In a previous paper, a time domain transformation useful for extrapolating 3-D near zone finite difference time domain (FDTD) results to the far zone was presented. In this paper, the corresponding 2-D transform is outlined. While the 3-D transformation produced a physically observable far zone time domain field, this is not convenient to do directly in 2-D, since a convolution would be required. However, a representative 2-D far zone time domain result can be obtained directly. This result can then be transformed to the frequency domain using a Fast Fourier Transform, corrected with a simple multiplicative factor, and used, for example, to calculate the complex wideband scattering width of a target. If an actual time domain far zone result is required it can be obtained by inverse Fourier transform of the final frequency domain result

Beggs, John H.

Kunz, Karl S.

Luebbers, Raymond J.

Ryan, Deirdre

English

NASA Technical Reports Server

N92-19742
A TWO-DIMENSIONAL
TIME DOMAIN
NEAR ZONE TO FAR ZONE TRANSFORMATION
by
Raymond Luebbers, Deirdre Ryan, John Beggs
and Karl Kunz
Electrical and Computer Engineering Department
The Pennsylvania State University
University Park, PA 16802
May 1991
Abstract
In a previous paper [i] a time domain transformation useful
for extrapolating three dimensional near zone finite difference
time domain (FDTD) results to the far zone was presented. In
this paper the corresponding two dimensional transform is
outlined. While the three dimensional transformation produced a
physically observable far zone time domain field, this is not
convenient to do directly in two dimensions, since a convolution
would be required. However, a representative two dimensional far
zone time domain result can be obtained directly. This result
can then be transformed to the frequency domain using a Fast
Fourier Transform, corrected with a simple multiplicative factor,
and used, for example, to calculate the complex wideband
scattering width of a target. If an actual time domain far zone
result is required it can be obtained by inverse Fourier
transform of the final frequency domain result.
https://ntrs.nasa.gov/search.jsp?R=19920010500 2020-03-17T12:41:09+00:00Z
Introduction
A previous paper Ill described a method for transforming
near zone Finite Difference Time Domain (FDTD) results directly
to the far zone without first transforming to the frequency
domain. This far zone field could then be used to compute the
scattering cross section of an illuminated target or an antenna
radiation pattern over the entire frequency band of the FDTD
calculations. A similar result was published in [2]. In this
paper the corresponding two dimensional transform is presented,
and validated by comparison with calculated results for a
perfectly conducting circular cylinder.
Approach
In [i] the frequency domain far zone transformation
equations were Fourier transformed to the time domain and used in
that form to derive an approach to transform near zone FDTD
fields to the far zone directly in the time domain. Our approach
here will be to present the fundamental frequency domain
equations for both two and three dimensions, and by comparing
them obtain the factor needed to convert the three dimensional
far zone transform to function in two dimensions.
We again surround the scatterer with a closed surface S',
and consider that equivalent tangential electric and magnetic
time harmonic surface currents may exist on this surface.
Referring to [3], we obtain the vector potentials for the three
dimensional case as
IF -
e -j.%z
4_r f _" eJk_'" dS!
S _
(2)
with j=-_i, k = _ (the wave number), the unit vector
to the far zone field point,
--/
r the vector to the source point
of integration, r the distance to the far zone field point, and
S' the closed surface surrounding the scatterer.
The far zone frequency domain electric fields of the
scatterer are then obtained from
E 8 = -j_ A 8 - j_ F_ (3)
E, : - j_ A_ + j_ F 8 (4)
One can then easily convert to radar cross section (RCS), if
desired, by applying
S
lira (4_r 2 I E3DI 2
a30 = r_ jEll2 )
(5)
where
s
E30 is either E 0 or E_ of (2) or (3), and E i is the
incident plane wave electric field.
In [i] the equations corresponding to (1,2) were easily
transformed to the time domain since the exponential phase term
inside the integrals corresponds to a time shift relative to an
arbitrary time reference point. Equations corresponding to (2,3)
were also readily transformed to the time domain since the j_
factor in these equations corresponds to a time derivative. Thus
the resulting time domain fields can be obtained conveniently
directly in the time domain, as shown in [i].
Now consider the corresponding equations in two dimensions.
The vector potentials are given by
-jkp
= e ! Js eJkP'C°s(_') ds! (6)
_8jk_p
-jkp
-- e I eJ kp'c°s(O-_')
F - Ms ds/ (7)
where p' and _' are the coordinates of the source point of
integration, and p and _ the coordinates of the far zone field
point. The corresponding far zone radiated fields are obtained
from
E z = -j_ A z + j_ F_ (B)
(9)
E_ = - j_ _ A_ - j_ F z
Also, the two dimensional scattering width is defined as
s 2
lira I E2DI
°2D = p--_o (2n'p_)
(i0)
where
$
E2D is either E z or E¢ of (8,9).
The approach applied in [i] cannot be conveniently applied
in the two dimensional case due to the factor of I/_
(actually I/j_-_-_ ) in (6,7). In order to evaluate the
Fourier transform of (6,7) directly in the time domain a
convolution operation would be required. To avoid this
complication our approach will be to modify the results in [i] to
provide representative two dimensional time domain far zone
fields which can then be converted to the actual frequency domain
fields by a multiplication in the frequency domain rather than
the time domain convolution. Should the actual time domain far
zone fields be required, they can then be obtained by an
additional Fourier transform of these results back to the time
domain.
In order to convert our previous three dimensional results
to two dimensions, we compare the two sets of equations. First,
comparing (3,4) with (8,9), since the spherical unit @ vector is
equal to the negative of the cylindrical unit z vector, (3,4) and
(8,9) correspond exactly, and no adjustment between two and three
dimensional transforms is needed.
Next, comparing (1,2) with (6,7) the r 2 and p factors are
compensated by the definitions in (5) and (I0) respectively, as
expected, and no compensation is needed here either.
Finally, consider equations (1,2) vs (6,7). The additional
dimension of integration in (1,2) is compensated for by defining
a scattering width per unit length (in z) in (i0). This
corresponds in (1,2) to having no z variation and integrating the
z' variable over a unit distance. The exponents provide
equivalent phase (time) delays and need not be compensated for.
Considering the remaining factors, it is easily determined that
in the frequency domain, the relationship between far zone
electric fields obtained from a three dimensional far zone
transformation with no z variation and the two dimensional far
zone fields is
!
s | 2_C s
E2D = _ _ E3D
(Ii)
where c = i/_
With these results the time domain far zone transform given
in [i] can be easily adapted to two dimensions as follows:
i) Consider only the field components and corresponding surface
currents excited in the two dimensional problem. For example,
for a TE z computation only Hz, Ex, Ey, and the corresponding
surface currents are included.
2) Calculate the far zone time domain fields using the method
described in [i], but for a two dimensional integration surface
which encloses the scatterer. Let 6z, the z coordinate unit cell
dimension used in [I], equal 1 (meter). (This field is not a
physically observable field. It represents the radiation from a
unit length of the scatterer in the time domain.)
3) Fourier transform the result of step 2) and multiply the
result by the factor in (ii). This result is the frequency
domain two dimensional far zone field, which can then be used in
_i0) to calculate the scattering width as a function of
frequency.
4) If the actual time domain two dimensional far zone field is
desired, it can be obtained by an additional Fourier
transformation of the result obtained in 3) back to the time
domain.
Demonstration
In order to demonstrate the capabilities of the above
approach a pair (TE z and TMz) of FDTD codes were developed from
the three dimensional FDTD code described in [i]. These codes
utilize second order Mur absorbing boundary conditions acting on
the electric fields. The test geometry was a circular perfectly
conducting cylinder of radius 0.25 meters.
Both TE and TM polarization was considered, and two sets of
calculations were made for each polarization. For the first set
the FDTD cells were 1 cm squares, with a problem space 200x200
cells. For the second set the cells were 0.5 cm squares in a
500x500 cell problem space. On a 25 MHz 486 PC (approximately 1
MFLOP) each of the first set required about 20 minutes to
compute, with each of the second set requiring a few minutes less
than 2 hours.
For all cases the incident plane wave traveled in the x
direction, backscatter was calculated, and 2048 time steps were
evaluated. In order to clearly show the response, not all time
steps are included in the Figures.
Figure i shows the relative far zone fields computed
directly in the time domain as outlined above for the TM
polarization. The small ripple at approximately 15 ns on Figure
1 is due to reflections from the Mur outer boundary. Figures 2
and 3 show comparison with the exact solution for the scattering
width amplitude and phase. The upper frequency limit of 3.0 GHz
corresponds to i0 cells per wavelength. The agreement is quite
good.
A similar set of data for TE polarization is shown in
Figures 4-6. The time domain far zone results in Figure 4 has
ripples in the 6-8 ns range due to the staircasing of the round
cylinder with square FDTD cells. The small negative pulse at I0
ns is the creeping wave which has traveled around the cylinder.
Again, there is a small ripple at 14 ns due to the Mur boundary
reflection. This is the difficult polarization for approximating
a smooth surface with a "staircased" FDTD code, yet the agreement
in Figures 5 and 6 is reasonably good, reproducing the first 6
1/2 ripples in the scattering width.
The above calculations are repeated in Figures 7-13 with a
greater expenditure of computer resources. For these results the
cell size of 0.5 cm changes the 3.0 GHz upper frequency limit of
the plots to correspond to 20 cells per wavelength. The
improvement in the agreement with the exact solution is clear,
indicating the accuracy that can be obtained from this approach.
Note especially Figure 12, which shows on a expanded dB scale
agreement with the exact solution within a fraction of a dB for
the first 9 lobes of the response.
Conclusions
A simple approach to calculating a wide bandwidth time
domain transformation of near zone FDTD fields to the far zone
has been presented. It is based on simple modifications to the
previous three dimensional method presented in [i]. Results
obtained using this transformation show good agreement with the
exact solution for a circular cylinder for both polarizations.
References
i.
R. J. Luebbers, K. S. Kunz, M. Schneider, and F.
Hunsberger, "A Finite Difference Time-Domain Near Zone to
Far Zone Transformation," IEEE Trans. Ant. Prop., Vol. 39,
No. 4, April 1991, pp 429-433.
•
K. S. Yee, D. Ingham, and K. Shlager, ,,Time-Domain
Extrapolation to the Far Field Based on FDTD Calculations,"
IEEE Trans. Ant. Prop,, Vol. 39, No. 3, March 1991, pp 410-
412.
•
R. F. Harrington, Time-Harmonic Electromagnetic Fields,
McGraw-Hill, 1961.
Fiqure Titles
Fig i. Far Zone relative electric field vs time for TM z
polarized incident Gaussian pulsed plane wave
illuminating a perfectly conducting circular cylinder.
FDTD cells are i cm squares.
Fig 2. Scattering width amplitude obtained from far zone time
domain results and compared with exact solution. FDTD
cells are 1 cm squares.
Fig 3. Scattering width phase obtained from far zone time
domain results and compared with exact solution. FDTD
cells are 1 cm squares.
Fig 4. Far Zone relative electric field vs time for TE z
polarized incident Gaussian pulsed plane wave
illuminating a perfectly conducting circular cylinder.
FDTD cells are 1 cm squares.
Fig 5. Scattering width amplitude obtained from far zone time
domain results and compared with exact solution. FDTD
cells are 1 cm squares.
Fig 6. Scattering width phase obtained from far zone time
domain results and compared with exact solution. FDTD
cells are 1 cm squares.
Fig 7. Far Zone relative electric field vs time for TM z
polarized incident Gaussian pulsed plane wave
illuminating a perfectly conducting circular cylinder.
FDTD cells are 0.5 cm squares.
Fig 8. Scattering width amplitude obtained from far zone time
domain results and compared with exact solution. FDTD
cells are 0.5 cm squares.
Fig 9. Scattering width phase obtained from far zone time
domain results and compared with exact solution. FDTD
cells are 0.5 cm squares.
Fig i0. Far Zone relative electric field vs time for TEz
polarized incident Gaussian pulsed plane wave
illuminating a perfectly conducting circular cylinder.
FDTD cells are 0.5 cm squares.
Fig Ii. Scattering width amplitude obtained from far zone time
domain results and compared with exact solution. FDTD
cells are 0.5 cm squares.
Fig 12. Scattering width amplitude obtained from far zone time
domain results and compared with exact solution on an
expanded dB scale. FDTD cells are 0.5 cm squares.
Fig 13. Scattering width phase obtained from far zone time
domain results and compared with exact solution. FDTD
cells are 0.5 cm squares.
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A two-dimensional time domain near zone to far zone transformation

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