We present a nonlinear mixing study of Ince–Gaussian (IG) beams in non-collinear sum-frequency generation, completing the fundamental analysis of all three families of paraxial wave equation solutions under this nonlinear process. Through theoretical and experimental results, we consider the nonlinear mixing of even and odd IG modes, and transition between the paraxial modes by controlling the eccentricity parameter $\epsilon$ and helical IG modes possessing arbitrary orbital angular momentum. Additionally, the selection rule of IG modes in the nonlinear mixing processes is determined. This work is important for optical communications and quantum protocols.

中文翻译：

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通过和频生成混合Ince-Gaussian模式

我们介绍了非共线和频生成中Ince-Gaussian（IG）光束的非线性混合研究，完成了在该非线性过程下所有近轴波方程解的所有三个族的基础分析。通过理论和实验结果，我们考虑了偶数和奇数IG模式的非线性混合，以及通过控制偏心率参数$ \ epsilon $和具有任意轨道角动量的螺旋IG模式在近轴模式之间的过渡。另外，确定了非线性混合过程中IG模式的选择规则。这项工作对于光通信和量子协议很重要。