We developed a generalized version of the invariant imbedding method, which enables us to solve the electromagnetic wave equations in arbitrarily inhomogeneous stratified media, where both the dielectric permittivity and magnetic permeability depend on the strengths of the electric and magnetic fields respectively, in a numerically accurate and efficient manner.
Using our version of the invariant imbedding theory, we study several problems based on three geometrical structures. First, the structures based on one-dimensional nonlinear photonic crystals, we studied both s and p wave cases of the electromagnetic wave propagation in one-dimensional nonlinear photonic crystals with and without a defect layer. Moreover we have calculates the transmission spectra and the electric field distribution inside nonlinear photonic crystals and we found that the electric field inside the defect layer is strongly enhanced. Additionally we have also observed that strong the optical bistability near the defect frequency. The strongest enhancement in both s and p wave cases of the electromagnetic field at defect layer in the linear calculation is found in some unsymmetry configurations.
We also investigated the optical diode properties of one-dimensional photonic crystals, which is made of alternating linear and nonlinear dielectric layers. More specifically, we have considered a photonic crystal consisting of twelve periods of polydiacetylene and titanium dioxide bilayers, where polydiacetylene layers are assumed to have Kerr-type focusing nonlinearity. The ratio of the transmittances for visible electromagnetic waves incident from the left and the right side of the structure were calculated and found that at the frequencies close to the band edge of the transmission spectrum, the ratio of the transmittances can be as high as 10 at the intensity is about 10 MW/cm^2 even the total thickness of the medium is about 200 nm. Hence, we expected that this phenomenon can be applied to design optical device such as optical diode.
We are interested in the second structure based on a bi-layer consisting a metal and a nonlinear dielectric layer. When the surface plasmons are excited at a metal-dielectric interface, the electromagnetic field takes a very large value near the interface. If the dielectric is a nonlinear Kerr medium, then the effect of nonlinearity can be greatly amplified due to the field enhancement. In this thesis, we calculate the lateral shift of p wave beams incident on metal-dielectric multilayer systems in both the Otto and Kretschmann configuration in a numerically exact manner. In the linear case, we find that the lateral shift becomes very large at the incident angles where the surface plasmons are excited. Whereas, the nonlinearity is turned on, the value of the lateral shift changes rapidly and found that even a small change of the intensity of the incident wave can cause a huge change of the lateral shift. Hence, We proposed that this phenomenon can be applied to designing precise optical switches operating at small powers.
We investigated the influence of nonlinear gain on the surface plasmon excitations in the Otto configuration and found that when the gain coefficient is large (about -0.1) the influence of the nonlinear gain on the reflectivity is small. When the value of the gain coefficient is small (about -0.01) there is a strong influence of the nonlinear gain on the reflectivity spectrum. This analysis clearly indicated that when the nonlinear gain is turned on, the strongest effect of nonlinear gain occurs at a small nonlinear gain parameter (about quarter of the nonlinear parameter) but not too small, and the effect of nonlinear gain becomes weaker at a large nonlinear gain parameter (about the nonlinear parameter) Hence, we assumed that the effect on nonlinear gain can be applied to design optical device at small powers.
We are interested in the Goos-Hanchen (GH) shift of a Gaussian incident beam. We generalize the invariant imbedding theory to a Gaussian incident beam and study the GH shift associate with surface plasmon polarization in the Otto configuration and found that the GH shift is still larger more than ten times of the incident wavelength in the finite half width of a Gaussian beam. The GH shift is the same as the value for a single plane wave when the half width of Gaussian incident beam is large. We found that the Gaussian beam is considered as a plane wave when the half width is larger than 200 nm.
Finally, we are interested in a media with nonlinearity in both ？ and ？ and the negative index material. We have applied our method to a uniform nonlinear slab and found that, in the presence of strong external radiation, an initially uniform medium of positive refractive index can spontaneously change into a highly inhomogeneous medium, where regions of positive or negative refractive index as well as metallic regions appear. We also studied the wave transmission properties of periodic nonlinear media. The current study argue that our theory is very useful in the study of the optical properties of a variety of nonlinear media including nonlinear negative index media fabricated using wires and split-ring resonators.
Moreover, we study a layered structure consisting of a slab of a positive index material with Kerr-type nonlinearity and a thin film which is made of a negative index material (NIM) or a positive index material (PIM). The structure is sandwiched between semi-infinite linear dielectrics. The influence of a thin film on the transmittance of a single nonlinear slab was observed and found that the peak of the transmission spectrum is shifted to a larger (smaller) angle due to a PIM (NIM) thin film in p wave case. Finally, we studied the influence of a thin film on the transmittance of a single nonlinear slab in both the focusing and defocusing cases and in both s and p-polarized waves. The detail study was used to identify the influence of the variations of the thin film parameters on the transmittance of a single slab.