Nonlinear Wave Propagation and Imaging in Deterministic and Random Media

Abstract

This thesis consists of three projects that attempt to understand and identify applications for optical scattering from small nonlinear scatterers.

In the first part of the thesis we consider the direct scattering problem from a collection of small nonlinear scatterers. We considered all common types of quadratic and cubic nonlinearities within the scalar wave theory. We assume that the scatterers are small compared to the incident wavelength, thus the Lippman-Schwinger integral equations can be converted to algebraic equations. We further assume that the nonlinearity is weak, thus the scattering amplitudes can be calculated by solving the algebraic equations perturbatively. We apply this method to explore the redistribution of energy among the frequency components of the field, the modifications of scattering resonances and the mechanism of optical bistability for the Kerr nonlinearity.

In the second part of the thesis we generalized the optical theorem to nonlinear scattering processes. The optical theorem is a conservation law which has only been shown to hold in linear media. We show that the optical theorem holds exactly for polarizations as arbitrary functions of the electric field, which includes nonlinear media as a special case. As an application, we develop a model for apertureless near-field scanning optical microscopy. We model the sample as a collection of small linear scatterers, and introduce a nonlinear metallic scatterer as the near-field tip. We show that this imaging method is background-free and achieves subwavelength resolution. This work is done for the full Maxwell model.

In the third part of the thesis we consider the imaging of small nonlinear scatterers in random media. We analyze the problem of locating small nonlinear scatterers in weakly scattering random media which respond linearly to light. We show that for propagation distances within a few transport mean free paths, we can obtain robust images using the coherent interferometry (CINT) imaging functions. We also show that imaging the quadratic susceptibility with CINT yields better result, because that the CINT imaging function for the linear susceptibility has noisy peaks in a region that depends on the geometry of the aperture and the cone of incident directions.