Radar Cross Section (RCS) - MATLAB Simulations for Radar Systems Design - page 512

11.5.1. Sphere

Due to symmetry, waves scattered from a perfectly conducting sphere are
co-polarized (have the same polarization) with the incident waves. This means
that the cross-polarized backscattered waves are practically zero. For example,
if the incident waves were Left Circularly Polarized (LCP), then the backscat-
tered waves will also be LCP. However, because of the opposite direction of
propagation of the backscattered waves, they are considered to be Right Circu-
larly Polarized (RCP) by the receiving antenna. Therefore, the PP backscat-
tered waves from a sphere are LCP, while the OP backscattered waves are
negligible.

The normalized exact backscattered RCS for a perfectly conducting sphere
is a Mie series given by

∞
∑

 krJ n k () nJ n – k () 

σ
π r 2

j
kr

----   –( n

– 1

-------- = 1 ( 2 n + 1 ) -----------------------------------------------------------

 

(11.27)

() k () nH n () k ()

 krH n – 

– 1

n = 1

 J n k ()

H n () k ()



– --------------------

 

 

where r is the radius of the sphere, k = 2πλ ⁄ , λ is the wavelength, J n is the

H n ()

spherical Bessel of the first kind of order n, and is the Hankel function of

order n, and is given by

H n () k () J n

= k () jY n + k ()

(11.28)

is the spherical Bessel function of the second kind of order n. Plots of the
normalized perfectly conducting sphere RCS as a function of its circumference
in wavelength units are shown in Figs. 11.16a and 11.16b. These plots can be
reproduced using the function Ðrcs_sphere.mÑ given in Listing 11.4 in Section
11.9.

Y n

In Fig. 11.16, three regions are identified. First is the optical region (corre-
sponds to a large sphere). In this case,

σ r 2

= r λ »

(11.29)

Second is the Rayleigh region (small sphere). In this case,

σ 9π r 2 k ( 4 ≈ r λ «

(11.30)
The region between the optical and Rayleigh regions is oscillatory in nature
and is called the Mie or resonance region.