## 3 Results
### Beam Overlap Characterization  
In this section, we analyze the beam overlap between the pump and signal beams within the nonlinear crystal, aiming to identify potential efficiency improvements. Based on Equation (7), the pump focus factor, \(\xi_{p}\), is held constant at 0.394, corresponding to a 100 \(\upmu\)m beam radius from the cavity simulation for the pump at 1064 nm. We plot the overlap integral \(h\) versus the signal focus factor \(\xi_{s}\), as shown in Figure 4. The simulated overlap integral \(h\) reaches higher values at lower \(\xi_{s}\) values, with an optimized focus lens achieving a simulated \(h\) value of 0.09282 at a \(\xi_{s}\) value of 0.8922. Experimentally, the red triangle indicates the previous setup [14] using an F280 collimator, where \(h=0.0652\) at \(\xi_{s}=3.23\), reflecting the limitations of the prior configuration. The red square represents the new setup with an optimized F110 lens, achieving an \(h\) value of 0.09282 at \(\xi_{s}=0.8922\), a value close to the simulation but slightly lower due to minor alignment and modeling discrepancies in thermal lensing for the pump beam. Figure 3: Measured and simulated DFG black-box efficiency versus 808 nm pump power for the same PPLN lengths: 29 mm (purple triangles), 40 mm (green squares), and 50 mm (red circles). The solid lines represent the simulated DFG black-box efficiency, while the markers indicate the experimental data. Data were obtained under constant temperature conditions, with significant findings demonstrating that longer PPLN lengths result in enhanced DFG power and black-box efficiency. The inset illustrates the three PPLN lengths tested, emphasizing their role in influencing DFG performance metrics. To quantify the expected efficiency gain, we calculate an overall theoretical efficiency boost based on the ratio of overlap integrals and input powers:  
\[\left(h_{new}/h_{old}\right)\times\left(400\ mW/330\ mW\right)_{1541 nm\ power}\times\left(4.7/4.5\ W\right)_{808 nm\ pump\ power}\]  
This theoretical boost results in an estimated 88% increase, while the experimental value reaches 80%, aligning closely with the simulated expectations. This 80% efficiency increase, derived as 63 mW/35 mW, verifies the effectiveness of the optimized configuration. Focusing the pump beam more tightly to improve overlap even further in the current setup presents challenges due to the hemispherical cavity design. The pump beam size is limited by the strength of the thermal lens, with stronger thermal lenses leading to theoretically smaller pump beam radii, which would improve the efficiency up to the theoretical maximum at a pump and signal focus factor approaching 2.2-2.8. However, at a certain point, we will reach the limit of cavity stability, limiting how much we can further focus the pump beam over the current radius. The hemispherical cavity was selected to keep the design compact, but to truly optimize the pump beam overlap as well as the signal beam overlap, a new cavity design is likely needed. Nonetheless, the current cavity design provides a reasonable efficiency that agrees well with theory, and the compact form factor provided by the hemispherical cavity means the device is useful for keeping the device's weight and volume down. A previous study from our group on a similar SFG-based design showed that by optimizing the pump and signal beam fully, the efficiency increase over the previous design could be >90%, so it is worth pursuing in future work [28]. Figure 4: A plot of the overlap integral for the current device, represented by the blue curve. The square dot indicates the optimized overlap efficiency for the current device, while the triangular dot represents the non-optimized overlap from previous work [14]. This comparison highlights the improvements in overlap efficiency achieved with the current design. The horizontal axis represents the signal focus factors (\(\xi_{s}\)), calculated by dividing the crystal length by twice the Rayleigh range of the signal beam.