Biomedical Engineering Reference
In-Depth Information
wherethelowerlimitofthesummationover
q
mustbeequalto1in
thecase
m
=
0,andequalto
|
m
|
inothercases(
m
=
0);thefunctions
V
mnq
and
W
mnq
aredefinedinAppendix4.B;thefunctions
α
n
,
β
n
,and
δ
n
are defined by (4.39).
To find the radiative decay rate of the spontaneous emission of
the chiral molecule located near a cluster of two spherical particles,
it is necessary to calculate the total power of the radiation of
the system “molecule and cluster” [see (4.41)]. If we use local
coordinates related to the first particle, the radiative part of the
spontaneous emission decay rate [cf. (4.43)] will have the following
form:
∞
n
γ
rad
γ
0
=
+
(
n
+
m
)!
(
n
3
n
(
n
1)
2
k
0
|
d
0
|
2
2
+
−
+ |
m
0
|
2
n
1
m
)!
=−
n
=
1
m
n
2
,
1
C
mn
1
D
mn
2
1
C
(0)
mn
1
D
(0)
mn
×
+
+
+
(4.57)
∞
V
mnq
2
C
mq
+
W
mnq
2
D
mq
,
1
C
mn
=
1
C
mn
+
(4.58)
q
=|
m
|
∞
V
mnq
2
D
mq
+
W
mnq
2
C
mq
.
1
D
mn
=
1
D
mn
+
q
=|
m
|
In (4.58), the lower limit of the summation over
q
must be equal to
1inthecase
m
=
0,andequalto
|
m
|
inanothercasewith
m
=
0;the
coe
cients
1
C
(0)
mn
,
V
mnq
,and
W
mnq
canbefoundinAppendix4.B
If there are losses in the chiral particle, there is a probability of the
nonradiativetransitionfromtheexcitedstatetothegroundone.The
generalexpressionforthetotal(radiative
mn
,
1
D
(0)
nonradiative)decayrate
ofthespontaneousemissioninthecaseofaclusteroftwospherical
particles has the form [cf. (4.28)]:
+
⎧
⎨
⎫
⎬
s
=
1
d
0
·
s
H
sc
(
s
r
0
)
2
s
E
sc
(
s
r
0
)
+
i
m
0
·
γ
γ
0
=
1
+
3
2
Im
k
0
|
d
0
|
2
, (4.59)
⎩
2
⎭
+ |
m
0
|
wherethereflectedfields
s
E
sc
(
s
r
0
)and
s
H
sc
(
s
r
0
)[see(4.54)]should
be calculated at the position of the chiral molecule
s
r
0
. Note that in
the case of chiral spherical particles without losses, the expression
(4.59) gives the same results as the expression (4.57). In what
follows, for simplicity, we will not study the nonradiative channel of
the spontaneous decay.
