5 Ultrafast Nonlinear Plasmonics
201
5.6 Conclusion
The third order nonlinear optical response of metals and plasmonic nanomaterials has
been described focusing on the resonant incoherent mechanisms associated to energy
absorption. Because of ultrafast dephasing of coherent electron-light polarization and
of the presence of electronic absorption over a very broad spectral range, these mechanisms usually dominate the Kerr-type nonlinearity of metals. They are directly connected to electronic and vibrational excitation and relaxation of the material and are
thus more conveniently modeled in the time-domain, describing the time-dependent
changes of the metal dielectric function due to electron excitation and lattice heating.
This time dependence reflects electron excitation and energy redistribution processes
within the metal (i.e., between the electrons and between the electrons and the lattice)
and to its environment (metal cooling). These processes can be reliably modeled in
noble metals, of key interest for plasmonic applications, using an approach developed for bulk metals. After computing modifications of the metal dielectric function,
the observable nonlinear optical response is obtained connecting the material linear
optical responses to the constituting metal dielectric function and analyzing impact
of its modification.
This modeling of the ultrafast nonlinear response has been successfully applied to
nanoparticles and nanomaterials. For not too small particles, down to about 10 nm,
the time and spectrally dependent nonlinear response can be fairly well reproduced
using the computed nonlinearity of bulk metals and taking into account the plasmonic
effects. Excellent agreement has also been found for individual metal nanoparticles,
for which both the spectral shape and amplitude of the nonlinear response have been
reproduced. As for their linear absorption, the fundamental mechanisms at the origin
of the optical nonlinearity are identical in bulk and confined metals, the main differences lying in enhancement effects around specific wavelength due to plasmonic
resonances. The dominant nonlinear mechanism depends on the investigated time
scale. It is mostly due to modification of the interband transition when the electrons
are out of equilibrium with the lattice, i.e., one to a few picoseconds, depending on
the transient temperature of the electron gas. Modification of the lattice temperature
yields the dominant contribution on a longer time-scale. These nonlinear mechanisms
are associated to metal heating and thus to losses for the incident pulse.
For smaller sizes, down to 2-3 nm, a similar model can be used introducing
modification of the electronic and vibrational kinetics [52]. Quantum effect and
increased interaction with the environment are expected to deeply modify the above
picture for even smaller sizes, with a transition from a solid state to a molecular type
of behavior in the 2-3 nm size range as reported in thiol stabilized clusters [140].
Though we have focused on description of the nonlinear response of simple plasmonic systems, i.e., individual metal nanoparticle and diluted ensemble of nanoparticles, the above model can be extended to any kind of plasmonic materials and metamaterials, such as dense ensemble of nanoparticles, arrays of particles or holes, or
waveguide stripes. The main point is then to properly connect the observable optical
properties of the plasmonic system with the dielectric functions of the constituting
201
5.6 Conclusion
The third order nonlinear optical response of metals and plasmonic nanomaterials has
been described focusing on the resonant incoherent mechanisms associated to energy
absorption. Because of ultrafast dephasing of coherent electron-light polarization and
of the presence of electronic absorption over a very broad spectral range, these mechanisms usually dominate the Kerr-type nonlinearity of metals. They are directly connected to electronic and vibrational excitation and relaxation of the material and are
thus more conveniently modeled in the time-domain, describing the time-dependent
changes of the metal dielectric function due to electron excitation and lattice heating.
This time dependence reflects electron excitation and energy redistribution processes
within the metal (i.e., between the electrons and between the electrons and the lattice)
and to its environment (metal cooling). These processes can be reliably modeled in
noble metals, of key interest for plasmonic applications, using an approach developed for bulk metals. After computing modifications of the metal dielectric function,
the observable nonlinear optical response is obtained connecting the material linear
optical responses to the constituting metal dielectric function and analyzing impact
of its modification.
This modeling of the ultrafast nonlinear response has been successfully applied to
nanoparticles and nanomaterials. For not too small particles, down to about 10 nm,
the time and spectrally dependent nonlinear response can be fairly well reproduced
using the computed nonlinearity of bulk metals and taking into account the plasmonic
effects. Excellent agreement has also been found for individual metal nanoparticles,
for which both the spectral shape and amplitude of the nonlinear response have been
reproduced. As for their linear absorption, the fundamental mechanisms at the origin
of the optical nonlinearity are identical in bulk and confined metals, the main differences lying in enhancement effects around specific wavelength due to plasmonic
resonances. The dominant nonlinear mechanism depends on the investigated time
scale. It is mostly due to modification of the interband transition when the electrons
are out of equilibrium with the lattice, i.e., one to a few picoseconds, depending on
the transient temperature of the electron gas. Modification of the lattice temperature
yields the dominant contribution on a longer time-scale. These nonlinear mechanisms
are associated to metal heating and thus to losses for the incident pulse.
For smaller sizes, down to 2-3 nm, a similar model can be used introducing
modification of the electronic and vibrational kinetics [52]. Quantum effect and
increased interaction with the environment are expected to deeply modify the above
picture for even smaller sizes, with a transition from a solid state to a molecular type
of behavior in the 2-3 nm size range as reported in thiol stabilized clusters [140].
Though we have focused on description of the nonlinear response of simple plasmonic systems, i.e., individual metal nanoparticle and diluted ensemble of nanoparticles, the above model can be extended to any kind of plasmonic materials and metamaterials, such as dense ensemble of nanoparticles, arrays of particles or holes, or
waveguide stripes. The main point is then to properly connect the observable optical
properties of the plasmonic system with the dielectric functions of the constituting