Biomedical Engineering Reference
In-Depth Information
The models of coupled oscillators can be employed for the
qualitative explanation of many experimental results on the Fano
resonancesinplasmonicstructures(see,e.g.,Refs.[9,10]).However,
for more adequate comparison of theoretical and experimental
results,thenumericalsolutionsofMaxwell'sequationsarerequired.
Although the basic physics of Fano resonances in plasmonic
materials is well understood, some important issues are not
resolved yet. We believe that the scaling of the Fano resonances for
small structures is among those important unsolvedproblems.
9.3 Scaling of the Fano Resonances within the Mie
Theory
The basic problem with scaling of Fano resonances can be observed
fornondissipativeplasmonicnanoparticleswithintheMietheory.As
mentioned above, the problem of the Mie scattering can be solved
analytically for a metallic sphere in which we can employ the exact
Mie solutions [3, 4]. We find the Fano resonance in the directional
scattering e
ciencies for the forward scattering (FS) and backward
scattering (BS). The scattering e
ciencies are presented by [1]
2
∞
1
q
2
1)
[
a
−
Q
BS
=
+
−
(2
1)(
b
]
,
=
1
2
∞
1
q
2
Q
FS
=
(2
+
1)[
a
+
b
]
.
(9.4)
=
1
<<
1 weexpand the scattering amplitudes up to
q
5
For
q
2
i
3
ε
−
1
2
i
5
(
ε
−
1)(
ε
−
2)
15
ε
−
1
i
ε
+
2
q
3
q
5
,
a
2
=−
2
ε
+
3
q
5
,
(9.5)
a
1
=−
−
2)
2
(
ε
+
i
45
q
5
(
b
1
=−
ε
−
1),
b
2
=
0.
(9.6)
Importantly, the expansions contain singularities in the electric
amplitudes, for example, in the dipole amplitude
a
1
at
ε
=−
2(this
singularity is seen in the formula for the Rayleigh scattering), in the
quadrupole amplitude
a
2
at
2, and so on. At the same time,
the magnetic amplitudes
b
have no singularities.
ε
=−
3
/
