Date of Award


Degree Name

Doctor of Philosophy



First Advisor

Dr. Arthur R. McGurn

Second Advisor

Dr. Thomas Gorczyca

Third Advisor

Dr. Alvin Rosenthal

Fourth Advisor

Dr. Dennis Pence


Photonic crystal waveguides, Kerr nonlinearity, difference equation waveguide, numerical simulation techniques, nonlinear optical media, waveguide and resonator modes


Nonlinear modes of electromagnetic fields propagating in photonic crystal systems have been studied by implementing various computer simulation techniques using electromagnetic theory. The fundamentals of simulation of photonic crystals are analyzed using general purpose methodologies such as the FDTD or PWE methods. Information derived from the underlying physical insights into the systems could be utilized to describe the control mechanisms over the propagation of the modes around impurities in the photonic crystal lattice. The impurities trap the resonantly localized electromagnetic modes having a frequency in a stop band of the photonic crystal, suggesting novel optical controls in the photonic crystal systems. Our focus is on deepening understanding of the nonlinear modes existing within photonic crystal waveguides which interact with the Kerr nonlinearity of the controlling medium; this enables us to reveal specific mechanisms of the nonlinear systems and their potential nonlinear functions and applications. To gain full generality of the nonlinear modes of increasing complexity, we propose a novel theoretical approach, the difference equation waveguide, using Green’s function theory of the non-homogeneous system. The recursive difference equations for the photonic crystal waveguides are solved for the guided modes interacting with multiple bound modes localized on the impurity features. The interest is on the transmissivity of resonant scattering of the nonlinear modes arising from in- or off-channel features formed in Kerr dielectrics in a 2D photonic crystal. Here, the modes display the wide and interesting varieties of behavior present in the system including optical bistability and induced transparency. More interestingly, the transmissions are fully treated for the case in which the field dependence of Kerr dielectric properties, with enhancing nonlinear effects, allows two different frequency waveguide modes to interact with one another by a modulation of the Kerr properties. In this regard, we show the observation of one mode used to model numerically the transmission characteristics of the other, e.g., the propagation of one frequency mode can turn on and off the resonant transmission of the other along the waveguide channel.

Access Setting

Dissertation-Open Access