Radar Cross Section of the Metal Sphere from Microwave to the
Optical Frequency
Ruijun Wang, Yuliang Qin, Bin Deng, and Hongqiang Wang
College of Electronic Science and Engineering, National University of Defense Technology, Changsha, Hunan,
410073, China
II. RESULTS
Take the metal Cu as an example and assume it is nonmagnetic material, the complex dielectric permittivity at
1
10
10
RCS
Reflectivity
0
1
-1
10
0.1
-2
10
Reflectivity
10
Normalized RCS
R
I. INTRODUCTION
ECENTLY, the understanding of the interaction
between the terahertz wave and the target, especially for
metal targets, is urgently and necessary for the terahertz
applications both in the civil and the military fields. At
microwave frequencies, the normal metal is conductor and the
back scattering field of the metallic target can be computed by
treating the target as perfectly electrical conducting target. At
terahertz frequencies and above, there is anomalous intrinsic
frequency dispersion within the metal and the classical
assumption of dc conductance becomes failed [1].
As known to all, the permittivity plays a vital role on the
scattering characteristic of a target. With the metallic target,
very accurate frequency dispersion models for the permittivity
are required. Stepan Lucyszyn’s investigation suggests that
Drude’s relaxation-effect model works well from DC to the
edge of the mid-infrared frequency range for normal metals [1].
M. A. Ordal had measured the optical constants of fourteen
metals in the infrared and far infrared and the data can be
reasonably fit using the Drude model [2]. Additionally, the
optical constants of metals above the infrared are sufficient.
Then, the scattering characteristic of the metallic target can be
shown through the whole electromagnetic spectrum.
Electromagnetic scattering from spheres has provided a
continuous stream of applications and scientific interest in part
due to its inherent symmetry, as well as its relative agreement
with the geometries of many naturally occurring objects. For
example, the metallic sphere is always used for the radar cross
section measurement for its symmetry and known RCS value.
At terahertz frequencies, the exact RCS of a metal sphere is still
important for the scattering measurement of targets.
The scattered field of a sphere can be given by the analytical
solution in the form of Mie’s series [3]. However, the RiccatiBessel functions and its derivative appearing in the coefficients
of scattered fields diverge for lossy media, and the effect is
especially strong for metals which have large imaginary parts
of the complex permittivity at terahertz frequencies. Further, it
is observed that the effect of divergence can be eliminated
through dividing nominator and denominator of the coefficients
by the Riccati-Bessel function.
different frequencies is given by the Drude model [2] and the
CRC Handbook of Chemistry and Physics. Fig.1 gives the
normalized RCS versus frequencies for a copper sphere with
the radius a=5mm. It can be seen that when the sphere becomes
electrically large, the normalized RCS keeps consistent with the
reflectivity of Cu which had been proved to be near unity below
the infrared. Thus the RCS of an electrically large copper
sphere is approximately thought to be πa2 at terahertz
frequencies. In order to evaluate the RCS carefully at terahertz
frequencies, Fig.2 gives the normalized RCS versus the size of
sphere for different frequencies which are corresponding to the
different complex permittivity. In contrast, The RCS of a
perfectly electrical conducting sphere is also given. One can see
that the sphere’s RCS curve at terahertz frequencies is similar
to the situation of PEC. From the embedded figure, we find that
the copper sphere deviates from the PEC assumption at
terahertz frequencies because the normalized RCS becomes
smaller when the electrical sizes of the sphere are the same
(ka=43). But the decrease in the sphere’s RCS is less than
0.1dB. Thus the polished copper sphere can still be assumed to
be PEC at terahertz frequencies. The conclusion can also be
applied to the metals such as Al, Ag and Au.
0.01
-3
10 10
10
11
12
10
10
13
14
10
10
Frequency (Hz)
15
10
16
10
Fig. 1. Comparison between the normalized RCS of a copper sphere and the
normal incidence reflectivity of copper versus frequencies.
4
PEC
εr at 0.3THz
3.5
1.02
εr at 1THz
3
Normalized RCS
Abstract—Considering the frequency dispersion of the complex
dielectric permittivity for metals, the accurate radar cross section
(RCS) of a copper sphere is given from microwave to the optical
frequency. By investigate the effects on the RCS caused by the
dispersion of metals, we show that polished copper sphere can be
treated as perfectly electrical conductor (PEC) for the RCS
prediction at terahertz frequencies.
εr at 5THz
1
2.5
εr at 10THz
0.98
2
41.5
42
42.5
43
43.5
1.5
1
0.5
0
0
10
20
30
40
50
ka
Fig. 2. The normalized RCS versus the size of the copper sphere.
III. SUMMARY
The analytical solution of the metal sphere which considers
the complex permittivity by the Drude model shows that the
normalized RCS of the electrically large copper sphere keeps
consistent with the reflectivity of Cu. Then, the polished copper
targets can be treated as PEC for the RCS prediction at terahertz
frequencies. This indicates that a polished copper sphere can be
used as a reference in the terahertz RCS measurement.
REFERENCES
[1]
[2]
[3]
S. Lucyszyn, "Investigation of anomalous room temperature conduction
losses in normal metals at terahertz frequencies" IEE Proc. Microw.,
Antennas and Propag, vol.151, no.4, pp. 321-329, 2004.
M. A. Ordal, R. J. Bell, R. W. Alexander and L. L. Long, "Optical
properties of fourteen metals in the infrared and far infrared Al, Co, Cu,
Au, Fe, Pb, Mo, Ni, Pd, Pt, Ag, Ti, V, and W" Appl. Optics, vol. 24, no.
24, pp. 4493-4499, 1985.
J. M. Jin, Theory and computation of electromagnetic fields. Hoboken, NJ,
USA: John Wiley & Sons, 2010.