2.2 General Theory of diffractionBasic theoryFresnel diffraction
gives correct results for planar, thin and absorbing obstacles only. Diffraction phenomena on
threedimensional objects have to be therefore computed a different way. Hence, we have to study
the basic steps of the general solution of diffraction tasks.

Fig. 2.2A.1  Diffraction on cylinder 

An objects of a known shape and of known electrical properties (permittivity, conductivity, etc.)
is illuminated by a wave, which intensity is known in every point of the environment. This is so
called primary wave. Reacting to the primary wave, the object is
polarized
(if the object is of dielectric nature) or
currents are induced
on the surface of the object (if the object is
conductive). As a cause of polarization or current induction, the object becomes a source
of a new wave (secondary wave). The secondary
wave propagates in all the directions from the object. The total field intensity in the
object surrounding (even behind the object) is then given by the summation of the primary
wave and secondary one. Mathematical description of the secondary wave can be obtained
by solving homogeneous wave equation.
Magnitude of the secondary wave can be determined from the requirement that the total field intensity on the object surface (summation of
the primary wave and the secondary one) meets the
boundary condition.
The algorithm of the solution is shown on an example. We are going to compute planewave
diffraction on an infinitely long perfectly conducting circular cylinder.
The crosssection of the cylinder is depicted in figure 2.2A.1.
The cylinder axis identical with the z axis of the
Cartesian coordinate system,
the cylinder radius is denoted as a. The cylinder is illuminated by the
plane wave
propagating in the direction x and consisting of the component: E_{z}: E_{z prim} = E_{0}exp(jkx) only.
Since the situation is identical in all the planes parallel to the plane xy, we solve the
task in two dimensions. In the next step, we transform the problem from the Cartesian
coordinate system x, y to the polar
one r, φ (fig. 2.2A.1). Therefore, we
consider the substitution x = r cos(φ). Field intensity of the primary wave is then

${E}_{z\text{\hspace{0.17em}}prim}={E}_{0}{e}^{jkr\mathrm{cos}\phi},\text{\hspace{1em}}\frac{\partial {E}_{z\text{\hspace{0.17em}}prim}}{\partial z}=0$.

( 2.2A.1 )

This wave excites currents on the surface of the cylinder,
which act as sources of the secondary wave. Dealing with induced currents, we do not have
any information about them now. We know only that the secondary wave has to meet
the
homogeneous wave equation
. Its general solution is of the form

${E}_{z\text{\hspace{0.17em}}sek}={\displaystyle \sum _{m=0}^{\infty}{A}_{m}{H}_{m}^{\left(2\right)}\left(kr\right)\mathrm{cos}\left(m\phi \right)}$.

( 2.2A.2 )

Here, m is a separation constant, A_{m} are integration constants and H_{m}^{(2)}(kr) is
Hankel function of order m of
second kind and of the argument (kr). After the rotation for 360 degrees in the direction φ, values of
the field intensity have to repeat, and therefore, the constant m is
integer number and summation (2.2A.2) is performed for all the integer m. Derivation
of (2.2A.2) is given in the layer B.
Finally, values of the integration constants A_{m}
have to be found. To reach that, the
boundary condition
has to be applied: tangential component of the total
field intensity on the surface of perfectly conducting cylinder has to be zero.
Since E_{z}
is tangential, we get

${E}_{z\text{\hspace{0.17em}}prim}+{E}_{z\text{\hspace{0.17em}}sek}=0$ on condition r = a.

( 2.2A.3 )

Substituting (2.2A.1) and (2.2A.2) to (2.2A.3) and performing
several mathematical rearrangements, we get relations for computing integration constants A_{m}:

${A}_{0}={E}_{0}\frac{{J}_{0}(ka)}{{H}_{0}^{(2)}(ka)}$,
${A}_{m}=2{j}^{m}{E}_{0}\frac{{J}_{m}(ka)}{{H}_{m}^{(2)}(ka)}$.

( 2.2A.4 )

In the abovegiven relation, E_{0} denotes the primary wave amplitude, k is wavenumber of the environment containing the cylinder, a
is radius of the cylinder, J_{m}(ka) is
Bessel function of order m and of the argument ka,
H_{m}^{(2)}(kr) is
Hankel function of order m of second kind and of the
argument (ka) and j denotes imaginary unit.
Derivation is given in the layer B.
The total intensity in the surrounding of the cylinder
equals to the summation of the primary wave intensity and the secondary one. The result
is obtained in the form of an infinite series  the secondary wave (2.2A.2) summed with
the primary one (2.2A.1). That way, the solution is finished. In applications, even the
speedingup the convergence of the final series has to be solved.
Now, turn our attention to the structure of the wave in the surrounding of the cylinder. The structure is complicated. In the radial direction,
cylindrical waves
given by (2.2A.2) propagate. These waves interfere with the primary wave, and therefore, both the
traveling wave,
and the
standing one
exist in the surrounding of the cylinder. Since each radial direction contains a different angle
with the propagation direction of the primary wave, the standingwave wavelength is different in
every direction. As a result, directivity patterns in different distances are different as shown
in fig. 2.2A.2. In this figure, the propagation direction of an incident wave is indicated by
an arrow. In the left, directivity patterns of the total wave is depicted, in the right, directivity
patterns of only secondary wave are shown. Surprisingly, the maximum of the secondary wave is in
the direction behind the cylinder.

a)  b) 
Fig. 2.2A.2  a) Intensity of the total field (primary + secondary) of an infinitely long cylinder of the radius a observed in various distances r
b) Patterns of a single secondary field of a infinitely long cylinder of the radius a observed in a very long distance 

The wave structure near the conducting cylinder can be observed
using matlab programs. The first program displays directivity patterns of both the single secondary
wave and the total wave, which are observed in various distances from the cylinder axis. The second
program shows the total (standing) wave in various radial directions. Programs are described in
the layer C.
Today, solution of the
diffraction task
is known for various geometrically simple objects. From the practical point of view,
the solution for a general ellipsoid is of great importance because the proper choice of the
halfaxis length enables to approximate technically useful shapes. For a = b = c
the ellipsoid becomes a sphere. For a = b << c the ellipsoid approximates a cylindrical conductor of a finite length. For
a = b >> c the ellipsoid approaches a circular slab.
Finally, a small note. During the solution of the diffraction on the cylinder, the primary wave was assumed to exist
everywhere in the space, i.e. even behind the cylinder. I.e. even behind the cylinder on the reverse side, the
boundary condition
(2.2A.3) has to be fulfilled. This fact enabled to solve many
diffraction problems
at the beginning of the 20^{th} century, which was very important for the development of radio electronics.
