# Direct Observation of Exceptional Points in Photonic Crystal by Cross-
Polarization Imaging in Momentum Space

Viet Anh Nguyen  The authors are equally contributed.  College of Engineering
and Computer Science, VinUniversity, Gia Lam district, Hanoi 14000, Vietnam
Viet Hoang Le  The authors are equally contributed.  College of Engineering
and Computer Science, VinUniversity, Gia Lam district, Hanoi 14000, Vietnam
Loïc Malgrey  Univ Lyon, Ecole Centrale de Lyon, CNRS, INSA Lyon, Université
Claude Bernard Lyon 1, CPE Lyon, CNRS, INL, UMR5270, 69130 Ecully, France
Département de Physique, Ecole Normale Supérieure de Lyon, 46 Allée d’Italie,
F 69342, Cedex 07 Lyon, France  Eirini Sarelli  Univ Lyon, Ecole Centrale de
Lyon, CNRS, INSA Lyon, Université Claude Bernard Lyon 1, CPE Lyon, CNRS, INL,
UMR5270, 69130 Ecully, France  Dang-Khue Luu  College of Engineering and
Computer Science, VinUniversity, Gia Lam district, Hanoi 14000, Vietnam  Ha
Linh Chu  College of Engineering and Computer Science, VinUniversity, Gia Lam
district, Hanoi 14000, Vietnam  Truong Tuan Vu  College of Engineering and
Computer Science, VinUniversity, Gia Lam district, Hanoi 14000, Vietnam  Cong
Quang Tong  Institute of Materials Science, VAST, 18 Hoang Quoc Viet, Ha Noi
11300, Vietnam  Vu Dinh Lam  Graduate University of Science and Technology,
Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, 11300,
Vietnam  Christian Seassal  Univ Lyon, Ecole Centrale de Lyon, CNRS, INSA
Lyon, Université Claude Bernard Lyon 1, CPE Lyon, CNRS, INL, UMR5270, 69130
Ecully, France  Quynh Le-Van<EMAIL_ADDRESS>College of Engineering
and Computer Science, VinUniversity, Gia Lam district, Hanoi 14000, Vietnam
Hai-Son Nguyen<EMAIL_ADDRESS>Univ Lyon, Ecole Centrale de Lyon,
CNRS, INSA Lyon, Université Claude Bernard Lyon 1, CPE Lyon, CNRS, INL,
UMR5270, 69130 Ecully, France  Institut Universitaire de France (IUF), Paris,
France

###### Abstract

This study explores exceptional points (EPs) in photonic crystals (PhCs) and
introduces a novel method for their single-shot observation. Exceptional
points are spectral singularities found in non-Hermitian systems, such as
leaky PhC slabs. However, directly observing EPs in PhC systems using regular
reflectivity spectroscopy is a considerable challenge due to interference
between guided resonances and background signals. In this work, we present a
simple, nondestructive technique that employs crossed polarizations to
directly observe EPs in momentum-resolved resonant scattering. This approach
effectively suppresses the background signal, enabling exclusive probing of
the guided resonances where EPs manifest. Our results demonstrate the
formation of EPs in both energy-momentum mapping and isofrequency imaging. All
experimental findings align seamlessly with numerical simulations and
analytical models. Our approach holds great potential as a robust tool for
studying non-Hermitian physics in PhC platform.

Over the past decade, there have been remarkable developments in non-Hermitian
physics within photonic platforms, encompassing both purely lossy systems and
parity-time symmetric systems that exhibit a balance between optical gain and
losses[1, 2, 3, 4]. These advancements have been possible due to the expertise
in crafting synthetic optical materials at the subwavelength scale, which
allows for precise modulation/tailoring of their gain and loss properties. The
physics of these open systems can be depicted using effective non-Hermitian
Hamiltonians[5], characterized by complex eigenvalues and non-orthogonal
eigenvectors. Intriguingly, the degeneracies in a non-Hermitian Hamiltonian
are termed Exceptional Points (EPs), where both the eigenvectors and
eigenvalues converge. EPs represent singularities in non-Hermitian topology[6,
7, 8, 4], and their unique attributes have been instrumental in illustrating
numerous non-trivial physical phenomena, including coupled waveguides [9, 10,
11, 12], unidirectional reflectionless light propagation[13, 14], loss-gain
microresonators for lasers [15, 16, 17, 18, 19, 20], improved sensing
sensitivity [21, 22, 23], enhanced local density of optical states and
spontaneous emission [24, 25].

Leaky resonances in Photonic Crystals (PhCs) serve as a versatile platform for
studying non-Hermitian band structures and EP physics[26, 27, 28]. By
engineering energy-momentum dispersion and radiative losses in PhC systems, it
becomes possible to generate EPs even in the absence of gain medium and
material losses[29, 30, 31, 32, 33, 34, 35, 25]. The emergence of EPs within
passive photonic crystals has paved new paths for investigating the
captivating physics concealed within these structures. These include EP
rings[29, 30], bulk Fermi arc[29], non-Hermitian band inversion[33, 34, 25],
non-Hermitian coupling between orthogonal polarization modes[32, 35], and non-
trivial propagation decay[35]. However, demonstrating EPs in such PhCs is not
straightforward, as shown by the detailed analysis required in angle-resolved
reflectivity measurements[29]. This complexity stems from the Fano-shaped
profile in the reflectivity spectra, which arises from the coupling between
guided resonances primarily confined within the PhC slab and background
optical signals emanating from the substrate and the interfaces between
layers. As a result, the presence of EPs tends to be obscured in angle-
resolved reflectivity spectra, necessitating advanced analytical techniques.
As a potential solution, coupling the EPs to a luminescent layer allows the
detection of EPs through an enhancement of fluorescence in the energy-momentum
spectra[25]. While this approach facilitates direct observation of EPs, the
inclusion of an active material modifies the optical environment. It also
introduces nonradiative losses, thereby significantly altering the properties
of EPs from the pristine sample. Another tactic involves using a dielectric or
metallic mirror as the sample substrate[34, 25]. This leads to a Lorentz-
shaped profile, as opposed to Fano ones, enabling direct observation of EPs.
Nevertheless, requiring a mirror substrate imposes significant constraints and
limits the design flexibility of the PhC. Therefore, it is imperative to
develop a method for the direct observation of EPs in passive PhCs that
maintains their inherent characteristics and imposes no restrictions on the
stack design.

In this letter, we demonstrate that EPs in a passive PhC structure can be
directly observed in angle-resolved reflectivity using a crossed-polarization
setup. This technique, termed resonant scattering, effectively suppresses the
specular reflection, thereby eliminating Fano interference and allowing for
the exclusive probing of the PhC’s resonant modes. EPs are discerned in
single-shot imaging without any post-processing in both energy-momentum
dispersion measurements and isofrequency contour measurements. The
experimental findings align well with the results from numerical simulations.
Our work introduces a straightforward and non-destructive method to monitor
EPs in momentum space, further broadening the potential for resonance-enhanced
scattering in photonic systems.

Figure 1: Sample design and photonic band structure. a) Sketch of the sample
design. The inset shows the Brillouin zone of the reciprocal lattice. b) The
field intensity of the bright and dark modes at $k=0$. c) The energy of bright
and dark modes at $k$ = 0 for different hole radius. d) Real and e) imaginary
parts of the energy eigenvalues along $\Gamma$M direction for structure with
$r=r_{EP}=160$ nm. The green color corresponds to the bright mode while the
red represents the dark modes. Analytical calculations are shown in solid line
(pale red and pale green) and the numerical simulations are presented with
square symbols (dark red and dark green). The refractive indices of
${\text{SiN}}_{\text{x}}$ and ${\text{SiO}}_{\text{2}}$ used in the FEM
simulations are 2.02 and 1.46, respectively. Analytical fits are obtained with
$E_{0}=2.0$ eV, $\delta$ = 0.59 meV, $\gamma$ = 19.9 meV, $v=76.6$ meV.µm and
$\sigma=-35$ meV.µm-2.

The sample design, depicted in Fig.1a, comprises a 158 nm thick
SiN${}_{\text{x}}$ layer, partially etched to form a square lattice of holes
(96 nm depth and radius $r=160$ nm) with a period of 396 nm. This PhC slab is
positioned on a silicon substrate with a 2µm-thick layer of SiO2 serving as an
optical spacer. In passive PhC slabs, EPs arise when two resonances of
differing radiative losses coalesce[29, 32, 25]. In our design, these
resonances encompass a leaky mode, termed a bright mode, and a non-radiating
mode, named the dark mode[25, 29, 30]. The bright (dark) mode is permitted
(forbidden) to radiate into the far field due to its symmetric (anti-
symmetric) near-field pattern. Figure 1b displays the electric field
distribution of these modes, derived from Finite Element Method (FEM)
simulations using Comsol Multiphysics. It reveals that the electric field of
the bright mode radiates into free space in the far field, whereas the field
distribution of the dark mode is predominantly confined to the
${\text{SiN}}_{\text{x}}$ layer, not radiating into the far field. Notably,
these modes resemble those previously documented for engineering exceptional
rings[29, 30]. At an oblique angle (i.e., nonzero momentum $k$), mirror
symmetry is disrupted, causing the modes to hybridize. The effective
Hamiltonian of the band structure near the $\Gamma$ point is given by[25]:

$H(k)=\begin{pmatrix}{E_{0}+\frac{\delta+\sigma k^{2}}{2}}&{v.k}\\\
{v.k}&{E_{0}-\frac{\delta+\sigma
k^{2}}{2}}\end{pmatrix}+\begin{pmatrix}i\gamma&0\\\ 0&0\end{pmatrix}$ (1)

In the Hermitian term (i.e. first term) of (1), $E_{0}$ is the offset energy
of the bands, $\delta$ is the frequency detuning between the bright mode and
the dark mode at $k=0$; and $v,\sigma$ are coefficients of second-order $k.p$
perturbation theory. In the non-Hermitian term (i.e. second term) of (1),
$\gamma$ corresponds to the radiative losses of the bright mode, given by:

$E_{\pm}(k)=E_{0}+i\frac{\gamma}{2}\pm\sqrt{\left(\frac{\delta+\sigma
k^{2}+i\gamma}{2}\right)^{2}+v^{2}k^{2}}$ (2)

Interestingly, when the detuning $\delta$ is tuned to
$\delta_{EP}=-\frac{\sigma\gamma^{2}}{4v^{2}}\approx 0$, the two eigenvalues
coalesce to $E_{EP}=E_{0}+i\frac{\gamma}{2}$ at $k_{EP}=\pm\frac{\gamma}{2v}$
and EPs are formed. This configuration can be achieved in our design since
$\delta$ can be finely tuned by sweeping the air-radius of the hole lattice.
Indeed, Figure 1c presents FEM results of the energy of the two modes at $k=0$
as functions of the air radius $r$. It shows that the zero-detuning
configuration is achieved for radius $r_{EP}=160$ nm. Figures 1d,e presents
the real (Fig. 1d) and imaginary (Fig. 1e) parts of two complex eigenvalues
obtained by FEM simulations along $\Gamma$M direction of the Brillouin zone
when $r=r_{EP}$. These numerical results are perfectly fitted by the
analytical model (2). They clearly evidence the formation of EPs on
$\Gamma{\text{M}}$ path at $k_{EP}=\pm$0.13 µm -1. The dark and bright modes
can also be inferred in Fig. 1e in which at $k=0$, the imaginary of the dark
mode reduces to zero whereas the bright mode exhibits the highest loss. The
calculated band structures in both directions $\Gamma$M and $\Gamma$X can be
found in supplemental information (SI).

The sample is fabricated using Plasma Enhanced Chemical Vapor Deposition to
deposit SiN${}_{\text{x}}$. This is followed by laser interference lithography
and dry etching to pattern the SiN${}_{\text{x}}$ layer into PhC structures.
Further fabrication details are provided in Section I of the SI. A scanning
electron microscopy (SEM) image of the final sample is presented in the inset
of Fig.2.

To solely probe the guided resonances in our PhC structure, we employ the
resonant scattering measurement technique that involves
reflectivity/transmission spectroscopy in a cross-polarization setup[36, 37,
38, 39, 40, 41]. While no previous studies have utilized this method to unveil
EPs, it has been effectively used to discern bare photonic modes for localized
states in PhC cavities[36, 37, 42, 43] and the band structures of guided mode
resonances[39, 44, 45]. The underlying principles of the resonant scattering
technique, used to determine the spectral position of guided resonances in
PhC, are discussed extensively in earlier reports[39, 40, 41]. In essence, the
cross-polarization arrangement in resonant scattering spectroscopy results in
the suppression of specular reflection/transmission and its interference with
light scattered from the guided resonances. Consequently, the spectra exhibit
a zero baseline and symmetric profile peaks, diverging from the Fano profile.
This facilitates direct insight into the spectral position and line-width of
the guided resonance.

The optical setup for the resonant scattering measurement in momentum space is
illustrated in Fig. 2. Broadband light (Thorlabs SLS202C) is focused onto the
sample through a microscope objective (OptoSigma 50X, NA = 0.55). Polarization
is controlled using a linear polarizer (LP1) and a half-wave plate (HWP1).
Reflected light is collected by the same objective and separated by using a
beam splitter (BS). A Fourier lens (L1) images the objective’s back focal
plane. For resonant scattering, background signals are filtered with another
polarization set (LP2 and HWP2), while regular reflectivity omits this step.
For momentum-resolved spectroscopy, light is focused via lens (L2) onto a
monochromator (Princeton Instrument HRS-300) slit, then relayed to a 2D
spectroscopic camera (Kuro model). This setup maps the energy-momentum band
structure in one shot, with the direction determined by the monochromator
slit. For isofrequency Fourier imaging, a band-pass filter (BF) sets the
wavelength, and a flipped mirror (FM) projects the back-focal plane to a CMOS
camera (Caltex). This maps photonic modes in momentum space ($k_{x}$, $k_{y}$)
at specific energy with a single-shot measurement.

Figure 2: Experimental setup. Illustration of the optical measurement system.
OL: objective lens. BS: beamsplitter. LP: linear polarizer. HWP: half-wave
plate. L1, L2, L3: lens system. FM: flippable mirror. BF: band-pass filter
($610\text{nm}\pm 10\text{nm}$). Inset: SEM image of the sample’s surface.

Four linear polarizations are defined: horizontal
($\mathrel{\leftrightarrow}$), vertical ($\mathrel{\updownarrow}$), diagonal
($\mathrel{\text{$\nearrow$\hbox to0.0pt{\hss$\swarrow$}}}$), and antidiagonal
($\mathrel{\text{$\nwarrow$\hbox to0.0pt{\hss$\searrow$}}}$). These
polarizations correspond to angles of 0${}^{\text{o}}$, 90${}^{\text{o}}$,
45${}^{\text{o}}$, and -45${}^{\text{o}}$, respectively, between the axes of
the polarizers (LP1, LP2) and the x-axis of the sample. Resonant scattering
experiments employ two configurations: the crossed polarizations of
$\left[\mathrel{\updownarrow},\mathrel{\leftrightarrow}\right]$ and
$\left[\mathrel{\text{$\nwarrow$\hbox
to0.0pt{\hss$\searrow$}}},\mathrel{\text{$\nearrow$\hbox
to0.0pt{\hss$\swarrow$}}}\right]$. In the
$\left[\mathrel{\updownarrow},\mathrel{\leftrightarrow}\right]$ setup, light
polarized in $\mathrel{\updownarrow}$ is incident on the sample, and the
reflected light is analyzed in $\mathrel{\leftrightarrow}$ polarization.
Conversely, in the $\left[\mathrel{\text{$\nwarrow$\hbox
to0.0pt{\hss$\searrow$}}},\mathrel{\text{$\nearrow$\hbox
to0.0pt{\hss$\swarrow$}}}\right]$ configuration, the sample is illuminated
with $\mathrel{\text{$\nwarrow$\hbox to0.0pt{\hss$\searrow$}}}$-polarized
light and the reflection is analyzed in $\mathrel{\text{$\nearrow$\hbox
to0.0pt{\hss$\swarrow$}}}$ polarization.

Figure 3: Angle-resolved reflectivity measurement. a) Experimental and b) FEM
simulation results of regular reflectivity with
$\mathrel{\text{$\nearrow$\hbox to0.0pt{\hss$\swarrow$}}}$-polarized
excitation. c) Experimental results of resonant scattering measurement in the
$\left[\mathrel{\updownarrow},\mathrel{\leftrightarrow}\right]$ configuration.
d) FEM simulation results of absorption with $\mathrel{\text{$\nearrow$\hbox
to0.0pt{\hss$\swarrow$}}}$-polarized excitation. The blue solid line indicates
analytical calculation of the band structure with $\gamma=0$ in (a,b), and
$\gamma=19.9$ meV in (c,d). Each single shot of the measured spectrum is
integrated for 100 ms and the monochromator slit is set at 100 µm. For the
absorption simulation, an imaginary component of 0.005 has been added to the
refractive index of SiN${}_{\text{x}}$.

We first perform a regular energy-momentum-resolved reflectivity experiment.
The sample is oriented so that its x-axis is 45${}^{\text{o}}$ tilted with
respect to the slit of the monochromator, so the collection plane corresponds
to the $\Gamma$M direction. Since the two modes of interest are of s-polarized
light[29, 30], their polarization for this collection plane is purely
$\mathrel{\text{$\nearrow$\hbox to0.0pt{\hss$\swarrow$}}}$. Figure 3a presents
the experimental result of regular reflectivity for
$\mathrel{\text{$\nearrow$\hbox to0.0pt{\hss$\swarrow$}}}$ excitation. The
corresponding numerical result by FEM is presented in Fig.3b. Following the
reflectivity peak, both experimental and numerical results show Dirac cone
degeneracy at $k=0$ instead of branch degeneracy of EPs. Indeed, they are
nicely fitted by the band structure of the Hermitian term of (1), given by
$E_{0}\pm v.k$ at zero-detuning and in the vicinity of $k=0$ (see solid blue
curves in Figs.3a,b). Therefore, regular reflectivity cannot provide a direct
observation of non-Hermitian degeneracy. These findings are corroborated by
the studies of Zhen et al.[29]. In their earlier research, reflectivity
mapping only shows Hermitian degeneracy (i.e. Dirac cones) and EPs only came
to light after tedious analysis of reflectivity spectra and fitting with a
theoretical model.

Next, we undertake resonant scattering experiments in the
$\left[\mathrel{\updownarrow},\mathrel{\leftrightarrow}\right]$ configuration.
This configuration is chosen since it enables the probing of our photonic
modes of pure $\mathrel{\text{$\nearrow$\hbox to0.0pt{\hss$\swarrow$}}}$
polarization (i.e., $\mathrel{\text{$\nearrow$\hbox to0.0pt{\hss$\swarrow$}}}$
has a non-zero projection in both $\mathrel{\updownarrow}$ and
$\mathrel{\leftrightarrow}$). As mentioned above, the crossed polarization
setup suppresses specular signals and Fano features that hindered the EPs
observation in the reflectivity measurements of Figs.3a,b. Therefore, EPs can
be directly revealed in the mapping of the resonant scattering experiment.
Indeed, Figure3c presents the experimental results of the energy-momentum-
resolved resonant scattering measurement. The real part of eigenmodes from (2)
is plotted over the experimental data. It shows that the non-Hermitian bands
are revealed in this single-shot experiment by simply monitoring the intensity
peaks. In particular, the branch points and momentum gap, hallmarks of EPs,
are clearly evidenced, without using any post-treatment. We note that the
dominant scattering signal at EPs (Figs.3c) is due to the enhancement of local
density of states at these spectral singularities[24, 25].These features are
in agreement with the simulation results of absorption when the structure is
excited with $\mathrel{\text{$\nearrow$\hbox
to0.0pt{\hss$\swarrow$}}}$-polarized light (Fig.3d). It confirms that resonant
scattering techniques can be used to probe EPs in PhC in a similar fashion as
active measurements (i.e., photoluminescence or absorption). The broadening of
experimental data (Figs.3c) with respect to the FEM results (Figs.3d) can be
attributed to an inhomogeneous broadening effect, as discussed in the
following.

Figure 4: Quantitative analysis of EP formation. a) Real and b) imaginary part
of the eigenmodes. Symbols correspond to experimental data extracted from the
fitting of resonant scattering spectra. Solid lines correspond to analytical
calculation with inhomogenous broadening $\gamma_{i}=4$ meV included. The
green color corresponds to the bright mode while the red represents the dark
modes.

To analyze quantitatively the formation of EPs in Fig.3c, we extract the real
and imaginary parts of the eigenvalues from resonant scattering measurements.
This is achieved by fitting the resonances in the spectrum for each
wavevector. It is important to note the presence of an inhomogeneous
broadening in the measured spectra. This broadening is due to spatial
variation of the hole radius, an undesired feature that is typical of laser
interference lithography[46]. To account for this inhomogeneous broadening, we
used Gaussian profiles instead of Lorentzian ones to fit the resonant
scattering spectra. Examples of fitted spectra are shown in Fig. S4 of the SI.
Figures 4a and 4b show the real (determined by the position of the resonance)
and imaginary (extracted from the linewidth of the resonance) parts of the
eigenvalues of the band diagram, respectively. Importantly, a simultaneous
degeneracy of real and imaginary parts is evidenced at $k$ = $\pm$0.13 µm-1,
thus confirming EPs formation at these wavevectors. We note that the extracted
imaginary part, shown in Fig.4b, exhibits an offset of 4 meV compared to that
of the numerical design (Fig.1e). This offset accounts for the inhomogeneous
broadening discussed above. One may include this broadening into the
analytical model by adding an offset $i\gamma_{i}$ to the diagonal
coefficients of the non-Hermitian term in (1), with $\gamma_{i}=4$ meV. Once
this offset is included, the experimental values align closely with the
analytical model (see Figs.4a,b). Further details of the analytical model,
including inhomogeneous broadening, are discussed in Fig. S5 of the SI.

Figure 5: Fourier imaging at the EPs energy. a-d) Experimental results of the
regular reflectivity image for four different polarizations of excitation
light. e,f) Experimental results of the resonant scattering image in two
different polarization configurations. The magnified views close to the
$\Gamma$ point of corresponding Fourier images are shown right below (a-d) and
on their right side (e,f). The black dashed circle indicates the limit of the
field of view, which corresponds to the numerical aperture NA = 0.55 of the
microscope objective. The scale in $k_{x}$ is equivalent to $k_{y}$ for all
the images. The EPs are points indicated by the black arrow in the magnified
picture of (e).

Finally, we perform Fourier imaging in the ($k_{x}$,$k_{y}$) plane at the EPs
energy for different polarization configurations for regular reflectivity
(Figs.5a-d) and resonant scattering (Figs.5e,f) experiments. This is achieved
by adding a bandpass filter ($\lambda$=610 nm or $E=2.0$ eV) in the collection
path to extract only spectral signals in the vicinity of the EPs and imaging
the back focal plane in various polarization configurations. We now focus on
the zoom-in images in the vicinity of the $\Gamma$ point (bottom panels of
Figs.5a-d, and right panels of Figs.5e,f). These results clearly show that
regular reflectivity mappings exhibit non-zero background from specular
reflection, while the background is completely suppressed in resonant
scattering mappings. Moreover, all isofrequency maxima curves in Figs.5a-d go
through the $\Gamma$ point, suggesting a degeneracy at $k=0$. This is
consistent with the results from the energy-momentum measurements of Fig.3a
previously discussed: regular reflectivity only reveals the Hermitian version
of the band structures. On the other hand, four local maxima corresponding to
four EPs along $\Gamma$M directions ($\pm 45^{\text{o}}$ planes) are
prominently observed in the resonant scattering mapping of the
$[\mathrel{\updownarrow},\mathrel{\leftrightarrow}]$ configuration (Fig.5e).
Such an observation is possible thanks to the local enhancement of the
scattering signal at EPs. We also note that horizontal and vertical
isofrequency segments are evident in the resonant scattering mapping of the
$[\mathrel{\text{$\nwarrow$\hbox
to0.0pt{\hss$\searrow$}}},\mathrel{\text{$\nearrow$\hbox
to0.0pt{\hss$\swarrow$}}}]$ configuration (Fig.5f). These segments correspond
to the slow-light upper-band along the $\Gamma$X direction (more details of
the band structure along $\Gamma$X are provided in Fig.S2 of the SI).

In conclusion, we have introduced a new approach for directly observing EPs in
PhC structures of hole lattice. By leveraging the crossed polarization
technique in momentum-resolved reflectivity, we effectively circumvented the
challenges that predominantly stem from the complex interplay between guided
resonances and background signals. Our experimental results of momentum-
resolved resonant scattering and Fourier images validate the formation of EPs
in a patterned SiN${}_{\text{x}}$ slab, perfectly aligning with both
theoretical predictions and simulations. This streamlined technique, adaptable
across various PhC stacks and geometries, promises a powerful asset for diving
deeper into non-Hermitian photonics and topology, particularly expanding the
horizon of discoveries and applications in advanced photonic platforms.

###### Acknowledgements.

The authors acknowledge the financial support of VinUniversity’s seed grant
under Grant No. VUNI.2021.SG10, VinUniversity-UIUC Smarthealth Center under
Grant No. 8647, and Vingroup Innovation Foundation under Grant
VINIF.2021.DA00169.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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— SUPPLEMENTAL INFORMATION —

##  I Sample fabrication

The designed structure is fabricated by depositing a 158 nm-thick layer of
SiNx directly onto a 2 $\mu\text{m}$-thick ${\text{SiO}}_{\text{2}}$ layer on
a Si substrate using the Plasma Enhanced Chemical Vapor Deposition technique.
We employ laser interference lithography to pattern the negative resist film
(maN-2043). Post-development, the structure is transferred to SiNx via
Reactive Ion Etching using the Corial 200FA system, leveraging a gas mixture
of CH4, SF6, and CHF3. This process lasts for nine minutes under a pressure of
100 mTorr at 20${}^{\text{o}}$C. Subsequently, the partly etched
${\text{SiN}}_{\text{x}}$ final structure is attained by removing the resist
using oxygen plasma. A scanning electron microscope (SEM) image of the
fabricated sample is shown in Fig. S1

Figure S1: Scanning Electron Microscopy image of the fabricated sample. A crop
version of this image is shown in the inset of Fig.2 in the main text.

##  II Photonic band structure

Figure S2 represents the results of the real (Fig. S2a) and imaginary (Fig.
S2b) part of the photonic modes along both $\Gamma$X and $\Gamma$M directions.
These results are obtained via FEM simulations for s-polarized modes (i.e.
transverse electric modes, corresponding to $\mathrel{\leftrightarrow}$ for
$\Gamma$X direction, and $\mathrel{\text{$\nearrow$\hbox
to0.0pt{\hss$\swarrow$}}}$ for $\Gamma$M direction ). It shows that the EPs
(degeneracy of both real and imaginary part of eigenvalues at the same point)
occur only in $\Gamma$M direction. Interestingly, the band structure along
$\Gamma$M is symmetric with respect to $E_{EP}$ for the real part and
$\gamma_{EP}$ for the imaginary part. The energy-momentum dispersion along
$\Gamma$M exhibits linear dispersion of opposite group velocities as soon as
$|k|$>$|k_{EP}$. On the other hand, the band structure along $\Gamma$X does
not possess the "particle-hole"-like symmetry as the one along $\Gamma$M
direction. In particular, the group velocity and the line-width of the upper-
band decrease rapidly when moving out of the $\Gamma$ point along $\Gamma$X
direction. Therefore, one may consider the upper-band as a slow-light mode
with low-loss. The isofrequency segments along $\Gamma$X directions observed
in Fig.5f of the main text correspond to this low-dispersive mode.

Figure S2: Band diagram of the structure. a) and b) are real and imaginary
parts of bright mode (green) and dark mode (red). The real and imaginary parts
are coalesced in $\Gamma$M direction through a mark of open dash circle.
Figure S3: Angle-resolved reflectivity and absorption spectra for different
hole radius. a,f) $r=0.8r_{\text{EP}}$, b,g) $r=0.9r_{\text{EP}}$, c,h)
$r=r_{\text{EP}}$, d,i) $r=1.1r_{\text{EP}}$ and e,j) $r=1.2r_{\text{EP}}$.The
upper panel shows regular reflectivity simulations and the lower panel shows
angle-resolved absorption spectra simulations.

##  III Angle-resolved absorption simulation with different hole radius

As discussed in the main text, the detuning between the dark and bright mode
can be finely tuned by sweeping the hole radius $r$. EPs configuration
corresponds to $r=r_{EP}=160$ nm. This effect is illustrated in Fig. S3,
showing energy-momentum mappings of the regular reflectivity (Fig. S3a-e) and
absorption (Fig. S3f-j) along $\Gamma$M direction with
$\mathrel{\text{$\nearrow$\hbox to0.0pt{\hss$\swarrow$}}}$-polarized
excitation for five different designs: $r=0.8r_{EP}$ (Fig. S3a,f),
$r=0.9r_{EP}$ (Fig. S3b,g), $r=r_{EP}$ (Fig. S3c,h), $r=1.1r_{EP}$ (Fig.
S3d,i), and $r=1.2r_{EP}$ (Fig. S3e,j). These results are obtained by FEM
simulations. The band inversion effect is clearly evidenced: the dark mode is
in lower-band when $r<r_{EP}$, coalesces with the bright mode to make EPs when
$r=<r_{EP}$, then is in the upper-band when $r>r_{EP}$. Interestingly, these
results show that while the EPs cannot be revealed by regular reflectivity
(Fig. S3c) and can only be observed in absorption (Fig. S3h), we can
distinguish the dark and bright modes when the are not coalesce in both
regular reflectivity and absorption spectra.

##  IV Gaussian fittings for experimental resonant scattering spectra

Spatial variation of patterned features is a well-known undesired effect of
laser inteferental lithography[46]. Such a slow variation leads to an
inhomogenous broadening of photonic resonances. Thus Gaussian profiles are
more adapted to fit the resonant scattering spectra instead of Lorentzian
ones. Figure S4 present resonant-scattering spectra at three different
wavectors. It shows a very good aggrement between the experimental data and
Gaussian fittings.

Figure S4: Gaussian Fitting of resonant scattering spectrum. Data measurement
(black) and Gaussian fitting (red) for a) $k=$ -0.12 µm-1, b) $k=$ -0.28 µm-1,
c) $k=$ -0.34 µm-1 and d) $k=$ -0.46 µm-1. e) The position of these slices on
the resonant scattering spectrum.

##  V Inhomogeneous broadening of the photonic modes

The inhomogeneous broadening can be included in the non-Hermitian effective
Hamiltonian by adding an offset loss $\gamma_{i}$ in the anti-Hermitian term:

$H(k)=\begin{pmatrix}{E_{0}+\frac{\delta+\sigma k^{2}}{2}}&{v.k}\\\
{v.k}&{E_{0}-\frac{\delta+\sigma
k^{2}}{2}}\end{pmatrix}+\begin{pmatrix}i(\gamma+\gamma_{i})&0\\\
0&i\gamma_{i}\end{pmatrix}$ (S1)

The corresponding complex eigenvalues are given by:

$E_{\pm}(k)=E_{0}+i\frac{\gamma}{2}+i\gamma_{i}\pm\sqrt{\left(\frac{\delta+\sigma
k^{2}+i\gamma}{2}\right)^{2}+v^{2}k^{2}}$ (S2)

In the main text, the FEM simulations (Figs. 1d,e) and the experimental data
(Figs. 4a,b) of the real and imaginary parts are all fitted by (S2) with the
same set of parameters: $E_{0}=2.0$ eV, $\delta$ = 0.59 meV, $\gamma$ = 19.9
meV, $v=76.6$ meV.µm and $\sigma=-35$ meV.µm-2. The only difference is that we
use $\gamma_{i}=0$ to fit FEM results of ideal structure since there is no
inhomogeneous broadening. However for experimental data, a non-zero value of
$\gamma_{i}=4$ meV is necessary to reproduce the observed broadening offset.
The comparison between the two fittings, as well as the FEM and experimental
results, are reported together in Fig. S5.

Figure S5: Inhomogeneous broadening of the photonic modes. Comparison between
imaginary parts obtained by analytical model, FEM simulation, and experimental
results. Green and red bands are the results of the experimental results and
analytical model with $\gamma_{i}$ = 4 meV. Pale red and pale green bands are
FEM simulation results and analytical model with $\gamma_{i}$ = 0 meV. Solid
lines correspond to analytical models and symbols correspond to experimental
and simulation results.