# Effects of CPT-odd terms of dimensions three and five on electromagnetic
propagation in continuous matter

Pedro D. S. Silvaa<EMAIL_ADDRESS>Letícia Lisboa-Santosa
<EMAIL_ADDRESS>aDepartamento de Física, Universidade Federal
do Maranhão, Campus Universitário do Bacanga, São Luís (MA), 65080-805, Brazil
Manoel M. Ferreira Jr.a<EMAIL_ADDRESS><EMAIL_ADDRESS>aDepartamento de Física, Universidade Federal do Maranhão, Campus
Universitário do Bacanga, São Luís (MA), 65080-805, Brazil  Marco Schrecka
<EMAIL_ADDRESS>aDepartamento de Física, Universidade Federal do
Maranhão, Campus Universitário do Bacanga, São Luís (MA), 65080-805, Brazil

###### Abstract

In this work we study how CPT-odd Maxwell-Carroll-Field-Jackiw (MCFJ)
electrodynamics as well as a dimension-5 extension of it affect the optical
activity of continuous media. The starting point is dimension-3 MCFJ
electrodynamics in matter whose modified Maxwell equations, permittivity
tensor, and dispersion relations are recapitulated. Corresponding refractive
indices are achieved in terms of the frequency and the vector-valued
background field. For a purely timelike background, the refractive indices are
real. Their associated propagation modes are circularly polarized and exhibit
birefringence. For a purely spacelike background, one refractive index is
always real and the other can be complex. The circularly polarized propagating
modes may exhibit birefringence and dichroism (associated with absorption).
Subsequently, we examine a dimension-five MCFJ-type electrodynamics,
previously scrutinized in the literature, in a continuous medium. Following
the same procedure, we find the refractive indices from a sixth-order
dispersion equation. For a purely timelike background, three distinct
refractive indices are obtained, one of them being real and two being complex.
They are associated with two circularly polarized propagating modes that
exhibit birefringence or dichroism, depending on the frequency range.
Scenarios of propagation and absorption analogous to those found in dispersive
dielectrics are also observed for purely spacelike background configurations.
We conclude by comparing the dimension-three and five results and by
emphasizing the richer phenomenology of the propagating modes in the higher-
derivative model. Our results are applicable in the realm of Weyl semimetals.

Electromagnetic wave propagation; Maxwell equations; Lorentz invariance
violation

###### pacs:

41.20.Jb, 03.50.De, 03.50.-z, 11.30.Cp

##  I INTRODUCTION

The dynamics of electromagnetic fields in continuous media is governed by the
Maxwell equations, supplemented by constitutive relations Jackson ; Zangwill
that describe the response of the medium to external, applied electromagnetic
fields. In vacuo these relations simply read ${\bf{D}}=\epsilon_{0}{\bf{E}}$
and ${\bf{H}}=\mu_{0}^{-1}{\bf{B}}$, where $\epsilon_{0}$ and $\mu_{0}$ are
the electric vacuum permittivity and magnetic permeability constant,
respectively. The first constitutive relation takes into account the electric
polarization in a dielectric medium, while the latter includes magnetization
effects. For an isotropic medium, the constitutive relations are
${\bf{D}}=\epsilon{\bf{E}}$ and ${\bf{H}}=\mu^{-1}{\bf{B}}$ with scalar
material parameters $\epsilon,\mu$ replacing $\epsilon_{0},\mu_{0}$ governing
the vacuum properties. More involved constitutive relations appear in two main
scenarios: (i) anisotropic media, where electric permittivity and magnetic
permeability become tensors (cf. uniaxial and biaxial crystals Landau ; Bain ;
Fowles ; Hecht ; Kurmanov ; Yakov , Weyl semimetals Halterman ; Zu , and
magnetized materials Krupka1 ; Krupka2 ); (ii) novel effects in matter
described by extended constitutive relations that introduce additional
electric and magnetic responses, encoded as linear functions of the type
$\mathbf{D}=\mathbf{D}(\mathbf{E},\mathbf{B})$ and
$\mathbf{H}=\mathbf{H}(\mathbf{E},\mathbf{B})$, in general. This happens, for
instance, in bi-isotropic media Aladadi ; Sihvola ; Sihvola2 ; Nieves , chiral
materials Hillion , topological insulators Li1 ; Urrutia ; Urrutia2 ;
Lakhtakia ; Winder ; Li , relativistic electron gases Carvalho , axion
electrodynamics Sekine ; Tobar2 ; BorgesAxion , and Lorentz-violating
electrodynamics Tobar1 ; Bailey , as well.

Generalizations of electrodynamics including higher-derivative terms have also
been conceived in the literature. First studies of electrodynamics in the
presence of higher derivatives are ascribed to Bopp Bopp  in 1940, and to
Podolsky Podolsky1 ; Podolsky2  in 1942. This model implements a second-order
derivative term
$\theta^{2}\partial_{\alpha}F^{\alpha\beta}\partial_{\sigma}F^{\sigma\beta}$
into the Maxwell Lagrangian in vacuo. The modified Maxwell-Podolsky equations,
sometimes called Bopp-Podolsky equations, yield a photon mass term,
proportional to the inverse of the Podolsky parameter $\theta$. Furthermore,
this extension exhibits two dispersion relations, the usual one from Maxwell
theory and a second one ascribed to a massive mode.

The constraint structure of this theory was investigated in Galvao  and its
quantization was performed in Barcelos . Further aspects of the Maxwell-
Podolsky model were examined, including the problems of self-force and self-
interaction Gratus ; Zayats , Green functions and classical solutions Lazar ,
multipole expansion for fields in the static regime Bonin ,
symmetrization/conservation of the energy-momentum tensor Fan , its
consistency based on the BRST approach Dai , quantum field theoretic
properties Bufalo ; Zambrano  as well as other aspects Granado .

A further example for a generalization of Maxwell electrodynamics is provided
by Lee-Wick electrodynamics Lee-Wick-1 ; Lee-Wick-2 , which introduces the
modification $F^{\mu\nu}\square F_{\mu\nu}$ Turcati ; Turcati2 ; Turcati3 .
The higher-derivative Lee-Wick term can arise as a quantum correction in
models with a nonminimal coupling between the gauge and fermionic fields
Borges .

In the past years, higher-derivative contributions have also been examined in
the context of Lorentz-violating theories. The possibility of Lorentz
invariance violation (LV) was proposed in the context of physics at the Planck
scale such as strings Kostelecky:1988zi . Presently, the Standard-Model
Extension (SME) Colladay , where fixed background tensor fields are coupled to
the dynamical Standard-Model fields, is usually employed as the main framework
to parameterize it. Lorentz violation in the electromagnetic sector of the SME
occurs by means of a CPT-odd or a CPT-even term KM ; Escobar ; Santos ; Belich
. The CPT-odd part is represented by the Carroll-Field-Jackiw (CFJ)
contribution CFJ ; CFJ2 ; CFJ3 ; CFJ4 ; CFJ5 ; CFJ6 ; CFJ7 , which has found
applications in condensed-matter systems that violate parity (P) and time
reversal (T) symmetry Qiu  as well as those endowed with the chiral magnetic
effect Fukushima ; Kharzeev ; Kharzeev1 ; Kharzeev2  and the anomalous Hall
effect Zyuzin .

Nonminimal extensions of the SME were proposed including higher-derivative
terms with mass dimensions greater than four (in natural units) Kostelecky ;
Mewes ; Schreck . In this context, the Myers-Pospelov model Myers ; Marat  was
a pioneering proposal focusing on a dimension-five contribution. Recently,
classical aspects of a modified, higher-derivative electrodynamics in vacuo
were discussed in Leticia1 ; Leticia2 , including the derivation of the gauge
propagator, the dispersion relations as well as an analysis of causality,
unitary, and stability of the modes. Profound analyses were accomplished for
the Maxwell-Podolsky electrodynamics modified by CPT-even, dimension-six terms
Leticia2  and for a CPT-odd, dimension-five electrodynamics Leticia1 . Some
results of Ref. Leticia1  were revisited and discussed in Ref. Passos2 .
Nonminimal higher-derivative models have also been used to study the
interaction energy between electromagnetic sources Borges-Ferrari  and the
thermodynamic properties of electrodynamic systems Filho-Maluf  as well as in
the context of Horǎva-Lifshitz electrodynamics Passos  and radiative
corrections Ferrari .

The plethora of nonminimal LV theories on the one hand and the optical
properties of new materials Aladadi ; Shibata  on the other hand is a strong
motivation for investigating higher-derivative effects on the propagation of
electromagnetic waves in dielectric substrates, including aspects of optical
activity and dichroism. In this sense, the present work is devoted to
analyzing the behavior of a continuous medium governed by a MCFJ-type
electrodynamics in the absence and presence of higher-derivative terms.

This paper is outlined as follows. In Sec. II, we briefly review the covariant
description of electrodynamics in macroscopic materials recapitulating the
definition of birefringence (double refraction) and dichroism. We will be
considering simple matter as opposed to designed materials with highly
peculiar properties such as metamaterials Shelby ; Valanju ; Kshetrimayum . In
Sec. III, we present aspects of the MCFJ electrodynamics in a ponderable
medium, showing that the timelike CFJ background yields birefringence, while
the spacelike one provides birefringence and dichroism. In Sec. IV, we discuss
the higher-derivative MCFJ electrodynamics in continuous matter based on more
involved scenarios. Finally, we present our main findings in Sec. V.
Throughout the paper, we employ natural units with $\hbar=c=1$ unless
otherwise stated. Furthermore, our signature choice for the Minkowski metric
$\eta_{\mu\nu}$ is $(+,-,-,-)$.

##  II  Electrodynamics in simple matter

In a continuous medium, the electromagnetic properties are described by the
Maxwell equations Jackson ; Zangwill  combined with the constitutive
relations. For a general linear, homogeneous, and anisotropic medium, the
constitutive relations can be written as

$\displaystyle\mathbf{D}$
$\displaystyle=\epsilon\cdot\mathbf{E}+\gamma\cdot\mathbf{B}\,,$ (1a)
$\displaystyle\mathbf{H}$
$\displaystyle=\tilde{\gamma}\cdot\mathbf{E}+\mu^{-1}\cdot\mathbf{B}\,,$ (1b)

where $\epsilon$ and $\mu$ represent the electric permittivity and magnetic
permeability tensors Yakov ; Aladadi ; Sihvola ; Sihvola2 ; Nieves ,
respectively. The tensor $\gamma$ measures the magnetic contribution to the
electric displacement field $\mathbf{D}$, while $\tilde{\gamma}$ represents
the electric contribution to the magnetic field $\mathbf{H}$. Regarding the
structure of constitutive relations (1a) and (1b), interesting scenarios of
electromagnetic behavior may occur, e.g., in anisotropic media Aladadi ; Bain
; Fowles ; Kurmanov ; Yakov ; Sihvola ; Sihvola2 ; Nieves , Weyl semimetals
Halterman ; Zu ; Grushin:2012mt , magnetized ferrites Krupka1 ; Krupka2 , and
in chiral media and topological insulators Urrutia ; Urrutia2 ; Lakhtakia .
Besides Eq. (1), general constitutive relations for the current density,
${\bf{J}}={\bf{J}}({\bf{E}},{\bf{B}})$, can also be considered. As an example,
a dielectric system endowed with a magnetic conductivity has recently been
examined at the classical level Pedro , reporting interesting effects such as
an induced electric conductivity, isotropic birefringence, and parity
violation. A physical realization of the antisymmetric magnetic current
examined in Pedro  was addressed in Ref. Kaushik .

The constitutive relations in Eq. (1) can be naturally encoded in the Lagrange
density formalism via

$\mathcal{L}=-\frac{1}{4}G^{\mu\nu}F_{\mu\nu}-A_{\mu}J^{\mu}\,,$ (2a) with the
four-potential $A_{\mu}$, the electromagnetic field strength tensor
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$, and an external,
conserved four-current $J^{\mu}$. Furthermore, the antisymmetric tensor
$G^{\mu\nu}$ is defined as Yakov
$G^{\mu\nu}\equiv\frac{1}{2}\chi^{\mu\nu\alpha\beta}F_{\alpha\beta}\,,$ (2b)

with $\chi^{\mu\nu\alpha\beta}$ being the constitutive tensor that
parameterizes the medium’s response to the applied electromagnetic fields Post
. The constitutive tensor satisfies the following symmetry properties:

$\displaystyle\chi^{\mu\nu\alpha\beta}$
$\displaystyle=-\chi^{\nu\mu\alpha\beta}\,,$ (3a)
$\displaystyle\chi^{\mu\nu\alpha\beta}$
$\displaystyle=-\chi^{\mu\nu\beta\alpha}\,,$ (3b)
$\displaystyle\chi^{\mu\nu\alpha\beta}$
$\displaystyle=\chi^{\alpha\beta\mu\nu}\,,$ (3c)

compatible with the symmetries of the field strength tensor. The Euler-
Lagrange equation applied to the Lagrangian of Eq. (2a) (a complete derivation
is presented in Appendix A) yields

$\partial_{\mu}G^{\mu\nu}=J^{\nu}\,.$ (4)

The homogenous Maxwell equations are obtained from the Bianchi identity valid
for the curvature $F_{\mu\nu}$ of the principal U(1) fiber bundle:

$\partial_{\mu}\tilde{F}^{\mu\nu}=0\,,\quad\tilde{F}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}\,,$
(5)

which is why the latter are not affected by the presence of the medium. A
straightforward calculation from Eqs. (4), (5) leads to the well-known Maxwell
equations in simple matter, namely:

$\displaystyle\nabla\cdot{\bf{D}}$ $\displaystyle=\rho\,,$ (6a)
$\displaystyle\nabla\times{\bf{H}}-\partial_{t}{\bf{D}}$
$\displaystyle={\bf{J}}\,,$ (6b) $\displaystyle\nabla\cdot{\bf{B}}$
$\displaystyle=0\,,$ (6c)
$\displaystyle\nabla\times{\bf{E}}+\partial_{t}{\bf{B}}$ $\displaystyle=0\,,$
(6d)

where the constitutive relations for the electric displacement field
${\bf{D}}$ and magnetic field ${\bf{H}}$ are given in Eq. (1). The Maxwell
equations of Eq. (6) and the constitutive relations given by Eq. (1) allow us
to describe the dynamics of electromagnetic fields in simple matter.

In crystals and generic anisotropic media, the dispersion relations are given
by the Fresnel equation Zangwill . The latter is obtained from algebraic
manipulations of the Maxwell equations for continuous media whose properties
are encoded in the permittivity tensor. In such a medium, the dielectric
permittivity tensor is a function of the frequency $\omega$, the wave vector
$\mathbf{k}$, and the (external) magnetic field $\mathbf{B}$ (or the
magnetization, alternatively),
$\epsilon_{ij}=\epsilon_{ij}(\omega,{\bf{k}},{\bf{B}})$, and can be expanded
as Shibata

$\epsilon_{ij}(\omega,{\bf{k}},{\bf{B}})=\epsilon^{0}_{ij}+\alpha_{ijl}k_{l}+\beta_{ijl}B_{l}+\gamma_{ijab}k_{a}B_{b}+\dots\,.$
(7)

The first term, $\epsilon^{0}_{ij}$, is the usual permittivity tensor of a
dielectric. The second term, $\alpha_{ijl}k_{l}$, is a signature of spatial
inversion symmetry breaking. It implies an optical activity that becomes
manifest via linear birefringence or a rotation of the oscillation plane of
linearly polarized light Bain ; Fowles . The third term, $\beta_{ijl}B_{l}$,
is associated with a fixed external magnetic field or magnetization and leads
to a violation of time reversal symmetry. It gives rise to a magneto-optical
activity via the Faraday or the Cotton-Mouton effect Shibata .

As already mentioned, a known consequence of the optical activity of a medium
is linear birefringence, occurring when two circularly polarized modes of
opposite chiralities, with refractive indices $n_{+}$ and $n_{-}$,
respectively, have different phase velocities, $c/n_{+}$ and $c/n_{-}$. This
property implies a rotation of the polarization plane of a linearly polarized
wave. The latter phenomenon is quantified by the specific rotatory power
$\delta$, which measures the rotation of the oscillation plane of linearly
polarized light per unit traversed length in the medium. It is defined as

$\delta=-\frac{\omega}{2}[\mathrm{Re}(n_{+})-\mathrm{Re}(n_{-})]\,,$ (8)

where $n_{+}$ and $n_{-}$ are associated with left and right-handed circularly
polarized waves, respectively.

Another interesting effect occurring in anisotropic crystals, dichroism, takes
place when one polarization component is more strongly absorbed than the
other. Obviously, this property is linked to the imaginary part of the
refractive index. The difference of absorption of left and right-handed
circularly polarized modes Shibata  is given by the dichroism coefficient [see
Appendix B for the derivation of Eqs. (8), (9)]:

${\delta}_{\mathrm{d}}=-\frac{\omega}{2}[\mathrm{Im}(n_{+})-\mathrm{Im}(n_{-})]\,.$
(9)

In the following, we examine two models of modified electrodynamics in
continuous matter: the first one governed by the MCFJ Lagrangian and the
second one by a higher-derivative MCFJ-type Lagrangian.

##  III MCFJ model in a continuous medium

In principle, MCFJ electrodynamics has connections with systems of chiral
fermions, in particular, the chiral magnetic effect (CME), the anomalous Hall
effect (AHE), and the anomalous generation of charge, besides birefringence
effects. In Ref. Qiu , the Maxwell equations in vacuo modified by the CFJ
background were obtained, which yields the terms ascribed to the CME and AHE.
In this section, we examine aspects of the MCFJ electrodynamics embedded in a
continuous medium. The latter plays a role for condensed-matter systems such
as Weyl semimetals Yan:2016euz . These novel materials are characterized by an
even number of Weyl cones separated from each other in momentum space. In the
vicinity of these cones, electrons behave as massless particles and have a
certain Fermi velocity associated with them, whereupon their description via
the Weyl equation is admissible. Having a microscopic realization of such a
material at hand, it can be consistently described in the context of effective
field theory via a $b$ term of the minimal SME fermion sector Colladay . The
modified Dirac theory is frequently recast into the form Grushin:2012mt ;
Behrends:2018qkj

$\mathcal{L}=\overline{\psi}\left[\gamma^{\mu}(\mathrm{i}\partial_{\mu}-b_{\mu}\gamma^{5})-m\right]\psi\,,$
(10)

where $\psi$ is a Dirac spinor field, $\overline{\psi}$ its Dirac conjugate,
$m$ the electron mass, $\gamma^{\mu}$ the standard Dirac matrices, and
$\gamma^{5}$ is the chiral Dirac matrix. The vector-valued background field
$b_{\mu}$ is known to catch the essential properties of a certain class of
these materials. Integrating out the fermion fields implies an action for the
electromagnetic fields. The latter decomposes into a CPT-even part, which
gives rise to a nontrivial permittivity and permeability of the system, as
well as a CPT-odd part corresponding to a CFJ term. We will come back to this
point below.

The MCFJ Lagrange density in matter has the form

$\mathcal{L}=-\frac{1}{4}G^{\mu\nu}F_{\mu\nu}-\frac{1}{4}\epsilon^{\beta\lambda\mu\nu}V_{\beta}A_{\lambda}F_{\mu\nu}-A_{\mu}J^{\mu}\,,$
(11)

with the intrinsic vector-valued field $V^{\mu}$. Furthermore,
$\epsilon^{\beta\lambda\mu\nu}$ is the Levi-Civita symbol in Minkowski
spacetime fulfilling
$\epsilon^{\beta\lambda\mu\nu}=-\epsilon_{\beta\lambda\mu\nu}$ and
$\epsilon_{0123}=1$. The latter Lagrange density yields the following modified
inhomogeneous Maxwell equations:

$\displaystyle\nabla\cdot{\bf{D}}-{\bf{V}}\cdot{\bf{B}}$
$\displaystyle=\rho\,,$ (12a)
$\displaystyle\nabla\times{\bf{H}}-\partial_{t}{\bf{D}}-V_{0}{\bf{B}}+{\bf{V}}\times{\bf{E}}$
$\displaystyle={\bf{J}}\,,$ (12b)

where $\mathbf{V}$ is the spatial part of $V^{\mu}$ and the fields
$\bf{D},\bf{H}$ fulfill the linear constitutive relations of Eq. (1). Notice
that the presence of the tensor $G^{\mu\nu}$ renders the Lagrangian (11)
different from the one of Ref. Qiu , meaning that the modified Maxwell
equations of Eq. (12) apply to a ponderable medium. Concerning the discrete
symmetries, C (charge conjugation), P, and T, it is worthwhile to recall that
the CFJ term is CPT-odd and the free part of the Lagrangian in Eq. (11) can be
written as:

$\displaystyle\mathrm{{\mathcal{L}}}$
$\displaystyle\supset\frac{1}{2}{\left(\mathbf{E}\cdot\mathbf{D}-\mathbf{B}\cdot\mathbf{H}\right)}$
$\displaystyle\phantom{{}\supset{}}+\frac{1}{2}\left[{{V^{0}\left(\mathbf{A\cdot
B}\right)}}-{{A^{0}\left(\mathbf{V\cdot
B}\right)}}+{{\mathbf{V\cdot}\left(\mathbf{A\times E}\right)}}\right]\,.$ (13)

In this sense, the pieces involving $V_{0}$ are P-odd and T-even, while the
terms composed of $\bf{V}$ are P-even and T-odd, as properly shown in Tab. 1.
Thus, these terms can induce an optical activity of the medium (in the form of
birefringence or dichroism).

Table 1: Behavior of the LV terms of the Lagrangian given by Eq. (13) under C (charge conjugation), P (parity), and T (time reversal). |  E |  B |  $A_{0}$ |  ${\bf{A}}$ |  ${V}_{0}({\bf{A}}\cdot{\bf{B}})$ |  $A_{0}({\bf{V}}\cdot{\bf{B}})$ |  $\mathbf{V\cdot}\left(\mathbf{A\times E}\right)$ |
---|---|---|---|---|---|---|---|---
C |  $-$ |  $-$ |  $-$ |  $-$ |  $+$ |  $+$ |  $+$ |
P |  $-$ |  $+$ |  $+$ |  $-$ |  $-$ |  $+$ |  $+$ |
T |  $+$ |  $-$ |  $+$ |  $-$ |  $+$ |  $-$ |  $-$ |

Another interesting aspect of MCFJ electrodynamics is that the term
$V_{0}{\bf{B}}$ plays a significant role in a chiral magnetic current,

${\bf{J}}_{\mathrm{CME}}={\frac{e^{2}}{{4{\pi}^{2}}}}(\Delta\mu){\bf{B}}\equiv\Sigma{\bf{B}}\,,$
(14)

usually generated in chiral fermion systems Pedro ; Fukushima ; Kharzeev ;
Kharzeev1 ; Kharzeev2 . Here, $\Delta\mu\equiv{\mu}_{R}-\mu_{L}$ is also known
as the chiral chemical potential and fermions of electric charge $\pm e$ are
considered. In Eq. (14), $\Sigma$ represents a chiral magnetic isotropic
conductivity and plays a role equivalent to that of the timelike component
$V_{0}$ in Eq. (12b), as pointed out in Ref. Qiu . For the present analysis,
we consider the usual constitutive relations. We set
$\gamma_{ij}=\tilde{\gamma}_{ij}=0$ in Eq. (1), implying

$D^{i}=\epsilon_{ij}E^{j}\,,\qquad H^{i}=(\mu^{-1})_{ij}B^{j}\,,$ (15)

which can be restricted to the special scenario of an isotropic medium by
choosing configurations proportional to the identity,

$\epsilon_{ij}=\epsilon\delta_{ij}\,,\qquad(\mu^{-1})_{ij}=\mu^{-1}\delta_{ij}\,,$
(16)

where $\epsilon$ and $\mu$ are the electric permittivity and magnetic
permeability constants, respectively. This approach is equivalent to taking
the constitutive relations ${\bf{D}}=\epsilon{\bf{E}}$, and
${\bf{H}}=\mu^{-1}{\bf{B}}$.

In what follows, we implement the latter relations as well as
${\bf{J}}=\sigma{\bf{E}}$, where $\sigma$ is the Ohmic conductivity.
Furthermore, we employ a plane-wave ansatz for the fields,
$\mathbf{E}=\mathbf{E}_{0}e^{\mathrm{i}(\mathbf{k}\cdot\mathbf{r}-\omega t)}$,
$\mathbf{B}=\mathbf{B}_{0}e^{\mathrm{i}(\mathbf{k}\cdot\mathbf{r}-\omega t)}$,
and similarly for ${\bf{D}}$ and ${\bf{H}}$ in Eq. (12b). One then gets

$\mathrm{i}{\bf{k}}\times{\bf{B}}+\mathrm{i}\mu\epsilon\omega{\bf{E}}-\mu
V_{0}{\bf{B}}+\mu{\bf{V}}\times{\bf{E}}-\mu\sigma{\bf{E}}=0\,,$ (17)

which can be simplified by using Faraday’s law, yielding

$\left[\mathbf{k}^{2}\delta_{ij}-k_{i}k_{j}-\omega^{2}\mu\bar{\epsilon}_{ij}\right]E^{j}=0\,,$
(18a) where we have defined
$\bar{\epsilon}_{ij}\equiv\left(\epsilon+\mathrm{i}\frac{\sigma}{\omega}\right)\delta_{ij}-\frac{\mathrm{i}}{\omega^{2}}\epsilon_{iaj}\left(k_{a}V_{0}-\omega
V_{a}\right)\,,$ (18b)

as an effective electric permittivity tensor. Here, $\epsilon_{ijk}$ is the
Levi-Civita symbol in three dimensions. The second term on the right-hand side
of Eq. (18b), $\epsilon_{iaj}k_{a}V_{0}$, is analogous to the tensor
$\alpha_{ijl}k_{l}$ of Eq. (7), which breaks parity invariance. The third
term, $\epsilon_{iaj}\omega V_{a}$, breaks time reversal invariance. Both are
responsible for the optical activity of the medium, becoming manifest in
birefringence, as we shall see.

For a continuous medium, we write $\mathbf{k}=\omega\mathbf{n}$, where
$\mathbf{n}$ is a vector pointing along the direction of the wave vector and
yielding the refractive index $n=+\sqrt{\mathbf{n}^{2}}$. To permit complex
refractive indices, we take $+\sqrt{\mathbf{n}^{2}}$ instead of
$|\mathbf{n}|$. The plus sign indicates, in principle, that we discard
refractive indices with negative real parts, whenever such could occur.
Composites with negative real parts of their refractive indices are called
metamaterials Shelby ; Valanju ; Kshetrimayum  and such possibilities will not
be taken into account. Then, Eq. (18a) becomes

$M_{ij}E^{j}=0\,,$ (19a) with the tensor
$M_{ij}=n^{2}\delta_{ij}-n_{i}n_{j}-\mu\bar{\epsilon}_{ij}\,,$ (19b) where
$\bar{\epsilon}_{ij}=\left(\epsilon+\mathrm{i}\frac{\sigma}{\omega}\right)\delta_{ij}-\frac{\mathrm{i}}{\omega}\epsilon_{iaj}\left(n_{a}V_{0}-V_{a}\right)\,.$
(19c)

The nontrivial solutions for the electric field are obtained by requiring that
the determinant of the matrix $M_{ij}$ vanish, $\mathrm{det}[M_{ij}]=0$, which
yields the dispersion relations that describe wave propagation in the medium.
The matrix $M_{ij}$ is explicitly given by

$\displaystyle[M_{ij}]$
$\displaystyle=\mathcal{M}+\mathrm{i}\frac{\mu}{\omega}\mathcal{V}\,,$ (20a)
$\displaystyle\mathcal{M}$
$\displaystyle=\left[n^{2}-\mu\left(\epsilon+\mathrm{i}\frac{\sigma}{\omega}\right)\right]\mathds{1}_{3}-\begin{pmatrix}n_{1}^{2}&n_{1}n_{2}&n_{1}n_{3}\\\
n_{1}n_{2}&n_{2}^{2}&n_{2}n_{3}\\\ n_{1}n_{3}&n_{2}n_{3}&n_{3}^{2}\\\
\end{pmatrix}\,,$ (20b) $\displaystyle\mathcal{V}$
$\displaystyle=\begin{pmatrix}0&V_{3}-V_{0}n_{3}&V_{0}n_{2}-V_{2}\\\
V_{0}n_{3}-V_{3}&0&V_{1}-V_{0}n_{1}\\\ V_{2}-V_{0}n_{2}&V_{0}n_{1}-V_{1}&0\\\
\end{pmatrix}\,,$ (20c)

with the $(3\times 3)$ identity matrix $\mathds{1}_{3}$. The dispersion
equation follows from $\mathrm{det}[M_{ij}]=0$:

$\displaystyle 0$
$\displaystyle=\tilde{\epsilon}\left(n^{2}-\mu\tilde{\epsilon}\right)^{2}-\frac{\mu}{\omega^{2}}\Big{\\{}\mu\tilde{\epsilon}\left[V_{0}^{2}n^{2}+{\bf{V}}^{2}-2V_{0}({\bf{n}}\cdot{\bf{V}})\right]$
$\displaystyle\phantom{{}={}}-{\bf{V}}^{2}n^{2}+({\bf{n}}\cdot{\bf{V}})^{2}\Big{\\}}\,,$
(21a) where $\tilde{\epsilon}=\epsilon+\mathrm{i}\frac{\sigma}{\omega}\,.$
(21b)

We note that via the choices

$\overline{p}^{\mu}\equiv\left(\sqrt{\tilde{\epsilon}}\,\omega,\frac{\mathbf{k}}{\sqrt{\mu}}\right)\,,\quad\overline{V}^{\mu}\equiv\left(\sqrt{\mu}\,V^{0},\frac{\mathbf{V}}{\sqrt{\tilde{\epsilon}}}\right)\,,$
(22)

our Eq. (21) is equivalent to

$\overline{p}^{4}+\overline{p}^{2}\overline{V}^{2}-(\overline{p}\cdot\overline{V})^{2}=0\,.$
(23)

Alternatively, we can introduce an effective metric of the form

$\tilde{\eta}_{\mu\nu}\equiv\mathrm{diag}\left(\tilde{\epsilon},-\frac{1}{\mu},-\frac{1}{\mu},-\frac{1}{\mu}\right)\,,$
(24)

and write the dispersion equation as

$\displaystyle 0$ $\displaystyle=(p\cdot\tilde{\eta}\cdot p)^{2}$
$\displaystyle\phantom{{}={}}+\frac{\mu}{\tilde{\epsilon}}\left[(p\cdot\tilde{\eta}\cdot
p)(V\cdot\tilde{\eta}\cdot V)-(p\cdot\tilde{\eta}\cdot V)^{2}\right]\,.$ (25)

In vacuo, where $\epsilon=\mu=1$ and $\sigma=0$, the conventional four-
momentum $p^{\mu}=(\omega,{\bf{k}})$ and the preferred spacetime direction
$V^{\mu}=(V^{0},\bf{V})$ satisfy

$p^{4}+p^{2}V^{2}-(p\cdot V)^{2}=0\,,$ (26)

as expected. Equation (26) is the well-known dispersion equation of the MCFJ
model in vacuo CFJ ; CFJ2 . Therefore, we interpret Eq. (23) as the dispersion
equation for a generalized MCFJ theory in media. The four-vector
$\overline{p}^{\mu}$ of Eq. (22) plays the role of an effective four-momentum
that formally satisfies an analogous dispersion equation as in vacuo when the
preferred direction is replaced by $\overline{V}^{\mu}$. The possibility of
expressing the dispersion equation in terms of the effective metric in Eq.
(24) and the conventional four-momentum $p^{\mu}$ is a different way of
understanding this result. The presence of a medium described by the material
parameters $\epsilon,\mu$, and $\sigma$ leads to electromagnetic waves obeying
an analogous dispersion equation as in vacuum, but with the Minkowski metric
replaced by an effective metric.

Now, let us have another look at MCFJ theory in the context of Weyl
semimetals. As described in Grushin:2012mt , integrating out the fermion
fields of the effective field theory stated in Eq. (10) implies a modified
electrodynamics described by the first two terms in Eq. (11). Considering the
realization of a Weyl semimetal studied in the latter reference leads to a
particular choice of $V^{\mu}$ with $V^{0}=0$ and $\mathbf{V}$ pointing along
the third spatial axis (compare the modified inhomogeneous Maxwell equations
of their Eqs. (31), (34) to our Eqs. (12a), (12b)). Although the CFJ term
incorporates the most intriguing properties of such materials, we must also
take into account that a Weyl semimetal (like any material) is characterized
by a nontrivial permittivity (whereas the permeability is often simply to set
1). Thus, the optical response of such a material is very well described by a
dispersion equation of the form of Eq. (III) (cf. Eq. (36) in Grushin:2012mt )
being a formidable motivation for considering theories such as Eq. (11). The
author of the latter paper emphasizes that the presence of the CFJ term
implies birefringence in Weyl semimetals.

In order to further understand some properties of MCFJ electrodynamics in a
continuous dielectric medium with magnetic properties, we address two main
scenarios: (i) a timelike and (ii) a spacelike background field $V^{\mu}$. We
choose a non-Ohmic dielectric as a substrate, which implies $\sigma\mapsto 0$
in Eq. (21).

###  III.1 Purely timelike case

For the purely timelike scenario, $V^{\mu}=(V^{0},\bf{0})$, Eq. (21) reduces
to

$\left(n^{2}-\mu\epsilon\right)^{2}-\frac{\mu^{2}V_{0}^{2}}{\omega^{2}}n^{2}=0\,,$
(27)

which yields two distinct refractive indices:

$n_{\pm}^{2}=\mu\epsilon+\frac{\mu^{2}V_{0}^{2}}{2\omega^{2}}\pm\frac{\mu
V_{0}}{\omega}\sqrt{\mu\epsilon+\frac{\mu^{2}V_{0}^{2}}{4\omega^{2}}}\,,$
(28a) or equivalently
$n_{\pm}=\sqrt{\mu\epsilon+\frac{\mu^{2}V_{0}^{2}}{4\omega^{2}}}\pm\frac{\mu
V_{0}}{2\omega}\,.$ (28b)

The latter result is in accordance with the refractive index given in Eq. (25)
of Ref. Pedro  for the diagonal isotropic magnetic conductivity tensor. This
is an expected correspondence, since one knows that $V_{0}$ plays the role of
a “magnetic conductivity,” as remarked below Eq. (14). Note that $n_{\pm}$ are
real and positive, allowing both modes to propagate for any frequency, so that
an absorbing behavior is not observed here. Furthermore, in the limit of high
frequencies, Eq. (28b) provides $n_{\pm}\mapsto\sqrt{\mu\epsilon}$, recovering
the refractive index of a medium with electric permittivity $\epsilon$ and
magnetic permeability $\mu$, as described in the context of Maxwell
electrodynamics. This behavior is illustrated in Fig. 1, which depicts the
refractive indices (28b) in terms of the dimensionless parameter
$\omega/V_{0}$ for some values of $\mu$ and $\epsilon$. The mode associated
with $n_{+}$ exhibits anomalous dispersion, meaning that
$\mathrm{d}n_{+}/\mathrm{d}\omega<0$, while $n_{-}$ is characterized by normal
dispersion.

Figure 1: Refractive indices $n_{\pm}(\omega)$ of Eq. (28b) in terms of
$\omega/V_{0}$. Blue (monotonically decreasing) lines represent $n_{+}$, while
red (monotonically increasing) lines depict $n_{-}$. For solid lines, $\mu=1$
and $\epsilon=2$; for dashed lines, $\mu=1$ and $\epsilon=3$; for dashed-
dotted lines, $\mu=2$ and $\epsilon=2$.

In order to examine the polarization state of the propagation modes, we first
rewrite Eq. (27),

$n^{2}-\mu\epsilon=\pm\frac{\mu V_{0}}{\omega}n\,.$ (29)

We employ the latter in Eq. (20), whereupon the condition $M_{ij}E^{j}=0$
yields

${\bf{E}}_{\pm}=\frac{1}{\sqrt{2}n\sqrt{n^{2}-n_{1}^{2}}}\begin{pmatrix}n^{2}-n_{1}^{2}\\\
\mp\mathrm{i}n_{3}n-n_{1}n_{2}\\\ \pm\mathrm{i}n_{2}n-n_{1}n_{3}\\\
\end{pmatrix}\,.$ (30)

Considering the special choice

$\bf{n}=\begin{pmatrix}0\\\ 0\\\ n_{3}\\\ \end{pmatrix}\,,$ (31)

the normalized electric fields obtained from Eq. (30) are

$\displaystyle{\bf{E}}_{\pm}=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\
\mp\mathrm{i}\\\ 0\end{pmatrix}\,.$ (32)

A polarization is defined to be right-handed (left-handed) if the polarization
vector of a plane wave rotates along a circle in clockwise (counterclockwise)
direction when the observer is facing into the incoming wave Jackson ;
Zangwill . Therefore, $\mathbf{E}_{-}$ is interpreted as a left-handed and
$\mathbf{E}_{+}$ as a right-handed circular polarization vector, respectively.
These are associated with the distinct refractive indices $n_{-}$ and $n_{+}$
of Eq. (28b) that imply different phase velocities of the physical modes
giving rise to a rotation of the polarization plane of a linearly polarized
wave. The implied birefringence is measured by the specific rotatory power
[see the definition of Eq. (8) and Appendix B], here written as

$\delta=-\frac{\mu V_{0}}{2}\,,$ (33)

which is a frequency-independent result dependent on the timelike component
$V_{0}$ of the LV background. This nondispersive rotatory power differs from
the rotatory power of a typical birefringent crystal, which increases with the
frequency, as indicated by Eq. (8) for constant refractive indices. As the
refractive indices of Eq. (28b) are real, there is no optical dichroism caused
by $V_{0}$.

###  III.2 Purely spacelike case

For the purely spacelike case, $V^{\mu}=(0,{\bf{V}})$, that is, $V^{0}=0$ and
${\bf{V}}\neq 0$, Eq. (21) yields

$\displaystyle\epsilon(n^{2}-\mu\epsilon)^{2}-\frac{\mu}{\omega^{2}}\left[\mu\epsilon{{\bf{V}}}^{2}-n^{2}{\bf{V}}^{2}+({\bf{n}\cdot\bf{V}})^{2}\right]=0\,.$
(34)

Implementing ${\bf{n}}\cdot{\bf{V}}=n|{\bf{V}}|\cos\theta$, one finds

$(n^{2}-\mu\epsilon)^{2}-\frac{\mu^{2}}{\omega^{2}}|{\bf{V}}|^{2}\alpha^{2}=0\,,$
(35a) where we defined $\alpha^{2}\equiv
1-\frac{n^{2}}{\mu\epsilon}\sin^{2}\theta\,.$ (35b)

The two refractive indices (squared) read

$\displaystyle n_{\pm}^{2}$
$\displaystyle=\mu\epsilon-\frac{\mu\mathbf{V}^{2}}{2\epsilon\omega^{2}}\sin^{2}\theta$
$\displaystyle\phantom{{}={}}\pm\frac{\mu|\mathbf{V}|}{2\epsilon\omega^{2}}\sqrt{4\epsilon^{2}\omega^{2}\cos^{2}\theta+\mathbf{V}^{2}\sin^{4}\theta}\,.$
(36)

It is useful to analyze two special configurations: (i) the perpendicular case
where ${\bf{n}}\cdot{\bf{V}}=0$ and $\sin^{2}\theta=1$, (ii) the longitudinal
case where ${\bf{n}}\cdot{\bf{V}}=\pm n|{\bf{V}}|$ and $\sin^{2}\theta=0$.

In order to examine the propagation modes, let us choose coordinates such that
Eq. (31) holds, whereupon Eq. (20) simplifies as

$[M_{ij}]=\begin{pmatrix}n^{2}-\mu\epsilon&&\mathrm{i}\frac{\mu}{\omega}V_{3}&&-\mathrm{i}\frac{\mu}{\omega}V_{2}\\\
\\\
-\mathrm{i}\frac{\mu}{\omega}V_{3}&&n^{2}-\mu\epsilon&&\mathrm{i}\frac{\mu}{\omega}V_{1}\\\
\\\
\mathrm{i}\frac{\mu}{\omega}V_{2}&&-\mathrm{i}\frac{\mu}{\omega}V_{1}&&-\mu\epsilon\end{pmatrix}\,.$
(37)

Solving $M_{ij}E^{j}=0$, one obtains

$\displaystyle{\bf{E}}_{\pm}=E_{0}\begin{pmatrix}V_{1}V_{2}-\mathrm{i}\epsilon\omega
V_{3}\\\ V_{2}^{2}-\epsilon^{2}\omega^{2}f(\alpha_{\pm})\\\
V_{2}V_{3}+\mathrm{i}V_{1}\epsilon\omega f(\alpha_{\pm})\\\ \end{pmatrix}\,,$
(38a) where
$f(\alpha)=1+\frac{\alpha^{2}-1}{\sin^{2}\theta}\,,\quad\alpha^{2}_{\pm}=1-\frac{n^{2}_{\pm}}{\mu\epsilon}\sin^{2}\theta\,,$
(38b)

and $E_{0}$ is an appropriate normalization.

####  III.2.1 $\mathbf{V}$-perpendicular configuration

Considering the perpendicular configuration with $\sin^{2}\theta=1$, the
solutions of Eq. (34) for $n^{2}$ according to Eq. (III.2) are

$\displaystyle
n^{2}_{\pm}=\mu\epsilon+\frac{\mu{\bf{V}}^{2}}{2\epsilon\omega^{2}}(-1\pm
1)\,,$ (39)

that is

$\displaystyle n_{+}=\sqrt{\mu\epsilon}\,,\qquad
n_{-}=\sqrt{\mu\epsilon-\frac{\mu\mathbf{V}^{2}}{\epsilon\omega^{2}}}\,.$ (40)

While $n_{+}$ is the standard refractive index of Maxwell electrodynamics in
media, corresponding to $\alpha_{+}=0$, the refractive index $n_{-}$ is
associated with $\alpha_{-}=|{\bf{V}}|/(\epsilon\omega)$, whereupon it is
affected by the background. For $\omega<\omega_{-}$, we have $n_{-}^{2}<0$ and
$n_{-}$ becomes purely imaginary, so that the corresponding mode no longer
propagates. This defines the cutoff frequency,

$\displaystyle\omega_{-}=\frac{|{\bf{V}}|}{\epsilon}\,.$ (41)

The general behavior of the refractive indices is depicted in Fig. 2, where
the squared refractive indices (39) are plotted in terms of the dimensionless
parameter $\omega/|{\bf{V}}|$. The horizontal dashed lines stand for
$n^{2}_{+}$, which is constant for all frequencies. As for the mode associated
with $n_{-}^{2}$, the vertical gray dashed lines (located at different
$\omega_{-}/|{\bf{V}}|$ for each case) separate the absorption regime,
$\omega<\omega_{-}$, from the propagation regime, $\omega>\omega_{-}$.
Furthermore, in the limit of high frequencies, $n_{-}^{2}\mapsto
n_{+}^{2}=\mu\epsilon$.

Figure 2:  Refractive indices $n^{2}_{\pm}(\omega)$ of Eq. (39) in terms of
$\omega/V$ where $V=|{\bf{V}}|$. Blue (horizontal) lines represent
$n^{2}_{+}$, while red (curved) lines depict $n^{2}_{-}$. For solid lines,
$\mu=1$ and $\epsilon=2$; for dashed lines, $\mu=1$ and $\epsilon=3$; for
dashed-dotted lines, $\mu=2$ and $\epsilon=2$.

In order to examine the propagation modes, let us choose coordinates such that
Eq. (31) holds. Then, a perpendicular background configuration is
${\bf{V}}=(V_{1},V_{2},0)$. Due to $\alpha_{+}=0$ and $f(\alpha_{+})=0$, Eq.
(38) yields a linearly polarized, transverse mode,

$\mathbf{E}_{+}=\frac{1}{|\mathbf{V}|}\begin{pmatrix}V_{1}\\\ V_{2}\\\
0\end{pmatrix}\equiv\hat{\mathbf{V}}\,,$ (42)

where $\hat{\mathbf{V}}$ is a unit vector pointing along the direction of
$\mathbf{V}$. Also, inserting $\alpha_{-}$ into Eq. (38) provides another
linearly polarized mode that has an additional longitudinal component:

$\displaystyle\bf{E}_{-}$ $\displaystyle=E_{0}^{(-)}\begin{pmatrix}V_{2}\\\
-V_{1}\\\ \mathrm{i}(V_{1}^{2}+V_{2}^{2})/(\epsilon\omega)\\\ \end{pmatrix}$
$\displaystyle=\tilde{E}_{0}^{(-)}\left(\hat{\mathbf{V}}\times\hat{\mathbf{n}}+\mathrm{i}\frac{|\mathbf{V}|}{\epsilon\omega}\hat{\mathbf{n}}\right)\,,$
(43)

where $E_{0}^{(-)},\tilde{E}_{0}^{(-)}$ are properly chosen amplitudes and
$\hat{\mathbf{n}}$ is the unit vector pointing along the propagation direction
of Eq. (31). Note that the longitudinal component is suppressed by the
magnitude of the preferred direction $\mathbf{V}$ in comparison to the
transverse part. For $V_{2}=0$ the behavior is even more transparent:

$\displaystyle{\bf{E}}_{+}=\begin{pmatrix}1\\\ 0\\\
0\end{pmatrix}\,,\qquad{\bf{E}}_{-}=\tilde{E}_{0}^{(-)}\begin{pmatrix}0\\\
-1\\\ \mathrm{i}V_{1}/(\epsilon\omega)\end{pmatrix}\,.$ (44)

The structure of Eqs. (42), (43) reveals immediately that
$\mathbf{E}_{+}\cdot\mathbf{E}_{-}^{*}=\mathbf{E}_{+}^{*}\cdot\mathbf{E}_{-}=0$,
i.e., both polarization vectors are orthogonal to each other. The refractive
indices (40) are associated with the linearly polarized modes of Eq. (44).
Although the vector $\mathbf{E}_{-}$ is composed of a transverse and a
longitudinal component, as for polarization properties, it is interpreted as a
linearly polarized mode and only its transverse component is taken into
account.

If birefringence originates from two linearly polarized modes having different
phase velocities, this property is not suitably characterized in terms of the
usual rotatory power given by Eq. (8). Note that the latter is based on a
decomposition of a linearly polarized mode into two circularly polarized ones
of different chirality (see App. B). Instead, in the propagation regime,
$\omega>\omega_{-}$, the phase shift developed between the propagating modes
as a consequence of the distinct phase velocities is valuable to characterize
birefringence (see Eq. (8.32) in Hecht ):

$\Delta=\frac{2\pi}{\lambda_{0}}d(n_{+}-n_{-})\,.$ (45)

Here, $\lambda_{0}$ is the wavelength of the electromagnetic radiation in
vacuo and $d$ corresponds to the thickness of the medium or the distance the
wave travels in the medium. Starting from the refractive indices of Eq. (40),
the phase shift per unit length is

$\frac{\Delta}{d}=\frac{2\pi}{\lambda_{0}}\sqrt{\mu\epsilon}\left(1-\sqrt{1-\frac{\mathbf{V}^{2}}{\epsilon^{2}\omega^{2}}}\right)\,,$
(46)

which simplifies to

$\displaystyle\frac{\Delta}{d}$
$\displaystyle\simeq\frac{\pi\sqrt{\mu\epsilon}}{\lambda_{0}\epsilon^{2}\omega^{2}}{\bf{V}}^{2}\,,$
(47)

in the limit $|{\bf{V}}|/\omega\ll 1$. Notice that $n_{-}$ is real for
$\omega>\omega_{-}$ or $|{\bf{V}}|/\omega<\epsilon$, so that in the limit
$|{\bf{V}}|/\omega\ll 1$, the expression (46) remains real, justifying the
result (47). These findings indicate that birefringence is governed by the
norm squared of the LV background vector ${\bf{V}}$ and depends quadratically
on the inverse of the frequency $\omega$, as well. That dependence is neither
observed in the purely timelike case (see Eq. (33) for comparison) nor in
usual crystals (see Eq. (45)).

For $\omega<\omega_{-}$, $n_{-}$ becomes complex while $n_{+}$ remains real.
Thus, absorption (only) occurs for the mode labeled with a minus sign. In this
case, the absorption coefficient Zangwill , $\gamma=2\omega\mathrm{Im}(n)$,
reads

$\gamma=2\sqrt{\mu\epsilon}\omega\sqrt{-1+\frac{|{\bf{V}}|^{2}}{\omega^{2}\epsilon^{2}}}\,.$
(48)

So the mode associated with Eq. (43) is absorbed, whereas the remaining mode
given by Eq. (42) propagates without attenuation. Therefore, after traveling a
certain distance in such a medium, only the mode of Eq. (42) will survive.

####  III.2.2 $\mathbf{V}$-longitudinal configuration

We now consider configurations where ${\bf{n}}\cdot{\bf{V}}=\pm n|{\bf{V}}|$
implying $\sin^{2}\theta=0$ and $\alpha^{2}=1$ in Eq. (35a). This means that
${\bf{n}}$ and ${\bf{V}}$ point along the same direction, i.e., for
$\mathbf{n}$ given by Eq. (31) we choose ${\bf{V}}=(0,0,{V}_{3})$. Hence,
based on Eq. (III.2), the solutions of Eq. (35a) for $n^{2}$ in this case are

$n^{2}_{\pm}=\mu\epsilon\pm\frac{\mu|{\bf{V}}|}{\omega}\,.$ (49)

Note that $n_{+}^{2}>0$, meaning that the mode associated with the refractive
index $n_{+}$ propagates within the full frequency domain. On the other hand,
the mode associated with $n_{-}$ just propagates for $\omega>\omega_{-}$, for
which $n_{-}^{2}>0$. Here, $\omega_{-}$ is the cutoff frequency of Eq. (41).
This description is verified in Fig. 3, where the refractive indices (49) are
depicted as functions of the dimensionless parameter $\omega/|{\bf{V}}|$. The
modes associated with $n^{2}_{+}$ and $n^{2}_{-}$ exhibit anomalous and normal
dispersion, respectively, recovering the standard value
$n_{\pm}^{2}\rightarrow\mu\epsilon$ in the regime of high frequencies.

For the special choice of Eq. (31), the longitudinal background is of the form
${\bf{V}}=(0,0,{V}_{3})$, so that $M_{ij}E^{j}=0$ provides

$\displaystyle{\bf{E}}_{\pm}$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\ \pm\mathrm{i}\\\
0\end{pmatrix}\,.$ (50)

Here, ${\bf{E}}_{+}$ and ${\bf{E}}_{-}$ represent polarization vectors for
left-handed and right-handed circularly polarized modes, respectively.

Figure 3:  Refractive indices $n^{2}_{\pm}(\omega)$ of Eq. (49) in terms of
$\omega/V$ where $V=|{\bf{V}}|$. Blue (monotonically decreasing) lines
represent $n^{2}_{+}$, while red (monotonically increasing) lines depict
$n^{2}_{-}$. For solid lines, $\mu=1$ and $\epsilon=2$; for dashed lines,
$\mu=1$ and $\epsilon=3$; for dotted-dashed lines, $\mu=2$ and $\epsilon=2$.
The vertical dashed lines, from left to right, are given by
$\omega_{-}/{V}=1/3$ and $\omega_{-}/V=1/2$, respectively, with $\omega_{-}$
from Eq. (41).

The two refractive indices of Eq. (49) also imply birefringence providing the
following rotatory power:

$\displaystyle\delta=-\frac{\sqrt{\mu\epsilon}}{2}\omega\left(\sqrt{1+\frac{|{\bf{V}}|}{\omega\epsilon}}-\sqrt{1-\frac{|{\bf{V}}|}{\omega\epsilon}}\right)\,.$
(51)

In the limit $|{\bf{V}}|/\omega\ll 1$, the quantity $n_{-}$ remains real,
which also implies a real rotatory power,

$\displaystyle\delta\simeq-\frac{1}{2}\sqrt{\frac{\mu}{\epsilon}}|{\bf{V}}|\,,$
(52)

representing frequency-independent birefringence, similarly to Eq. (33).

On the other hand, for $|{\bf{V}}|/\omega>\epsilon$, $n_{-}$ becomes purely
imaginary, while $n_{+}$ remains real. In this frequency regime, both modes
are absorbed to a different degree. The latter is characterized by the
dichroism coefficient defined in Eq. (9), which yields (cf. Eq. (48)):

$\displaystyle\delta_{\mathrm{d}}$
$\displaystyle=\frac{\sqrt{\mu\epsilon}}{2}\omega\sqrt{-1+\frac{|{\bf{V}}|}{\omega\epsilon}}\,.$
(53)

With the latter finding at hand, we bring our study of the essential
properties of MCFJ theory in continuous media to a close.

##  IV  Higher-derivative dimension-five electrodynamics in matter

After analyzing the properties of MCFJ theory in a material (see Eq. (11)),
the next logical step is to construct and investigate an extension involving
additional four-derivatives. Such extensions are naturally contained in the
nonminimal (nongravitational) SME Kostelecky ; Mewes ; Schreck , which is a
comprehensive framework for the parameterization of Lorentz and CPT violation
in effective field theory in Minkowski spacetime. For the past two decades it
has been the foundation of various experiments testing the fundamental
spacetime symmetries Kostelecky:2008ts . No signal of Lorentz violation in
vacuo has been found, so far. However, Lorentz violation can be considered as
an intrinsic property of material media, which is why the SME is more than
suitable as a base for representing certain material properties within a field
theory setting and to even propose novel materials with unusual
characteristics.

The electromagnetic sector of the nonminimal SME gives rise to a modified
electrodynamics and is given by

$\displaystyle\mathcal{L}$
$\displaystyle=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}\epsilon^{\kappa\lambda\mu\nu}A_{\lambda}(\hat{k}_{AF})_{\kappa}F_{\mu\nu}$
$\displaystyle\phantom{{}={}}-\frac{1}{4}F_{\kappa\lambda}(\hat{k}_{F})^{\kappa\lambda\mu\nu}F_{\mu\nu}\,.$
(54)

The CPT-odd and CPT-even operators, $(\hat{k}_{AF})_{\kappa}$ and
$(\hat{k}_{F})^{\kappa\lambda\mu\nu}$, respectively, are the analogs of
$(k_{AF})_{\kappa}$ and $(k_{F})^{\kappa\lambda\mu\nu}$ of the minimal SME.
However, they involve nonminimal coefficients contracted with additional four-
derivatives in the form of the following infinite operator series:

$\displaystyle(\hat{k}_{AF})_{\kappa}$ $\displaystyle=\sum_{d\text{
odd}}(k_{AF}^{(d)})_{\kappa}^{\phantom{1}\alpha_{1}\dots\alpha_{(d-3)}}\partial_{\alpha_{1}}\dots\partial_{\alpha_{(d-3)}}\,,$
(55a) $\displaystyle(\hat{k}_{F})^{\kappa\lambda\mu\nu}$
$\displaystyle=\sum_{d\text{
even}}(k_{F}^{(d)})^{\kappa\lambda\mu\nu\alpha_{1}\dots\alpha_{(d-4)}}\partial_{\alpha_{1}}\dots\partial_{\alpha_{(d-4)}}\,,$
(55b)

where $d$ is the mass dimension of the tensor field operator that a certain
coefficient is contracted with. Besides, $(4-d)$ is the mass dimension of the
associated controlling coefficients
$(k_{AF}^{(d)})_{\kappa}^{\phantom{1}\alpha_{1}\dots\alpha_{(d-3)}}$ and
$(k_{F}^{(d)})^{\kappa\lambda\mu\nu\alpha_{1}\dots\alpha_{(d-4)}}$. The
Lorentz indices $\alpha_{i}$ are contracted with additional spacetime
derivatives.

We are interested in the CPT-odd dimension-five ($d=5$) extension, which is
represented by a CFJ-like term of the form

$\frac{1}{2}\epsilon^{\kappa\lambda\mu\nu}A_{\lambda}(\hat{k}_{AF})_{\kappa}F_{\mu\nu}\,,$
(56a) with
$\displaystyle(\hat{k}_{AF})_{\kappa}=(k_{AF}^{(5)})_{\kappa}^{\phantom{\kappa}\alpha_{1}\alpha_{2}}\partial_{\alpha_{1}}\partial_{\alpha_{2}}\,.$
(56b)

For our investigation, we will use the parameterization

$\displaystyle(k_{AF}^{(5)})_{\kappa}^{\phantom{\kappa}\alpha_{1}\alpha_{2}}={U}_{\kappa}\eta^{\alpha_{1}\alpha_{2}}\,,$
(57)

with the Lorentz-violating four-vector, $U_{\kappa}$, and the Minkowski metric
tensor, $\eta^{\mu\nu}$. Using Eq. (57), the higher-derivative term becomes

$\displaystyle\frac{1}{2}\epsilon^{\kappa\lambda\mu\nu}A_{\lambda}U_{\kappa}\square\,,$
(58)

where we have introduced the d’Alembertian
$\square=\eta^{\alpha_{1}\alpha_{2}}\partial_{\alpha_{1}}\partial_{\alpha_{2}}$.
The resulting higher-derivative Lagrangian,

$\mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\frac{1}{2}\epsilon^{\beta\lambda\mu\nu}U_{\beta}A_{\lambda}\square
F_{\mu\nu}-A_{\mu}J^{\mu}\,,$ (59)

involves LV parameterized by the background vector,
$U^{\mu}=(U^{0},{\bf{U}})$. Some classical aspects of this model were examined
in Refs. Leticia1 ; Marat .

In order to study the effects of this higher-derivative term on
electromagnetic propagation in continuous matter, we take as a starting point
the Lagrangian (59), but employ the field strength tensor $G^{\mu\nu}$ in its
kinetic term, as it occurs in Eq. (2a). Thus, the Lagrangian of this new model
is

$\mathcal{L}=-\frac{1}{4}G^{\mu\nu}F_{\mu\nu}+\frac{1}{2}\epsilon^{\beta\lambda\mu\nu}U_{\beta}A_{\lambda}\square
F_{\mu\nu}-A_{\mu}J^{\mu}\,,$ (60)

where the tensor $G^{\mu\nu}$ is written in terms of the constitutive tensor
$\chi^{\mu\nu\alpha\beta}$, defined in Eqs. (2b), (3). The latter provides a
generalization of the electrodynamics of Eq. (59) in matter. One may expect a
connection between this theory and a generalization of the modified Dirac
theory given by Eq. (10) where additional derivatives are included in the
second contribution. However, it is beyond the scope of the current paper to
demonstrate such a connection explicitly. Thus, by using Eq. (59) we can take
into consideration an additional energy-momentum dependence that goes beyond
that of the CFJ term in matter.

The Lagrangian of Eq. (60) involves a third-order derivative of the four-
potential, which requires an associated Euler-Lagrange equation endowed with
derivatives for field derivatives that are of the same order. In principle,
the derivative order can be decreased by rewriting Eq. (60) in the form

$\mathcal{L}=-\frac{1}{4}G^{\mu\nu}F_{\mu\nu}-\frac{1}{2}\epsilon^{\beta\lambda\mu\nu}U_{\beta}(\partial_{\eta}A_{\lambda})\partial^{\eta}F_{\mu\nu}-A_{\mu}J^{\mu}\,.$
(61)

As for the Lagrangian of Eq. (61), it is enough to consider the Euler-Lagrange
equation involving derivatives for second-order derivatives of the fields,
that is,

$\frac{\partial\mathcal{L}}{\partial
A_{\kappa}}-\partial_{\rho}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{\rho}A_{\kappa})}\right)+\partial_{\alpha}\partial_{\rho}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{\rho}\partial_{\alpha}A_{\kappa})}\right)=0\,.$
(62)

Applying the latter to Eq. (61) yields

$\partial_{\rho}G^{\rho\kappa}+\epsilon^{\beta\kappa\mu\nu}U_{\beta}\square
F_{\mu\nu}=J^{\kappa}\,.$ (63)

In this scenario, the modified Gauss’s and Ampère’s laws are

$\displaystyle\nabla\cdot{\bf{D}}+2\square({\bf{U}}\cdot{\bf{B}})$
$\displaystyle=\rho\,,$ (64a)
$\displaystyle\nabla\times{\bf{H}}-\partial_{t}{\bf{D}}+2\square{U}_{0}{\bf{B}}-2\square({\bf{U}}\times{\bf{E}})$
$\displaystyle={\bf{J}}\,,$ (64b)

respectively. These modified inhomogeneous Maxwell equations can describe new
effects on the propagation of electromagnetic waves in continuous media
characterized by the constitutive tensor $\chi^{\mu\nu\alpha\beta}$. In the
forthcoming sections we obtain the dispersion relations and study the behavior
of refractive indices and propagating modes for a medium characterized by the
usual constitutive relations, ${\bf{D}}=\epsilon{\bf{E}}$ and
${\bf{H}}=\mu^{-1}{\bf{B}}$.

With regards to the discrete symmetries, the background ${U}^{\mu}$ in the
Lagrangian of Eq. (60) behaves in the very exact way as $V^{\mu}$ does in Eq.
(11), since the two CPT-odd terms differ from each other by the presence of
the second-order differential operator, $\square$, that is even under the
discrete symmetries P and T. In fact, by simple inspection, one finds that the
terms involving the timelike coefficient, $U_{0}$, are P-odd, C-even, T-even,
and PT-odd, while the contributions with ${\bf{U}}$ are P-even, C-even, T-odd,
and PT-odd, as shown in Tab. 2. This means that the terms proportional to
$U_{0}$ and ${\bf{U}}$ will act as a source for optical activity (as well as
birefringence) of the medium under study.

Table 2: Behavior of the LV terms in the Lagrangian of Eq. (59) under charge conjugation, parity transformation, and time reversal. |  E |  B |  $A_{0}$ |  ${\bf{A}}$ |  ${U}_{0}({\bf{A}}\cdot\square{\bf{B}})$ |  $A_{0}({\bf{U}}\cdot\square{\bf{B}})$ |  ${\bf{U}}\cdot({\bf{A}}\times\square{\bf{E}})$ |
---|---|---|---|---|---|---|---|---
C |  $-$ |  $-$ |  $-$ |  $-$ |  $+$ |  $+$ |  $+$ |
P |  $-$ |  $+$ |  $+$ |  $-$ |  $-$ |  $+$ |  $+$ |
T |  $+$ |  $-$ |  $+$ |  $-$ |  $+$ |  $-$ |  $-$ |

###  IV.1 Dispersion relations

As is commonly known, the Maxwell equations constitute one starting point for
achieving the dispersion relations in electrodynamics. Taking the time
derivative of Eq. (64b) and employing Eq. (6d), one obtains

$\partial_{t}\nabla\times{\bf{H}}-\partial_{t}^{2}{\bf{D}}-2\square
U_{0}(\nabla\times{\bf{E}})-2\partial_{t}\square({\bf{U}}\times{\bf{E}})=\partial_{t}{\bf{J}}\,.$
(65)

Using now the constitutive relations given in Eqs. (15) and (16) as well as
${\bf{J}}=\sigma{\bf{E}}$ and the plane-wave ansatz for the fields, Eq. (65)
yields

$\displaystyle\left[\mathbf{k}^{2}\delta_{ij}-k_{i}k_{j}-\omega^{2}\mu\bar{\epsilon}_{ij}(\omega)\right]E^{j}=0\,,$
(66a) where we have defined the effective permittivity tensor (cf. Eq. (18b))
$\displaystyle\bar{\epsilon}_{ij}(\omega)$
$\displaystyle\equiv\left(\epsilon+\mathrm{i}\frac{\sigma}{\omega}\right)\delta_{ij}-\frac{2\mathrm{i}}{\omega^{2}}(k^{2}-\omega^{2})$
$\displaystyle\phantom{{}={}}\times\epsilon_{iaj}\left(\omega
U_{a}-k_{a}U_{0}\right)\,.$ (66b)

The latter quantity is interpreted as an extended frequency-dependent electric
permittivity, which contains contributions stemming from the higher-derivative
term. On the right-hand side of Eq. (66b), the contribution involving
$\epsilon_{iaj}\omega U_{a}$ violates time reversal invariance, while the term
$\epsilon_{iaj}k_{a}U_{0}$ breaks parity invariance. Using
$\mathbf{k}=\omega\mathbf{n}$, Eq. (66a) can now be cast into the form:

$M_{ij}E^{j}=0\,,$ (67a) with the tensor $M_{ij}$ given by
$M_{ij}=n^{2}\delta_{ij}-n_{i}n_{j}-\mu\bar{\epsilon}_{ij}(\omega)\,,$ (67b)
while the effective permittivity tensor now reads
$\displaystyle\bar{\epsilon}_{ij}(\omega)$
$\displaystyle=\left(\epsilon+\mathrm{i}\frac{\sigma}{\omega}\right)\delta_{ij}-2\mathrm{i}\omega(n^{2}-1)$
$\displaystyle\phantom{{}={}}\times\epsilon_{iaj}\left(U_{a}-n_{a}U_{0}\right)\,.$
(67c)

It is important to note that although the medium has an isotropic electric
permittivity $\epsilon\delta_{ij}$, anisotropy effects are generated by the
background $U_{\mu}$, present in the off-diagonal components of
$\bar{\epsilon}_{ij}(\omega)$ in Eq. (67).

The matrix $M_{ij}$ in Eq. (67b) has the explicit form

$[M_{ij}]=\mathcal{M}+2\mathrm{i}\mu\omega(n^{2}-1)\mathcal{W}\,,$ (68a) with
$\mathcal{M}$ given by Eq. (20b) and
$\mathcal{W}=\begin{pmatrix}0&U_{0}n_{3}-U_{3}&U_{2}-U_{0}n_{2}\\\\[4.30554pt]
U_{3}-U_{0}n_{3}&0&U_{0}n_{1}-U_{1}\\\\[4.30554pt]
U_{0}n_{2}-U_{2}&U_{1}-U_{0}n_{1}&0\end{pmatrix}\,.$ (68b)

Evaluating $\mathrm{det}[M_{ij}]=0$ implies the dispersion equation

$\displaystyle 0$
$\displaystyle=\tilde{\epsilon}(n^{2}-\mu\tilde{\epsilon})^{2}-4(n^{2}-1)^{2}\mu\omega^{2}$
$\displaystyle\phantom{{}={}}\times\left\\{\mu\tilde{\epsilon}\left[U_{0}^{2}n^{2}+{\bf{U}}^{2}-2U_{0}({\bf{n}}\cdot{\bf{U}})\right]\right.$
$\displaystyle\phantom{{}={}}\left.\hskip
17.07182pt{}-{\bf{U}}^{2}{\bf{n}}^{2}+({\bf{n}}\cdot{\bf{U}})^{2}\right\\}\,,$
(69)

with $\tilde{\epsilon}$ stated in Eq. (21b). We point out that by employing
the four-momentum of Eq. (22) as well as

$\overline{U}^{\mu}\equiv\left(\sqrt{\mu}\,{U}^{0},\frac{{\mathbf{U}}}{\sqrt{\tilde{\epsilon}}}\right)\,,$
(70)

we can cast the dispersion equation into the form

$\overline{p}^{4}+4p^{4}\left[\overline{U}^{2}\overline{p}^{2}-(\overline{U}\cdot\overline{p})^{2}\right]=0\,.$
(71)

By consulting the effective metric of Eq. (24), the latter can also be
expressed in terms of the conventional four-momentum $p^{\mu}$ and the
preferred direction ${U}^{\mu}$ as follows:

$\displaystyle 0$ $\displaystyle=(p\cdot\tilde{\eta}\cdot
p)^{2}+4(p\cdot\eta\cdot p)^{2}\frac{\mu}{\tilde{\epsilon}}$
$\displaystyle\phantom{{}={}}\times\left[({U}\cdot\tilde{\eta}\cdot{U})(p\cdot\tilde{\eta}\cdot
p)-({U}\cdot\tilde{\eta}\cdot p)^{2}\right]\,.$ (72)

Note that in contrast to the dispersion equation of MCFJ theory stated in Eq.
(23), the recent Eq. (71) cannot be written in terms of the effective four-
momentum $\overline{p}^{\mu}$ only, but $p^{\mu}$ is necessary, as well. The
reason for $p^{\mu}$ playing a role are the two additional four-derivatives
contracted with the dimension-5 coefficients in Eq. (57). Equation (IV.1) also
allows us to say that the propagation of modified electromagnetic waves in
media is governed by two metrics: the Minkowski metric $\eta_{\mu\nu}$ and the
effective metric $\tilde{\eta}_{\mu\nu}$ of Eq. (24). Thus, the dimension-5
MCFJ-type theory defined by Eq. (60) could be called bimetric in this sense.
We conclude that the structure of the dimension-5 MCFJ-type theory in media is
quite different from that of the generalized MCFJ model in Eq. (60).

In vacuo, the constitutive parameters read $\epsilon=1$, $\mu=1$, and
$\sigma=0$. In this case, the dispersion equation in Eq. (IV.1) reduces to

$\displaystyle 0$
$\displaystyle=\left(n^{2}-1\right)^{2}\Big{\\{}1-4\omega^{2}\left[{U}_{0}^{2}n^{2}-{\bf{U}}^{2}n^{2}+{\bf{U}}^{2}\right.$
$\displaystyle\phantom{{}={}}\left.\hskip
45.52458pt{}+({\bf{n}}\cdot{\bf{U}})^{2}-2{U}_{0}({\bf{n}}\cdot{\bf{U}})\right]\Big{\\}}\,,$
(73)

being conveniently simplified as

$p^{4}\left\\{1+4\left[p^{2}{U}^{2}-({U}\cdot p)^{2}\right]\right\\}=0\,,$
(74)

with the four-momentum $p^{\mu}$ and the preferred direction
${U}^{\mu}=({U}^{0},{\bf{U}})$. Notice that Eq. (74) recovers the dispersion
equation obtained in Eq. (23) of Ref. Leticia1 , where this higher-derivative
electrodynamics was examined in vacuo. It is important to point out that the
remarkable difference between Eq. (71) and Eq. (74) is ascribed to the
presence of the continuous medium, since the dimension-five higher-derivative
terms in the Lagrangians of Eqs. (59), (60) correspond to each other.

In what follows, we analyze the dispersion equation (IV.1) for the timelike
and spacelike configurations of the vectorial background, ${U}^{\mu}$.

###  IV.2 Purely timelike case

Considering the purely timelike scenario for the background vector,
${U}_{0}\neq 0$ and ${\bf{U}}={\bf{0}}$, and also
$\tilde{\epsilon}\mapsto\epsilon$, which means that the medium does not have
Ohmic conductivity (whereupon $\sigma=0$), Eq. (IV.1) is reduced to the form

${\epsilon}(n^{2}-\mu{\epsilon})^{2}-4\mu^{2}\omega^{2}{U}_{0}^{2}{\epsilon}n^{2}(n^{2}-1)^{2}=0\,,$
(75a) implying $n^{2}-\mu{\epsilon}=\pm 2\mu\omega{U}_{0}n(n^{2}-1)\,,$ (75b)
or equivalently $\pm 2\mu\omega{U}_{0}n^{3}-n^{2}\mp
2\mu\omega{U}_{0}n+\mu\epsilon=0\,.$ (75c)

The latter equation is cubic in $n$ and has 3 (complex) solutions, in general,
given as functions $n=n(\omega)$. These solutions extend to frequency domains
defined in accordance with the sign of the discriminant of the cubic equation,
written as

$\Delta=\frac{S}{2^{4}3^{3}\mu^{3}\omega^{4}{U}_{0}^{4}}\,,$ (76a) with
$S=-\epsilon-\mu\omega^{2}{U}_{0}^{2}\left[1+9\mu\epsilon\left(2-3\mu\epsilon\right)+16\mu^{2}\omega^{2}{U}_{0}^{2}\right]\,.$
(76b)

For a cubic polynomial equation, the sign of $\Delta$ helps us to identify the
nature (real or complex) of the 3 solutions, in accordance with Tab. 3.

Table 3: Sign of discriminant $\Delta$ of Eq. (76) and the nature of the roots (solutions) of Eq. (75c). Sign |  Solutions |
---|---|---
$\Delta>0$ |  one real root and two complex conjugate roots |
$\Delta\leq 0$ |  three real roots (with two or all three equal to each other if $\Delta=0$) |
Figure 4:  Plot of one real root of Eq. (75c), the refractive index
$n_{1}(\omega)$, in terms of $\omega{U}_{0}$. It is obtained by choosing the
lower signs of Eq. (75c). The solid (dotted) line represents
$\mathrm{Re}[n_{1}(\omega)]$ ($\mathrm{Im}[n_{1}(\omega)]$) where the latter
vanishes.

Since the denominator of Eq. (76a) is positive, we only need to analyze the
sign of the numerator, $S$. As $S$ is a function quartic in $\omega$, it is
possible to find two roots that provide three frequency ranges for positive or
negative values of $\Delta$. In this way, the relation $S=0$ establishes the
critical values of frequencies (roots) that separate the absorption domain
$S>0$ from the propagation domain $S<0$. Solving $S=0$, one achieves two roots
for $\omega^{2}$ given by

$\displaystyle\omega^{2}_{\pm}$
$\displaystyle=\frac{1}{32\mu^{2}{U}_{0}^{2}}\bigg{\\{}9\mu\epsilon\left(3\mu\epsilon-2\right)-1$
$\displaystyle\phantom{{}={}}\hskip
42.67912pt\pm\sqrt{\mu\epsilon-1}(9\mu\epsilon-1)^{3/2}\bigg{\\}}\,.$ (77)

Thus, the three frequency ranges associated with two distinct scenarios are as
follows:

  * i)

For $\omega_{-}<\omega<\omega_{+}$, one has $S>0$ and $\Delta>0$, so that Eq.
(75c) yields one real function $n(\omega)$ and two complex functions
$n(\omega)$.

  * ii)

For $\omega<\omega_{-}$ or $\omega>\omega_{+}$, one has $S<0$ and $\Delta<0$,
so that there are three real refractive indices $n(\omega)$.

The first domain describes absorption effects, whereas electromagnetic waves
can freely propagate without attenuation in the second domain. The sign of $S$
determines the real or complex nature of $n(\omega)$ in the corresponding
frequency range. For a complex refractive index, we can write
$n(\omega)=n^{\prime}(\omega)+\mathrm{i}n^{\prime\prime}(\omega)$, where
$\mathrm{Re}[n(\omega)]=n^{\prime}(\omega)$ is the refractive index of the
medium, and $\mathrm{Im}[n(\omega)]=n^{\prime\prime}(\omega)$ is associated
with the medium’s absorption coefficient $\alpha=2\omega
n^{\prime\prime}(\omega)$ Zangwill .

Figure 5:  Plot of one complex root of Eq. (75c), the refractive index
$n_{2}(\omega)$, in terms of $\omega{U}_{0}$. It follows from Eq. (75c) with
the upper signs taken into account. The solid (dotted) line represents
$\mathrm{Re}[n_{2}(\omega)]$ ($\mathrm{Im}[n_{2}(\omega)]$).

Joining the above domains, we can conclude that:

  * a)

For $\omega<\omega_{-}$ there are three real solutions.

  * b)

For $\omega_{-}<\omega<\omega_{+}$ two solutions become complex and the
remaining one stays real.

  * c)

For $\omega>\omega_{+}$ the three solutions become real again.

In general, propagation without attenuation is associated with real (positive)
refractive indices, whereas absorption effects (damping of the amplitude of
electromagnetic waves) are related to complex refractive indices. The modified
electrodynamics defined by Eq. (61) ascribes a conducting behavior to a
dielectric substrate (with additional magnetic properties). For the particular
scenario studied previously, electromagnetic waves propagate without being
damped in the frequency range where the three solutions are real. In the range
where complex solutions for $n(\omega)$ occur, both propagation and absorption
(attenuation) is observed. These novel effects stem from the higher-derivative
coupling of the background coefficient ${U}_{0}$ with the electromagnetic
fields.

The refractive indices for a continuous medium with signal propagation
described by Eq. (75c) are given by very intricate expressions (the roots of
Eq. (75c)), which will not be stated here explicitly. We depict these three
functions, $n_{i}(\omega)$, for $i=1,2,3$ in terms of the dimensionless
parameter $\omega{U}_{0}$ for the special values $\mu=1$ and $\epsilon=2$.
These plots are presented in Figs. 4, 5, and 6, where the solid (dotted) lines
stand for the real (imaginary) part of $n(\omega)$. The refractive indices
shown in the previous figures are characterized by positive real parts. The
remaining three refractive indices, which follow from the generic sixth-order
polynomial of Eq. (75a), have negative real parts.

We notice that $n_{1}(\omega)$ is always real for all frequency ranges. The
functions $n_{2}(\omega)$ and $n_{3}(\omega)$ become complex in the range
$\omega_{-}<\omega<\omega_{+}$, in agreement with the previous analysis.

Figure 6:  Plot of one complex root of Eq. (75c), the refractive index
$n_{3}(\omega)$, in terms of $\omega{U}_{0}$. It results from choosing the
upper signs of Eq. (75c). The solid (dotted) line depicts
$\mathrm{Re}[n_{3}(\omega)]$ ($\mathrm{Im}[n_{3}(\omega)]$).

Combining all three plots in Fig. 7, we realize the full scenario described in
items (a) – (c) previously stated. The vertical dashed lines indicate the
critical frequency values of Eq. (77), namely $\omega_{-}{U}_{0}$ and
$\omega_{+}{U}_{0}$, which define the transition between the ranges given in
(a) – (c). Another characteristic of Figs. 5 and 6 are the discontinuities in
the real parts of $n_{2}(\omega)$ and $n_{3}(\omega)$, at the frequencies
$\omega_{\pm}$. Note that $n_{2}(\omega)$ and $n_{3}(\omega)$ become purely
imaginary at these values.

Figure 7:  Compilation of complex refractive indices $n(\omega)$ from Figs. 4,
5, and 6. The solid lines illustrate the real parts of $n_{1}(\omega)$ (blue),
$n_{2}(\omega)$ (red), and $n_{3}(\omega)$ (green). Their corresponding
imaginary pieces are represented by dotted lines with the same colors. The
solid brown line indicates that $\mathrm{Re}[n_{2}(\omega)]$ and
$\mathrm{Re}[n_{3}(\omega)]$ lie on top of each other. Black dotted lines are
employed whenever all three imaginary parts merge.

As a final comment, we point out that the physical behavior described above
only occurs for the higher-derivative electrodynamics of Eq. (61) in matter.
In fact, in vacuo, Eq. (75a) would provide

$(n^{2}-1)^{2}(1-4\omega^{2}{U}_{0}^{2}n^{2})=0\,,$ (78)

whose solutions are real, namely:

$n=1,\quad n=\frac{1}{2\omega|{U}_{0}|}\,,$ (79)

meaning the absence of absorption effects in vacuo (for this dimension-5
theory). This behavior can also be inferred directly from Eq. (77), since
$\Delta\omega=\omega_{+}-\omega_{-}=0$, for $\mu=\epsilon=1$, corresponding to
the disappearance of frequency ranges where absorption occurs.

Furthermore, the second refractive index of Eq. (79) does not have a well-
defined limit for $U_{0}\mapsto 0$. In vacuo, such modes are sometimes called
spurious and their occurrence is characteristic for higher-derivative theories
(see, e.g., Schreck ; Leticia2 ; Leticia1  for detailed investigations in the
nonminimal electromagnetic sector of the SME). They can be interpreted as
high-energy effects decoupling from the theory at low energies. However, a
finite $U_{0}$ in macroscopic media, that is, $m|U_{0}|\sim\mathcal{O}(1)$
(with the electron mass $m$), is realistic. Then, the second refractive index
is not necessarily suppressed for low energies in continuous media, but must
be considered on an equal footing with the remaining modes. This behavior will
become more transparent for the purely spacelike case to be investigated
below.

####  IV.2.1 Propagation modes

In order to examine the propagation modes for the purely timelike sector, we
can employ Eq. (75b) in the matrix of Eq. (68), yielding

$\displaystyle\left[M_{ij}\right]$
$\displaystyle=-\begin{pmatrix}n_{1}^{2}&n_{1}n_{2}&n_{1}n_{3}\\\
n_{1}n_{2}&n_{2}^{2}&n_{2}n_{3}\\\ n_{1}n_{3}&n_{2}n_{3}&n_{3}^{2}\\\
\end{pmatrix}$
$\displaystyle\phantom{{}={}}+2\mu\omega{U}_{0}(n^{2}-1)\begin{pmatrix}\pm
n&\mathrm{i}n_{3}&-\mathrm{i}n_{2}\\\ -\mathrm{i}n_{3}&\pm
n&\mathrm{i}n_{1}\\\ \mathrm{i}n_{2}&-\mathrm{i}n_{1}&\pm n\\\
\end{pmatrix}\,.$ (80)

Solving $M_{ij}E^{j}=0$, one finds

$\displaystyle E_{y}$
$\displaystyle=\frac{\pm\mathrm{i}n_{3}n-n_{1}n_{2}}{n^{2}-n_{1}^{2}}E_{x}\,,$
(81a) $\displaystyle E_{z}$
$\displaystyle=\frac{\mp\mathrm{i}n_{2}n-n_{1}n_{3}}{n^{2}-n_{1}^{2}}E_{x}\,,$
(81b)

such that the normalized electric fields ${\bf{E}}_{\pm}$ of the propagating
waves are given by

${\bf{E}}_{\pm}=\frac{1}{\sqrt{2}n\sqrt{n^{2}-n_{1}^{2}}}\begin{pmatrix}n^{2}-n_{1}^{2}\\\
\pm\mathrm{i}n_{3}n-n_{1}n_{2}\\\ \mp\mathrm{i}n_{2}n-n_{1}n_{3}\\\
\end{pmatrix}\,.$ (82)

The latter coincide exactly with those of Eq. (30) except of the labels being
switched. Basically, for the timelike configuration, the electric-field modes
of the MCFJ and MCFJ-type higher-derivative electrodynamics are the same,
despite the different refractive indices of these theories. Note that
$\mathbf{E}_{\pm}$ of Eqs. (30), (82) do not depend on $V_{0}$ and $U_{0}$,
respectively. The refractive index illustrated in Fig. 4 is associated with
the electric field $\mathbf{E}_{-}$ of Eq. (82), whereas those of Figs. 5, 6
are linked to $\mathbf{E}_{+}$.

For a direct physical interpretation of these propagating modes, let us choose
again a convenient coordinate system where propagation occurs along the $z$
axis, i.e., let $\mathbf{n}$ be given by Eq. (31). In this system, the
normalized electric fields are

$\displaystyle{\bf{E}}_{\pm}$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\ \pm\mathrm{i}\\\
0\end{pmatrix}\,,$ (83)

which are the same as those stated in Eq. (32). These are polarization vectors
for a left-handed and right-handed circular polarization, respectively,
typical of optically active media. Such an optical activity can be expressed
in terms of the rotatory power of Eq. (8), if the refractive indices $n_{+}$
and $n_{-}$ are known. It is worthwhile to note that, although Eq. (75c)
provides, in general, three refractive indices, there are only two distinct
electric-field configurations, those of Eq. (83). There are still three
propagating modes, one associated with each refractive index. We will come
back to this aspect in the forthcoming section, too.

###  IV.3 Purely spacelike case

Let us now consider the purely spacelike scenario for the background vector,
${U}_{0}=0$ and ${\bf{U}}\neq{\bf{0}}$, and also
$\tilde{\epsilon}\mapsto\epsilon$ (setting $\sigma=0$). Then Eq. (IV.1) yields

$\displaystyle 0$
$\displaystyle=\epsilon(n^{2}-\mu\epsilon)^{2}-4(n^{2}-1)^{2}\mu\omega^{2}$
$\displaystyle\phantom{{}={}}\times\left[(\mu\epsilon-n^{2}){\bf{U}}^{2}+({\bf{n}}\cdot{\bf{U}})^{2}\right]\,.$
(84)

Implementing ${\bf{n}}\cdot{\bf{U}}=n|{\bf{U}}|\cos\theta$ in Eq. (84), we
obtain

$n^{2}-\mu\epsilon=\pm 2\mu\omega(n^{2}-1)|{\bf{U}}|\alpha\,,$ (85a) where we
have defined $\alpha^{2}\equiv 1-\frac{n^{2}}{\mu\epsilon}\sin^{2}\theta\,.$
(85b)

With this parameterization, we can straightforwardly analyze two special
cases: (i) the perpendicular configuration where ${\bf{n}}\cdot{\bf{U}}=0$ and
$\sin^{2}\theta=1$; (ii) the longitudinal configuration with
$\sin^{2}\theta=0$ and $\bf{n}\cdot{\bf{U}}=\pm|{\bf{n}}||{\bf{U}}|$ where the
plus (minus) sign holds for ${\bf{n}}$ parallel (antiparallel) to ${\bf{U}}$.
These choices can provide some physical insights on the behavior of
electromagnetic-wave propagation.

To obtain the propagation modes, we again work in a coordinate system where
Eq. (31) holds. Then the matrix (68) simplifies as

$\displaystyle[M_{ij}]$
$\displaystyle=\begin{pmatrix}n_{3}^{2}-\mu\epsilon&&-\mathrm{i}\mu\epsilon\beta(\omega)U_{3}&&\mathrm{i}\mu\epsilon\beta(\omega)U_{2}\\\
\mathrm{i}\mu\epsilon\beta(\omega)U_{3}&&n_{3}^{2}-\mu\epsilon&&-\mathrm{i}\mu\epsilon\beta(\omega)U_{1}\\\
-\mathrm{i}\mu\epsilon\beta(\omega)U_{2}&&\mathrm{i}\mu\epsilon\beta(\omega)U_{1}&&-\mu\epsilon\end{pmatrix}\,,$
(86)

where $\beta(\omega)=2\omega(n_{3}^{2}-1)/\epsilon$. For each case
parameterized with $\alpha$, we can insert Eq. (85a) into Eq. (86) and solve
$M_{ij}E^{j}=0$ to achieve the electric fields of the corresponding modes.

####  IV.3.1 $\mathbf{U}$-perpendicular configuration

First, we consider the orthogonal configuration, i.e., ${\bf{U}}\bot{\bf{n}}$
and $\sin^{2}\theta=1$, so that Eq. (85) becomes

$n^{2}-\mu\epsilon=\pm
2\mu\omega(n^{2}-1)|{\bf{U}}|\sqrt{1-\frac{n^{2}}{\mu\epsilon}}\,,$ (87)

which can be written as

$(n^{2}-\mu\epsilon)\left[4\mu^{2}\omega^{2}{\bf{U}}^{2}(n^{2}-1)^{2}+\mu\epsilon(n^{2}-\mu\epsilon)\right]=0\,,$
(88)

implying $n^{2}=\mu\epsilon$ and

$\displaystyle 0$
$\displaystyle=4\mu^{2}\omega^{2}{\bf{U}}^{2}n^{4}+(\mu\epsilon-8\mu^{2}\omega^{2}{\bf{U}}^{2})n^{2}$
$\displaystyle\phantom{{}={}}-\mu^{2}\epsilon^{2}+4\mu^{2}\omega^{2}{\bf{U}}^{2}\,.$
(89)

The first solution, $n^{2}=\mu\epsilon$, corresponds to the ordinary
refractive index of Maxwell electrodynamics in macroscopic media that we
denote as $n_{0}=\sqrt{\mu\epsilon}$. On the other hand, Eq. (89) captures
information stemming from the higher-derivative term of Eq. (58) and the
background ${\bf{U}}$, leading to the following solutions:

$n^{2}_{\pm}=1+f_{\pm}\,,$ (90a) where $\displaystyle f_{\pm}$
$\displaystyle=\frac{\epsilon}{8\mu\omega^{2}{\bf{U}}^{2}}\left(-1\pm\sqrt{1+\Upsilon}\,\right)\,,$
(90b) $\displaystyle\Upsilon$
$\displaystyle=16\mu^{2}\omega^{2}{\bf{U}}^{2}\left(1-\frac{1}{\mu\epsilon}\right)\,.$
(90c)

The behavior of $n_{\pm}^{2}$ in terms of the dimensionless parameter
$\omega|{\bf{U}}|$ is presented in Fig. 8. We notice that $n_{+}$ is real in
the entire frequency domain and exhibits anomalous dispersion. Furthermore,
the function $n^{2}_{-}$ has a simple root,

$\omega_{-}=\frac{\epsilon}{2|{\bf{U}}|}\,.$ (91)

The latter is interpreted as a critical value (cf. Eq. (41)), below which
$n_{-}$ is purely imaginary, whereupon no propagation occurs. Above
$\omega_{-}$, the refractive index $n_{-}$ becomes real. As a consequence,
electromagnetic waves can propagate in this regime.

Figure 8:  Behavior of the refractive indices $n^{2}_{\pm}$ of Eq. (90) in
terms of $\omega U$ where $U=|{\bf{U}}|$. Blue curves (above the horizontal
dashed line) represent $n^{2}_{+}$, while red curves (below the horizontal
dashed line) depict $n^{2}_{-}$. For solid lines, $\mu=2$ and $\epsilon=2$;
for dashed lines, $\mu=2$ and $\epsilon=4$; for dashed-dotted lines, $\mu=1$
and $\epsilon=4$. Gray vertical dashed lines indicate $\omega_{-}U=1$ and
$\omega_{-}U=2$, respectively, where $\omega_{-}$ is given by Eq. (91).

The first vertical dashed line, located at the value $\omega_{-}|{\bf{U}}|=1$,
separates the absorption and propagation zones for the mode represented by the
solid red line. The second vertical dashed line, in $\omega_{-}|{\bf{U}}|=2$,
does so for the modes depicted by the dashed and dashed-dotted lines. In
detail, we observe that:

  * •

For $0<\omega<\omega_{-}$: the refractive index $n_{+}$ is real and $n_{-}$ is
purely imaginary; thus, only the mode associated with $n_{+}$ propagates in
this range.

  * •

For $\omega>\omega_{-}$: one has $n_{\pm}^{2}>0$ and both modes propagate.

  * •

In the limit of very low frequencies, ${\omega|{\bf{U}}|\mapsto 0}$, it holds
that $n_{+}=\sqrt{\mu\epsilon}$, recovering the usual refractive index of a
simple continuous medium in standard electrodynamics.

  * •

In the limit of very high frequencies, $\omega|\mathbf{U}|\mapsto\infty$, the
behavior of the refractive indices is

$n_{\pm}=1\pm\frac{1}{4\omega|\mathbf{U}|}\sqrt{\epsilon\left(\epsilon-\frac{1}{\mu}\right)}\,.$
(92)

Unsurprisingly, the high-frequency behavior of Eq. (92), $n_{\pm}\mapsto 1$,
differs from that of the refractive indices of MCFJ theory of Eq. (40) in
macroscopic media, which is given by $n_{\pm}\mapsto\sqrt{\mu\epsilon}$. Thus,
the impact of a nontrivial permeability and permittivity is suppressed in this
regime of the MCFJ-type theory in Eq. (60) endowed with higher-derivative
operators.

With regards to the propagating modes, Eq. (85b) yields

$\displaystyle\alpha_{\pm}=\sqrt{1-\frac{n_{\pm}^{2}}{\mu\epsilon}}=\sqrt{1-\frac{1+f_{\pm}}{\mu\epsilon}}\,,$
(93)

indicating different values of $\alpha$ for the distinct refractive indices
$n_{\pm}$ of Eq. (90). Taking $\mathbf{n}$ as given in Eq. (31), the
background has the form ${\bf{U}}=(U_{1},U_{2},0)$ such that
${\bf{n}}\cdot{\bf{U}}=0$. The following propagation modes are then achieved:

$\displaystyle{\bf{E}}_{\pm}$
$\displaystyle=E_{0}^{\prime}\begin{pmatrix}U_{2}\\\ -U_{1}\\\
-2\mathrm{i}\omega f_{\pm}(U_{1}^{2}+U_{2}^{2})/\epsilon\end{pmatrix}$
$\displaystyle=\tilde{E}_{0}^{\prime}\left(\hat{\mathbf{U}}\times\hat{\mathbf{n}}-2\mathrm{i}\omega
f_{\pm}\frac{|\mathbf{U}|}{\epsilon}\hat{\mathbf{n}}\right)\,,$ (94)

with $f_{\pm}$ given by Eq. (90b) and the unit vector $\hat{\mathbf{U}}$
pointing along the direction of $\mathbf{U}$. In case we choose the background
vector of the simple form ${\bf{U}}=(0,U_{2},0)$, Eq. (IV.3.1) provides

${\bf{E}}_{\pm}=E_{0}^{\prime}\begin{pmatrix}1\\\ 0\\\ -2\mathrm{i}\omega
f_{\pm}U_{2}/\epsilon\end{pmatrix}\,.$ (95)

The latter correspond to transverse, linear polarization modes with additional
longitudinal components, in analogy to the mode $\mathbf{E}_{-}$ of Eq. (44).
Now, by comparing Eq. (IV.3.1) to Eq. (43) obtained for the MCFJ theory in
macroscopic matter, we spot intriguing similarities. Our interpretation is
that the single mode of Eq. (43) splits into the two of Eq. (IV.3.1) as a
result of the higher-derivative nature of this theory. To understand these
modes better, it is reasonable to perform Taylor expansions for
$\mathbf{U}\mapsto\mathbf{0}$. Investigating the behavior of $f_{\pm}$ in Eq.
(90b) provides

$f_{+}\simeq\epsilon\mu-1\,,\quad
f_{-}\simeq-\frac{\epsilon}{4\mu\omega^{2}\mathbf{U}^{2}}+1-\epsilon\mu\,,$
(96)

giving rise to

$n_{+}\mapsto\sqrt{\mu\epsilon}\,,\quad
n_{-}\mapsto\sqrt{2-\frac{\epsilon}{4\mu\omega^{2}\mathbf{U}^{2}}-\epsilon\mu}\,.$
(97)

As a consequence, the mode described by $\mathbf{E}_{+}$ has a well-defined
limit for $\mathbf{U}\mapsto 0$, whereas the second mode associated with
$\mathbf{E}_{-}$ does not. Here it is also evident that $n_{-}$ becomes
complex in this regime. So such as for the purely timelike sector, we again
encounter a mode whose counterpart in vacuo would frequently be denoted as
spurious. The situation is different in macroscopic matter, though, because
$m|\mathbf{U}|\sim\mathcal{O}(1)$ can be realistic. As before, the second mode
must be interpreted as a regular, propagating mode.

Finally, we discuss the first solution $n_{0}=\sqrt{\mu\epsilon}$ of Eq. (88).
In this case, Eq. (85b) provides $\alpha=0$. Hence, $M_{ij}E^{j}=0$ implies

$\displaystyle\begin{pmatrix}0&&0&&\mathrm{i}U_{2}\\\
0&&0&&-\mathrm{i}U_{1}\\\
-\mathrm{i}U_{2}&&\mathrm{i}U_{1}&&-1/\beta\end{pmatrix}\begin{pmatrix}E_{x}\\\
E_{y}\\\ E_{z}\\\ \end{pmatrix}=0\,,$ (98)

yielding the following propagating mode:

$\displaystyle{\bf{E}}_{0}$
$\displaystyle=\frac{1}{|\mathbf{U}|}\begin{pmatrix}U_{1}\\\ U_{2}\\\
0\end{pmatrix}=\hat{\mathbf{U}}\,.$ (99)

The latter is a linearly polarized mode related to the refractive index
$n_{0}=\sqrt{\mu\epsilon}$ and it is perpendicular to the propagation
direction of Eq. (31). Also, one finds
${\bf{E}}_{0}\cdot{\bf{E}}_{\pm}^{*}=0$, with ${\bf{E}}_{\pm}$ given by Eq.
(IV.3.1). The mode of Eq. (99) is equivalent to that of Eq. (42) found for
MCFJ theory in macroscopic matter. Thus, this particular mode remains
unaffected by the presence of the additional derivatives in the CFJ-type field
operator of Eq. (58). Also, even in the limit $\mathbf{U}\mapsto 0$, the
electric fields $\mathbf{E}_{0},\mathbf{E}_{\pm}$ are still governed by the
direction $\hat{\mathbf{U}}$. However, $\hat{\mathbf{U}}$ does then not
indicate a preferred direction, anymore. Instead, the components $U_{1},U_{2}$
take the role of parameterizing the plane orthogonal to the propagation
direction $\hat{\mathbf{n}}$.

In total, the number of the physical modes in the MCFJ-type theory defined by
Eq. (60) amounts to 3. Two of these approach the behavior of a standard
isotropic medium in the limit $\mathbf{U}\mapsto\mathbf{0}$. In particular, it
is the mode associated with $n_{+}$ in Eq. (97) and that linked to
$n_{0}=\sqrt{\mu\epsilon}$ of Eq. (88). Having three propagating modes does
not indicate a breakdown of gauge invariance of the theory defined by Eq.
(60). The operator of Eq. (58) is clearly gauge-invariant. The third mode
originates from the presence of the d’Alembertian in Eq. (58) increasing the
polynomial order of the dispersion equation. In vacuo, the third mode could be
denoted as spurious, but this technical term is misleading in macroscopic
matter where the coefficients $m\mathbf{U}$ can take values of
$\mathcal{O}(1)$.

As the associated modes are not circularly polarized, birefringence for this
case is better characterized in terms of the phase shift per unit length given
by Eq. (45) instead of the rotatory power in Eq. (8). We introduce

$\frac{\Delta_{a,b}}{d}\equiv\frac{2\pi}{\lambda_{0}}(n_{a}-n_{b})\,,$ (100)

where $a,b\in\\{0,+,-\\}$. Since there are three propagating modes for
$\omega>\omega_{-}$, we can define the following phase shifts acquired after
propagation (divided by the propagation distance $d$):

$\displaystyle\frac{\Delta_{\pm,0}}{d}$
$\displaystyle=\frac{2\pi}{\lambda_{0}}\left[\sqrt{1+f_{\pm}}-\sqrt{\mu\epsilon}\right]\,,$
(101a) $\displaystyle\frac{\Delta_{+,-}}{d}$
$\displaystyle=\frac{2\pi}{\lambda_{0}}\left[\sqrt{1+f_{+}}-\sqrt{1+f_{-}}\right]\,,$
(101b)

which are valid in the range where $n_{-}$ is real, i.e.,
$\omega|{\bf{U}}|>\epsilon/2$. In the limit of high frequencies,
$(\omega|{\bf{U}}|)^{-1}\ll 1$, Eq. (101) yields

$\displaystyle\frac{\Delta_{\pm,0}}{d}$
$\displaystyle=\frac{2\pi}{\lambda_{0}}(1-\sqrt{\mu\epsilon})\pm\frac{\Delta}{2d}\,,$
(102a) $\displaystyle\frac{\Delta_{+,-}}{d}$
$\displaystyle=\frac{\Delta}{d}\,,$ (102b) with
$\frac{\Delta}{d}\equiv\frac{\pi}{\lambda_{0}\omega|{\bf{U}}|}\sqrt{\epsilon\left(\epsilon-\frac{1}{\mu}\right)}\,.$
(102c)

Comparing the modes labeled with $\pm$ to the standard mode, there is a
zeroth-order contribution that only involves the permittivity and permeability
of the medium.

For $\omega<\omega_{-}$ (or $\omega|{\bf{U}}|<\epsilon/2$), $n_{-}$ is purely
imaginary. Then from Eq. (90), $n_{-}$ is rewritten as

$n_{-}=\mathrm{i}\sqrt{-1-f_{-}}\,.$ (103)

Since $\mathrm{Im}(n_{+})=0$ for the full frequency domain, only the mode
labeled with the minus sign undergoes attenuation, which is quantified by the
absorption coefficient, $\gamma=2\omega\mathrm{Im}(n_{-})$, that is

$\gamma=2\omega\sqrt{\frac{\epsilon}{8\mu\omega^{2}{\bf{U}}^{2}}\left(1+\sqrt{1+\Upsilon}\right)-1}\,,$
(104)

with $\Upsilon$ given by Eq. (90c). In the limit of low frequencies,
$\omega|{\bf{U}}|\ll 1$, Eq. (104) can be expanded as

$\gamma\simeq\frac{1}{|{\bf{U}}|}\sqrt{\frac{\epsilon}{\mu}}\left[1+2\left(1-\frac{2}{\mu\epsilon}\right)\omega^{2}\mu^{2}|{\bf{U}}|^{2}\right]\,.$
(105)

It is important to note that the absorption coefficient of Eq. (105) is
evaluated in the limit $\omega|{\bf{U}}|\ll 1$, while the phase shift in Eq.
(102) is determined in the opposite limit $(\omega|{\bf{U}}|)^{-1}\ll 1$.
Attenuation takes place for a purely imaginary $n_{-}$ and birefringence
occurs when $n_{-}$ is real. The condition $\omega|{\bf{U}}|=\epsilon/2$
states a clear cutoff separating the frequency regimes for each effect from
each other.

####  IV.3.2 $\mathbf{U}$-longitudinal configuration

Let us now consider the configurations where $\sin\theta=0$, i.e., ${\bf{n}}$
and ${\bf{U}}$ are parallel or antiparallel, for which Eq. (84) is equivalent
to

$\displaystyle 0$
$\displaystyle=(1-4\mu^{2}\omega^{2}{\bf{U}}^{2})n^{4}-2(\mu\epsilon-4\mu^{2}\omega^{2}{\bf{U}}^{2})n^{2}$
$\displaystyle\phantom{{}={}}+\mu^{2}\epsilon^{2}-4\mu^{2}\omega^{2}{\bf{U}}^{2}\,,$
(106)

whose solutions for $n^{2}$ are

$n_{\pm}^{2}=\frac{\mu(\epsilon\pm 2\omega|{\bf{U}}|)}{1\pm
2\mu\omega|{\bf{U}}|}\,.$ (107)

The behavior of $n^{2}_{\pm}$ in terms of the dimensionless parameter
$\omega|{\bf{U}}|$ is displayed in Fig. 9, for some parameter values.

Figure 9:  Plot of $n^{2}_{\pm}$ of Eq. (107) in terms of $\omega U$ with
$U=|{\bf{U}}|$. The blue curves, which are positive and monotonically
decreasing for the entire frequency range, represent $n_{+}^{2}$. The red
lines, constituted by positive upper and negative lower branches, illustrate
$n^{2}_{-}$. Solid lines: $\mu=2$ and $\epsilon=2$; dashed lines: $\mu=2$ and
$\epsilon=4$; dashed-dotted lines; $\mu=1$ and $\epsilon=3$. Gray vertical
dashed lines indicate $\omega{U}\in\\{1/4,1/2,1,3/2,2\\}$.

In this scenario, the mode associated with $n_{+}$ exhibits anomalous
dispersion and propagates in the full frequency range, since $n_{+}^{2}>0$.

The mode associated with $n_{-}^{2}$ has two branches. In the superior branch,
defined in the frequency range $0<\omega<\omega_{0}$, the mode propagates,
with $n^{2}_{-}$ increasing very rapidly with $\omega$. Here,

$\omega_{0}=\frac{1}{2\mu|{\bf{U}}|}\,,$ (108)

is the value for which $n^{2}_{-}$ diverges. In Fig. 9, the first vertical
dashed line, given by $\omega_{0}|{\bf{U}}|=1/4$, is asymptotic to the red
solid as well as the red dashed curve where the associated functions have
singularities at this point and change their signs. The second vertical dashed
line is in $\omega_{0}|{\bf{U}}|=1/2$, being asymptotic to both the red upper
and lower dashed-dotted curves. When $\omega>\omega_{0}$ one has $n^{2}_{-}<0$
whose lower branch becomes a purely imaginary refractive index $n_{-}$,
representing a nonpropagating mode. This behavior is characteristic in the
range $\omega_{0}<\omega<\omega_{-}$, with

$\omega_{-}=\frac{\epsilon}{2|{\bf{U}}|}\,,$ (109)

being the root of Eq. (107). Equation (109) stands for the cutoff frequency
above which the mode associated with $n_{-}$ propagates. The third and fourth
vertical dashed lines, given by $\omega_{-}|{\bf{U}}|=1$ and
$\omega_{-}|{\bf{U}}|=1.5$, indicate the beginning of the propagation regime
for the red solid curve and the red dashed curve, respectively.

Regarding the $\mathbf{U}$-longitudinal propagation modes, for which
${\bf{n}}\cdot{\bf{U}}=\pm n|{\bf{U}}|$, one takes $\alpha=1$ as well as
${\bf{U}}=(0,0,{U}_{3})$ for $\mathbf{n}$ given by Eq. (31). In this case, the
resulting modes are

$\mathbf{E}_{\pm}=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\ \mp\mathrm{i}\\\
0\end{pmatrix}\,,$ (110)

representing right-handed and left-handed circularly polarized waves,
respectively. Hence, when ${\bf{U}}$ and ${\bf{n}}$ point along the same
direction, the modes become transverse again such that their polarizations are
perpendicular to ${\bf{n}}$.

Now, in order to describe birefringence effects, we evaluate the rotatory
power by inserting Eq. (107) into Eq. (8), that is,

$\delta=-\frac{\sqrt{\mu\epsilon}}{2}\omega\left[\frac{g_{+}}{{\sqrt{1+2\mu\omega|{\bf{U}}|}}}-\frac{g_{-}}{{\sqrt{1-2\mu\omega|{\bf{U}}|}}}\right]\,,$
(111a) where $g_{\pm}=\sqrt{1\pm\frac{2\omega}{\epsilon}|{\bf{U}}|}\,.$ (111b)

The latter result holds for the regions where $n_{-}$ is real, that is, for
$\omega<\omega_{0}$ and $\omega>\omega_{-}$, according to Fig. 9. In the limit
$\omega|{\bf{U}}|\ll 1$, Eq. (111) provides a rotatory power nonlinear in the
frequency, namely:

$\delta\simeq\sqrt{\frac{\mu}{\epsilon}}(\mu\epsilon-1)\omega^{2}|{\bf{U}}|\,.$
(112)

As already mentioned, the refractive index $n_{-}$ is purely imaginary in the
range $\omega_{0}<\omega<\omega_{-}$, constituting an absorption zone, which
is explicitly given by

$\displaystyle\frac{1}{2\mu|{\bf{U}}|}<\omega<\frac{\epsilon}{2|{\bf{U}}|}\,.$
(113)

In this regime the refractive index reads

$\displaystyle n_{-}$
$\displaystyle=\mathrm{i}\sqrt{\mu\epsilon}\sqrt{\frac{1-2\omega|{\bf{U}}|/\epsilon}{2\mu\omega|{\bf{U}}|-1}}\,,$
(114)

and the corresponding dichroism coefficient is

$\displaystyle\delta_{\mathrm{d}}=\frac{\sqrt{\mu\epsilon}}{2}\omega\sqrt{\frac{1-2\omega|{\bf{U}}|/\epsilon}{2\mu\omega|{\bf{U}}|-1}}\,.$
(115)

##  V Final Remarks

In this work, we examined an electrodynamics of continuous media based on
Maxwell equations modified by CPT-odd terms, whereas the usual constitutive
relations ${\bf{D}}=\epsilon{\bf{E}}$ and ${\bf{H}}=\mu^{-1}{\bf{B}}$ were
assumed to hold. At first, we reviewed some basic properties of the MCFJ
model, followed by an analysis of the dimension-five higher-derivative
extension of MCFJ electrodynamics. Our general focus was on describing
electromagnetic-wave propagation in matter governed by these CPT-odd
modifications.

In Sec. III, we examined MCFJ electrodynamics, given by the Lagrangian of Eq.
(11), in a continuous medium with a fixed background
$V^{\mu}=(V^{0},{\bf{V}})$ present. To analyze the propagation behavior of
electromagnetic waves, we obtained the dispersion relations and the refractive
indices for two scenarios: (i) a timelike background,
$V^{\mu}=(V^{0},{\bf{0}})$ and (ii) a spacelike background,
$V^{\mu}=(0,{\bf{V}})$. For scenario (i), the refractive indices are always
real, giving rise to propagation without losses as well as birefringence. The
corresponding rotatory power, $\delta=-\mu V_{0}/2$, is frequency-independent.
For scenario (ii), one refractive index, $n_{+}$, is always real, while the
other, $n_{-}$, may be complex, corresponding to an absorption regime. In this
case, birefringence and dichroism occur in different frequency ranges. The
rotatory power and dichroism coefficient are both frequency-dependent.

In Sec. IV, we considered an electrodynamics in a ponderable medium modified
by a MCFJ-type higher-derivative term of dimension five, given by the
Lagrangian of Eq. (60). After writing up the altered Maxwell equations, a
sixth-order dispersion equation was achieved. In the purely timelike scenario,
${U}^{\mu}=({U}^{0},{\bf{0}})$, studied in Sec. IV.2, we obtained a third-
order equation in the refractive index $n$ providing three solutions. One
solution is real for any frequency, while the remaining two are complex for
some frequency range (absorption range). This behavior occurs even for a
dielectric nonconducting substrate. Such an effect is represented, for
example, by Figs. (5) and (6), where the graphs indicate that
$\mathrm{Im}[n(\omega)]\neq 0$ in the absorption range
$\omega_{-}<\omega<\omega_{+}$, with $\omega_{\pm}$ given in Eq. (77). That
property is entirely ascribed to the higher-derivative coupling term, since
the usual MCFJ electrodynamics in ponderable media does not exhibit an
absorption regime for a purely timelike background. Comparing Figs. (1) and
(7) with each other allows us to notice the differences between the
propagating modes in the usual and higher-derivative timelike case.
Furthermore, the propagation modes obtained correspond to left-handed and
right-handed circular polarizations [see Sec. IV.2.1].

In Sec. IV.3, we addressed the purely spacelike scenario, governed by an
involved dispersion relation. It was analyzed for two particular cases: (a)
the perpendicular configuration, where ${\bf{n}}\cdot{\bf{U}}=0$, and (b) the
longitudinal configurations, ${\bf{n}}\cdot{\bf{U}}=\pm n|{\bf{U}}|$. In
scenario (a), one finds $n_{+}^{2}>0$ for all frequencies, which indicates the
absence of absorption for this mode. On the other hand, $n_{-}$ becomes purely
imaginary for $\omega<\omega_{-}$, with $\omega_{-}$ defined in Eq. (91).
Absorption occurs in this range for the mode associated with $n_{-}$ (see Fig.
8). Hence, attenuation and birefringence are expected in the regions
$\omega<\omega_{-}$ and $\omega>\omega_{-}$, respectively. For scenario (b),
the refractive index $n_{+}$ is always real, as well, while $n_{-}$ exhibits
two distinct branches separated by the frequency $\omega_{0}$ given in Eq.
(108). The upper branch, defined for $\omega<\omega_{0}$, is characterized by
a region of sharp normal dispersion. In $\omega=\omega_{0}$, the refractive
index $n_{-}$ diverges. In the lower branch, the mode associated with $n_{-}$
turns complex and returns to the propagation regime for $\omega>\omega_{-}$.
This mode possesses different physical behaviors (propagation or absorption).
Birefringence occurs for $\omega<\omega_{0}$ and $\omega>\omega_{-}$, while
absorption takes place for $\omega_{0}<\omega<\omega_{-}$. In both ranges, the
acquired phaseshift between different modes and the absorption coefficient are
frequency-dependent.

In order to compare the spacelike configurations of the dimension-three and
five MCFJ electrodynamics, we examined Figs. 2 and 8. The dimension-three
model shows normal dispersion, while in the dimension-five framework modes
emerge that exhibit both anomalous and normal dispersion. The absorption zones
are qualitatively analogous to each other in both cases. Comparing Figs. 3 and
9, we notice that dimension-three and five modes are characterized by normal
and anomalous dispersion, while only the higher-derivative model exhibits two
branches of normal dispersion. In the limit of high frequencies, one has
$n_{\pm}^{2}\mapsto\mu\epsilon$ based on Eq. (III.2) for dimension-three MCFJ
electrodynamics and $n_{\pm}^{2}\mapsto 1$ inferred from Eqs. (90), (107) for
dimension-five MCFJ-type electrodynamics. These findings allow us to
distinguish between the two models. Therefore, the presence of higher
derivatives implies a richer plethora of frequency-dependent propagating
modes.

## Acknowledgments

The authors thank the anonymous referee for helpful comments that contributed
to improving the paper and clarifying some results. The authors also express
their gratitude to FAPEMA, CNPq and CAPES (Brazilian research agencies) for
invaluable financial support. In particular, M.M.F. is supported by FAPEMA
Universal 01187/18 and CNPq Produtividade 311220/2019-3. M.S. appreciates
support by FAPEMA Universal 00830/19 and CNPq Produtividade 312201/2018-4.
Furthermore, the authors are indebted to CAPES/Finance Code 001.

##  Appendix A Covariant Maxwell equations in matter

Here we derive the Maxwell equations and the constitutive relations from Eqs.
(2a) and (2b). Equation (2b) implies

$\displaystyle G^{0i}$
$\displaystyle=\frac{1}{2}\chi^{0i\alpha\beta}F_{\alpha\beta}\,,$ (116a)
$\displaystyle G^{0i}$
$\displaystyle=\frac{1}{2}\chi^{0i0j}F_{0j}+\frac{1}{2}\chi^{0ij0}F_{j0}+\frac{1}{2}\chi^{0imn}F_{mn}\,,$
(116b)

which can be simplified by using the symmetry properties of the tensor
$\chi^{\mu\nu\varrho\sigma}$, i.e., Eq. (3b). We also implement

$F_{0i}=-F_{i0}=E^{i}\,,\quad F_{mn}=-\epsilon_{mnk}B^{k}\,,$ (117)

where $\epsilon_{mnk}$ is the three-dimensional Levi-Civita symbol. Thus, Eq.
(116b) becomes

$\displaystyle G^{0i}$
$\displaystyle=-\chi^{0ij0}E^{j}-\frac{1}{2}\chi^{0imn}\epsilon_{mnk}B^{k}\,,$
(118a) $\displaystyle G^{0i}$ $\displaystyle=-D^{i}\,,$ (118b) where we have
defined the electric displacement field $D^{i}$, which involves the medium’s
response to applied electromagnetic fields, as
$D^{i}=\chi^{0ij0}E^{j}+\frac{1}{2}\chi^{0imn}\epsilon_{mnk}B^{k}\,.$ (118c)

From Eq. (118c) we can define the electric permittivity $\epsilon_{ij}$ as
well as the tensor $\gamma_{ij}$ describing the magnetic contribution to
electric displacement field (see Eq. (1a)) as

$\epsilon_{ij}\equiv\chi^{0ij0},\quad\gamma_{ik}\equiv\frac{\chi^{0imn}\epsilon_{mnk}}{2}\,.$
(119)

The antisymmetric nature of $\chi^{\mu\nu\alpha\beta}$ allows us to write

$G^{\mu\nu}=-G^{\nu\mu}\,.$ (120)

Now we can evaluate the components $G^{ij}$. In doing so, we get

$\displaystyle G^{ij}$
$\displaystyle=\frac{1}{2}\chi^{ij\alpha\beta}F_{\alpha\beta}\,,$ (121a)
$\displaystyle G^{ij}$
$\displaystyle=\frac{1}{2}\chi^{ij0k}F_{0k}+\frac{1}{2}\chi^{ijk0}F_{k0}+\frac{1}{2}\chi^{ijmn}F_{mn}\,,$
(121b)

which is recast by using Eq. (3b) as well as Eq. (117). Then,

$G^{ij}=-\chi^{ijk0}E^{k}-\frac{1}{2}\chi^{ijmn}\epsilon_{mnk}B^{k}\,.$ (122)

In order to obtain a relation between $G_{ij}$ and $H^{i}$ similar to that
between $F_{ij}$ and $B^{i}$, let us now contract Eq. (122) with
$\epsilon_{ijl}$ such that

$\epsilon_{ijl}G^{ij}=-\frac{2}{2}\epsilon_{ijl}\chi^{ijk0}E^{k}-\frac{2}{4}\epsilon_{ijl}\chi^{ijmn}\epsilon_{mnk}B^{k}\,,$
(123)

where we introduced the factor $(2/2)$ in each term of Eq. (123). The
motivation for doing so will become clear shortly, as this manipulation allows
us to write down an expression very similar to $F_{mn}=-\epsilon_{mnk}B^{k}$,
but for the components $G_{ij}$ and $H^{i}$. Thus, we define the magnetic
permeability $\mu_{ij}$ as well as $\tilde{\gamma}_{ij}$ governing the
electric contribution to the magnetic field (see Eq. (1b)) as

$\displaystyle(\mu^{-1})_{lk}\equiv\frac{1}{4}\epsilon_{ijl}\chi^{ijmn}\epsilon_{mnk},\quad\tilde{\gamma}_{lk}\equiv\frac{\epsilon_{ijl}\chi^{ijk0}}{2}\,.$
(124)

Then Eq. (123) simplifies as

$\displaystyle\epsilon_{ijl}G^{ij}$
$\displaystyle=-2(\mu^{-1})_{lk}B^{k}-2\tilde{\gamma}_{lk}E^{k}\,,$ (125a)
$\displaystyle\epsilon_{ijl}G^{ij}$ $\displaystyle=-2H^{l}\,,$ (125b) where we
have defined the magnetic field $H^{l}$, which describes the medium’s response
to applied electromagnetic fields via
$H^{l}=(\mu^{-1})_{lk}B^{k}+\tilde{\gamma}_{lk}E^{k}\,.$ (125c)

Let us contract Eq. (125b) with $\epsilon_{lmn}$, whereupon

$\displaystyle\epsilon_{lmn}\epsilon_{ijl}G^{ij}$
$\displaystyle=-2\epsilon_{lmn}H^{l}\,,$ (126a) $\displaystyle G^{mn}$
$\displaystyle=-\epsilon_{mnl}H^{l}\,,$ (126b)

where we have used Eq. (120).

Now that we have expressed the constitutive relations in terms of the
constitutive tensor $\chi^{\mu\nu\alpha\beta}$, we can derive the field
equations associated with the Lagrange density of Eq. (2a). Thus, we start by
rewriting Eq. (2a):

$\displaystyle\mathcal{L}$
$\displaystyle=-\frac{1}{8}\chi^{\mu\nu\alpha\beta}F_{\alpha\beta}F_{\mu\nu}-A_{\mu}J^{\mu},$
$\displaystyle=-\frac{1}{8}\chi^{\mu\nu\alpha\beta}\partial_{\alpha}A_{\beta}\partial_{\mu}A_{\nu}+\frac{1}{8}\chi^{\mu\nu\alpha\beta}\partial_{\alpha}A_{\beta}\partial_{\nu}A_{\mu}$
$\displaystyle\phantom{{}={}}+\frac{1}{8}\chi^{\mu\nu\alpha\beta}\partial_{\beta}A_{\alpha}\partial_{\mu}A_{\nu}-\frac{1}{8}\chi^{\mu\nu\alpha\beta}\partial_{\beta}A_{\alpha}\partial_{\nu}A_{\mu}$
$\displaystyle\phantom{{}={}}-A_{\mu}J^{\mu}\,.$ (127)

We rename the indices ($\nu\leftrightarrow\mu$) in the second and fourth term
of Eq. (127) and after that we employ the symmetry property of Eq. (3a). This
gives us

$\mathcal{L}=-\frac{1}{4}\chi^{\mu\nu\alpha\beta}\partial_{\alpha}A_{\beta}\partial_{\mu}A_{\nu}+\frac{1}{4}\chi^{\mu\nu\alpha\beta}\partial_{\beta}A_{\alpha}\partial_{\mu}A_{\nu}-A_{\mu}J^{\mu}\,,$
(128)

which can be simplified by replacing $\alpha\leftrightarrow\beta$ in the
second term and using Eq. (3b). Hence, we finally obtain

$\displaystyle\mathcal{L}$
$\displaystyle=-\frac{1}{2}\chi^{\mu\nu\alpha\beta}\partial_{\alpha}A_{\beta}\partial_{\mu}A_{\nu}-A_{\mu}J^{\mu}\,.$
(129)

Using the Euler-Lagrange equations

$\frac{\partial\mathcal{L}}{\partial
A_{\kappa}}-\partial_{\rho}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{\rho}A_{\kappa})}\right)=0\,,$
(130)

one arrives at

$\displaystyle\frac{\partial\mathcal{L}}{\partial(\partial_{\rho}A_{\kappa})}$
$\displaystyle=-\frac{1}{2}\left(\chi^{\beta\alpha\rho\kappa}\partial_{\beta}A_{\alpha}+\chi^{\rho\kappa\alpha\beta}\partial_{\alpha}A_{\beta}\right)\,,$
(131)

where we have relabeled $\mu\rightarrow\beta$, $\nu\rightarrow\alpha$ in the
first term on the right-hand side. Now, we also implement Eq. (3c) in the
first contribution, and in the second term we take advantage of Eq. (3b).
Therefore, Eq. (131) provides

$\frac{\partial\mathcal{L}}{\partial(\partial_{\rho}A_{\kappa})}=-\frac{1}{2}\chi^{\rho\kappa\beta\alpha}F_{\beta\alpha}=-G^{\rho\kappa}\,,$
(132)

and one also finds

$\frac{\partial\mathcal{L}}{\partial
A_{\kappa}}=-J^{\mu}\delta_{\mu\kappa}=-J^{\kappa}\,.$ (133)

So using Eqs. (132) and (133) in Eq. (130), we finally get the covariant form
of the Maxwell equations in simple matter:

$\partial_{\rho}G^{\rho\kappa}=J^{\kappa}\,.$ (134)

Taking $\kappa=0$, one finds Gauss’s law:

$\displaystyle\partial_{i}G^{i0}$ $\displaystyle=J^{0}\,,$ (135a)
$\displaystyle\nabla\cdot{\bf{D}}$ $\displaystyle=\rho\,,$ (135b)

where we have employed Eq. (118b) and $J^{\mu}=(\rho,{\bf{J}})$. Ampère’s law
is obtained by taking $\kappa=i$ in Eq. (134), that is

$\displaystyle\partial_{0}G^{0i}+\partial_{j}G^{ji}$ $\displaystyle=J^{i}\,,$
(136a) $\displaystyle\partial_{t}(-D^{i})-\partial_{j}(\epsilon_{jik}H^{k})$
$\displaystyle=J^{i}\,,$ (136b)

where we have used Eqs. (118b), (120), and (126b). Applying further
simplifications to Eq. (136b), yields

$\displaystyle\epsilon_{ijk}\partial_{j}H^{k}-\partial_{t}D^{i}$
$\displaystyle=J^{i}\,,$ (137a)
$\displaystyle\nabla\times{\bf{H}}-\partial_{t}{\bf{D}}$
$\displaystyle={\bf{J}}\,.$ (137b)

##  Appendix B  Rotatory power and dichroism coefficient

As mentioned at the end of Sec. II, when the propagating modes resulting from
an electromagnetic theory are left-handed and right-handed circularly
polarized waves, birefringence is characterized in terms of the rotatory power
while absorption is described via the dichroism coefficient, presented in Eqs.
(8) and (9), respectively. Such relations can be derived by means of the
polarization vectors of a wave traveling through a medium. Consider, for
instance, a linearly polarized wave propagating through a medium along the $z$
axis. Hence, the initial electric field can be written as

${\bf{E}}_{i}={\bf{E}}_{0i}\mathrm{e}^{\mathrm{i}(kz-\omega t)}\,,$ (138a)
with the polarization vector (for an electric field pointing along the $x$
axis): $\displaystyle{\bf{E}}_{0i}=\begin{pmatrix}1\\\ 0\\\
0\end{pmatrix}=\frac{1}{2}\begin{pmatrix}1\\\ -\mathrm{i}\\\
0\end{pmatrix}+\frac{1}{2}\begin{pmatrix}1\\\ \mathrm{i}\\\ 0\end{pmatrix}\,,$
(138b)

which corresponds to the sum of polarization vectors associated with left-
handed and right-handed circular polarizations, respectively. After the wave
passes through a distance $z$ in the medium, the final electric field is a
linear combination of two components, ${\bf{E}}_{+}$ and ${\bf{E}}_{-}$, with
the wave vectors ${\bf{k}}_{+}$ and ${\bf{k}}_{-}$, respectively. One then has

$\displaystyle{\bf{E}}_{f}$
$\displaystyle={\bf{E}}_{+}\mathrm{e}^{\mathrm{i}(k_{+}z-\omega
t)}+{\bf{E}}_{-}\mathrm{e}^{\mathrm{i}(k_{-}z-\omega t)},$
$\displaystyle{\bf{E}}_{f}$ $\displaystyle=\frac{1}{2}\begin{pmatrix}1\\\
\mathrm{i}\\\
0\end{pmatrix}\mathrm{e}^{\mathrm{i}k_{+}z}\mathrm{e}^{-\mathrm{i}\omega
t}+\frac{1}{2}\begin{pmatrix}1\\\ -\mathrm{i}\\\
0\end{pmatrix}\mathrm{e}^{\mathrm{i}k_{-}z}\mathrm{e}^{-\mathrm{i}\omega
t}\,,$ (139)

which can be cast into the form

$\displaystyle{\bf{E}}_{f}$
$\displaystyle=\frac{1}{2}\mathrm{e}^{\mathrm{i}\psi}\mathrm{e}^{-\mathrm{i}\omega
t}\left[\mathrm{e}^{-\mathrm{i}\theta}\begin{pmatrix}1\\\ \mathrm{i}\\\
0\end{pmatrix}+\mathrm{e}^{\mathrm{i}\theta}\begin{pmatrix}1\\\ -\mathrm{i}\\\
0\end{pmatrix}\right],$ $\displaystyle{\bf{E}}_{f}$
$\displaystyle=\mathrm{e}^{\mathrm{i}\psi}\mathrm{e}^{-\mathrm{i}\omega
t}\begin{pmatrix}\cos\theta\\\ \sin\theta\\\ 0\end{pmatrix}\,,$ (140a) with
the quantities $\displaystyle\theta$
$\displaystyle\equiv-\frac{(k_{+}-k_{-})z}{2}\,,$ (140b) $\displaystyle\psi$
$\displaystyle\equiv\frac{(k_{+}+k_{-})z}{2}\,.$ (140c)

Notice that Eq. (140a) describes a linearly polarized wave whose polarization
vector is rotated by an angle $\theta$. From Eq. (140b), one obtains

$\theta=-\frac{(n_{+}-n_{-})z\omega}{2}\,,$ (141)

where we have used ${\bf{k}}=\omega{\bf{n}}$. In general, the refractive
indices can be complex quantities. Because of this, one can infer from Eq.
(141)

$\frac{\theta}{z}=-\frac{\omega}{2}\left[\mathrm{Re}(n_{+})+\mathrm{i}\mathrm{Im}(n_{+})-\mathrm{Re}(n_{-})-\mathrm{i}\mathrm{Im}(n_{-})\right]\,,$
(142)

from which we define the specific rotatory power stated in Eq. (8) as well as
the dichroism coefficient of Eq. (9). Notice that when the medium is
nonbirefringent, $\theta=0$ and $\psi=kz$. Then, the form of Eq. (138a) is
recovered from Eq. (140a).

## References

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