9 Symmetry Approach to Chiral Optomagnonics in Antiferromagnetic Insulators
217
Lifted degeneracy between left ( p − ) and right ( p + ) polarized eigenmodes in
(9.34) leads to the reduction of the eight-dimensional invariance algebra to four
basis elements, which commute with each other
Q 2 = iσ 2 ⊗ ˆ
I , Q 5 = σ 0 ⊗ ( ˆ
S · ˜
p)
Q 7 = σ 0 ⊗ ˆ
I , Q 8 = iσ 2 ⊗ ( ˆ
S · ˜
p).
(9.35)
These symmetries, however, still contain the duality transformation Q 2 , which means
that the medium is dual-symmetric and supports the conservation of the electromagnetic helicity [49] and, as a consequence, optical zilches.
Definition of the optical chirality density in chiral media requires some attention.
This situation is similar to the definition of the electromagnetic energy density where
one should take care of the continuity of the energy flow at the boundary between two
chiral media [51]. It can be demonstrated that chirality density in the medium with
constituent relations (9.31) and (9.32) that provides continuity of chirality flow at
the boundary of two chiral media with different κ can be introduced in the following
way [50]
ρ χ =
εε 0
2
B
∗
·
∂ E
∂t
−
μμ 0
2
D
∗
·
∂ H
∂t
,
(9.36)
This expression remains valid even if ε(r), μ(r), and κ(r) become position dependent. In this case, it satisfies the continuity equation with a source term on the
right-hand side
∂ t ρ χ + ∇ · J χ = F(t, r).
(9.37)
where
J χ =
ε 0 ε
2
E
∗
× ∂ t E +
μ 0 μ
2
H
∗
× ∂ t H,
(9.38)
and the source term contains only gradients of ε and μ, but does not depend on the
gradient of κ
F(t, r) =
ε 0
2
∇ε · E
∗
× ∂ t E +
μ 0
2
∇μ · H
∗
× ∂ t H.
(9.39)
In order to understand the physical meaning of ρ χ , let us look at energy absorption
in a dissipative gyrotropic medium with the constituent relations (9.31) and (9.32).
As was demonstrated in [27], the electromagnetic energy absorption rate in this
case has an asymmetric part, which has opposite signs for left and right polarized
electromagnetic waves. This part is proportional the product between the chirality
of the material, given by the imaginary part of κ, and the chirality density of the
electromagnetic field ρ χ . The flow of optical chirality in (9.3), in this situation, can
be associated with the asymmetric components of the electromagnetic forces in the
medium, which can be used, for example, for optical separation of chiral molecules
[37].
217
Lifted degeneracy between left ( p − ) and right ( p + ) polarized eigenmodes in
(9.34) leads to the reduction of the eight-dimensional invariance algebra to four
basis elements, which commute with each other
Q 2 = iσ 2 ⊗ ˆ
I , Q 5 = σ 0 ⊗ ( ˆ
S · ˜
p)
Q 7 = σ 0 ⊗ ˆ
I , Q 8 = iσ 2 ⊗ ( ˆ
S · ˜
p).
(9.35)
These symmetries, however, still contain the duality transformation Q 2 , which means
that the medium is dual-symmetric and supports the conservation of the electromagnetic helicity [49] and, as a consequence, optical zilches.
Definition of the optical chirality density in chiral media requires some attention.
This situation is similar to the definition of the electromagnetic energy density where
one should take care of the continuity of the energy flow at the boundary between two
chiral media [51]. It can be demonstrated that chirality density in the medium with
constituent relations (9.31) and (9.32) that provides continuity of chirality flow at
the boundary of two chiral media with different κ can be introduced in the following
way [50]
ρ χ =
εε 0
2
B
∗
·
∂ E
∂t
−
μμ 0
2
D
∗
·
∂ H
∂t
,
(9.36)
This expression remains valid even if ε(r), μ(r), and κ(r) become position dependent. In this case, it satisfies the continuity equation with a source term on the
right-hand side
∂ t ρ χ + ∇ · J χ = F(t, r).
(9.37)
where
J χ =
ε 0 ε
2
E
∗
× ∂ t E +
μ 0 μ
2
H
∗
× ∂ t H,
(9.38)
and the source term contains only gradients of ε and μ, but does not depend on the
gradient of κ
F(t, r) =
ε 0
2
∇ε · E
∗
× ∂ t E +
μ 0
2
∇μ · H
∗
× ∂ t H.
(9.39)
In order to understand the physical meaning of ρ χ , let us look at energy absorption
in a dissipative gyrotropic medium with the constituent relations (9.31) and (9.32).
As was demonstrated in [27], the electromagnetic energy absorption rate in this
case has an asymmetric part, which has opposite signs for left and right polarized
electromagnetic waves. This part is proportional the product between the chirality
of the material, given by the imaginary part of κ, and the chirality density of the
electromagnetic field ρ χ . The flow of optical chirality in (9.3), in this situation, can
be associated with the asymmetric components of the electromagnetic forces in the
medium, which can be used, for example, for optical separation of chiral molecules
[37].