Solid-state lasers pumped by laser diodes have advantages including high conversion efficiency, excellent beam quality, and stable laser output. As the demand for laser power in various applications keeps increasing, the thermal effects inside the laser cavity have become a crucial factor affecting laser mode, efficiency, and beam quality. For end-pumped solid-state lasers, a good match between the pump light and laser mode is beneficial for better control of the laser beam quality. In contrast, side-pumping allows for higher pump powers but has poorer pump uniformity, which can easily lead to simultaneous oscillations of multiple transverse modes. Therefore, besides homogenizing the pump distribution as much as possible, it is necessary to study the effect of uneven pump distributions and the resulting thermal effects on the resonator modes. Since side-pumping and its thermal effects may have complex and asymmetric spatial distributions, and potentially excite higher-order transverse modes, in this study, we adopt a diffraction-based method for solving the resonator modes for analysis. This method has developed rapidly since the beginning of this century, and it can simultaneously calculate a large number of eigenmodes inside the resonator and handle resonators affected by complex physical fields, thus providing abundant physical information.

In this study, we first introduce the transmission matrix eigenvalue method for solving the resonant cavity mode, including its mathematical form and corresponding physical meaning. Using this method, we analyze the influence of thermal effects on the modes of a laser resonant cavity that utilizes a double-end side-pumped Nd∶YAG cylindrical crystal as the gain medium. In this process, a mathematical model for the double-end side-pumped power distribution is established, and the temperature and strain distributions within the gain crystal are obtained through finite element analysis (FEA). Then, the spatial phase modulation of the laser caused by the gain crystal, which results from end-surface deformation and refractive index variation under different pump powers, is derived. Finally, by integrating the FEA results with the transmission matrix method, the distributions and loss variations of both the transverse and longitudinal modes within the resonant cavity are traced, respectively.

We study the influence of thermal effects on both the transverse and longitudinal modes of the laser resonator at different pump power levels. Our results show that, in the presence of thermal effects, the losses of each transverse mode are higher than those of a resonator without such effects. As the pump power increases, there is a consistent trend in the loss variation of transverse modes, but there is a significant reordering of loss magnitudes among modes. Specifically, higher-order transverse modes can have lower losses than lower-order ones, making them more likely to oscillate (Figs. 12 and 13). In contrast, for longitudinal modes, thermal effects have a minimal effect on the mode spacing but cause a noticeable shift in the resonator’s characteristic frequencies (Figs. 14 and 15). In this study, we first use finite element analysis and ray tracing methods to analyze the intracavity phase modulation caused by thermal effects in doubly side-pumped solid-state lasers. To address thermal effects, we adopt the eigenvalue approach of the diffraction transfer matrix method, replacing the conventional thermal effect simulation based only on the thermal lens focal length. This effectively utilizes the rich spatial information obtained from finite element analysis, enabling the modeling and simulation of complex thermal phase modulations that are difficult to accurately describe only with the thermal lens focal length. This approach facilitates the simultaneous calculation of all eigenmodes and their respective losses within the resonator under the influence of thermal effects.

To study the effect of non-uniform and asymmetric thermal effects on resonator modes, we build a mathematical model for the power distribution of the pump light in an LD side-pumped solid-state laser. Using the eigenvalue method of the diffraction transmission matrix, we quantitatively study the influence of thermal effects within the laser medium on the transverse and longitudinal modes of the resonator in the case of this solid-state laser. Our research reveals that as the thermal effects increase with the increase in pump power, the overall loss of each transverse mode increases, causing the loss of some higher-order transverse modes to be lower than that of lower-order modes, thus making them more likely to oscillate. The amplitude and phase distributions of the solved transverse modes also indicate that distortions occur at the edges of different transverse modes, suggesting a degradation in beam quality. At the same time, while the longitudinal mode spacing of the resonator remains unchanged, the positions of the longitudinal modes shift.