Photonic Crystal Calculation Methods And Design Research

A photonic crystal is an artificial structure which has a periodic arrangement of dielectric or metallic materials. In the past decade, it has become a new fast-developing research field due to its unique properties and many potential applications.As the structure of a photonic crystal is complex, it is relatively hard to analyze a photonic crystal in an explicit way. People usually analyze a photonic crystal through numerical simulations. Therefore, it is essential to study numerical methods in the theoretical research of photonic crystals. In this thesis, a new finite difference time domain (FDTD) method is developed to treat a two-dimensional photonic crystal consisting of nearly-free-electron metals. The method is used to calculate the band structures and investigate defect modes and guide modes in such a photonic crystal.Among several theoretical methods proposed for photonic crystals, theplane-wave expansion method (PWM) is the one that was proposed first and is usedmost commonly. However, it has the disadvantage of slow convergence. This ismainly due to the discontinuity in the permittivity at the boundaries of the inclusions.In this thesis we have introduced the effective medium theory to overcome effectivelythis problem. A plane-wave expansion method based on the effective medium theoryis firstly introduced, and the new method has a better convergence than that of thePWM. As a new method for calculating the band structure of a photonic crystal, afinite difference scheme is also introduced to the governing differential equationbased on the effective medium theory. Compared with the conventional PWM, thepresent finite difference method improves the convergence of the solution and thusprovides a fast and accurate algorithm.For many applications of photonic crystals, it is essential to design structures with large band gaps. Using a general approach for designing two-dimensional photonic crystals of square lattice, we have found many structures whose absolute band gap hi the low or high frequency range is much larger than what has been reported in the previous literature. To speed up the computation, a fast plane-wave expansion method is introduced for calculating the band structures for such special photonic crystals.Recently, much attention has been attracted by the negative refraction in the so-called left-handed media (LHM). In this thesis, a novel directional coupler with a LHM layer between single-mode waveguides of usual material is introduced. It is shown that the coupling length for such a waveguide system can be shortenedsignificantly by putting a LHM layer since the LHM can amplify evanescent waves.Negative refraction can also occur in some special photonic crystals. As the last part of this thesis, subwavelength imaging by photonic crystals is theoretically investigated. Using the FDTD method, the imaging by photonic crystals is demonstrated and the transfer function of this imaging system is given, which illustrates the influence of surface terminations to the imaging quality.In conclusion, the work in the present thesis is devoted to developing new numerical methods, designing new structures with large absolute bandgap and studying the negative refraction for photonic crystals. Much progress has been obtained in each of the above-mentioned aspects.