Asymptotic Behavior of the TE and TM Approximations to Second Harmonic Generation (Journal of Computational Mathematics, 2018)
We consider a Second Harmonic Generation (SHG) problem of an optical signal wave with an optical pump in a medium represented by a smooth bounded domain $Ω\subset \mathbb{R}^d $, which is assumed to contain a heterogeneous material: a compactly imbedded subdomain $B^r\subset \subset Ω$ in the shape of a small ball contains a nonlinear material, while $Ω\setminus \overline {B^r}$ is filled with a linear material. We begin by proving existence and uniqueness of the solution to the TE approximation of SHG for arbitrary bounded susceptibilities, thus improving the result obtained by Bao and Dobson ( Eur. J. Appl. Math. 6 (1995), 573-590) under small enough susceptibilities assuption. We then establish an existence and uniqueness result of a solution to the TM approximation problem. In both parts we study the asymptotic behavior of the system as the size of the nonlinear material vanishes: error estimates and asymptotic expansion of the solution are derived for both TE and TM approximations.
Keywords: Second harmonic generation; TE Approximation; TM Approximation.