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M. Kauranen et al.
In geometries, where the Helmholtz equation is not separable, more general, but in
certain sense approximate, methods must be employed. The numerical methods can
be divided into differential-operator and integral-operator methods. In the latter, the
Helmholtz equations are first transformed into integral equations by the use of Green’s
functions [99, 100]. This allows one to impose the radiation conditions analytically
and the solution can be represented in a compact subset of the original domain.
Although the integral-operator approach is not generally applicable to nonlinear
problems, it can be used to model harmonic generation within the undepleted pump
approximation.
6.8.3 Recent Approaches for Modeling SHG
The second-harmonic scattering problem has been addressed only recently in general geometries, although it has been treated in simple geometries for a long time.
SHG from plane surfaces and stratified media has been done using the plane-wave
Green’s function formalism [101]. Recently, SHG from nanodefects on a planar surface has also been modeled [102, 103]. Multipole solution of SHG from spherical
particles has been derived for arbitrary size parameters and especially the properties
of spheres with a small size parameter have been studied [104–106]. Multipole expansion has also been used to model sum-frequency generation in spheres [107] and to
model SHG from collections of spheres [108]. The multiple scattering matrix method
has been applied for modeling SHG from cylindrical nanowires and from photonic
crystals [109, 110]. For particles whose permittivity is close to that of the surrounding
medium, the Rayligh-Gans-Debye approximation provides a good approximation,
also for modeling SHG [111].
SHG from metal nanostructure arrays has been modeled by using the finitedifference time-domain method [60, 112] and the Fourier modal method [113, 114].
The finite element method has been used to study second-harmonic scattering and
multipolar response from particles that slightly differ from spheres [31, 32].
In recent years, the integral-operator methods have gained popularity due to rapid
developments in the theory and since computational power has increased. The volume integral method has been applied to model second-harmonic microscopy of
nanoparticles [115]. Recently it has also been applied to model SHG from nanowires,
nanoantennas and from nano-resonator chains [116, 117]. We have recently applied
the surface integral-operator boundary-element method (BEM) for modeling SHG
from arbitrary nanoparticles [118], which will be described in the following section.
6.8.4 Boundary-Element Method
BEM has been widely used for modeling scattering from ideally conducting and
dielectric bodies as well as in antenna problems. More recently, it has also gained
M. Kauranen et al.
In geometries, where the Helmholtz equation is not separable, more general, but in
certain sense approximate, methods must be employed. The numerical methods can
be divided into differential-operator and integral-operator methods. In the latter, the
Helmholtz equations are first transformed into integral equations by the use of Green’s
functions [99, 100]. This allows one to impose the radiation conditions analytically
and the solution can be represented in a compact subset of the original domain.
Although the integral-operator approach is not generally applicable to nonlinear
problems, it can be used to model harmonic generation within the undepleted pump
approximation.
6.8.3 Recent Approaches for Modeling SHG
The second-harmonic scattering problem has been addressed only recently in general geometries, although it has been treated in simple geometries for a long time.
SHG from plane surfaces and stratified media has been done using the plane-wave
Green’s function formalism [101]. Recently, SHG from nanodefects on a planar surface has also been modeled [102, 103]. Multipole solution of SHG from spherical
particles has been derived for arbitrary size parameters and especially the properties
of spheres with a small size parameter have been studied [104–106]. Multipole expansion has also been used to model sum-frequency generation in spheres [107] and to
model SHG from collections of spheres [108]. The multiple scattering matrix method
has been applied for modeling SHG from cylindrical nanowires and from photonic
crystals [109, 110]. For particles whose permittivity is close to that of the surrounding
medium, the Rayligh-Gans-Debye approximation provides a good approximation,
also for modeling SHG [111].
SHG from metal nanostructure arrays has been modeled by using the finitedifference time-domain method [60, 112] and the Fourier modal method [113, 114].
The finite element method has been used to study second-harmonic scattering and
multipolar response from particles that slightly differ from spheres [31, 32].
In recent years, the integral-operator methods have gained popularity due to rapid
developments in the theory and since computational power has increased. The volume integral method has been applied to model second-harmonic microscopy of
nanoparticles [115]. Recently it has also been applied to model SHG from nanowires,
nanoantennas and from nano-resonator chains [116, 117]. We have recently applied
the surface integral-operator boundary-element method (BEM) for modeling SHG
from arbitrary nanoparticles [118], which will be described in the following section.
6.8.4 Boundary-Element Method
BEM has been widely used for modeling scattering from ideally conducting and
dielectric bodies as well as in antenna problems. More recently, it has also gained