Perturbation theory of transverse modes in the off-axis pumping cavity

Off-axis pumping scheme has been widely used in experiments to generate high-order transverse modes, and how to characterize the evolution principle of transverse modes in the off-axis pumping cavity has received widespread attention. In the present paper, we introduce an operator to convert the diffraction integral equation into the eigenequation and then exploit the perturbation theory of the eigenequation to perform a theoretical analysis for exploring the influence of off-axis pumping on transverse modes. Using operator, we can calculate the eigenequation of the off-axis pumping cavity without complicated integral calculation. Theoretical analysis shows that off-axis pumping can increase and decrease the order of transverse mode. It can provide a complete theoretical model for the off-axis pumping experiment and explain the phenomena in the experiment.