## 2 Method and Theory
### Overlap Efficiency  
The overlap efficiency of the DFG process is also a critical factor in optimizing DFG processes, determining how effectively the interacting pump and signal beams combine within the nonlinear medium. Higher overlap efficiency translates to improved energy conversion and higher output power. Following the methodologies outlined by [PERSON] and [PERSON] [13; 18], we employed overlap integrals to quantify system efficiency, emphasizing \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Ref.** \\ \hline Nonlinear coefficient (\(d_{33}\)) & 27 pm/V & [14] \\ Nd: YVO\({}_{4}\) length (I) & 8 mm & This work \\ Output coupler transmission (T) at 1064 nm & 0.15\% & This work \\ Round-trip losses (L) & 3\% & This work \\ Absorption efficiency \(\eta_{a}\) & 99\% & This work \\ Boltzmann factor (f) & 0.6 & [14] \\ Upper manifold lifetime (\(\tau\)) & 90 \(\upmu\)s & [14] \\
1064 nm lasing beam radius (\(\omega_{L}\)) at the output coupler & 190 \(\upmu\)m & This work \\
808 nm pump beam radius (\(\omega_{P}\)) & 225 \(\upmu\)m & This work \\
1064 nm pump beam radius in PPLN & 100 \(\upmu\)m & This work \\ Signal beam radius in PPLN & 80 \(\upmu\)m & This work \\ Nd doping & 0.54 \(\upmu\)m & This work \\ PPLN refractive index & -2.2 & [14] \\ Stimulated emission cross-section (\(\sigma\)) & \(25\times 10^{-19}\)cm\({}^{2}\) & [14] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation input parameters for the DFG model, with experimentally determined or simulated values noted as “This work” and referenced values sourced from the literature. the importance of engineering beam diameters to enhance overlap efficiency. While their paper outlines a single-pass approach, we have found, using the previously described cavity simulation to estimate the circulating 1064 nm pump power and using ABCD matrix resonator modeling software (reZonator version. 2.0.13-beta9) to estimate the pump and signal beam sizes in the PPLN, that the simulation results are a good match for our intracavity device as well [27]. By carefully adjusting the beam sizes and geometries, we can significantly improve the overall DFG efficiency. The general idea is to integrate over the entire plane of the output crystal facet to obtain the output idler power for a given pump and signal power. The result of the derivation is the complex double integral \(h\) with many variable terms, found below [18; 13; 28]:  
\[h=\left(\frac{1}{4\xi_{4}}\right)II=\int\limits_{0}^{l}\int\limits_{0}^{l}dz_{ 1}dz_{2}\times\frac{exp[i\Delta kl(z_{1}-z_{2})]exp\left[-\left(\frac{4B^{2}}{ \alpha}\right)(z_{1}-z_{2})^{2}f\right]}{(z_{1}-A)\left(z_{2}-A_{1}^{*}\right) +C_{1}} \tag{6}\]  
where  
\[ \begin{array}{c}k=\frac{k_{1}}{k_{2}}\\ B=\frac{\rho}{2}\left(\frac{l(k_{1}+k_{2})}{2}\right)^{\frac{1}{2}}\\ \xi_{1}=l/(2z_{R1})\\ \xi_{2}=l/(2z_{R2})\\ \xi_{3}=\frac{\xi_{1}\xi_{2}(1+k)}{\xi_{2}+k_{1}^{*}}\\ \xi_{4}=\frac{k_{1}+k_{2}}{1+k_{1}^{*}}\\ \alpha=\frac{\xi_{1}+k_{2}}{(1+k)\xi_{2}}\\ A_{1}=\frac{1}{2}+\frac{i}{4}\left(\frac{1}{\xi_{3}}+\frac{1}{\xi_{4}}\right) \\ C_{1}=-\frac{1}{16}\left(\frac{1}{\xi_{3}}-\frac{1}{\xi_{4}}\right)^{2}\\ z_{3}^{\prime}=\frac{1}{2}+\frac{i}{2\xi_{3}}\end{array}  \tag{7}\]  
and  
\[f=\frac{(z_{1}-z_{3}^{\prime})(z_{2}-z_{3}^{\prime}{}^{*})}{(z_{1}-A_{1}) \left(z_{2}-A_{1}^{*}\right)+C_{1}} \tag{8}\]  
\(z_{1}\) and \(z_{2}\) are the positions of the waves inside of the nonlinear medium along the optical axis of propagation. The integral result, \(h(\Delta kl),\xi_{1},\xi_{2},k,B)\), incorporates several variables: \(\Delta kl\) (phase mismatch across crystal length \(l\)), the focus factors \(\xi_{i}\), calculated by dividing the crystal length by twice the Rayleigh range of each beam, and k, the pump-to-idler wavenumber ratio. For our setup with PPLN, the phase walk-off term \(B\) is zero. By tuning the phase mismatch through PPLN temperature adjustments, we maximized \(h\) for each crystal.