Lasers with different wavelengths have crucial applications such as optoelectronic countermeasures, lidar, precision measurement, and medical treatment. Compared to traditional population inversion lasers, stimulated Raman scattering (SRS) technology featuring a large Raman frequency shift is more conducive to the wavelength expansion of lasers and can achieve wavelength outputs that are difficult for conventional lasers to obtain. Additionally, unlike other second-order nonlinear effects, the self-phase matching characteristic of SRS allows for the generation of Stokes optical outputs at multiple frequencies via cascaded conversion in a gain medium. Diamond is an excellent Raman crystal providing ultra-high thermal conductivity (>2000 W·m^-1·K^-1) and a high Raman gain coefficient (~10 cm/GW@ 1 μm), and outperforming other Raman crystals such as YVO₄, BaWO₄, KGW, and Ba (NO₃)₂. The diamond Raman laser (DRL) based on cascaded conversion is an effective method for generating multi-wavelength laser output. Meanwhile, a theoretical output characteristic analysis of high-order Stokes light is necessary to achieve efficient multi-wavelength Raman laser conversion and adjustable output with various parameters.

We derive the theoretical equation for the third-order Stokes output characteristics in continuous wave pumping conditions based on the first-order and second-order external cavity Raman steady-state models. The steady-state equations for first-order and second-order Stokes while generating third-order Stokes light are obtained. The Raman cavity mode is simulated based on the designed diamond Raman resonator. The beam radii of the pump and first-order, second-order, and third-order Stokes light are determined. By employing a steady-state model and resonant cavity parameters of the designed DRL, we predict the output characteristics of a continuous wave external cavity pumped DRL. Additionally, we simulate the interaction between the output transmittance, maximum conversion efficiency, and Raman generation threshold of the third-order Stokes light. Additionally, we analyze the dependence of output power on parameters such as pump beam waist size, output transmittance, crystal length, and other related parameters under different pump powers.

By adopting the diamond Raman steady-state model, we calculate the variation of first-order, second-order, and third-order Stokes optical output power with pump power. The results demonstrated that output of different wavelengths with arbitrary power ratios can be achieved by controlling the output transmittance at different orders. We calculate the generation threshold and maximum conversion efficiency of third-order Stokes light (1851 nm) under various output transmittance. The simulation results indicate that optimal third-order threshold and conversion efficiency can be achieved by designing suitable output transmittance according to experimental pump conditions. We study the relationship between Stokes output power and output transmittance, pump beam waist size, and crystal length under different pump powers. The simulations reveal that optimal output power can be obtained by adjusting the pump beam waist size, output transmittance, crystal length, and other related parameters, thus providing theoretical guidance for future experimental research.

To deal with the current lack of parameter calculation models for cascaded Raman oscillators, we develop a third-order Stokes theoretical model based on first-order and second-order theoretical models. We provide the theoretical framework for generating third-order Stokes light and employ the diamond Raman steady-state model to calculate the variation of first-order, second-order, and third-order Stokes optical output power with pump power. The results indicate that outputs of different wavelengths and arbitrary power ratios can be achieved by controlling the output transmittance at different orders. Furthermore, the steady-state equation for any Stokes order laser can be derived by adopting the same deviation method. The theoretical derivation and numerical calculations presented in our study serve as valuable tools for designing and analyzing external cavity dual-pass pumped Raman lasers, providing references for experimental research in this field.