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Figure 1: The accelerating electric field in the cross-section of a cavity in a photonic crystal made of sapphire rods; here, the first ring of rods
has been moved inwards to the electric field minimum, thus increasing the Q factor and decreasing the possibility for breakdown.

Photonic crystals are crystal structures with "dielectric atoms" (imagine: a regular lattice of glass spheres, or a lattice of air bubbles inside a block of glass). Some photonic crystals reflect all incident light within a certain frequency range, regardless of incidence angle. Carefully designed, photonic crystals (PhCs) make excellent mirrors-but only for a narrow frequency range (unlike metals, which also make good mirrors, but indiscriminatingly reflect all light up to visible frequencies).

Dielectric photonic crystals could therefore replace metal in many applications requiring reflection within a narrow frequency band, such as in walls of waveguides wanted to support only a single frequency; moreover, photonic crystals may be preferable to metals: (1) at frequencies higher than several GHz, PhCs can be less lossy than metals; (2) at near-optical frequencies, dielectrics can withstand higher fields than metals; (3) narrow-bandwidth reflectivity is desirable where higher frequency fields are undesirable (like higher order modes in an accelerating cavity); (4) PhCs have many more adjustable properties than metals (for instance, the frequency window in which reflection occurs can be shifted or narrowed).

Figure 2

Although the relatively large number of variable parameters determining a PhC structure may prove beneficial (Figure 1 shows how a slight adjustment can improve the Q factor of a PhC cavity), optimization requires computer simulation of potential designs. Because a PhC structure tends to be much larger than its metal counterpart, this simulation requires significant computation time. Adding to the difficulty, curved dielectric surfaces are difficult to simulate. We have been searching for an algorithm to simulate electromagnetics in the presence of dielectric objects with curved surfaces; while frequency domain codes can achieve second-order error (in the grid cell size), time domain codes have so far achieved only first-order error. Frustratingly, there seems to be a trade-off between stability and accuracy of dielectric algorithms. We have demonstrated a stable FDTD algorithm for tensor dielectrics that has second-order error for continuously-varying dielectrics, but first-order error in the presence of sharp dielectric interfaces. This algorithm has been implemented into the VORPAL simulation framework.

To complement VORPAL's ability to simulate dielectrics, we have added other capabilities to help design PhC structures. For example, to test how well VORPAL can trap a mode within a finite PhC crystal, we added uniaxial perfectly-matched-layer absorbing boundary conditions. Figure 2 shows the reflection from such a boundary for waves incident at different angles. We continue to explore methods for improving the accuracy of dielectric simulation and speeding up the computation and analysis of dielectric PhC structures.

Figure 3

In 2006, we continued to work on the computational simulation of interactions of RF waves and plasmas. We have improved the computational method, and made it more accurate and efficient. With this simulation tool, we explored the linear mode conversion between an extraordinary wave and an electron Bernstein wave, and the work has been published.

Secondly, we demonstrated a new nonlinear phenomenon during the propagation of electron Bernstein waves. If the amplitude of an electron Bernstein wave is sufficiently large, it can generate another electron Bernstein wave (EBW) at the second harmonic frequency due to the self-interaction of the fundamental wave. This nonlinear process can change the wave propagation and power deposition significantly. Figure 3 shows the linear dispersion curves for the fundamental and second harmonic EBWs, and they are compared with the ones obtained from delta-f simulations. Good agreement indicates that the second harmonic EBW is generated. In Figure 4, the fundamental, second and third harmonic EBWs from a full-PIC simulation are shown at time t=630/ω0. the spectrum of the electrostatic wave is also plotted. Clearly, the second, third and fourth harmonic modes are excited.

Figure 4

We also have proposed a theory to describe the nonlinear self-interaction of an electrostatic wave. The simulation results have been compared to the predictions of the theory, and good agreement is found. We also develop a theory to describe the variation of a wave amplitude in an inhomogeneous plasma. A wave equation with a compact form is obtained which can describe the wave evolution in a complex kinetic plasma system. We compared the solutions of the wave equation with the results from delta-f simulations, and good agreement is reached.