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Continuous symmetries and
conservation laws in chiral media
Crimin, Frances, Mackinnon, Neel, Götte, Jörg, Barnett,
Stephen
Frances Crimin, Neel Mackinnon, Jörg Götte, Stephen M. Barnett,
"Continuous symmetries and conservation laws in chiral media," Proc. SPIE
11297, Complex Light and Optical Forces XIV, 112970J (24 February 2020);
doi: 10.1117/12.2550815
Event: SPIE OPTO, 2020, San Francisco, California, United States
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Continuous symmetries and conservation laws in chiral media
Frances Crimin
a
, Neel Mackinnon
a
, J¨org B. G¨otte
a,b
, and Stephen M. Barnett
a
a
Department of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, UK
b
College of Engineering and Applied Science, Nanjing University, Nanjing 210093, China
ABSTRACT
Locally conserved quantities of the electromagnetic field in lossless chiral media are derived from Noether’s
theorem, including helicity, chirality, momentum, and angular momentum, as well as the separate spin and orbital
components of this last quantity. We discuss sources and sinks of each in the presence of current densities within
the material, and in some cases, as also generated by inhomogeneity of the medium. A previously obtained result
connecting sources of helicity and energy within chiral materials is explored, revealing that association between
the two quantities is not restricted to chiral media alone. Rather, it is analogous to the connection between the
momentum, and the spin and orbital components of the total angular momentum. The analysis reveals a new
quantity, appearing as the “orbital” counterpart of the helicity density in classical electromagnetism.
Keywords: Chiral media, conserved quantities, Noether’s theorem, optical angular momentum, gauge transfor-
mations
1. INTRODUCTION
The propagation of electromagnetic fields in chiral media has long been a topic of interest in classical elec-
tromagnetism.
1–4
In the simplest case, a chiral medium is a macroscopic, continuous material made up of
indistinguishable chiral objects that are distributed homogeneously throughout, but with random orientation. A
chiral object is one which cannot be superimposed upon its mirror image by rotation or translation: such objects
exhibit handedness. When a linearly polarised field passes through a chiral medium, the left- and right-circularly
polarised components of the field travel at different group velocities,
4, 5
so that, upon leaving the medium and
recombining, the polarisation of the transmitted field is rotated with respect to the incident one.
5–8
The angle,
θ, through which this polarisation is rotated, differs in sign depending upon whether the constituent objects in
the medium are left- or right-handed. In this way, chiral matter demonstrates handedness.
The simplest example of a chiral object is a helix, a geometric figure which is also traced by the electric and
magnetic field vector of circularly polarised light. There has been a recent resurgence of activity in the study
of the chiral content of light,
9–13
and the optical helicity
11, 14–20
has been advocated as an appropriate measure
of this.
11, 13, 21
It has been shown
22
that the helicity density, in particular, takes on a peculiar form in chiral
media, acquiring an additional term which is proportional to the energy density, with a sign dependent upon the
handedness of the material: sources of energy therefore become equivalent to sources of helicity in such media.
This was previously observed by Lakhtakia et al.,
23
who found that the far-field radiation emitted by a point
electric dipole in a chiral sphere is identical to that of an electric-magnetic dipole pair radiating into vacuum,
which itself is a simple model of a source of helicity.
21, 24
The relationship between the helicity and the energy
thus warrants further investigation, and so this curious result prompts us to consider whether such a relationship
holds between other conserved quantities in chiral media, particularly those naturally associated with the angular
momentum content of the field, in the hope that such findings may shed some light upon it.
In this article, we derive the form of some locally conserved quantities in chiral media from Noether’s theorem,
including the helicity, chirality, momentum and (spin and orbital) angular momentum densities. The forms of
sources of these quantities in the presence of a local current are found, and the results discussed. Throughout,
we consider only homogeneous, isotropic, and lossless media, unless otherwise stated.
Further author information: (Send correspondence to F.C.)
F.C.: E-mail: frances.crimin@glasgow.ac.uk
Complex Light and Optical Forces XIV, edited by David L. Andrews, Enrique J. Galvez,
Halina Rubinsztein-Dunlop, Proc. of SPIE Vol. 11297, 112970J · © 2020 SPIE
CCC code: 0277-786X/20/$21 · doi: 10.1117/12.2550815
Proc. of SPIE Vol. 11297 112970J-1
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2. THE ENERGY DENSITY AND LAGRANGIAN DENSITY IN CHIRAL MEDIA
In this section, we use the known form of the energy density in chiral media
25, 26
to postulate the form of the
Lagrangian density, and then go on test this quantity by various means. The energy density in a chiral medium
can be written in the exact form
25, 26
w
β
=
1
2
1
D · D +
1
µ
B · B
. (1)
We use the Drude-Born-Fedorov (DBF) constitutive relations,
27
where the electric and magnetic inductions D
and B have the form
D = (E + β∇ × E),
B = µ(H + β∇ × H). (2)
Here, the pseudoscalar β describes the chiroptical response of the medium, with and µ acting as the permittivity
and permeability. Using (2), it is straightforward to check the local conservation of the energy density (1). In
the presence of a current density j, we obtain
∂
t
w
β
+ ∇ ·(E ×H) = −
1
j · D. (3)
Expansion of the energy density (1) using (2) highlights its peculiar form in a chiral medium: for each of the
electric and magnetic contributions, it contains two terms which do not depend on the handedness of the material,
and one, E ·(∇ ×E) and H ·(∇ ×H), which does, as it is proportional to β. This structure, the dot product of
a field with its curl, is echoed by the form of both the optical chirality density,
28
which in vacuum is given by
χ =
0
2
h
E · (∇ × E) + c
2
B · (∇ × B)
i
, (4)
and the helicity density in vacuum
11, 18
h =
1
2
"
r
0
µ
0
A
⊥
· (∇ ×A) +
r
µ
0
0
C
⊥
· (∇ ×C)
#
, (5)
where the transverse part of C defines the transverse part of the displacement field, D
⊥
= −∇ × C.
29, 30
This
already hints at the connection between measures of the chiral content of a field and the energy density,
22
which
only becomes apparent if the medium is chiral.
From (1), we introduce the following form of the Lagrangian density of the free field in a chiral medium:
L =
1
2
1
D · D −
1
µ
B · B
. (6)
Again using the DBF constitutive relations, this is written in terms of the vector and scalar coordinates of the
electromagnetic potential, A and A
0
, in an arbitrary gauge:
L =
1
2
˙
A + ∇A
0
+ β∇ ×
˙
A
2
−
1
µ
(∇ × A)
2
. (7)
The momentum conjugate to A is
Π =
∂L
∂
˙
A
= −(1 + β∇×)D, (8)
and the form of (6) may be verified by the resulting Euler-Lagrange equations, since
d
dt
∂L
∂
˙
A
= −(1 + β∇×)
˙
D (9)
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and
∂L
∂A
= −(1 + β∇×)∇ × H, (10)
such that the Maxwell equation
˙
D = ∇ × H is recovered. It is worthwhile to note that the Lagrangian density
(6) can also be written in terms of the electric scalar and vector potentials, C
0
and C. Following the above
treatment for the generalised coordinate A, the Euler-Lagrange equations in the present case reveal Faraday’s
law,
˙
B = −∇ × E. As a further test of (6), we find that the energy density (1) can be recovered by finding
H ≡ w
β
= Π
˙
A − L. Analogous results follow using the coordinate C and associated momentum.
3. CONSERVED QUANTITIES
We wish to test whether various electromagnetic quantities, conserved in vacuum, are also conserved in chiral
media, and to find the explicit form of these quantities. To this end, we apply the infinitesimal transformation
known to generate each conserved quantitiy in vacuum to the Lagrangian density (6). Should it remain invariant,
we then use Noether’s theorem
31
to find the form of these quantities in chiral media.
3.1 The helicity density in a chiral medium
It is now well-known that the duality transformation is generated by the conservation of helicity in vacuum,
16, 17
and in lossless media in which dual-symmetry is preserved.
19, 22
The form of the helicity density in chiral media
has been found previously,
21, 22
by considering its continuity equation directly. Here, we verify these results, and
indeed the form of the Lagrangian density (6), by deriving it instead from Noether’s theorem. We proceed by
applying the infinitesimal form of the duality transformation
32
A → A + θ
r
µ
C,
C → C − θ
r
µ
A, (11)
to (6). Verifying that the Euler-Lagrange equations still hold upon this transformation, it is straightforward to
find:
∂L
∂
˙
A
δA +
∂L
∂
˙
C
δC = θ
−
r
µ
C · (1 + β∇×) D +
r
µ
A · (1 + β∇×) B
, (12)
such that the conserved quantity is given in the square brackets. After integration by parts, setting
Z
∇
i
r
µ
(A × B)
i
−
r
µ
(C × D)
i
d
3
r ≡ 0, (13)
and dividing the whole quantity by a factor of two, the helicity density in a chiral medium is identified:
h
β
=
1
2
r
µ
B · (1 + β∇×) A
⊥
−
r
µ
D · (1 + β∇×) C
⊥
=
1
2
r
µ
A
⊥
· B −
r
µ
C
⊥
· D
+ β
√
µ
1
2
1
µ
B · B +
1
D · D
. (14)
We have further specified that only the transverse parts of the vector potentials are included in (14): this enables
the helicity density to be written in a manifestly gauge-independent form,
16
and indeed, these are the only parts
of the vector potentials picked out by an integration over all space in calculating the total helicity. Eq. (14) is in
agreement with the form of the helicity density in a chiral medium found in earlier work.
22
It is worthwhile to
examine this equation in the presence of a current density j within the medium. Using the helicity flux density
v =
1
2
r
µ
E × A
⊥
+
r
µ
H × C
⊥
, (15)
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and defining j
⊥
= ∇ × g,
18
the continuity equation for the helicity density produces
∂
t
h
β
+ ∇ ·v =
1
2
r
µ
[g · (∇ × C) + C · (∇ × g)] −
r
µ
βj
⊥
· D. (16)
This is exact to all O(β), i.e. there are no higher order source terms as previously expected.
21
This result is
comparable to the analogous continuity equation for the energy density, as shown in (3).
3.2 Chirality density in a chiral medium
Following the above treatment, we proceed to find the form of other well-known quantities of the electromagnetic
field in a chiral medium, beginning with the chirality density. We consider the following infinitesimal transfor-
mation of the magnetic vector potentials A and C, known to be genrated by the conservation of chirality in
vacuum:
33
A → A + η∇ ×
˙
A,
C → C + η∇ ×
˙
C. (17)
It is straightforward to check by substitution into the macroscopic Maxwell equations that these constitute a
symmetry transformation of the system only when the values of µ, , and β are constant. We would therefore
expect chirality, and not helicity,
19
to be generated at the interface between vacuum and a dual-symmetric chiral
medium where the ratio /µ is constant, but the separate permittivity and permeability may vary. In analogy
with the treatment for the helicity density above, we identify, up to surface terms, the following conserved
quantity as the chirality density:
χ
β
=
1
2
1
D · (∇ × D) +
1
µ
B · (∇ × B)
. (18)
We proceed to find an exact expression for the source terms on the right-hand side of the chirality continuity
equation (up to surface terms) in the presence of a current density j:
∂
t
χ
β
+
1
2
∇ · [E × (∇ × H) − H × (∇ × E)] = −
1
2
[j · (∇ × E) + E · (∇ × j)] − β(∇ × j) · (∇ × E). (19)
Setting β = 0 reduces this result to that obtained for achiral media,
18, 34
and it is clear that it parallels the result
for the helicity density (16).
3.3 The canonical and kinetic momentum density in a chiral medium
The conservation of electromagnetic momentum is associated with symmetry under translation of the vector
potentials:
35
A(r, t) → A(r + η, t),
C(r, t) → C(r + η, t). (20)
We Taylor expand to first order, to write
A(r, t) → A(r, t) + (η ·∇)A(r, t),
C(r, t) → C(r, t) + (η ·∇)C(r, t), (21)
which is equivalent to
A
i
→ A
i
+ η
j
∇
j
A
i
,
C
i
→ C
i
+ η
j
∇
j
C
i
, (22)
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