5.2.3.1 Reflection/Scattering from a PEC Sphere
The plane wave scattering from a sphere is a classic problem and is often used as the reference because of
its symmetry. This is used to measure the scattering properties like RCS (Radar Cross Section) of other
targets. As the problem has been extensively studied by many authors, only an outline of the problem is
sketched and references will be provided for the interested reader.
For a sphere of radius ‘a’, it can be shown that assuming geometrical optics and a perfectly reflecting
sphere, the incident radiation is uniformly scattered in all directions (Isotropic scattering) and the RCS is
given as,
σ = π a2
When the wavelength is not small compared with the sphere radius, the calculation is much more
complicated. This region where the wavelength is not small compared to the radius is called the Mie region
or resonance region. After several developments from different authors [6] [7] [8] [9], the solution is given
as
σ
1
= 2
2
πa
ρ
Where,
ρ = 2π a λ ; an
∞
∑ ( −1) ( 2n + 1)( a
n
n
n =1
and bn (for a perfectly conducting sphere) are given by
an =
bn =
( 2)
Where, jn and hn
+ bn )
2
jn ( ρ )
hn(
2)
(ρ)
−  ρ jn ( ρ ) ′
 ρ hn( 2) ( ρ ) ′


are the spherical Bessel function of the first kind and the spherical Hankel function of
the second kind, respectively. The primes denote the differentiation with respect to ρ . The plot of the RCS
is given in below (Figure 11).
Radar cross section of a sphere
10
5
σ/πa^2 (dB)
0
-5
-10
-15
-20
-25
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
(2*π*a)/λ
Figure 5.2.3.1 RCS of a perfectly conducting sphere (Reproduced from [2])
The Radar Cross Section (RCS) for a sphere of radius ‘a’ is calculated and the results are reported in the
excel file “rcs_sphere.xls”. The behavior is indeed oscillatory with a peak of approximately 5.4dB at
a = λ / 2π .
References
[1] Roger J. Sullivan, Microwave Radar imaging and advanced concepts, Norwood, MA; Artech House,
2000.
[2] Barton. D. K., Modern Radar System Analysis, Norwood, MA; Artech House, 1988.
[3] Knott, E. F., J. F. Shaffer and M. T. Tuley, Radar Cross Section, 2nd edition, Norwood, MA; Artech
House, 1993.
[4] Ruck, G. T., et al., Radar Cross Section Handbook, New York: Plenum, 1970 (2 Volumes)
[5] Cirspin, J. W. and K. M. Siegel, Methods of Radar Cross Section Analysis, New York: Academic, 1968
[6] Me, G., “A Contribution to the Optics of Turbid Media, Especially Colloidal Metallic Suspensions”,
Ann. Physik, Vol. 25, 1908, pp. 377-445
[7] Stratton, J. A., Electromagnetic Theory, New York: McGraw-Hill, 1941
[8] Kerr, D. E., et al., Propagation of Short Radio Waves, New York: Dover Publications, 1965
[9] Blake, L. V., Radar Range-Performance Analysis, Norwood, MA; Artech House, 1986.