2.4 Geometric theory of diffractionBasic theoryGeometrical optics
is a proper method for computing wave propagation in homogeneous media or in the environment
exhibiting a continuous variation of parameters. If the environment contains objects, the method fails.

Fig. 2.4A.1  Illuminated reflecting halfplane from point of view of geometric optics 

Look at the fig. 2.4A.1. We can observe the situation in the
surrounding of an illuminated reflecting halfplane from the point of view of
geometrical optics. The
space around the halfplane can be subdivided into three sections: into the section
a, where both the incident wave and the
reflected one are present, into the section b,
which contains the incident wave only, and to the section l,
where no wave (from the point of view of geometrical optics) propagates. Moreover, exactly
defined planes H_{0} (border of reflection)
and H_{S} (border of shade) separate the sections. From the point of view of
geometrical optics, a discontinuity of the field distribution appears on those planes because on
different sides of H_{0} and H_{S}, intensity is computed in a different way. In real
world, the field distribution is continuous, and in the section of shade, intensity is nonzero. In the sixties
of 20^{th} century, Keller proposed a specific correction of geometrical optics in
order to eliminate field discontinuities in the surrounding of objects. That way, the basics of
the geometric theory of diffractionwere built.
Geometric theory of diffraction (GTD) is an extension of geometrical
optics to inhomogeneous media. GTD solves interaction of beams
and objects according to the laws of geometrical optics, which is completed that way to eliminate
obvious incorrectness (intensity discontinuities at the border of the reflection section and the
shade one) and to preserve the main advantages of geometrical optics (conception of beams). The
incorrectness of geometric optics is eliminated following the belowdescribed postulates:
 Beams of the incident wave excite new waves on
the illuminated object. Those waves propagate away from the object (not from the
primary source). Those waves are called diffraction waves.
Diffraction waves can be described (computed) using beams. Diffraction waves
(diffraction beams are of such amplitudes and
phases to eliminate field discontinuities on boundaries.
Diffraction waves are excited only by the beams, which
 "fall to the edges of the object, to the boundaries of different curvature radii of the surface, etc.
 "touch the surface of the object.
 Amplitude of diffraction waves is proportional to the amplitude of the incident wave. Proportionality coefficient is called
diffraction factor.
 Diffraction constant is given by local (!) properties of the object surface.
In fig. 2.4A.2a, arrows indicate diffraction waves excited by an incident wave illuminating the edge of the halfplane. Diffraction waves create the set of
cylindrical waves, which common source
is the halfplane edge. Adding incident and reflected wave together, continuous intensity
transitions on the border of the reflection region and the shade one is ensured.
In fig. 2.4A.2b, beams of diffraction waves
excited by the incident wave beam are depicted for
the situation when the beam touches the object surface. In the moment when the beam of the diffraction
wave leaves the object surface, it starts to behave according to the
geometric optics rules.
Finally, the postulate about the local parameters of objects
is very important. In the situation of fig. 2.4A.2b, the diffraction factor is given by parameters e, m
of the object surface and by the curvature radius of the surface in the region, where the beam slides
(over the surface. Parameters of the rest of the object do not play any role. Thanks to this property, even
complicated diffraction situations appearing in a real terrain can be solved (fig. 2.4A.2c).

a)  b)  c) 
Fig. 2.4A.2  Geometric theory of diffraction. Diffraction waves beams a) on the halfplane edge, b) on the spherical surface, c) on obstacles in the real terrain 

Thanks to the abovementioned postulate, values of
diffraction factors can be computed. E.g., all
the oblong objects, which have the same curvature radius of a certain part of their surface, have
the same diffraction factor. The factor has to be the same as for the cylinder of the same radius.
And the diffraction on a cylinder can be solved exactly. We only have to rearrange the results of
the exact theory (see the chapter 2.2) to the form applicable to the theory of
geometric optics. This is solved by so called
canonical problems.
GTD is together with geometrical optics an efficient tool
for solving diffraction problems. Preserving the conception of beams and computing with local
object properties are the main advantages of the method. The method has to be implemented
using computer.
