# Observation of Berry curvature in non-Hermitian system from far-field
radiation

Xuefan Yin1,2† Ye Chen2† Xiaoyu Zhang2 Zixuan Zhang2 Susumu Noda1 Chao
Peng2,3∗

###### Abstract

Berry curvature that describes local geometrical properties of energy bands
can elucidate many fascinating phenomena in solid-state, photonic, and
phononic systems, given its connection to global topological invariants such
as the Chern number. Despite its significance, the observation of Berry
curvature poses a substantial challenging since wavefunctions are deeply
embedded within the system. Here, we theoretically propose a correspondence
between the geometry of far-field radiation and the underneath band topology
of non-Hermitian systems, thus providing a general method to fully capture the
Berry curvature without strongly disturbing the eigenstates. We further
experimentally observe the Berry curvature in a honeycomb photonic crystal
slab from polarimetry measurements and quantitatively obtain the non-trivial
valley Chern number. Our work reveals the feasibility of retrieving the bulk
band topology from escaping photons and paves the way to exploring intriguing
topological landscapes in non-Hermitian systems.

  1. 1.

Department of Electronic Science and Engineering, Kyoto University, Kyoto-
Daigaku-Katsura, Nishikyo-ku, Kyoto 615-8510, Japan

  2. 2.

State Key Laboratory of Advanced Optical Communication Systems and Networks,
School of Electronics, $\&$ Frontiers Science Center for Nano-optoelectronics,
Peking University, Beijing, 100871, China

  3. 3.

Peng Cheng Laboratory, Shenzhen 518055, China

†These authors contributed equally to this work
∗To whom correspondence should be addressed; E-mail<EMAIL_ADDRESS>
Topology, namely the mathematics of conserved properties under continuous
deformations, is creating a range of new opportunities throughout matters,
photonics, phononics, and other wave systems [?,?,?,?,?,?]. To characterize
the topology in physics, the Berry curvature [?,?] is an essential concept
that describes the gauge-invariant, local, geometric manifestation of the
wavefunctions in the parameter space, which is closely related to the global
topological invariants such as a variety of Chern numbers [?,?,?,?,?].
However, since Berry curvature belongs to the intrinsic topological property
of wavefunctions, it is usually deeply bound inside the system and difficult
to observe. Although using tomography can reconstruct the wavefunctions in
some particular scenarios [?,?], much effort has been devoted to retrieving
the Berry curvature from its external consequence in physics. Examples
includes Hall drift in driving optical lattice [?,?,?] or synthetic gauge
field [?,?], Aharonov-Bohm interference of magnetic-controlled ultracold atom
[?,?,?,?], and pseudospin [?,?,?,?,?,?] or dichroism [?,?] in exciton-
polariton-correlated material. Even though the specific physics varies, the
mentioned observations of Berry curvature generally rely on the strong light-
matter interaction to imprint the topological features of bulk wavefunction to
external observables, and thus they should be categorized into the class of
“strong measurement” [?] that the observation strongly interferes with the
system. In comparison, the method of measuring the Berry curvature without
much disturbing the eigenstates [?] remains absent. Recall the fact that non-
Hermitian photonic systems [?,?,?,?,?] necessarily lose photons. The escaping
photons, namely the far-field radiation, naturally carry information about the
wavefunctions, thus allowing direct access to the intrinsic bulk topology that
would be conventionally thought impossible. The escaping photons simply act as
the “messengers” that weakly interact with the system, but they could bridge
the band topology and radiation topology to enable direct observation of Berry
curvature from the far field. In recent years, radiation geometry
[?,?,?,?,?,?,?] that concerns the non-trivial geometric structures of far-
field polarization has attracted much attention because they can give rise to
interesting physical consequences such as polarization half-charge around
paired exceptional point [?,?], vortex beam [?], chiral devices with circular
dichroism [?,?], bound states in the continuum (BICs) [?,?,?,?,?,?,?,?,?] and
unidirectional guided resonances (UGRs) [?,?,?]. However, whether the
geometric features in the radiation originated from the band topology and how
to retrieve the Berry curvature from the far field radiation still remain as
elusive questions.

Here we theoretically establish a correspondence between the band topology and
radiation geometry to experimentally observe the Berry curvature by
characterizing the escaping photons from a non-Hermitian photonic crystal
(PhC) slab system. Specifically, we prove that a full tomography of the Berry
curvature can be realized by simply measuring a number of radiation channels,
while for a two-level system, only one radiation channel is sufficient.
Accordingly, we experimentally observe the nontrivial Berry curvatures
originated from diabolic points (DPs) in a honeycomb latticed PhC slab by
using a polarimetry measurement [?]. The Berry phases [?,?,?] of
$\gamma\sim\pm\pi$ are obtained in an individual valley by employing integrals
of the observed Berry curvature, showing nontrivial valley Chern numbers [?,?]
of $C_{v}^{\mathcal{K}}\sim\pm 1/2$ as expected, and thus quantitatively
validates our method. The theory and measurement also clarify that the band
topology only manifests on the “left-right” curvature [?] upon a bi-orthogonal
basis of a non-Hermitian system, while a “right-right” curvature [?]
represents the geometry of radiation itself and connects to Pancharatnam-Berry
(PB) phase [?,?,?,?] of far-field polarization.

The bulk-radiation correspondence of Berry curvature — We start from a PhC
slab operating in the radiation continuum as schematically illustrated in Fig.
1A, in which the $n$th photonic eigenstate $|\psi_{n}\rangle$ of a Hamiltonian
$\hat{H}$ resides in the continuum. The eigenstate would radiate towards some
specific directions owing to the diffraction of periodically modulated
permittivity, and each gives rise to a radiation vector of $|\Psi_{n}\rangle$
in the far field. In a given diffraction direction, the radiation vector
$|\Psi_{n}\rangle$ can be described by the polarization vector field of
$|\Psi_{n}\rangle=[c_{x;n},~{}c_{y;n}]^{T}$, where $c_{x,y;n}$ are complex-
valued electric-field components in $x$ and $y$ directions, respectively.
Components in $s$ and $p$ directions are discussed in Supplementary materials
[?]. Consequently, the radiation process can be understood as a linear mapping
of
$\mathcal{P}:|\psi_{n}\rangle\mapsto|\Psi_{n}\rangle=\hat{P}|\psi_{n}\rangle$,
showing a direct connection between the bulk wavefunction and its radiation
far-field, governed by a projection matrix denoted as $\hat{P}$.

It is well known that the Berry curvature of bulk bands can be calculated from
wavefunctions as $B_{n}=i\nabla\times\langle\phi_{n}|\nabla\psi_{n}\rangle$
(bottom panel, Fig. 1B), with $|\phi_{n}\rangle$ representing the left vector
of $|\psi_{n}\rangle$. Since our system is intrinsically non-Hermitian due to
the existence of radiation losses, $|\psi_{n}\rangle$ and $|\phi_{n}\rangle$
should be bi-orthogonal to each other. If $\hat{P}$ is an invertible matrix,
we found that $|\Psi_{n}\rangle$ and the accompanied left vector
$|\Phi_{n}\rangle=(\hat{P}^{-1})^{\dagger}|\phi_{n}\rangle$ also form a bi-
orthogonal basis for the non-Hermitian Hamiltonian of
$\hat{H}^{r}=\hat{P}\hat{H}\hat{P}^{-1}$, leading to another Berry curvature
of $B_{n}^{r}=i\nabla\times\langle\Phi_{n}|\nabla\Psi_{n}\rangle$ defined in
the radiation field (top panel, Fig. 1B). We further prove that [?], in the
case that the matrix $\hat{P}$ is smooth ($\nabla\hat{P}\sim 0$) and doesn’t
give rise to extra vortexes such as BICs that carry vanished amplitudes
$\langle\Psi_{n}|\Psi_{n}\rangle=0$, the system follows a simple
correspondence between the band topology and radiation topology, given by:

$\displaystyle B_{n}^{r}\approx B_{n}$ (1)

The above equation reveals that the escaping photons act as “messengers” that
project the bulk Berry curvature onto the far field through the matrix
$\hat{P}$ (mid panel, Fig. 1B). Although the wavefunctions $|\psi_{n}\rangle$
and $|\phi_{n}\rangle$ are difficult to access because they belong to the
near-field features of the eigenstates, the radiation vectors
$|\Psi_{n}\rangle$ and $|\Phi_{n}\rangle$ are directly observable and can be
characterized by standard optical measurement.

In theory, the radiation in a particular diffraction direction gives a
perspective projection of the wavefunctions, and thus it is noteworthy to
discuss whether the projection is complete. For a general system that has $N$
internal degrees of freedom (DOFs), $\hat{P}$ is in the form of $2\times N$
matrix. Since one diffraction direction can only characterize two DOFs
($[c_{x;n},c_{y;n}]^{T}$), we probably need to measure multiple radiation
channels simultaneously — observe the same object from different views, to
fully capture the information about the bulk wavefunctions [?]. As a specific
case, if the system is a simple two-level one with $N=2$, the measurement of
only one radiation channel can provide sufficient DOFs to make the projection
of wavefunctions complete. Namely, we can reverse the projection process to
directly determine the bulk wavefunctions from far-field radiations if
$\hat{P}$ is a non-singular $2\times 2$ matrix.

To elaborate the correspondence, we consider a two-dimensional (2D) PhC slab
of Si3N4 material with circular air hole patterns on a honeycomb lattice (Fig.
2A). The lattice constant and slab thickness are denoted as $a$ and $h$,
respectively, which give a reciprocal lattice as shown in Fig. 2B. The grey
shading area denotes the first Brillouin zone (BZ). According to the Bloch
theorem, the bulk wavefunctions can be depicted by a superposition of a series
of quasi-plane-waves with discrete momentum, represented by the dots in the
reciprocal lattice and we refer them as “diffraction orders” [?]. Around the
second $\mathcal{K}$ point that resides in the continuum, several diffraction
orders fall into the light cone and thus open radiation channels. We take the
$\mathcal{K}_{1}$ point of $(-4\pi/3a,0)$ as a specific example, and there
exist three radiation channels $C_{1-3}$ that cause the non-Hermiticity (red
arrows, Fig. 2B). Accordingly, their in-plane momentum at $\mathcal{K}_{1}$
point are $\beta_{1}=(\sqrt{3}\beta_{0}/3\hat{x},0)$,
$\beta_{2}=(-\sqrt{3}\beta_{0}/6\hat{x},-\beta_{0}/2\hat{y})$ and
$\beta_{3}=(-\sqrt{3}\beta_{0}/6\hat{x},\beta_{0}/2\hat{y})$, respectively,
with denoting $\beta_{0}=4\pi/\sqrt{3}a$. Three transverse-electric (TE)
polarized modes can be found around the $\mathcal{K}_{1}$ point, and we mark
them as TEA,B,C. We assume the adjacent air holes have different radii of
$r_{1}$ and $r_{2}$. For $\delta_{r}=r_{2}-r_{1}=0$ that preserves the
$C_{6v}$ symmetry (left panel, Fig. 2C), it would result in a two-fold
degeneracy of TEA and TEB right at the $\mathcal{K}_{1}$ point. As a contrary,
if $r_{1}\neq r_{2}$ (right panel, Fig. 2D), the $C_{6v}$ symmetry degrades
into the $C_{3v}$ symmetry that lifts the degeneracy. In the case that the in-
plane symmetry breaking is sufficiently weak that allows the coupling from the
TEC mode to be neglectable, the TEA and TEB modes form a two-level system near
the $\mathcal{K}_{1}$ point that is described by the Hamiltonian of:

$\displaystyle\hat{H}=\omega+\delta\sigma_{z}+\eta k_{x}\sigma_{x}+\eta
k_{y}\sigma$ (2)

in which $\omega$ is the degenerate frequency at $\mathcal{K}_{1}$ point;
$\delta$ is related to the in-plane asymmetry and $\eta$ is the group
velocity; $k_{x}$ and $k_{y}$ are dimensionless numbers to describe the
momentum deviation from the $\mathcal{K}_{1}$ point of
$\mathbf{k}=k_{x}\beta_{0}\hat{x}+k_{y}\beta_{0}\hat{y}$. Here we only
consider the radiation loss but omit the material dissipation or gain, so the
non-Hermiticity of $\hat{H}$ is represented by the complex degenerate
frequency $\omega=\omega_{r}+i\gamma_{0}$, where $\gamma_{0}$ is the radiation
decay rate. When $C_{6v}$ symmetry is preserved ($\delta=0$), the eigenvectors
can be derived as $|\psi_{n}\rangle=[1,\pm|\eta|e^{i\theta}/\eta]^{T}$ where
$e^{i\theta}=(k_{x}+ik_{y})/|\mathbf{k}|$, creating a diabolic point at the
$\mathcal{K}_{1}$ point which is exactly a non-Hermitian counterpart of the
Dirac point in Hermitian case, and we still denote it as the DP for short
without any confusion.

At the DP, the far-field polarizations of TEA,B bands are ill-defined since
they can be mixed in arbitrary weights. Correspondingly, polarization vortexes
can be found in the momentum space that each carries a half-integer
topological charge (Fig. 2E), showing as a geometric feature of the DP in the
far-field radiation. Once the symmetry-breaking lifts the degeneracy, the
polarization vortexes degrade to a meron and anti-meron configuration [?] with
circular-polarization (CPs) in opposite helicities around the
$\mathcal{K}_{1}$ point (Fig. 2F) for TEA,B modes, respectively, which can be
an observable signature to validate the theory. By taking the degeneracy
lifting into account, the close form of theoretical Berry curvature in such a
two-level system follows:

$\displaystyle
B_{n;t}=\pm\frac{4\delta\eta^{2}}{(4\delta^{2}+4\eta^{2}|\mathbf{k}|^{2})^{3/2}}$
(3)

where the subscript “$t$” distinguishes $B_{n;t}$ from the $B_{n}$ while the
later one takes TEC into account; the signs “$\pm$” denote the two bands
$n=A,B$, respectively. Since TEC mode can be neglected in our case, we have
$B_{n}\approx B_{n;t}$. For a $C_{6}$ symmetric system ($\delta=0$), the Berry
curvature shows as a $\delta$-function peaked at the $\mathcal{K}_{1}$ point.
While for $\delta\neq 0$, the DP splits and opens a topologically nontrivial
bandgap, leading to a nontrivial Berry curvature in the vicinity of
$\mathcal{K}_{1}$ point that gives rise to non-zero valley Chern number. As we
stated, such a nontrivial Berry curvature can be directly observed from the
far-field radiation by employing the bulk-radiation correspondence of Eq. 1.
Specifically, the radiation vector $|\Psi_{n}\rangle$ exactly corresponds to
the far-field polarization:
$\vec{S}_{n}(k_{x},k_{y})=[s_{1},s_{2},s_{3}]^{T}/s_{0}=\langle\Psi_{n}|\hat{\mathbf{\sigma}}|\Psi_{n}\rangle/s_{0}$,
where
$\hat{\mathbf{\sigma}}=[\hat{\sigma}_{z},\hat{\sigma}_{x},\hat{\sigma}_{y}]^{T}$
are the Pauli’s matrices and $\vec{S}_{n}$ refers to the Stokes’ vector in
Poincare sphere which can be measured by using standard polarimetry method.
Besides, the left radiation vector $|\Phi_{n}\rangle$ of $|\Psi_{n}\rangle$
can be determined from the bi-orthogonal normalization relation of
$\langle\Phi_{m}|\Psi_{n}\rangle=\delta_{mn}$. As a result, we can observe the
intrinsic band Berry curvature $B_{n;t}$ directly from measuring $B_{n}^{r}$.

Experimental observation of Berry curvature — To experimentally observe the
Berry curvature, we first fabricate the PhC sample by using e-beam lithography
(EBL) and inductively coupled plasma etching (ICP) processes on a Si3N4 slab
of thickness $h=180$ nm on silica substrate (see Methods section for details).
The air holes are arranged as a honeycomb lattice of $a=440$ nm with two
slightly different hole radii of $r_{1}=50$ and $r_{2}=54$ nm, respectively,
as the scanning electron microscope (SEM) images shown in Fig. 2A. The angle-
resolved measurement system is schematically illustrated in Fig. 3A, in which
a supercontinuum white light source is first sent through an acoustic-optic
tunable filter (AOTF) and then linearly polarized by POL1 to generate
incoherent light in a wavelength range from $550$ nm to $580$ nm. After
passing through a quarter-wave plate (QWP1), the light is focused by a lens
(L1) onto the rear focal plane (RFP) of an infinity-corrected objective lens
($NA=0.95$), and then illuminates the sample to excite the optical modes. POL1
and QWP1 are used for adjusting the incident polarization for better
excitation. The radiations from the PhC sample are collected by the same
objective lens and imaged by a charge-coupled device (CCD) camera which is co-
focused with the RFP of the objective lens. By inserting another polarizer
(POL2) and another quarter-wave plate (QWP2) before the CCD, we can fully
characterize the Stokes’ vector of radiation through a polarimetry method.

As shown in Fig. 3B, we found three scattered beams in the aperture of the
objective lens, which correspond to the radiation channels $C_{1-3}$ plotted
in Fig. 2B. To achieve the best excitation, we fine-tune the incident angle by
moving L1 lens in the $x-y$ plane to illuminate the PhC sample through the
channel $C_{1}$. According to Fig. 2F, POL1 and QWP1 adjust the incident
polarization to be left-handed circular polarized (LCP) to excite TEA mode and
right-handed circular polarized (RCP) to excite TEB mode, respectively. When
the AOTF selects a specific wavelength, the scatterings originating from
fabrication disorders would make the iso-frequency contour at such a
wavelength visible upon every radiation channel due to the on-resonance
pumping mechanism [?,?,?] (Fig. 3C). Accordingly, we observe the $C_{3}$
channel which is far from the direct reflected light and apply a cascaded $4f$
system to zoom-in the far-field pattern at a magnification rate of $\times 6$.
Further, by recording the CCD images at particular arrangements of POL2 and
QWP2 as polarimetry measurements, we can decompose the radiation into Stokes’
vector. The result at the wavelength of $\lambda_{i}=570.328$ nm is presented
in Fig. 3D, in which the dashed line is the iso-frequency contour calculated
from numerical simulation for visual guidance. By overlapping all the iso-
frequency contours in the wavelength range of $559.383\sim 564.861$ nm for the
TEA band and $565.203\sim 571.352$ nm for the TEB band, we obtain the
polarization vector field in the momentum space for both TEA,B bands (Fig.
3E). In the vicinity of the $\mathcal{K}_{1}$ point, a LCP and a RCP can be
found on the TEA and TEB bands, respectively (red circles, Fig. 3E), which
agree well with our prediction in theory as shown in Fig. 2F.

Berry curvature $B^{r}_{n}$ defined in far-field radiation can be directly
obtained from the full polarization vector field. Here we consider four
samples with radius differences of $\delta_{r}=0,~{}~{}4,~{}~{}7,~{}~{}10$ nm,
and measure the $B_{A,B}^{r}$ of each sample, respectively (top panels, Fig.
4). Accordingly, we calculate the numerical Berry curvature $B_{A,B}$ by
employing the semi-analytical coupled-wave theory framework (CWT) [?,?,?] for
a comparison (bottom panels, Fig.4). To better show the evolution of Berry
curvatures, we plot the unit-cell geometry of each sample and the
corresponding band structures of TEA,B as the insets in the top and bottom
panels of Fig. 4. Specifically, we start from a realistic sample with
$\delta_{r}=0$ (Fig. 4A), where the fabrication imperfections would inevitably
break the DP degeneracy at $\mathcal{K}_{1}$ point and give rise to a very
small bandgap. We estimate that the bandgap is equivalent to the case of
$\delta_{r}=2$ nm. In this case, we found that the $B_{A,B}^{r}$ show as
bright spots centered at $\mathcal{K}_{1}$ point — quite like
$\delta$-functions (top panels, Fig. 4A), agreeing well with the numerical
result $B_{A,B}$ (bottom panels, Fig. 4A). Note that the signs of the
$B_{A}^{r}$ and $B_{B}^{r}$ are exactly opposite, agreeing well with the
theoretical prediction in Eq. 3. Further, we gradually open the bandgap by
increasing $\delta_{r}$ from $4$ nm to $10$ nm (Fig. 4B). During this process,
the Berry curvatures $B_{B}^{r}$ and $B_{B}$ gradually diffuse to a larger
region in the momentum space while their peak absolute values decrease. At
$\delta_{r}=10$ nm, the bandgap becomes quite large, and thus both $B_{n}^{r}$
and $B_{n}$ become fully dispersed and no longer congregate around
$\mathcal{K}_{1}$ point. $B_{A}^{r}$ and $B_{A}$ also match well with each
other and show the similar behavior (see Methods section). The great agreement
between the observed Berry curvatures $B_{A,B}^{r}$ and the numerical Berry
curvature $B_{A,B}$ validates the bulk-radiation correspondence of band
topology we propose in Eq. 2. Noteworthy that Berry curvatures are generally
complex values in a non-Hermitian system. For our PhC slab in which only
radiation contributes to the non-Hermiticity, the imaginary parts of Berry
curvature are quite small compared to their real parts [?].

To quantitatively validate the correspondence of Berry curvature, we calculate
the geometric phases (Berry phases) $\gamma_{n}^{r}$ and $\gamma_{n}$ by
applying 2D integrals over the measured and numerical Berry curvatures of
$B_{n}^{r}$ and $B_{n}$ in Fig. 4, respectively. As a reference, we also
derive a close-form of the theoretical geometric phase $\gamma_{n;t}$ in the
two-level model without the contribution of TEC according to Eq. 3 as

$\displaystyle\gamma_{n;t}=\pm\left(\pi-\frac{2\delta}{\sqrt{4\delta^{2}+4\eta^{2}k_{s}^{2}}}\pi\right)$
(4)

According to the valleytronics [?], the integral upon the individual valley
region can determine the valley-Chern number (blue shading, Fig. 5A).
Considering that the nontrivial Berry curvatures congregate around the
$\mathcal{K}_{1}$ point, here we choose a circular integral region with radius
of $k_{s}=0.03$ for simplicity and thus calculate $\gamma_{n}^{r}$ (circle),
$\gamma_{n}$ (triangle), and $\gamma_{n;t}$ (solid line) shown in Fig. 5B and
C. According to Eq.4, the close-form geometric phases $\gamma_{n;t}$ exactly
equals to $\pm\pi$ at $\delta_{r}=0$ for TEA,B bands, respectively [?],
corresponding to the nontrivial, quantized, valley-Chern number of
$C_{v}^{\mathcal{K}}=\pm 1/2$ in an individual valley [?]. When
$\delta_{r}\neq 0$, the open bandgap (nonzero $\delta$) would make the
geometric phases deviate from $\pm\pi$, unless the integral region $k_{s}$
tends to be infinite [?]. Such a behavior can be verified by our experimental
observation. Specifically, the three geometric phases $\gamma_{n}^{r}$,
$\gamma_{n}$ and $\gamma_{n;t}$ of both TEA,B bands agree well with each
other, quantitatively confirming the validity of bulk-radiation correspondence
we propose in Eq. 2. We also found
$\gamma_{A,B}^{r}\approx\gamma_{A,B}\approx\gamma_{A,B;t}\approx\pm 0.6\pi$ at
$\delta_{r}=10$ nm, clearly showing the impact of nonzero bandgap. Besides,
when $\delta_{r}$ is considerable large, we notice that the theoretical phase
$\gamma_{n;t}$ from the two-level model slightly deviates from the numerical
phase $\gamma_{n}$. This is because the influence of the TEC band is neglected
in the derivation of $\gamma_{n;t}$. The discussion on measuring Berry
curvature in a system with higher DOFs ($N>2$) is presented in Supplementary
materials [?].

In the experiment stated above, we obtain the left vector $|\Phi_{n}\rangle$
by using the bi-orthogonal normalization relation of
$\langle\Phi_{n}|\Psi_{n}\rangle=1$. However, the left vector can also be
directly measured. According to the reciprocity, the left vector at the
$\mathbf{k}$ point corresponds to the right vector at the $-\mathbf{k}$ point,
namely $|\Phi_{n}(\mathbf{k})\rangle=|\Psi^{*}_{n}(-\mathbf{k})\rangle$ [?].
Therefore, we can also observe one band at both the $\mathbf{k}$ and
$-\mathbf{k}$ points to retrieve the system’s topology, instead of measuring
two bands simultaneously. We also emphasize that the radiation Berry curvature
$B_{n}^{r}$ directly corresponds to the bulk topology of $B_{n}$ only when the
projection matrix $\hat{P}$ doesn’t cause extra geometric phases, which is the
case of our experiment ($\nabla\hat{P}\approx 0$ around the $\mathcal{K}$
point).

We note that other than the aforementioned “left-right” curvature $B_{n}^{r}$
which depicts the bulk band topology, we can also define a “right-right”
curvature of $B_{n;rr}^{r}=i\nabla\times\langle\Psi_{n}|\nabla\Psi_{n}\rangle$
to capture the geometric features of the polarization field itself. The
integral over “right-right” curvature directly presents the PB phase of far-
field polarization, showing the swirling structure of Stokes’ vector
$\vec{S}_{n}$. For instance, as shown in Fig. 3E, we can find meron and anti-
meron features that are originated from the DP. Such nontrivial polarization
features don’t directly correspond to the valley-Chern number defined on bulk
Berry curvature (“left-right” curvature). Instead, they are related to the
Skyrmion number [?,?,?] that is associated with the PB phase and “right-right”
curvature (See Methods section for details).

Conclusion — In summary, our findings of the “bulk-radiation correspondence”
of Berry curvature reveal the feasibility of retrieving the band topology from
characterizing escaping photons in far-field radiation. We prove in theory and
demonstrate in experiments that characterizing the polarizations of one
radiation channel can capture a complete map of the eigenstates in a two-level
non-Hermitian system, to directly access the Berry curvature and Chern number
without strongly disturbing the system. The proposed method can also be
extended to multi-level systems by measuring more radiation channels
simultaneously [?], and utilized to extract other topological features such as
quantum geometric tensor [?,?,?,?,?,?]. Our work demonstrates a simple and
effective way of directly observing the Berry curvature in non-Hermitian
systems and thus could shed light on the exploration of the intriguing phases
in topological systems.

## References

  * 1. Hasan, M. Z. & Kane, C. L.  Colloquium: topological insulators.  _Rev. Mod. Phys._ 82, 3045  (2010).
  * 2. Xiao, D., Chang, M.-C. & Niu, Q.  Berry phase effects on electronic properties.  _Rev. Mod. Phys._ 82, 1959  (2010).
  * 3. Lu, L., Joannopoulos, J. D. & Soljačić, M.  Topological photonics.  _Nature Photon._ 8, 821–829  (2014).
  * 4. Khanikaev, A. B. & Shvets, G.  Two-dimensional topological photonics.  _Nature Photon._ 11, 763–773  (2017).
  * 5. Ozawa, T. _et al._ Topological photonics.  _Rev. Mod. Phys._ 91, 015006  (2019).
  * 6. Bergholtz, E. J., Budich, J. C. & Kunst, F. K.  Exceptional topology of non-hermitian systems.  _Rev. Mod. Phys._ 93, 015005  (2021).
  * 7. Berry, M. V.  Quantal phase factors accompanying adiabatic changes.  _Proc. R. Soc. A_ 392, 45–57  (1984).
  * 8. Haldane, F. D. M.  Berry curvature on the fermi surface: Anomalous hall effect as a topological fermi-liquid property.  _Phys. Rev. Lett._ 93, 206602  (2004).
  * 9. Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M.  Quantized hall conductance in a two-dimensional periodic potential.  _Phys. Rev. Lett._ 49, 405  (1982).
  * 10. Sheng, D., Weng, Z., Sheng, L. & Haldane, F.  Quantum spin-hall effect and topologically invariant chern numbers.  _Phys. Rev. Lett._ 97, 036808  (2006).
  * 11. Xiao, D., Yao, W. & Niu, Q.  Valley-contrasting physics in graphene: magnetic moment and topological transport.  _Phys. Rev. Lett._ 99, 236809  (2007).
  * 12. Zhang, F., Jung, J., Fiete, G. A., Niu, Q. & MacDonald, A. H.  Spontaneous quantum hall states in chirally stacked few-layer graphene systems.  _Phys. Rev. Lett._ 106, 156801  (2011).
  * 13. Zhang, F., MacDonald, A. H. & Mele, E. J.  Valley chern numbers and boundary modes in gapped bilayer graphene.  _Proc. Natl. Acad. Sci. U.S.A._ 110, 10546–10551  (2013).
  * 14. Hauke, P., Lewenstein, M. & Eckardt, A.  Tomography of band insulators from quench dynamics.  _Phys. Rev. Lett._ 113, 045303  (2014).
  * 15. Fläschner, N. _et al._ Experimental reconstruction of the berry curvature in a floquet bloch band.  _Science_ 352, 1091–1094  (2016).
  * 16. Price, H. M. & Cooper, N.  Mapping the berry curvature from semiclassical dynamics in optical lattices.  _Phys. Rev. A_ 85, 033620  (2012).
  * 17. Jotzu, G. _et al._ Experimental realization of the topological haldane model with ultracold fermions.  _Nature_ 515, 237–240  (2014).
  * 18. Aidelsburger, M. _et al._ Measuring the chern number of hofstadter bands with ultracold bosonic atoms.  _Nat. Phys._ 11, 162–166  (2015).
  * 19. Ozawa, T. & Carusotto, I.  Anomalous and quantum hall effects in lossy photonic lattices.  _Phys. Rev. Lett._ 112, 133902  (2014).
  * 20. Wimmer, M., Price, H. M., Carusotto, I. & Peschel, U.  Experimental measurement of the berry curvature from anomalous transport.  _Nat. Physics_ 13, 545–550  (2017).
  * 21. Abanin, D. A., Kitagawa, T., Bloch, I. & Demler, E.  Interferometric approach to measuring band topology in 2d optical lattices.  _Phys. Rev. Lett._ 110, 165304  (2013).
  * 22. Atala, M. _et al._ Direct measurement of the zak phase in topological bloch bands.  _Nature Physics_ 9, 795–800  (2013).
  * 23. Duca, L. _et al._ An aharonov-bohm interferometer for determining bloch band topology.  _Science_ 347, 288–292  (2015).
  * 24. Li, T. _et al._ Bloch state tomography using wilson lines.  _Science_ 352, 1094–1097  (2016).
  * 25. Bleu, O., Solnyshkov, D. & Malpuech, G.  Measuring the quantum geometric tensor in two-dimensional photonic and exciton-polariton systems.  _Phys. Rev. B_ 97, 195422  (2018).
  * 26. Gianfrate, A. _et al._ Measurement of the quantum geometric tensor and of the anomalous hall drift.  _Nature_ 578, 381–385  (2020).
  * 27. Ren, J. _et al._ Nontrivial band geometry in an optically active system.  _Nat. commun._ 12, 689  (2021).
  * 28. Liao, Q. _et al._ Experimental measurement of the divergent quantum metric of an exceptional point.  _Phys. Rev. Lett._ 127, 107402  (2021).
  * 29. Polimeno, L. _et al._ Tuning of the berry curvature in 2d perovskite polaritons.  _Nat. Nanotech._ 16, 1349–1354  (2021).
  * 30. Łempicka-Mirek, K. _et al._ Electrically tunable berry curvature and strong light-matter coupling in liquid crystal microcavities with 2d perovskite.  _Sci. Adv._ 8, eabq7533  (2022).
  * 31. Wu, S. _et al._ Electrical tuning of valley magnetic moment through symmetry control in bilayer mos2.  _Nat. Phys._ 9, 149–153  (2013).
  * 32. Cho, S. _et al._ Experimental observation of hidden berry curvature in inversion-symmetric bulk 2 h- wse 2.  _Phys. Rev. Lett._ 121, 186401  (2018).
  * 33. Vallone, G. & Dequal, D.  Strong measurements give a better direct measurement of the quantum wave function.  _Phys. Rev. Lett._ 116, 040502  (2016).
  * 34. Dressel, J., Malik, M., Miatto, F. M., Jordan, A. N. & Boyd, R. W.  Colloquium: Understanding quantum weak values: Basics and applications.  _Rev. Mod. Phys._ 86, 307  (2014).
  * 35. Feng, L., El-Ganainy, R. & Ge, L.  Non-hermitian photonics based on parity–time symmetry.  _Nat. Photonics_ 11, 752–762  (2017).
  * 36. Leykam, D., Bliokh, K. Y., Huang, C., Chong, Y. D. & Nori, F.  Edge modes, degeneracies, and topological numbers in non-hermitian systems.  _Phys. Rev. Lett._ 118, 040401  (2017).
  * 37. El-Ganainy, R. _et al._ Non-hermitian physics and pt symmetry.  _Nat. Phys._ 14, 11–19  (2018).
  * 38. Shen, H., Zhen, B. & Fu, L.  Topological band theory for non-hermitian hamiltonians.  _Phys. Rev. Lett._ 120, 146402  (2018).
  * 39. Zhen, B., Hsu, C. W., Lu, L., Stone, A. D. & Soljačić, M.  Topological nature of optical bound states in the continuum.  _Phys. Rev. Lett._ 113, 257401  (2014).
  * 40. Doeleman, H. M., Monticone, F., den Hollander, W., Alu, A. & Koenderink, A. F.  Experimental observation of a polarization vortex at an optical bound state in the continuum.  _Nat. Photonics_ 12, 397–401  (2018).
  * 41. Zhang, Y. _et al._ Observation of polarization vortices in momentum space.  _Phys. Rev. Lett._ 120, 186103  (2018).
  * 42. Chen, W., Chen, Y. & Liu, W.  Singularities and poincaré indices of electromagnetic multipoles.  _Phys. Rev. Lett._ 122, 153907  (2019).
  * 43. Yin, X. & Peng, C.  Manipulating light radiation from a topological perspective.  _Photonics Res._ 8, B25–B38  (2020).
  * 44. Liu, W., Liu, W., Shi, L. & Kivshar, Y.  Topological polarization singularities in metaphotonics.  _Nanophotonics_ 10, 1469–1486  (2021).
  * 45. Che, Z. _et al._ Polarization singularities of photonic quasicrystals in momentum space.  _Phys. Rev. Lett._ 127, 043901  (2021).
  * 46. Zhou, H. _et al._ Observation of bulk fermi arc and polarization half charge from paired exceptional points.  _Science_ 359, 1009–1012  (2018).
  * 47. Chen, W., Yang, Q., Chen, Y. & Liu, W.  Evolution and global charge conservation for polarization singularities emerging from non-hermitian degeneracies.  _Proc. Natl. Acad. Sci. U.S.A._ 118, e2019578118  (2021).
  * 48. Huang, C. _et al._ Ultrafast control of vortex microlasers.  _Science_ 367, 1018–1021  (2020).
  * 49. Zhang, X., Liu, Y., Han, J., Kivshar, Y. & Song, Q.  Chiral emission from resonant metasurfaces.  _Science_ 377, 1215–1218  (2022).
  * 50. Chen, Y. _et al._ Observation of intrinsic chiral bound states in the continuum.  _Nature_ 613, 474–478  (2023).
  * 51. von Neuman, J. & Wigner, E.  Über merkwürdige diskrete Eigenwerte. Uber das Verhalten von Eigenwerten bei adiabatischen Prozessen.  _Physikalische Zeitschrift_ 30, 467–470  (1929).
  * 52. Friedrich, H. & Wintgen, D.  Interfering resonances and bound states in the continuum.  _Phys. Rev. A_ 32, 3231–3242  (1985).
  * 53. Hsu, C. W. _et al._ Observation of trapped light within the radiation continuum.  _Nature_ 499, 188–191  (2013).
  * 54. Yang, Y., Peng, C., Liang, Y., Li, Z. & Noda, S.  Analytical perspective for bound states in the continuum in photonic crystal slabs.  _Phys. Rev. Lett._ 113, 037401  (2014).
  * 55. Hsu, C. W., Zhen, B., Stone, A. D., Joannopoulos, J. D. & Soljačić, M.  Bound states in the continuum.  _Nat. Rev. Mater._ 1, 16048  (2016).
  * 56. Jin, J. _et al._ Topologically enabled ultrahigh-q guided resonances robust to out-of-plane scattering.  _Nature_ 574, 501–504  (2019).
  * 57. Sadreev, A. F.  Interference traps waves in an open system: bound states in the continuum.  _Rep. Prog. Phys._ 84, 055901  (2021).
  * 58. Kang, M., Zhang, S., Xiao, M. & Xu, H.  Merging bound states in the continuum at off-high symmetry points.  _Phys. Rev. Lett._ 126, 117402  (2021).
  * 59. Hu, P. _et al._ Global phase diagram of bound states in the continuum.  _Optica_ 9, 1353–1361  (2022).
  * 60. Yin, X., Jin, J., Soljačić, M., Peng, C. & Zhen, B.  Observation of topologically enabled unidirectional guided resonances.  _Nature_ 580, 467–471  (2020).
  * 61. Zeng, Y., Hu, G., Liu, K., Tang, Z. & Qiu, C.-W.  Dynamics of topological polarization singularity in momentum space.  _Phys. Rev. Lett._ 127, 176101  (2021).
  * 62. Yin, X., Inoue, T., Peng, C. & Noda, S.  Topological unidirectional guided resonances emerged from interband coupling.  _Phys. Rev. Lett._ 130, 056401  (2023).
  * 63. Mcmaster, W. H.  Polarization and the stokes parameters.  _Am. J. Phys._ 22, 351–362  (1954).
  * 64. Simon, B.  Holonomy, the quantum adiabatic theorem, and berry’s phase.  _Phys. Rev. Lett._ 51, 2167  (1983).
  * 65. Pancharatnam, S.  Generalized theory of interference, and its applications: Part i. coherent pencils.  In _Proc. Indian Acad. Sci. A_ , vol. 44, 247–262  (1956).
  * 66. Berry, M. V.  The adiabatic phase and pancharatnam’s phase for polarized light.  _J. Mod. Opt._ 34, 1401–1407  (1987).
  * 67. Lee, Y.-H. _et al._ Recent progress in pancharatnam–berry phase optical elements and the applications for virtual/augmented realities.  _Opt. Data Process. Storage_ 3, 79–88  (2017).
  * 68. Xie, X. _et al._ Generalized pancharatnam-berry phase in rotationally symmetric meta-atoms.  _Phys. Rev. Lett._ 126, 183902  (2021).
  * 69. supplementary materials on Science Online, S.
  * 70. Kogelnik, H. & Shank, C. V.  Coupled-wave theory of distributed feedback lasers.  _J. Appl. Phys._ 43, 2327–2335  (1972).
  * 71. Guo, C., Xiao, M., Guo, Y., Yuan, L. & Fan, S.  Meron spin textures in momentum space.  _Phys. Rev. Lett._ 124, 106103  (2020).
  * 72. Regan, E. C. _et al._ Direct imaging of isofrequency contours in photonic structures.  _Sci. Adv._ 2, e1601591  (2016).
  * 73. Liang, Y., Peng, C., Sakai, K., Iwahashi, S. & Noda, S.  Three-dimensional coupled-wave model for square-lattice photonic crystal lasers with transverse electric polarization: A general approach.  _Phys. Rev. B_ 84, 195119  (2011).
  * 74. Peng, C., Liang, Y., Sakai, K., Iwahashi, S. & Noda, S.  Three-dimensional coupled-wave theory analysis of a centered-rectangular lattice photonic crystal laser with a transverse-electric-like mode.  _Phys. Rev. B_ 86, 035108  (2012).
  * 75. Zhang, Y., Tan, Y.-W., Stormer, H. L. & Kim, P.  Experimental observation of the quantum hall effect and berry’s phase in graphene.  _Nature_ 438, 201–204  (2005).
  * 76. Skyrme, T.  Particle states of a quantized meson field.  _Proc. Math. Phys. Eng._ 262, 237–245  (1961).
  * 77. Skyrme, T. H. R.  A unified field theory of mesons and baryons.  _Nucl. Phys._ 31, 556–569  (1962).
  * 78. Fert, A., Reyren, N. & Cros, V.  Magnetic skyrmions: advances in physics and potential applications.  _Nat. Rev. Mat._ 2, 1–15  (2017).
  * 79. Provost, J. & Vallee, G.  Riemannian structure on manifolds of quantum states.  _Commun. Math. Phys._ 76, 289–301  (1980).
  * 80. Anandan, J. & Aharonov, Y.  Geometry of quantum evolution.  _Phys. Rev. Lett._ 65, 1697  (1990).

Figure 1:  Correspondence between bulk band topology and far-field radiation.
(A) The schematic of radiation process from PhC slab to the far field in real
space. The wavefunction of the optical eigenmodes $|\psi_{n}\rangle$ in PhC
slab is diffracted by the periodic lattice into several specific diffraction
directions $C_{1-3}$, acting as the radiation channels. For one channel (i.e.
$C_{3}$), the radiation vector of $|\Psi_{n}\rangle$ can be defined in the
polarization of the diffracted wave, marked as the spiral arrows. (B) The
“bulk-radiation correspondence” of Berry curvature in momentum space. The
radiation polarization field (middle panel) can bridge the Berry curvature
$B_{n}$ defined in wavefunction $|\psi_{n}\rangle$ (bottom panel) with the
Berry curvature $B_{n}^{r}$ defined in far-field radiation vector
$|\Psi_{n}\rangle$ (top panel). $c_{x,y}$ are complex amplitudes of the
radiative waves in $x-y$ plane.

Figure 2: Demonstration of Berry curvature observation on a honeycomb latticed
PhC slab. (A) SEM image of the fabricated PhC sample, showing a honeycomb
latticed structure of SiN slab on SiO2 substrate with different hole radii.
Inset: the side view of the air hole. The structural parameters are given as
$a=440$ nm, $r_{1}=50$ nm, $r_{2}=54$ nm, $h=180$ nm, respectively.
$\delta_{r}$ is defined as $r_{2}-r_{1}$. (B) The reciprocal lattice of the
PhC sample. Grey shading area: the first BZ; purple dot: the $\Gamma$ point;
blue dot: the $\mathcal{K}_{1}$ point; red vectors: three diffraction orders
acting as radiation channels $C_{1-3}$; green vectors: non-radiative basic
diffraction orders. (C, D) The band structures of PhCs around
$\mathcal{K}_{1}$ point with ($\delta_{r}=0$) and without ($\delta_{r}=4$ nm)
the inversion symmetry. Owing to the $C_{6}$ symmetry with $\delta_{r}=0$, TEA
mode (purple sheet) and TEB mode (blue sheet) are degenerate at
$\mathcal{K}_{1}$ point, giving rise to a diabolic point. When $\delta_{r}\neq
0$, the $C_{6}$ symmetry degrades to $C_{3}$ symmetry, and thus the DP split
to open a non-trivial band gap between TEA,B modes. (E, F) The polarization
fields in the momentum space of TEA (purple, the top panels) and TEB (blue,
the bottom panels) modes around $\mathcal{K}_{1}$ point with ($\delta_{r}=0$)
or without ($\delta_{r}=4$nm) the inversion symmetry, respectively. For
$\delta_{r}=0$ that maintains the $C_{6}$ symmetry, half charges emerge at
$\mathcal{K}_{1}$ point due to the DP for both TEA,B modes. When
$\delta_{r}\neq 0$, two CPs with opposite handness emerge around the
$\mathcal{K}_{1}$ point instead. Black dot: DP; red marks: the quasi-CPs. All
data are calculated by numerical simulations (COMSOL Multiphysics).

Figure 3: Polarimetry measurement of far-field polarization fields. (A)
Schematic of the measurements setup. AOTF: acoustic-optic tunable filter; POL1
and POL2: polarizers; QWP1 and QWP2: quarter-wave plates; L1: convex lens with
$20$ cm focal length; BS: beam splitter; RFP: rear focal plane; Obj: Objective
lens with NA of 0.95 and working distance of 150 $\mu$m; $4f$: lens system
with magnification rate of $\times$6\. (B) The observed image of three
scattered beams from radiation channels $C_{1-3}$ within the NA range. In the
experiment, we excite the optical modes from channel $C_{1}$ by moving lens L1
to a proper position and then collect the diffracted lights in channel $C_{3}$
after it is magnified by the $4f$ system. (C) Schematic of isofrequency
contours of the two-level system near the $\mathcal{K}_{1}$ point. Yellow
plane denotes the wavelength of $\lambda_{i}=570.328$ nm. Purple sheet: TEA
mode; blue sheet: TEB mode. (D) The measured isofrequency contour $S_{0}$ and
Stokes’ parameters $S_{1-4}$ at wavelength of $\lambda_{i}=570.328$ nm. The
Stokes’ parameters are determined through different configurations of POL2 and
QWP2. Dashed line: the simulated iso-frequency contour at $\lambda_{i}$. (E)
The measured polarization distributions in the momentum space around the
$\mathcal{K}_{1}$ point, by overlapping several isofrequency contours and
evaluating the overall Stokes’ parameters. Red marks: the quasi-CPs.

Figure 4: Experimental observation of Berry curvatures. (A) The measured Berry
curvatures $B_{n}^{r}$ from far-field radiation for the PhC sample with
$\delta_{r}\approx 0$ (top panels) and the numerically calculated (semi-
analytical CWT) Berry curvatures $B_{n}$ with $\delta_{r}=2$ nm (bottom
panels) for a comparison. For a realistic sample with $\delta_{r}=0$, the
fabrication errors would slightly lift the DP at $\mathcal{K}_{1}$ point to
create a small bandgap. We estimate that the bandgap is equivalent to the case
of $\delta_{r}=2$ nm. In this case, the Berry curvatures congregate around the
$\mathcal{K}_{1}$ point since the bandgap is very small, showing opposite
signs for TEA and TEB modes. (B) The measured Berry curvatures $B_{B}^{r}$
from far-field radiation (top panels) and the numerically calculated bulk
Berry curvatures $B_{B}$ (bottom panels) for TEB mode when $\delta_{r}=4$ nm
(left), $7$ nm (middle) and $10$ nm (right). Along with the increasing
$\delta_{r}$, the bandgap gradually opens and the Berry curvature gradually
diffuses to a larger region in the momentum space. Insets in top panels: SEM
images of unit cell of each PhC sample; insets in bottom panels: band
structures of the two-level system accordingly.

Figure 5: Geometric phases obtained from measured Berry curvatures. (A)
Schematic of an individual valley (blue shading) around the $\mathcal{K}_{1}$
point (blue dot) in the reciprocal lattice. The integral on Berry curvature
over the individual valley gives the geometric (Berry) phases. Considering
that the Berry curvatures congregate around the $\mathcal{K}_{1}$ point when
$\delta_{r}$ is relatively small, we perform the integral on a circular region
(black circle) to simplify the calculation. Grey shading: the first BZ; purple
dot: the $\Gamma$ point. (B, C) The geometric phases for TEA,B modes. Blue
solid line: theoretical Berry phase $\gamma_{n;t}$ of the two-level model
according to Eq. 4; red circles: numerical Berry phase $\gamma_{n}$ obtained
from the integral of Berry curvatures $B_{n}$ that are calculated by the CWT;
yellow triangles: geometric phase $\gamma_{n}^{r}$ obtained from measured
Berry curvature $B_{n}^{r}$ shown in Fig. 4. When $\delta_{r}=0$, the Berry
phases are exactly $\pm\pi$ for TEA,B modes owing to the existence of DP,
corresponding to quantized valley-Chern numbers of $\pm 1/2$. When
$\delta_{r}$ gradually increases, the calculated and measured Berry phases
both gradually deviate from the quantized $\pm\pi$. Moreover, when
$\delta_{r}$ becomes relatively large, the theoretical $B_{n;t}$ slightly
deviates from the numerical $B_{n}$, due to the impact of TEC mode.