计算光子晶体和衍射光栅折射和反射谱的几种新方法-数学专业论文.docx

摘要数。对于由交错圆柱组成的三维光子晶体，以woodpile材料作为其特例，我们 摘要 数。对于由交错圆柱组成的三维光子晶体，以woodpile材料作为其特例，我们 研制了一个十分有效和精确的计算方法。该方法依赖于将一系列算子从材料 的_端推进到另一端。推进步骤中包括在每一排圆柱轴不变层上使用二维单 元上的DtN映射。进一步的简化方法可以通过Tangential．to—Tangential(T2T)算 子来实现，其在2D单元上，将波场的两个横向分量映射为另外两个横向分量。 这些算子通过柱面波展开方法来逼近成矩阵。 ，对衍射光栅，一个新的方法被研制出来，它将一个周期上的衍射光栅区域 分隔成一系列齐次子区域并在子区域上通过边界积分方程方法来计算NtD映 射。对每一个子区域，NtD算子在区域边界上将波场法向导数映射为波场本 身。该方法保留已有的边界积分方程方法优点，但它避免了较难计算的拟周 期Green’s函数。对用倾斜平面波照射在衍射光栅材料上的情形，NtD算子在子 区域边界上将波场的两个纵向分量映射为另外两个纵向分量。边界面处需要 对微分算子合理逼近来满足边界面上的条件。该方法对绝缘和金属构成的衍 射光栅均有效。 关键字：光子晶体，衍射光栅，Dirichlet．to．Neumann映射，算子推进，Neumann． to-Dirichlet映射，边界积分方程 Ⅱ ABSTRACTABSTRACT ABSTRACT ABSTRACT Due tO their unusual optical properties and significant potentials in applications， photonic crystals(PhCs)have been extensively studied both theoretically and experi- mentally．Many PhCs devices，such as waveguide bends，branches，frequency filters and waveguide couplers，are important building blocks of integrated optical circuits． Three-dimensional(3D)PhCs with a complete bandgap in the optical wavelength re— gion have potential applications，such as ultrahi gh quality—factor cavities and zero— threshold lasers．The woodpile structures composed of alternating layers of rods have attracted much attention due tO their relatively simple fabrication process compared with other 3D PhCs．Diffraction gratings have been extensively studied for many years，and they are important in practical applications，such as monochromators，spec· trometers，lasers，wavelength division multiplexing devices，optical pulse compressing devices，etc． Numerical methods are essential in the design，analysis and optimization of pho— tonic crystals and diffraction gratings．Some of these methods，such as the finite— difference time—domain(FDTD)method，are general methods that can be used tO study various aspects of PhCs and diffraction gratings，but their accuracy and efficiency are often limited．FDTD requires a small mesh size to