Theoretical And Numerical Studies On Scattering And Inverse Scattering By A Random Periodic Structure

Consider the scattering of a time-harmonic electromagnetic plane wave by a periodic structure.The scattering problem is to determine the scattered field from the incident field,the scatterer and properties of the medium,while the inverse scattering problem is to determine the wave source,the scatterer or material properties of the medium from the scattered field.In recent years,the scattering and inverse scattering by periodic structures has received extensive attention due to its diverse applications,particularly in the design and fabrication of optical elements such as corrective lenses,antireflective interfaces,beam splitters,and sensors.Existing studies of scattering problems mostly assume that the periodic structure is deterministic and only the noise level of the measured data is considered for the inverse problem.In practice,however,there is a level of uncertainty for the scattering surface,e.g.the grating structure may have manufacturing defects or it may suffer other possible damages from regular usage.Therefore,in addition to the noise level of measurements,the random surface itself also influences the measured scattered fields.Taking into account the uncertainty of the grating structure,this thesis will study the scattering and inverse scattering problems by random periodic structures which are more in line with the actual cases.The research content in this thesis is divided into two parts: theoretical analysis and numerical calculation.Theoretically,this thesis proves the well-posedness and a priori bounds of the Helmholtz equation on a random periodic Lipschitz surface,which is the first such results for the scattering problem by a random periodic structure.Since Lax-Milgram theorem cannot be used to obtain well-posedness and a priori bounds for the variational form of Helmholtz equation,and Fredholm theory is no longer applicable to stochastic Helmholtz equations,the proof method in this thesis is based on the well-posedness and a prior bounds of the solutions to the deterministic Helmholtz equation which describes the interaction between waves and deterministic periodic surfaces,and the equivalence of the three variational forms of the stochastic Helmholtz equation.The specific proof method is described as follows: First,determine the well-posedness and a priori bounds that is explicitly dependent on the large wavenumber for the deterministic periodic structure scattering problem? Then,integrate the deterministic results in the probability space and prove the measurability of each quantity?Finally,according to the equivalence of the variational forms of the random scattering problem,the corresponding well-posedness results and a priori bounds explicitly dependent on the large wavenumber are obtained.It is worth noting that our results do not depend on LaxMilgram theorem or Fredholm theory,which will help establish the theoretical framework of random scattering problems.Numerically,this thesis studies the modeling and calculation of the inverse scattering of a time-harmonic electromagnetic plane wave incident on a perfect conducting random periodic structure in the two fundamental polarizations of electromagnetic waves(i.e.,transverse electric(TE)polarization and transverse magnetic(TM)polarization).One difficulty lies in the nonlinearity between the boundary scattered fields and the scattering surface,and the ill-posedness of the inverse problem itself.Another difficulty is to understand to what extend the reconstruction could be made and how to recover statistics based on the boundary measured data to characterize the grating profile with uncertainties appropriately.Until now,only some initial progress has been made.A new and efficient numerical method,namely Monte Carlo-continuation-uncertainty quantification(MCCUQ)reconstruction method,is proposed in this thesis for both TE and TM polarizations,which is the first attempt to fully realize the reconstruction of the random periodic structure from noisy boundary measurements of the scattered fields away from the structure.The method is based on a novel combination of the Monte Carlo technique for sampling the probability space,a recursively linearized continuation method with respect to the wavenumber,and the Karhunen-Loève expansion for the uncertainty quantification of the random structure,which reconstructs key statistical parameters of the profile for the unknown random periodic structure.Numerical results are presented to demonstrate the reliability and efficiency of the MCCUQ reconstruction method proposed in this thesis for both TE polarization under Dirichlet boundary condition and TM polarization,with worse convergence and stability,under Neumann boundary condition.