# Anisotropic scaling non-relativistic holography: a symmetry perspective

H. Lü, Pujian Mao, and Jun-Bao Wu

USTC-ICTS/PCFT-23-19

Anisotropic scaling non-relativistic holography: a symmetry perspective

H. Lüa,b,c, Pujian Maoa, and Jun-Bao Wua,b

a Center for Joint Quantum Studies and Department of Physics,

School of Science, Tianjin University, 135 Yaguan Road, Tianjin 300350, China

b Peng Huanwu Center for Fundamental Theory,

Hefei, Anhui 230026, China

c Joint School of National University of Singapore and Tianjin University,

International Campus of Tianjin University, Binhai New City, Fuzhou 350207,
China

Abstract. We study the holographic dual of the two dimensional non-
relativistic field theory with anisotropic scaling from a symmetry
perspective. We construct a new four dimensional metric with two dimensional
global anisotropic scaling isometry. The four dimensional spacetime is
homogeneous and is a solution of Einstein gravity with quadratic-curvature
extension. We consider this spacetime dual to the vacuum of the boundary field
theory. By introducing proper solution phase space, we find that the
asymptotic symmetry of the gravity theory is the two dimensional local
anisotropic conformal symmetry, which recovers precisely the results from the
dual non-relativistic field theory side.

##  1 Introduction

An interesting feature of quantum field theory (QFT) with scaling symmetry in
two dimensional (2D) spacetime is that the global symmetry can be enhanced to
an infinite dimensional local symmetry. The best known example is revealed by
Polchinski in [1] that a local, unitary Poincare-invariant 2D QFT with a
global scaling symmetry and a discrete non-negative spectrum of scaling
dimensions must have both a left and a right local conformal symmetry. More
than ten years ago, Hofman and Strominger showed that, for a chiral situation,
the local conformal symmetry is still implied [2] which leads to two kinds of
minimal theories, namely the 2D conformal field theory (CFT) [3] or the 2D
warped conformal field theory (WCFT) [4]. The global symmetry of the WCFT is
$SL(2,R)\times U(1)$ and its local enhancement is the Virasoro-Kac-Moody
algebra. The enhanced local symmetries have a clear dual interpretation from
the gravity side in the context of AdS3/CFT2 correspondence. They reveal the
enhancement of the asymptotic symmetry from the isometry of the AdS spacetime.
More precisely, for AdS3 gravity, the asymptotic symmetry group under Brown-
Henneaux boundary conditions contains two copies of Virasoro symmetry [5].
Meanwhile, for waped AdS3 case [6, 7], the algebra of asymptotic symmetries is
isomorphic to the semi-direct product of a Virasoro algebra and an algebra of
currents under Compère-Detournay boundary condition [8].111The local symmetry
of WCFT can be also realized in AdS3 gravity under the Compère-Song-Strominger
boundary conditions [9].

Recently, the enhanced symmetry was revealed for 2D Galilean field theories
with anisotropic scaling symmetry [10]. The 2D Galilean field theories with
global translations and anisotropic scaling symmetries are shown to have
enhanced local symmetries which are generated by the infinite dimensional
spin-$k$ Galilean algebra. For 2D Galilean field theories with isotropic
scaling symmetry, the dual gravity theory is proposed to be three dimensional
and asymptotically flat [11, 12] where the asymptotic symmetry is enhanced
from the global 3D Poincare group to infinite dimensional Bondi-Metzner-Sachs
(BMS) group. For the anisotropic case, the dual gravity theory is proposed
[13] to be higher-dimensional Schrödinger geometry [14]. But the asymptotic
symmetry of these geometries has not been addressed. Whether it can recover
the enhanced local symmetry from the field theory side is not yet known.

The main purpose of this work is to find the gravitational duality for the
enhancement of symmetry in [10]. We find a new 4D metric with the isometry
group isomorphic to the global symmetry of the 2D Galilean field theories with
anisotropic scaling, which presents a different realization of gravity dual
for the 2D theory from the one in [14, 13]. This metric describes a
homogeneous spacetime with constant curvature tensor. The 4D spacetime is
supposed to be dual to the vacuum of the field theory. We show that Einstein
gravity with quadratic-curvature extension admits the 4D spacetime as vacuum
solution when the coupling constants of the higher derivative terms are
specially adapted to the dynamical exponent $z$. The 4D spacetime, with
restricted range of the dynamical exponent $z$, can also be obtained from the
Einstein-Proca theory. Then, we find a solution phase space of the higher
derivative theory, which admits the vacuum metric and yields precisely the
infinite dimensional spin-$k$ Galilean algebra in [10] as asymptotic symmetry.
Since the duality is between 4D and 2D, we adopt a double expansion of inverse
spatial directions. Correspondingly, the 2D Galilean theory is defined on the
corner of the 4D spacetime boundary.

The organization of this paper is as follows. In Section 2, we present the
vacuum solution and show that it has constant curvature tensor. In Section 3,
we comment on the gravitational theory that admits the vacuum solution. In
Section 4, we show a solution phase space and derive the asymptotic symmetry
of the phase space, namely the most generic residual gauge transformations
that preserve the solution phase space. In Section 5, we derive the asymptotic
symmetry algebra. We conclude in the last section.

##  2 Spacetime with global anisotropic scaling symmetry

The global symmetry of 2D Galilean field theory with anisotropic scaling
consists of translations along two directions, the Galilean boost

$x\rightarrow x-vt\,,$ (2.1)

and the dilations222We set the parameter $c$ in [10] to be 1. This can always
be realized by the rescaling $\lambda^{c}\rightarrow\tilde{\lambda}$, hence
$\frac{d}{c}\rightarrow k$.

$t\rightarrow\lambda t,\quad x\rightarrow\lambda^{k}x\,,$ (2.2)

In [10], it is shown that symmetry of the 2D Galilean field theory with the
above global symmetry is enhanced to an infinite dimensional spin-$k$ Galilean
algebra. In plane modes, the algebra is given by

$\begin{split}&[l_{n},l_{m}]=(n-m)l_{n+m}\,,\\\
&[l_{n},m_{m}]=(kn-m)m_{n+m}\,,\\\ &[m_{n},m_{m}]=0\,.\end{split}$ (2.3)

This algebra with $k=1$ is precisely the BMS3 algebra derived in [15, 16].

From the holographic principle, the field theory is defined on the boundary of
the dual gravity theory. The global symmetry of the field theory is the
isometry of the bulk spacetime. For the 2D Galilean field theory with
anisotropic scaling, the dual gravity theory is four dimensional. The 4D
metric which we find with global anisotropic scaling isometry is

$ds^{2}={\frac{\ell^{2}dr^{2}}{r^{2}}}+r^{2z}dy^{2}+r^{2}(2dxdt-ydt^{2})\,.$
(2.4)

The metric is invariant under the translations along $t$ and $x$ directions
plus the following global transformations,

$\displaystyle\hbox{Galilean embedding}:$ $\displaystyle x\rightarrow
x-vt\,,\quad t\rightarrow t\,,\quad y\rightarrow y-2v\,,$
$\displaystyle\hbox{scaling embedding}:$ $\displaystyle
x\rightarrow\lambda^{k}x\,,\quad t\rightarrow\lambda t\,,\quad
y\rightarrow\lambda^{k-1}y\,,\quad r\rightarrow\lambda^{-\frac{k+1}{2}}r\,,$
(2.5)

with

$k={\frac{2+z}{2-z}}.$ (2.6)

The spacetime described by the metric (2.4) is homogeneous and has constant
curvature tensor. For constant $y$, the metric is the AdS3 in planar
coordinate with 2D Minkowski boundary. The introduction of $y$ serves the
purpose of breaking the 2D conformal group to Gallilean symmetry with
anistropic scaling. Our construction of the background metric (2.4) is very
different from the proposal in [13], where an extra null Killing direction
associated with the coordinate $\xi$ was introduced. In our case, however, $y$
is not a Killing direction, but is analogous to the radial coordinate $r$,
which allows us to take an additional $y\rightarrow\infty$ limit in the
asymptotic expansion. This makes the analysis of the asymptotic symmetry
simpler, well controlled by both the $(r,y)$ coordinates. We shall return to
this in the next sections.

The constant curvature tensor can be easily obtained from the vielbein
formalism. A natural vielbein choice that respects the global isometry is

$e^{0}={\frac{\ell dr}{r}}\,,\qquad e^{1}=r^{z}dy\,,\qquad
e^{+}=r^{1-{\frac{1}{2}}z}dt\,,\qquad
e^{-}=r^{1+{\frac{1}{2}}z}(dx-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}ydt)\,,$ (2.7)

such that $ds^{2}=e^{0}e^{0}+e^{1}e^{1}+2e^{+}e^{-}$. Thus we have

$\displaystyle de^{0}=0\,,\qquad de^{1}={\frac{z}{\ell}}e^{0}\wedge
e^{1}\,,\qquad de^{+}={\frac{1-{\frac{1}{2}}z}{\ell}}e^{0}\wedge e^{+}\,,$
$\displaystyle de^{-}={\frac{1+{\frac{1}{2}}z}{\ell}}e^{0}\wedge
e^{-}-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle 2}}}e^{1}\wedge e^{+}\,.$
(2.8)

The spin connections are

$\displaystyle\omega^{\pm
0}={\frac{e^{\pm}}{\ell}}\,,\quad\omega^{+-}={\frac{z}{2\ell}}e^{0}\,,\quad\omega^{10}={\frac{ze^{1}}{\ell}}\,,\quad\omega^{1-}={\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2}}}e^{+}\,.$ (2.9)

We thus have the curvature tensor 2-form
$\Theta^{a}{}_{b}={\frac{1}{2}}R^{a}{}_{bcd}e^{c}\wedge e^{d}$ as

$\displaystyle\Theta^{+}{}_{+}$
$\displaystyle=-{\frac{1}{\ell^{2}}}e^{+}\wedge
e^{-}\,,\quad\Theta^{+}{}_{0}={\frac{1}{\ell^{2}}}e^{0}\wedge
e^{+}\,,\quad\Theta^{-}{}_{0}={\frac{1}{\ell^{2}}}e^{0}\wedge
e^{-}+{\frac{1-z}{2\ell}}e^{+}\wedge e^{1}\,,$ $\displaystyle\Theta^{+}{}_{1}$
$\displaystyle=-{\frac{z}{\ell^{2}}}e^{+}\wedge
e^{1}\,,\quad\Theta^{-}{}_{1}={\frac{z-1}{2\ell}}e^{0}\wedge
e^{+}-{\frac{z}{\ell^{2}}}e^{-}\wedge
e^{1}\,,\quad\Theta^{0}{}_{1}=-{\frac{z^{2}}{\ell^{2}}}e^{0}\wedge e^{1}\,.$
(2.10)

Hence, the independent non-vanishing components of the Riemann tensor are

$\displaystyle R^{+}{}_{++-}=-{\frac{1}{\ell^{2}}}\,,\qquad
R^{+}{}_{0+0}=-{\frac{1}{\ell^{2}}}\,,\qquad
R^{-}{}_{0+1}={\frac{1-z}{2\ell}}\,,$ $\displaystyle
R^{+}{}_{1+1}=-{\frac{z}{\ell^{2}}}\,,\qquad
R^{0}{}_{101}=-{\frac{z^{2}}{\ell^{2}}}\,.$ (2.11)

The Ricci tensor is given by

$R_{+-}=-{\frac{z+2}{\ell^{2}}}\,,\qquad
R_{00}=-{\frac{z^{2}+2}{\ell^{2}}}\,,\qquad
R_{11}=-{\frac{z(z+2)}{\ell^{2}}}\,.$ (2.12)

and the Ricci scalar is

$R=-{\frac{2(z^{2}+2z+3)}{\ell^{2}}}\,.$ (2.13)

When $z=1$, the metric (2.4) is just the 4D AdS spacetime.

As one can see from (2.6) that $k=-1$ requires $z\rightarrow\pm\infty$. So the
metric is not well defined for this particular choice. To include this
limiting case, we make coordinate transformation and redefine parameter as
follows,

$r=\tilde{r}^{{\frac{1}{z}}}\,,\qquad\ell=z\tilde{\ell}\,.$ (2.14)

The new metric admits a $z\rightarrow\pm\infty$ limit, and we obtain

$ds^{2}={\frac{\tilde{\ell}^{2}d\tilde{r}^{2}}{\tilde{r}^{2}}}+\tilde{r}^{2}dy^{2}-ydt^{2}+2dtdx\,.$
(2.15)

The vielbein, spin connections and curvature can simply obtained from the same
treatment when taking the limit $z\rightarrow\pm\infty$.

##  3 Dual gravity theories

Einstein gravity with the most general quadratic-curvature extension in four
dimensions is

${\cal L}=\sqrt{-g}(R-2\Lambda_{0}+\alpha R^{2}+\beta
R^{\mu\nu}R_{\mu\nu})\,.$ (3.1)

Since the Gauss-Bonnet term is a total derivative, we do not add the Riemann
squared term. The metric (2.4) is a solution of the theory (3.1) when the
coupling constants $\alpha$, $\beta$ and the cosmological constant
$\Lambda_{0}$ are specially chosen with respect to the dynamical exponent $z$,

$\displaystyle z=1:$
$\displaystyle\Lambda_{0}=-{\frac{3}{\ell^{2}}}\,,\qquad\hbox{no constraint on
$\alpha$ and $\beta$,}$ (3.2) $\displaystyle z\neq 1:$
$\displaystyle\Lambda_{0}=-\frac{(z^{2}+2z+3)}{2\ell^{2}}\,,\quad
4\alpha\left(z^{2}+2z+3\right)+2\beta\left(z^{2}+2\right)=\ell^{2}\,.$ (3.3)

Note that, when setting $\beta=0$, the generic theory (3.1) is reduced to the
ghost free theory ${\cal L}=\sqrt{-g}(R-2\Lambda_{0}+\alpha R^{2})$, which
still admits the vacuum solution (2.4) and the coupling constant is completely
fixed by

$\alpha=\frac{\ell^{2}}{4\left(z^{2}+2z+3\right)}\,.$ (3.4)

It is also of interest to examine whether our metric (2.4) can arise from
Einstein theory with minimally-coupled matter. The curvature tensor in the
vielbein base implies that

$G^{00}={\frac{1+2z}{\ell^{2}}}\,,\qquad G^{11}={\frac{3}{\ell^{2}}}\,,\qquad
G^{+-}={\frac{z^{2}+z+1}{\ell^{2}}}\,.$ (3.5)

If the solutions are constructed from the Einstein theory with minimally
coupled matter fields, then we can deduce that $T_{\rm tot}^{ab}=G^{ab}$. In
the diagonal base, we have

$\rho=-\frac{z^{2}+z+1}{\ell^{2}}=-p_{1}\,,\qquad
p_{2}=\frac{3}{\ell^{2}}\,,\qquad p_{3}={\frac{1+2z}{\ell^{2}}}\,.$ (3.6)

The null energy conditions (NEC) $\rho+p_{i}\geq 0$ imposes on the following
conditions

$2-z-z^{2}\geq 0\,,\qquad z(1-z)\geq 0\,.$ (3.7)

This requires that $0\leq z\leq 1$. As a concrete example, we consider
Einstein gravity coupled to massive vector theory (Proca theory)

${\cal L}=\sqrt{-g}(R-2\Lambda-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
4}}}F^{2}-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle 2}}}\mu^{2}A^{2})\,.$
(3.8)

To admit metric (2.4) as a solution, the cosmological constant $\Lambda$, the
coupling constant $\mu$, and the vector field $A$ in the Proca theory should
have the following forms,

$\Lambda=-\frac{z^{2}+z+4}{2\ell^{2}}\,,\qquad\mu^{2}=\frac{2z}{\ell^{2}}\,,\qquad
A=\sqrt{{\frac{2(1-z)}{z}}}\,r^{z}\,dy\,.$ (3.9)

Thus, the reality condition requires that $0<z\leq 1$.

In fact, although the total energy momentum tensor in the Einstein-
Proca-$\Lambda$ theory satisfies only the NEC, but violates both the weak and
strong energy condition, the energy-momentum tensor of the Proca field

$\rho^{A}={\frac{(1-z)(2+z)}{\ell^{2}}}=-p_{1}^{A}=p_{3}^{A}\,,\qquad
p_{2}^{A}=-{\frac{(1-z)(2-z)}{2\ell^{2}}}\,,$ (3.10)

satisfies all the energy conditions, namely strong energy condition and
dominant energy condition, since $0<z\leq 1$. It is well known that the
culprit of violating weak or strong energy condition in these cases is the
cosmological constant.

##  4 Asymptotic symmetries

The complete set of gauge transformation of Einstein gravity with quadratic-
curvature extension is generated by infinitesimal diffeomorphism,333Since the
Proca theory admits metric (2.4) as a solution only when $0<z\leq 1$, we
consider Einstein gravity with quadratic-curvature extension for deriving the
asymptotic symmetry to have a consistent study for generic value of $z$.

$\delta_{\xi}g_{\mu\nu}=\mathcal{L}_{\xi}g_{\mu\nu}\,.$ (4.1)

The asymptotic symmetry is the residual gauge transformation that preserves
required gauge and boundary conditions. We will follow the Fefferman-Graham
gauge,

$g_{rr}={\frac{\ell^{2}}{r^{2}}}\,,\quad g_{rA}=0\,,$ (4.2)

where $A=(y,t,x)$. The residual gauge transformation preserving the Fefferman-
Graham gauge can be solved as follows:

  * •

$\mathcal{L}_{\xi}g_{rr}=0\quad\Longrightarrow\quad\xi^{r}=-\frac{1}{2}r\Psi(y,t,x)$.

  * •

$\mathcal{L}_{\xi}g_{rA}=0\quad\Longrightarrow\quad\xi^{A}=Y^{A}(y,t,x)-\frac{\ell^{2}}{2}\partial_{B}\Psi\int_{r}^{\infty}\frac{dr^{\prime}}{r^{\prime}}g^{AB}$,

where $g^{AB}$ is the inverse metric. For a generic $z$, it is very hard to
impose boundary conditions to study asymptotic symmetries in a unified way.
Alternatively, we apply the solution phase space method [17, 18, 19] to
investigate the asymptotic (symplectic) symmetry of the system. The solution
phase space method was originally introduced for the Near-Horizon Extremal
Geometries (NHEG) [17, 18]. Since the NHEG is absent of dynamical physical
perturbations [20], it is naturally to consider the action of diffeomorphisms
on the NHEG to construct the classical phase space. In our study, the theory
(3.1) depends on the dynamical exponent $z$, namely for each choice of $z$,
there are a corresponding theory and fall-off conditions. So the solution
phase space method is particularly useful for us to study asymptotic
symmetries for generic choice of $z$.

We find a solution phase space of the theory (3.1), which is given by

$ds^{2}=\frac{\ell^{2}}{r^{2}}dr^{2}+r^{2z}\Phi(t)^{2}dy^{2}+2r^{2}dtdx\\\
-r^{2}\left[yf_{1}(t)+f_{2}(t)+xf_{3}(t)+\frac{\ell^{2}f_{3}(t)\Phi^{\prime}(t)}{2\Phi(t)r^{2}}-\frac{\ell^{2}\Phi^{\prime\prime}(t)}{\Phi(t)r^{2}}\right]dt^{2}\,,$
(4.3)

where a prime denotes a derivative on $t$. There are four arbitrary functions
of time $t$, namely, $\Phi(t),f_{1}(t),f_{2}(t),f_{3}(t)$ which represent four
types of independent diffeomorphisms. They are dynamical fields of the phase
space. However they only represent boundary dynamics as they are introduced
from the action of diffeomorphisms on the homogeneous spacetime (2.4). There
is no propagating degree of freedom in the phase space. The dual theory of the
2D Galilean field theory is from boundary gravity in the context of AdS/CFT
[5, 21].

The most generic residual gauge transformation preserving the phase space is
characterized by

$\begin{split}&\Psi(y,t,x)=\psi(t)\,,\;\;\;\;\;\;Y^{y}(y,t,x)=Y_{1}y+Y_{2}\,,\\\
&Y^{t}(y,t,x)=L(t)\,,\;\;\;\;Y^{x}(y,t,x)=[\psi(t)-\partial_{t}L(t)]x+M(t)\,,\end{split}$
(4.4)

where $Y_{1}$ and $Y_{2}$ are real constant. Note that we choose all symmetry
parameters field independent.

To manifest the fact that the spacetime is scaling invariant asymptotically,
we need to perform a double expansion in terms of both $\frac{1}{r}$ and
$\frac{1}{y}$ in the region of large $r$ and large $y$ and set
$f_{1}(t)=\Phi(t)$. The asymptotic form of the metric is

$ds^{2}=\frac{\ell^{2}}{r^{2}}dr^{2}+r^{2z}\Phi(t)^{2}dy^{2}+2r^{2}dtdx-r^{2}y\Phi(t)dt^{2}+\text{sub-
leading terms}\,,$ (4.5)

The leading part of the metric is invariant under the scaling transformation,

$\begin{split}&x\rightarrow\lambda^{k}x\,,\qquad t\rightarrow\lambda
t\,,\qquad r\rightarrow\lambda^{-\frac{k+1}{2}}r\,,\\\
&y\rightarrow\lambda^{k-1-a}y\,,\quad\Phi(t)\rightarrow\lambda^{a}\Phi(t)\,.\end{split}$
(4.6)

The condition $f_{1}(t)=\Phi(t)$ will further yield that

$\psi(t)=(k+1)L^{\prime}(t)\,.$ (4.7)

To summarize, the asymptotic symmetry that preserves the scaling invariant
phase space (4.3) is generated by

$\begin{split}&\xi^{r}=-\frac{1}{2}(k+1)L^{\prime}(t)r\,,\\\
&\xi^{y}=Y_{1}y+Y_{2}\,,\\\ &\xi^{t}=L(t)\,,\\\
&\xi^{x}=M(t)+kxL^{\prime}(t)-\frac{\ell^{2}}{4}(k+1)\frac{L^{\prime\prime}(t)}{r^{2}}\,.\end{split}$
(4.8)

##  5 Asymptotic symmetry algebra

The asymptotic Killing vectors (LABEL:asg) satisfy the standard Lie algebra

$[\xi_{1},\xi_{2}]^{\mu}=\xi_{1}^{\nu}\partial_{\nu}\xi_{2}^{\mu}-\xi_{2}^{\nu}\partial_{\nu}\xi_{1}^{\mu}\,.$
(5.1)

The algebra is closed and the Jacobi identity of the symmetry algebra is
guaranteed by the Jacobi identity of the Lie algebra of three vectors. In
terms of the basis vectors

$\begin{split}&L_{m}=\frac{1}{2}(k+1)(m+1)t^{m}r\partial_{r}-t^{m+1}\partial_{t}-k(m+1)t^{m}x\partial_{x}\\\
&\hskip
56.9055pt+\frac{\ell^{2}}{4}(k+1)m(m+1)t^{m-1}\frac{1}{r^{2}}\partial_{x}\,,\\\
&M_{m}=-t^{m+k}\partial_{x}\,,\qquad B=\partial_{y}\,,\qquad
D=y\partial_{y}\,,\end{split}$ (5.2)

the asymptotic algebra is

$\displaystyle[L_{n},L_{m}]=(n-m)L_{m+n}\,,$ (5.3)
$\displaystyle[L_{n},M_{m}]=(kn-m)M_{m+n}\,,$ (5.4)
$\displaystyle[M_{n},M_{m}]=0\,,$ (5.5)
$\displaystyle[B,B]=0\,,\;\;\;\;[D,D]=0\,,\;\;\;\;[D,B]=-\frac{k-1}{k+1}B\,,$
(5.6) $\displaystyle[B,L_{m}]=0\,,\;\;\;\;\;\;[B,M_{m}]=0,$ (5.7)
$\displaystyle[D,L_{m}]=0\,\,,\;\;\;\;\;\;[D,M_{m}]=0\,.$ (5.8)

The first three lines of above equations are precisely the symmetry algebra
(2.3) derived from the dual 2D field theory side [10]. We have two more
generators $D$ and $B$ from the extra dimension $y$ which commute all the
modes $L_{m}$ and $M_{m}$.

The Killing vector that generates Galilean transformation of the background
metric (2.4) is

$G\equiv\partial_{y}+\frac{1}{2}t\partial_{x}\,.$ (5.9)

It is of course included in the asymptotic Killing vectors which in mode
expansion (LABEL:mode) is given by

$G=B-\frac{1}{2}M_{1-k}\,.$ (5.10)

The scaling symmetry

$\Delta\equiv-\frac{1}{2}r\partial_{r}+\frac{k-1}{k+1}y\partial_{y}+\frac{t}{k+1}\partial_{t}+\frac{kx}{k+1}\partial_{x}\,,$
(5.11)

in mode expansion (LABEL:mode) is given by

$\Delta=\frac{k-1}{k+1}D-\frac{1}{k+1}L_{0}.$ (5.12)

Two translations along $t$ and $x$ directions are $L_{-1}$ and $M_{-k}$. The
four Killing vectors form a closed subalgebra of the asymptotic symmetry
algebra,

$\displaystyle[G,G]=0\,,\;\;\;\;[\Delta,\Delta]=0\,,\;\;\;\;[\Delta,G]=-\frac{k-1}{k+1}G\,,$
(5.13)
$\displaystyle[G,L_{-1}]=-\frac{1}{2}M_{-k}\,,\qquad[\Delta,L_{-1}]=-\frac{1}{k+1}L_{-1}\,,$
(5.14)
$\displaystyle[G,M_{-k}]=0\,,\qquad[\Delta,M_{-k}]=-\frac{k}{k+1}M_{-k}\,.$
(5.15)

The commutators of $G$ and $\Delta$ with other modes are

$\displaystyle[G,L_{m}]=\frac{1}{2}[(m+1)k-1]M_{m+1-k}\,,$ (5.16)
$\displaystyle[\Delta,L_{m}]=\frac{m}{k+1}L_{m}\,,$ (5.17)
$\displaystyle[G,M_{m}]=0\,,$ (5.18)
$\displaystyle[\Delta,M_{m}]=\frac{m}{k+1}M_{m}\,.$ (5.19)

##  6 Conclusion

In this paper, we present an alternative realization of the holographic dual
for the 2D Galilean field theory with anisotropic scaling. The isometry of the
bulk spacetime is isomorphic to the global symmetry of the 2D Galilean field
theory. And the bulk spacetime is a vacuum solution of the Einstein gravity
with quadratic-curvature extension. For some restricted range of the dynamical
exponent $z$, the bulk spacetime is also a solution of the Einstein-Proca
theory. We find a solution phase space which admits the vacuum solution. The
residual gauge transformation preserving the solution phase space, namely the
asymptotic symmetry of this system, recovers precisely the enhanced local
symmetry of the 2D Galilean field theory in [10]. Since the dual gravity
theory is in 4D, our proposal is an example of a codimension-2 holography and
we have two more symmetry transformations from the extra dimensions. We show
that the algebra of the two extra symmetry generators is closed and the
generators of the two extra symmetry commute with all the generators of the 2D
enhanced local symmetry. In our construction, the extra dimension is not a
Killing direction and there is no conserved quantity associated to this extra
direction. So it can not be interpreted as particle-number circle, such as in
the Schrodinger theories [22] arisen from TsT transformation [23] or Melvin
twist of AdS space [24]. The extra dimension $y$ in our case is not compacted,
so one can not get rid of it by dimensional reduction [25]. However, in our
construction, one can realize the enhanced local symmetry of the boundary
field theory as asymptotic symmetry of the dual gravity system in a double
expansion in large $r$ and large $y$. Correspondingly the dual field theory is
supposed to be defined on the corner of the 4D boundary, see, e.g., in [26]
for a comprehensive introduction of the corner proposal and references
therein.

As an ending remark, it is worth mentioning that our construction is
temporarily restricted in 4D with a ground state. It is definitely of interest
to generalize our construction to higher dimension or black hole solutions
(thermal state) to incorporate various anisotropic invariant field theories in
different perspectives, such as theories studied in [27, 28, 29, 30, 31, 32].

## Acknowledgments

This work is supported in part by the National Natural Science Foundation of
China under Grants No. 11905156, No. 11875200, No. 11975164, No. 11935009, No.
11947301, No. 12047502 and Natural Science Foundation of Tianjin under Grant
No. 20JCYBJC00910.

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# Broadband Bragg Phenomenon in a Uniform Medium

Martin W. McCall and Stefanos Fr. Koufidis
Blackett Laboratory, Department of Physics,
Imperial College of Science, Technology and Medicine,
Prince Consort Road, London SW7 2AZ, United Kingdom
<EMAIL_ADDRESS>

###### Abstract

A new mechanism of Bragg reflection is identified, one that, remarkably,
occurs in a uniform medium and relies on resonant tuning of the medium’s
parameters. Due to uniformity, reflection ensues over a broad wavelength
range, much like a metal, but it is polarization dependent: one circular state
is reflected, whilst the other is transmitted. Such a medium can thus provide
a broadband, low-loss polarization divider/combiner. Assessment of the
required parameters suggests that manufacturing is within range of current
metamaterials technology. By externally regulating the medium’s parameters, a
highly efficient optical modulator is possible with potential applications
across optics, optoelectronics, and photonics.

_K_ eywords Birefringence $\cdot$ Broadband devices $\cdot$ Circular Bragg
phenomenon $\cdot$ Metamaterials $\cdot$ Negative refraction $\cdot$ Optical
activity $\cdot$ Polarization selectivity

##  1 Introduction

Bragg reflection is said to occur when successive reflections of monochromatic
light from a spatially modulated dielectric add up coherently. Such photonic
structures are extensively used in optics, finding application, for example,
in distributed feedback lasers [1], in wavelength division multiplexers (WDM)
[2], or as dispersion cancellation filters in optical waveguides [3].
Fabrication of Bragg gratings [4, 5] is often challenging owing to the
necessity of matching the scale of the medium’s variation to the wavelength of
light, say on the order of microns or less. In this letter, we identify a
fundamentally new mechanism to achieve Bragg-like reflection that, remarkably,
is independent of matching the wavelength to a spatial period, and relies
solely on resonant tuning of the medium’s parameters. The optical response is
similar to the well-studied circular Bragg phenomenon (CBP) in structurally
chiral media [6], whereby circularly polarized incident light that is co-
handed with the medium’s helicity is reflected, whilst contra-handed light is
transmitted. The CBP has been observed naturally in iridescent beetle shells
[7], as well as in nano-fabricated sculptured thin films [8]. However, unlike
the traditional CBP, in our identified mechanism there is no “Bragg
wavelength” as such, and polarization selective response occurs over a broad
wavelength band, limited only by material dispersion. Evaluating the
parameters necessary to achieve such optical response, we confirm that
practical realisations are indeed within reach of current metamaterials
technology.

##  2 Combining birefringence with optical activity

For a birefringent and optically active reciprocal medium, the temporal
frequency domain constitutive relations combine a dielectric tensor
${\boldsymbol{\epsilon}}$ with the Drude-Born-Fedorov model [9]

$\displaystyle{\bf D}=\epsilon_{0}\left({\boldsymbol{\epsilon}}\cdot{\bf
E}+i\alpha\eta_{0}{\bf H}\right)\ \ \text{and}\ \ {\bf
B}=\mu_{0}\left[-i\left({\alpha}/{\eta_{0}}\right){\bf E}+\mu{\bf
H}\right]\,,$ (1)

where $\bf E$, $\bf B$ are the fundamental electromagnetic fields, $\bf D$,
$\bf H$ are the excitation fields, $\epsilon_{0}$, $\mu_{0}$, and
$\eta_{0}=\left(\mu_{0}/\epsilon_{0}\right)^{1/2}$ are the free-space
permittivity, permeability, and impedance, respectively, and $\mu$ is the
relative permeability. The chirality parameter $\alpha$ measures the length,
in wavelengths, that a linearly polarized wave propagates before its
$\bf{E}$-vector rotates through $2\pi$. Defining the auxiliary fields ${\bf
h}=\eta_{0}{\bf H}$, ${\bf b}=(\eta_{0}/\mu_{0}){\bf B}$, and ${\bf
d}=\epsilon_{0}^{-1}{\bf D}$, the constitutive relations of Eq. (1) simplify
to

$\displaystyle{\bf d}$ $\displaystyle={\boldsymbol{\epsilon}}\cdot{\bf
E}+i\alpha{\bf h}\ \ \text{and}\ \ {\bf b}=-i\alpha{\bf E}+\mu{\bf h}\,.$ (2)

This notation assures that all fields are similarly dimensioned, so that all
the medium’s parameters are relative. We may now express the fields as ${\bf
E}={E}_{\parallel}{\hat{\bf k}}+{\bf E}_{\perp}$ and ${\bf
h}={h}_{\parallel}{\hat{\bf k}}+{\bf h}_{\perp}$, with ${\hat{\bf k}}$ a unit
vector in the direction of the wavevector $\bf k$ being perpendicular to
$\left({\bf E}_{\perp},{\bf h}_{\perp}\right)$, and partition
${\boldsymbol{\epsilon}}$ so that

${\boldsymbol{\epsilon}}\cdot{\bf
E}=\left(\begin{array}[]{c}{\hat{\boldsymbol{\epsilon}}}_{\perp}\cdot{\bf
E}_{\perp}+{\boldsymbol{\epsilon}}_{\parallel}{E}_{\parallel}\\\
{\boldsymbol{\epsilon}}_{\parallel}^{T}\cdot{\bf
E}_{\perp}+\epsilon_{s}{E}_{\parallel}\end{array}\right)\,,$ (3)

where
${\hat{\boldsymbol{\epsilon}}}_{\perp}:{\mathbb{C}}^{2}\rightarrow{\mathbb{C}}^{2}$,
${\boldsymbol{\epsilon}}_{\parallel}\in{\mathbb{C}}^{2}$, and
$\epsilon_{s}\in{\mathbb{C}}$. Then, for a monochromatic plane wave ${\bf
E}e^{i\left({\bf k}\cdot{\bf r}-\omega t\right)}$, combining the projections
of Maxwell’s macroscopic source-free curl relations yields

$k^{2}{\bf E}_{\perp}-2i\alpha k_{0}k(\times){\bf
E}_{\perp}-\mu^{2}\left.{\hat{{\boldsymbol{\epsilon}}}}\right|\cdot{\bf
E}_{\perp}+\alpha^{2}k_{0}^{2}{\bf E}_{\perp}=0\,,$ (4)

where
$\left.{\hat{{\boldsymbol{\epsilon}}}}\right|={\hat{\boldsymbol{\epsilon}}}_{\perp}-\left(\epsilon_{s}-{\alpha^{2}}/{\mu}\right)^{-1}{\boldsymbol{\epsilon}}_{\parallel}{\boldsymbol{\epsilon}}_{\parallel}^{T}$
and in Cartesian coordinates $\left(\times\right)$ is interpreted as
$\left(\begin{array}[]{cc}0&-1\\\ 1&0\end{array}\right)$. The free-space
wavenumber is $k_{0}={\omega}/{c}$, where $c$ is the speed of light in vacuum,
and the wavenumber in the medium is $k$. In a propagation coordinate system
(${{\bf k}}=k{\hat{\bf z}}$) with transverse coordinates ($x,y$) chosen to
diagonalize $\left.{\hat{{\boldsymbol{\epsilon}}}}\right|$, as
$\left.{\hat{{\boldsymbol{\epsilon}}}}\right|=\text{diag}\left(\epsilon_{1},\epsilon_{2}\right)$,
the supported by the medium refractive indices $n={k}/{k_{0}}$, turn out to be

$\pm
n^{(\pm)}\equiv\pm\left[{\bar{\epsilon}}\mu+\alpha^{2}\pm\left({\Delta\epsilon}^{2}\mu^{2}+4\alpha^{2}{\bar{\epsilon}}\mu\right)^{1/2}\right]^{1/2}\,,$
(5)

where ${\bar{\epsilon}}=(\epsilon_{1}+\epsilon_{2})/2$ is the average
transverse dielectric constant and
${\Delta\epsilon}=(\epsilon_{2}-\epsilon_{1})/2$ measures the birefringence.

##  3 Wavelength-independent Bragg zones

For an idealized lossless medium with real parameters, Eq. (5) shows that $n$
is purely imaginary whenever

$\left({\bar{\epsilon}}-\Delta\epsilon\right)\mu<\alpha^{2}<\left({\bar{\epsilon}}+\Delta\epsilon\right)\mu\,,$
(6)

assuming ordering such that ${\Delta\epsilon}>0$. Within these bands, centered
at $\alpha=\pm({\bar{\epsilon}}\mu)^{1/2}$, one index is real and related to
forward propagation, whilst the other is imaginary and related to evanescent
waves. The sign of $\alpha$ determines which is which: for $\alpha>0$, as
illustrated in Fig. 1, $n^{\left(+\right)}$ is associated with a left-handed
propagating eigenmode, while $n^{\left(-\right)}$ is associated with a right-
handed eigenmode which becomes evanescent in the stopband region
$(\epsilon_{1}\mu)^{1/2}<\alpha<(\epsilon_{2}\mu)^{1/2}$. In this band, the
eigenmode corresponding to $n^{\left(-\right)}$ becomes linear, while the
eigenmode corresponding to $n^{\left(+\right)}$ remains circular. Accordingly,
for $\alpha<0$, as also depicted in Fig. 1, the emerging stopband refers to a
left-handed linear eigenmode.

Both stopband edges signify extreme values of chirality, with $\alpha$ being
comparable to the average refractive index,
$\left(\bar{\epsilon}\mu\right)^{1/2}$. This resonance condition triggers the
onset of negative refraction in a purely optically active medium [10, 11]. In
fact, solving for optical activity, with a scalar $\epsilon$, the
characteristic eigenvalues are $\pm\gamma_{\pm}=\pm
k_{0}\left[\alpha\pm\left(\epsilon\mu\right)^{{1}/{2}}\right]$. For forward
(backward) propagation, we must either choose $+\gamma_{+}$ or $-\gamma_{-}$
($-\gamma_{+}$ or $+\gamma_{-}$). Then, the corresponding eigenmodes describe,
nominally, right- and left- handed spatial helices. For
$\left|\alpha\right|>\left(\bar{\epsilon}\mu\right)^{{1}/{2}}$, the direction
of phase advance and handedness of two of the modes are interchanged, but the
Poynting vector does not change [12].

Figure 1:  Dispersion of a uniform birefringent and optically active medium in
the chirality domain: the refractive indices of Eq. (5) as functions of the
chirality parameter, each associated with the polarization (linear, circular,
or elliptical) of the corresponding eigenmode. Stopbands emerge around
$\alpha\approx\pm\left(\bar{\epsilon}\mu\right)$. The base parameters are:
$\epsilon_{1}=1.69$, $\epsilon_{2}=2.89$, and $\mu=\ 2.29$.

An intuitive picture of how such a stopband occurs is illustrated in Fig. 2.
In the absence of optical activity ($\alpha=0$), the polarization eigenstates
are linear and propagate with speeds
${c}/{\left(\epsilon_{1,2}\mu\right)^{1/2}}$, or on average
${c}/{\left(\bar{\epsilon}\mu\right)^{1/2}}$. With optical activity present
($\alpha>0$, say), a circularly polarized wave propagating with speed
$c/\alpha$ will synchronously and alternately sample the co-propagating
eigenaxes, provided that $\alpha\approx\left(\bar{\epsilon}\mu\right)^{1/2}$.
For $\Delta\epsilon\ll\bar{\epsilon}$, the purely imaginary refractive index
of the resonant polarization is $n_{R}^{\text{res}}\approx
i\left({\Delta\epsilon}/{2}\right)\left({\mu}/{\bar{\epsilon}}\right)^{{1}/{2}}$,
while the opposite-handed polarization, with refractive index
$n_{L}\approx\left(2\bar{\epsilon}\mu\right)^{1/2}$, propagates too slowly to
sample the $\epsilon_{1,2}$ alternation synchronously.

Figure 2:  Heuristic picture: chiral circularly polarized eigenstates,
propagating with speed $c/\alpha$, will synchronously and alternately sample
the co-propagating eigenaxes if
$\alpha\approx\left(\bar{\epsilon}\mu\right)^{1/2}$.

##  4 Coupled wave theory description of birefringent and optically active
media

This interpretation is confirmed via coupled wave theory (CWT) [13]. To
demonstrate this, we expand $\bf{E}$ in terms of circularly polarized states,
$L$-left and $R$-right, for which the forward and backward propagating
electric field amplitudes $A_{L,R}^{\pm}$ are perturbed along the direction of
propagation $\hat{\bf{z}}$ by the presence of birefringence, i.e.,

${\bf
E}_{\bot}=\left(A_{L}^{+}e^{ik_{0}\left[\left(\bar{\epsilon}\mu\right)^{1/2}+\alpha\right]z}+A_{R}^{-}e^{-ik_{0}\left[\left(\bar{\epsilon}\mu\right)^{1/2}-\alpha\right]z}\right){\bf
e}_{1}+\left(A_{L}^{-}e^{-ik_{0}\left[\left(\bar{\epsilon}\mu\right)^{1/2}+\alpha\right]z}+A_{R}^{+}e^{ik_{0}\left[\left(\bar{\epsilon}\mu\right)^{1/2}-\alpha\right]z}\right){\bf
e}_{2}\,,$ (7)

where

${\bf e}_{1}=\frac{1}{\sqrt{2}}\left(\begin{matrix}1\\\ i\\\
\end{matrix}\right)\ \ \text{and}\ \ {\bf
e}_{2}=\frac{1}{\sqrt{2}}\left(\begin{matrix}1\\\ -i\\\
\end{matrix}\right)\,.$

The relevant Helmholtz wave equation is found to be

$\frac{{\rm d}^{2}{\bf E}}{{\rm
d}z^{2}}+k_{0}^{2}\left(\mu{{{{\left.{\hat{{\boldsymbol{\epsilon}}}}\right|}}}}-a^{2}\right)\cdot{\bf
E}+2ak_{0}\hat{{\bf z}}\left(\times\right)\frac{{\rm d}{\bf E}}{{\rm d}z}={\bf
0}\,.$ (8)

Substituting Eq. (7) into Eq. (8), resolving along ${\bf e}_{1}$ and ${\bf
e}_{2}$, and phase-matching potentially synchronous terms, we obtain two
systems of coupled wave equations, one for each polarization state, namely

$\frac{\rm{d}}{{\rm d}z}\left(\begin{matrix}A_{L,R}^{+}\\\ A_{L,R}^{-}\\\
\end{matrix}\right)=i\kappa\left(\begin{matrix}0&e^{-ik_{0}\delta_{L,R}z}\\\
-e^{ik_{0}\delta_{L,R}z}&0\\\
\end{matrix}\right)\left(\begin{matrix}A_{L,R}^{+}\\\ A_{L,R}^{-}\\\
\end{matrix}\right)\,,$ (9)

with detuning parameters and coupling constant

$\delta_{L,R}=2\left[\left(\bar{\epsilon}\mu\right)^{1/2}\pm\alpha\right]\ \
\text{and}\ \
\kappa\approx\frac{\pi}{\lambda_{0}}\left(\epsilon_{1}^{{1}/{2}}-\epsilon_{2}^{{1}/{2}}\right)\,,$
(10)

respectively, where $\lambda_{0}$ is the design wavelength. Crucially, on
neglecting the dispersion of the average dielectric constant, the detuning
parameters $\delta_{L,R}$ are _wavelength-independent_ being solely determined
by the medium’s parameters. The location of the resonances, each related to a
specific polarization, can then be found by setting $\delta_{L,R}=0$, leading
to the regimes of Eq. (6).

When $\alpha\approx\left(\bar{\epsilon}\mu\right)^{1/2}$
($\alpha\approx-\left(\bar{\epsilon}\mu\right)^{1/2}$), the $A_{L}^{\pm}$
($A_{R}^{\pm}$) amplitudes remain constant, while the $A_{R}^{\pm}$
($A_{L}^{\pm}$) evolve exactly as for a Bragg grating (see, e.g., [14]). Given
a medium of length $z$, these evolutions are described by the solutions to Eq.
(9)

$\left(\begin{matrix}A_{L,R}^{+}\\\ A_{L,R}^{-}\\\
\end{matrix}\right)_{z}=\left(\begin{matrix}e^{-i\frac{k_{0}\delta_{L,R}z}{2}}P_{L,R}^{+}&e^{-i\frac{k_{0}\delta_{L,R}z}{2}}Q_{L,R}^{+}\\\
e^{i\frac{k_{0}\delta_{L,R}z}{2}}Q_{L,R}^{-}&e^{i\frac{k_{0}\delta_{L,R}z}{2}}P_{L,R}^{-}\\\
\end{matrix}\right)\left(\begin{matrix}A_{L,R}^{+}\\\ A_{L,R}^{-}\\\
\end{matrix}\right)_{0}\,,$ (11)

where

$\displaystyle P_{L,R}^{\pm}$ $\displaystyle=\cosh{\left(q_{L,R}z\right)}\pm
i\frac{k_{0}\delta_{L,R}}{2q_{L,R}}\sinh{\left(q_{L,R}z\right)}\,,$
$\displaystyle Q_{L,R}^{\pm}$ $\displaystyle=\frac{\pm
i\kappa}{q_{L,R}}\sinh{\left(q_{L,R}z\right)\,,\ \ \text{and}\ \
q_{L,R}=\left[\kappa^{2}-\left(\frac{k_{0}\delta_{L,R}}{2}\right)^{2}\right]^{1/2}\,.}$

The optical response of a thin film of the considered medium to a normally
incident plane wave may then be computed analytically (via the eigenmodes of
Eq. (4) and field-matching at both interfaces of the film, noting the subtlety
that for $\alpha>\left(\epsilon\mu\right)^{{1}/{2}}$
($\alpha<-\left(\epsilon\mu\right)^{{1}/{2}}$), it is the eigenmode associated
with $n_{R}^{-}$ ($n_{L}^{+}$) that carries energy away from the interface) or
approximately (via CWT, as in [6]). The intensity reflectances $R_{ij}$ and
transmittances $T_{ij}$, where $\\{i,j\\}=\\{L,R\\}$ indicate reflection or
transmission of the $i$ polarization for incident $j$ polarization, are
demonstrated in Figs. 3 and 4, respectively. Bragg-like spectral features are
evident at $\alpha\approx\pm\left(\bar{\epsilon}\mu\right)^{1/2}$, whereby for
broadband incident light of equal amounts of right and left circular
polarizations (RCP and LCP), half is strongly reflected for one polarization,
while the other half is transmitted for the orthogonal one. Therefore, the
medium acts as a broadband polarization divider. For
$\alpha\approx\left(\bar{\epsilon}\mu\right)^{1/2}$
($\alpha\approx-\left(\bar{\epsilon}\mu\right)^{1/2}$), the range over which
the fraction of reflected intensity into RCP (LCP) for incident RCP (LCP),
$R_{RR}$ ($R_{LL}$), is high ($>0.5$, say) is
$\Delta\alpha=\left|\epsilon_{2}-\epsilon_{1}\right|\mu$. Within this range,
unit reflectance of incident RCP (LCP) results in the amplitudes of forward
and backward RCP (LCP) light being equal within the medium, the superposition
of these giving a net linear polarization. Indeed, for
$\alpha=(\bar{\epsilon}\mu)^{{1}/{2}}$ and just RCP light, combining Eq. (7)
with Eq. (11) for $A_{R}^{-}\left(z\right)=0$, yields

$\frac{{\bf
E}_{\bot}\left(0\right)}{A_{R}^{+}\left(0\right)}=\frac{i}{\sqrt{2}}\left(\begin{matrix}1\\\
i\\\ \end{matrix}\right)+\frac{1}{\sqrt{2}}\left(\begin{matrix}1\\\ -i\\\
\end{matrix}\right)=\frac{1+i}{\sqrt{2}}\left(\begin{matrix}1\\\ -1\\\
\end{matrix}\right)\,.$ (12)

Hence, the polarization is linear, which is logical as once $\alpha$ exceeds
$(\epsilon_{1}\mu)^{1/2}$, the wave is evanescent and must match the backward
propagating reflected field.

Interestingly, for $\alpha\approx 0$, resolving along ${\bf e}_{1}$ and ${\bf
e}_{2}$, yet another pair of coupled wave equations emerges for the contra-
handed forward (or backward) propagating waves. In terms of the circular
components of the electric fields, and precisely at $\alpha=0$, both the exact
and the CWT solutions agree to

$\left(\begin{matrix}E_{L}^{+}\\\ E_{R}^{+}\\\
\end{matrix}\right)_{z}=e^{ik_{0}(\bar{\epsilon}\mu)^{1/2}z}\left(\begin{matrix}\cos{\left(\kappa
z\right)}&i\sin{\left(\kappa z\right)}\\\ i\sin{\left(\kappa
z\right)}&\cos{\left(\kappa z\right)}\\\
\end{matrix}\right)\left(\begin{matrix}E_{L}^{+}\\\ E_{R}^{+}\\\
\end{matrix}\right)_{0}\,.$ (13)

This is a manifestation of the evolution of circularly polarized light in a
linearly birefringent medium whereby $T_{RR}$ and $T_{LL}$ evolve,
respectively, into $T_{RL}$ and $T_{LR}$, being ‘unnaturally’ expanded on a
circular basis. This continuous exchange of circular states between these
transmissions with increasing film thickness is illustrated in Fig. 5. By
contrast, Fig. 6 shows the characteristic Bragg-like evolution of $T_{RR}$
into $R_{RR}$ with increasing slab thickness, at
$\alpha=\left(\bar{\epsilon}\mu\right)^{1/2}$, corroborating Eq. (11).

Figure 3:  Intensity reflectances as functions of the chirality parameter for
a thin film of a uniform birefringent and optically active medium. Light is
normally incident and the surrounding medium is vacuum. The parameters are
those of Fig. 1 and the normalized film thickness is $k_{0}z=20$. Figure 4:
Intensity transmittances as functions of the chirality parameter for the
scenario of Fig. 3. Around $\alpha\approx 0$, an exchange between circular
states ensues and at $\alpha=0$, the medium is linearly birefringent with
purely linear eigenmodes. Figure 5:  Continuous exchange between the $T_{RL}$
and $T_{RR}$ transmittances as a function of the normalized film thickness.
The medium is uniform and birefringent with the parameters of Fig. 1, but with
optical activity absent (cf. Fig. 4, $\alpha\approx 0$ region). Figure 6:
Bragg-like evolution of the $T_{RR}$ transmittance into the $R_{RR}$
reflectance as a function of film thickness in a uniform birefringent and
optically active medium when on-resonance, at
$\alpha=\left(\bar{\epsilon}\mu\right)^{1/2}$. The parameters are those of
Fig. 1.

##  5 Material dispersion and practical realizations

The foundations of the proposed Bragg-like phenomenon lie in the stopbands
occurring in the medium’s chirality domain dispersion, rather than that of
wavelength. A natural question to address is whether considering material
dispersion will affect the main features. Filling the medium under discussion
with metallic spherical Mie resonators, uniformly distributed in space, as in
[15], and assuming the relative permittivities to be equal and dispersive in
both birefringent axes, the stopbands appear, shifted, as long as the electron
density $N$,

$N\ll
N_{\text{max}}=\left(\epsilon_{0}m_{e}\right)^{{1}/{2}}q_{e}^{-1}\left|\omega^{2}-\omega_{0}^{2}\right|^{{1}/{2}}\,,$
(14)

where $m_{e}$, $q_{e}$ are the electron mass and charge, respectively,
$\omega_{0}$ refers to the design frequency, and $\omega$ to the operational
half-bandwidth. If, for example, the design wavelength is $\lambda_{0}=5$ µm,
and the desired bandwidth $\Delta\lambda=4$ µm, then by setting
$N=N_{\text{max}}\cdot{10}^{-3}$, the stopbands will be centred at $\alpha=\pm
2.0602$ with $\Delta\alpha=0.7362$. Evidently, controlling $\omega_{p}$ via
$N$, provides a _switching mechanism_ between resonant polarizations.

Although natural media that are simultaneously birefringent and optically
active are known [16, 17], one with such extreme optical activity to satisfy
$\alpha=\left(\bar{\epsilon}\mu\right)^{1/2}$ is unlikely to be found in
Nature. Turning to metamaterials, the giant optical rotatory power of
$\approx{10}^{4}$ ${}^{\circ}/{\rm mm}$, observed in [18], corresponds to a
required refractive index of just $\left(\bar{\epsilon}\mu\right)^{1/2}=0.04$
at $\lambda_{0}=1.5$ µm. Such extreme values have been observed in
metamaterials at THz frequencies ($\lambda_{0}\approx 0.46$ mm) with the
remarkably high index of $\left(\bar{\epsilon}\mu\right)^{1/2}=7$, achieving
an optical rotatory power of $5.46\cdot{10}^{3}$ ${}^{\circ}/{\rm mm}$ in
[11]. Recently, embedding a chiral medium into a non-chiral gold nanorod
metamaterial, enhanced the circular dichroism [19]. Hence, the phenomena
predicted here are within reach of a carefully designed meta-medium. To reduce
the required optical activity we may lower the average refractive index.
Actually, one of the founding meta-media consisted of orthogonal wire
inclusions that present a lower effective refractive index through control of
the plasma frequency [20, 21]. Utilizing wires of different diameters in the
orthogonal directions will also provide the necessary birefringence.

##  6 Conclusion

Concluding, we have demonstrated that a uniform birefringent and optically
active medium exhibits a broadband and polarization selective Bragg
phenomenon, exclusively relying on matching the medium’s parameters rather
than on wavelength matching. Current metamaterials technology provides the
necessary parameters that are, moreover, in principle modulatable, leading to
a very flexible broadband switch. Potential applications include: lasers
(broadband, short-pulsed), optoelectronics (OLED displays, electron transport
in organic semiconductors), waveguiding (add-drop WDM), or multi-use fiber
sensors.

## Acknowledgments

The authors acknowledge fruitful discussions with Dr. M. Foreman and Dr. K.
Weir of Imperial College London.

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