Biomedical Engineering Reference
In-Depth Information
Appendix 4.A
Chiral Molecule near a Chiral Half-Space
In Appendix 4.A, mathematical expressions necessary to solve the
problemofradiationofachiralmoleculenearachiralhalf-spacewill
be presented.
4.A.1 Vector Cylindrical Harmonics
Vector cylindrical harmonics describing electric and magnetic fields
insideachiralhalf-space(
z
<
0,seeFig.4.4)havethefollowingform
=
(
n
0, 1, 2, . . .;
q
>
0):
rot
e
z
J
n
(
q
)exp
q
2
z
,
i
k
J
−
m
(
J
)
=
ρ
)cos(
n
ϕ
−
nqe
rot
e
z
J
n
(
q
)exp
q
2
z
,
i
k
J
−
m
(
J
)
=
ρ
ϕ
−
)sin(
n
nqo
1
1
n
(
J
)
k
J
rot
m
(
J
)
nqe
,
n
(
J
)
k
J
rot
m
(
J
)
=
=
nqo
,
(4.A.1)
nqe
nqo
where 0
are polar coordinates;
e
z
is the unit
vector directed along the Cartesian z axis;
k
J
is the wavenumber of
left (
J
≤
ρ<
∞
,0
≤
ϕ<
2
π
=
L
)andright(
J
=
R
) polarized waves in a chiral medium,
and
J
n
(
q
) is the Bessel function [1].
Vector cylindrical harmonics describing electric and magnetic
fields outside the chiral medium (
n
ρ
=
>
0, 1, 2, . . .;
q
0) have the
following form:
rot
e
z
J
n
(
q
)exp
q
2
z
,
i
k
0
−
m
(
±
)
=
ρ
ϕ
±
)cos(
n
nqe
rot
e
z
J
n
(
q
ρ
)sin(
n
ϕ
)exp
q
2
z
,
±
i
k
0
−
m
(
±
)
=
nqo
1
1
n
(
±
)
k
0
rot
m
(
±
)
nqe
,
n
(
±
)
k
0
rot
m
(
±
)
=
=
nqo
,
(4.A.2)
nqe
nqo
where the “
” signs correspond to outgoing and incoming
waves and
k
0
is the wavenumber outside the chiral medium (in
vacuum).
+
”and“
−
