200
F. Vallée and N. Del Fatti
with
¯
β N L =
1
3
i
β
i
N L .
(5.62)
The transmission change of a sample of thickness L is then given by:
ωT
T
(ω, t) ≈ − ω ε
1/2
m N np I m { ¯
β N L (ω, t)} L/c = −N np ω ¯
η abs (ω, t)L , (5.63)
where ω ¯
η abs is the mean absorption change per particle. It can be rewritten:
ωT
T
(ω, t d ) ≈ a 1 (ω)ωε 1 (ω, t) + a 2 (ω)ωε 2 (ω, t),
(5.64)
where the coefficients a 1 and a 2 are directly related to the equilibrium absorption
coefficient of the sample: A(ω) = N abs ¯
η abs (ω):
a 1,2 = −
γ A
γε i
(ω)L ,
(5.65)
and are entirely determined by the linear absorption properties.
Knowing the dielectric function change of a metal, the transmission change can
thus be computed for different composite nanomaterials made of nanoparticles of
different shapes dispersed in different environment using the relevant a 1 and a 2
coefficients (at least for not too small objects so that the bulk electron kinetics can
be used). It is illustrated in Fig. 5.8f, in the case of an ensemble of randomly oriented
identical gold ellipsoids (prolate spheroids with long and short axis 40 nm and 10 nm,
respectively) dispersed in water. The same excitation temperature has been assumed
in each particle with ωT me
e
= 100 K. The computed features are similar to those
observed in single nanoellipsoids, the main difference lying in the absence of light
polarization effect.
This approach permits comparison of the experimental and theoretical data
obtained in ensemble of metal nanoparticles. In particular it has been extensively
used in the case of nanospheres for which excellent agreement between the measured
and computed spectral shapes have been obtained both for low and high excitation
of the electrons [37, 52, 60, 137]. Note that in very diluted samples, the nonlinear
response of the matrix can also contribute, yielding additional non-resonant contribution to the observed nonlinearity. Comparison of the amplitude of the signals are
however more difficult as it requires taking into account the size and shape dispersions of the particles and to determine their absorbed energy. Though this can be
done summing up the nonlinear contribution of the different particles, taking into
account their possibly different absorption, the precision is limited by the knowledge
of these dispersions. This problem is overcome performing experiments in single
nanoparticle, whose morphology and size can be either optically characterized [136]
or independently measured by electron microscopy [138, 139].
F. Vallée and N. Del Fatti
with
¯
β N L =
1
3
i
β
i
N L .
(5.62)
The transmission change of a sample of thickness L is then given by:
ωT
T
(ω, t) ≈ − ω ε
1/2
m N np I m { ¯
β N L (ω, t)} L/c = −N np ω ¯
η abs (ω, t)L , (5.63)
where ω ¯
η abs is the mean absorption change per particle. It can be rewritten:
ωT
T
(ω, t d ) ≈ a 1 (ω)ωε 1 (ω, t) + a 2 (ω)ωε 2 (ω, t),
(5.64)
where the coefficients a 1 and a 2 are directly related to the equilibrium absorption
coefficient of the sample: A(ω) = N abs ¯
η abs (ω):
a 1,2 = −
γ A
γε i
(ω)L ,
(5.65)
and are entirely determined by the linear absorption properties.
Knowing the dielectric function change of a metal, the transmission change can
thus be computed for different composite nanomaterials made of nanoparticles of
different shapes dispersed in different environment using the relevant a 1 and a 2
coefficients (at least for not too small objects so that the bulk electron kinetics can
be used). It is illustrated in Fig. 5.8f, in the case of an ensemble of randomly oriented
identical gold ellipsoids (prolate spheroids with long and short axis 40 nm and 10 nm,
respectively) dispersed in water. The same excitation temperature has been assumed
in each particle with ωT me
e
= 100 K. The computed features are similar to those
observed in single nanoellipsoids, the main difference lying in the absence of light
polarization effect.
This approach permits comparison of the experimental and theoretical data
obtained in ensemble of metal nanoparticles. In particular it has been extensively
used in the case of nanospheres for which excellent agreement between the measured
and computed spectral shapes have been obtained both for low and high excitation
of the electrons [37, 52, 60, 137]. Note that in very diluted samples, the nonlinear
response of the matrix can also contribute, yielding additional non-resonant contribution to the observed nonlinearity. Comparison of the amplitude of the signals are
however more difficult as it requires taking into account the size and shape dispersions of the particles and to determine their absorbed energy. Though this can be
done summing up the nonlinear contribution of the different particles, taking into
account their possibly different absorption, the precision is limited by the knowledge
of these dispersions. This problem is overcome performing experiments in single
nanoparticle, whose morphology and size can be either optically characterized [136]
or independently measured by electron microscopy [138, 139].