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Spatio-temporal instabilities for counter-propagating waves in periodic media

J. W. Haus, B. Y. Soon, M. Scalora, M. J. Bloemer, C. M. Bowden, C. Sibilia, and A. Zheltikov

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Abstract

Nonlinear evolution of coupled forward and backward fields in a multi-layered film is numerically investigated. We examine the role of longitudinal and transverse modulation instabilities in media of finite length with a homogeneous nonlinear susceptibility χ(3). The numerical solution of the nonlinear equations by a beam-propagation method that handles backward waves is described.

©2002 Optical Society of America

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Supplementary Material (3)

Media 1: GIF (197 KB)     
Media 2: GIF (216 KB)     
Media 3: GIF (156 KB)     

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Figures (6)
Fig. 1.
Fig. 1. Transmission curve and input pulse spectrum. The parameters have been scaled as discussed in the text. The frequency is shifted relative to the laser line, which is placed at the first transmission maximum on the high frequency side.

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Fig. 2.
Fig. 2. Animation of the pulse evolution through the nonlinear medium with initial detuning δ = 1.12, i.e. the central frequency of the laser is tuned to the first transmission maximum on the high frequency branch. The initial amplitude is A=0.2. Top panel is the pulse amplitude launched outside the medium. The center panel is the forward-propagating pulse amplitude and the bottom panel is the backward propagating pulse amplitude. The numbers in brackets denote the maximum value of the intensity in that plot (196 KB gif animation).

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Figure 3:
Figure 3: The pulse amplitude launched in the forward direction is shown in the top panel. The bottom panel is the backward propagating pulse amplitude. See Figure 2 for a discussion of parameters. For this case A=0.6 (216 KB gif animation).

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Figure 4:
Figure 4: A three-dimensional representation of the final frame in Fig. 3. The pulse amplitude in the forward direction is shown in the top panel. The bottom panel is the backward propagating pulse amplitude.

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Fig. 5:
Fig. 5: The spatial spectrum, H(q), of the pulse after traversing the PBG, as defined in Eq. (11) and the peak is normalized to unity. The three curves correspond to the amplitudes A=0.2 (solid blue line), 0.3 (dash-dotted red line), and 0.4 (dashed black line).

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Fig. 6.
Fig. 6. Top panel is the pulse amplitude in the forward-propagating direction and the bottom panel is the backward propagating pulse amplitude. The parameters are the same as in discussed earlier, except δ = -1.12 and A= 0.7 (156 KB gif animation).

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Equations (16)

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1 v ∂ E f ∂ t = − ∂ E f ∂ z + i F ∇ ⊥ 2 E f + iδ E f + i κ E b + iη ( ∣ E f ∣ 2 + 2 ∣ E b ∣ 2 ) E f
1 v ∂ E b ∂ t = + ∂ E b ∂ z + i F ∇ ⊥ 2 E b + iδ E b + i κ E f + iη ( ∣ E b ∣ 2 + 2 ∣ E f ∣ 2 ) E b
E f ( x , y , 0 , t ) = S ( x , y , t ) ,
E b ( x , y , L , t ) = 0 .
E f ( q , z , ω ) = S ( q , ω ) ( cos ⁡ ( Δ z ) + i Ω Δ sin ⁡ ( Δ z ) + κ 2 Δ sin ⁡ ( Δ L ) sin ⁡ ( Δ z ) [ Δ cos ⁡ ( Δ L ) + i Ω sin ⁡ ( Δ L ) ] )
E b ( q , z , ω ) = S ( q , ω ) i κ sin ⁡ ( Δ ( z − L ) ) [ Δ cos ⁡ ( Δ L ) + i Ω sin ⁡ ( Δ L ) ]
L ff = v ( − ∂ ∂ z + i F ∇ ⊥ 2 + iδ ) ; ,
L bb = v ( ∂ ∂ z + i F ∇ ⊥ 2 + iδ ) ; .
N ff = ivη ( ∣ E f ∣ 2 + 2 ∣ E b ∣ 2 ) ; N bb = ivη ( ∣ E b ∣ 2 + 2 ∣ E f ∣ 2 ) ;
K fb = ivκ ; K bf = ivκ .
∂ U ∂ t = ( L + V ) U .
U ( t + Δ t ) = exp ⁡ ( Δ tL 2 ) exp ⁡ ( Δ tV ) exp ⁡ ( Δ tL 2 ) U ( t ) .
exp ⁡ ( Δ tV ) = exp ⁡ ( Δ tK 2 ) exp ⁡ ( Δ tN ) exp ⁡ ( Δ tK 2 ) .
K 2 = δ 2 − κ 2 .
F ( x , z , 0 ) = A exp ⁡ ( − ( z − z 0 ) 2 σ z 2 ) exp ⁡ ( − x 2 σ x 2 ) .
H ( q ) = ∣ ∫∫ e iqx E f ( x , z , t ) dzdx ∣ ,
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