In this paper, we propose new numerical methods for scattering problems in periodic waveguides. Based on [20], the “physically meaningful” solution, which is obtained via the Limiting Absorption Principle (LAP) and is called an LAP solution, is written as an integral of quasi-periodic solutions on a contour. The definition of the contour depends both on the wavenumber and the periodic structure. The contour integral is then written as the combination of finite propagation modes and a contour integral on a small circle. Numerical methods are developed and based on the two representations. Compared with other numerical methods, we do not need the LAP process during numerical approximations, thus a standard error estimation is easily carried out. Based on this method, we also develop a numerical solver for halfguide problems. The method is based on the result that any LAP solution of a halfguide problem can be extended to the LAP solution of a fullguide problem. At the end of this paper, we also give some numerical results to show the efficiency of our numerical methods.

在本文中，我们提出了解决周期性波导中散射问题的新数值方法。基于[20]，通过极限吸收原理（LAP）获得的“物理上有意义的”解被称为LAP解，写成等高线上准周期解的积分。轮廓的定义取决于波数和周期结构。然后将轮廓积分写为有限传播模式和小圆上的轮廓积分的组合。数值方法是基于这两种表示法开发的。与其他数值方法相比，我们在数值逼近过程中不需要 LAP 过程，因此很容易进行标准误差估计。基于这种方法，我们还开发了半导问题的数值求解器。该方法基于以下结果：半导问题的任何 LAP 解决方案都可以扩展到全导问题的 LAP 解决方案。在本文的最后，我们还给出了一些数值结果来展示我们的数值方法的有效性。