Optimal pump wavelength for frequency doubling in periodically poled media
Background
I am currently working on achieving second harmonic generation (SHG) in periodically poled lithium niobate (PPLN) using a 1560 nm pump laser. PPLN makes use of quasi-phase-matching of the pump and frequency doubled wave to achieve high conversion efficiencies. The phase matching is usually achieved by varying the crystal temperature and manufacturers provide phase matching curves like figure 1.
Problem
I was trying to calculate one of these curves myself to find the optimum parameters for my system. For quasi-phase matching in SHG the condition
$k_{shg} - 2k_{pump} - K_{QPM} = 0$
has to be met, where $k_{shg} = \frac{2\pi n_e(\lambda_{shg},T)}{\lambda_{shg}}$, $k_{pump} = \frac{2\pi n_e(\lambda_{pump},T)}{\lambda_{pump}}$ and $K_{QPM} = \frac{2\pi}{\Lambda(T)}$. For the temperature and wavelength dependent refractive indices I used empirical data from this paper (https://www.sciencedirect.com/science/article/pii/0031916366905919), which provides the <PERSON> equation for the extraordinary refractive index. 394
Similarly the periodicity $\Lambda(T)$ varies with temperature due to thermal expansion of the crystal (https://aip.scitation.org/doi/pdf/10.1063/1.1657244?class=pdf).
I attempted substituting the relevant <PERSON> equations into the quasi-phase matching condition to then solve for $\lambda_{shg}=\frac{1}{2}\lambda_{pump}=\lambda$. This gives me an equation of the form
$\frac{1}{\Lambda(T)} = \sqrt{\frac{h(T^2)}{\lambda^2}+\frac{k(T^2)}{\lambda^2(\lambda^2 - m(T^2)^2)}-l}-\sqrt{\frac{h(T^2)}{\lambda^2}+\frac{k(T^2)}{\lambda^2(4\lambda^2 - m(T^2)^2)}-4l}$
where $h(T^2), k(T^2), l$ and $m(T^2)$ are wavelength independent terms of the <PERSON><PHONE_NUMBER>), which provides the Sellmeier equation for the extraordinary refractive index. Similarly the periodicity $\Lambda(T)$ varies with temperature due to thermal expansion of the crystal (https://aip.scitation.org/doi/pdf/10.1063/1.1657244?class=pdf).
I attempted substituting the relevant Sellmeier equations into the quasi-phase matching condition to then solve for $\lambda_{shg}=\frac{1}{2}\lambda_{pump}=\lambda$. This gives me an equation of the form
$\frac{1}{\Lambda(T)} = \sqrt{\frac{h(T^2)}{\lambda^2}+\frac{k(T^2)}{\lambda^2(\lambda^2 - m(T^2)^2)}-l}-\sqrt{\frac{h(T^2)}{\lambda^2}+\frac{k(T^2)}{\lambda^2(4\lambda^2 - m(T^2)^2)}-4l}$
where $h(T^2), k(T^2), l$ and $m(T^2)$ are wavelength independent terms of the sellmeier equation.
So this is ultimately what I tried to use to solve for $\lambda$ but I can't see any obvious way of doing this analytically. Is there a way to do this (either by approximation or pure algebra) or do I have to find a numeric solution? Also how do the manufacturers of these crystals compute the tuning curves? Is my equation just wrong? 592