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Numerical method for an analysis of nonlinear light propagation in photorefractive media - time nonlocal approach

Andrzej Ziółkowski

Open Access Open Access

Abstract

Nonlinear light propagation in photorefractive media can be analyzed by numerical methods. The presented numerical approach has regard to the effects of time nonlocality. Two algorithms are presented, and compared in terms of physical results and computing times. The possibility to address the issue of time nonlocality in two ways is attributed to the fact that, it is possible to completely separate carrier dynamics evaluation and wave equation calculation. This in turn, allows to choose a short integration time for carrier dynamics and a longer one to solve the wave equation. The tests of the methods were carried out for a one-carrier model that describes most of photorefractive media, and for a model with bipolar transport and hot electron effect, used in descriptions of semiconductor materials.

© 2014 Optical Society of America

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Supplementary Material (2)

Media 1: MOV (332 KB)     
Media 2: MOV (1851 KB)     

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Figures (7)
Fig. 1
Fig. 1 Computing algorithm: time local version; numerical procedure I.

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Fig. 2
Fig. 2 Block diagram of a computing algorithm: time nonlocal version; a) numerical procedure II, b) numerical procedure III.

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Fig. 3
Fig. 3 Time evolution of a soliton beam in a material described by the model (1a)-(1d), determined by procedure II. The results present transition states for a) 0 ms, b) 0.3 ms, c) 0.9 ms, d) 1.65 ms, e) 2.55 ms, f) 9 ms. For more effective illustration, the process has been recorded (Media 1).

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Fig. 4
Fig. 4 a) Comparison of diffraction coefficient changes during self-trapping of light beam, determined by algorithms I and II. b) Comparison of time evolution of the diffraction coefficient for low dynamics of self-trapping. Computed results produced by algorithms I and II for propagation lengths Z = 3.5 mm (Z/LD = 1.6).

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Fig. 5
Fig. 5 a) Comparison of the diffraction coefficient obtained by algorithm II for various time steps dt = 1x10−7s and 3x10−7s. b) Comparison of the diffraction coefficient obtained by algorithm III for various values of parameter H to results obtained by numerical procedures I and II.

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Fig. 6
Fig. 6 Time evolution of a Gaussian beam in a material described by the model (5a)- (5f), determined by algorithm III for H = 20. The results present transition states for a) 0 μs, b) 2 μs, c) 4.8 μs, d) 12.8 μs, e) 25.2 μs, f) 41 μs. For better illustration, the process is shown in a movie (Media 2).

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Fig. 7
Fig. 7 Time evolution of light intensity distribution collected at the output of computing field for algorithm I and III (H = 20). The computation results present transition states for a) 0 μs, b) 2 μs, c) 4 μs, d) 8 μs, e) 16 μs, f) 20 μs, g) 24 μs, h) 32 μs, i) 48 μs.

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Tables (3)

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Table 1 Material parameters used in computations based on model (1a)-(1d).

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Table 2 Material parameters used in computations based on model (5a) – (5f).

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Table 3 Computing times determined by the numerical procedure III.

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Equations (33)

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∂ N D + ∂ t = [ β n + S n ( I + I B ) ] ( N D − N D + ) − γ n n N D + ,
J n = q μ n n E + μ n k B T   grad   n ,
∂ ∂ t ( N D + − N A − − n ) = 1 q div   J n ,
div   E = −   div ( grad φ ) = q ε ε o ( N D + − N A − − n ) ,
T n ( E ) = T L + 2 q τ r v n ( E ) 3 k B E ,
μ n ( E ) = μ n l f ( E ) + μ n u [ 1 − f ( E ) ] ,
f ( E ) = { 1 + R   exp [ − Δ U k B T n ( E ) ] } − 1 ,
∂ n ∂ t − 1 q div   J n = [ S n ( I + I B ) + β n ] ( N D − N D + ) − γ n n N D +   ,
∂ p ∂ t   + 1 q div   J p = [ S p ( I + I B ) + β p ] N D + − γ p p ( N D − N D + )   ,
J n = q μ n ( E ) n E + k B grad [ μ n ( E ) T n ( E ) n   ] ,
J p = q μ p p E − k B μ p T L grad   p   ,
∂ ∂ t ( n + N A − − p − N D + ) = 1 q div ( J n + J p ) ,
div E = −   div ( grad φ ) = q ε ε o ( N D + + p − n − N A − ) ,
( ∂ ∂ z + i 2 k ∂ 2 ∂ x 2 ) A ( x , z ) + i k n b Δ n ( E ) A ( x , z ) = 0   ,
μ n = μ n l ,     T n = T L .
n = [ S n ( I + I B ) + β n ] ( N D − N D + ) γ n N D + ,   p = [ S p ( I + I B ) + β p ] N D + γ p ( N D − N D + ) .
∂ ∂ x ( J n + J p ) = − ε ε 0 ∂ 2 E ∂ t ∂ x .
∂ E ∂ t + 1 τ D ( 1 + I I B + I d ) E = E 0 τ D .
τ D = ε ε 0 q ( μ n a n + μ p a p ) I B + q ( μ n b n + μ p b p ) ,
a n = S n ( N D − N A ) γ n N A ,   a p = S p N A γ p ( N D − N A ) , b n = β n ( N D − N A ) γ n N A ,   b p = β p N A γ p ( N D − N A ) .
I d = μ n b n + μ p b p μ n a n + μ p a p .
E ( t ) = E 0 exp [ − 1 τ D ∫ 0 t ( 1 + I I B + I D ) d t ′ ] { 1 + 1 τ D ∫ 0 t d t ′ exp [ 1 τ D ∫ 0 t ′ ( 1 + I I B + I D ) d t ″ ] } .
E ( t ) = E 0 exp [ − t τ D ( 1 + I I B + I D ) ] + E 0 ( 1 + I I B + I D ) − 1 { 1 − exp [ − t τ D ( 1 + I I B + I D ) ] } ,
τ n = 1 γ n N A ,   τ p = 1 γ p ( N D − N A ) .
n 0 = ( S n I B + β n ) ( N D − N A ) γ n N A ,   N D + 0 = N A + n 0 ,   E 0 = φ 0 L .
∂ N D + ∂ t = ( ∂ n ∂ t − 1 q ∂ J n ∂ x ) − ( ∂ p ∂ t + 1 q ∂ J p ∂ x ) .
N D + j s + 1 = N D + j s + N D [ ​ S n ( I j s + I B ) + β n + γ p p j s ] d t ​ − ​ N D + j s [ S p ( I j s + I B ) + β p + γ n n j s ] d t 1 + [ S n ( I j s + I B ) + β n + γ p p j s ] d t ,
A + p j + 1 s + 1 + B + p j s + 1 + C + p j − 1 s + 1 = A   p j + 1 s + B   p j s + C   p j − 1 s + D ,
A + = ξ ( μ p 2 d x E j + 1 s − D p d x 2 ) ,   B + = 1 d t + ξ 2 D p d x 2 ,   C + = ξ ( − μ p 2 d x E j − 1 s − D p d x 2 ) , A = ( ξ − 1 ) ( μ p 2 d x E j + 1 s − D p d x 2 ) ,   B = 1 d t + ( ξ − 1 ) 2 D p d x 2 − γ p ( N D − N D + j s ) , C = ( ξ − 1 ) ( − μ p 2 d x E j − 1 s − D p d x 2 ) ,   D = N D + [ S p ( I j s + I B ) + β p ] ,
F + n j + 1 s + 1 + G + n j s + 1 + H + n j − 1 s + 1 = F   n j + 1 s + G   n j s + H   n j − 1 s + M ,
F + = ξ ( − μ n j + 1 s 2 d x E j + 1 s + D n ( j + 1 / 2 ) d x 2 ) ,   G + = 1 d t − ξ D n ( j + 1 / 2 ) + D n ( j − 1 / 2 ) d x 2 , H + = ξ ( μ n j − 1 s 2 d x E j − 1 s + D n ( j − 1 / 2 ) d x 2 ) ,   F = ( μ n j + 1 s 2 d x E j + 1 s − D n ( j + 1 / 2 ) d x 2 ) , G = 1 d t − ( 1 − ξ ) D n ( j + 1 / 2 ) + D n ( j − 1 / 2 ) d x 2 − γ n N D + j s ,   H = ( ξ − 1 ) ( μ n j − 1 s 2 d x E j − 1 s + D n ( j − 1 / 2 ) d x 2 ) , M = [ S n ( I j s + I B ) + β n ] ( N D − N D + j s ) ,   D n ( j + 1 / 2 ) ≈ D n j + 1 + D n j 2 ,   D n ( j − 1 / 2 ) ≈ D n j + D n j − 1 2 .
f ( E ) = [ 1 + ( E E C ) 5.6 ] − 1 4 ,
φ j + 1 s + 1 − 2 φ j s + 1 + φ j − 1 s + 1 = − q d x 2 ε ε 0 ( N D j s + 1 + p j s + 1 − n j s + 1 − N A ) ,
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