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.

离轴泵浦方案已广泛应用于产生高阶横模的实验中，如何表征离轴泵浦腔内横模的演化原理受到广泛关注。在本文中，我们引入算子将衍射积分方程转换为本征方程，然后利用本征方程的摄动理论进行理论分析，以探讨离轴泵浦对横模的影响。利用算子，我们可以计算出离轴泵腔的特征方程，无需复杂的积分计算。理论分析表明，离轴泵浦可以增加和减少横模的阶数。它可以为离轴泵送实验提供完整的理论模型并解释实验中的现象。