A path integral time-domain method for electromagnetic scattering

Abstract

A new full wave time-domain formulation for the electromagnetic field is obtained by means of a path integral. The path integral propagator is derived via a state variable approach starting with Maxwell's differential equations in tensor form. A numerical method for evaluating the path integral is presented and numerical dispersion and stability conditions are derived and numerical error is discussed. An absorbing boundary condition is demonstrated for the one-dimensional (1-D) case. It is shown that this time domain method is characterized by the unconditional stability of the path integral equations and by its ability to propagate an electromagnetic wave at the Nyquist limit, two numerical points per wavelength. As a consequence the calculated fields are not subject to numerical dispersion. Other advantages in comparison to presently popular time-domain techniques are that it avoids time interval interleaving and it does not require the methods of linear algebra such as basis function selection or matrix methods.

A probabilistic approach to wave propagation and scattering

“…In the first papers exploring this direction [Buslaev, 1967;Keller and KcLaughlin, 1975] the ray approximation of the wave field was derived from the path integral solution of the Helmholtz equation. Later, the path integrals were used for numerical simulations of acoustical [Schlottmann, 1999] and electromagnetic [Nevels et al, 2000] waves, but as mentioned in the survey [Galdi et al, 2000], the perspectives of broader application of the path integrals to wave propagation were limited, presumably because of the notorious difficulty of computation of the Feynman path integrals.…”

Time-Domain Analysis of a Lossy Nonuniform Transmission Line

“…After the term-by-term operations in (6) are carried out, the series is returned to exponential form, producing…”

“…Recently, an unconditionally stable implicit path integral time-domain method for the electromagnetic field was introduced [6]. In a subsequent explicit version, which is referred to here as the explicit propagator (EP) method, it was shown that numerical dispersion is virtually eliminated on a homogeneous transmission line [7].…”

Random walk approach to wave propagation in wedges and cones

“…16 These theories have a long record of successful applications to numerical simulation of evolutions of quantum systems, and quite recently attempts have been made to apply path integral methods to acoustics 17 and electromagnetics. 18 Probabilistic methods have also been applied for the analysis of transport of energy by waves propagating in random media. 19,20 In our previous papers it was shown that the probabilistic solutions of the Helmholtz equation, as the exact solutions, make it possible to provide a unified description of different phenomena of wave propagation.…”