## 1 Introduction  
There has been a surge in demand for compact and efficient mid-infrared (mid-IR) sources delivering several milliwatts to hundreds of milliwatts. These mid-IR lasers play vital roles in fields such as environmental monitoring for greenhouse gas (GHG) detection, free-space optical communication, medical diagnostics, and spectroscopy. Accurately detecting and analyzing chemical compounds in the mid-IR range is of particular interest for addressing pressing environmental challenges. Additionally, the reduced attenuation and scintillation experienced by mid-infrared wavelengths make these wavelengths an interesting option for terrestrial and long-distance satellite free-space optical communication [1, 2, 3, 4, 5, 6]. Mid-IR laser technology has seen significant progress recently, particularly in developing intracavity difference-frequency generation (IC-DFG) systems that leverage the nonlinear, optical properties of materials for efficient mid-IR wavelength generation [7, 8, 9]. However, it can be difficult to fully optimize all aspects of the system that affect device efficiency. A critical aspect is the overlap efficiency, which requires both longitudinal and transverse alignment of interacting waves within the optical cavity to maximize the output power. Improving the overlap efficiency can substantially enhance the output power and stability when using focused Gaussian beams. In our results, we observe a strong correlation between focused Gaussian beam theory and experimental data regarding the overlap efficiency, highlighting its importance in system optimization. Parameters such as the length and temperature dependence of the chosen nonlinear medium, which is periodically poled lithium niobated (PPLN) in this instance, also significantly affect IC-DFG system performance [9; 10; 11; 12; 13]. Addressing these challenges is vital for maximizing the overall efficiency and ensuring reliable DFG-based mid-IR lasers in practical applications. A common approach to evaluating DFG frequency converters involves several efficiency metrics. The primary metric, DFG efficiency (\(\eta_{DFG}\)), is expressed as follows:  
\[\eta_{DFG}=\frac{P_{DFG}}{P_{pump}P_{signal}}\times 100\% \tag{1}\]  
This efficiency, typically given in units of %W\({}^{-1}\), is calculated using the output DFG power, \(P_{DFG}\), measured after the Ge filter, relative to the input pump power, \(P_{pump}\), measured before the gain medium (in this case, before entering the Nd:YVO\({}_{4}\)), and the input signal power, \(P_{signal}\), measured at the input facet of the nonlinear medium [14]. While this metric provides a general efficiency measure, it does not account for coupling losses or the efficiencies of the pump and signal sources themselves, which can be crucial in power-limited scenarios, where maximizing the output power for a given input power is essential. To address this practical limitation, we include black-box efficiency (\(\eta_{BB}\)), shown as follows:  
\[\eta_{BB}=\frac{\eta_{pump}\eta_{signal}P_{DFG}}{P_{pump}P_{signal}}\times 1 00\% \tag{2}\]  
where \(\eta_{pump}\) is the optical-to-optical efficiency of the fiber pump source, and \(\eta_{signal}\) is the efficiency of the fiber signal source. For setups using fiber lasers, such as 1064 nm Yb-doped fiber lasers (pumped by 980 nm diodes), typical efficiencies are around 60%, so we assume \(\eta_{pump}\) = 0.6 [15]. For direct diode-pumped systems, we set \(\eta_{pump}\) = 1 due to the lack of an additional optical conversion step. We also assume \(\eta_{signal}\) = 1 for erbium-doped fiber amplifiers (EDFAs), which are commonly used in DFG setups. Although some EDFAs can achieve up to 80% efficiency [16], we omit this factor for simplicity, as all setups use EDFAs-based signal lasers, meaning that this would equally lower the efficiency for all laser sources. The third metric, internal efficiency (\(\eta_{internal}\)), provides insight into how well the system converts signal power specifically.