5.2 Scattering of a Plane Wave from the Rough Surface
91
Let us write the expressions for the field scattered by a certain smooth rough
surface z = H (x, y) in the Kirchhoff approximation. We select a certain region S of
this surface, whose linear size is much larger than the mean size of the roughness,
which in turn is much larger than the wavelength. We assume that there are no
elements of the surface shadowed from the incident wave or scattered wave.
Let us suppose that a plane s- or p- monochromatic wave is incident on a rough
surface; the unit wave amplitude has the form
E inc (r ) = e
−ik 1 r
.
The observation will be carried out in the Fraunhofer zone of domain S and in the
direction of wave vector k. In this zone, the elementary waves of all elements of the
scattering domain can be treated as plane waves.
Fields E and H can be expressed in terms of certain scalar function (e.g., E) that
satisfies the equation
E + k
2 E = 0
(5.1)
with boundary conditions of the form [7, 8]
E re f | z=H (x,y) = (1 + V )E inc | z=H (x,y) ,
(5.2)
∂ E re f
∂n
| z=H (x,y) = (1 − V )
∂ E inc
∂n
| z=H (x,y) ,
(5.3)
where k
2
= ω
2
ε 0 μ 0 , V is the reflection coefficient depending on physical parameters
of the medium, and n is the unit vector of the outward normal. It should be borne in
mind that the formulas for the reflection coefficient for the s- or p- polarization are
different.
It should be noted that using the Kirchhoff method, we solve not the boundaryvalue problem of diffraction, but a simpler problem that basically differs from it (i.e.,
the problem with a preset discontinuity of the field and of its normal derivative on
the surface). Thus, in contrast to the perturbation method considered in Chap. 9, in
which the applicability limits of the results can be indicated for a wide class of special
cases and the next terms of the expansion can be calculated from the known small
parameters, the results obtained using the Kirchhoff method cannot be treated as the
expansion of the exact solution into a series in a small parameter (e.g., the ratio of
the wavelength to the characteristic linear size of the body at which diffraction takes
place).
It is well known that the values of E inside the domain are connected with E and
∂ E
∂n
on the surface bounding this domain by the Green formula
E(r) =
S
E(r
)
∂G(r, r
)
∂n
−
∂ E(r
)
∂n
G(r, r
)
d S,
(5.4)
where G(r, r
)is the Green function, which has the form
91
Let us write the expressions for the field scattered by a certain smooth rough
surface z = H (x, y) in the Kirchhoff approximation. We select a certain region S of
this surface, whose linear size is much larger than the mean size of the roughness,
which in turn is much larger than the wavelength. We assume that there are no
elements of the surface shadowed from the incident wave or scattered wave.
Let us suppose that a plane s- or p- monochromatic wave is incident on a rough
surface; the unit wave amplitude has the form
E inc (r ) = e
−ik 1 r
.
The observation will be carried out in the Fraunhofer zone of domain S and in the
direction of wave vector k. In this zone, the elementary waves of all elements of the
scattering domain can be treated as plane waves.
Fields E and H can be expressed in terms of certain scalar function (e.g., E) that
satisfies the equation
E + k
2 E = 0
(5.1)
with boundary conditions of the form [7, 8]
E re f | z=H (x,y) = (1 + V )E inc | z=H (x,y) ,
(5.2)
∂ E re f
∂n
| z=H (x,y) = (1 − V )
∂ E inc
∂n
| z=H (x,y) ,
(5.3)
where k
2
= ω
2
ε 0 μ 0 , V is the reflection coefficient depending on physical parameters
of the medium, and n is the unit vector of the outward normal. It should be borne in
mind that the formulas for the reflection coefficient for the s- or p- polarization are
different.
It should be noted that using the Kirchhoff method, we solve not the boundaryvalue problem of diffraction, but a simpler problem that basically differs from it (i.e.,
the problem with a preset discontinuity of the field and of its normal derivative on
the surface). Thus, in contrast to the perturbation method considered in Chap. 9, in
which the applicability limits of the results can be indicated for a wide class of special
cases and the next terms of the expansion can be calculated from the known small
parameters, the results obtained using the Kirchhoff method cannot be treated as the
expansion of the exact solution into a series in a small parameter (e.g., the ratio of
the wavelength to the characteristic linear size of the body at which diffraction takes
place).
It is well known that the values of E inside the domain are connected with E and
∂ E
∂n
on the surface bounding this domain by the Green formula
E(r) =
S
E(r
)
∂G(r, r
)
∂n
−
∂ E(r
)
∂n
G(r, r
)
d S,
(5.4)
where G(r, r
)is the Green function, which has the form