13 Theoretical Generalization of the Optical Chirality to Arbitrary Optical Media
325
where ε 0 and μ 0 are, respectively, the permittivity and permeability of the vacuum, and E = E(r, t) and H = H(r, t) are the local, time-dependent electric and
magnetic fields. This definition for the optical chirality as given here, in the timedependent representation, has been successfully used in enhanced circular dichroism
spectroscopic measurements for the experimental detection and characterization of
chiral biomolecules [20], thus confirming its physical significance, and highlighting its feasibility for practical applications. Notwithstanding, its applicability can
be highly questionable beyond the simplest case of the vacuum, even further when
used for looking into chiral effects either in metallic nanostructures or metamaterials, often regarded as the paradigmatic examples of dispersive media. That in itself
is reason enough for wanting to address a generalization of such a quantity. Furthermore, recent advances in nanofabrication are opening up new possibilities for
the experimental measurement and investigation of dynamical properties such as the
EM energy-momentum, the optical orbital and spin angular momentum, and the EM
helicity, so far only accessible theoretically.
2 This fact has likely led to reexamine both
the theoretical treatment and the formulation of these dynamical properties taking
into account their dispersive features [39–42]. As for the optical chirality, it should
be noted that there are also few works attempting to extend its original definition to
arbitrarily defined linear [19], gyrotropic [43], or dispersive media [44]. However, as
will be shown throughout this chapter, at the anomalous dispersion region, i.e., the
spectral range where the real part of the permittivity decreases abruptly, or, to put it
simply, near the resonance frequencies, the optical chirality can be strongly enhanced.
This evidences the important role played by the absorption losses of a given medium
in the analysis of the optical response by a dispersive material,
3 in particular, as far
as chiral light-matter interactions are concerned. Hence, a full description of optical
chirality in dispersive media, extending it to account for the medium’s dissipative
effects as well would be highly valuable and enlightening.
There are many different ways in which one might want to conduct such a generalization; either from phenomenological aspects, just by fitting experimental outcomes or via arguments stemming from the chiroptical responses; or directly through
a thorough theoretical analysis. The latter will be the approach followed up herein,
specifically, we shall take advantage of the underlying mathematical structure of
the corresponding continuity equation (or conservation law) for the optical chirality
2 In this regard, it is noteworthy to mention the so-called Abraham-Minkowski dilemma, a longstanding problem concerning with an ambiguity that arises from the real definition of the linear and
angular momentum for optical radiation in media [34]. Even though there are a number of influential
papers claiming to have solved it (see, e.g., [35, 36]), this challenging problem still remains as a
subject of current interest and debate [37, 38].
3 Notice that, strictly speaking, dispersion is necessarily tied to dissipation. This connection is well
established by the so-called Kramers-Kronig relations [28], according to which the real and imaginary parts of the material parameters, i.e., the electric permittivity and the magnetic permeability
(ε(ω) = ε + iε and μ(ω) = μ + iμ ), appear to be coupled together. In addition, it has been
demonstrated that Kramers-Kronig relations underpin the fundamental principle of causality [45],
and hence, initial assumptions regarding dispersion and dissipation have to be carefully considered,
otherwise they may lead to misleading outcomes. Still, one can find many cases where is assumed
a dispersive medium neglecting the losses.
325
where ε 0 and μ 0 are, respectively, the permittivity and permeability of the vacuum, and E = E(r, t) and H = H(r, t) are the local, time-dependent electric and
magnetic fields. This definition for the optical chirality as given here, in the timedependent representation, has been successfully used in enhanced circular dichroism
spectroscopic measurements for the experimental detection and characterization of
chiral biomolecules [20], thus confirming its physical significance, and highlighting its feasibility for practical applications. Notwithstanding, its applicability can
be highly questionable beyond the simplest case of the vacuum, even further when
used for looking into chiral effects either in metallic nanostructures or metamaterials, often regarded as the paradigmatic examples of dispersive media. That in itself
is reason enough for wanting to address a generalization of such a quantity. Furthermore, recent advances in nanofabrication are opening up new possibilities for
the experimental measurement and investigation of dynamical properties such as the
EM energy-momentum, the optical orbital and spin angular momentum, and the EM
helicity, so far only accessible theoretically.
2 This fact has likely led to reexamine both
the theoretical treatment and the formulation of these dynamical properties taking
into account their dispersive features [39–42]. As for the optical chirality, it should
be noted that there are also few works attempting to extend its original definition to
arbitrarily defined linear [19], gyrotropic [43], or dispersive media [44]. However, as
will be shown throughout this chapter, at the anomalous dispersion region, i.e., the
spectral range where the real part of the permittivity decreases abruptly, or, to put it
simply, near the resonance frequencies, the optical chirality can be strongly enhanced.
This evidences the important role played by the absorption losses of a given medium
in the analysis of the optical response by a dispersive material,
3 in particular, as far
as chiral light-matter interactions are concerned. Hence, a full description of optical
chirality in dispersive media, extending it to account for the medium’s dissipative
effects as well would be highly valuable and enlightening.
There are many different ways in which one might want to conduct such a generalization; either from phenomenological aspects, just by fitting experimental outcomes or via arguments stemming from the chiroptical responses; or directly through
a thorough theoretical analysis. The latter will be the approach followed up herein,
specifically, we shall take advantage of the underlying mathematical structure of
the corresponding continuity equation (or conservation law) for the optical chirality
2 In this regard, it is noteworthy to mention the so-called Abraham-Minkowski dilemma, a longstanding problem concerning with an ambiguity that arises from the real definition of the linear and
angular momentum for optical radiation in media [34]. Even though there are a number of influential
papers claiming to have solved it (see, e.g., [35, 36]), this challenging problem still remains as a
subject of current interest and debate [37, 38].
3 Notice that, strictly speaking, dispersion is necessarily tied to dissipation. This connection is well
established by the so-called Kramers-Kronig relations [28], according to which the real and imaginary parts of the material parameters, i.e., the electric permittivity and the magnetic permeability
(ε(ω) = ε + iε and μ(ω) = μ + iμ ), appear to be coupled together. In addition, it has been
demonstrated that Kramers-Kronig relations underpin the fundamental principle of causality [45],
and hence, initial assumptions regarding dispersion and dissipation have to be carefully considered,
otherwise they may lead to misleading outcomes. Still, one can find many cases where is assumed
a dispersive medium neglecting the losses.