In this dissertation, a set of numerical simulation tools are developed under previous work to efficiently and accurately study one-dimensional (1D), two-dimensional (2D), 2D slab and three-dimensional (3D) photonic crystal structures and their defects effects by means of spectrum (transmission, reflection, absorption), band structure (dispersion relation), and electric and/or magnetic fields distribution (mode profiles). Further more, the lasing property and spontaneous emission behaviors are studied when active gain materials are presented in the photonic crystal structures.;First, the planewave based transfer (scattering) matrix method (TMM) is described in every detail along with a brief review of photonic crystal history (Chapter 1 and 2). As a frequency domain method, TMM has the following major advantages over other numerical methods: (1) the planewave basis makes Maxwell's Equations a linear algebra problem and there are mature numerical package to solve linear algebra problem such as Lapack and Scalapack (for parallel computation). (2) Transfer (scattering) matrix method make 3D problem into 2D slices and link all slices together via the scattering matrix (S matrix) which reduces computation time and memory usage dramatically and makes 3D real photonic crystal devices design possible; and this also makes the simulated domain no length limitation along the propagation direction (ideal for waveguide simulation). (3) It is a frequency domain method and calculation results are all for steady state, without the influences of finite time span convolution effects and/or transient effects. (4) TMM can treat dispersive material (such as metal at visible light) naturally without introducing any additional computation; and meanwhile TMM can also deal with anisotropic material and magnetic material (such as perfectly matched layer) naturally from its algorithms. (5) Extension of TMM to deal with active gain material can be done through an iteration procedure with gain material expressed by electric field dependent dielectric constant.;Next, the concepts of spectrum interpolation (Chapter 3), higher-order incident (Chapter 4) and perfectly matched layer (Chapter 5) are introduced and applied to TMM, with detailed simulation for 1D, 2D, and 3D photonic crystal examples. Curvilinear coordinate transform is applied to the Maxwell's Equations to study waveguide bend (Chapter 6). By finding the phase difference along propagation direction at various XY plane locations, the behaviors of electromagnetic wave propagation (such as light bending, focusing etc) can be studied (Chapter 7), which can be applied to diffractive optics for new devices design.;Numerical simulation tools for lasing devices are usually based on rate equations which are not accurate above the threshold and for small scale lasing cavities (such as nano-scale cavities). Recently, we extend the TMM package function to include the capacity of dealing active gain materials. Both lasing (above threshold) and spontaneous emission (below threshold) can be studied in the frame work of our Gain-TMM algorithm. Chapter 8 will illustrate the algorithm in detail and show the simulation results for 3D photonic crystal lasing devices.;Then, microwave experiments (mainly resonant cavity embedded at layer-by-layer woodpile structures) are performed at Chapter 9 as an efficient practical way to study photonic crystal devices. The size of photonic crystal under microwave region is at the order of centimeter which makes the fabrication easier to realize. At the same time due to the scaling property, the result of microwave experiments can be applied directly to optical or infrared frequency regions. The systematic TMM simulations for various resonant cavities are performed and consistent results are obtained when compared with microwave experiments. Besides scaling the experimental results to much smaller wavelength, designing potential photonic crystal devices for application at microwave is also an interesting and important topic.;Finally, we describe the future development of TMM algorithm such as using localized functions as basis to more efficiently simulate disorder problems (Chapter 10). Future applications of photonic crystal concepts are also discussed at Chapter 10.;Along with this dissertation, TMM Photonic Crystal Package User Manual and Gain TMM Photonic Crystal Package User Manual written by me, Dr. Jiangrong Cao (Canon USA) and Dr. Xinhua Hu (Ames Lab) focus more on the programming detail, software user interface, trouble shooting, and step-by-step instructions. This dissertation and the two user manuals are essential documents for TMM software package beginners and advanced users. Future software developments, new version releases and FAQs can be tracked through my web page: http://www.public.iastate.edu/~mli/;In summary, this dissertation has extended the planewave based transfer (scattering) matrix method in many aspects which make the TMM and Gain-TMM software package a powerful simulation tool in photonic crystal study. Comparisons of TMM and GTMM results with other published numerical results and experimental results indicate that TMM and GTMM is accurate and highly efficient in photonic crystal device simulation and design. (Abstract shortened by UMI.)