5 Ultrafast Nonlinear Plasmonics
193
L x =
1 − e 2
e 2
−1 +
1
2e
ln
1 + e
1 − e
,
(5.47)
where e is the ellipsoid eccentricity: e 2 = 1 − (l y /l x ) 2 , and L y = L z = (1 − L x )/2.
The extinction cross-sections take similar forms in both types of nanoparticles,
(with L i = 1/3 for a sphere). For a weakly dispersed ε 2 , it is enhanced close to the
frequency minimizing the denominator in Eq. 5.44 or 5.46, which corresponds to the
condition for the localized surface plasmon resonance (SPR). This is concomitant
with enhancement of the electromagnetic field in the particle and its close environment as compared to the applied one, i.e., corresponds to a local field or dielectric
confinement effect [5]. For a sphere, the field is uniform in the particle and enhanced
as compared to the applied field by the plasmonic factor:
f pl (ω) =
3ε d
ε(ω) + 2ε d
.
(5.48)
For a nanosphere, the resonance condition is independent of the light polarization
direction (Eq. 5.44) and is associated to resonant collective oscillation of the electrons
driven by the electromagnetic field along its polarization direction. This degeneracy
is lifted in an ellipsoid, three different resonance conditions being obtained along
its three main axis (Eq. 5.46). It reduced to two in a spheroid, two SPR showing-up
associated to electron oscillations along its long and short axis. The SPR frequency
splitting depending of its aspect ratio Φ = l x /l y [1, 5, 128].
The extinction cross-sections of a gold nano-sphere and of a gold prolate spheroid
of long axis x and short axis y, z are shown in Figs. 5.8a and 5.9a and d, respectively
(the bulk ε data [83] were corrected for surface effect in the sphere case using g e f f =
0.7, Eq. 5.8). The SPR of the sphere overlap the interband transition (Fig. 5.8a), while
the longitudinal SPR (along its x axis) of the spheroid is strongly red shifted away
from ω ib , (Fig. 5.9a). Its transverse one (along its y or z axis) is slightly blue shifted (as
compared to the sphere SPR) and overlaps with the interband transitions (Fig. 5.9d).
The corresponding coefficients a i
1,2 (ω) computed using Eqs. 5.40, 5.44 and 5.46 are
shown in Figs. 5.8b and 5.9b and e. The plasmonic effect is at the origin of their
large amplitude and dispersion around the longitudinal SPR frequency ω x
R . As for
the linear response, it reflect local field effect enhancement. Similar effects show-up
around the transverse SPR, but with weaker amplitude as expected because of smaller
amplitude of this SPR.
Using these coefficients together with the computed changes of ε (Fig. 5.5a, b,
and f), one obtains the amplitude and time and spectral dependence of the extinction
cross-section change ωη i
ext (ω) for a given excitation ωT me
e (Figs. 5.8c, and 5.9c and
f). Modification of the surface scattering term has been disregarded for the considered
sizes [60]. For the gold nanosphere, the SPR being around ω ib , the plasmonic effect
only leads to modification of the spectral shape and enhancement of the amplitude
of the nonlinear response as compared to the film one around ω ib (Figs. 5.7 and 5.8).
These two effects are separated in a prolate spheroid for light polarized along
its long axis. Two distinct spectral features around ω ib and ω x
R are thus obtained.
193
L x =
1 − e 2
e 2
−1 +
1
2e
ln
1 + e
1 − e
,
(5.47)
where e is the ellipsoid eccentricity: e 2 = 1 − (l y /l x ) 2 , and L y = L z = (1 − L x )/2.
The extinction cross-sections take similar forms in both types of nanoparticles,
(with L i = 1/3 for a sphere). For a weakly dispersed ε 2 , it is enhanced close to the
frequency minimizing the denominator in Eq. 5.44 or 5.46, which corresponds to the
condition for the localized surface plasmon resonance (SPR). This is concomitant
with enhancement of the electromagnetic field in the particle and its close environment as compared to the applied one, i.e., corresponds to a local field or dielectric
confinement effect [5]. For a sphere, the field is uniform in the particle and enhanced
as compared to the applied field by the plasmonic factor:
f pl (ω) =
3ε d
ε(ω) + 2ε d
.
(5.48)
For a nanosphere, the resonance condition is independent of the light polarization
direction (Eq. 5.44) and is associated to resonant collective oscillation of the electrons
driven by the electromagnetic field along its polarization direction. This degeneracy
is lifted in an ellipsoid, three different resonance conditions being obtained along
its three main axis (Eq. 5.46). It reduced to two in a spheroid, two SPR showing-up
associated to electron oscillations along its long and short axis. The SPR frequency
splitting depending of its aspect ratio Φ = l x /l y [1, 5, 128].
The extinction cross-sections of a gold nano-sphere and of a gold prolate spheroid
of long axis x and short axis y, z are shown in Figs. 5.8a and 5.9a and d, respectively
(the bulk ε data [83] were corrected for surface effect in the sphere case using g e f f =
0.7, Eq. 5.8). The SPR of the sphere overlap the interband transition (Fig. 5.8a), while
the longitudinal SPR (along its x axis) of the spheroid is strongly red shifted away
from ω ib , (Fig. 5.9a). Its transverse one (along its y or z axis) is slightly blue shifted (as
compared to the sphere SPR) and overlaps with the interband transitions (Fig. 5.9d).
The corresponding coefficients a i
1,2 (ω) computed using Eqs. 5.40, 5.44 and 5.46 are
shown in Figs. 5.8b and 5.9b and e. The plasmonic effect is at the origin of their
large amplitude and dispersion around the longitudinal SPR frequency ω x
R . As for
the linear response, it reflect local field effect enhancement. Similar effects show-up
around the transverse SPR, but with weaker amplitude as expected because of smaller
amplitude of this SPR.
Using these coefficients together with the computed changes of ε (Fig. 5.5a, b,
and f), one obtains the amplitude and time and spectral dependence of the extinction
cross-section change ωη i
ext (ω) for a given excitation ωT me
e (Figs. 5.8c, and 5.9c and
f). Modification of the surface scattering term has been disregarded for the considered
sizes [60]. For the gold nanosphere, the SPR being around ω ib , the plasmonic effect
only leads to modification of the spectral shape and enhancement of the amplitude
of the nonlinear response as compared to the film one around ω ib (Figs. 5.7 and 5.8).
These two effects are separated in a prolate spheroid for light polarized along
its long axis. Two distinct spectral features around ω ib and ω x
R are thus obtained.