Phd in MATHEMATICAL MODELS FOR ENGINEERING, ELECTROMAGNETICS AND
NANOSCIENCES

Thesis title: Multi-sphere scattering and Fourier Modal method in periodic structures

The thesis has two-parts: The first part: The single-sphere and multi-sphere scattering is an important topic in many fields, such as plasmonics and photonics. Here I base on the Mie theory and transfer the Maxwell equations from the local Cartesian coordinate system to the spherical coordinate. In spherical coordinate systems, all the variables are described in , calculating the single-sphere scattering,usinge the Lentz method to get the Riccati-Bessel equations instead of the recurrence method, which removes the incorrect of recurrence method. After that we get the electrical field and the absorption coefficients in the calculation domain. Get gold and silver sphere scattering and compare the result with Finite Different Time Domains (FDTD). After getting the idea of sphere scattering, expand to multi-sphere scattering. In multi-sphere scattering, used the factorial primer method calculates the Wigner-3j functions, Gaunt-coefficient which is used in multi-sphere scattering, this method can make the calculation much faster and more precise than the former method, and compare them with Xu’s publications. Get the field and absorption coefficient from my code. Comparing them with the FDTD method, the results show the Mie theory is much faster and more precise than the FDTD method. Especially for the surface plasmon polariton which scientists have recently found stusing the Lentz method to get the Riccati-Bessel equations instead of the recurrence method, which removes the incorrect . I transfer the Maxwell e,quations from the local real space to Fourier Space, the advantage of Fourier space is all the available are described in Fourier Form, which transfers the location to the Harmonic numbers. For one direction periodic structure, gets the matrix differential equations in Fourier space, shows the disadvantage of classical FMM (Laurent’s method), when the incident wave is Transverse Magnetic (TM) wave and with complex refractive index in the structure, use the Fourier Factorization method instead of Laurent’s method, When the profile of the structure is not a rectangle, the normal vector method shows the advantages over than former method, when the conductivity is very high, in order to remove instability of over and low Gibbs phenomenon, a spatial adaptive resolution is used. For multilayer structure, I have studied the Transfer Matrix method, which is unstable in most situations. So, use the Enhance Transfer Matrix (ETM) method and S-matrix method, and show both methods are stable in the calculation, calculating reflection coefficients, transmission coefficients and the field in the structure. and getting the same results as the FDTD method, found out FMM method is much faster than the FDTD method. Expand it to two directions periodic structure. Compare the Fourier Factorization method and the normal vector method, if the structure surface of the structure is a rectangle the FF method is much better than the NV method. After that, get the forward of the FMM, especially the ASR method in two directions periodic structure.