# Chiral modes near exceptional points in symmetry broken H1 photonic crystal
cavities

C. F. Fong<EMAIL_ADDRESS>Nanoscale Quantum Photonics Laboratory,
RIKEN Cluster for Pioneering Research, Saitama 351-0198, Japan  Quantum
Optoelectronics Research Team, RIKEN Center for Advanced Photonics, Saitama
351-0198, Japan  Institute for Nano Quantum Information Electronics, The
University of Tokyo, Tokyo 153-8505, Japan  Y. Ota  Department of Applied
Physics and Physico-Informatics, Keio University, Kanagawa 223-8522, Japan
Institute for Nano Quantum Information Electronics, The University of Tokyo,
Tokyo 153-8505, Japan  Y. Arakawa  Institute for Nano Quantum Information
Electronics, The University of Tokyo, Tokyo 153-8505, Japan  S. Iwamoto
Institute for Nano Quantum Information Electronics, The University of Tokyo,
Tokyo 153-8505, Japan  Institute of Industrial Science, The University of
Tokyo, Tokyo 153-8505, Japan  Research Center for Advanced Science and
Technology, The University of Tokyo, Tokyo 153-0041, Japan  Y. K. Kato
<EMAIL_ADDRESS>Nanoscale Quantum Photonics Laboratory, RIKEN Cluster
for Pioneering Research, Saitama 351-0198, Japan  Quantum Optoelectronics
Research Team, RIKEN Center for Advanced Photonics, Saitama 351-0198, Japan

###### Abstract

The H1 photonic crystal cavity supports two degenerate dipole modes of
orthogonal linear polarization which could give rise to circularly polarized
fields when driven with a $\pi$/$2$ phase difference. However, fabrication
errors tend to break the symmetry of the cavity which lifts the degeneracy of
the modes, rendering the cavity unsuitable for supporting circular
polarization. We demonstrate numerically, a scheme that induces chirality in
the cavity modes, thereby achieving a cavity that supports intrinsic circular
polarization. By selectively modifying two air holes around the cavity, the
dipole modes could interact via asymmetric coherent backscattering which is a
non-Hermitian process. With suitable air hole parameters, the cavity modes
approach the exceptional point, coalescing in frequencies and linewidths as
well as giving rise to significant circular polarization close to unity. The
handedness of the chirality can be selected depending on the choice of the
modified air holes. Our results highlight the prospect of using the H1
photonic crystal cavity for chiral-light matter coupling in applications such
as valleytronics, spin-photon interfaces and the generation of single photons
with well-defined spins.

##  I Introduction

The dynamics of physical systems with open boundaries that could exchange
energy with their surrounding environment can be described by non-Hermitian
Hamiltonians. Such systems, in general, do not have an orthogonal set of
eigenstates and there exist non-trivial degeneracies known as exceptional
points (EPs) at which both the eigenfrequencies and eigenstates coalesce to
become one and the same. There is a surge in interest in non-Hermitian
photonics and optical systems [1, 2, 3] due to the relative ease to implement
the complex potential required for non-Hermiticity in terms of the refractive
index by incorporating gain and/or loss. For example, non-hermiticity have
been successfully introduced in the whispering gallery resonators [4, 5, 6, 7,
8, 9, 10] via various means such as the manipulation of excitation geometry
[4, 11], heating [9], patterned metal depositions [5, 10], patterned defect
scatterers [12] and the usage of nanotip scatterers [6, 7]. Intriguing
phenomena have been reported in these whispering gallery resonators including
single mode lasing [4, 5], enhanced sensitivity to perturbations [7, 9, 13,
14], directional coupling [6, 15], enhanced spontaneous emission [12, 16] and
the generation of a vortex laser beam [10].

Analogous to the whispering gallery resonator is the H1 photonic crystal (PhC)
cavity due to the $C_{6v}$ rotational symmetry of the cavity. The H1 PhC
cavity modes can be approximated using the cylindrical harmonics as in the
whispering gallery resonator [17, 18]. A notable feature of the H1 PhC cavity
is that it could support two degenerate and orthogonal linearly polarized
dipole modes. When the two modes are driven with a $\pi$/$2$ phase difference,
they give rise to circularly polarized cavity fields. Such a nanocavity —
given its small mode volume and prospects for high $Q$-factor — would be an
important component for chiral quantum optics [19], photonic circuits [20,
21], spin-nanolasers, optical sensors [22] and other applications. However,
these functionalities are usually hindered by fabrication errors of the
nanocavity which tend to lift the mode degeneracies, making it incapable of
supporting circular polarization. Previous attempts to restore mode
degeneracies relied on implementing perturbation by straining the cavity [23],
by purposefully designing a cavity in a “stretched lattice” [24] or
nanooxidation [25]. Nonetheless, an intrinsically circularly polarized
nanocavity has yet to be realized.

In this work, based on the non-Hermitian effects in an H1 PhC cavity, we
propose a scheme to achieve chiral cavity modes that could support intrinsic
circular polarization. In our scheme, two air holes around the cavity are
selectively modified. The modifications of the first air hole give rise to a
mode splitting, while the second air hole is then modified to bring the system
towards or away from the EP. From our numerical finite-difference time-domain
(FDTD) simulation results, we observe the characteristic branching in the
surfaces of the complex eigenfrequencies in the parameter space, indicating
the presence of an EP. Near the EP, there are chiral modes with corresponding
high degrees of circular polarization. The chiral modes in turn emit
circularly polarized light into the farfield, shown schematically in Fig.
1(a). By selecting different pairs of air holes, modifications can be done in
a controllable manner to obtain either right or left circularly polarized
chiral modes. The coalescence of the modes near the EP also promises the
enhancement of spontaneous emission [26, 27] of the chiral modes.

##  II H1 photonic crystal cavity

We consider a PhC which consists of a triangular lattice of air holes with a
lattice period of $a$ and radius $r$, in a slab of material of refractive
index $n$. The H1 cavity is formed by removing an air hole within the lattice
[Fig. 1(b)]. We first describe the relevant features and quantities of the H1
PhC cavity based on 2D FDTD simulations. Figure 1(c) shows the near-field
profiles of the relevant transverse electric field components of both dipole
modes D1 and D2 in which the $E_{x}$ and $E_{y}$ fields are dominant,
respectively.

Each dipole mode in the H1 PhC cavity can be thought to be constituted of two
travelling wave components rotating in the opposite directions [18], analogous
to whispering gallery modes [17]. In each dipole mode, the counter rotation of
the two travelling components cancels out to result in a stationary mode. On
the contrary, a superposition of the dipole modes with a $\pm\pi$/$2$ phase
difference essentially recovers one of the travelling modes that rotates in
the clockwise (counterclockwise) direction, producing a chiral mode with a
dominant right (left) circularly polarized field [Fig. 1(d)]. This phenomenon
is known as spin-momentum locking, in which the direction of rotation of the
travelling mode is directly correlated to the handedness of its chirality.

Figure 1:  (a) Schematic showing the emission of circularly polarized light
from a symmetry-broken H1 photonic crystal cavity. (b) Schematic of the cavity
in the $xy$-plane with the key air holes and parameters labelled. See main
text for the descriptions of the parameters. The first nearest air holes are
outlined with green circles while the relevant second nearest air holes are
outlined with black circles. The coordinate origin is at the center of the
cavity. The purple dotted lines intersecting at the cavity center mark the
direction of high symmetry along the $\Gamma-K$ directions while the blue
dashed lines mark the hexagonal path, along which the air holes are shifted.
(c) The relevant field distributions of the two dipole modes in an unmodified
cavity based on 2D FDTD simulations. (d) The distribution of the difference in
the field intensities with orthogonal circular polarization (spin angular
momentum density, $\bm{s_{d}}$ $\hat{\bm{z}}$) when the dipole modes are
superposed with $\pm\pi$/2 phase difference, giving predominantly RCP (LCP)
field respectively. The colorbar is shared for (c) and (d).

In order to quantify chirality, one could first calculate the spin angular
momentum density [28], $\bm{s_{d}}=$ Im
$(\epsilon_{o}\epsilon_{r}\bm{E}^{*}\times\bm{E}+\mu_{o}\mu_{r}\bm{H}^{*}\times\bm{H})/4$
which can be regarded as the difference in the intensity of right (RCP) and
left circularly polarized (LCP) fields. $\bm{E}$ ($\bm{H}$) represents the
electric (magnetic) field vectors, the asterisk (*) indicates complex
conjugation and Im() means taking the imaginary part. $\epsilon_{o}$
($\epsilon_{r}$) and $\mu_{o}$ ($\mu_{r}$) are the vacuum (relative)
permittivity and vacuum (relative) permeability respectively. The degree of
circular polarization (DCP) can then be obtained by dividing $\bm{s_{d}}$ with
the total field energy density,
$W=(\epsilon_{o}\epsilon_{r}\bm{E}^{*}\cdot\bm{E}+\mu_{o}\mu_{r}\bm{H}^{*}\cdot\bm{H})/4$,
giving a value between $\pm 1$. Since the cavity modes are transverse electric
in nature with the electric fields oscillating in the plane of the PhC slab,
$\bm{s_{d}}$ points in the out-of-plane $\pm\hat{\bm{z}}$ direction and thus
we only consider the $z$-component of $\bm{s_{d}}$ and its associated DCP.
Furthermore, as the dominant electric field components and the $z$-component
of $\bm{s_{d}}$ have the strongest intensity at the cavity center, the DCP at
the cavity center can be taken as a representative measure of the chirality of
the cavity.

2D FDTD simulations are performed using a PhC cavity with an overall hexagonal
shape so that it is consistent with the $C_{6v}$ symmetry of the cavity in
order to avoid or reduce any unintended perturbation to the cavity modes. It
is important to note that the PhC is of a finite size with lateral loss such
that it forms a non-Hermitian open boundary system. Since the two modes need
to be overlapping to support chirality, we ensure that the spectral widths of
the modes are sufficiently broad by using a $9\times 9$ air hole lattice. The
FDTD simulations are performed using the opensource package MEEP [29], with
parameter values $a=300$ nm, $r/a=0.3$, and a grid resolution of ${\sim}a$/16
(54 pixels/micron). We consider a material of $n$ = 3.4, for example GaAs. A
Gaussian-pulse point current source with a sufficient spectral width is placed
at the center of the cavity as the excitation source. Given these parameters,
the dipole modes are obtained at a normalized frequency of
$a/\lambda_{cav}\sim 0.244$. The $Q$-factor, which is the ratio of the mode
center frequency to its linewidth, is ${\sim}450$. For an unmodified cavity,
the modes are not completely degenerate, with a splitting of ${\sim}1.5$
linewidth for the parameters given above. The non-degeneracy is possibly due
to the discretization of the simulation grid and it persists even at higher
grid resolutions. Nonetheless, this non-degeneracy poses no significant issues
to our scheme. On the contrary, it presents us with a system that more closely
resembles an actual sample which tend to have non-degenerate modes due to
fabrication errors.

##  III Surfaces of complex eigenfrequencies and exceptional point

Figure 2:  Surfaces of the (a) real and (b) imaginary eigenfrequencies
normalized to their midpoints $\omega_{m}=(\omega_{+}+\omega_{-})/2$. The
points of interest that gives the characteristic branching associated with the
presence of an EP is marked on the surfaces in blue dots and black squares for
the real and imaginary eigenfrequencies respectively. (Inset) Schematic
indicating the air hole with fixed modifications only (orange circle) and the
air hole with both fixed and parameter sweep modifications (red circle). The
red arrow indicates the direction that the air-hole is being shifted along the
hexagonal path. (c) The branching of the real (blue dots) and imaginary (black
squares) frequencies plotted against $p_{23}$. The region where the EP is
expected to occur is highlighted in yellow. Color plots showing the absolute
difference in the (d) real and (e) imaginary eigenfrequencies within the
parameter space spanned by $\Delta r_{23}$ and $p_{23}$. The cyan line traces
along the branching of the eigenfrequencies. The region where the EP is
expected to occur indicated with bigger data markers.

In our scheme to achieve a chiral cavity, two second nearest air holes to the
cavity in the $\Gamma-K$ directions are modified to give the necessary
perturbation to the modes. In principle, our scheme could work by modifying
either the first or the second nearest air holes to the cavity. The first
nearest air holes are in the direct vicinity of the modes and thus even small
modifications to these air holes could result in excessive perturbations.
Despite being further away from the cavity, modifications to the second
nearest air holes could still provide sufficient perturbation. In addition,
certain second nearest air holes are positioned closer to the antinodes of
either one of the two dipole modes. As such, each modified air hole would
predominantly perturb one of the dipole modes. For these reasons, our scheme
is based on the second nearest air holes as they allow for better control and
intuitive understanding over the effects of their modifications.

The relevant second nearest air holes for our scheme are labelled as 21 to 26
in Fig. 1(b). The radii and the positions of two air holes are modified at a
time. The positions of the air holes will be shifted along the direction of
high symmetry [purple lines in Fig. 1(b)] towards (negative shift) or away
(positive shift) from the cavity. The air holes could also be shifted along
the hexagonal path around the cavity [blue dashed lines in Fig. 1(b)] with
clockwise (counterclockwise) shift being the positive (negative) direction.
The hexagonal path is defined by the distance from the cavity center to the
center of the relevant air hole after the air hole has been shifted along the
direction of high symmetry. The change in the air hole radius, shift along the
direction of high symmetry and shift along the hexagonal path will be labelled
as $\Delta r$, $d$ and $p$, respectively, followed by the hole number in
subscript e.g., $\Delta r_{21}$. The different cases described in this work
will be referred to in accordance with the pair of air holes that are being
modified, for example h21h23 refers to the case in which air holes 21 and 23
are modified.

We begin with the h21h23 case, where the following fixed modifications are
applied to the air holes: radius of air hole 21 is enlarged ($\Delta
r_{21}=+0.03a$) and shifted towards the cavity ($d_{21}=-0.1a$), while air
hole 23 is also shifted towards the cavity ($d_{23}=-0.20a$). $p_{23}$ and
$\Delta r_{23}$ are then varied over a range of values for the parameter sweep
simulations. In particular, air hole 23 is shifted towards $y=0$ i.e.,
negative $p_{23}$ values. FDTD simulation is performed for each combination of
$p_{23}$ and $\Delta r_{23}$ to extract the complex eigenfrequencies of the
two modes $\omega_{\pm}$, as well as the temporal evolution of $\bm{s_{d}}$
$\hat{\bm{z}}$, $W$, and the DCP.

Presented in Fig. 2(a) and 2(b) are the surfaces of the extracted real and
imaginary eigenfrequencies, Re($\omega_{\pm}$) and Im($\omega_{\pm}$),
respectively, normalized to their midpoints
$\omega_{m}=(\omega_{+}+\omega_{-})/2$ within the parameter space spanned by
$\Delta r_{23}$ and $p_{23}$. The surfaces of the eigenfrequencies exhibit
characteristics of a non-Hermitian system. For both Re($\omega_{\pm}$) and
Im($\omega_{\pm}$), there are particular regions where the surfaces come close
together, indicating that the eigenfrequencies are close to degeneracy. The
points of interest are marked with blue dots (black squares) for the real
(imaginary) eigenfrequencies and their values plotted in Fig. 2(c). The
resulting plot shows the branching in the eigenfrequencies with $p_{23}$ as
the “tuning parameter”. To show where these points of interests lie within the
parameter space, Fig. 2(d) and 2(e) present a different visualization of the
eigenvalue surfaces, namely, the distributions of the absolute difference in
the real and imaginary eigenfrequencies, respectively. In the range of
$p_{23}=-0.30a$ to $-0.20a$, Re($\omega_{\pm}$) are close to being degenerate
over a range of $\Delta r_{21}$ values, while Im($\omega_{\pm}$) are split
into two branches. At around $p_{23}=-0.20a$, Re($\omega_{\pm}$) begins to
branch while the two branches of Im($\omega_{\pm}$) merge.

The branching of eigenfrequencies indicate the presence of an EP within the
parameter space defined by all the five parameters that we consider here —
$d_{21}$, $\Delta r_{21}$, $d_{23}$, $\Delta r_{23}$ and $p_{23}$ — and
possibly other parameters. At the exact EP, the branch points of
Re($\omega_{\pm}$) and Im($\omega_{\pm}$) will coincide. In our simulation
reults, the branch points of Re($\omega_{\pm}$) and Im($\omega_{\pm}$) do not
coincide exactly which could be related to the non-degeneracy of the dipole
modes of the unmodified H1 PhC cavity. Although the location of the EP becomes
less well-defined, the EP is expected to occur in the vicinity of
$p_{23}=-0.20a$ as marked by the yellow region in Fig. 2(c).

##  IV Chiral modes

To determine the chirality of the modes, we perform FDTD simulations with long
simulation time of about 2000 wave period propagation for each combination of
parameter values. The simulation time is dependent on the cavity field
lifetime which is associated with the $Q$-factor. The simulation time — of
more than five times the cavity field lifetime — used here is to ensure that
the cavity field evolves sufficiently to reflect its intrinsic polarization.
The DCP at the cavity center at every fixed timestep is recorded during each
simulation. From the temporal evolution of the DCP at the cavity center, we
calculate its time-averaged value, $\langle$DCP$\rangle$, taking care to
consider only the temporal evolution of DCP after the excitation source has
been turned off.

Figure 3(a) show the distribution of $\langle$DCP$\rangle$ under linearly
polarized excitation oriented at $45^{\circ}$ relative to the $x$-axis (LP45)
for the h21h23 case. Despite the linearly polarized excitation, we observed
RCP chiral eigenmodes near the EP, achieving the largest $\langle$DCP$\rangle$
of 0.94 at parameters $p_{23}=-0.20a$ and $\Delta r_{23}=0.01a$. At the chiral
mode, the cavity fields evolve temporally from its initial linearly polarized
state — as determined by the excitation polarization — to its intrinsic
polarization. A temporal slice of the spatial distribution of DCP around the
cavity at ${\sim}$1700 wave period propagation at the parameters with the
largest $\langle$DCP$\rangle$ is presented in Fig. 3(c). The DCP at the center
of the cavity at this instant can be taken as a measure of the intrinsic
polarization, indicating that the cavity modes are intrinsically RCP with a
DCP of 0.97. While there are other regions around the cavity with significant
DCP, the field intensity [Fig. 3(e)] at these regions is negligible and thus
irrelevant to the mode chirality.

Figure 3:  Distribution of $\langle$DCP$\rangle$ in the parameter space for
(a) the h21h23 case under LP45 excitation and (b) the h21h22 case under LP-45
excitation. (Insets) Schematics indicating the modified air holes. (c, d) The
distribution of the DCP around the cavity at ${\sim}1700$ wave period
propagation with the parameter values where the $\langle$DCP$\rangle$ is
maximally right and left circular polarized for the two respective cases. The
scale bar represents one lattice period. The color bar is shared for (a-d).
(e, f) The distribution of the difference in the field intensities (spin
angular momentum density) where the $\langle$DCP$\rangle$ is maximally right
and left circular polarized for the RCP and LCP chiral cases, respectively.
The field intensity and thus field energy density is centered in the cavity as
expected. The DCP at the center of the cavity can be taken as a measure of
chirality. The color bar is shared for (e) and (f).

A different example with LCP chiral modes is the h21h22 case [Fig. 3(b)]. The
fixed modifications to air holes 21 and 22 are similar to that in the h21h23
case: $\Delta r_{21}=+0.02a$, $d_{21}=-0.10a$ and $d_{22}=-0.20a$. For the
parameter sweep, the air hole 22 is shifted towards $y=0$ by applying positive
$p_{22}$ values. The distribution of the eigenfrequencies and the branching is
largely similar to that of the h21h23 case but flipped about the vertical axis
(not shown). Performing the simulations under LP-45 excitation, the
$\langle$DCP$\rangle$ in the parameter space indicates LCP chiral eigenmodes
in the vicinity of the EP [Fig. 3(d)], achieving the largest
$\langle$DCP$\rangle$ of $-0.80$ at parameters $p_{22}=-0.19a$ and $\Delta
r_{22}=0.01a$. The corresponding temporal slice of the DCP distribution around
the cavity [Fig. 3(d)] gives an intrinsic polarization of $-0.83$, with the
field intensities similarly centered at the cavity [Fig. 3(f)].

Despite using modifications of the same magnitude in the air holes in both the
h21h22 and h21h23 cases, the difference in the distribution of
$\langle$DCP$\rangle$ in the two cases could be due to the interaction between
the two modified air holes via scattering [17]. Such interactions between air
holes are more likely to occur for the h21h22 case since the modified air
holes are closer to each other. These findings suggest that the air hole
modifications of the h21h23 and h21h22 cases need to be optimized separately
to obtain the intended chiral modes with high DCP.

As an alternative to the h21h22 case, air holes 22 and 24 can be modified
instead, i.e., a mirror reflection of the h21h23 case about the $y$-axis,
which gives LCP chiral modes with the same magnitude in the DCP as that in the
h21h23 case. This shows the versatility of our scheme: when a chiral mode is
found, the mode with the opposite chirality can be obtained simply by symmetry
considerations, namely by reflection of the PhC cavity about the $x$\- or
$y$-axis.

We confirmed that the eigenmodes near the EP remain RCP or LCP for the
respective cases presented above regardless of the excitation polarization,
providing further support that the chirality is indeed an intrinsic property.
The excitation polarization will, however, affect the initial polarization of
the cavity modes and the subsequent temporal evolution of the cavity field
polarization, giving rise to varying $\langle$DCP$\rangle$. The
$\langle$DCP$\rangle$ values are slightly lower than the intrinsic DCP values
since the cavity modes evolve from the initial linearly polarized state with
zero DCP to its intrinsic chiral state over some time. We find that for
linearly polarization excitation, orientation at $45^{\circ}$($-45^{\circ}$)
give the largest achievable $\langle$DCP$\rangle$ for the RCP (LCP) chiral
modes (see Appendix B for further details). Our choice of excitation
polarization is based on this observation, which helps to simultaneously
highlight the intrinsic nature of the chirality as well as to show the large
DCP of the chiral modes.

As mentioned in section II, the dipole modes can be thought of as a
superposition of rotating components. As the eigenmodes approach the EP, they
coalesce and eventually become one and the same at the EP. In this regime, the
eigenmodes are no longer stationary. The asymmetric coherent backscattering
between the two eigenmodes causes unbalanced amplitudes of the constituent
counter-rotating components [17]. As a consequence, only one of the
constituent rotating components is dominant and both the eigenmodes co-rotate
in the same direction. Clockwise (counterclockwise) rotating mode will result
in RCP (LCP) mode due to spin-momentum locking. Such clockwise
(counterclockwise) rotation of the field profiles is indeed observed in the
FDTD simulations for chiral modes of the h21h23 (h21h22) cases (see Appendix
C). The choice of modified air holes — 22 or 23 — will mainly induce either a
“forward or backward” backscattering, allowing one to select a dominant
rotating mode and thus the handedness of the chirality (see Appendix A for
further details).

##  V 3D FDTD simulation

Figure 4:  Simulation results of 3D h21h23 case. Distribution of the absolute
difference in the (a) real and (b) imaginary eigenfrequencies within the
parameter space. Color plots showing (c) $\langle$DCP$\rangle$ at cavity
center and (d) the farfield $\langle$DCP$\rangle$ within NA = $0.65$ in the
parameter space. The colorbar is shared for (c-d). (e) Temporal slices of the
DCP around the cavity (top) and in the farfield (bottom) ${\sim}2000$ wave
period propagation for parameters $p_{23}=-0.20a$ and $\Delta r_{23}=-0.05a$.
At this instant, the DCP at the center of the cavity approaches 0.92 and the
mean DCP within NA = 0.65 (black circle) approaches 0.94. The colorbar is
shared for both plots. The scale bar indicates one lattice period.

Figure 5:  Simulation results of 3D h21h22 case. Distribution of the absolute
difference in the (a) real and (b) imaginary eigenfrequencies within the
parameter space. Color plots showing (c) $\langle$DCP$\rangle$ at cavity
center and (d) the farfield $\langle$DCP$\rangle$ within NA = $0.65$ in the
parameter space. (e) Temporal slices of the DCP around the cavity (top) and in
the farfield (bottom) ${\sim}2000$ wave period propagation for parameters
$p_{22}=-0.24a$ and $\Delta r_{22}=-0.05a$. At this instant, the DCP at the
center of the cavity approaches $-0.83$ and the mean DCP within NA = 0.65
(black circle) approaches $-0.90$. The colorbar is shared for both plots. The
scale bar indicates one lattice period.

To simulate practical 3D devices, we consider a PhC with the same $9\times 9$
air hole lattice, lattice period and air hole radius as in the 2D simulations,
with an additional dimension of slab thickness set to be $0.5a$. A grid
resolution of ${\sim}a/17$ ($57$ pixels/micron) is used in the simulations.
The $Q$-factor of an H1 PhC cavity mode tends to be low, as modifications to
the nearest air holes are usually required to achieve a high $Q$-factor [30].
In order to show that our scheme is compatible with other modifications, we
changed the radius of all the first nearest air holes ($r_{1}$) from $0.3a$ to
$0.28a$, increasing the $Q$-factor from ${\sim}200$ to ${\sim}320$. The dipole
modes of the $r_{1}$-modified H1 PhC cavity are at
$a/\lambda_{cav}{\sim}0.294$ with a mode volume of
${\sim}0.35(\lambda/n)^{3}$. The dipole modes are not completely degenerate
with a splitting of ${\sim}0.5$ times the linewidth.

To achieve chiral modes, we apply the following modifications to the selected
second nearest air holes for the h21h23 case: $\Delta r_{21}=+0.01a$,
$d_{21}=-0.115a$, and $d_{23}=-0.225a$. Figure 5(a) and 5(b) shows the
absolute difference of the Re($\omega_{\pm}$) and Im($\omega_{\pm}$)
respectively, with the distributions reflecting the complex eigenfrequency
surfaces with the characteristic branching indicating the presence of an EP as
expected.

In 3D simulations, we are able to extract the farfield emission. The farfield
— in the spherical coordinate — is calculated from the nearfield distribution
via Fourier transform [31, 32]. In addition to the DCP at the cavity center,
we monitor the temporal evolution of the DCP of the farfield emission. At
every fixed timestep, the mean farfield DCP within a numerical aperture (NA)
$\leq$ $0.65$ is calculated and recorded. The choice of NA $\leq$ $0.65$ is to
reflect realistic experimental conditions based on the commonly available and
often-used microscope objective lenses.

It is found that the temporal evolution of the DCP at the cavity center and
the mean farfield DCP follow each other closely, especially in terms of the
trend though the values may differ slightly. The time-averaged DCP and mean
farfield DCP in the parameter space obtained under LP45 excitation is
presented in Fig. 5(c) and 5(d) respectively. RCP chiral modes are observed
near the EP as expected, with the $\langle$DCP$\rangle$ reaching 0.82 at the
cavity center and 0.88 in the farfield at $p_{23}=-0.20a$ and $\Delta
r_{23}=-0.05a$. Figure 5(e) show the temporal slices of the DCP around the
cavity (top) and in the farfield (bottom) at ${\sim}$2000 wave period
propagation for parameters $p_{23}=-0.20a$ and $\Delta r_{23}=-0.05a$. The
farfield emission is RCP throughout. At this instant, the DCP at the center of
the cavity approaches 0.92 and the mean DCP within NA = 0.65 (black circle)
approaches 0.94 indicating the intrinsic DCP of the chiral modes.

For the 3D h21h22 case, slightly different fixed modifications are used:
$\Delta r_{21}=-0.01a$, $d_{21}=-0.145a$ and $d_{22}=-0.23a$. We obtained LCP
chiral modes with $\langle$DCP$\rangle$ of $-0.73$ and $-0.82$ at the cavity
center and in the farfield, respectively, at $p_{22}=-0.24a$ and $\Delta
r_{22}=-0.05a$ [Fig. 5]. The intrinsic DCP approaches $-0.83$ at the cavity
center and $-0.90$ in the farfield.

To achieve chiral modes in this scheme, one only needs to find suitable
parameter values to bring the system close to EP without having obtain the
exact EP. Strictly speaking, only the eigenmodes near the EP exhibit
significant chirality and the chirality goes to zero at the EP [6]. As such,
chiral modes can be achieved under less stringent conditions, further
highlighting the practicality of our scheme. In Appendix B, we present a
discussion of a two-mode approximation model that supports our key findings in
the FDTD simulations.

Comparing simulations results for the $r_{1}$-modified and unmodified H1 PhC
cavity (results not shown), we find that the $r_{1}$-modified PhC cavity
requires larger modifications to the second nearest air holes in order to
achieve chiral modes. Using the h21h23 case as an example, with $r_{1}$
modified, holes 21 and 23 need to be shifted closer to the cavity and the
radius of hole 23 needs to be reduced by larger magnitude. We also found that
larger changes to the first nearest holes to increase the $Q$-factor will
require larger modifications to the second nearest air holes to obtain chiral
modes. Should the required modifications to the second nearest air holes
become so significant such that the structure is rendered impractical — for
example, adjacent air holes are in contact or that the air holes become too
small for nanofabrication — one could instead modify two selected first
nearest air holes to achieve chiral modes. As such, by applying all necessary
modifications to the first nearest air holes, one could achieve a high-Q
chiral cavity.

A PhC nanocavity confines optical fields within a small mode volume
facilitating strong light-matter coupling as well as the Purcell effect which
describes the enhanced spontaneous emission. The local density of states near
the EP is altered due to the non-orthogonality of the modes which could in
turn give additional enhancement to the spontaneous emission [26, 33, 27].
This additional enhancement can be improved by increasing the $Q$-factor or by
introducing material gain [26] which are both applicable to the chiral PhC
cavity, highlighting the prospects of using such chiral PhC cavity for device
applications.

##  VI Conclusion

We have presented a scheme to design intrinsically circularly polarized chiral
H1 PhC cavities. In this scheme, the symmetry of the cavity is broken
intentionally by modifying two of the second-nearest air holes to the cavity
in the $\Gamma-K$ directions. The modifications induce an asymmetric coherent
backscattering between the two eigenmodes, which is a non-Hermitian process.
As a result, within the parameter space of the modified air holes, there are
EPs at which the eigenfrequencies and the eigenmodes coalesce. In the vicinity
of the EP, the eigenmodes are no longer stationary but are co-rotating modes
with an overall direction. This in turn gives rise to chiral modes in which
the DCP is correlated with the rotation direction of the mode profile. From
the 3D FDTD simulation results, we show that a practical device is achievable
in which both the nearfield and farfield emission exhibit near unity DCP. The
handedness of the chirality can be controlled by modifying selected air hole
pairs.

The PhC nanocavity localizes the optical fields in a small mode volume while
being able to maintain a relatively high $Q$-factor and thus capable of
achieving a high $Q$-to-mode volume ratio which is important for quantum
information technologies [34]. In addition, incorporating chirality in the
form of circular polarization in our proposed H1 PhC nanocavity will provide
additional degrees of freedom for optical control and information processing.
Our proposed chiral PhC nanocavity expands the toolbox of exceptional point
photonics and will complement existing chiral whispering gallery mode
resonators [6, 15, 16], chiral photonics structures [19, 35, 36] as well as
metamaterials [37, 38] for extended functionalities. By exploiting the
favorable properties of the chiral H1 PhC nanocavity, one could expect the
further development of chiral photonics application such as valleytronics with
2D materials [39, 40], spin-photon interfaces [41, 42], and the generation of
single photons with well-defined spins [43].

##  Appendix A Selecting chirality

In the case of a microdisk or microring resonator, two scatterers can be
placed close to the edge of the resonator to induce the necessary
backscattering to bring about EPs [6, 12, 17]. One of the scatterers is
usually fixed, for example at the field antinode of the whispering gallery
mode and the position of the second scatterer is varied. The backscattering
conditions to achieve an EP are met when the second scatterer is rightly
positioned close to an antinode and EPs would occur periodically as the second
scatterer is moved across consecutive antinodes. In addition, the system would
alternate between dominant forward and backward backscattering, therefore the
associated handedness of the chirality would also switch periodically.

Consequently, it should be possible to selectively induce an EP of a specific
chirality by a judicious placement of scatterers relative to the antinodes of
the mode profiles. Note that the antinode here refers to the antinode of a
nominally unperturbed mode profile as significant perturbation near the EP
could strongly affect the mode profile and reduce the visibility of the
antinodes [6, 17].

The h21h22 and h21h23 H1 PhC cavity cases described in the main text are
consistent with the previously reported observations in the whispering gallery
resonators. In the H1 PhC cavities, the modified air holes behave as the
scatterers. Only one EP exists within the parameter space in each of the two
cases. The resulting chiralities of the eigenmodes near the EPs are of
opposite handedness in the two cases depending on whether the modified air
hole is to the left or the right of the antinode of the D1 dipole mode
profile.

To further verify this idea, we perform simulations in which air hole 21 and
the air hole between 22 and 23 — which we refer to as air hole 23m — are
modified. Both the air holes are originally aligned to the antinodes of the D1
and D2 dipole mode profiles. As such, moving either one of the air holes along
the hexagonal path should, in principle, give 2 EPs with associated
chiralities of opposite handedness within the parameter space. In the
simulations, we use LP45 excitation in the simulations to demonstrate the
independence of the mode chirality on the excitation polarization. The
expected behavior is indeed observed in the 2D FDTD simulation results [Fig.
6]. Under co-circular polarization excitation, the $\langle$DCP$\rangle$ of
the chiral modes approaches $0.95$.

Figure 6:  h21h23m cases in which parameter sweep is performed on air hole 23m
(top row) and air hole 21 (bottom row) respectively. The color plots present
the absolute difference in the eigenfrequencies and the ⟨DCP⟩ at cavity center
as labelled. A $45^{\circ}$ linearly polarized source is used for excitation.
In both cases, the distribution of the absolute difference in the
eigenfrequencies indicate the presence of two EPs. The chirality associated
with each EP is of opposite handedness, giving RCP (LCP) eigenmodes when
respective air holes are shifted in the clockwise (counterclockwise) direction
along the hexagonal path. For the simulations in the top row, the fixed air
hole modifications are $\Delta r_{21}=+0.02a$, $d_{21}=-0.08a$ and
$d_{23m}=-0.17a$. For the bottom row, the fixed air hole modifications are
given by $\Delta r_{23m}=+0.04a$, $d_{23m}=-0.10a$ and $d_{21}=-0.22a$.

For the h21h22 case described in the main text [Fig. 3(b)], there is a region
around $p_{22}=0.25a$ to $0.275a$ in which the modes have a modest degree of
RCP instead of LCP. The appearance of such a region suggests the onset of
formation of the second EP with the opposite chirality for the particular set
of $\Delta r_{21}$, $d_{21}$ and $d_{22}$ parameter values used. Such a
corresponding opposite chirality region is also present, though much less
prominent, in the h21h23 case [Fig. 3(a), $p_{23}=-0.275a$ to $-0.25a$]. By
further modifying the air hole parameter values, it is possible to clearly
observe two EPs within the parameter space.

##  Appendix B Two-mode approximation model

Figure 7:  (Top row) The distribution of the absolute differences in the real
and imaginary eigenfrequencies within the parameter space. (Bottom row) The
distribution of the $\langle$DCP$\rangle$ under RCP, LCP and linearly
polarized excitation oriented at $45^{\circ}$. The colorbar is shared for the
plots in the bottom row. The following values are used for these modelling
results:
$\omega_{1}=1.0027-0.0022i,\omega_{2}=1-0.0020i,f_{1}=0.025,f_{2}=0.04,s_{1}=0.195i$
and $s_{2}=0.200i$.  Figure 8:  Temporal evolution of DCP calculated using
two-mode approximation model (left column) and FDTD simulations (right
column). The top (bottom) row compares the temporal evolution of DCP under
different excitation polarization for the corresponding RCP (LCP) chiral modes
obtained in the two-mode approximation and 3D FDTD simulations. The relevant
parameter values are noted on top of each plot. The grey areas in the plots of
the FDTD simulation results indicate the period of time when the excitation is
on. The temporal evolution of DCP under vertical linear polarization, LPy
excitation (not shown) is exactly the same as that under horizontal linear
polarization, LPx for both the two-mode approximation model and FDTD
simulations. The temporal evolution of DCP in the farfield from the FDTD
simulations is largely similar to that at the cavity center and thus not
presented here.

For a non-Hermitian PhC nanocavity system that is sufficiently close to the
EP, it can be described by a 2-dimensional Hamiltonian associated with the two
coalescing states [44]. To approximate a H1 PhC cavity, we can define a
Hamiltonian as follows, in which the basis states are the $E_{x}$ and $E_{y}$
electric fields of the two respective linearly polarized dipole modes:

$H(\Gamma)=\begin{bmatrix}\omega_{1}&0\\\
0&\omega_{2}\end{bmatrix}+\Gamma\begin{bmatrix}f_{1}&s_{1}\\\
s_{2}&f_{2}\end{bmatrix}$

The second term on the right-hand side can be thought of as the perturbation.
The terms $\omega_{1,2}$ and $f_{1,2}$ determine the non-interacting resonance
frequencies $\omega_{1,2}+\Gamma f_{1,2}$. The terms $s_{1}(s_{2})$
representing the coherent backscattering from mode D1 to D2 (D2 to D1) and
$\Gamma$ being the tuning parameter. For $s_{1,2}\neq 0$, the eigenfrequencies
are

$\begin{split}\omega_{\pm}(\Gamma)=&1/2\Bigl{(}\omega_{1}+\omega_{2}+\Gamma(f_{1}+f_{2})\\\
&\pm\sqrt{(f_{1}-f_{2})^{2}+4s_{1}s_{2}(\Gamma-\Gamma_{1})(\Gamma-\Gamma_{2})}\Bigr{)}\end{split}$
(1)

where $\Gamma_{1}$ and $\Gamma_{2}$ are the values at which the EP occurs,
given by
$\Gamma_{1}=\frac{-i(\omega_{1}-\omega_{2})}{i(f_{1}-f_{2})+2\sqrt{s_{1}s_{2}}}$
and
$\Gamma_{2}=\frac{-i(\omega_{1}-\omega_{2})}{i(f_{1}-f_{2})-2\sqrt{s_{1}s_{2}}}$.
The square root terms on the right-hand side of Eqn. (2) gives rise to the
characteristic distribution of eigenfrequencies with the branching in the
parameter space. The Hamiltonian can be solved to obtain two eigenvectors
(eigenmodes) $\Phi_{\pm}$ and a general wavefunction can be defined to
describe the temporal evolution of the eigenvectors, $\psi(t)=a_{+}\Phi_{+}$
exp($-i\omega_{+}t)+a_{-}\Phi_{-}$ exp($-i\omega_{-}t)$. The amplitudes
$a_{\pm}$ can be solved by considering initial conditions at $t=0$ based on
the excitation polarization, for example RCP excitation gives
$\psi(0)=\bigl{(}1/\sqrt{2},i/\sqrt{2}\bigr{)}$ and so on. From the
wavefunction, one could then calculate the temporal evolution of the spin
angular momentum density, field energy density, as well as the DCP as is done
for the FDTD simulations.

Figure 7 shows the distribution of the absolute difference in the real and
imaginary eigenfrequencies calculated using the two-mode approximation model.
There are clear signatures of branching and, there are two EPs within the
parameter space spanned by Re($\Gamma$) and Im($\Gamma$) as expected from the
Hamiltonian. The distribution of the eigenfrequencies do not match with that
from FDTD simulations since Re($\Gamma$) and Im($\Gamma$) do not correspond
directly to the air hole sweep parameters. Nonetheless, using the model, the
eigenmodes in the vicinity of both the EPs are chiral with high DCP but of
different handedness. It can be seen from the figure that the handedness of
the chirality is independent of the excitation polarization.

Figure 9:  Field profiles showing the rotation of the eigenmodes near the EPs
for the h21h23 (top) and the h21h22 (bottom) cases. Each timestep $\Delta t$
is $10$ wave period propagation.

The temporal evolution of DCP calculated using the two-mode approximation is
highly consistent with that from the FDTD simulations (after the excitation
has been turned off) for both the RCP [Fig. 8 (top row)] and LCP [Fig. 8
(bottom row)] chiral modes. The initial polarization of the cavity fields is
essentially determined by the excitation polarization. The cavity fields then
proceed to evolve to their intrinsic polarization after the excitation is
turned off. The temporal evolution of the polarization is, however, less
intuitive for the LP±45 excitations. After the LP45(-45) excitation, the
cavity fields tend to subsequently become RCP (LCP) regardless of the
intrinsic polarization. This suggests that when oriented at $\pm 45^{\circ}$,
the linearly polarized excitation imposes a relative $\pm\pi/2$ phase between
the two orthogonal linear basis modes before allowing them to reach their
intrinsic polarization. As such, for linearly polarized excitation,
orientation at $45^{\circ}(-45^{\circ})$ is favored to obtain a larger ⟨DCP⟩
for the RCP (LCP) chiral modes. Under co-polarized excitation, the
instantaneous DCP of the chiral modes remains high throughout the simulation
time, giving the maximum $\langle$DCP$\rangle$ which can also be seen in Fig.
7. Despite the simplicity of the model, it captures the key features of the
FDTD simulation results of the H1 PhC cavity.

##  Appendix C Rotating modes

Near the EPs, the eigenmodes rotate in the clockwise (counterclockwise)
direction for the h21h23 (h21h22) case as exemplified by the change in the
$H_{z}$ field profile with time in Fig. 9.

##  Appendix D Further simulation details

We note that our simulations do not consider any material dispersion. Given
that we are dealing with resonant modes, the effect of material dispersion is
expected to be minimal. Furthermore, for the wavelength of around 1 $\mu$m in
GaAs that we are considering here, including the correction for material
dispersion [45] should have a negligible effect on the calculation of the DCP.

###### Acknowledgements.

This work is partly supported by MIC (SCOPE 191503001). Most FDTD simulations
for this work are performed on the Supercomputer HOKUSAI BigWaterfall at
RIKEN. C.F. Fong is supported by the RIKEN Special Postdoctoral Program. S.
Iwamoto acknowledges funding from JST-CREST (JPMJCR19T1) and JSPS KAKENHI
(17H06138).

## References

  * El-Ganainy  _et al._ [2019] R. El-Ganainy, M. Khajavikhan, D. N. Christodoulides, and S. K. Ozdemir, The dawn of non-Hermitian optics, Commun. Phys. 2, 37 (2019).
  * Miri and Alù [2019] M.-A. Miri and A. Alù, Exceptional points in optics and photonics, Science 363, eaar7709 (2019).
  * Parto  _et al._ [2020] M. Parto, Y. G. N. Liu, B. Bahari, M. Khajavikhan, and D. N. Christodoulides, Non-Hermitian and topological photonics: Optics at an exceptional point, Nanophotonics 10, 403 (2020).
  * Hodaei  _et al._ [2014] H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, Parity-time–symmetric microring lasers, Science 346, 975 (2014).
  * Feng  _et al._ [2014] L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, Single-mode laser by parity-time symmetry breaking, Science 346, 972 (2014).
  * Peng  _et al._ [2016] B. Peng, Ş. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yılmaz, J. Wiersig, S. Rotter, and L. Yang, Chiral modes and directional lasing at exceptional points, Proc. Natl. Acad. Sci. 113, 6845 (2016).
  * Chen  _et al._ [2017] W. Chen, Ş. Kaya Özdemir, G. Zhao, J. Wiersig, and L. Yang, Exceptional points enhance sensing in an optical microcavity, Nature 548, 192 (2017).
  * Chang  _et al._ [2014] L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, Parity–time symmetry and variable optical isolation in active–passive-coupled microresonators, Nat. Photonics 8, 524 (2014).
  * Hodaei  _et al._ [2017] H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, Enhanced sensitivity at higher-order exceptional points, Nature 548, 187 (2017).
  * Miao  _et al._ [2016] P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, Orbital angular momentum microlaser, Science 353, 464 (2016).
  * Yang  _et al._ [2020] J. Yang, C. Qian, X. Xie, K. Peng, S. Wu, F. Song, S. Sun, J. Dang, Y. Yu, S. Shi, J. He, M. J. Steer, I. G. Thayne, B.-B. Li, F. Bo, Y.-F. Xiao, Z. Zuo, K. Jin, C. Gu, and X. Xu, Diabolical points in coupled active cavities with quantum emitters, Light Sci. Appl. 9, 6 (2020).
  * Yang  _et al._ [2021] J. Yang, S. Shi, X. Xie, S. Wu, S. Xiao, F. Song, J. Dang, S. Sun, L. Yang, Y. Wang, Z.-Y. Ge, B.-B. Li, Z. Zuo, K. Jin, and X. Xu, Enhanced emission from a single quantum dot in a microdisk at a deterministic diabolical point, Opt. Express 29, 14231 (2021).
  * Liu  _et al._ [2016] Z.-P. Liu, J. Zhang, Ş. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu, Metrology with PT-Symmetric Cavities: Enhanced Sensitivity near the PT-Phase Transition, Phys. Rev. Lett. 117, 110802 (2016).
  * Wiersig [2016] J. Wiersig, Sensors operating at exceptional points: General theory, Phys. Rev. A 93, 033809 (2016).
  * Kim  _et al._ [2014] M. Kim, K. Kwon, J. Shim, Y. Jung, and K. Yu, Partially directional microdisk laser with two Rayleigh scatterers, Opt. Lett. 39, 2423 (2014).
  * Zhong  _et al._ [2021] Q. Zhong, A. Hashemi, Ş. K. Özdemir, and R. El-Ganainy, Control of spontaneous emission dynamics in microcavities with chiral exceptional surfaces, Phys. Rev. Research 3, 013220 (2021).
  * Wiersig [2011] J. Wiersig, Structure of whispering-gallery modes in optical microdisks perturbed by nanoparticles, Phys. Rev. A 84, 063828 (2011).
  * Fong  _et al._ [2018] C. F. Fong, Y. Ota, S. Iwamoto, and Y. Arakawa, Scheme for media conversion between electronic spin and photonic orbital angular momentum based on photonic nanocavity, Opt. Express 26, 21219 (2018).
  * Lodahl  _et al._ [2017] P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, Chiral quantum optics, Nature 541, 473 (2017).
  * Coles  _et al._ [2014] R. J. Coles, N. Prtljaga, B. Royall, I. J. Luxmoore, A. M. Fox, and M. S. Skolnick, Waveguide-coupled photonic crystal cavity for quantum dot spin readout, Opt. Express 22, 2376 (2014).
  * Söllner  _et al._ [2015] I. Söllner, S. Mahmoodian, S. L. Hansen, L. Midolo, A. Javadi, G. Kiršanske, T. Pregnolato, H. El-Ella, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, Deterministic photon-emitter coupling in chiral photonic circuits, Nat. Nanotechnol. 10, 775 (2015).
  * Troia  _et al._ [2013] B. Troia, A. Paolicelli, F. Leonardis, and V. Passaro, Photonic Crystals for Optical Sensing: A Review, in  _Advances in Photonic Crystals_ (IntechOpen, 2013).
  * Luxmoore  _et al._ [2012] I. J. Luxmoore, E. D. Ahmadi, B. J. Luxmoore, N. A. Wasley, A. I. Tartakovskii, M. Hugues, M. S. Skolnick, and A. M. Fox, Restoring mode degeneracy in H1 photonic crystal cavities by uniaxial strain tuning, Appl. Phys. Lett. 100, 121116 (2012).
  * Luxmoore  _et al._ [2011] I. J. Luxmoore, E. D. Ahmadi, A. M. Fox, M. Hugues, and M. S. Skolnick, Unpolarized H1 photonic crystal nanocavities fabricated by stretched lattice design, Appl. Phys. Lett. 98, 041101 (2011).
  * Hennessy  _et al._ [2006] K. Hennessy, C. Högerle, E. Hu, A. Badolato, and A. Imamoğlu, Tuning photonic nanocavities by atomic force microscope nano-oxidation, Appl. Phys. Lett. 89, 041118 (2006).
  * Pick  _et al._ [2017a] A. Pick, B. Zhen, O. D. Miller, C. W. Hsu, F. Hernandez, A. W. Rodriguez, M. Soljačić, and S. G. Johnson, General theory of spontaneous emission near exceptional points, Opt. Express 25, 12325 (2017a).
  * Takata  _et al._ [2021] K. Takata, K. Nozaki, E. Kuramochi, S. Matsuo, K. Takeda, T. Fujii, S. Kita, A. Shinya, and M. Notomi, Observing exceptional point degeneracy of radiation with electrically pumped photonic crystal coupled-nanocavity lasers, Optica 8, 184 (2021).
  * Aiello  _et al._ [2015] A. Aiello, P. Banzer, M. Neugebauer, and G. Leuchs, From transverse angular momentum to photonic wheels, Nat. Photonics 9, 789 (2015).
  * Oskooi  _et al._ [2010] A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. Joannopoulos, and S. G. Johnson, Meep: A flexible free-software package for electromagnetic simulations by the FDTD method, Comput. Phys. Commun. 181, 687 (2010).
  * Takagi  _et al._ [2012] H. Takagi, Y. Ota, N. Kumagai, S. Ishida, S. Iwamoto, and Y. Arakawa, High Q H1 photonic crystal nanocavities with efficient vertical emission, Opt. Express 20, 28292 (2012).
  * Vučković  _et al._ [2002] J. Vučković, M. Lončar, H. Mabuchi, and A. Scherer, Optimization of the Q factor in photonic crystal microcavities, IEEE J. Quantum Electron. 38, 850 (2002).
  * Kim  _et al._ [2006] S.-H. Kim, S.-K. Kim, and Y.-H. Lee, Vertical beaming of wavelength-scale photonic crystal resonators, Phys. Rev. B 73, 235117 (2006).
  * Pick  _et al._ [2017b] A. Pick, Z. Lin, W. Jin, and A. W. Rodriguez, Enhanced nonlinear frequency conversion and Purcell enhancement at exceptional points, Phys. Rev. B 96, 224303 (2017b).
  * Hennessy  _et al._ [2007] K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atature, S. Gulde, S. Fält, E. Hu, and A. Imamoglu, Quantum nature of a strongly coupled single quantum dot-cavity system, Nature 445, 896 (2007).
  * Parappurath  _et al._ [2020] N. Parappurath, F. Alpeggiani, L. Kuipers, and E. Verhagen, Direct observation of topological edge states in silicon photonic crystals: Spin, dispersion, and chiral routing, Sci. Adv. 6, aaw4137 (2020).
  * Mehrabad  _et al._ [2020] M. J. Mehrabad, A. P. Foster, R. Dost, E. Clarke, P. K. Patil, A. M. Fox, M. S. Skolnick, and L. R. Wilson, Chiral topological photonics with an embedded quantum emitter, Optica 7, 1690 (2020).
  * Wang  _et al._ [2016] Z. Wang, F. Cheng, T. Winsor, and Y. Liu, Optical chiral metamaterials: A review of the fundamentals, fabrication methods and applications, Nanotechnology 27, 412001 (2016).
  * Konishi  _et al._ [2020] K. Konishi, T. Kan, and M. Kuwata-Gonokami, Tunable and nonlinear metamaterials for controlling circular polarization, J. Appl. Phys 127, 230902 (2020).
  * Gong  _et al._ [2018] S. H. Gong, F. Alpeggiani, B. Sciacca, E. C. Garnett, and L. Kuipers, Nanoscale chiral valley-photon interface through optical spin-orbit coupling, Science 359, 443 (2018).
  * Rong  _et al._ [2020] K. Rong, B. Wang, A. Reuven, E. Maguid, B. Cohn, V. Kleiner, S. Katznelson, E. Koren, and E. Hasman, Photonic Rashba effect from quantum emitters mediated by a Berry-phase defective photonic crystal, Nat. Nanotechnol. 15, 927 (2020).
  * Carter  _et al._ [2013] S. G. Carter, T. M. Sweeney, M. Kim, C. S. Kim, D. Solenov, S. E. Economou, T. L. Reinecke, L. Yang, A. S. Bracker, and D. Gammon, Quantum control of a spin qubit coupled to a photonic crystal cavity, Nat. Photonics 7, 329 (2013).
  * Lodahl  _et al._ [2015] P. Lodahl, S. Mahmoodian, and S. Stobbe, Interfacing single photons and single quantum dots with photonic nanostructures, Rev. Mod. Phys 87, 347 (2015).
  * Kan  _et al._ [2020] Y. Kan, S. K. H. Andersen, F. Ding, S. Kumar, C. Zhao, and S. I. Bozhevolnyi, Metasurface-Enabled Generation of Circularly Polarized Single Photons, Adv. Mater. 32, 1907832 (2020).
  * Heiss [2012] W. D. Heiss, The physics of exceptional points, J. Phys. A: Math. Theor. 45, 444016 (2012).
  * Bliokh  _et al._ [2017] K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, Optical momentum and angular momentum in complex media: From the Abraham–Minkowski debate to unusual properties of surface plasmon-polaritons, New J. Phys. 19, 123014 (2017).