L ̃ = ω c 2 j [ i ψ j ψ j * t + α 0 ( ψ j 1 j j j ψ j ) ψ j * α 1 2 ψ j * ψ j α 2 16 ( ψ j * ψ j + ψ j ψ j * ) 2 + c.c. ] .
H = d V [ j , μ = 0 , 1 , 2 , 3 ( ψ j L ̃ \ μ ψ j * + ψ j * L ̃ \ μ ψ j ) L ̃ ]
ψ j ( t , r ) = c 2 ω V 0 k a j k e i ( ω + Δ ω ) t exp ( i kr ) ,
ψ j * ( t , r ) = c 2 ω V 0 k a j k + e i ( ω + Δ ω ) t exp ( i kr ) .
H ̂ = j , k { [ w ̃ ( k ) + α ̃ j , k ( a j k + a j k + 1 2 ) ] ( a j k + a j k + 1 2 ) α 0 2 k j ( j k k j ( a j k + a j k + a j k a j k + ) ) } ,
N = j N J 0 + 0 j , k 0 a j k + a j k 0 , μ = E ̃ 0 N
H ̂ = H ̂ 0 + H ̂ 1 + H ̂ 2 ,
H ̂ 0 = [ ω ̃ ( 0 ) ( a 0 + a 0 + 1 2 ) + α ̃ ( a 0 + a 0 + 1 2 ) 2 ] ,
H ̂ 1 = k 0 ω ̃ ( k ) ( a k + a k + 1 2 ) ,
H ̂ 2 = α ̃ k 0 k 0 ( a k + a k + 1 2 ) ( a k + a k + 1 2 ) ,
ω ̃ ( k ) = ω + Δ ω + α 1 + α 0 ( k 2 2 + k 3 2 ) .
H ̂ = E ̃ ( 0 ) + k 0 [ Ω ( k ) ( a k + a k + a k + a k + 1 ) + α ̃ N 0 ( a k + a k + + a k a k ) ] ,
u k v k = α ̃ N 0 2 Ω ̃ ( k ) , u k 2 = 1 2 ( Ω ( k ) Ω ̃ ( k ) + 1 ) , v k 2 = 1 2 ( Ω ( k ) Ω ̃ ( k ) 1 ) .
Ω ̃ ( k ) = [ ( ω ̃ ) 2 + 2 α ̃ ω ̃ ( N 0 + 1 2 ) + α ̃ 2 ( N 0 + 1 4 ) ] 1 2 ,
E ̃ 0 = E ̃ ( 0 ) + k 0 Ω ̃ ( k ) .
i ψ ( t , r ) t = ( 2 2 m e f 2 + μ ) ψ ( t , r ) + [ ψ * ( t , r ) ψ ( t , r ) U ( r r ) d V ] ψ ( t , r ) .
i ψ ( t , r ) t = ( 2 2 m e f 2 + μ ) ψ ( t , r ) + α ̃ ψ * ( t , r ) ψ ( t , r ) ψ ( t , r ) .
2 ψ ̃ + g 1 ψ ̃ g 2 ψ ̃ * ψ ̃ ψ ̃ = 0 ,
d 2 f d X 2 + f f 3 = 0 ,
ψ 1 = g ̃ g 2 b 2 sn ( b 1 g ̃ 2 x , k ̃ ) exp ( i Ω ̃ t + i π L z ) ,
i ψ 2 ( t , r ) t = 2 m 2 e f 2 ψ 2 ( t , r ) + α ̃ ψ 1 * ( t , r ) ψ 1 ( t , r ) ψ 2 ( t , r ) ,
d 2 ψ ̃ 2 d x 2 + [ 2 ω 2 2 c 2 2 π 2 L 2 g ̃ ω 2 b 2 2 Ω ̃ sn 2 ( b 1 g ̃ 2 x , k ̃ ) ] ψ ̃ 2 = 0 .
H ̂ 0 ψ ̃ 2 + ( η V ) ψ ̃ 2 = 0 ,
A ( k 2 x 2 + η V ) exp ( i k 2 x x ) + B ( k 2 x 2 + η V ) exp ( i k 2 x x ) = 0 .
A ( k 2 x 2 + η V 11 ) + B V 12 = 0 ,
A V 21 + B ( k 2 x 2 + η V 22 ) = 0 ,
V 11 = V 22 = ω 2 g ̃ b 2 2 Ω ̃ Λ 2 Λ 2 d X sn 2 ( X , k ̃ ) ;
V 12 = ω 2 g ̃ b 2 2 Ω ̃ Λ 2 Λ 2 d X [ exp ( i 2 2 k 2 x b 1 g ̃ X ) × sn 2 ( X , k ̃ ) ] , V 21 = V 12 *
( k 2 x 2 + η V 11 ) V 12 V 12 * ( k 2 x 2 + η V 11 ) = 0 ,
η = k 2 x 2 + V 11 ± V 12 ( k 2 x ) 2 ,
V 12 ( k 2 x ) 2 = ω 2 g ̃ b 2 2 Ω ̃ [ ( Λ 2 Λ 2 d X sn ( 2 2 k 2 x b 1 g ̃ X ) sn 2 ( X , k ̃ ) ) 2 + ( Λ 2 Λ 2 d X cos ( 2 2 k 2 x b 1 g ̃ X ) sn 2 ( X , k ̃ ) ) 2 ] 1 2 .
V n = g ̃ ω 2 b 2 2 Ω ̃ Λ 2 Λ 2 d X sn 2 ( X , k ̃ ) exp ( i n K x X )
ω 2 = c 2 4 { V 11 ± V 12 ( k 2 x ) 2 + [ ( V 11 ± V 12 ( k 2 x ) 2 ) 2 + 2 8 π 2 c 2 L 2 + 8 c 2 k 2 x 2 ] 1 2 } ,

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