Biomedical Engineering Reference
In-Depth Information
dipole scattering is dominating,
|
a
1
|
>>
|
a
2
|
. Amplitudes of
|
a
1
|
and
|
a
2
|
become comparable when the size parameter becomes of
the order of unity (see Fig. 9.1a). At the same time, for the Fano
resonance, we need these two amplitudes being comparable. Thus,
with the small size parameter, one of the scattering amplitudes
with necessity is very small. As a result, the Fano resonance is
“not pronounced” for
q
1. However, two scattering modes can
interfere either constructively or destructively.
The basic features of the Fano resonance can be understood
from a simple classical problem of two coupled oscillators. The
existence of interference effects in such a system has been known
foralongtime,andithasbeenextensivelyemployed,forexample,in
mechanical systems for dynamic damping [6]. The dynamics of two
coupledoscillatorscanbedescribedintermsoftheirdisplacements
x
1
and
x
2
from theequilibrium positions.
x
1
+
γ
1
x
1
+
ω
<<
2
2
(
x
2
−
f
1
e
−
i
ω
t
,
1
x
1
=
x
1
)
+
2
x
2
=
2
(
x
1
−
x
2
)
+
f
2
e
−
i
ω
t
.
x
2
+
γ
2
x
2
+
ω
(9.2)
ω
2
are the
eigenfrequencies,
describes the coupling between the oscillators,
and
γ
1
and
γ
2
are the dissipation coe
cients. The steady-state
solutions for the displacement of the oscillators are periodic,
x
1
=
x
10
e
−
i
ω
t
,
x
2
ω
ω
1
and
Here,
is the frequency of an external force,
x
20
e
−
i
ω
t
, where the amplitudes are given by the
=
expressions:
f
1
i
γ
2
ω
+
ω
2
2
2
(
f
1
+
f
2
)
−
−
ω
x
10
=
ω
−
i
γ
1
ω
ω
−
i
γ
2
ω
−
4
,
1
2
−
ω
2
−
ω
2
f
2
i
1
2
2
(
f
1
+
f
2
)
−
γ
1
ω
+
ω
−
ω
ω
−
i
γ
1
ω
ω
−
i
γ
2
ω
−
x
20
=
4
.
(9.3)
1
−
ω
2
2
2
−
ω
2
2
The paradigm of the classical analog of the Fano resonance is
that light excites only the broad mode, for example
x
1
, while the
narrow resonance mode
x
2
(dark mode) is excited just only due to
the coupling [7, 8]. In this sense, only one oscillator is driven by a
harmonic force, so that we may put
f
2
=
0. A particular example
of this situation is shown in Fig. 9.2. In the resonance region, we
identifybothconstructive(at
ω
=
ω
10
)anddestructive(at
ω
=
ω
20
)
interferences.
