In this work we show an error estimate for a first order Gaussian beam at a fold caustic, approximating time-harmonic waves governed by the Helmholtz equation. For the caustic that we study the exact solution can be constructed using Airy functions and there are explicit formulae for the Gaussian beam parameters. Via precise comparisons we show that the pointwise error on the caustic is of the order $O(k^{-5/6})$ where $k$ is the wave number in Helmholtz.

翻译：在本研究中，我们展示了一个一阶高斯光束在折叠焦散处的误差估计，这个光束近似于由亥姆霍兹方程的时间谐波控制的波。对于我们研究的焦散，精确解可以使用艾里函数构造，而高斯光束参数有明确的公式。通过精确的比较，我们表明焦散点的逐点误差是$O(k^{-5/6})$的，其中$k$是亥姆霍兹中的波数。