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M. Kauranen et al.
ial parameters in a scale that typically varies in the range of a few nm to 100s of
nm. Hence, the susceptibility is a locally varying quantity. Furthermore, the local
electromagnetic fields at the fundamental frequency are strongly varying in space
in metal nanostructures. A detailed approach would therefore require taking into
account the local-field variations, the spatially varying susceptibility tensors, the
locally-generated nonlinear sources, and also, the coupling of the incoming and outgoing radiation fields to the local quantities. Such an approach would give detailed
knowledge on the nonlinear response of the nanostructures, but it is very challenging
even computationally.
We therefore use a simplified approach that allows us to treat the experiments in
a convenient way. The sample is treated as a “black box”, and only the input and
output fields are considered. The relation between the incoming and outgoing fields
is described by the nonlinear response tensor (NRT) components A jkl defined as [28]
E j (2ω) =
kl
A jkl E k (ω)E l (ω),
(6.1)
where E j (2ω) is the outgoing field at the second-harmonic (SH) frequency, and
E k (ω) and E l (ω) are the incoming fields at the fundamental frequency. The fields
are assumed to be plane waves, which is the case also in all our experiments.
The nonlinear response tensor is a macroscopic parameter, which implicitly takes
into account all the effects in the nanoscale, while avoiding the difficulties in their
explicit treatment. The main disadvantage is that the tensor components can depend
significantly on the experimental setup, however, even this can be used to advantage.
Comparison of NRTs under different experimental conditions provides important
insight to the nanoscopic origin of the nonlinear response of the sample. Another
benefit is that, due to its macroscopic character, NRT includes implicitly contributions
from different multipolar sources.
The nonlinear response tensor has in total 3 3 = 27 complex valued components.
In our measurements, the sample is always characterized at normal or near-normal
incidence using plane waves. In consequence, only the in-plane field components
need to be included in the analysis, which significantly reduces the number of components. The coordinate system is chosen so that the fields are polarized in the
(x, y)-plane and the propagation is in the z direction. By using the described coordinate system, the x- and y-polarized second-harmonic fields can be written in terms
of the fundamental field components as
E j (2ω) = A j xx E
2
x (ω) + A j yy E
2
y (ω) + 2 A j xy E x (ω)E y (ω),
(6.2)
where j is either x or y and the factor of two comes from the fact that for
second-harmonic generation the latter two indices are interchangeable. Only six
tensor components therefore need to be considered.
Second-harmonic generation, as well as other even-order processes, is very
sensitive to the symmetry of the nanostructures [29]. A typical example is a
M. Kauranen et al.
ial parameters in a scale that typically varies in the range of a few nm to 100s of
nm. Hence, the susceptibility is a locally varying quantity. Furthermore, the local
electromagnetic fields at the fundamental frequency are strongly varying in space
in metal nanostructures. A detailed approach would therefore require taking into
account the local-field variations, the spatially varying susceptibility tensors, the
locally-generated nonlinear sources, and also, the coupling of the incoming and outgoing radiation fields to the local quantities. Such an approach would give detailed
knowledge on the nonlinear response of the nanostructures, but it is very challenging
even computationally.
We therefore use a simplified approach that allows us to treat the experiments in
a convenient way. The sample is treated as a “black box”, and only the input and
output fields are considered. The relation between the incoming and outgoing fields
is described by the nonlinear response tensor (NRT) components A jkl defined as [28]
E j (2ω) =
kl
A jkl E k (ω)E l (ω),
(6.1)
where E j (2ω) is the outgoing field at the second-harmonic (SH) frequency, and
E k (ω) and E l (ω) are the incoming fields at the fundamental frequency. The fields
are assumed to be plane waves, which is the case also in all our experiments.
The nonlinear response tensor is a macroscopic parameter, which implicitly takes
into account all the effects in the nanoscale, while avoiding the difficulties in their
explicit treatment. The main disadvantage is that the tensor components can depend
significantly on the experimental setup, however, even this can be used to advantage.
Comparison of NRTs under different experimental conditions provides important
insight to the nanoscopic origin of the nonlinear response of the sample. Another
benefit is that, due to its macroscopic character, NRT includes implicitly contributions
from different multipolar sources.
The nonlinear response tensor has in total 3 3 = 27 complex valued components.
In our measurements, the sample is always characterized at normal or near-normal
incidence using plane waves. In consequence, only the in-plane field components
need to be included in the analysis, which significantly reduces the number of components. The coordinate system is chosen so that the fields are polarized in the
(x, y)-plane and the propagation is in the z direction. By using the described coordinate system, the x- and y-polarized second-harmonic fields can be written in terms
of the fundamental field components as
E j (2ω) = A j xx E
2
x (ω) + A j yy E
2
y (ω) + 2 A j xy E x (ω)E y (ω),
(6.2)
where j is either x or y and the factor of two comes from the fact that for
second-harmonic generation the latter two indices are interchangeable. Only six
tensor components therefore need to be considered.
Second-harmonic generation, as well as other even-order processes, is very
sensitive to the symmetry of the nanostructures [29]. A typical example is a