Numerical Modeling of Photonic Crystal Circuits Using Fourier Series Expansion Method Based on Floquet-Modes

Abstract

Microwave waveguides are usually composed of metals and dielectrics. The high contrast of their electromagnetic parameters benefits that the electromagnetic energy is strongly confined near the waveguiding structures and propagates along them. However, metal waveguides in millimeter and sub-millimeter wave regions are often suffered from large attenuations because the conductor loss of metal rapidly increases with the operating frequency. This is a reason why optical fiber, which is a waveguide for much higher frequency ranges, is composed of dielectrics only and avoids the use of metals. Then, the dielectric waveguides are widely investigated in millimeter and sub-millimeter wave regions to get rid of the conductor loss. They are typically composed of the core and the surrounding cladding. The refractive index of the core is some larger than that of the cladding. The fields propagating along a dielectric waveguide are not totally confined as with a metallic waveguide, and they are composed of the guided modes and the radiation modes. These modes are coupled to each other when the waveguide structure is not uniform along the wave propagation. Then the bending loss tends to be large in the dielectric waveguides. This is an obstacle to realize integrated circuits. The photonic crystal waveguide attracts attention as a waveguiding structure that resolves this problem. The photonic crystals are periodic dielectric structures that are designed to reject the propagation of electromagnetic waves at certain wavelength range. Local collapses of the periodicity supply significant advantages for field confinement, wave guiding, and directing radiation. Especially, defects introduced into the photonic crystals compose electromagnetic wave devices such as cavities, waveguides, splitter, coupler, etc. and they constitute photonic crystal circuits. The photonic crystals are composed of dielectrics only and the conductor loss is negligible in many cases. Also, the electromagnetic fields are strongly confined around the defects because any energy cannot escape through the surrounding photonic crystal. These features provide a significant progress towards reducing the size of electromagnetic wave circuits. The electromagnetic wave propagation along the photonic crystal circuits has been simulated using various numerical methods such as the beam propagation method(Koshiba et al., 2000), the finite difference time domain (FDTD) method(Taflove, 1995), and the plane wave expansion method(Benisty, 1996). These methods require adequate treatments of terminating conditions for the waves at the output ends of the circuits. However, the structure of photonic 5