Biomedical Engineering Reference
In-Depth Information
However, there are no real singularities at exact plasmon
resonances where
(
a
)
=
0. Electrical amplitudes
a
tend to unity
at the corresponding resonant frequencies, as can be seen clearly
from the formula (9.1). This means that the Rayleigh approximation
is not applicable at the points of the plasmon resonances where
the Rayleigh scattering is replaced by the so-called anomalous light
scattering [5]. To escape singularities at the scattering amplitudes,
it is su
cient to expand numerator and denominator in (9.1)
independently.However,itisnotsoimportantforthedetermination
of anisotropy in scattering where the positions of the forward
scattering and back scattering resonances can be found from Eqs.
(9.5) and (9.6) with su
cient accuracy. Applying expansions (9.5)
and (9.6) to Eq. (9.4), one can find that the solution of the equations
Q
FS
=
0and
Q
BS
=
0 yields the size parameters
3)
(1
−
ε
)
38
+
27
ε
+
ε
(
ε
+
2)(2
ε
+
q
2
q
FS
=
2
,
=
15
ε
+
ε
+
(
2)(2
3)
q
2
q
BS
=
=
15
3
.
(9.7)
+
ε
−
ε
+
ε
70
29
10
2
Both quantities
q
FS
(
ε
) and
q
BS
(
ε
) vanish in the vicinity of the
dipole (
1.5) resonances, see Fig.
9.4. In Fig. 9.4, we do not show nonphysical brunches of Eq. (9.7)
corresponding to negative values of
q
2
.
At the corresponding frequencies, one can observe the trans-
formation of the far-field scattering diagrams typical for the Fano
resonance [1]. These polar diagrams are calculated by a standard
way, see Ref. [4]. Corresponding scattering intensitiesare
ε
→−
2) and quadrupole (
ε
→−
e
B
P
(1)
2
∞
P
(1)
(cos
θ
)
sin
θ
I
(
s
)
(
−
1)
m
B
=
C
(cos
θ
)sin
θ
−
,
II
=
1
e
B
2
∞
P
(1)
(cos
θ
)
m
B
P
(1)
I
(
s
)
1)
⊥
=
C
(
−
−
(cos
θ
)sin
θ
.
sin
θ
=
1
(9.8)
In Eq. (9.8),
C
is the normalization coe
cient; other values are
the same as in Ref. [4]. For small values of the size parameter
q
<<
1, we observe the Fano resonance in weakly dissipated plasmonic
