13 Theoretical Generalization of the Optical Chirality to Arbitrary Optical Media
347
media, such as plasmonic nanostructures, and metamaterials [1]. It is important to
realize that this expression differs significantly from the standard formula for fields
in free space [31, 32], and even from other derivations claiming to work out the
optical chirality in dispersive media [44, 80]. The main difference essentially raises
on account of considering properly the dynamic response of the time-dependent EM
fields within a dispersive medium. In this regard, it should be noted that the time
derivative of the fields D and B, must be expressed as convolution integrals in the
time domain [28–30], as pointed out in (13.11a) and (13.11b). Notwithstanding the
foregoing, departing from this general expression (13.68), it is, of course, possible to
particularize it to the lossless case (13.62), as well as to the original definition for freespace EM fields (13.52), just by relaxing successively the corresponding conditions,
i.e., by imposing γ n = ˜
γ n = 0, and ε(ω) = ε and μ(ω) = μ, respectively. Hence,
it can be concluded that this approach, relying upon the underlying mathematical
structure of the continuity equation, leads to a sound and self-consistent definition
for the optical chirality density in dispersive and lossy media [46, 48].
13.4.3 Brillouin’s Approach Vs Loudon’s Approach
The classical approaches put forward by Brillouin and Landau enable one to obtain
a closed expression for the EM energy density, (13.37a) and (13.37b), and likewise
for the optical chirality density, (13.62), which are only valid under the slowly varying amplitude approximation [29], i.e., in a relatively narrow frequency range where
the effects of material absorption can be considered as negligible. Therefore, those
expressions are suitable for describing such dynamical properties solely in lossless
dispersive media. In order to further include the dissipation, one should perform a
careful analysis from a material standpoint. This can be carried out by means of the
corresponding dynamic equations characterizing the polarization and the magnetization fields, thereby introducing additional terms that must be explicitly accounted
for. By doing so, one finally is able to find the most general definition for the EM
energy density, (13.28a) and (13.28b), and the optical chirality density (13.68), thus
capturing analytically the effects due to dispersion, as well as the absorption, for a
complete and proper description of chiroptical interaction between light and matter.
Naturally, both approaches yield different behaviors, see Fig. 13.1. Indeed, assuming a nonmagnetic medium (i.e., such that μ = 1) whose permittivity is described by
a single Lorentz pole with ω p = ω 0 , one can observe that, whereas C lossless displays
both positive and negative values, the general expression C lossy remains always positive, with a minimum value of C vacuum , that is reached in the high-frequency limit.
Importantly, the largest discrepancies are occurring close to the resonance frequency,
as expected. Still, the peaks for both approaches are almost equal in absolute value.
These signatures are actually better appreciated in Fig. 13.2, where we compare the
dispersion-dependent features of the optical chirality density in both approaches, the
lossless and lossy cases, for silver and silicon. Both materials have been modeled
using (13.12) with parameters taken from [72] and [73] (see the specific values listed
347
media, such as plasmonic nanostructures, and metamaterials [1]. It is important to
realize that this expression differs significantly from the standard formula for fields
in free space [31, 32], and even from other derivations claiming to work out the
optical chirality in dispersive media [44, 80]. The main difference essentially raises
on account of considering properly the dynamic response of the time-dependent EM
fields within a dispersive medium. In this regard, it should be noted that the time
derivative of the fields D and B, must be expressed as convolution integrals in the
time domain [28–30], as pointed out in (13.11a) and (13.11b). Notwithstanding the
foregoing, departing from this general expression (13.68), it is, of course, possible to
particularize it to the lossless case (13.62), as well as to the original definition for freespace EM fields (13.52), just by relaxing successively the corresponding conditions,
i.e., by imposing γ n = ˜
γ n = 0, and ε(ω) = ε and μ(ω) = μ, respectively. Hence,
it can be concluded that this approach, relying upon the underlying mathematical
structure of the continuity equation, leads to a sound and self-consistent definition
for the optical chirality density in dispersive and lossy media [46, 48].
13.4.3 Brillouin’s Approach Vs Loudon’s Approach
The classical approaches put forward by Brillouin and Landau enable one to obtain
a closed expression for the EM energy density, (13.37a) and (13.37b), and likewise
for the optical chirality density, (13.62), which are only valid under the slowly varying amplitude approximation [29], i.e., in a relatively narrow frequency range where
the effects of material absorption can be considered as negligible. Therefore, those
expressions are suitable for describing such dynamical properties solely in lossless
dispersive media. In order to further include the dissipation, one should perform a
careful analysis from a material standpoint. This can be carried out by means of the
corresponding dynamic equations characterizing the polarization and the magnetization fields, thereby introducing additional terms that must be explicitly accounted
for. By doing so, one finally is able to find the most general definition for the EM
energy density, (13.28a) and (13.28b), and the optical chirality density (13.68), thus
capturing analytically the effects due to dispersion, as well as the absorption, for a
complete and proper description of chiroptical interaction between light and matter.
Naturally, both approaches yield different behaviors, see Fig. 13.1. Indeed, assuming a nonmagnetic medium (i.e., such that μ = 1) whose permittivity is described by
a single Lorentz pole with ω p = ω 0 , one can observe that, whereas C lossless displays
both positive and negative values, the general expression C lossy remains always positive, with a minimum value of C vacuum , that is reached in the high-frequency limit.
Importantly, the largest discrepancies are occurring close to the resonance frequency,
as expected. Still, the peaks for both approaches are almost equal in absolute value.
These signatures are actually better appreciated in Fig. 13.2, where we compare the
dispersion-dependent features of the optical chirality density in both approaches, the
lossless and lossy cases, for silver and silicon. Both materials have been modeled
using (13.12) with parameters taken from [72] and [73] (see the specific values listed