Biomedical Engineering Reference
In-Depth Information
following form (
n
=
1, 2, 3, . . .;
m
=
0,
±
1,
±
2, ...,
±
n
):
k
0
r
rot
r
ζ
n
(
k
0
r
)
P
n
(cos
θ
)
e
im
ϕ
,
N
ζ
mn
=
1
1
k
0
rot
M
ζ
mn
,
(4.B.2)
M
ζ
mn
=
k
0
rh
(1
n
(
k
0
r
);
h
(1
n
(
k
0
r
) is the spherical Hankel
function of the first kind[1], and
k
0
is the wavenumberin vacuum.
Informationaboutthepropertiesofvectorcylindricalharmonics
can be also found, for example, in [69].
To obtain the spherical vector harmonics
s
N
where
ζ
n
(
k
0
r
)
=
ψ
mn
,
s
M
(
J
)
ψ
(
J
)
mn
,
s
N
ζ
mn
,
and
s
M
ζ
mn
inthelocalcoordinatesofthe
s
-thparticle(
s
=
1,2),itis
necessary to replace the coordinates
r
,
θ
,
ϕ
by the local coordinates
s
r
,
s
θ
,
s
ϕ
in the expressions (4.B.1) and (4.B.2).
4.B.2 Electromagnetic Field of a Chiral Molecule in
Spherical Coordinates
(0)
(0)
mn
can be
In (4.36), the vector spherical harmonics
N
ψ
mn
and
M
ψ
(
J
)
mn
[see
(4.B.1)] by changing the index
J
→
0. The coe
cients of expansion
in (4.36) can be written down as
(
J
)
obtained from the expressions for harmonics
N
ψ
mn
and
M
ψ
mn
(
r
0
)
=
ik
0
[
F
mn
(
r
0
,
d
0
)
+
iG
mn
(
r
0
,
−
i
m
0
)],
B
(0)
A
(0)
mn
(
r
0
)
=
ik
0
[
G
mn
(
r
0
,
d
0
)
+
iF
mn
(
r
0
,
−
i
m
0
)], (4.B.3)
where
F
mn
(
r
0
,
d
0
) and
G
mn
(
r
0
,
d
0
) have the form:
2
d
0
x
−
id
0
y
1
1
F
mn
(
r
0
,
d
0
)
=−
n
b
m
−
1,
n
−
1
(
r
0
)
2
d
0
x
−
id
0
y
1
1
+
n
+
1
b
m
−
1,
n
+
1
(
r
0
)
2
d
0
x
+
id
0
y
1
1
+
n
(
n
−
m
−
1)(
n
−
m
)
b
m
+
1,
n
−
1
(
r
0
)
2
d
0
x
+
id
0
y
1
1
−
n
+
1
(
n
+
m
+
1)(
n
+
m
+
2)
b
m
+
1,
n
+
1
(
r
0
)
+
d
0
z
1
1
n
+
1
(
n
+
m
+
1)
b
m
,
n
+
1
(
r
0
),
(4.B.4)
n
(
n
−
m
)
b
m
,
n
−
1
(
r
0
)
+
d
0
z
