Biomedical Engineering Reference
In-Depth Information
c
=
4 applies for
r
>
r
s
. The unknown scattered fields due to the
CNP, that is, the fields inside and outside the CNP, are also expanded
intermsofTMandTEsphericalwaves.Theseexpansionsinvolvethe
unknown TM and TE expansion coe
cients denoted by
A
i
,
nm
and
B
i
,
nm
, respectively, where
i
=
1 for the fields inside the nano-core,
i
4forthe
fields outside the CNP. These expansion coe
cients depend on the
EHD location and orientation; they are easily obtained by enforcing
the boundary conditions on the two spherical interfaces,
r
=
2 and 3 for the fields inside the nano-shell, and
i
=
=
r
1
and
r
2
. Once these coe
cients are known, we have the complete
knowledge of the fields in all regions. These fields will, in general,
possess all three components. We have for the electric field:
E
=
r
=
a
r
E
r
+
a
θ
E
θ
+
a
φ
E
φ
,where
E
r
,
E
θ
,and
E
φ
are,respectively,its
r
-,
θ
-,
and
φ
-components. In the same manner, we have for the magnetic
field:
H
=
a
r
H
r
+
a
θ
H
θ
+
a
φ
H
φ
,where
H
r
H
r
,
H
θ
,and
H
φ
areits
r
-,
θ
-,
and
φ
-components. With the fields in place, we can derive the
expressionforthetotalpowerradiatedbytheCNPwhenitisexcited
by the EHD, and this is given by the expression:
|
α
nm
|
,
N
max
n
2
2
π
ω
k
0
2
n
(
n
+
1)
2
n
+
1
(
n
+|
m
|
)!
(
n
−|
m
|
)!
+
|
β
nm
|
P
t
=
ε
0
μ
0
=
m
=−
n
n
1
(14.5)
where the coe
cients
α
nm
=
a
(4)
nm
+
A
4,
nm
,
β
nm
=
b
(4)
nm
+
B
4,
nm
if the
EHD is outside the CNP, and
α
nm
=
A
4,
nm
,
β
nm
=
B
4,
nm
if the EHD is
insidetheCNP.Thesymbol
N
max
isthetruncationlimitinapractical
numerical implementation of the infinite summation in the exact
solution and is chosen in a manner that ensures the convergence of
this expansion.
The power radiated by the EHD, when situated alone in free
space, is given by (14.5) with
=
a
(4)
β
nm
=
b
(4)
α
nm
nm
and
nm
,which
reduces to the simple expression:
P
EHD
=
η
0
3
2
pk
0
2
. (14.6)
In our investigations, the so-called normalized radiation resistance
(NRR) is examined; this is the radiation resistance of the dipoles
radiating in the presence of the CNP normalized by the radiation
resistance ofthe dipoles radiating in free space, andin dB it reads:
π
NRR(dB)
=
10
·
log
10
P
t
P
EHD
.
(14.7)
