Abstract

We consider a passive optical cavity containing a photonic crystal and a purely absorptive two-level medium. The cavity is driven by a superposition of two coherent beams forming a periodically modulated pump. Using a coupled mode reduction and direct numerical modeling of the full system we demonstrate the existence of bistability between uniformly periodic states, modulational instabilities and localized structures of light. All are found to exist within the conduction band of the photonic material. Moreover, contrary to similar previously found intra-band structures, we show that these localized structures can be truly stationary states.

© 2006 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  8. A. Schreiber, B. Thuring, M. Kreuzer, and T. Tschudi, "Experimental investigation of solitary structures in a nonlinear optical feedback system," Opt. Commun. 136, 415-418 (1997).
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  9. P. L. Ramazza, S. Ducci, S. Boccaletti, and F. T. Arecchi, "Localized versus delocalized patterns in a nonlinear optical interferometer," J. Opt. B: Quantum and semiclass. Opt. 2, 399-405 (2000).
    [CrossRef]
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  15. N.K. Efremidis and D.N. Christodoulides, "Discrete Ginzburg-Landau solitons," Phys. Rev. E 67, 026606 (2003).
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661 (1)

N. Akhmediev and A. Ankiewicz, Dissipative Solitons, Series: Lecture Notes in Physics, Vol. 661 (Springer, 2005).

J. Opt. B: Quantum and semiclass. Opt. (2)

P. Mandel and M. Tlidi, "Transverse dynamics in cavity nonlinear optics (2000-2003)," J. Opt. B: Quantum and semiclass. Opt. 6, R60-R75 (2004).
[CrossRef]

P. L. Ramazza, S. Ducci, S. Boccaletti, and F. T. Arecchi, "Localized versus delocalized patterns in a nonlinear optical interferometer," J. Opt. B: Quantum and semiclass. Opt. 2, 399-405 (2000).
[CrossRef]

Nature (London) (1)

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödl, M. Miller, and R. Jäger "Cavity solitons as pixels in semiconductor microcavities," Nature (London) 419, 699-702 (2002).
[CrossRef]

Opt. Commun. (2)

K. Maruno, A. Ankiewicz, and N. Akhmediev, "Exact localized and periodic solutions of the discrete complex Ginzburg-Landau equation," Opt. Commun. 221, 199-209 (2003).
[CrossRef]

A. Schreiber, B. Thuring, M. Kreuzer, and T. Tschudi, "Experimental investigation of solitary structures in a nonlinear optical feedback system," Opt. Commun. 136, 415-418 (1997).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (2)

V. B. Taranenko, K. Staliunas, and C. O. Weiss, "Spatial soliton laser: Localized structures in a laser with a saturable absorber in a self-imaging resonator," Phys. Rev. A 56, 1582-1591 (1997).
[CrossRef]

L. Spinelli, G. Tissoni, M. Brambilla, F. Prati, and L.A. Lugiato, "Spatial solitons in semiconductor microcavities," Phys. Rev. A 58, 2542-2559 (1998).
[CrossRef]

Phys. Rev. E (3)

D. Gomila and G.L. Oppo, "Coupled-mode theory for photonic band-gap inhibition of spatial instabilities," Phys. Rev. E 72, 016614 (2005).
[CrossRef]

K. Staliunas, "Midband solitons in nonlinear photonic crystal resonators," Phys. Rev. E 70, 016602 (2004).
[CrossRef]

N.K. Efremidis and D.N. Christodoulides, "Discrete Ginzburg-Landau solitons," Phys. Rev. E 67, 026606 (2003).
[CrossRef]

Phys. Rev. Lett. (7)

K. Staliunas, "Midband dissipative spatial solitons," Phys. Rev. Lett. 91, 053901 (2003).
[CrossRef] [PubMed]

D. Gomila, R. Zambrini, and G.L. Oppo, "Photonic band-gap inhibition of modulational instabilities," Phys. Rev. Lett. 92, 253904 (2004).
[CrossRef] [PubMed]

Y. F. Chen and Y. P. Lan, "Formation of repetitively nanosecond spatial solitons in a saturable absorber Q-switched laser," Phys. Rev. Lett. 93, 013901 (2004).
[CrossRef]

J. Tredicce, M. Guidici, and P. Glorieux, "Comment on "Formation of repetitively nanosecond spatial solitons in a saturable absorber Q-switched laser," Phys. Rev. Lett. 94, 249401 (2005).
[CrossRef]

U. Bortolozzo, L. Pastur, P. L. Ramazza, M. Tlidi, and G. Kozyreff "Bistability between different localized structures in nonlinear optics," Phys. Rev. Lett. 93, 253901 (2004).
[CrossRef]

M. Pesch, E. Große Westhoff, T. Ackemann, and W. Lange, "Observation of a discrete family of dissipative solitons in a nonlinear optical system," Phys. Rev. Lett. 95, 143906 (2005). #9444 - $15.00 USD Received 10 November 2005; revised 19 December 2005; accepted 19 December 2005.
[CrossRef] [PubMed]

L. A. Lugiato and R. Lefever, "Spatial dissipative structures in passive optical systems," Phys. Rev. Lett. 58, 2209-2211 (1987).
[CrossRef] [PubMed]

Other (2)

N. N. Rosanov, Spatial hysteresis and optical patterns (Springer, Berlin 2002).

Y.S. Kivshar and G.P. Agrawal, Optical solitons: from a fiber to photonic crystals (Academic Press, 2003).

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Figures (5)

Fig. 1.
Fig. 1.

Schematic setup of the nonlinear cavity filled with a passive two-level medium (PM) and a photonic crystal film (PCF). The Fabry-Perot cavity with flat Mirrors (M) is driven by two pumping beam P 1,2.

Fig. 2.
Fig. 2.

Instability boundaries as a function of the effective detuning parameter Ω. (a) Pump P 1 = P 2 = P versus Ω. The solid curve is the modulational instability boundary. The bistability region is delimited by the three dashed curves. Grey region indicates photonic band gap (BG). (b) Critical wavenumber at the modulational instability versus Ω. Parameters are γ = 0.01, C = 0.4, δ = 0, and q = 0.

Fig. 3.
Fig. 3.

Stationary localized structures. Ω=1.05. Other parameters are the same as in Fig. 2. (a) Real and imaginary parts of the field amplitudes A 1,2 for P 1 = P 2 = 0.225. Solid (broken) lines correspond to A 1 (A 2). (b) Bifurcation diagram. LS: localized structures, HSS: homogeneous steady state. Broken lines correspond to unstable solutions.

Fig. 4.
Fig. 4.

Stationary localized structure obtained by direct numerical simulation of Eqs. (1,2). Parameters are γ = 0.05, C = 2.0, δ = 0, ϕ = 0, P 1 = P 2 = 1.2, km = 2.5 √ Δε k 0, and ωc −ω0 = -0.3125 ω0 Δε.

Fig. 5.
Fig. 5.

Transverse velocity v = dξ/dt of a localized structure as a function of (a) the phase shift δ, (b) pump imbalance δP= (P 2-P 1)/(P 2+P 1), where P 2+P 1 = 0.45, (c) incidence angle ϕ. Other parameters are the same as in Fig. 3.

Equations (12)

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F t = P ( x ) κF i ( ω c ω 0 ) F αcNF + ic 2 k 0 2 F x 2 + i ω 0 Δ ε cos ( k m x ) F ,
N t = Γ [ 1 N ( 1 + F 2 ) ] .
P ( x ) = ω 0 Δ ε 2 [ P 1 e i 2 ( k m ( 1 + ϕ ) x δ ) + P 2 e i 2 ( k m ( 1 ϕ ) x δ ) ] ,
Δ x ~ 1 k 0 ω 0 2 κ = 2 L k 0 T .
F ( x , t ) = A 1 ( x , t ) e i 2 ( k m x δ ) + A 2 ( x , t ) e i 2 ( k m x δ )
N 0 = 1 + A 1 2 + A 2 2 S , N 1 = A 1 A 2 * S , N 2 = A 1 * A 2 S ,
S = ( 1 + A 1 2 + A 2 2 ) 2 2 A 1 2 A 2 2 .
A 1 τ = P 1 exp iqξ ( γ + i Ω ) A 1 + i A 2 e + A 1 ξ 2 C 1 + A 1 2 S A 1 ,
A 2 τ = P 2 exp iqξ ( γ + i Ω ) A 2 + i A 1 e A 2 ξ 2 C 1 + A 2 2 S A 2 ,
γ = 2 κ ω 0 Δ ε , Ω = 1 ω 0 Δ ε ( ω c ω 0 + c k m 2 4 k 0 ) , C = αc ω 0 Δ ε , q = k m 2 ϕ 2 k 0 2 Δ ε .
Δ x B = k m k 0 2 Δ ε Δ ξ B ~ ω 0 2 κ Δ ε k m k 0 2 = k m k 0 Δ ε Δ x .
Δ εγ k m 2 k 0 2 1 ,

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