Biomedical Engineering Reference
In-Depth Information
(
r
s
,
q
s
,
f
s
)
Free space
z
EHD
(
r,
q
,
f
)
r
2
Observation
point
r
1
x
y
Nano-core
Nano-shell
Figure 14.8
Single EHD illumination of a CNP.
μ
0
; thus, its wav
e num
ber is
k
0
=
ω
√
ε
0
μ
0
, while its intrinsic
impedance is
and
η
0
=
√
μ
0
/ε
0
.
The CNP is excited by an arbitrarily oriented and located EHDs
with the dipole moment being equal to
=
p
a
p
, where ˆ
p
is its
orientationand
p
[Am]isitscomplexamplitude.ThisEHDamplitude
is typically expressed as the product of the constant current
I
e
[A]
applied to it and its length
l
[m], that is,
p
=
I
e
l
. The EHDs are
driven by a time harmonic source with the frequency
f
;the
corresponding free-space wavelength is
λ
=
2
π/
k
0
.Weintroduce
a spherical coordinate system, with the coordinates (
r
,
θ
,
φ
)and
unit vectors (
a
r
,
a
θ
,
a
φ
), and a Cartesian coordinate system, with
the coordinates (
x
,
y
,
z
) and unit vectors (
a
x
,
a
y
,
a
z
), such that
the origin coincides with the center of the CNP. The coordinates
of the observation point and the EHD are (
r
,
θ
,
φ
)and(
r
s
,
θ
s
,
φ
s
),
respectively. In the two coordinate systems, the dipole moment
can be expressed, respectively, as
p
p
=
a
r
p
r
+
a
θ
p
θ
+
a
φ
p
φ
and
p
=
a
x
p
x
+
a
y
p
y
+
a
z
p
z
.
14.3.2
Theoretical Considerations
The analytical solution for the single-EHD problem in Fig. 14.8 was
derived in [7], and here we only summarize its main points. The
known electromagnetic field due to the EHD is expanded in terms
of transverse magnetic (TM) and transverse electric (TE) spherical
waves with the known expansion coe
cients
a
(
c
)
nm
(TM coe
cients),
and
b
(
c
)
nm
, (TE coe
cients). The index
c
=
1 applies for
r
<
r
s
while
