192
F. Vallée and N. Del Fatti
whose optical response can be computed analytically, as well as that of a nanorod,
are illustrated below.
5.5.2.1 Small Metal Nanosphere and Nanoellipsoid
The optical response, absorption or scattering, of a sphere of dielectric constant
ε and diameter D embedded in a dielectric medium of dielectric constant ε d has
been described by Mie using a multipolar expansion [1, 3]. For a nano-sphere much
smaller than the optical wavelength σ only the dipolar term has to be retained [1,
128]. The sphere response can then be described as that of a dipole p L at its center
and oriented along the electric field E of the incident electromagnetic wave:
p L (ω) = ε 0 ε d β L (ω)E(ω).
(5.42)
where β L is the nanosphere linear polarizability:
β L (ω) = 3V np
ε(ω) − ε d
ε(ω) + 2ε d
(5.43)
Its extinction cross-section can be identified with its absorption cross-section in
this regime, scattering being negligible as compared to absorption in the dipolar
approximation, η ext ≈ η abs . >> η sca . It is given by the imaginary part of β L and is
independent of the polarization direction:
η ext (ω) ≈ η abs (ω) = ω ε
1/2
d I m(β L )/c =
9 ω V np ε
3/2
d
c
ε 2 (ω)
|ε(ω) + 2ε d |
2
, (5.44)
where V np is the nanoparticle volume. Similar calculations were performed for ellipsoidal shapes, yielding similar simple expression in the small size limit [1]. For light
polarized along one of the ellipsoid main axis i the optical response is equivalent to
that of a i direction dipole p i
L (ω) = ε 0 ε d β i
L (ω)E i (ω) at the ellipsoid center with:
β
i
L (ω) =
V np
L i
ε(ω) − ε d
ε(ω) + (1/L i − 1)ε d
,
(5.45)
yielding the extinction cross section for light polarized along the i direction
η
i
ext (ω) ≈ η
i
abs (ω) = ω ε
1/2
d I m(β
i
L )/c =
ω V np ε
3/2
d
cL 2
i
ε 2 (ω)
|ε(ω) + (1/L i − 1)ε d |
2
,
(5.46)
where L i are geometrical factors. For instance, for a prolate spheroid of length l x
along its long axis direction x, and l y = l z , along its short axis directions y,z, they
are given by:
F. Vallée and N. Del Fatti
whose optical response can be computed analytically, as well as that of a nanorod,
are illustrated below.
5.5.2.1 Small Metal Nanosphere and Nanoellipsoid
The optical response, absorption or scattering, of a sphere of dielectric constant
ε and diameter D embedded in a dielectric medium of dielectric constant ε d has
been described by Mie using a multipolar expansion [1, 3]. For a nano-sphere much
smaller than the optical wavelength σ only the dipolar term has to be retained [1,
128]. The sphere response can then be described as that of a dipole p L at its center
and oriented along the electric field E of the incident electromagnetic wave:
p L (ω) = ε 0 ε d β L (ω)E(ω).
(5.42)
where β L is the nanosphere linear polarizability:
β L (ω) = 3V np
ε(ω) − ε d
ε(ω) + 2ε d
(5.43)
Its extinction cross-section can be identified with its absorption cross-section in
this regime, scattering being negligible as compared to absorption in the dipolar
approximation, η ext ≈ η abs . >> η sca . It is given by the imaginary part of β L and is
independent of the polarization direction:
η ext (ω) ≈ η abs (ω) = ω ε
1/2
d I m(β L )/c =
9 ω V np ε
3/2
d
c
ε 2 (ω)
|ε(ω) + 2ε d |
2
, (5.44)
where V np is the nanoparticle volume. Similar calculations were performed for ellipsoidal shapes, yielding similar simple expression in the small size limit [1]. For light
polarized along one of the ellipsoid main axis i the optical response is equivalent to
that of a i direction dipole p i
L (ω) = ε 0 ε d β i
L (ω)E i (ω) at the ellipsoid center with:
β
i
L (ω) =
V np
L i
ε(ω) − ε d
ε(ω) + (1/L i − 1)ε d
,
(5.45)
yielding the extinction cross section for light polarized along the i direction
η
i
ext (ω) ≈ η
i
abs (ω) = ω ε
1/2
d I m(β
i
L )/c =
ω V np ε
3/2
d
cL 2
i
ε 2 (ω)
|ε(ω) + (1/L i − 1)ε d |
2
,
(5.46)
where L i are geometrical factors. For instance, for a prolate spheroid of length l x
along its long axis direction x, and l y = l z , along its short axis directions y,z, they
are given by: