6 Second-Order Nonlinear Optical Properties of Plasmonic Nanostructures
223
however, the particle geometry is usually more complicated [41, 59, 83] and such
methods, relying on the separation of variables, are not feasible. Most numerical
methods then divide the solution domain into simple primitives, such as n-simplexes,
and represent the fields by some low-degree polynomials [95]. In addition, plasmon
resonances usually lead to field variations on a sub-wavelength scale, which makes
accurate field representation a necessity [83]. The resonances can also render the
problem sensitive, so that small variations in the excitation (also due to round-off
errors) can give rise to abrupt changes in the solution.
In plasmonic scattering problems, the division of methods into time-domain and
frequency-domain methods is also crucial. This is because the material properties
are highly frequency-dependent [1] and taking into account the dispersive material
response in the time-domain requires either time-consuming evaluation of the response convolution or the use of simplified models obtained from first-principles
description of the response. Most often, frequency-domain descriptions are more
convenient, because they offer more flexibility. If desired, the frequency-dependent
material parameters can be calculated from first principles. On the other hand, the
parameters can also be described phenomenologically or obtained directly from experiments.
These challenges are further augmented for nonlinear problems. The second-order
response can originate from the dipolar response at material surfaces [96], requiring
accurate representation of the surfaces. The response, however, can also arise from
the atomic-level higher multipoles in the bulk medium [97]. Consequently, a volume
source may exist also in the case where the solution domain is divided into parts
homogeneous in material properties. This source involves derivatives of the fields,
which can be problematic for numerical schemes near the interfaces between different
materials.
6.8.2 General Approach for Nonlinear Problems
The three-dimensional electromagnetic scattering problem is described in the
frequency-domain by the vectorial Helmholtz equations for the electric and magnetic fields and by the Silver-Müller radiation conditions [98]. To study SHG in
the undepleted-pump approximation, one can then first solve the fundamental fields
described by source-free Helmholtz equations, and then solve the second-harmonic
fields described by Helmholtz equations with a polarization source, which is related
to the fundamental fields by the second-order susceptibility.
The nanostructure is often composed of piece-wise homogeneous media and it is
most convenient to divide the domain of the boundary value problem accordingly. The
electromagnetic interface conditions are posed at the resulting domain interfaces. In
traditional linear problems, the tangential components of the electric and magnetic
fields are continuous at the interfaces. In the presence of a dipolar second-order
surface source, the tangential fields have jump discontinuities [24]. The nonlinear
source is then invoked via the interface conditions.
223
however, the particle geometry is usually more complicated [41, 59, 83] and such
methods, relying on the separation of variables, are not feasible. Most numerical
methods then divide the solution domain into simple primitives, such as n-simplexes,
and represent the fields by some low-degree polynomials [95]. In addition, plasmon
resonances usually lead to field variations on a sub-wavelength scale, which makes
accurate field representation a necessity [83]. The resonances can also render the
problem sensitive, so that small variations in the excitation (also due to round-off
errors) can give rise to abrupt changes in the solution.
In plasmonic scattering problems, the division of methods into time-domain and
frequency-domain methods is also crucial. This is because the material properties
are highly frequency-dependent [1] and taking into account the dispersive material
response in the time-domain requires either time-consuming evaluation of the response convolution or the use of simplified models obtained from first-principles
description of the response. Most often, frequency-domain descriptions are more
convenient, because they offer more flexibility. If desired, the frequency-dependent
material parameters can be calculated from first principles. On the other hand, the
parameters can also be described phenomenologically or obtained directly from experiments.
These challenges are further augmented for nonlinear problems. The second-order
response can originate from the dipolar response at material surfaces [96], requiring
accurate representation of the surfaces. The response, however, can also arise from
the atomic-level higher multipoles in the bulk medium [97]. Consequently, a volume
source may exist also in the case where the solution domain is divided into parts
homogeneous in material properties. This source involves derivatives of the fields,
which can be problematic for numerical schemes near the interfaces between different
materials.
6.8.2 General Approach for Nonlinear Problems
The three-dimensional electromagnetic scattering problem is described in the
frequency-domain by the vectorial Helmholtz equations for the electric and magnetic fields and by the Silver-Müller radiation conditions [98]. To study SHG in
the undepleted-pump approximation, one can then first solve the fundamental fields
described by source-free Helmholtz equations, and then solve the second-harmonic
fields described by Helmholtz equations with a polarization source, which is related
to the fundamental fields by the second-order susceptibility.
The nanostructure is often composed of piece-wise homogeneous media and it is
most convenient to divide the domain of the boundary value problem accordingly. The
electromagnetic interface conditions are posed at the resulting domain interfaces. In
traditional linear problems, the tangential components of the electric and magnetic
fields are continuous at the interfaces. In the presence of a dipolar second-order
surface source, the tangential fields have jump discontinuities [24]. The nonlinear
source is then invoked via the interface conditions.