Researching | One-Dimensional Modulational Instability of Broad Optical Beams In Photorefractive Crystals with Both Linear and Quadratic Electro-Optic Effects

Journals >Laser & Optoelectronics Progress >Volume 61 >Issue 5 >Page 0519001 > Article

Laser & Optoelectronics Progress
Vol. 61, Issue 5, 0519001 (2024)

One-Dimensional Modulational Instability of Broad Optical Beams In Photorefractive Crystals with Both Linear and Quadratic Electro-Optic Effects

Lili Hao¹, Zhen Wang¹, Hongxia Tang², Xiaoyang Zhang¹..., Qi Yang¹ and Qiang Wang^1,*|Show fewer author(s)

Author Affiliations

¹Department of Physics, College of physics and Electronic Engineering, Northeast Petroleum University, Daqing 163318, Heilongjiang , China

²Department of Physics, College of Electrical Engineering, Suihua University, Suihua 152000, Heilongjiang , China

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DOI: 10.3788/LOP230900 Cite this Article Set citation alerts

Lili Hao, Zhen Wang, Hongxia Tang, Xiaoyang Zhang, Qi Yang, Qiang Wang. One-Dimensional Modulational Instability of Broad Optical Beams In Photorefractive Crystals with Both Linear and Quadratic Electro-Optic Effects[J]. Laser & Optoelectronics Progress, 2024, 61(5): 0519001 Copy Citation Text

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Abstract

We present a theoretical study of the one-dimensional modulational instability of a broad optical beam propagating in a biased photorefractive crystal with both linear and quadratic electro-optic effects (Kerr effect) under steady-state conditions. One-dimensional modulational instability growth rates are obtained by treating the space-charge field equation globally and locally. Both theoretical reasoning and numerical simulation show that both the global and local modulational instability gains are governed simultaneously by the strength and the polarity of external bias field and by the ratio of the intensity of the broad beam to that of the dark irradiance. Under a strong bias field, the results obtained using these two methods are in good agreement in the low spatial frequency regime. Moreover, the instability growth rate increases with the bias field, and the maximum instability growth occurs when ratio of light intensity to dark irradiance is 0.88.

Keywords

modulational instability nonlinear optics photorefractive effects spatial soliton

(i \frac{\partial}{\partial z} + \frac{1}{2 k} \cdot \frac{\partial^{2}}{\partial x^{2}} + \frac{k}{n_{e}} Δ n) φ (x, z) = 0

.（1）

Δ n = - \frac{n_{e}^{3} r_{33} E_{s c}}{2} - \frac{n_{e}^{3} g_{e f f} ε_{0}^{2} {(ε_{r} - 1)}^{2} E_{s c}^{2}}{2}

，（2）

i \frac{\partial U}{\partial z} + \frac{1}{2 k} \frac{\partial^{2} U}{\partial x^{2}} - β_{1} E_{s c} U - β_{2} E_{s c}^{2} U = 0

，（3）

\begin{matrix} E_{s c} = E_{0} \frac{1}{[1 + {|U|}^{2}]} (1 + \frac{ε_{0} ε_{r}}{e N_{A}} \cdot \frac{\partial E_{s c}}{\partial x}) - \frac{K_{B} T}{e} \frac{[\partial {|U|}^{2} / \partial x]}{[1 + {|U|}^{2}]} + \frac{K_{B} T}{e} \cdot \frac{ε_{0} ε_{r}}{e N_{A}} \cdot \\ {(1 + \frac{ε_{0} ε_{r}}{e N_{A}} \frac{\partial E_{s c}}{\partial x})}^{- 1} \frac{\partial^{2} E_{s c}}{\partial x^{2}} \end{matrix}

.（4）

E_{s c} = \frac{E_{0}}{1 + {|U|}^{2}}

.（5）

U = r^{1 / 2} e x p \{- i \{[β_{1} E_{0} / (1 + r)] + [β_{2} E_{0}^{2} / {(1 + r)}^{2}]\} z\}

，（6）

U = [r^{1 / 2} + ε (x, z)] e x p \{- i \{[β_{1} E_{0} / (1 + r)] + [β_{2} E_{0}^{2} / {(1 + r)}^{2}]\} z\}

，（7）

i \frac{\partial ε}{\partial z} + \frac{1}{2 k} \frac{\partial^{2} ε}{\partial x^{2}} - [β_{1} + β_{2} (E + 2 E_{01})] (r^{1 / 2} + ε) E = 0

，（8）

E - υ \frac{\partial E}{\partial x} - Δ \frac{\partial^{2} E}{\partial x^{2}} = - \frac{r^{1 / 2}}{(1 + r)} [E_{01} (ε + ε^{*}) + \frac{K_{B} T}{e} (\frac{\partial ε}{\partial x} + \frac{\partial ε^{•}}{\partial x})]

，（9）

\begin{matrix} \hat{E} = - \frac{r^{1 / 2}}{(1 + r)} \cdot \\ \{\frac{E_{01} + i [υ E_{01} k_{x} + k_{x} (K_{B} T / e) (1 + Δ k_{x}^{2})]}{{(1 + Δ k_{x}^{2})}^{2} + k_{x}^{2} υ^{2}}\} \cdot \\ (\hat{ε} + {\hat{ε}}^{•}) \end{matrix}

，（10）

ε = a (z) e x p (i p x) + b (z) e x p (- i p x)

.（11）

(\hat{ε} + {\hat{ε}}^{•}) = 2 π [(a + b^{*}) δ (k_{x} - p) + (b + a^{*}) \cdot δ (k_{x} + p)]

，（12）

\begin{matrix} E = - \frac{r^{1 / 2}}{1 + r} (a + b^{*}) \cdot \\ \frac{E_{01} + i [υ E_{01} p + p (K_{B} T / e) (1 + Δ p^{2})]}{{(1 + Δ p^{2})}^{2} + p^{2} υ^{2}} e x p (i p x) - \\ \frac{r^{1 / 2}}{1 + r} (a^{*} + b) \frac{E_{01} - i [υ E_{01} p + p (K_{B} T / e) (1 + Δ p^{2})]}{{(1 + Δ p^{2})}^{2} + p^{2} υ^{2}} \cdot \\ e x p (- i p x) \end{matrix}

.（13）

G (p) = \frac{r}{1 + r} \frac{E_{01} + i [υ E_{01} p + p (K_{B} T / e) (1 + Δ p^{2})]}{{(1 + Δ p^{2})}^{2} + p^{2} υ^{2}}

.（14）

E = - r^{- 1 / 2} [G (p) (a + b^{*}) e x p (i p x) + G^{*} (p) (a^{*} + b) e x p (- i p x)]

.（15）

i \frac{d a}{d z} - \frac{p^{2}}{2 k} a + (β_{1} + 2 β_{2} E_{01}) G (p) (a + b^{*}) = 0

，（16a）

i \frac{d b}{d z} - \frac{p^{2}}{2 k} b + (β_{1} + 2 β_{2} E_{01}) G^{*} (p) (a^{*} + b) = 0

.（16b）

\frac{d^{2} a}{d z^{2}} = [\frac{p^{2}}{k} (β_{1} + 2 β_{2} E_{01}) G (p) - \frac{p^{4}}{4 k^{2}}] a

，（17a）

\frac{d^{2} b}{d z^{2}} = [\frac{p^{2}}{k} (β_{1} + 2 β_{2} E_{01}) G^{*} (p) - \frac{p^{4}}{4 k^{2}}] b

.（17b）

g_{g l} = R e \{{[\frac{p^{2}}{k} (β_{1} + 2 β_{2} E_{01}) G (p) - \frac{p^{4}}{4 k^{2}}]}^{1 / 2}\}

，（18）

i \frac{\partial U}{\partial z} + \frac{1}{2 k} \frac{\partial^{2} U}{\partial x^{2}} - β_{1} E_{0} \frac{U}{1 + {|U|}^{2}} - β_{2} E_{0}^{2} \frac{U}{{(1 + {|U|}^{2})}^{2}} = 0

，（19）

i \frac{\partial ε}{\partial z} + \frac{1}{2 k} \frac{\partial^{2} ε}{\partial x^{2}} + β_{1} E_{0} \frac{r}{{(1 + r)}^{2}} (ε + ε^{•}) + 2 β_{2} E_{0}^{2} \frac{r}{{(1 + r)}^{3}} (ε + ε^{•}) = 0

.（20）

i \frac{d a}{d z} - \frac{p^{2}}{2 k} a + β_{1} E_{0} \frac{r}{{(1 + r)}^{2}} (a + b^{*}) + 2 β_{2} E_{0}^{2} \frac{r}{{(1 + r)}^{3}} (a + b^{*}) = 0

，（21a）

i \frac{d b}{d z} - \frac{p^{2}}{2 k} b + β_{1} E_{0} \frac{r}{{(1 + r)}^{2}} (a^{*} + b) + 2 β_{2} E_{0}^{2} \frac{r}{{(1 + r)}^{3}} (a^{*} + b) = 0

.（21b）

\frac{d^{2} a}{d z^{2}} = \{- \frac{p^{4}}{4 k^{2}} + [β_{1} E_{0} \frac{r}{{(1 + r)}^{2}} + 2 β_{2} E_{0}^{2} \frac{r}{{(1 + r)}^{3}}] \frac{p^{2}}{k}\} a

，（22a）

\frac{d^{2} b}{d z^{2}} = \{- \frac{p^{4}}{4 k^{2}} + [β_{1} E_{0} \frac{r}{{(1 + r)}^{2}} + 2 β_{2} E_{0}^{2} \frac{r}{{(1 + r)}^{3}}] \frac{p^{2}}{k}\} b

，（22b）

g_{l c} = R e \{{[- \frac{p^{4}}{4 k^{2}} + (β_{1} E_{0} \frac{r}{{(1 + r)}^{2}} + 2 β_{2} E_{0}^{2} \frac{r}{{(1 + r)}^{3}}) \frac{p^{2}}{k}]}^{1 / 2}\}

.（23）

g_{m a x} = \frac{k_{0} n_{e}^{3} r_{33} E_{0}}{2} \frac{r}{{(1 + r)}^{2}} + k_{0} n_{e}^{3} g_{e f f} ε_{0}^{2} {(ε_{r} - 1)}^{2} E_{0}^{2} \frac{r}{{(1 + r)}^{3}}

，（24）

p_{m a x} = \frac{k_{0} n_{e}^{2}}{1 + r} {\{E_{0} r [r_{33} + \frac{2 g_{e f f} ε_{0}^{2} {(ε_{r} - 1)}^{2} E_{0}}{1 + r}]\}}^{1 / 2}

.（25）

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Lili Hao, Zhen Wang, Hongxia Tang, Xiaoyang Zhang, Qi Yang, Qiang Wang. One-Dimensional Modulational Instability of Broad Optical Beams In Photorefractive Crystals with Both Linear and Quadratic Electro-Optic Effects[J]. Laser & Optoelectronics Progress, 2024, 61(5): 0519001

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