5 Ultrafast Nonlinear Plasmonics
189
frequency ω, T (ω) and R(ω), are determined by its geometry and dielectric function
ε(ω). Their time-dependent changes after excitation by a short pulse of frequency
ω 0 thus reflects that of ε. If the optical property changes are sufficiently weak, the
relative transmission and reflection changes can be connected to those of the real,
ωε 1 , and imaginary, ωε 2 , parts of ωε, using a lowest order development:
ωT
T (ω, t) = t 1 (ω)ωε 1 (ω, t) + t 2 (ω)ωε 2 (ω, t)
ωR
R (ω, t) = r 1 (ω)ωε 1 (ω, t) + r 2 (ω)ωε 2 (ω, t)
(5.36)
where ωT (ω, t) = T (ω, t) − T (ω) and ωR(ω, t) = R(ω, t) − R(ω), t being the
time delay after the maximum of the excitation pulse. The coefficients t 1,2 (ω) =
γ ln T /γε 1,2
ω
and r 1,2 (ω) =
γ ln R/γε 1,2
ω
can be analytically or numerically
computed from the dependence of T (ω) and R(ω) on the sample dielectric function.
These general expressions will be used below for discussing the nonlinear response
of different metal materials.
In the following, weak metal excitation will be assumed (ωT me
e
= 100 K), and
only the dominant contributions to ωε due to modification of the interband term by
electron heating (Sect. 4.1.1) and of the intraband one by lattice heating (Eq. 5.34)
will be included. In particular, dilation effect will be disregarded, their inclusion also
requiring to take into account the direct dependences of T or R on the metal object
size, e.g., film thickness or nanoparticle volume (Eq. 5.44 or 5.46, for instance).
5.5.1 Metal Film
In the case of a homogeneous metal film deposited on a substrate, the coefficient
linking ωT /T and ωR/R to ωε can be analytically computed using the expression
T (ε) and R(ε) for a thin film of thickness L [127], ε being the metal dielectric
function, assumed to be identical to the bulk one [83]. The dispersion of T, R, t 1,2 ,
and r 1,2 are shown in Figs. 5.7a and b, in the case of an optically thin gold film
(L = 20 nm). The temporal and spectral dependences of ωT (ω, t) and ωR(ω, t)
calculated from the computed ωε (Fig. 5.5a, b and f) are also shown. They exhibit
similar amplitudes with extrema for frequencies ω close to the interband threshold
ω ib , around the L point of the Brillouin zone (the X point yields smaller structures
as for ωε ib , Fig. 5.5a and b). The spectral shapes are mostly set by the ωε ib spectra
modified by t 1,2 or r 1,2, the ωε in contribution being weaker.
The computed spectral profiles have been found to be in excellent agreement with
the measured ones in optically thin films when the electronic response dominates
[26, 29, 30, 61, 65]. Note that simultaneous measurement of ωT (ω, t) and ωR(ω, t)
also permits experimental estimation of ωε bulk
1
(ω, t) and ωε bulk
2
(ω, t) using Eq. 5.36
[26]. Though this agreement validates the above theoretical modeling, one has to keep
in mind that quantitative comparison, i.e., of the ωT /T and ωR/R amplitudes, is
however limited by the experimental difficulty in estimating the temperature rise of
the metal film.
189
frequency ω, T (ω) and R(ω), are determined by its geometry and dielectric function
ε(ω). Their time-dependent changes after excitation by a short pulse of frequency
ω 0 thus reflects that of ε. If the optical property changes are sufficiently weak, the
relative transmission and reflection changes can be connected to those of the real,
ωε 1 , and imaginary, ωε 2 , parts of ωε, using a lowest order development:
ωT
T (ω, t) = t 1 (ω)ωε 1 (ω, t) + t 2 (ω)ωε 2 (ω, t)
ωR
R (ω, t) = r 1 (ω)ωε 1 (ω, t) + r 2 (ω)ωε 2 (ω, t)
(5.36)
where ωT (ω, t) = T (ω, t) − T (ω) and ωR(ω, t) = R(ω, t) − R(ω), t being the
time delay after the maximum of the excitation pulse. The coefficients t 1,2 (ω) =
γ ln T /γε 1,2
ω
and r 1,2 (ω) =
γ ln R/γε 1,2
ω
can be analytically or numerically
computed from the dependence of T (ω) and R(ω) on the sample dielectric function.
These general expressions will be used below for discussing the nonlinear response
of different metal materials.
In the following, weak metal excitation will be assumed (ωT me
e
= 100 K), and
only the dominant contributions to ωε due to modification of the interband term by
electron heating (Sect. 4.1.1) and of the intraband one by lattice heating (Eq. 5.34)
will be included. In particular, dilation effect will be disregarded, their inclusion also
requiring to take into account the direct dependences of T or R on the metal object
size, e.g., film thickness or nanoparticle volume (Eq. 5.44 or 5.46, for instance).
5.5.1 Metal Film
In the case of a homogeneous metal film deposited on a substrate, the coefficient
linking ωT /T and ωR/R to ωε can be analytically computed using the expression
T (ε) and R(ε) for a thin film of thickness L [127], ε being the metal dielectric
function, assumed to be identical to the bulk one [83]. The dispersion of T, R, t 1,2 ,
and r 1,2 are shown in Figs. 5.7a and b, in the case of an optically thin gold film
(L = 20 nm). The temporal and spectral dependences of ωT (ω, t) and ωR(ω, t)
calculated from the computed ωε (Fig. 5.5a, b and f) are also shown. They exhibit
similar amplitudes with extrema for frequencies ω close to the interband threshold
ω ib , around the L point of the Brillouin zone (the X point yields smaller structures
as for ωε ib , Fig. 5.5a and b). The spectral shapes are mostly set by the ωε ib spectra
modified by t 1,2 or r 1,2, the ωε in contribution being weaker.
The computed spectral profiles have been found to be in excellent agreement with
the measured ones in optically thin films when the electronic response dominates
[26, 29, 30, 61, 65]. Note that simultaneous measurement of ωT (ω, t) and ωR(ω, t)
also permits experimental estimation of ωε bulk
1
(ω, t) and ωε bulk
2
(ω, t) using Eq. 5.36
[26]. Though this agreement validates the above theoretical modeling, one has to keep
in mind that quantitative comparison, i.e., of the ωT /T and ωR/R amplitudes, is
however limited by the experimental difficulty in estimating the temperature rise of
the metal film.