13 Theoretical Generalization of the Optical Chirality to Arbitrary Optical Media
337
13.3 Generalizing the Conservation Law for the Optical
Chirality
As we have just seen, conservation laws, in particular, that for the EM energy, can
be straightforwardly treated from its corresponding continuity equation. From this
viewpoint, we have shown an insightful manner of identifying the corresponding
conserved quantity, the EM energy density. Yet, this task of digging out conserved
quantities typically relies upon the search of the underlying symmetry properties of
the physical system [75]. Indeed, as established by the Noether’s theorem, conserved
quantities and symmetries can be regarded as equivalent features [76]. For example,
the well-known conservation laws of energy, linear momentum and angular momentum, are actually associated with continuous symmetries of the system, namely, they
follows from the invariance under the universal space-time transformations [77].
These theoretical concepts can be mathematically described by means of symmetry
groups, which are in turn related to the corresponding physical transformations [78].
Furthermore, drawing on a formalism resembling that often used in quantum mechanics [79], one can deal with the conserved quantities through differential operators
representing the generators of the corresponding infinitesimal symmetry transformations. For the above dynamical properties, i.e., the energy, the linear momentum,
and the orbital angular momentum, these generators simply involve first derivatives
with respect to the space-time coordinates acting on the EM fields, and are given by
{i∂ t , i∇}, for the space-time translations [39], and i (r × ∇), for the spatial rotations
[40]. As for the optical chirality, it was demonstrated that its conservation is underpinned by i (∂ t ∇×) [44], but noticing that it must be applied to the vector potentials
[51]. Importantly, these generators allow us to find the corresponding eigenstates
associated to the conserved quantities. In this regard, one can find that the plane
waves are the eigenstates of the energy-momentum differential operator. Similarly,
for the optical chirality, it is found that the associated eigenstates are the circularly
polarized plane waves [44].
Noether’s theorem, therefore, constitutes a powerful tool for identifying and analyzing conserved quantities from the symmetries of the physical system. However,
this approach only holds in the absence of external sources. In the presence of sources
(i.e., external charges and/or currents), though, conservation laws and the subsequent
identification and analysis of the conserved quantities ought to be addressed through
the continuity equations [51, 52]. This would be the case if we wanted to analyze
a given dynamical property in a dispersive medium, wherein we also include the
presence of absorption losses. Hence, by building upon the sound and self-consistent
framework described above for the EM energy conservation law, we are now ready
to proceed with the generalization of the conservation law for the optical chirality.
Akin to the EM energy continuity equation, the procedure to derive the most
general form of the optical chirality conservation law starts from the definition of
the corresponding optical chirality flux density. In this regard, it is worth recalling
that the literature concerning optical chirality and its interaction with matter has
mostly dealt with EM fields in free space [31–33]. That is why one can often find
337
13.3 Generalizing the Conservation Law for the Optical
Chirality
As we have just seen, conservation laws, in particular, that for the EM energy, can
be straightforwardly treated from its corresponding continuity equation. From this
viewpoint, we have shown an insightful manner of identifying the corresponding
conserved quantity, the EM energy density. Yet, this task of digging out conserved
quantities typically relies upon the search of the underlying symmetry properties of
the physical system [75]. Indeed, as established by the Noether’s theorem, conserved
quantities and symmetries can be regarded as equivalent features [76]. For example,
the well-known conservation laws of energy, linear momentum and angular momentum, are actually associated with continuous symmetries of the system, namely, they
follows from the invariance under the universal space-time transformations [77].
These theoretical concepts can be mathematically described by means of symmetry
groups, which are in turn related to the corresponding physical transformations [78].
Furthermore, drawing on a formalism resembling that often used in quantum mechanics [79], one can deal with the conserved quantities through differential operators
representing the generators of the corresponding infinitesimal symmetry transformations. For the above dynamical properties, i.e., the energy, the linear momentum,
and the orbital angular momentum, these generators simply involve first derivatives
with respect to the space-time coordinates acting on the EM fields, and are given by
{i∂ t , i∇}, for the space-time translations [39], and i (r × ∇), for the spatial rotations
[40]. As for the optical chirality, it was demonstrated that its conservation is underpinned by i (∂ t ∇×) [44], but noticing that it must be applied to the vector potentials
[51]. Importantly, these generators allow us to find the corresponding eigenstates
associated to the conserved quantities. In this regard, one can find that the plane
waves are the eigenstates of the energy-momentum differential operator. Similarly,
for the optical chirality, it is found that the associated eigenstates are the circularly
polarized plane waves [44].
Noether’s theorem, therefore, constitutes a powerful tool for identifying and analyzing conserved quantities from the symmetries of the physical system. However,
this approach only holds in the absence of external sources. In the presence of sources
(i.e., external charges and/or currents), though, conservation laws and the subsequent
identification and analysis of the conserved quantities ought to be addressed through
the continuity equations [51, 52]. This would be the case if we wanted to analyze
a given dynamical property in a dispersive medium, wherein we also include the
presence of absorption losses. Hence, by building upon the sound and self-consistent
framework described above for the EM energy conservation law, we are now ready
to proceed with the generalization of the conservation law for the optical chirality.
Akin to the EM energy continuity equation, the procedure to derive the most
general form of the optical chirality conservation law starts from the definition of
the corresponding optical chirality flux density. In this regard, it is worth recalling
that the literature concerning optical chirality and its interaction with matter has
mostly dealt with EM fields in free space [31–33]. That is why one can often find