196
F. Vallée and N. Del Fatti
with energy injection in the electrons and decays with the electron energy losses to
the lattice. On a longer time scale, after electron-lattice thermalization, the spectral
profile of ωη x
ext is strongly modified. The ωε ib
1 (ω x
R , t) contribution is then negligible
as compared to the ωε in
2 (ω x
R , t) one due to heating of the lattice (Fig. 5.5f). ωη x
ext then
essentially reflects modification of the electron-phonon scattering rate (Eq. 5.34). Its
spectral shape is set by the dispersion of a x
2 (Fig. 5.9b and c), ωε in
2 (ω x
R ) being weekly
dispersed on the relevant spectral range around ω x
R (Fig. 5.5f).
For light polarized along the short direction of a prolate spheroid, the transverse
SPR and interband transitions overlap leading to a small SPR amplitude (Fig. 5.9d).
The situation is very similar to the gold sphere one, and ωη
y,z
ext around the transverse
SPR frequency ω
y,z
R shows similar spectral and temporal features as ωη x
ext in gold
nanosphere (Figs. 5.9f and 5.8c).
It is interesting to note that the characteristic spectral shape of ωη x
ext imposed by
the a x
1 and a x
2 dispersion closely corresponds to that obtained for a SPR spectral shift
and broadening, respectively (assuming ωε undispersed). This is simply related to
the fact that the longitudinal SPR exhibits a quasi-Lorentzian shape with a frequency
determined by ε 1 (ω x
R ) and a width by ε 2 (ω x
R ). The full approach used here, can then
be reduced to a simple analysis in term of SPR frequency shift (for t ≈ 500 fs) and
broadening (for t ≥ 500 fs), related to ωε 1 (ω x
R , t) and ωε 2 (ω x
R , t), respectively.
This approach can be used as long as the SPR is away from the interband transitions,
as for the longitudinal SPR of a prolate spheroid or nanorod or for the SPR of silver
nanospheres [37, 60, 132]. In contrast, it cannot be used when the SPR overlap
the interband transition as in gold or copper nanospheres or for the transverse SPR
of a prolate spheroid, ωε ib
1,2 showing a large dispersion on the relevant frequency
range around their SPR (Fig. 5.5, note that similarly because of the dispersion of ε ib ,
Fig. 5.1a, the SPR width can also not be correctly defined). The ωη ext spectral profile
is then related to both ωε 1 and ωε 2 with contributions weighted by the enhancement
of the nonlinear response around the SPR (Figs. 5.8c and 5.9f).
The above simulations have been performed assuming a given excitation of the
nanoparticles, imposing the maximum electron temperature rise ωT me
e
or, equivalently, the injected total energy u abs
e
(Eq. 5.19). More generally, ωT me
e
or u abs
e
are
imposed by absorption of the exciting pulse of frequency ω e . Assuming it is polarized
along the j direction with a time-dependent intensity I
j
p (t), it is given by:
u
abs
e
=
η
j
abs (ω e )
V np
+∞
−∞
I
j
p (t)dt =
η
j
abs (ω e )
V np
F
j
p (ω e ),
(5.49)
where F
j
p (ω e ) is the pulse fluence. Assuming ωε linearly depends on u abs
e , an approximation verified in the weak excitation regime, ωT me
e ≤ 200 − 300 K, ωη i
ext (ω) can
thus be written:
ωη
i
ext (ω, t) =
a
i
1 (ω)ω¯ ε 1 (ω, t) + a
i
2 (ω)ω¯ ε 2 (ω, t)
η
j
abs (ω e )F
j
p (ω e )/V np .
(5.50)
F. Vallée and N. Del Fatti
with energy injection in the electrons and decays with the electron energy losses to
the lattice. On a longer time scale, after electron-lattice thermalization, the spectral
profile of ωη x
ext is strongly modified. The ωε ib
1 (ω x
R , t) contribution is then negligible
as compared to the ωε in
2 (ω x
R , t) one due to heating of the lattice (Fig. 5.5f). ωη x
ext then
essentially reflects modification of the electron-phonon scattering rate (Eq. 5.34). Its
spectral shape is set by the dispersion of a x
2 (Fig. 5.9b and c), ωε in
2 (ω x
R ) being weekly
dispersed on the relevant spectral range around ω x
R (Fig. 5.5f).
For light polarized along the short direction of a prolate spheroid, the transverse
SPR and interband transitions overlap leading to a small SPR amplitude (Fig. 5.9d).
The situation is very similar to the gold sphere one, and ωη
y,z
ext around the transverse
SPR frequency ω
y,z
R shows similar spectral and temporal features as ωη x
ext in gold
nanosphere (Figs. 5.9f and 5.8c).
It is interesting to note that the characteristic spectral shape of ωη x
ext imposed by
the a x
1 and a x
2 dispersion closely corresponds to that obtained for a SPR spectral shift
and broadening, respectively (assuming ωε undispersed). This is simply related to
the fact that the longitudinal SPR exhibits a quasi-Lorentzian shape with a frequency
determined by ε 1 (ω x
R ) and a width by ε 2 (ω x
R ). The full approach used here, can then
be reduced to a simple analysis in term of SPR frequency shift (for t ≈ 500 fs) and
broadening (for t ≥ 500 fs), related to ωε 1 (ω x
R , t) and ωε 2 (ω x
R , t), respectively.
This approach can be used as long as the SPR is away from the interband transitions,
as for the longitudinal SPR of a prolate spheroid or nanorod or for the SPR of silver
nanospheres [37, 60, 132]. In contrast, it cannot be used when the SPR overlap
the interband transition as in gold or copper nanospheres or for the transverse SPR
of a prolate spheroid, ωε ib
1,2 showing a large dispersion on the relevant frequency
range around their SPR (Fig. 5.5, note that similarly because of the dispersion of ε ib ,
Fig. 5.1a, the SPR width can also not be correctly defined). The ωη ext spectral profile
is then related to both ωε 1 and ωε 2 with contributions weighted by the enhancement
of the nonlinear response around the SPR (Figs. 5.8c and 5.9f).
The above simulations have been performed assuming a given excitation of the
nanoparticles, imposing the maximum electron temperature rise ωT me
e
or, equivalently, the injected total energy u abs
e
(Eq. 5.19). More generally, ωT me
e
or u abs
e
are
imposed by absorption of the exciting pulse of frequency ω e . Assuming it is polarized
along the j direction with a time-dependent intensity I
j
p (t), it is given by:
u
abs
e
=
η
j
abs (ω e )
V np
+∞
−∞
I
j
p (t)dt =
η
j
abs (ω e )
V np
F
j
p (ω e ),
(5.49)
where F
j
p (ω e ) is the pulse fluence. Assuming ωε linearly depends on u abs
e , an approximation verified in the weak excitation regime, ωT me
e ≤ 200 − 300 K, ωη i
ext (ω) can
thus be written:
ωη
i
ext (ω, t) =
a
i
1 (ω)ω¯ ε 1 (ω, t) + a
i
2 (ω)ω¯ ε 2 (ω, t)
η
j
abs (ω e )F
j
p (ω e )/V np .
(5.50)