High Accuracy Algorithms for Computing Light Propagation in Dielectrics, Dispersive Matierials and Metamaterials

由超材料和色散材料制成的结构(如衍射栅)的光学性质的计算是非常困难的。当结构既不是对称的也不是无限周期的时，麦克斯韦方程不存在解析解，需要使用数值方法。时域有限差分法(FDTD)可以处理任意结构，但对于色散材料，必须在FDTD算法中加入递归卷积(RC)。这不仅大大增加了计算成本，而且算法可能会变得数值不稳定。对于器件的设计来说，进行二维计算通常是足够的，二维计算的计算成本也很高，但在TE模式(电场E，垂直于无限大平面波的入射面)下，电场和磁场可以独立计算。即使当电介电常数随位置变化时，E也控制…更多地受到波动方程的影响。我们提出了一种新的波动方程RC-FDTD算法，极大地降低了TE模式的计算量。当在FDTD算法中加入RC时，必须使用色散的物理模型，但除非仔细选择模型参数，否则FDTD算法可能会变得数值不稳定。我们仔细分析了FDTD算法的稳定性，找到了保证稳定性的条件。对于TE模和TM模(E垂直于入射平面)，我们都这样做了。此外，我们还发现了如何根据频率来优化色散模型参数，以提高计算的精度。我们的新算法比通常的算法更准确，而且更稳定。通过计算大型周期结构(有效无限)的光学特性，并与解析解进行比较，验证了新算法的准确性。较少

It is very difficult to compute the optical properties of structures (such as diffraction gratings) made out of metamaterials and dispersive materials. When the structure is neither symmetric nor infinitely periodic, there are no analytic solutions of Maxwell's equations, and it is necessary to use a numerical method. The finite difference time domain algorithm (FDTD) can handle arbitrary structures, but for dispersive materials recursive convolution (RC) must be added to the FDTD algorithm. This not only greatly increases the computational cost, but also the algorithm may become numerically unstable. For designing devices it is often sufficient to do a two-dimensional calculation.The high computational cost of two-dimensional calculations is also high, but in the TE mode (electric field, E, perpendicular to the incidence plane of an infinite plane wave) the electric and magnetic fields can be computed independently. Even when the electric permittivity varies with position, E is govern … More ed by the wave equation. We have developed a new RC-FDTD algorithm for the wave equation, which greatly reduces the computational cost of the TE mode. When RC is added to the FDTD algorithm, a physical model of the dispersion must be used, but unless the model parameters are carefully chosen the FDTD algorithm may become numerically unstable. We have carefully analyzed the stability of the FDTD algorithm, and have found the conditions that guarantee stability. We have done this for both the TE and TM mode (E perpendicular to the plane of incidence). In addition we have found out how to optimize the dispersion model parameters according to frequency to improve the accuracy of the calculation. Our new algorithm is more accurate than the usual one and is stable. We verified the accuracy of our new algorithm by computing the optical characteristics of large periodic structures 8 effectively infinite) and comparing with analytic solutions.We used our algorithm to study metallic gratings made of gold and silver in the infrared region. Less