Numerical Methods For Forward And Inverse Problems Of Diffraction Grating

The study of scatterings by periodic structures is very important in science and engineering,where periodic structures are generally called diffraction gratings.As a new technique,diffraction gratings are extensively used in optical elements such as corrective lenses,beam splitters,and sensors.When time-harmonic electric-magnetic waves propagate in homogenous media,d-iffraction phenomena will happen and polarizations will take place on the grating surface.Standard polarization generally include TE(transverse electric field)polarization and TM(transverse magnetic field)polarization.Thus the model problem of the three-dimensional Maxwell equations can be reduced to a simpler model problem of the two-dimensional Helmholtz equation.The TM polarization causes the electric field component at the grating interface to meet the Dirichlet boundary condition,and the TE polarization leads to the magnetic field component at the grating interface to satisfy the Neumann boundary condition.What are first considered is the problem of a class of diffraction grating.This kind of direct problem is solving diffraction field(or total)by using the integral equation method given plane incident wave and grating shape function.The diffraction field is rep-resented by the single layer potential method.For solving the ill-posed integral equation,we give the numerical implementation process of Nystrom method.For the several cases of grating shape,such as smooth curve and polygon,we give numerical experiments.Additionally,we are concerned with the numerical solution of the inverse diffraction problem under the Neumann boundary condition,which may be described as follows.Given an incident field,one try to determine the grating structure from a measured reflected field at a straight line away from the structure.In order to fit the grating function more accurately,we use the multi frequency data,and use the Fourier series expansion of the periodic function to approximate the grating function.For the nonlinear problem,we use the nonlinear Landweber iterative method and obtain the approximate Fourier coefficient of the grating function.Some regularization methods are used to ensure the stability of the numerical algorithm.Finally,a large number of numerical experiments have been presented to demonstrate the effectiveness of the algorithm.