Sun Qiang, Klaseboer Evert, Yuffa Alex J, Chan Derek Y C
J Opt Soc Am A Opt Image Sci Vis. 2020 Feb 1;37(2):284-293. doi: 10.1364/JOSAA.37.000284.
An efficient field-only nonsingular surface integral method to solve Maxwell's equations for the components of the electric field on the surface of a dielectric scatterer is introduced. In this method, both the vector wave equation and the divergence-free constraint are satisfied inside and outside the scatterer. The divergence-free condition is replaced by an equivalent boundary condition that relates the normal derivatives of the electric field across the surface of the scatterer. Also, the continuity and jump conditions on the electric and magnetic fields are expressed in terms of the electric field across the surface of the scatterer. Together with these boundary conditions, the scalar Helmholtz equation for the components of the electric field inside and outside the scatterer is solved by a fully desingularized surface integral method. Compared with the most popular surface integral methods based on the Stratton-Chu formulation or the Poggio-Miller-Chew-Harrington-Wu-Tsai (PMCHWT) formulation, our method is conceptually simpler and numerically straightforward because there is no need to introduce intermediate quantities such as surface currents, and the use of complicated vector basis functions can be avoided altogether. Also, our method is not affected by numerical issues such as the zero-frequency catastrophe and does not contain integrals with (strong) singularities. To illustrate the robustness and versatility of our method, we show examples in the Rayleigh, Mie, and geometrical optics scattering regimes. Given the symmetry between the electric field and the magnetic field, our theoretical framework can also be used to solve for the magnetic field.
介绍了一种高效的仅基于场的非奇异表面积分方法，用于求解介质散射体表面电场分量的麦克斯韦方程组。在该方法中，散射体内外均满足矢量波动方程和无散度约束。无散度条件被一个等效边界条件所取代，该边界条件关联了散射体表面电场的法向导数。此外，电场和磁场的连续性和跃变条件通过散射体表面的电场来表示。结合这些边界条件，采用完全去奇异化的表面积分方法求解散射体内外电场分量的标量亥姆霍兹方程。与基于斯特拉顿 - 朱公式或波吉奥 - 米勒 - 丘 - 哈林顿 - 吴 - 蔡（PMCHWT）公式的最流行表面积分方法相比，我们的方法概念上更简单，数值计算更直接，因为无需引入诸如表面电流等中间量，并且可以完全避免使用复杂的矢量基函数。此外，我们的方法不受诸如零频率灾难等数值问题的影响，并且不包含具有（强）奇异性的积分。为了说明我们方法的稳健性和通用性，我们给出了瑞利、米氏和几何光学散射区域的示例。鉴于电场和磁场之间的对称性，我们的理论框架也可用于求解磁场。
