| Thirty years have passed since that time when the analogy between solid-sate and optics led to the concept of photonic crystals(PhCs), which is first termed by Yablonovitch and S. John. Fast progress in theory research and appli-cations of PhCs has been stimulated to a large extent by their particular features that allow increasing the potential of confining and controlling the electromag-netic wave propagation. A large variety of the fascinating physical phenomena have been demonstrated in the structure of PhCs with linear lattice, which in-clude superprism, subwavelength imaging, splitters, negative refraction, etc.Nonlinear PhCs have motivated extensive interest due to the important central role of the wavelength converters in all optical networks over recent decades.Therefore, it will be significant for the wavelength conversion technique and wavelength division multiplexing technology to study the harmonic generations in nonlinear PhCs and optimal designs for some PhCs devices. In this thesis, the main innovative research efforts and achievements are summarized as follows:(1) A systematical approach to obtain the optimal design for some non-linear effects in nonlinear PhCs is developed. A computational study of some harmonic generations and enhancements through X(2) or X(3) processes are p-resented. The mathematical models partly overcome the shortcoming of some existing models based on the undepleted pump approximation, which is directly derived from a nonlinear system of Maxwell equations, together with boundary conditions obtained from jump conditions.(2) A variational approach that combined the finite element methods and the fixed-point iteration is developed. It can be used to study the second har-monic generation or third harmonic generation through different nonlinear pro-cesses. This finite element fixed-point iteration algorithm (FEFPIA) can accu-rately predict the field intensity distribution of both reflected and transmitted waves when an intense pump beam is incident on the surface of a PhC. To over-come the shortcoming that FEFPIA may fail to produce convergent solutions when the nonlinearity is very strong, we introduce a continuation method into FEFPIA, which can be called CFEFPIA.(3) Another widely application of PhCs, photonic crystal waveguide bend,is also involved to study. A typical sharp waveguide bend is built by remov-ing one line of holes from a specified PhC on triangle lattice air holes arranged in Silicon background. By modifying the radii of air holes, both 120° and 60° PhC waveguide bend are designed to offer a maximum photonic bandgap around the wavelength ? = 1550nm. Based on the analysis of transmission properties of the conventional waveguide bend, we have optimized a sharp waveguide bend by simply changing the position and radius of air holes near the bending corner. The field computation is govern by a scattering problem and approached by finite element techniques. It involves in the design optimization problem, and usually several hundred solutions need to be computed in a single optimization procedure. Compared with some other optimization results, our optimized waveguide bend avoids using the more complicated shapes and has very lower fabrication complexity.Finally, all the results of numerical experiments show that the accuracy and efficiency of the designed methods and algorithms, which can help researcher design some optical devices. |
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