FDTD - NFFFT

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 46, NO. 9, SEPTEMBER 1998

1263

An Improved Near- to Far-Zone Transformation for
the Finite-Difference Time-Domain Method
Torleif Martin, Student Member, IEEE

Abstract— Near- to far-zone transformation for the finitedifference time-domain (FDTD) method can be performed by
integration of the equivalent electric and magnetic currentsoriginating from scattered electric and magnetic fields on a surface
enclosing the object. Normally, when calculating the surface
integrals, either the electric or magnetic fields are averaged
since the electric and magnetic fields are spatially shifted in the
FDTD grid. It is shown that this interpolation is unnecessary
and also less accurate than if an integration is performed on
two differentsurfaces. It is also shown that the accuracy of
the far-zone transformation can be further improved if the
phase is compensated with respect to a second-order dispersion
corrected wavenumber. For validation, scattering results for an
empty volume, a circular disk, and a sphere are compared with
analytical solutions.
Index Terms— FDTD methods.

I. INTRODUCTION

N

EAR- to far-zonetransformation in conjunction with
finite-difference time-domain (FDTD) has lately become
more frequently used. Due to the fast development of computers and the ability to model complex metal or dielectric objects,
FDTD has become a powerful tool for antenna and radar
cross-section (RCS) calculations. To the author’s knowledge,
the first published papers on near- to far-zone transformation
for FDTDappeared in the early eighties by Umashankar and
Taflove [1], [2]. Since then a number of such calculations
have been published (for example, see [3]–[8], [10], [11]).
The near- to far-zone transformation is normally performed
in the frequency domain, which requires that the scattered
fields are transformed into the frequency domain either by
a discrete Fourier transform (DFT) or a fastFourier transform
(FFT). Luebbers et al. [4] and Yee et al. [5] derived similar
time-domain far-zone transformations, which is well suited
for problems where time-domain results or a large number of
frequencies are required. The FDTD near- to far-zone transformation technique has also been extended to objects over lossy
dielectric half-planes [6], [7]. Recently, Ramahi presented a
near- tofar-zone transformation using the Kirchhoff’s surfaceintegral representation [8].
In an analytical case, equivalent electric and magnetic
currents multiplied with a Green’s function can be integrated
on a surface enclosing the object yielding the far-zone fields.
The analytical surface-equivalence theorem requires that both
the electric and magnetic surface currents lies on the same

surface. Whenapplying this to FDTD, which has spatially
shifted - and -fields, the normal procedure is to spatially
average the tangential -fields (or the -fields) from two
adjacent planes, so that the averaged fields represents the
values at the grid plane where the tangential -fields (or
the -fields) are positioned. This procedure is commonly
used but seldom discussed in detail and among the published
workon the subject referenced in this paper, it is only
explicitly mentioned in [4]–[6] and in [11]. In other published
work, the details of integrating the surface currents are not
treated although, in some cases, the derivations indicate that
this approach is used. Ramahi has shown that by using
the Kirchhoff’s surface integral representation instead of the
traditional vector potentials, thefield interpolation can be
avoided [8].
In this paper, an integral procedure is presented which is
based on the integration of equivalent surface currents but
where field interpolation is avoided. This resides from the fact
that the numerical interpretation of the equivalence theorem
for FDTD uses the tangential electric and magnetic fields on
two spatially shifted surfaces in order to...