Modal Computation Of Helmholtz Equation With Variable Coefficient And It’s Application

Based on the waveguide in physics, in the thesis, modal solution of Helmholtz equation with variable coefficients in unbounded domain is firstly considered, then the numerical algorithm of some inverse scattering problem is studied. For the first kind of problem, by using the perfectly matched layer (PML) to terminate the unbounded domain, two asymptotic dispersion relations for TM case are proposed, for the waveguides with slowly varied refractive index in the transverse direction. A more general situation, when the refractive index profile in the core is a continuous and derivable function, is also considered. Just as the above situation, the thesis gives the dispersion relation with higher precision for both TM and TE cases. Moreover, the asymptotic solutions for Berenger modes and leaky modes are deduced. Meanwhile, for the first kind of problem, a numerical reconstruction algorithm of a defect in an open waveguide with multiple frequency data is given.Specifically speaking, for the open waveguide, the PML is used to truncate the infinite domain. Due that the waveguide considered in the thesis has varying refractive index, the differential transfer matrix(DTM) is used to deduce the transfer matrix for TM case in the core and then operate integral operation and matrix exponential computing on the DTM. Matrix exponential computing is complex and it is difficult to give a simple and clear expression. So for gradually varied waveguide, the DTM is approximated by upper triangular matrix or lower triangular matrix separately. Two asymptotic dispersion relations for the TM case are proposed for varying refractive-index waveguide terminated by PML. After that, asymptotic formulae of the eigenmode are obtained. To increase the precision of eigenvalue, a more general case is considered in the thesis, Namely nonhomogeneous waveguide in which the refractive index is an continuous and derivable function in the core. The thesis don’t use any approximation for DTM and use its exact expression of it directly. Operate integral operation on the DTM and make matrix similarity transformation on the integral form of DTM. Through matrix exponential computing approximately, for a general waveguide with PML, the dispersion relations with higher precision for TE case and TM case are deduced finally. The general waveguide just need that the refractive index is derivative. The dispersion relations with higher precision can be applied in a general condition, without the need that waveguide is gradually varied. Moreover, under certain conditions, the above matrix exponential can be computed exactly, thus the dispersion relation obtained above is exact.For the numerical reconstruction of a defect in open waveguide, based on the existing theoretical method, a new numerical method, for the inverse medium scattering problem, is developed. At every fixed frequency, by using numerical searching and Newton’s iteration, a series of high precision eigenvalues are got. Then we use multiple frequency information and analyze the feature of the theo-retical formula. A linear system with relatively small size is established. Finally, by analyzing the distribution of singular value for the linear system, singular value decomposition and iterative regularization techniques are mainly used for regularization inversion to solve linear systems. Different defects (defects contain-ing fourier modes from low frequency to high frequency) are used in the numerical experiments. The numerical results show that the proposed numerical inversion algorithm is effective.