Research On Mode Reduction In Vertical Mode Expansion Method For Computing Photonic Crystal Slab Structures

Due to the unique bandgaps property,photonic crystals(Ph Cs)have been ex-tensively studied in the last three decades,especially one type called Ph Cs slabs.Since they are easy to fabricate and have a lot of applications,the Ph C slabs have become an important kind of fundamental photonic structures.For the need of de-sign and application,for many years researchers have been working on develop-ing and applying numerical methods to solve the computational problems of Ph C slabs.However,the complicate three-dimensional geometry makes it difficult to develop an efficient numerical scheme.Among the existing numerical methods,there is a semi-analytical method known as a vertical mode expansion method(VMEM).It takes advantage of the cylindrical feature of Ph C slabs,and reduces the three-dimensional Maxwell’s equations to problems of a bunch of two-dimensional Helmholtz equations.For the VMEM,its key step is to expand the electromagnetic fields with one-dimensional eigenmodes in the vertical direction,and then it constructs the so-called Dirichlet-to-Neumann(Dt N)map so as to build a linear system for computing the unknown expansion coefficients.Generally,in order to get results with high ac-curacy,it is necessary in the VMEM to construct a large matrix approximation of the Dt N map,which limits the applications of the VMEM for more complicated structures.In this thesis,a new mode reduction scheme is developed to reduce the size of the matrix approximation of the Dt N map in the VMEM.The new scheme takes into consideration the effects of different vertical modes,and then arrange these modes in descending order according to their impacts.For the sorted vertical modes,the number of discrete points related to each mode in the horizontal plane is set to decrease as the order of vertical modes decreases.Particularly,there is only one point on each edge in the horizontal plane for vertical modes placed in the back of the sequence.Thus,the size of the Dt N map matrix is significantly reduced.In the numerical examples presented in this thesis,the size of the new Dt N map matrix obtained by applying the mode reduction scheme is just 2.4 percent of that of the old Dt N map matrix,while the error is less than 10-~3.The result here shows that the new mode reduction scheme is able to effectively improve the efficiency of the VMEM.