Nanotechnology is a scientific frontier with enormous possibilities. Reducing the size of objects to nanometer scale to physically manipulate the electronic or structural properties offers a fabrication challenge with a large payoff. Nanophotonics is a subfleld of nanotechnology and a part of nanophotonics includes photonic band gap structures, which manipulate the properties of light to enable new applications by periodically modulating the relative permittivity. In photonic band gap (PBG) structures, the electromagnetic properties of materials, such as the electromagnetic density of states, phase, group velocities, signal velocities, field confinement, and field polarization are precisely controlled. The size scale of interest in PBG structures is typically of the order of a wavelength, which is not quite as demanding as required to observe quantum confinement effects in electronic materials. Nevertheless, photonic devices designed with nanophotonic technology enable new technology for devices and applications in sensing, characterization, and fabrication.
Even though PBG photonic devices are complex and the fabrication is often expensive, rapid progress on PBG structures has been possible because of the development of powerful numerical computation tools that provide a detailed analysis of the electromagnetic properties of the system prior to fabrication. To design photonic devices we use a variety of computational techniques that help in evaluating performance.
Several books have already been written about the optical properties of PBGs. A classic book on ID periodic structures was written by Brillouin. Yariv and Yeh's book is an excellent resource on many aspects of periodic optical media. Recent books devoted to the subject include the books by Joannopoulos et al., the very thorough book by Sakoda, and a recent book on nonlinear optics of PBGs by Slusher and Eggleton that features results of several researchers who have contributed to the subject. In addition, many good articles on PBG structures can be found in special issues or in summer school proceedings.
Numerical approaches are available to completely describe the properties of electromagnetic wave propagation in PBG structures. Three methods are of general use; they are the plane wave, the transfer matrix, and the finite-difference time-domain (FDTD) methods. The results of the plane wave method with the latter two are to some degree complementary, as is demonstrated and discussed later in this chapter.
Analytical methods are also available and have been especially useful for 1D systems. For instance, the development of coupled-mode equations for propagation by using multiple scales or slowly varying amplitude methods has given researchers powerful tools for studying nonlinear effects and designing new electro-optic (EO) devices, such as tunable optical sources from the ultraviolet to the terahertz regime, EO modulators, and a new generation of sensitive bioâchem sensors.
In this chapter, several basic algorithms are introduced and results exemplifying each numerical method are presented.