# Chiral-Gain Photonics

Sylvain Lannebère1 David E. Fernandes2 Tiago A. Morgado1 Mário G. Silveirinha2
To whom correspondence should be addressed:
<EMAIL_ADDRESS>1 Instituto de Telecomunicações and
Department of Electrical Engineering, University of Coimbra, 3030-290 Coimbra,
Portugal  2University of Lisbon – Instituto Superior Técnico and Instituto de
Telecomunicações, 1049-001 Lisboa, Portugal

###### Abstract

Here, we present an exploratory study of the potential applications of
electrically biased materials that possess a nonreciprocal and non-Hermitian
electromagnetic response analogous to the electronic response of field-effect
transistors. The most distinctive feature of such materials is their chiral-
gain, meaning that their response can be active or dissipative depending on
the handedness of the wave polarization. Here, we show how the chiral-gain can
be harnessed to develop novel electromagnetic devices with unique properties
such as chiral lasers, polarization-dependent mirrors, and coherent-perfect-
absorber lasers. Furthermore, it is demonstrated that materials with chiral-
gain can bypass a reciprocity constraint that typically limits the external
coupling strength, thus facilitating the excitation of cavities with extremely
large quality factors.

##  I Introduction

Photonic integrated circuits have the potential to revolutionize information
processing devices by enhancing their speed, bandwidth, and power efficiency
Chrostowski and Hochberg (2015); Komljenovic  _et al._ (2016); Karabchevsky
_et al._ (2020). However, their development has been slowed down by several
challenges, particularly the absence of integrated light sources and
amplifiers suitable for large scale production Zhou  _et al._ (2015); Wang
_et al._ (2017); Zhou  _et al._ (2023); Azzam  _et al._ (2020), as well as the
lack of efficient integrated nonreciprocal components Levy (2002); Caloz  _et
al._ (2018); Asadchy  _et al._ (2020).

In a recent article Lannebère  _et al._ (2022), we introduced a nonlinear
(meta-)material that mimics the response of semiconductor transistors. Such an
idealized platform, which we designated by MOSFET-metamaterial, is
characterized by a linearized optical response that is both nonreciprocal and
non-Hermitian. These effects arise from the the interplay between material
nonlinearities with a static electric bias. The most remarkable property of
the metamaterial is that its non-Hermitian response is controlled by an
_indefinite_ matrix (see below). Consequently, the interactions of a wave with
the material can be either dissipative or exhibit gain, depending on the
field-polarization. In a series of subsequent articles, we demonstrated that
transistor-like electromagnetic responses may be implemented in different
ways, using transmission lines coupled with FETs Fernandes  _et al._ (2024),
in 2D engineered materials (strained twisted bilayer graphene) Rappoport  _et
al._ (2023), and in 3D low symmetry natural materials with a large Berry
curvature dipole Morgado  _et al._ (2024). Remarkably, the gain response of
such platforms has chiral-properties, such that the material exhibits gain for
a particular handedness of the optical field, while it is dissipative for the
opposite polarization handedness Lannebère  _et al._ (2022); Rappoport  _et
al._ (2023); Hakimi  _et al._ (2023); Morgado  _et al._ (2024).

The objective of this article is to present an exploratory study of the
potential applications of such “chiral-gain” materials in the realm of
nanophotonics. Here, similar to Lannebère  _et al._ (2022), we restrict our
attention to the case of an idealized nonlinear material characterized by the
constitutive relation
$\mathbf{P}=\varepsilon_{0}\overline{\chi}(\mathbf{E})\cdot\mathbf{E}$ with a
nonlinear susceptibility of the form

$\displaystyle\overline{\chi}(\mathbf{E})$
$\displaystyle=\begin{pmatrix}\chi_{xx}(E_{z})&0&0\\\ 0&\chi_{yy}&0\\\
0&0&\chi_{zz}\end{pmatrix}$ (1)

where $\mathbf{E}$ is the electric field and $\mathbf{P}$ is the polarization
vector. As shown in Lannebère  _et al._ (2022), under a static electric bias
with a nonzero $x$ component, the linearization of $\mathbf{P}$ leads to a
(linearized) permittivity tensor of the form

$\displaystyle\overline{\varepsilon}=\begin{pmatrix}\varepsilon_{xx}&0&\varepsilon_{xz}\\\
0&\varepsilon_{yy}&0\\\ 0&0&\varepsilon_{zz}\end{pmatrix}.$ (2)

Remarkably, this permittivity tensor describes a nonreciprocal
($\overline{\varepsilon}^{T}\neq\overline{\varepsilon}$) Altman and Suchy
(2011) and non-Hermitian
($\overline{\varepsilon}^{\dagger}\neq\overline{\varepsilon}$) material
response. Additionally, in the special case where the permittivity tensor is
real-valued and frequency independent, the linearized response is also time-
reversal ($\mathcal{T}$) invariant
($\overline{\varepsilon}^{\ast}=\overline{\varepsilon}$). Here, $T$, $\ast$
and $\dagger$ represent the transpose, the complex conjugate and the
Hermitian-conjugate operations, respectively.

The non-Hermitian response of the material is ruled by the matrix
$\overline{\varepsilon}^{\prime\prime}\equiv\frac{\overline{\varepsilon}-\overline{\varepsilon}^{\dagger}}{2i}$
Lannebère  _et al._ (2022). For the model of Eq. (2), supposing for simplicity
that the diagonal elements are real-valued, the matrix
$\overline{\varepsilon}^{\prime\prime}$ is given by:

$\displaystyle\overline{\varepsilon}^{\prime\prime}$
$\displaystyle=\frac{1}{2i}\begin{pmatrix}0&0&\varepsilon_{xz}\\\ 0&0&0\\\
-\varepsilon_{xz}^{\ast}&0&0\end{pmatrix}.$ (3)

As the matrix trace vanishes, the matrix is indefinite. This means that some
of its eigenvalues are positive, corresponding to a dissipative response,
whereas other eigenvalues are negative corresponding to a gain response. It
was shown in Lannebère  _et al._ (2022) that when $\varepsilon_{xz}$ is real-
valued the relevant eigenvectors of the matrix
$\overline{\varepsilon}^{\prime\prime}$ are $\hat{\mathbf{x}}\pm
i\hat{\mathbf{z}}$. Thus, the polarization states associated with the
dissipative/gain responses are circular and have opposite handedness. The
polarization that activates the gain (dissipation) is controlled by the
orientation of the electric field static bias.

In our previous work, we showed that a material with chiral-gain can be used
to combine optical isolation with amplification within the same device
Lannebère  _et al._ (2022). Here, we analyze how the chiral-gain can be
harnessed for several potential nanophotonic applications. First, in section
II we characterize the spectrum of generalized indefinite-gain materials and
discuss the conditions for their stability and the role of parity-time
($\mathcal{PT}$) symmetry. In section III, we investigate the application of
chiral-gain materials in lasing. In section IV, we theoretically demonstrate
that chiral-gain materials can be used as polarization-dependent amplifying
mirrors. Next, in section V, it is shown that open cavities formed by chiral-
gain media may be engineered to exhibit an arbitrarily high quality factor.
Finally, in section VI, we demonstrate that such cavities are ideal for light-
storage and that their operation is not limited by the reciprocity constraint.
This distinctive behavior is traced back to the unique analytical structure of
the scattering matrix, which simultaneously accommodates a zero and a pole at
the same frequency. This property delineates a regime where coherent perfect
absorption (CPA or anti-lasing) Chong  _et al._ (2010); Wan  _et al._ (2011)
and “quasi-lasing” are seamlessly integrated within the same resonator,
similar to the behavior of “CPA-lasers” Longhi (2010); Chong  _et al._ (2011);
Longhi and Feng (2014); Wong  _et al._ (2016) and their nonlinear counterparts
Longhi (2011); Schackert  _et al._ (2013). We find that, under optimal
conditions, the energy retention within the cavity is primarily influenced by
the excitation duration rather than by the incoming wave amplitude.

##  II Bulk stability and $\mathcal{PT}$-symmetry

Indefinite-gain materials are characterized by a matrix
$\overline{\varepsilon}^{\prime\prime}=\frac{\overline{\varepsilon}-\overline{\varepsilon}^{\dagger}}{2i}$
that is indefinite, meaning that the interactions between the material and a
wave can be either dissipative or exhibit gain. Next, we study the stability
of a bulk material described by the relative permittivity tensor

$\displaystyle\overline{\varepsilon}=\begin{pmatrix}\varepsilon_{xx}&0&\varepsilon_{xz}\\\
0&\varepsilon_{yy}&0\\\ \varepsilon_{zx}&0&\varepsilon_{zz}\end{pmatrix}.$ (4)

This corresponds to a generalization of the MOSFET-metamaterial model
introduced in Lannebère  _et al._ (2022). When
$\varepsilon_{xz}\neq\varepsilon_{zx}$
($\varepsilon_{xz}\neq\varepsilon_{zx}^{\ast}$) the response is nonreciprocal
(non-Hermitian). Furthermore, when the diagonal-elements are real-valued (as
assumed below) the matrix $\overline{\varepsilon}^{\prime\prime}$ is
indefinite. For simplicity, in this article we neglect the frequency
dispersion.

The natural modes of an indefinite-gain medium can be obtained by solving
Maxwell equations in the frequency domain

$\displaystyle\nabla\times\mathbf{E}$ $\displaystyle=i\omega\mu_{0}\mathbf{H}$
(5a) $\displaystyle\nabla\times\mathbf{H}$
$\displaystyle=-i\omega\varepsilon_{0}\overline{\varepsilon}\cdot\mathbf{E}$
(5b)

where $\omega$ is the oscillation frequency. A time variation of the type
$\text{e}^{-i\omega t}$ is implicit. For continuous material invariant under
arbitrary space translations, the bulk normal modes are plane waves with a
spatial variation of the form $\text{e}^{i\mathbf{k}\cdot\mathbf{r}}$, where
$\mathbf{k}$ is the wave vector and $\mathbf{r}$ is the position vector. To
assess the material stability we characterize the plane waves with a real-
valued wave vector and complex frequency
$\omega=\omega^{\prime}+i\omega^{\prime\prime}$. The real part of $\omega$
determines the time period $T=2\pi/\omega^{\prime}$ of the natural
oscillations and the imaginary part determines the amplitude variation with
time. Since $\text{e}^{-i\omega
t}=\text{e}^{-i\omega^{\prime}t}\text{e}^{\omega^{\prime\prime}t}$, it follows
that $\omega^{\prime\prime}>0$ corresponds to a wave growing exponentially
with time (unstable response) while $\omega^{\prime\prime}<0$ to a wave
decaying with time (stable response). The plane wave solutions satisfy the
wave equation
$\mathbf{k}\times(\mathbf{k}\times\mathbf{E})+\left(\omega^{2}/c^{2}\right)\overline{\varepsilon}\cdot\mathbf{E}=0$
where $c$ is the speed of light in vacuum.

The non-Hermitian physics is ruled by the nontrivial eigenvectors of
$\overline{\varepsilon}^{\prime\prime}$, which are in the $xoz$ plane.
Thereby, we focus our attention in the propagation along the $y$ direction, as
it maximizes the gain or loss interactions. For
$\mathbf{k}=k\hat{\mathbf{y}}$, there are two transverse modes characterized
by

$\displaystyle\omega$
$\displaystyle=\omega_{\pm}\equiv\frac{kc}{\sqrt{\varepsilon_{\pm}}}$ (6a)
$\displaystyle\mathbf{E}$
$\displaystyle\sim\mathbf{E}_{\pm}\equiv\varepsilon_{xz}\hat{\mathbf{x}}+\left(\varepsilon_{\pm}-\varepsilon_{xx}\right)\hat{\mathbf{z}}$
(6b) $\displaystyle\varepsilon_{\pm}$
$\displaystyle=\frac{\varepsilon_{xx}+\varepsilon_{zz}\pm\sqrt{\gamma}}{2}$
(6c)

where
$\gamma=(\varepsilon_{xx}-\varepsilon_{zz})^{2}+4\varepsilon_{xz}\varepsilon_{zx}$.

Consider first that the permittivity tensor (4) is real-valued so that the
response is invariant under a time-reversal. In this situation, $\gamma$ is
also real-valued. When $\gamma>0$, both $\varepsilon_{\pm}$ [Eq. (6c)] and the
natural frequencies $\omega_{\pm}$ [Eq.(6a)] are real-valued and non-
degenerate. In this scenario $\omega^{\prime\prime}=0$ and the bulk material
is stable. On the other hand, when $\gamma<0$, both $\varepsilon_{\pm}$ and
$\omega_{\pm}$ become pairs of complex conjugates numbers, meaning that one
mode is attenuated ($\omega^{\prime\prime}<0$) while the other is amplified
($\omega^{\prime\prime}>0$). In this case, which arises when
$\varepsilon_{xz}$ and $\varepsilon_{zx}$ have opposite signs and their
product satisfies
$|\varepsilon_{xz}\varepsilon_{zx}|>\frac{1}{4}(\varepsilon_{xx}-\varepsilon_{zz})^{2}$,
the bulk material is unstable. The transition from the stable to the unstable
occurs when $\gamma=0$, which corresponds to an exceptional point where
$\varepsilon_{\pm}$, the eigenfrequencies [Eq. (6a)] and the eigenvectors [Eq.
(6b)] coalesce and become degenerate. It is worth noting that when
$\varepsilon_{zz}=\varepsilon_{xx}$, the analysis presented here reduces to
the case discussed in Buddhiraju  _et al._ (2020).
The characteristics of the natural modes can be understood by noting that the
material is invariant under a parity transformation $\mathcal{P}$ that flips
the $y$ spatial coordinate $\mathcal{P}:(x,y,z)\to(x,-y,z)$. The permittivity
tensor of a nonmagnetic and non-bianisotropic material invariant under
$\mathcal{P}$ must satisfy
$\overline{\varepsilon}(y)=\overline{V}\cdot\overline{\varepsilon}(-y)\cdot\overline{V}$,
where
$\overline{V}=\hat{\mathbf{x}}\otimes\hat{\mathbf{x}}-\hat{\mathbf{y}}\otimes\hat{\mathbf{y}}+\hat{\mathbf{z}}\otimes\hat{\mathbf{z}}$.
This condition is satisfied by the permittivity tensor (4). Therefore, since
the material is invariant under both $\mathcal{P}$ and $\mathcal{T}$
transformations, it is also invariant under a $\mathcal{PT}$ transformation,
the composition of the parity and time-reversal transformations.
$\mathcal{PT}$ symmetric systems are known to possess a spectrum with mirror
symmetry with respect to the real-frequency axis and to exhibit stable and
unstable phases separated by an exceptional point Bender  _et al._ (1999);
Bender (2007). When $\gamma>0$, the system is in the so-called unbroken
$\mathcal{PT}$ symmetric phase, where all the eigenmodes are real-valued and
the bulk material is stable. When $\gamma<0$, the system is in the broken
$\mathcal{PT}$ symmetric phase, where the eigenmodes appear in complex
conjugated pairs, corresponding to an unstable phase.
This behavior is illustrated in Fig. 1 where the locus of the normalized
natural frequencies $\omega_{\pm}/(kc)$ given by (6a) is represented in the
complex plane as a function of $\varepsilon_{zx}$.

Figure 1: Locus in the complex plane of the normalized bulk natural
frequencies $\omega_{\pm}/(kc)$ , when $\varepsilon_{zx}$ varies in the range
$0<\varepsilon_{zx}<0.1$. The insets indicate the value of $\varepsilon_{zx}$
at the points marked with crosses. The remaining parameters used in the
simulation are $\varepsilon_{xx}=1$, $\varepsilon_{zz}=1.1$ and
$\varepsilon_{xz}=-0.05$. The exceptional point ($\gamma=0$) is reached when
$\varepsilon_{zx}=-\varepsilon_{xz}=0.05$.

For the considered range of parameters, it is seen that when
$0<\varepsilon_{zx}<0.05$ the parameter $\gamma$ is positive and both
frequencies $\omega_{\pm}$ lie on the real axis, implying that the system is
stable and is in the unbroken $\mathcal{PT}$ symmetric phase. As
$\varepsilon_{zx}$ increases from 0 to 0.05, it can be observed that the
eigenmodes gradually converge towards each other until they coalesce at the
exceptional point associated with $\varepsilon_{zx}=0.05$. For larger values
of $\varepsilon_{zx}$, the eigenmodes leave the real axis and become complex
conjugates. In this case, the system enters the broken $\mathcal{PT}$
symmetric phase, where it becomes unstable, with waves growing exponentially
in time until some nonlinear saturation mechanism limits the gain.

Both for the stable ($\gamma>0$) and unstable ($\gamma<0$) situations, the
eigenvectors of the indefinite matrix $\overline{\varepsilon}^{\prime\prime}$
that controls the non-Hermitian interactions are $\hat{\mathbf{x}}\pm
i\hat{\mathbf{z}}$. Thus, when $\overline{\varepsilon}$ is real-valued, the
wave polarizations that activate the gain/loss are always circularly polarized
(chiral-gain). On the other hand, if either $\varepsilon_{xz}$ or
$\varepsilon_{zx}$ are allowed to be complex-valued the eigenpolarizations
that control the non-Hermitian interactions are in general elliptically
polarized, corresponding to a more general form of indefinite-gain. For
example, when $\varepsilon_{zx}=0$ and $\varepsilon_{xz}$ is pure-imaginary
(as considered later in the article) the relevant eigenpolarizations are
linearly polarized and are of the form $\hat{\mathbf{x}}\pm\hat{\mathbf{z}}$.

In the rest of the article, we focus on the permittivity tensor given by (2)
with $\varepsilon_{zx}=0$. From the previous stability analysis, it is evident
that such a response is always stable ($\gamma\geq 0$), even when
$\varepsilon_{xz}$ is allowed to be complex-valued. Furthermore, the
exceptional point ($\gamma=0$), where the eigenvectors of the two transverse
natural modes coalesce into a single linear polarization, is reached when
$\varepsilon_{xx}=\varepsilon_{zz}$. In this regime the indefinite-gain medium
behaves as an ideal travelling wave amplifier, as discussed in detail in
Fernandes  _et al._ (2024). Throughout the remainder of the paper, we avoid
this scenario by assuming that $\varepsilon_{xx}\neq\varepsilon_{zz}$, so that
the material response is “bounded”.

##  III Lasing

In this section, we illustrate how the non-Hermitian response provided by
chiral-gain media can be harnessed for the creation of an electromagnetic
oscillator, i.e. a laser. A related study based on a stack of 2D materials was
recently put forward in Hakimi  _et al._ (2023). Next, we study laser-
configurations based on 3D materials.

A laser is an oscillator that supports exponentially growing natural modes
($\omega^{\prime\prime}>0$) before it reaches the saturation (nonlinear)
regime. Here, we consider a one dimensional closed cavity consisting of a
chiral-gain medium slab positioned in between two mirrors. We investigate the
natural modes of such a cavity for wave propagation along the $y$ direction.

The complex eigenmodes of the cavity can be determined by solving Maxwell
equations (5). For propagation along the $y$ direction, we have
$\nabla=\partial/\partial y\,\hat{\mathbf{y}}$, and these equations can be
written in a matrix form as

$\displaystyle i\frac{\partial}{\partial
y}\mathbf{f}(y)=\overline{\mathbf{M}}\cdot\mathbf{f}(y)$ (7)

where the four-component state vector $\mathbf{f}$ is defined by

$\displaystyle\mathbf{f}(y)=\begin{pmatrix}E_{x}(y)&E_{z}(y)&H_{x}(y)&H_{z}(y)\end{pmatrix}^{T}$
(8)

with $E_{x}$, $E_{z}$, $H_{x}$, $H_{z}$ the $x$ and $z$ components of the
electric and magnetic fields at position $y$. The $4\times 4$ matrix
$\overline{\mathbf{M}}$ is given by

$\displaystyle\overline{\mathbf{M}}=\begin{pmatrix}0&0&0&\omega\mu_{0}\\\
0&0&-\omega\mu_{0}&0\\\ 0&-\omega\varepsilon_{0}\varepsilon_{zz}&0&0\\\
\omega\varepsilon_{0}\varepsilon_{xx}&\omega\varepsilon_{0}\varepsilon_{xz}&0&0\end{pmatrix}.$
(9)

Similar to a Schrödinger equation, the solution of Eq. (7) is

$\displaystyle\mathbf{f}(y)=\text{e}^{-iy\overline{\mathbf{M}}}\cdot\mathbf{f}(0),$
(10)

where the exponential $\text{e}^{-iy\overline{\mathbf{M}}}$ is a $4\times 4$
matrix that plays the role of a transfer matrix. The matrix exponential can be
evaluated analytically (not shown here), or alternatively it can be
numerically evaluated. Thus, the fields at a generic position $y$ within the
material depend only on the fields calculated at $y=0$.

###  III.1 Cavity closed by two PECs

To begin, we consider the scenario where the two reflectors are perfect
electric conducting (PEC) walls as represented in Fig. 2.

Figure 2:  Indefinite-gain medium cavity closed with two PEC walls at $y=0$
and $y=d$.

In this case, the fields at each boundary of the cavity satisfy

$\displaystyle\mathbf{f}(0)$
$\displaystyle=H_{x}(0)\hat{\mathbf{u}}_{3}+H_{z}(0)\hat{\mathbf{u}}_{4}$
(11a) $\displaystyle\mathbf{f}(d)$
$\displaystyle=H_{x}(d)\hat{\mathbf{u}}_{3}+H_{z}(d)\hat{\mathbf{u}}_{4}$
(11b)

where
$\hat{\mathbf{u}}_{i}=(\delta_{i1}\;\delta_{i2}\;\delta_{i3}\;\delta_{i4})^{T}$
with $\delta_{ij}$ the Kronecker delta symbol, is a vector with the entry ”1”
at the $i$-th column. Taking into account that
$\mathbf{f}(d)=\text{e}^{-id\overline{\mathbf{M}}}\cdot\mathbf{f}(0)$
[Eq.(10)] and that
$\mathbf{f}(d)\cdot\hat{\mathbf{u}}_{1}=\mathbf{f}(d)\cdot\hat{\mathbf{u}}_{2}=0$,
we find after straightforward manipulations the following system of equations:

$\displaystyle\begin{pmatrix}\hat{\mathbf{u}}_{1}\cdot\text{e}^{-id\overline{\mathbf{M}}}\cdot\hat{\mathbf{u}}_{3}&&\hat{\mathbf{u}}_{1}\cdot\text{e}^{-id\overline{\mathbf{M}}}\cdot\hat{\mathbf{u}}_{4}\\\
\hat{\mathbf{u}}_{2}\cdot\text{e}^{-id\overline{\mathbf{M}}}\cdot\hat{\mathbf{u}}_{3}&&\hat{\mathbf{u}}_{2}\cdot\text{e}^{-id\overline{\mathbf{M}}}\cdot\hat{\mathbf{u}}_{4}\end{pmatrix}\cdot\begin{pmatrix}H_{x}(0)\\\
H_{z}(0)\end{pmatrix}=\begin{pmatrix}0\\\ 0\end{pmatrix}$ (12)

The dispersion equation is found by setting the determinant of the matrix in
Eq. (12) equal to zero. After simplifications, we obtain the following
explicit result

$\displaystyle\sin\left(\frac{\omega}{c}d\sqrt{\varepsilon_{xx}}\right)\sin\left(\frac{\omega}{c}d\sqrt{\varepsilon_{zz}}\right)=0.$
(13)

Clearly, the natural frequencies of the cavity of Fig. 2 are
$\omega=\omega_{n}^{x}$ and $\omega=\omega_{n}^{z}$ with

$\displaystyle\omega_{n}^{x}$ $\displaystyle=\frac{n\pi
c}{d\sqrt{\varepsilon_{xx}}}$ (14a) $\displaystyle\omega_{n}^{z}$
$\displaystyle=\frac{n\pi c}{d\sqrt{\varepsilon_{zz}}}$ (14b)

and $n$ is an integer. Curiously, even though the bulk material can provide
optical gain Lannebère  _et al._ (2022), the cavity eigenmodes are real-valued
and consequently it is not possible to generate lasing relying simply on PEC
walls. To understand why, let us examine the time averaged net power exchanged
between the wave and the material. For time-harmonic fields
$\mathbf{E}(t)=\frac{1}{2}\left(\mathbf{E}_{\omega}\text{e}^{-i\omega
t}+\mathbf{E}_{\omega}^{\ast}\text{e}^{i\omega t}\right)$ it is given by
$\left\langle
Q\right\rangle=\frac{\omega\varepsilon_{0}}{2}\int\mathbf{E}_{\omega}^{\ast}\cdot\overline{\varepsilon}^{\prime\prime}\cdot\mathbf{E}_{\omega}~{}dV$
(see the supplemental material of Lannebère  _et al._ (2022)). For a chiral-
gain medium described by the permittivity tensor (2) with purely real diagonal
elements, one has

$\displaystyle\left\langle Q\right\rangle$
$\displaystyle=\frac{\omega\varepsilon_{0}}{2}\int\mathrm{Im}\left\\{\varepsilon_{xz}E_{\omega,x}^{\ast}E_{\omega,z}\right\\}~{}dV.$
(15)

From Eqs. (10), (11a) and (12) the eigenvectors $\mathbf{E}_{n}^{x}$ and
$\mathbf{E}_{n}^{z}$ associated with the natural modes $\omega_{n}^{x}$ and
$\omega_{n}^{z}$, respectively, are:

$\displaystyle\mathbf{E}_{n}^{x}$ $\displaystyle=A^{x}\sin\left(\frac{n\pi
y}{d}\right)\hat{\mathbf{x}}$ (16a) $\displaystyle\mathbf{E}_{n}^{z}$
$\displaystyle=A^{z}\sin\left(\frac{n\pi
y}{d}\right)\left(\hat{\mathbf{x}}+\frac{\varepsilon_{zz}-\varepsilon_{xx}}{\varepsilon_{xz}}\hat{\mathbf{z}}\right)$
(16b)

where $A^{x}$ and $A^{z}$ are normalization constants. As seen, the eigenmodes
are linearly polarized everywhere in the cavity when $\varepsilon_{xz}$ is
real-valued. When $\varepsilon_{xz}$ is complex-valued, the eigenmode
associated with $\omega_{n}^{z}$ becomes elliptically polarized. In either
case, by substituting the explicit formulas for the fields into Eq. (15)
($\mathbf{E}_{\omega}\to\mathbf{E}_{n}^{i}$) one finds that $\left\langle
Q\right\rangle=0$ for arbitrary $\varepsilon_{xz}$. Thus, in this
configuration the polarization of the eigenmodes is unable to unlock the gain
and consequently there is no net transfer of energy from the material to the
wave.
It is interesting to point out that the PEC mirrors preserve the
$\mathcal{PT}$-symmetry of the bulk material, and thereby for a real-valued
$\overline{\varepsilon}$ the spectrum must remain real-valued for an unbroken
$\mathcal{PT}$-phase.

###  III.2 Cavity closed by a PEC and a soft-hard boundary

To extract energy from the medium we need to engineer the polarization of the
cavity eigenmodes so that it “overlaps” the eigenvector of
$\overline{\varepsilon}^{\prime\prime}$ that triggers the gain response. This
can be done by replacing one of the PEC walls by a soft-hard boundary (or
Kildal’s mirror) Kildal (1990); Kildal and Kishk (2003) as represented in Fig.
3 (a). An ideal soft-hard boundary may be visualized as a set of alternating
strips made of PEC and perfect magnetic conductor (PMC) [Fig. 3 (b)]. In
practice, this type of boundary condition can be implemented using a
corrugated plate mirror Kildal (1990); Kildal and Kishk (2003). Such mirror
imposes that the electric and magnetic fields at the boundary satisfy

$\displaystyle\mathbf{E}\cdot\hat{\mathbf{s}}=\mathbf{H}\cdot\hat{\mathbf{s}}=0$
(17)

where
$\hat{\mathbf{s}}=\cos(\varphi)\hat{\mathbf{x}}+\sin(\varphi)\hat{\mathbf{z}}$
is a unit vector parallel to the direction of the strips as shown in Fig. 3
(b). The vector $\hat{\mathbf{s}}$ makes an angle $\varphi$ with the $x$ axis.
It is useful to introduce the unit vector
$\hat{\mathbf{t}}=-\sin(\varphi)\hat{\mathbf{x}}+\cos(\varphi)\hat{\mathbf{z}}$
that is orthogonal to $\hat{\mathbf{s}}$. Note that the combination of the PEC
boundary with the Kildal boundary breaks the $\mathcal{P}$ and $\mathcal{PT}$
symmetries of the system.

Figure 3: $\bf{(a)}$ Indefinite-gain medium cavity closed with a PEC wall at
$y=0$ and a soft-hard boundary condition at $y=d$. $\bf{(b)}$ Sketch of a
soft-hard boundary implemented by alternating thin strips of PEC and PMC
materials oriented along $\hat{\mathbf{s}}$. $\bf{(c)}$–$\bf{(e)}$ Locus of
the eigenfrequencies $\omega=\omega^{\prime}+i\omega^{\prime\prime}$ in the
complex-plane for the cavity represented in (a). The cross permittivity
component is $\varepsilon_{xz}=0.2$ in (c) and $\varepsilon_{xz}=0.1$ in (d).
The red dots (blue crosses) are calculated for an orientation of the soft-hard
mirror $\varphi=58.23\mathrm{\SIUnitSymbolDegree}$
($\varphi=45\mathrm{\SIUnitSymbolDegree}$). In (e)
$\varphi=45\mathrm{\SIUnitSymbolDegree}$ while $\varepsilon_{xz}=0.2i$ (red
dots) or $\varepsilon_{xz}=0.01i$ (blue crosses). $\bf{(f)}$ $y$-component of
the spin angular momentum (in arbitrary units) and its spatial average for the
mode with the highest value of $\omega^{\prime\prime}$ for the blue crosses in
(c). In all plots $\varepsilon_{xx}=1$ and $\varepsilon_{zz}=1.1$.

For the geometry of Fig. 3 (a), the PEC boundary condition imposes that the
state vector satisfies Eq. (11a) at $y=0$, whereas the soft-hard boundary
condition at $y=d$ imposes that

$\displaystyle\mathbf{f}(d)\cdot\hat{\mathbf{s}}_{E}=\mathbf{f}(d)\cdot\hat{\mathbf{s}}_{H}=0$
(18)

where $\hat{\mathbf{s}}_{E}=(s_{x}\;s_{z}\;0\;0)^{T}$ and
$\hat{\mathbf{s}}_{H}=(0\;0\;s_{x}\;s_{z})^{T}$ with
$s_{x}=\hat{\mathbf{s}}\cdot\hat{\mathbf{x}}$ and
$s_{z}=\hat{\mathbf{s}}\cdot\hat{\mathbf{z}}$. Using Eq. (10) it may be shown
that the cavity fields are constrained as

$\displaystyle\begin{pmatrix}\hat{\mathbf{s}}_{E}\cdot\text{e}^{-id\overline{\mathbf{M}}}\cdot\hat{\mathbf{u}}_{3}&&\hat{\mathbf{s}}_{E}\cdot\text{e}^{-id\overline{\mathbf{M}}}\cdot\hat{\mathbf{u}}_{4}\\\
\hat{\mathbf{s}}_{H}\cdot\text{e}^{-id\overline{\mathbf{M}}}\cdot\hat{\mathbf{u}}_{3}&&\hat{\mathbf{s}}_{H}\cdot\text{e}^{-id\overline{\mathbf{M}}}\cdot\hat{\mathbf{u}}_{4}\end{pmatrix}\cdot\begin{pmatrix}H_{x}(0)\\\
H_{z}(0)\end{pmatrix}=\begin{pmatrix}0\\\ 0\end{pmatrix}.$ (19)

Again, the dispersion equation is found by setting the determinant of the
matrix on the left-hand side equal to zero. After simplifications, we find
that for $\varepsilon_{xx}\neq\varepsilon_{zz}$, the natural frequencies are
given by the solutions of:

$\displaystyle\beta_{-}\sin\left(\frac{\omega
d}{c}\left[\sqrt{\varepsilon_{zz}}-\sqrt{\varepsilon_{xx}}\right]\right)-\beta_{+}\sin\left(\frac{\omega
d}{c}\left[\sqrt{\varepsilon_{zz}}+\sqrt{\varepsilon_{xx}}\right]\right)=0$
(20)

where

$\beta_{\pm}=\left[\left(\sqrt{\varepsilon_{zz}}\pm\sqrt{\varepsilon_{xx}}\right)\left[\sqrt{\varepsilon_{zz}}\cos^{2}\left(\varphi\right)\pm\sqrt{\varepsilon_{xx}}\sin^{2}\left(\varphi\right)\right]-\varepsilon_{xz}\cos\left(\varphi\right)\sin\left(\varphi\right)\right]\cdot\\\
\left(\sqrt{\varepsilon_{zz}}\mp\sqrt{\varepsilon_{xx}}\right).$ (21)

For $\varphi=0$ or $\varphi=\pi/2$, i.e., when the strips are parallel to a
coordinate axis, one has $\beta_{+}=\pm\beta_{-}$. As the sum (or the
difference) of two sine-functions can be written as the product of a sine
function and a cosine function, it is clear that for $\varphi=0$ or
$\varphi=\pi/2$ the natural frequencies of the cavity are purely real. For
these specific orientations, the corrugated grid cannot induce lasing in the
cavity. Interestingly, for other orientations of the corrugations, the
eigenfrequencies may become complex-valued depending on the value of
$\varepsilon_{xz}$.
We show in Fig. 3 (c)–(e) the solutions
$\omega=\omega^{\prime}+i\omega^{\prime\prime}$ of Eq. (20) in the complex
plane (eigenfrequencies) for different values of the cross-coupling term
$\varepsilon_{xz}$ and various orientations of the corrugations $\varphi$. For
all the examined cases, the eigenfrequencies form a discrete (countable) set
without any finite accumulation points in the complex plane.
For the reasons discussed above, the modal equation (20) has real-valued
solutions when $\beta_{+}=\pm\beta_{-}$. Solving the equations
$\beta_{+}=\pm\beta_{-}$ with respect to $\varepsilon_{xz}$ one finds that
$\varepsilon_{xz}=\varepsilon_{xz,1}$ or $\varepsilon_{xz}=\varepsilon_{xz,2}$
where
$\varepsilon_{xz,1}\equiv\left(\varepsilon_{xx}-\varepsilon_{zz}\right)\tan(\varphi)$
and
$\varepsilon_{xz,2}\equiv\left(\varepsilon_{zz}-\varepsilon_{xx}\right)\cot(\varphi)$.
Our numerical analysis shows that these values determine exactly the gain
thresholds required to have lasing. As $\varepsilon_{xz,1}$ and
$\varepsilon_{xz,2}$ have opposite signs, they determine the threshold for
positive and negative values of $\varepsilon_{xz}$. Specifically, to have
lasing one needs that $\varepsilon_{xz}>\varepsilon_{xz,\text{th}}^{+}$ or
$\varepsilon_{xz}<\varepsilon_{xz,\text{th}}^{-}$ with

$\displaystyle\varepsilon_{xz,\text{th}}^{+}\equiv\max\left\\{\varepsilon_{xz,1},\varepsilon_{xz,2}\right\\},$
(22a)
$\displaystyle\varepsilon_{xz,\text{th}}^{-}\equiv\min\left\\{\varepsilon_{xz,1},\varepsilon_{xz,2}\right\\}.$
(22b)

It can be checked that in these conditions $|\beta_{+}|<|\beta_{-}|$. It is
implicit that $\varepsilon_{xx}\neq\varepsilon_{zz}$, $\varphi\neq 0,\pi/2$.
Remarkably, for a fixed value of the permittivity, the gain threshold can be
made as small as desired by selecting the appropriate angle $\varphi$.
Moreover, as expected the thresholds are always different from zero,
indicating that in the absence of cross-coupling the resonant frequencies
remain real-valued.

For $\varepsilon_{xz}$ real, the best orientation for the corrugations, i.e.,
the orientation that yields the largest amplification rate
$\omega^{\prime\prime}$, depends on the specific elements of the permittivity
tensor. For small $\left|{{\varepsilon_{xz}}}\right|$, the optimal grid
orientation is close to either $\varphi=0$ or $\varphi=\
90\mathrm{\SIUnitSymbolDegree}$. On the other hand, for moderately large
values of $\left|{{\varepsilon_{xz}}}\right|$ the optimal grid orientation
approaches $\varphi=45\mathrm{\SIUnitSymbolDegree}$, as illustrated in the
following numerical example.
Figures 3 (c)–(d) depict the cavity spectra calculated for positive cross-
coupling with $\varepsilon_{xz}=0.2$ and $\varepsilon_{xz}=0.1$, respectively.
We consider two grid orientations $\varphi=58.23\mathrm{\SIUnitSymbolDegree}$
and $\varphi=45.0\mathrm{\SIUnitSymbolDegree}$. The corresponding positive
gain threshold values are $\varepsilon_{xz,\text{th}}^{+}\approx 0.062$ for
$\varphi=58.23\mathrm{\SIUnitSymbolDegree}$ and
$\varepsilon_{xz,\text{th}}^{+}=0.1$ for
$\varphi=45\mathrm{\SIUnitSymbolDegree}$. It can be seen in Fig. 3 (d) that
when $\varepsilon_{xz}=\varepsilon_{xz,\text{th}}^{+}$ all the
eigenfrequencies are real-valued (blue crosses). Conversely, for
$\varepsilon_{xz}>\varepsilon_{xz,\text{th}}^{+}$, some of the modes split
into two branches with opposite imaginary part of $\omega$ [red dots in Figs.
3 (c) and (d) and blue crosses in (c)]. This pattern is repeated periodically
in frequency with period $\Delta\omega_{1}\equiv\pi
c/\left(d\left|\sqrt{\varepsilon_{xx}}-\sqrt{\varepsilon_{zz}}\right|\right)$
independent of $\varphi$. For $\varepsilon_{xz}=0.2$, the optimal $\varphi$ is
close to the value $\varphi=58.23\mathrm{\SIUnitSymbolDegree}$ used in Fig. 3
(c), whereas for $\varepsilon_{xz}=0.1$ the optimal orientation is close to
$\varphi=67.5\mathrm{\SIUnitSymbolDegree}$ (not shown). In Fig. 3 (f), we
represent the wave spin angular momentum density (proportional to the vector
$i\mathbf{E}\times\mathbf{E}^{\ast}$ Bliokh  _et al._ (2014)), as a function
of the position in the cavity. We consider the mode with the highest
$\omega^{\prime\prime}$ in the example of Fig. 3 (c) corresponding to the blue
crosses. As seen, the spin angular momentum oscillates with the position in
the cavity and takes both positive and negative values (associated with
different handednesses). However, its average value aligns with the spin of
the eigenmode of $\overline{\varepsilon}^{\prime\prime}$ that controls the
gain (a specific circular polarization with handedness controlled by the sign
of $\varepsilon_{xz}$). We checked that this remains true for all the lasing
modes (with $\omega^{\prime\prime}>0$).
When $\varepsilon_{xz}$ is a pure-imaginary number, similar to the
implementation proposed in Fernandes  _et al._ (2024), $\beta_{\pm}$ are also
complex-valued. In this situation, our numerical study shows that the cavity
can always support solutions exponentially growing in time. Thus, when
$\varepsilon_{xz}$ is complex-valued, lasing is in principle possible as soon
as the cross-coupling is different from zero as illustrated in Fig. 3 (e). The
numerical simulations show that the best orientation for the corrugations is
$\varphi=45\mathrm{\SIUnitSymbolDegree}$. The polarization of lasing modes is
now approximately linear because the eigenmode of
$\overline{\varepsilon}^{\prime\prime}$ that controls the gain is of the type
$\hat{\mathbf{x}}\pm\hat{\mathbf{z}}$.
It is interesting to note that when the modes move away from the real axis,
the real-parts of adjacent eigenfrequencies are equally spaced by
$\Delta\omega^{\prime}=\Delta\omega_{2}$ with $\Delta\omega_{2}\equiv 2\pi
c/\left(d\left[\sqrt{\varepsilon_{xx}}+\sqrt{\varepsilon_{zz}}\right]\right)$
for modes in the same semi-plane (upper-half or lower-half frequency plane).
For a real-valued $\varepsilon_{xz}$, the spectrum has mirror symmetry about
the horizontal axis [Figs. 3 (c)–(d)], analogous to $\mathcal{P}\mathcal{T}$
symmetric systems. We will explain this property in the next subsection with
the help of image theory. In contrast, for a complex-valued
$\varepsilon_{xz}$, the eigenfrequencies in the different semi-planes are not
related by complex-conjugation [Fig. 3 (e)].

###  III.3 Cavity closed by two Kildal’s mirrors

Suppose now that the PEC wall at $y=0$ is replaced by a second soft-hard
mirror as illustrated in Fig. 4 (a). For simplicity, it is assumed that the
two soft-hard mirrors have the same orientation. When $\varepsilon_{xz}$ is
real-valued, the cavity is $\mathcal{PT}$-symmetric .

Interestingly, the modes of this cavity can be split into $\bf{E}$-even and
$\bf{E}$-odd modes, with the even (odd) solutions associated with a system
where a PMC (PEC) mirror is placed at the center of the cavity. Thereby, the
spectrum of this resonator is the direct sum of the spectrum of a PEC/Kildal
resonator with length $L/2$, already studied in subsection III.2, and the
spectrum of a PMC/Kildal resonator, also with length $L/2$. This observation
explains why the spectrum of the PEC/Kildal cavity exhibits mirror symmetry
when $\varepsilon_{xz}$ is real-valued. Indeed, due to image theory these
modes can be as well regarded as modes of a $\mathcal{PT}$-symmetric cavity
with two Kildal mirrors.

For the resonator with two Kildal mirrors, the boundary conditions at $y=0$
imply that

$\displaystyle\mathbf{f}(0)=A_{E}\hat{\mathbf{t}}_{E}+A_{H}\hat{\mathbf{t}}_{H}$
(23)

where $A_{E}$ and $A_{H}$ are some unknown coefficients,
$\hat{\mathbf{t}}_{E}=(t_{x}\;t_{z}\;0\;0)^{T}$ and
$\hat{\mathbf{t}}_{H}=(0\;0\;t_{x}\;t_{z})^{T}$ with
$t_{x}=\hat{\mathbf{t}}\cdot\hat{\mathbf{x}}$ and
$t_{z}=\hat{\mathbf{t}}\cdot\hat{\mathbf{z}}$.

Figure 4: $\bf{(a)}$ Indefinite-gain medium cavity closed with two soft-hard
mirrors at $y=0$ and $y=d$. $\bf{(b)}$–$\bf{(d)}$ Locus of the cavity
eigenfrequencies $\omega=\omega^{\prime}+i\omega^{\prime\prime}$ in the
complex plane. The cross permittivity is $\varepsilon_{xz}=0.2$ in (b) and
$\varepsilon_{xz}=0.1$ in (c). The grid orientation is
$\varphi=58.23\mathrm{\SIUnitSymbolDegree}$ (red dots) or
$\varphi=45\mathrm{\SIUnitSymbolDegree}$ (blue crosses). The gain thresholds
are the same as in Fig.3. In (d) $\varphi=45\mathrm{\SIUnitSymbolDegree}$ and
$\varepsilon_{xz}=0.2i$ (red dots) or $\varepsilon_{xz}=0.01i$ (blue crosses).
In all plots $\varepsilon_{xx}=1$ and $\varepsilon_{zz}=1.1$.

Imposing the soft-hard boundary conditions at $y=d$ one finds that:

$\displaystyle\begin{pmatrix}\hat{\mathbf{s}}_{E}\cdot\text{e}^{-id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{E}&&\hat{\mathbf{s}}_{E}\cdot\text{e}^{-id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{H}\\\
\hat{\mathbf{s}}_{H}\cdot\text{e}^{-id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{E}&&\hat{\mathbf{s}}_{H}\cdot\text{e}^{-id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{H}\end{pmatrix}\cdot\begin{pmatrix}A_{E}\\\
A_{H}\end{pmatrix}=\begin{pmatrix}0\\\ 0\end{pmatrix}.$ (24)

The dispersion equation is obtained using the same procedure as before.
Consistent with image theory, our numerical studies show that the resonant
frequencies for this configuration exhibit the same qualitative behavior as in
the PEC/Kildal cavity of Fig.3 (a). In particular, we find that resonant
frequencies with a positive imaginary part always exist for a pure imaginary
$\varepsilon_{xz}$ [see Fig. 4 (d)], and are only allowed above some positive
and negative thresholds $\varepsilon_{xz,\text{th}}^{\pm}$ for a real-valued
$\varepsilon_{xz}$ [see Fig. 4 (c)]. The gain thresholds for lasing are
coincident with the ones determined in the previous section. The optimal
orientation for the corrugations for a real-valued $\varepsilon_{xz}$ is also
unchanged as compared to the previous subsection. The main difference as
compared to the PEC/Kildal cavity is the peak gain rate of the lasing modes,
which is two times larger with the two Kildal mirrors. Furthermore, the
frequency period of the spectrum is also two times larger than before
($2\Delta\omega_{1}$). These properties are all a simple consequence of image
theory.

In summary, we demonstrated that optical cavities based on a indefinite-gain
media terminated by a Kildal’s mirror and a PEC or two Kildal’s mirrors may be
used to implement electrically pumped chiral lasers. The configuration with
two Kildal mirrors is the most advantageous as it provides the largest
amplification rate.

##  IV Chiral-gain Mirrors

In this section, we analyze another application of chiral-gain materials,
namely the realization of non-Hermitian mirrors that provide either
amplification or attenuation, depending on the wave polarization handedness.
To this end, we consider the plane wave excitation of a chiral-gain medium
slab backed by a mirror, as illustrated in Fig. 5 (a). We focus on the case
where $\varepsilon_{xz}$ is real-valued, so that the system is time-reversal
symmetric.

Figure 5: $\bf{(a)}$ A chiral-gain medium slab of thickness $d$ is terminated
by a Kildal’s mirror. The material slab is illuminated along the normal
direction by a plane wave $\mathbf{E}^{\text{inc}}$. $\bf{(b)}$-$\bf{(c)}$
Schematics depicting the working principle of the non-Hermitian mirror when
$|\rho_{1}|^{2}<1$ (panel (b), corresponding to attenuation) and
$|\rho_{2}|^{2}>1$ (panel (c), corresponding to gain). $\bf{(d)}$ Illustration
of the mirror response when a time-reversal transformation ($\mathcal{T}$) is
applied to scenario (b). In all plots, the incident (reflected) wave is
represented in blue (red).

The mirror is taken as a corrugated wall described by soft-hard boundary
conditions, while the surrounding medium is a dielectric of permittivity
$\varepsilon_{d}$, for example air. As illustrated in Fig. 5 (b)–(c), the non-
Hermitian mirror can be used to selectively amplify or attenuate the incoming
wave, depending on its polarization state. This can be useful, for instance,
to enhance the back-scattering of some object with a specific polarization
signature.
It is assumed that the plane wave propagates along the $+y$-direction and
impinges on the slab along the normal direction. Let
$\mathbf{E}_{t}^{\text{inc}}=E_{x}^{\text{inc}}\hat{\mathbf{x}}+E_{z}^{\text{inc}}\hat{\mathbf{z}}$
be the complex amplitude of the (transverse) incident field and
$\mathbf{E}_{t}^{\text{ref}}$ the complex amplitude of the (transverse)
reflected field. At the $y=0$ interface these fields are related by

$\displaystyle\mathbf{E}_{t}^{\text{ref}}(y=0)$
$\displaystyle=\overline{\mathbf{R}}\cdot\mathbf{E}_{t}^{\text{inc}}(y=0)$
(25)

where $\overline{\mathbf{R}}$ is the 2$\times$2 reflection matrix, whose
expression is derived in appendix A [Eq. (42)]. The power reflected by the
chiral-gain slab is proportional to

$\displaystyle|\mathbf{E}_{t}^{\text{ref}}|^{2}$
$\displaystyle=\mathbf{E}_{t}^{\text{inc},\ast}\cdot\overline{\mathbf{R}}^{\dagger}\cdot\overline{\mathbf{R}}\cdot\mathbf{E}_{t}^{\text{inc}}$
(26)

The reflectance matrix
$\overline{\mathcal{R}}=\overline{\mathbf{R}}^{\dagger}\cdot\overline{\mathbf{R}}$
plays a crucial role in our analysis. It is a positive definite Hermitian
matrix, meaning that its eigenvectors $\mathbf{E}_{\text{eig},1}$ and
$\mathbf{E}_{\text{eig},2}$ form an orthogonal basis, while the eigenvalues
$|\rho_{1}|^{2}$ and $|\rho_{2}|^{2}$ are positive numbers. It is assumed,
without loss of generality, that $|\rho_{1}|^{2}\leq|\rho_{2}|^{2}$. Clearly,
the reflectance
$|\rho|^{2}\equiv|\mathbf{E}_{t}^{\text{ref}}|^{2}/|\mathbf{E}_{t}^{\text{inc}}|^{2}$
is bounded by the eigenvalues of $\overline{\mathcal{R}}$, namely
$|\rho_{1}|^{2}\leq|\rho|^{2}\leq|\rho_{2}|^{2}$. Additionally, the
eigenvectors $\mathbf{E}_{\text{eig},1}$ and $\mathbf{E}_{\text{eig},2}$
determine the polarizations of the incident field that minimize and maximize
the reflected signal power, respectively. It is important to note that
$\mathbf{E}_{\text{eig},i}$ are not eigenvectors of the reflection matrix,
which means that the reflected wave may have a polarization different from
that of the incident wave when
$\mathbf{E}_{t}^{\text{inc}}=\mathbf{E}_{\text{eig},i}$.
In a lossless and passive system (two-port microwave network), energy
conservation imposes that the power reflected by the network must be identical
to the incident power. Thus, in the conservative case one has
$|\rho_{1}|^{2}=|\rho_{2}|^{2}=1$. On the other hand, non-Hermitian systems
are not constrained by energy conservation, and hence it is possible to have
$|\rho_{i}|^{2}>1$ ($|\rho_{i}|^{2}<1$), which corresponds to amplification
(attenuation) of the incident wave.

Figure 6: $\bf{(a)}$ Plot of the eigenvalues $|\rho_{1}|^{2}$,
$|\rho_{2}|^{2}$ of $\overline{\mathcal{R}}$ and their product
$|\rho_{1}|^{2}|\rho_{2}|^{2}$ (note the logarithmic scale), for
$\varepsilon_{d}=1$, $\varepsilon_{xx}=1$, $\varepsilon_{zz}=1.1$,
$\varepsilon_{xz}=0.1$, and $\varphi=45^{\mathrm{\SIUnitSymbolDegree}}$. A, B,
…, E represent the high-symmetry points within the plot. $\bf{(b)}$-$\bf{(d)}$
Polarization curves for the eigenvectors $\mathbf{E}_{\text{eig},1}$,
$\mathbf{E}_{\text{eig},2}$ in the $xoz$ plane for three different frequencies
marked with vertical dashed lines in (a).

Figure 6 shows a plot of the eigenvalues and eigenvectors of
$\overline{\mathcal{R}}$ for a ($\mathcal{T}$-invariant) chiral-gain medium
slab surrounded by air ($\varepsilon_{d}=1$). As seen, the eigenvalues are
different from unity and have values that oscillate periodically with the
frequency or with the slab thickness. As illustrated in Fig. 5 (b)–(c), in a
wide range of frequencies, polarization 1 (red lines) can be strongly absorbed
by the slab whereas in the same frequency range (the orthogonal) polarization
2 (blue lines) is strongly amplified leading to a reflected wave with
amplitude larger than the incoming wave. This behavior is possible as soon as
$\varepsilon_{xz}$ is different from zero (no threshold). Furthermore, it can
be checked that the orientation for the Kildal’s mirror corrugations that
provides the strongest amplification is the same as the orientation that gives
the largest $\omega^{\prime\prime}$ in the closed PEC/Kildal cavity. The
maximum power gain is 18.7 for the example of Fig. 6 (corresponding to point B
of the plot) with $\varphi=45^{\mathrm{\SIUnitSymbolDegree}}$, whereas it can
reach 37.5 when $\varphi=67.5\mathrm{\SIUnitSymbolDegree}$ (not shown).

Quite remarkably the product $|\rho_{1}|^{2}|\rho_{2}|^{2}$ is equal to the
unity meaning that when one of the eigenvectors is amplified the corresponding
orthogonal polarization is attenuated in the same proportion. The property
$|\rho_{1}|^{2}|\rho_{2}|^{2}=1$ is a consequence of time-reversal symmetry.
Indeed, under a time-reversal the roles of the incident and reflected waves
are swapped [see Fig. 5 (b) and (d)] Silveirinha (2019a); Fernandes and
Silveirinha (2022). Thereby, the maxima and the minima of the reflectance are
necessarily interchanged under a time-reversal $|\rho|^{2}\to 1/|\rho|^{2}$,
implying that $|\rho_{1}|^{2}=1/|\rho_{2}|^{2}$ . Note that this result
demonstrates that
$\det\overline{\mathcal{R}}=1=\left|\det\overline{\mathbf{R}}\right|$.

Furthermore, the previous argument also implies that if the field reflected by
wave 1, $\overline{\mathbf{R}}\cdot{{\mathbf{E}}_{{\rm{eig,1}}}}$, (i..e, by
the wave ${\mathbf{E}}_{{\rm{eig,1}}}$ that activates maximum attenuation) is
sent back to the material slab then it it will generate maximum gain. Thereby,
it follows that
${\left(\overline{\mathbf{R}}\cdot{\mathbf{E}_{\rm{eig,1}}}\right)^{\ast}}\sim{{\mathbf{E}}_{{\rm{eig,2}}}}$.
A similar identity holds true with the indices 1 and 2 interchanged. We took
into account that under a time-reversal the electric field is conjugated.
Thus, we have shown that the field reflected by one of the eigenvectors is
proportional to the time-reversed version of the other eigenvector. Formally,
this can be expressed as:

$\displaystyle\overline{\mathbf{R}}\cdot\mathbf{E}_{\text{eig},j}=|\rho_{j}|\text{e}^{i\theta_{j}}\mathbf{E}_{\text{eig},l}^{\ast},\qquad
j\neq l$ (27)

with $\theta_{j}$ some phase. It is implicit in this discussion that $\omega$
is real-valued and that the system is non-Hermitian
($|\rho_{1}|\neq|\rho_{2}|$). The eigenvectors amplitude is normalized to the
unity.

The polarization curves of the eigenvectors $\mathbf{E}_{\text{eig},1}$,
$\mathbf{E}_{\text{eig},2}$ are depicted in Fig. 6 (b)–(d) for different
values of the normalized frequency $\omega d/(\pi c)$ . The incident fields
that maximize the amplification/absorption are generally elliptically
polarized [see Fig. 6 (c)], except at some special high-symmetry points. Due
to the orthogonality of the eigenvectors, the handedness of the wave that
activates the dissipation, $\mathbf{E}_{\text{eig},1}$, is always the opposite
of the handedness of the wave that activates the gain,
$\mathbf{E}_{\text{eig},2}$. From Eq. (27), the polarization curve of the
field reflected by wave 1 is coincident (apart from a scale factor and the
absolute sense of rotation) with the polarization curve of wave 2. In
particular, the polarization curves of an incident wave (1 or 2) and of the
corresponding reflected wave always differ by a
90$\mathrm{\SIUnitSymbolDegree}$ rotation.

Counter-intuitively, the frequency for which the mirror
amplification/attenuation is maximized [points B, B’, D and D’ in Fig. 6 (a)],
corresponds to linearly polarized incident wave [Fig. 6 (b)], with the
electric field forming an angle of approximately
$30^{\mathrm{\SIUnitSymbolDegree}}$ with respect to the coordinate axes. This
is possible because the field polarization inside the material slab is
controlled not only by the polarization of the incident wave, but also by the
anisotropy of the material and by the anisotropy of the Kildal-mirror.
Consequently, there is no strict correlation between the polarization of the
incident wave that triggers the maximum gain and the eigenvector of the matrix
$\overline{\varepsilon}^{\prime\prime}$ that maximizes the gain response of
the medium. Thereby, an incident linearly polarized wave can unlock the
chiral-gain within the material. It is also important to note that any
incident polarization can be decomposed into the basis vectors
$\mathbf{E}_{\text{eig},1}$ and $\mathbf{E}_{\text{eig},2}$. In cases of very
strong gain, the component associated with $\mathbf{E}_{\text{eig},2}$
typically dominates the scattered field, even if it is weak in the incident
wave. Therefore, there is no universal relationship between the polarization
handedness of the incident field and the type of non-Hermitian (dissipative or
amplifying) mirror response.

While the configuration of Fig. 5 provides the strongest non-Hermitian
effects, it is also feasible to achieve chiral-amplification when the Kildal’s
mirror is replaced by a PEC mirror. For the parameters used in Fig. 6, the
chiral-gain slab backed by a PEC mirror yields a maximum power gain of 6 (not
shown), which is roughly six times smaller than the maximum amplification
achievable with the Kildal’s mirror. The fact that the slab backed by the PEC
wall can provide amplification while a cavity enclosed by two PEC mirrors
[Fig.2] cannot produce lasing does not result in any contradiction. This
difference of behavior can be attributed to the polarization of the field
inside the material slab. In the closed cavity, the polarization is controlled
only by the boundary conditions. On the other hand, in the open system, the
polarization of the incident field becomes a free parameter that can be
harnessed to achieve either amplification or absorption.
In summary, it was demonstrated that a chiral-gain medium can be used to
realize polarization-controlled non-Hermitian mirrors. The degree of
amplification/absorption is highly dependent on the polarization and frequency
of the incident wave as well as on the slab thickness.

##  V Modes of the open cavity

Next, we study the natural modes supported by an open cavity, i.e., by a
system with the same geometry as in section IV, without the incoming wave. The
natural modes of passive open systems are characterized by complex-valued
frequencies with $\omega^{\prime\prime}<0$, corresponding to an exponential
decay in time. The oscillation damping is caused either by material absorption
or by radiation loss Silveirinha (2014); Hsu  _et al._ (2016). Here, we want
to study if it is possible to exploit the chiral-gain to compensate these
effects and engineer open resonators with high-quality factors.
For an open-resonator, the field in the dielectric-region ($y<0$) is an
outgoing plane wave. In particular, at the interface ($y=0^{-}$) with the
chiral-gain medium the state vector is of the form:

$\displaystyle\mathbf{f}_{\text{out}}(0)=\begin{pmatrix}\mathbf{E}_{\text{out}}\\\
-\frac{\sqrt{\varepsilon_{d}}}{\eta_{0}}\overline{\mathbf{J}}\cdot\mathbf{E}_{\text{out}}\end{pmatrix}$
(28)

where $\mathbf{E}_{\text{out}}$ is the transverse outgoing electric field and
$\overline{\mathbf{J}}$ is a $2\times 2$ matrix defined as in Eq. (36). We
used Eq. (39) to obtain this result.

The dispersion of the natural modes can be found by matching
$\mathbf{f}_{\text{out}}$, with the field in the chiral-gain slab ($y=0^{+}$)
terminated with a Kildal’s mirror [Eq. (34)].This procedure yields the
following dispersion equation

$\displaystyle\det\left[\begin{pmatrix}\hat{\mathbf{u}}_{3}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{E}&\hat{\mathbf{u}}_{3}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{H}\\\
\hat{\mathbf{u}}_{4}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{E}&\hat{\mathbf{u}}_{4}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{H}\end{pmatrix}+\frac{\sqrt{\varepsilon_{d}}}{\eta_{0}}\overline{\mathbf{J}}\cdot\begin{pmatrix}\hat{\mathbf{u}}_{1}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{E}&\hat{\mathbf{u}}_{1}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{H}\\\
\hat{\mathbf{u}}_{2}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{E}&\hat{\mathbf{u}}_{2}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{H}\end{pmatrix}\right]=0$
(29)

The roots of this equation give the complex eigenfrequencies
$\omega=\omega^{\prime}+i\omega^{\prime\prime}$ of the open cavity. The
qualitative behavior of these solutions with respect to the cross-coupling
coefficient $\varepsilon_{xz}$ is analogous to the behavior observed in the
closed cavity of Sect. III.2. The main difference between the two
configurations can be understood by comparing the spectra obtained with
$\varepsilon_{xz}=0$. Evidently, a closed cavity with $\varepsilon_{xz}=0$ has
all the eigenfrequencies on the real-frequency axis.

Figure 7: $\bf{(a)}$ Locus of the eigenfrequencies of the open cavity
represented in Fig. 5 (a) for $\varepsilon_{d}=3$, $\varepsilon_{xx}=1$,
$\varepsilon_{zz}=1.1$, $\varphi=45^{\mathrm{\SIUnitSymbolDegree}}$ and no
cross-coupling $\varepsilon_{xz}=0$ (green diamonds), a real-valued cross-
coupling $\varepsilon_{xz}=0.2$ (red dots) and a pure-imaginary cross-coupling
$\varepsilon_{xz}=0.18i$ (blue crosses). $\bf{(b)}$ Zoom in view of the
spectrum near the real-frequency axis for $\varepsilon_{xz}=0.2$ and
$\varphi=45^{\mathrm{\SIUnitSymbolDegree}}$ (red dot), for
$\varepsilon_{xz}=0.2$ and $\varphi=45.0395^{\mathrm{\SIUnitSymbolDegree}}$
(green cross) and for $\varepsilon_{xz}=0.2001382$ and
$\varphi=45^{\mathrm{\SIUnitSymbolDegree}}$ (purple triangle).

In contrast, as shown in Fig. 7 (a) with green diamonds, in a passive open
cavity the eigenfrequencies are located in the lower-half frequency plane,
because of the energy leakage towards the exterior. To limit the radiation
loss of the open cavity we consider that the outside medium has a permittivity
of $\varepsilon_{d}=3$. The radiation loss causes the spectrum to be shifted
in the direction of the lower-half plane. The frequency shift (negative
imaginary part) is approximately constant for all modes.
This property still holds true when the gain parameter $\varepsilon_{xz}$ is
different from zero. For example, for an open cavity with a real-valued
$\varepsilon_{xz}$ smaller than the gain thresholds
$\varepsilon_{xz,\text{th}}^{-}<\varepsilon_{xz}<\varepsilon_{xz,\text{th}}^{+}$,
all the eigenfrequencies have roughly the same imaginary part (not shown).
Conversely, for $\varepsilon_{xz}>\varepsilon_{xz,\text{th}}^{+}$ or
$\varepsilon_{xz}<\varepsilon_{xz,\text{th}}^{-}$ the modes split into two
branches in some frequency ranges [red dots in Fig. 7 (a)]. For a pure-
imaginary $\varepsilon_{xz}$ there is no threshold, and consequently the modes
always split into two branches [blues crosses in Fig. 7 (a)], similar to the
closed cavity. Interestingly, for a sufficiently large $\varepsilon_{xz}$,
some modes exhibit a negative imaginary part very close to zero, meaning that
their lifetime in the cavity is very long [Fig. 7 (b)]. Thus, the chiral-gain
provided by the material can indeed compensate the radiation loss effects. By
finely adjusting the resonator parameters, it is possible to achieve a state
where the cavity gain compensates almost exactly the cavity leakage for the
eigenmode with the largest lifetime. This is the scenario represented in Fig.
7 (b), where the mode frequency is only slightly below the real-frequency
axis. This can be done either by tuning the corrugation orientation (green
cross) of the Kildal’s mirror or the strength of the gain parameter (purple
triangle). In this peculiar situation, that we will refer to as the “perfectly
tuned” case, the field inside the cavity oscillates in time without any
noticeable decay. It corresponds to a system that can store light for a long
period of time, despite being open to the exterior. Such systems may have
potential applications in optical memories Alexoudi  _et al._ (2020) or in
sensing. By further increasing the gain coefficient $\varepsilon_{xz}$ beyond
the perfectly tuned case, the imaginary part of the eigenfrequency of at least
one mode becomes positive, causing the open cavity to become an oscillator
(laser). It is relevant to note that in the perfectly tuned case (lasing
threshold) the field in the dielectric region corresponds to an outgoing
propagating plane wave, implying that the non-Hermitian material continuously
radiates energy towards the dielectric.

##  VI Lasing-Anti-Lasing Operation and the reciprocity constraint

In the following, we show that time-reversal invariant open-resonators relying
on materials with chiral-gain can be operated in a regime where they behave as
a CPA-laser Longhi (2010); Chong  _et al._ (2011); Longhi and Feng (2014);
Wong  _et al._ (2016); Longhi (2011); Schackert  _et al._ (2013). A CPA-laser
simultaneously behaves as an anti-laser – signifying that it can fully absorb
an incoming signal – and as “quasi-laser” signifying that it can interact
resonantly with an incoming wave and store efficiently its energy with no
sensible relaxation in time. Previous implementations of linear CPA-lasers
Longhi (2010); Chong  _et al._ (2011); Longhi and Feng (2014); Wong  _et al._
(2016) were based on combining different gain and lossy materials. In
contrast, our system integrates both gain and loss within the same material.
We show that unlike conventional passive resonators Haus (1983), the external
coupling in our resonator is not limited by reciprocity. Thereby our system
can efficiently harvest a significant amount of energy, even for incoming
pulses with relatively short-duration.

###  VI.1 Time-reversal symmetry and anti-lasing

In recent years, there has been significant interest in the analytical
properties of the scattering matrix of time-reversal invariant systems (e.g,
formed by lossless dielectrics) Chong  _et al._ (2010, 2011); Grigoriev  _et
al._ (2013); Baranov  _et al._ (2017). In such systems, the scattering matrix
is constrained by the symmetry
$\overline{\mathbf{S}}\left(\omega\right)=\left[\overline{\mathbf{S}}^{-1}\left(\omega^{\ast}\right)\right]^{\ast}$
Silveirinha (2019a). Thereby, the poles and zeros of $\overline{\mathbf{S}}$
are related by complex conjugation. Note that the poles of
$\overline{\mathbf{S}}$ determine the natural frequencies of the open-system,
and thereby for a passive system lie in the lower half-plane. On the other
hand, the zeros of $\overline{\mathbf{S}}$ determine the frequencies for which
the system does not generate scattering. The “zeros” are the frequencies
associated with a nontrivial null space of the scattering matrix. For a time-
reversal invariant passive platform, these zeros lie in the upper-half
frequency plane. We underline that the “zeros” do not refer to the individual
elements of the scattering matrix, but rather to the zeros of the matrix
determinant.

Several works have developed interesting strategies to harvest, absorb, and
store light by leveraging these analytical properties. Notably, it has been
suggested that introducing a suitable amount of loss into a system may enable
the displacement of a zero of $\overline{\mathbf{S}}$ to the real-frequency
axis, thereby allowing the full absorption of an incoming wave at that
specific frequency with no back-scattering. This phenomenon is known as
“coherent perfect absorption” (CPA) or “anti-lasing” Chong  _et al._ (2010);
Wan  _et al._ (2011). Moreover, it has been demonstrated that the complex
zeros of the scattering matrix can be utilized to store and release light in
an ideally lossless cavity for an arbitrarily long time period. This
mechanism, known as “coherent virtual absorption” Baranov  _et al._ (2017),
relies on exciting the system with a signal that exponentially grows over
time.
Conversely, by judiciously inserting gain into an open-resonator it is
possible to push one of the poles to the real-frequency axis, so that the
resonator can support oscillations that do not decay in time. We refer to this
mode of operation as a “quasi-laser”.
Furthermore, by suitably incorporating loss and gain – such as in
$\mathcal{PT}$ symmetric systems – it is feasible to align a pole and a zero
at the same real-frequency, enabling the system to function both as an anti-
laser and a quasi-laser, depending on the excitation. This dual functionality
defines what is known as a “CPA-laser” Longhi (2010); Chong  _et al._ (2011);
Longhi and Feng (2014); Wong  _et al._ (2016); Longhi (2011); Schackert  _et
al._ (2013). In the next subsections, we demonstrate that the perfectly tuned
resonator introduced in Sect. V can behave as a CPA-laser.

###  VI.2 Analytical structure of $\overline{\mathbf{R}}$ near a resonance

The geometry of our cavity is the same as in Sect. V. We suppose that the
permittivity tensor is real-valued to ensure that the system is time-reversal
symmetric. In order to understand how the chiral-gain affects the excitation
of the open cavity, next we study the analytical properties of the scattering
matrix, which in our system corresponds to the reflection matrix
$\overline{\mathbf{R}}$.

In Appendix B, it is demonstrated that in the vicinity of a resonance
($\omega_{0}=\omega^{\prime}_{0}+i\omega^{\prime\prime}_{0}$) of a open
resonator with high-quality factor, the reflection matrix has the following
structure:

$\displaystyle\overline{\mathbf{R}}\left(\omega\right)\approx\mathbf{U}^{\ast}\left(\omega_{0}^{\prime}\right)\cdot\begin{pmatrix}0&\frac{A}{\left(\omega-\omega_{0}\right)}\\\
\frac{1}{A^{\ast}}\left(\omega-\omega_{0}^{\ast}\right)&0\end{pmatrix}\cdot\mathbf{U}^{\dagger}\left(\omega_{0}^{\prime}\right),$
(30)

where $A$ is some constant and $\mathbf{U}$ is a unitary matrix.

Consistent with the principles of time-reversal symmetry, the pole of the
reflection matrix is accompanied by a complex conjugated zero, so that
$|\det(\overline{\mathbf{R}})|=1$ in the real-frequency axis, as it should be.
Notably, the pole and zero pertain to _distinct_ elements within the middle
matrix in the right-hand side of Eq. (30). This aspect becomes particularly
significant when the system gain is adjusted in such a way that $\omega_{0}$
approaches the real-axis. In fact, the structure of the reflection matrix
preserves the influences of both the pole and zero (associated with a
nontrivial nullspace), allowing them to coexist without cancellation for a
perfectly tuned resonator. Note that one of the eigenvalues of the reflection
matrix is controlled by the pole, whereas the second eigenvalue is controlled
by the zero. This remarkable property is rooted on the non-Hermitian response
of the material. Indeed, it is well known from the study of bound states in
the continuum (BIC), that the influence of a system pole approaching the real-
frequency axis through precise tuning is often completely canceled by a
corresponding zero. This phenomenon renders the excitation of the bound state
from an external region unfeasible, as highlighted in various studies
Silveirinha (2014); Monticone and Alù (2014); Lannebère and Silveirinha
(2015); Monticone  _et al._ (2019); Silva  _et al._ (2020); Prudêncio and
Silveirinha (2021). This underscores the exceptional nature of the chiral-gain
response in our context, where the non-Hermitian characteristics fundamentally
alter the expected analytical properties of the reflection matrix.

To illustrate the discussion, we show in Fig. 8 the magnitude of the elements
of the reflection coefficient for the perfectly tuned cavity corresponding to
the purple triangle in Fig. 7 (b).

Figure 8: $\bf{(a)}$-$\bf{(b)}$ Plot of the magnitude of all entries of the
reflection matrix $\overline{\mathbf{R}}$ as a function of the normalized
frequency for the perfectly tuned case of Fig. 7 (b) (purple triangle)
$\varepsilon_{d}=3$, $\varepsilon_{xx}=1$, $\varepsilon_{zz}=1.1$,
$\varepsilon_{xz}=0.2001382$, $\varphi=45^{\mathrm{\SIUnitSymbolDegree}}$. (b)
Zoom in of panel (a) near the normalized frequency 10.4942. $\bf{(c)}$ Plot of
the eigenvalues $|\rho_{1}|^{2}$ and $|\rho_{2}|^{2}$ of
$\overline{\mathcal{R}}$ as well as their product
$|\rho_{1}|^{2}|\rho_{2}|^{2}$ for the perfectly tuned cavity.

As seen in Fig. 8 (a), the elements of the reflection matrix have multiple
peaks whose positions match the eigenfrequencies of the open cavity. The peak
associated with the almost perfectly tuned case [Fig. 8 (b)] has a very large
magnitude that can be made as large as desired by fine tuning the system
parameters, in agreement with Eq. (30). This behavior suggests that different
from a perfectly tuned BIC resonator Silveirinha (2014); Monticone and Alù
(2014); Lannebère and Silveirinha (2015); Monticone  _et al._ (2019), the
trapped mode of our system can be excited from the outside. This property will
be analyzed in detail in the next subsection with time domain simulations.
Furthermore, when the cavity is finely tuned so that $\omega_{0}$ approaches
the real axis, the reflectivities provided by the open cavity vary as
$|\rho_{1}|^{2}\to 0$ and $|\rho_{2}|^{2}\to\infty$, confirming that the
effects of the pole and of the zero do not cancel out [see Fig. 8 (c)].

###  VI.3 Time excitation of the open cavity and the reciprocity constraint

Let us now analyze the excitation of a high-quality factor open cavity.
Consider first a time harmonic excitation with the incident field polarized as
$\mathbf{E}_{\text{eig},1}$. In the limit ($\omega^{\prime\prime}_{0}\to
0^{-}$), the reflected wave is vanishingly weak because $|\rho_{1}|\to 0$.
This field polarization is thereby completely absorbed by the resonator,
corresponding to a form of coherent perfect absorption (anti-lasing). This
effect is a consequence of the “zero” of the scattering matrix ($|\rho_{1}|\to
0$) on the real frequency axis. It is relevant to note that under a time-
reversal, the scattering problem just discussed, maps into a scenario where
the resonator radiates away a plane wave with frequency $\omega^{\prime}_{0}$
without any external excitation. This is precisely the perfectly tuned
eigenstate discussed in Sect. V, corresponding to the lasing threshold.
Consider now, a time harmonic excitation with the incident polarization
determined by $\mathbf{E}_{\text{eig},2}$. In this case, the response of the
open-resonator diverges $|\rho_{2}|\to\infty$, indicating that there is no
time-harmonic steady-state (quasi-laser resonance). This divergence is caused
by the fact that in the limit $\omega^{\prime\prime}_{0}\to 0^{-}$ the
excitation frequency matches a pole of the scattering matrix.
To gain a deeper insight of the consequences of the time-harmonic response
collapse, we conducted a numerical study of the dynamic response of the open-
resonator for an incident wave with a Gaussian profile

$\displaystyle\mathbf{E}^{\text{inc}}(y=0,t)=\mathrm{Re}\left\\{\mathbf{E}_{0}^{\text{inc}}\text{e}^{-i\omega_{0}t}\text{e}^{-\left(\frac{t-t_{0}}{\sigma}\right)^{2}}\right\\}.$
(31)

Here $\sigma$ is proportional to the full-width half-maximum (FWHM) of the
Gaussian pulse, $t_{0}$ is the time instant for which the incoming wave is
peaked at $y=0$, and $\mathbf{E}_{0}^{\text{inc}}$ is a vector in the $xoz$
plane that determines the polarization of the incident field. Using the
formalism described in Appendix C, we determined the time evolution of
incident Gaussian pulses with different polarizations (Fig. 9).

Figure 9: Time excitation of the open chiral-gain medium cavity with an
incident Gaussian pulse [Eq. (31)]. $\bf{(a)}$ Time evolution of the electric
field envelope in the middle of the cavity ($y=d/2$) as a function of the
normalized time for $\varepsilon_{xz}=0$, $\varepsilon_{d}=3$ and with the
field polarization along $x$. $\bf{(b)}$ Similar to (a) but for the perfectly
tuned case. $\bf{(c)}$ Similar to (b) but for an incident field polarized with
the optimal polarization [see (d)]. For clarity, the amplitude of the incident
field was multiplied by a factor of 20. $\bf{(d)}$ Polarization curves for the
optimal incident field (blue solid line) and for the output (reflected) field
for both (b) and (c) (dashed red line). In all the plots, the incident pulse
parameters are $\sigma=3d/c\approx 15.74\times 2\pi/\omega_{0}$, $t_{0}=10d/c$
and $\omega_{0}\approx 10.4942\pi c/d$.

To begin with, we consider a cavity with no gain ($\varepsilon_{xz}=0$) and an
incident field polarized along the $x$ direction [Fig. 9 (a)]. We examine the
behavior of the electric field envelope, calculated using Eq. (48), in the
middle of the cavity. We see that the incident field excites a wave that is
initially polarized along $x$, but that after hitting the Kildal mirror gains
a component along $z$. Both components of the fields inside the cavity quickly
decay after the end of the excitation, as expected from the value of the
imaginary part of the eigenfrequencies
($\omega^{\prime\prime}\approx-0.664c/d$) represented in Fig. 7 (a).
The time evolution of the incident pulse for a perfectly tuned cavity [with
the parameters corresponding to the purple triangle of Fig. 7 (b)] is shown in
Fig. 9 (b). The duration of the incident pulse was chosen large enough (about
eighty cycles of oscillation) so that only the trapped state can be excited
efficiently. Clearly, even though a zero and pole of the reflection matrix
coexist at the same frequency, the incident wave is able to efficiently pump
the eigenmode of the open resonator. As before, the Kildal’s mirror is
responsible for creating a field component along the $z$-direction. Since
$\varepsilon_{xz}\neq 0$, the energy associated with the $z$-field component
can be subsequently converted back into a $x$ component. Remarkably it is
observed that after the incident pulse terminates, the field inside the cavity
does not decay with time, confirming that the trapped mode can indeed be
excited by the incoming wave. The eigenstate does not relax despite the
constant field leakage towards the exterior (not shown; see also the
discussion in the end of Sect. V). In fact, the leakage through the open wall
is exactly compensated by the gain material. Consistent with the principles of
time-reversal symmetry, the polarization of the field radiated by the cavity
(the eigenstate) after the excitation terminates is coincident with
$\mathbf{E}_{\text{eig},1}^{\ast}$. As shown in Fig. 9 (d) (dashed red line)
the output polarization is elliptical and the major axis is slightly tilted
with respect to the $x$ axis.
The magnitude of the field stored inside the resonator depends strongly on the
polarization of the incident field as seen by comparing Figs.9 (b)–(c).
Indeed, our numerical study confirms that the optimal polarization, i.e. the
polarization that maximizes the energy transferred to the trapped state, is
determined by $\mathbf{E}_{\text{eig},2}$ evaluated at the eigenstate
frequency. As seen in Fig. 9 (d), this polarization is elliptical, rotating
clockwise, with the ellipse major axis making an angle of around 17∘ with
respect to the $z$ axis. For this optimal case the field stored in the cavity
is about 3 times larger than for a linear polarization along the $x$ axis
[Fig. 9 (b)], but only roughly 2% larger than a linear polarization parallel
to the major axis of the ellipse (not shown). Moreover, the polarization of
the output field [dashed red line in Fig. 9 (d)] is insensitive to the
polarization of the incoming wave. This confirms that the orthogonal
polarization described by $\mathbf{E}_{\text{eig},1}$ is fully absorbed (anti-
lasing) by the cavity and hence plays no role in determining the reflected
wave.
The magnitude of the field stored in the resonator also depends on the pulse
duration. Remarkably, despite the relatively short duration of the incoming
pulse in the example of Fig. 9 it is feasible to inject a substantial amount
of energy into the open cavity. We verified that for a passive and reciprocal
dielectric resonator with the same quality factor, the field amplitude inside
the cavity would be near zero (not shown), for the same excitation. This
happens due to the reciprocity constraint discussed in Refs. Haus (1983); Mann
_et al._ (2019) and appendix D, which imposes that the amount of energy that
can be pumped into a time-reversal symmetric and passive resonator is
proportional to the energy leakage towards the exterior [see Eq. (60)].
Interestingly, our cavity, despite being time-reversal invariant and thus
seemingly subject to the same limitations, behaves differently due to its non-
passive nature. Indeed, different from standard resonators, it is demonstrated
in appendix D that in our system the energy leakage towards the exterior is
not controlled by the decay rate of the eigenmode, because the cavity is
pumped by the material gain. Thereby, our system is not constrained by the
result of Haus (1983). In fact, the radiation leakage remains strong when the
quality factor of the eigenstate approaches infinity due to the gain effect.

For instance, in the perfectly tuned scenario depicted in Fig. 7 (b), the
leakage to the exterior is comparable to that in the scenario with
$\varepsilon_{xz}=0$ shown in Fig. 7 (a), yet the quality factors in these two
cases are dramatically different. This discrepancy elucidates the ability to
channel a significant amount of energy into the cavity within a limited number
of oscillation cycles, thereby surpassing the reciprocity constraint.
It is interesting to further investigate the time dynamics of the chiral-gain
open-resonator when it is subjected to a very long monochromatic excitation.
To this end, in the last example we suppose that the incident pulse for $t\geq
0$ is of the form

$\displaystyle\mathbf{E}^{\text{inc}}(y,t)=\mathrm{Re}\left\\{\mathbf{E}_{0}^{\text{inc}}\text{e}^{i\left(\frac{\omega_{0}}{c}\sqrt{\varepsilon_{d}}y-\omega_{0}t\right)}\tanh\left(\frac{1}{\alpha}\left[t-\frac{y\sqrt{\varepsilon_{d}}}{c}\right]\right)\right\\}$
(32)

where $\alpha$ determines the rising time of the pulse [see Fig. 10 (a)].

Figure 10:  Time excitation of the open-cavity with a continuous wave pulse
[Eq. (32)] polarized along $x$. $\bf{(a)}$ Envelope of the incident field for
a pulse with $\omega_{0}\approx 10.4942\pi c/d$, $\alpha=60d/c$ and
$\varepsilon_{d}=3$. $\bf{(b)}$ Time evolution of the envelope of the electric
field in the middle of the perfectly tuned cavity ($y=d/2$) as a function of
the normalized time.

In this case, one can see from Fig. 10 (b) that in the stationary regime the
magnitude of the field stored inside the resonator grows linearly with time.
This behavior is analogous to that of a lossless LC circuit pumped at the
resonance, i.e., a resonator with infinite quality factor under a time-
harmonic excitation. This divergent behavior arises due to the pole of the
reflection matrix in the real-frequency axis (quasi-laser), and explains the
collapse of the time-harmonic regime which predicts an infinite response.
Clearly, different from conventional resonators, the amount of energy stored
into the open-cavity depends solely on the duration of the excitation, not on
the amplitude of the incoming wave. In practice, the maximum amount of energy
that can be pumped into the cavity will be limited by nonlinear effects.
In summary, we showed that a chiral-gain cavity can be operated in a peculiar
regime corresponding to the threshold of lasing/anti-lasing, similar to a CPA-
laser. For an excitation polarized as $\mathbf{E}_{\text{eig},1}$ the incoming
wave is completely absorbed by the resonator (anti-lasing operation). On the
other hand, for an excitation polarized as $\mathbf{E}_{\text{eig},2}$ the
system exhibits a strongly resonant response due to the excitation of the
cavity mode (lasing threshold). The cavity is capable of efficiently
collecting and storing the energy of the incoming wave within it, even for
short-duration pulses, without being limited by a reciprocity constraint on
the external coupling strength. The field trapped inside the cavity does not
decay over time. The amount of energy stored in the cavity is proportional to
the duration of the excitation.

##  VII Conclusions

We investigated various applications facilitated by chiral-gain media,
revealing their potential in the development of electrically pumped chiral
lasers, polarization-controlled non-Hermitian mirrors, and light storage. In
particular, we demonstrated that the non-Hermitian properties of chiral-gain
media lay the groundwork for creating CPA-lasers using a single material that
integrates both gain and loss responses.

This capability sets our approach apart from traditional resonators, such as
those supporting BICs, which typically suffer from inefficient excitation due
to pole-zero cancellation. By contrast, the unique analytical features of our
system scattering matrix allow a zero and a pole to coexist at the same
frequency. Additionally, we established that our system transcends the
limitations imposed by reciprocity. As a result, our CPA-laser can be
efficiently excited with signals of relatively short duration and can retain
the captured energy without significant decay. Thus, chiral-gain materials
emerge as a promising solution for advancing nanophotonic technologies.

## Acknowledgments

This work was partially funded by the Institution of Engineering and
Technology (IET), by the Simons Foundation award 733700 (Simons Collaboration
in Mathematics and Physics, ”Harnessing Universal Symmetry Concepts for
Extreme Wave Phenomena”) and by Instituto de Telecomunicações under Project
N∘. UID/EEA/50008/2020. S.L. acknowledges FCT and IT-Coimbra for the research
financial support with reference DL 57/2016/CP1353/CT000. D.E.F. acknowledges
financial support by IT-Lisbon and FCT under a research contract Ref.
CEECINST/00058/2021/ CP2816/CT0003. T.A.M. acknowledges FCT for research
financial support with reference CEECIND/04530/2017/CP1393/CT0004 (DOI
10.54499/CEECIND/04530/2017/CP1393/CT0004) under the CEEC Individual 2017, and
IT-Coimbra for the contract as an assistant researcher with reference CT/N∘.
004/2019-F00069.

##  Appendix A Reflection matrix

In this Appendix, we derive the reflection matrix $\overline{\mathbf{R}}$ for
the configuration of Fig.5.
We start by noting that the fields at the Kildal mirror ($y=d$) are
necessarily of the form:

$\displaystyle\mathbf{f}(d)=A_{E}\hat{\mathbf{t}}_{E}+A_{H}\hat{\mathbf{t}}_{H}$
(33)

where $A_{E}$ and $A_{H}$ are some unknown coefficients. Using the transfer
matrix [Eq. (10)], the fields at $y=0$ can be written as
$\mathbf{f}(0)=\text{e}^{id\overline{\mathbf{M}}}\cdot\mathbf{f}(d)$, or
equivalently:

$\displaystyle\begin{pmatrix}E_{x}\\\ E_{z}\\\ H_{x}\\\
H_{z}\end{pmatrix}_{y=0}=\begin{pmatrix}\hat{\mathbf{u}}_{1}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{E}&&\hat{\mathbf{u}}_{1}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{H}\\\
\hat{\mathbf{u}}_{2}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{E}&&\hat{\mathbf{u}}_{2}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{H}\\\
\hat{\mathbf{u}}_{3}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{E}&&\hat{\mathbf{u}}_{3}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{H}\\\
\hat{\mathbf{u}}_{4}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{E}&&\hat{\mathbf{u}}_{4}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{H}\end{pmatrix}\cdot\begin{pmatrix}A_{E}\\\
A_{H}\end{pmatrix}$ (34)

Next, we introduce an impedance matrix $\overline{\mathbf{Z}}$ that relates
the transverse components of the electric field $\mathbf{E}_{t}$ and magnetic
field $\mathbf{H}_{t}$ at the immediate vicinity of the dielectric interface
($y=0^{+}$) as Morgado and Silveirinha (2016); Silveirinha  _et al._ (2018);
Latioui and Silveirinha (2019):

$\displaystyle\overline{\mathbf{J}}\cdot\mathbf{E}_{t}=\overline{\mathbf{Z}}\cdot\mathbf{H}_{t}$
(35)

where

$\displaystyle\overline{\mathbf{J}}=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$
(36)

In order to determine the impedance matrix, we use Eq. (34), to obtain

$\displaystyle\overline{\mathbf{J}}\cdot\mathbf{E}_{t}(0)$
$\displaystyle=\overline{\mathbf{J}}\cdot\begin{pmatrix}\hat{\mathbf{u}}_{1}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{E}&\hat{\mathbf{u}}_{1}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{H}\\\
\hat{\mathbf{u}}_{2}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{E}&\hat{\mathbf{u}}_{2}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{H}\end{pmatrix}\cdot\begin{pmatrix}A_{E}\\\
A_{H}\end{pmatrix}$
$\displaystyle=\overline{\mathbf{J}}\cdot\begin{pmatrix}\hat{\mathbf{u}}_{1}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{E}&\hat{\mathbf{u}}_{1}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{H}\\\
\hat{\mathbf{u}}_{2}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{E}&\hat{\mathbf{u}}_{2}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{H}\end{pmatrix}\cdot\begin{pmatrix}\hat{\mathbf{u}}_{3}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{E}&\hat{\mathbf{u}}_{3}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{H}\\\
\hat{\mathbf{u}}_{4}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{E}&\hat{\mathbf{u}}_{4}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{H}\end{pmatrix}^{-1}\cdot\begin{pmatrix}H_{x}\\\
H_{z}\end{pmatrix}_{y=0}.$ (37)

Therefore, comparing with Eq. (35) we see that the impedance matrix at $y=0$
is given by:

$\displaystyle\overline{\mathbf{Z}}$
$\displaystyle=\overline{\mathbf{J}}\cdot\begin{pmatrix}\hat{\mathbf{u}}_{1}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{E}&\hat{\mathbf{u}}_{1}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{H}\\\
\hat{\mathbf{u}}_{2}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{E}&\hat{\mathbf{u}}_{2}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{H}\end{pmatrix}\cdot\begin{pmatrix}\hat{\mathbf{u}}_{3}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{E}&\hat{\mathbf{u}}_{3}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{H}\\\
\hat{\mathbf{u}}_{4}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{E}&\hat{\mathbf{u}}_{4}\cdot\text{e}^{id\overline{\mathbf{M}}}\cdot\hat{\mathbf{t}}_{H}\end{pmatrix}^{-1}.$
(38)

On the other hand, for a plane wave propagating in a dielectric with
permittivity $\varepsilon_{d}$ the transverse electric and magnetic fields are
related as:

$\displaystyle\mathbf{H}_{t}^{\pm}=\pm\frac{\sqrt{\varepsilon_{d}}}{\eta_{0}}\overline{\mathbf{J}}\cdot\mathbf{E}_{t}^{\pm}$
(39)

where the $\pm$ sign determines whether the wave propagates towards the +$y$
or the $-y$ direction and $\eta_{0}=\sqrt{\mu_{0}/\varepsilon_{0}}$ is the
free-space impedance.
Hence, using Eqs. (25) and (39) it follows that the total field at the
dielectric side of the boundary ($y=0^{-}$) is related to the incident field
as:

$\displaystyle\mathbf{f}(0)=\begin{pmatrix}\left(\overline{\mathbf{1}}_{2\times
2}+\overline{\mathbf{R}}\right)\cdot\mathbf{E}_{t}^{\text{inc}}\\\
\frac{\sqrt{\varepsilon_{d}}}{\eta_{0}}\overline{\mathbf{J}}\cdot\left(\overline{\mathbf{1}}_{2\times
2}-\overline{\mathbf{R}}\right)\cdot\mathbf{E}_{t}^{\text{inc}}\end{pmatrix}$
(40)

where $\mathbf{1}_{2\times 2}$ is the $2\times 2$ identity matrix. At the
other side of the interface ($y=0^{+}$), the fields satisfy (35). Matching the
transverse components of the electric and magnetic fields at the two sides of
the boundary, it follows that:

$\displaystyle\overline{\mathbf{J}}\cdot\left(\overline{\mathbf{1}}_{2\times
2}+\overline{\mathbf{R}}\right)=\frac{\sqrt{\varepsilon_{d}}}{\eta_{0}}\overline{\mathbf{Z}}\cdot\overline{\mathbf{J}}\cdot\left(\overline{\mathbf{1}}_{2\times
2}-\overline{\mathbf{R}}\right)$ (41)

After some manipulations, it is readily found that the reflection coefficient
matrix is given by

$\displaystyle\overline{\mathbf{R}}$
$\displaystyle=\left(-\frac{\sqrt{\varepsilon_{d}}}{\eta_{0}}\overline{\mathbf{J}}\cdot\overline{\mathbf{Z}}\cdot\overline{\mathbf{J}}+\overline{\mathbf{1}}_{2\times
2}\right)^{-1}\cdot\left(-\frac{\sqrt{\varepsilon_{d}}}{\eta_{0}}\overline{\mathbf{J}}\cdot\overline{\mathbf{Z}}\cdot\overline{\mathbf{J}}-\overline{\mathbf{1}}_{2\times
2}\right)$ (42)

The derivation assumes that the inverse matrix in (37) is well defined. When
it is not the case, one can use the alternative formula that uses the inverse
of $\overline{\mathbf{Z}}$:

$\displaystyle\overline{\mathbf{R}}$
$\displaystyle=\left(\frac{\sqrt{\varepsilon_{d}}}{\eta_{0}}\overline{\mathbf{1}}_{2\times
2}-\overline{\mathbf{J}}\cdot\overline{\mathbf{Z}}^{-1}\cdot\overline{\mathbf{J}}\right)^{-1}\cdot\left(\frac{\sqrt{\varepsilon_{d}}}{\eta_{0}}\overline{\mathbf{1}}_{2\times
2}+\overline{\mathbf{J}}\cdot\overline{\mathbf{Z}}^{-1}\cdot\overline{\mathbf{J}}\right).$
(43)

The matrix $\overline{\mathbf{Z}}^{-1}$ is typically well defined at points
where $\overline{\mathbf{Z}}$ is not.

##  Appendix B Derivation of Eq. (30)

To characterize the analytical structure of the reflection matrix, first we
note that Eq. (27) can be rewritten as:

$\displaystyle\overline{\mathbf{R}}\cdot\mathbf{U}\left(\omega\right)=\mathbf{U}^{\ast}\left(\omega\right)\cdot\begin{pmatrix}0&a_{2}\left(\omega\right)\\\
a_{1}\left(\omega\right)&0\end{pmatrix}.$ (44)

The identity holds true in the real-frequency axis. Here, $\mathbf{U}$ is a
matrix whose columns are the normalized eigenvectors
($\mathbf{E}_{\text{eig},1}$ and $\mathbf{E}_{\text{eig},2}$) of the
reflectance matrix ($\overline{\mathcal{R}}$) and
${a_{j}}\left(\omega\right)=\left|{{\rho_{j}}}\right|{e^{i{\theta_{j}}}}$.
Since $\mathbf{U}$ is a unitary matrix and varies little near the resonance,
for a high-quality resonator it is possible to write:

$\displaystyle\overline{\mathbf{R}}\left(\omega\right)\approx\mathbf{U}^{\ast}\left(\omega_{0}^{\prime}\right)\cdot\begin{pmatrix}0&a_{2}\left(\omega\right)\\\
a_{1}\left(\omega\right)&0\end{pmatrix}\cdot\mathbf{U}^{\dagger}\left(\omega_{0}^{\prime}\right).$
(45)

The resonance corresponds to the pole of $a_{2}$, which thereby may be
approximated by $a_{2}\approx A/(\omega-\omega_{0})$, where $A$ is some
constant.

In order that our theory is applicable away from the real-frequency axis, we
enforce that the reflection matrix satisfies the time-reversal symmetry
condition:
$\overline{\mathbf{R}}\left(\omega\right)={\left[{{{\overline{\mathbf{R}}}^{-1}}\left({{\omega^{\ast}}}\right)}\right]^{\ast}}$.
It is easy to check that this possible only if
${a_{1}}\left(\omega\right)={1\mathord{\left/{\vphantom{1{{{\left[{{a_{2}}\left({{\omega^{\ast}}}\right)}\right]}^{\ast}}}}}\right.\kern-1.2pt}{{{\left[{{a_{2}}\left({{\omega^{\ast}}}\right)}\right]}^{\ast}}}}$.
These approximations for $a_{1}$ and $a_{2}$ yield Eq. (30).

##  Appendix C Time domain formalism

Here, we describe the formalism used to characterize the time domain electric
field $\mathbf{E}^{\text{tot}}(y,t)$ in the resonator problem.

We rely on Fourier theory. The Fourier transform $\psi(\mathbf{r},\omega)$ of
a time signal $\psi(\mathbf{r},t)$ is defined as
$\psi(\mathbf{r},\omega)=\int_{-\infty}^{\infty}dt\psi(\mathbf{r},t)\text{e}^{i\omega
t}$. Then, it follows that the complex amplitude of the incident field [Eq.
(31)] in the frequency domain is

$\displaystyle\mathbf{E}^{\text{inc}}(y,\omega)$
$\displaystyle=\sigma\sqrt{\pi}\mathbf{E}_{0}^{\text{inc}}\text{e}^{ik_{d}y}\text{e}^{i(\omega-\omega_{0})t_{0}}\text{e}^{-\left(\frac{\sigma[\omega-\omega_{0}]}{2}\right)^{2}}$
(46)

where $k_{d}=\sqrt{\varepsilon_{d}}\omega/c$. Using Eqs. (10), (25) and (39)
it can be shown that in the frequency domain the total field
$\mathbf{E}^{\text{tot}}$ is:

$\displaystyle\mathbf{E}^{\text{tot}}(y,\omega)$
$\displaystyle=\begin{cases}\mathbf{E}^{\text{inc}}(y,\omega)+\text{e}^{-ik_{d}y}\overline{\mathbf{R}}\cdot\mathbf{E}^{\text{inc}}(0,\omega)&y\leq
0\\\
\left(\hat{\mathbf{x}}\otimes\hat{\mathbf{u}}_{1}+\hat{\mathbf{z}}\otimes\hat{\mathbf{u}}_{2}\right)\cdot\text{e}^{-iy\mathbf{M}}\cdot\begin{pmatrix}\left(\mathbf{1}_{2\times
2}+\overline{\mathbf{R}}\right)\cdot\mathbf{E}^{\text{inc}}(0,\omega)\\\
\frac{\sqrt{\varepsilon_{d}}}{\eta_{0}}\mathbf{J}\cdot\left(\mathbf{1}_{2\times
2}-\overline{\mathbf{R}}\right)\cdot\mathbf{E}^{\text{inc}}(0,\omega)\end{pmatrix}&y\geq
0\\\ \end{cases}$ (47)

Note that the region $y<0$ corresponds to the dielectric region and $0<y<d$ is
the chiral-gain resonator. The field in the time domain is obtained with an
inverse Fourier transformation:

$\displaystyle\mathbf{E}^{\text{tot}}(y,t)$
$\displaystyle=\mathrm{Re}\left\\{\frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega\mathbf{E}^{\text{tot}}(y,\omega)\text{e}^{-i\omega
t}\right\\}.$ (48)

For systems with spectrum not confined to the lower-half frequency plane, the
Fourier transform is ill-defined and one should use instead

$\displaystyle\mathbf{E}^{\text{tot}}(y,t)=\mathrm{Re}\left\\{\frac{1}{2\pi}\text{e}^{\omega^{\prime\prime}t}\int_{-\infty}^{\infty}d\omega^{\prime}\mathbf{E}^{\text{tot}}(y,\omega^{\prime}+i\omega^{\prime\prime})\text{e}^{-i\omega^{\prime}t}\right\\}.$
(49)

where $\omega^{\prime\prime}$ is chosen above all the poles of
$\mathbf{E}^{\text{tot}}(y,\omega^{\prime}+i\omega^{\prime\prime})$.

##  Appendix D Constraint on the coupling strength of time-reversal symmetric
and non-conservative resonators

We consider a single mode resonator coupled to $n$ ports. The complex
amplitude $a(t)$ of the resonator’s mode is governed by Haus (1983); Suh  _et
al._ (2004); Mann  _et al._ (2019):

$\displaystyle\frac{da}{dt}$
$\displaystyle=-i\omega_{0}a+\mathbf{K}\cdot\mathbf{s}_{+}$ (50)

where $\omega_{0}$ is the complex resonant frequency,
$\mathbf{s}_{+}=\begin{pmatrix}s_{+1}&s_{+2}&\dots&s_{+n}\end{pmatrix}^{T}$ is
the vector describing the complex amplitude of the input waves and
$\mathbf{K}$ the vector describing the input coupling at each port. The
complex amplitude of the outgoing waves $\mathbf{s}_{-}$ is ruled by

$\displaystyle\mathbf{s}_{-}$
$\displaystyle=\overline{\mathbf{c}}\cdot\mathbf{s}_{+}+\mathbf{d}a$ (51)

where $\overline{\mathbf{c}}$ is the direct scattering matrix and $\mathbf{d}$
the vector describing the output coupling to the ports. The different
amplitudes are normalized in such a way that the stored energy is
${\left|a\right|^{2}}$, and the incoming and outgoing power fluxes are
controlled by ${\left|{{s_{+i}}}\right|^{2}}$ and
${\left|{{s_{-i}}}\right|^{2}}$, respectively.
For a natural mode ($\mathbf{s}_{+}=0$):

$\displaystyle\frac{da}{dt}$ $\displaystyle=-i\omega_{0}a$ (52a)
$\displaystyle\mathbf{s}_{-}$ $\displaystyle=\mathbf{d}a$ (52b)

and it follows that the complex amplitude of a natural mode is given by
$a(t)=a_{0}\text{e}^{-i\omega_{0}t}$ with $a_{0}$ a complex number. Under a
time-reversal transformation $\mathcal{T}$, $a(t)$ is transformed as

$\displaystyle\mathrm{Re}\left[a(t)\right]\xrightarrow{~{}\mathcal{T}~{}}\mathrm{Re}\left[a(-t)\right]=\mathrm{Re}\left[a_{0}^{\ast}\text{e}^{-i\omega_{0}^{\ast}t}\right],$
(53)

meaning that under a $\mathcal{T}$ transformation
$a(t)\xrightarrow{~{}\mathcal{T}~{}}a^{\ast}(-t)$ . If the system exhibits
time-reversal symmetry, then the time-reversed natural mode field
$a^{\ast}(-t)=a_{0}^{\ast}\text{e}^{-i\omega_{0}^{\ast}t}$ is also solution of
equations (52) with the incoming and outgoing waves time-reversed. Under a
time-reversal the incoming and outgoing waves are interchanged Silveirinha
(2019a, b). Thus, there is no outgoing wave and the input wave is the time-
reversal of $\mathbf{s}_{-}(t)$, given by $\mathbf{s}_{-}^{\ast}(-t)$. It
follows that:

$\displaystyle\frac{da^{\ast}(-t)}{dt}$
$\displaystyle=-i\omega_{0}a^{\ast}(-t)+\mathbf{K}\cdot\mathbf{s}_{-}^{\ast}(-t)$
(54)
$\displaystyle=-i\omega_{0}a^{\ast}(-t)+\mathbf{K}\cdot\mathbf{d}^{\ast}a^{\ast}(-t)$
(55)

where we used Eq. (52b). This is equivalent to

$\displaystyle i\left(\omega_{0}-\omega_{0}^{\ast}\right)a^{\ast}$
$\displaystyle=\mathbf{K}\cdot\mathbf{d}^{\ast}a^{\ast}$ (56)

So, we have proven that for systems invariant under time-reversal symmetry:

$\displaystyle-2\omega_{0}^{\prime\prime}$
$\displaystyle=\mathbf{K}\cdot\mathbf{d}^{\ast}$ (57)

From the definition of the radiative decay rate
$\omega_{\text{rad}}^{\prime\prime}$, vector $\bf{d}$ satisfies Haus (1983);
Suh  _et al._ (2004); Mann  _et al._ (2019):

$\displaystyle-2\omega_{\text{rad}}^{\prime\prime}$
$\displaystyle=\mathbf{d}\cdot\mathbf{d}^{\ast}$ (58)

For one-port systems, the combination of Eqs. (57) and (58) implies that the
magnitude of the input coupling is:

$\displaystyle|K|$
$\displaystyle=\frac{-2\omega_{0}^{\prime\prime}}{\sqrt{-2\omega_{\text{rad}}^{\prime\prime}}}$
(59)

For conservative systems,
$\omega_{0}^{\prime\prime}=\omega_{\text{rad}}^{\prime\prime}$, and the
previous equation reduces to the well-known result of Haus Haus (1983),

$\displaystyle|K|=\sqrt{-2\omega_{\text{rad}}^{\prime\prime}},$ (60)

which constraints the strength of the input coupling. It can be shown that the
above result remains valid for arbitrary (possibly dissipative) reciprocal
systems (such systems are constrained by $\bf{K}=\bf{d}$).
In contrast, our (two-port) non-Hermitian system is only constrained by Eqs.
(57) and (58). Applying the first equation to the CPA-laser of section VI
($\omega_{0}^{\prime\prime}\to 0$), it follows that:

$\displaystyle\mathbf{K}\cdot\mathbf{d}^{\ast}$ $\displaystyle=0$ (61)
$\displaystyle-2\omega_{\text{rad}}^{\prime\prime}$
$\displaystyle=\mathbf{d}\cdot\mathbf{d}^{\ast}$ (62)

Since the radiative decay rate does not vanish ($\mathbf{d}\neq 0$), it
follows that $\mathbf{K}$ is perpendicular to $\mathbf{d}$. For the problem of
section VI.3, $\mathbf{d}\sim\mathbf{E}_{\text{eig},1}^{\ast}$ , which
corresponds to the field radiated by the natural mode of the cavity (time-
reversal of the mode that is fully absorbed). This means that
$\mathbf{K}\sim\mathbf{E}_{\text{eig},2}^{\ast}$, as expected, so that the
optimal incident wave to excite the cavity is
$\mathbf{s}_{+}\sim\mathbf{E}_{\text{eig},2}$. The amplitude of $\mathbf{K}$
is not directly constrained by time-reversal symmetry.

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