Nonlinear Optics

Martin Schechter

Chapter Chapter 16 in Critical Point Theory, 2020, pp 261-276 from Springer

Abstract: Abstract Light waves propagating in a photo refractive crystal are governed by a nonlinear Schrödinger equation of the form i ∂ u ∂ z + D Δ u = g ( x , | u | 2 ) u , $$\displaystyle i\frac {\partial u}{\partial z} +D \Delta u=g(x, |u|{ }^2)u, $$ where D > 0 is the beam diffraction coefficient and the functions are periodic with respect to the variables x = ( x 1 , x 2 ) ∈ Ω ⊂ ℝ 2 . $$x=(x_1, x_2) \in \Omega \subset \mathbb R^2.$$ Here, g ( x , | u | 2 ) = K 1 + V ( x ) + | u | 2 , $$\displaystyle g(x, |u|{ }^2)=\frac {K}{ 1+V(x)+|u|{ }^2}, $$ where V (x) is a continuous, nonnegative function periodic in Ω ¯ . $$\overline \Omega .$$ Steady state solutions satisfy the following equation over a periodic domain Ω ⊂ ℝ 2 : $$\Omega \subset \mathbb R^2:$$ Δ u = P u 1 + V ( x ) + | u | 2 + λ u , $$\displaystyle \Delta u= \frac {Pu}{1+V(x)+|u|{ }^2} + \lambda u, $$ where P, λ are parameters. The solutions u are to be periodic in Ω with the same periods as those of Ω. This equation has the trivial solution u = 0.

Date: 2020
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DOI: 10.1007/978-3-030-45603-0_16

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