Modeling of quasi-phase-matched cavity-enhanced second-harmonic generation

We propose a mean-field model to describe second-harmonic generation in a resonator made of a material with zinc-blende crystalline structure. The model is obtained through an averaging of the propagation equations and boundary conditions. It considers the phase-mismatched terms, which act as an effective Kerr effect. We analyze the impact of the different terms on the steady state solutions, highlighting the competition between nonlinearities.

Figure 1

Sketch of the considered resonator. The coordinates $({\mathbf{u}}_x,{\mathbf{u}}_y,{\mathbf{u}}_z)$ stand for the crystal axis while (${\mathbf{u}}_{\rho },{\mathbf{u}}_y,{\mathbf{u}}_{\varphi }$) refer to the cylindrical basis of the resonator. The inset shows the transverse structure of the ring that is composed of a rectangular waveguide of height $h$ and width $w$.

Figure 2

(a) and (b) show the nonlinear coefficient as a function of $\varphi$. (a) shows $\kappa \left(\varphi \right)$ when the SH mode is ${\mathrm{TM}}_{00}$. (b) shows $\kappa \left(\varphi \right)$ when the SH mode is ${\mathrm{TM}}_{10}$. (c) and (d) are the coefficients of the Fourier series of the functions represented in (a) and (b), respectively. (e) and (f) show $|{\kappa }_{+}|$ and $|{\kappa }_{-}|$ as a function of the bending radius for a SH modes ${\mathrm{TM}}_{00}$ (e) and ${\mathrm{TM}}_{10}$ (f).

Figure 3

Evolution of power in the FW (a) and SH (b) modes. Black (solid) line corresponds to the map, blue (point-dashed) line corresponds to Eqs. (16) with $\gamma =0$, and green (dashed) line is the solution of Eqs. (16) with $\gamma \ne 0$. The parameters are $\mathrm{\Delta }=-0.4$, $|a_{\text{in}}{|}^2=0.2\text{W}$, $|{\kappa }_{+}|=38$ ${\mathrm{W}}^{-1/2}\mathrm{m}{}^{-1}$, $|{\kappa }_{-}|=1101$ ${\mathrm{W}}^{-1/2}\mathrm{m}{}^{-1}$, $R$ = 14.5 $\mu \mathrm{m}$.

Figure 4

Comparison between the steady states obtained with the exact map (circles) and the averaged equations Eqs. (17) (blue line for SH and red line for the FW). (a) The phase-matched term mode is ${\mathrm{TM}}_{00}$. Coefficients are ${\kappa }_{-}=$ 1305i ${\mathrm{W}}^{-1/2}\mathrm{m}{}^{-1}$, ${\kappa }_{+}=-1613i$ ${\mathrm{W}}^{-1/2}\mathrm{m}{}^{-1}$, $\overline{\gamma }=-3\times {10}^{-4}$ and $h=122$ nm. (b) The phase-matched mode is ${\mathrm{TM}}_{10}$. Coefficients are ${\kappa }_{-}=$ 1083i ${\mathrm{W}}^{-1/2}$ $\mathrm{m}{}^{-1}$, ${\kappa }_{+}$=57i ${\mathrm{W}}^{-1/2}\mathrm{m}{}^{-1}$, $\overline{\gamma }$ $=-0.16$ and $h=130$ nm.