Abstract
Two novel methods for implementing recursively the convolution between the electric field and a time dependent electric susceptibility function in the finite-difference time domain (FDTD) method are presented. Both resulting algorithms are straightforward to implement and employ an inclusive susceptibility function which holds as special cases the Lorentz, Debye, and Drude media relaxations. The accuracy of the new proposed algorithms is found to be systematically improved when compared to existing standard piecewise linear recursive convolution (PLRC) approaches, it is conjectured that the reason for this improvement is that the new proposed algorithms do not make any assumptions about the time variation of the polarization density in each time interval; no finite difference or semi-implicit schemes are used for the calculation of the polarization density. The only assumption that these two new methods make is that the first time derivative of the electric field is constant within each FDTD time interval.
Highlights
T HE finite-difference time domain method (FDTD) [1], [2] is a very popular numerical technique for solvingMaxwell’s equations for a wide range of different media, including materials with frequency dependent properties
piecewise linear recursive convolution (PLRC) assumes that the electric field has a piecewise linear behavior and uses a central difference scheme in order to calculate the derivative of the polarization density in time
We present two new novel recursive convolution based methodologies which in their development make only the assumption that the first time derivative of the electric field is constant in each time interval—in other words that the electric field is piecewise linear between time steps—and do not make any assumptions about the variation in time of the polarization density
Summary
T HE finite-difference time domain method (FDTD) [1], [2] is a very popular numerical technique for solving. PLRC assumes that the electric field has a piecewise linear behavior and uses a central difference scheme in order to calculate the derivative of the polarization density in time. PLRC is a widely used method, and one of the key reasons for its popularity, is that it is an accurate algorithm which can simulate materials with an inclusive susceptibility function which holds as special cases the Lorentz, Debye, and Drude media. This makes the implementation of dispersive materials in FDTD codes easy and practical and at the same time retains its computational efficiency [22], [36]. Using a central difference scheme to calculate the derivative of the electric field in time, (36) becomes (37) Substituting (37) to (34) and subsequently into (33) for and discretizing as was done in (15) yields (15) with (38)
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