Dispersion-Relation-Preserving FDTD Schemes

The design of optimal second-order and higher order FDTD algorithms with dispersion-relation-preserving (DRP) properties constitutes a promising tool for the drastic reduction of dispersion errors, as firstly presented and extensively investigated in [55, 56]. Actually, the key 

FIGURE 2.10 Relative error of the normalized phase velocity (a) for a 2-D angle-optimized FDTD scheme, optimized at 22.5° without filtering and (b) for a 3-D Chebyshev angle-optimized FDTD scheme at 9 = 90°, p = 0° with different pc values 

Expanding electric and magnetic fields into a discrete set of Fourier modes and substituting the outcomes into (2.94), along with E(/) = Se MnAt and H(/) = 7 efa,( +1/2)4 one acquires 

The dispersion relation of (2.94) can be extracted if either or n is eliminated from (2.95). To obtain coefficients and, the maximum dispersion error will be initially expanded in a rapidly convergent series as a function of propagation angles 9 and ip and then its dominant terms will be set to zero. Hence, the coefficients, so computed, minimize the maximum dispersion error at all angles in completely adjustable way. 

Consider a plane wave traveling in the (9, ip) direction with kx = k sin0 cos ip, ky = k sin 9 sin ip, and kz = k cos 9. Recalling that such a wave can be decomposed into a TE and a TM polarization, the former part is selected for our analysis, yielding 

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