We present a general framework capable of describing the nonlinear propagation of pulsed optical beams of arbitrary shapes and phase fronts inside a graded-index (GRIN) fiber. The main assumption made is that the spatial self-imaging features of the beam are not affected by the temporal evolution of optical pulses. A propagation kernel known from the work done in the 1970s is used to obtain a distance-dependent nonlinear coefficient that captures all spatial effects within an effective nonlinear Schrödinger equation. We consider three specific beam shapes (Gaussian, circular, and square) to study the impact of the shape, position, and curvature of optical beams on the complex spatiotemporal dynamics specific to GRIN fibers. In particular, we focus on the impact of an input beam’s shape on the modulation-instability sidebands and the generation of multiple dispersive waves when higher-order solitons form inside a GRIN fiber. The results of our numerical analysis indicate that for beam widths chosen to yield the same value of the effective mode area at the input end of the fiber, the nonlinear effects are pronounced considerably when a Gaussian beam is launched into the fiber. We also found that even though the self-imaging period is doubled when an off-centered Gaussian beam is launched into a GRIN fiber, it does not affect the nonlinear evolution because the effective beam area still maintains the same periodicity, as long as the shift in the beam’s center is not so large that it does not remain confined to the fiber’s core.

中文翻译：

---

输入光束的形状和曲率对渐变折射率光纤中非线性效应的影响

我们提出了一个通用的框架，该框架能够描述渐变折射率（GRIN）光纤内任意形状和相前的脉冲光束的非线性传播。做出的主要假设是，光束的空间自成像特性不受光脉冲的时间演变影响。从1970年代完成的工作中已知的传播核用于获得依赖于距离的非线性系数，该系数捕获有效非线性Schrödinger方程内的所有空间效应。我们考虑三种特定的光束形状（高斯，圆形和正方形），以研究光束的形状，位置和曲率对GRIN纤维特有的复杂时空动力学的影响。尤其是，我们重点研究当GRIN光纤内部形成高阶孤子时，输入光束形状对调制不稳定性边带的影响以及多重色散波的产生。我们的数值分析结果表明，对于光束宽度选择为在光纤输入端产生相同有效模式面积的值，当高斯光束发射到光纤中时，非线性效应显着。我们还发现，即使将偏心的高斯光束发射到GRIN光纤中，自成像周期也会加倍，但它不会影响非线性演化，因为只要移位，有效光束区域仍保持相同的周期性光束中心的直径不会太大，以至于它不会局限于光纤的纤芯。