Figure 1. Periodical conical gratings
Let
2
lN
=
and
d
be the gratings period and the angular width of every slot,
respectively. Value of the slot width equals to the value of the dihedral angle
,
which is
formed by the planes that intersect cone axis and edges of the next strips. As the circular
conical surface is a coordinate surface of the spherical coordinate system then for the
solution of the considered problem of wave diffraction it is convenient to use the
spherical coordinate system
,,
r
with the tip of the cone at the origin of coordinates
(
0
r=
). In the introduced coordinate system the conical gratings
is defined by the
equation
=
and the source of the field has coordinates
0
0
0
,,
r
.
It is required to find the scalar potential
()
ur
,
(
)
,,
rr
=
, that correspond to the
field in the presence of a source and a conical structure and satisfy:
1) the Helmholtz equation everywhere out of the conical strips and the source:
2
0
u
qu
−
=
,
q
is a wave parameter;
2) the boundary condition on the conical strips
1
1
1
2
2
1
1
2
2
,
u
u
u
u
g
n
n
n
n
−
−
+
−
+
−
−
−
−
−
−
−
−
+
−
=
(1)
(
)
3
,,
:
[0,
),
,
,
r
R
r
L
=
+
=
(
)
3
,,
:
[0,
),
0,
,
r
R
r
L
+
=
+
=+
(
)
3
,,
:
[0,
),
0,
,
r
R
r
L
−
=
+
=−
1
N
s
s
LL
=
=
,
(
)
(
1)
2,
2
s
L
s
l
d
sl
d
=
−
+
−
,
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