the boundary value problem for the simple periodic gratings that has tapes with stated
first and second boundary conditions are fundamental for the complicated conical
geometry. The change of the geometry of the conical diffraction gratings and also its
surface properties complicates singular integrated equations. Advantage of the SIE
method is that it can be used, in particular, for the solution of the boundary value
problem of waves scattering on multielement gratings when there are several tapes of
different width on the period. For a numerical experiment in such boundary value
problems the method of discrete singularities can be recommended. It is well-
established in studying problems of diffraction on simple geometry gratings [27]. One
of the limitations of the SIE method is that it is quite difficult to obtain the analytical
solution in the limiting particular cases of conical diffraction gratings.
As an application of the methods given in the paper we will obtain SIE for the
simple conical diffraction gratings. It is supposed that mixed boundary conditions are
given on the gratings tapes (the third boundary value problem of mathematical
physics). Such problem is a model problem for the problem of diffraction of
electromagnetic waves on the gratings with variable surface properties.
Mixed boundary condition on the strips. Let on the conical strips of the gratings
the following boundary conditions are given
1
0
uu
u
nn
+−
+
−
=
,
(50)
uu
+−
=
.
(51)
Assuming in (50)
1
0
=
we obtain the first boundary condition (4). Let us rewrite
(50) in the proposed spherical coordinate system:
00
0
uu
u
=
=+
=−
+
−
=
,
(52)
where
1
r
=
. Function
()
1
,
mi
U
(3) is retrieved in the form
()
0
1
()
1/2
,,
1/2
(cos)
()
.
(cos)
mnN
inNm
i
mi
mnm
mnN
n
i
P
U
x
e
P
+
+
+
−+
+
+
=−
−+
=
(53)
As a result of applying of boundary conditions (51), (52) we receive the following
system of the functional equations
(
)
0
(1)
,
,
,
2
1
(
)
1
,
sin
:,
imN
inN
mn
mn
mn
n
n
x
Nn
x
e
e
n
d
SN
l
+
=−
−
+
−
=
(54)
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