6 Second-Order Nonlinear Optical Properties of Plasmonic Nanostructures
209
the second-order nonlinear response, and they can arise from two complementary
mechanisms. First, higher-multipole interactions on the atomic level give rise to
new types of nonlinear responses that allow second-order effects even in the bulk of
centrosymmetric materials [21]. Second, field retardation across nanoparticles may
give rise to effective multipole effects, similar to the various multipole terms in the
Mie scattering theory [26, 27].
To theoretically understand the optical properties of nanoparticles, one usually
needs to solve an electromagnetic scattering problem. The geometry of the particles
is often so complicated that closed form solutions cannot be found and approximate
solutions must be sought numerically. Plasmonic structures pose challenges for numerical schemes, as resonances can make the problem sensitive to its mathematical
formulation and the structures tend to give rise to sub-wavelength features in the local
fields. Modeling nonlinear effects in nanoparticles can make these issues even more
pronounced, as multiple frequencies are present and the associated wavelengths can
be very short. The early approaches on modeling have focused on SHG in spheres
or other simple geometries that can be treated analytically [24]. Recently, various,
more general methods have been considered to address particles of arbitrary shapes.
This Chapter is structured as follows. In Sect. 6.2, we introduce a theoretical formalism based on effective multipole nonlinearities, which provides a convenient
way to describe the experimental results. The sample fabrication and experimental
setups for SHG measurements are described in Sect. 6.3. Section 6.4 then summarizes the early results on SHG from metal nanostructures where fabrication related
defects were found to play a disproportionate role in the nonlinear response, giving
rise to symmetry-forbidden signals and effective multipole effects in the response.
Section 6.5 discusses the role of the local-field distribution and its symmetry in the
nonlinear response. In Sect. 6.6, we show that recent progress in nanofabrication has
led to a significant improvement in the sample quality, allowing the desired dipole
limit in the nonlinear response to be reached. This result is an important milestone
for the development of metamaterials with tailored nonlinear properties, an example
of which is discussed in Sect. 6.7. We then discuss recent progress in the numerical
modeling of the nonlinear response in Sect. 6.8 and provide an outlook for the future
in Sect. 6.9.
6.2 Theoretical Background
6.2.1 Nonlinear Response Tensor
On the fundamental level, SHG arises from the interaction between light and matter
characterized by the second-order susceptibility tensor, which is a material quantity that can be derived using microscopic theories of the light-matter interaction.
In homogeneous media, the susceptibilities are spatially invariant over macroscopic
length scales. Metal nanostructures, however, have spatial variations of the mater-
209
the second-order nonlinear response, and they can arise from two complementary
mechanisms. First, higher-multipole interactions on the atomic level give rise to
new types of nonlinear responses that allow second-order effects even in the bulk of
centrosymmetric materials [21]. Second, field retardation across nanoparticles may
give rise to effective multipole effects, similar to the various multipole terms in the
Mie scattering theory [26, 27].
To theoretically understand the optical properties of nanoparticles, one usually
needs to solve an electromagnetic scattering problem. The geometry of the particles
is often so complicated that closed form solutions cannot be found and approximate
solutions must be sought numerically. Plasmonic structures pose challenges for numerical schemes, as resonances can make the problem sensitive to its mathematical
formulation and the structures tend to give rise to sub-wavelength features in the local
fields. Modeling nonlinear effects in nanoparticles can make these issues even more
pronounced, as multiple frequencies are present and the associated wavelengths can
be very short. The early approaches on modeling have focused on SHG in spheres
or other simple geometries that can be treated analytically [24]. Recently, various,
more general methods have been considered to address particles of arbitrary shapes.
This Chapter is structured as follows. In Sect. 6.2, we introduce a theoretical formalism based on effective multipole nonlinearities, which provides a convenient
way to describe the experimental results. The sample fabrication and experimental
setups for SHG measurements are described in Sect. 6.3. Section 6.4 then summarizes the early results on SHG from metal nanostructures where fabrication related
defects were found to play a disproportionate role in the nonlinear response, giving
rise to symmetry-forbidden signals and effective multipole effects in the response.
Section 6.5 discusses the role of the local-field distribution and its symmetry in the
nonlinear response. In Sect. 6.6, we show that recent progress in nanofabrication has
led to a significant improvement in the sample quality, allowing the desired dipole
limit in the nonlinear response to be reached. This result is an important milestone
for the development of metamaterials with tailored nonlinear properties, an example
of which is discussed in Sect. 6.7. We then discuss recent progress in the numerical
modeling of the nonlinear response in Sect. 6.8 and provide an outlook for the future
in Sect. 6.9.
6.2 Theoretical Background
6.2.1 Nonlinear Response Tensor
On the fundamental level, SHG arises from the interaction between light and matter
characterized by the second-order susceptibility tensor, which is a material quantity that can be derived using microscopic theories of the light-matter interaction.
In homogeneous media, the susceptibilities are spatially invariant over macroscopic
length scales. Metal nanostructures, however, have spatial variations of the mater-