# Observation of nonlinear disclination states

Boquan Ren  Key Laboratory for Physical Electronics and Devices of the
Ministry of Education & Shaanxi Key Lab of Information Photonic Technique,
School of Electronic and Information Engineering, Xi’an Jiaotong University,
Xi’an 710049, China  A. A. Arkhipova  Institute of Spectroscopy, Russian
Academy of Sciences, Troitsk, Moscow, 108840, Russia  Faculty of Physics,
Higher School of Economics, 105066 Moscow, Russia  Yiqi Zhang
<EMAIL_ADDRESS>Key Laboratory for Physical Electronics and Devices of
the Ministry of Education & Shaanxi Key Lab of Information Photonic Technique,
School of Electronic and Information Engineering, Xi’an Jiaotong University,
Xi’an 710049, China  Y. V. Kartashov<EMAIL_ADDRESS>Institute of
Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow, 108840, Russia
Hongguang Wang  Key Laboratory for Physical Electronics and Devices of the
Ministry of Education & Shaanxi Key Lab of Information Photonic Technique,
School of Electronic and Information Engineering, Xi’an Jiaotong University,
Xi’an 710049, China  S. A. Zhuravitskii  Institute of Spectroscopy, Russian
Academy of Sciences, Troitsk, Moscow, 108840, Russia  Quantum Technology
Centre, Faculty of Physics, M. V. Lomonosov Moscow State University, 119991,
Moscow, Russia  N. N. Skryabin  Institute of Spectroscopy, Russian Academy of
Sciences, Troitsk, Moscow, 108840, Russia  Quantum Technology Centre, Faculty
of Physics, M. V. Lomonosov Moscow State University, 119991, Moscow, Russia
I. V. Dyakonov  Quantum Technology Centre, Faculty of Physics, M. V. Lomonosov
Moscow State University, 119991, Moscow, Russia  A. A. Kalinkin  Institute of
Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow, 108840, Russia
Quantum Technology Centre, Faculty of Physics, M. V. Lomonosov Moscow State
University, 119991, Moscow, Russia  S. P. Kulik  Quantum Technology Centre,
Faculty of Physics, M. V. Lomonosov Moscow State University, 119991, Moscow,
Russia  V. O. Kompanets  Institute of Spectroscopy, Russian Academy of
Sciences, Troitsk, Moscow, 108840, Russia  S. V. Chekalin  Institute of
Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow, 108840, Russia  V.
N. Zadkov  Institute of Spectroscopy, Russian Academy of Sciences, Troitsk,
Moscow, 108840, Russia  Faculty of Physics, Higher School of Economics, 105066
Moscow, Russia

###### Abstract

Introduction of controllable deformations into periodic materials that lead to
disclinations in their structure opens novel routes for construction of
higher-order topological insulators hosting topological states at
disclinations. Appearance of these topological states is consistent with the
bulk-disclination correspondence principle, and is due to the filling anomaly
that results in fractional charges to the boundary unit cells. So far,
topological disclination states were observed only in the linear regime, while
the interplay between nonlinearity and topology in the systems with
disclinations has been never studied experimentally. We report here bon the
experimental observation of the nonlinear photonic disclination states in
waveguide arrays with pentagonal or heptagonal disclination cores inscribed in
transparent optical medium using the fs-laser writing technique. The
transition between nontopological and topological phases in such structures is
controlled by the Kekulé distortion coefficient $r$ with topological phase
hosting simultaneously disclination states at the inner disclination core and
spatially separated from them corner, zero-energy, and extended edge states at
the outer edge of the structure. We show that the robust nonlinear
disclination states bifurcate from their linear counterparts and that location
of their propagation constants in the gap and, hence, their spatial
localization can be controlled by their power. Nonlinear disclination states
can be efficiently excited by Gaussian input beams, but only if they are
focused into the waveguides belonging to the disclination core, where such
topological states reside.

##  I Introduction

Topological systems hosting topologically protected states at their edges or
in their corners are attracting considerable attention in diverse areas of
physics, including solid-state physics Hasan and Kane (2010); Qi and Zhang
(2011), mechanics Huber (2016), acoustics Xue  _et al._ (2022), physics of
matter waves Jotzu  _et al._ (2014), exciton-polariton condensates Klembt  _et
al._ (2018), and, particularly, in photonics Lu  _et al._ (2014); Ozawa  _et
al._ (2019); Smirnova  _et al._ (2020). This attention is connected, in part,
with considerable practical potential of topological systems for construction
of transmission lines, switching devices, routers, and lasers resilient to
disorder and edge deformations. With the development of topological photonics
Lu  _et al._ (2014); Ozawa  _et al._ (2019); Smirnova  _et al._ (2020), the
class of systems, where topologically nontrivial states can be encountered has
been substantially extended. While many works on photonic topological
insulators employed periodic in the bulk structures for demonstration of
topologically protected edge states Rechtsman  _et al._ (2013); Liang and
Chong (2013); Wu and Hu (2015); Yang  _et al._ (2019); Maczewsky  _et al._
(2017); Noh  _et al._ (2018a, b); Pyrialakos  _et al._ (2022), it is now
realized that the topological insulators can also be created using aperiodic
structures that possess discrete rotational symmetry, but lack periodicity,
such as quasi-crystals Bandres  _et al._ (2016), fractal structures Yang  _et
al._ (2020); Biesenthal  _et al._ (2022), and structures with disclinations
Liu  _et al._ (2021); Peterson  _et al._ (2021); Wu  _et al._ (2021).

The concept of topological insulators with disclinations originates from solid
state physics Rüegg and Lin (2013); Teo and Hughes (2013); Benalcazar  _et
al._ (2014, 2019); Li  _et al._ (2020), where it was predicted that
disclinations — crystallographic defects disrupting lattice structure — can
trap fractional “spectral” charges (connected with the local density of states
Liu  _et al._ (2021); Wang  _et al._ (2021a)) and support localized states of
the topological origin. Such systems can also be used Peterson  _et al._
(2020) for realization of the higher-order topological insulators hosting so-
called zero-dimensional states Xie  _et al._ (2021). The bulk-disclination
correspondence principle proposed for these systems that links the appearance
of the disclination states with the topological properties of the spectrum,
illustrates the importance of fractional spectral charges as a probe of
“crystalline” topology of these systems Peterson  _et al._ (2021); Liu  _et
al._ (2021); Chen  _et al._ (2022). Higher-order topological disclination
states typically form at the boundary of the hollow disclination core of the
structure. It has been demonstrated that linear lattices with disclinations
offer new opportunities for the control of confinement and internal structure
of the field, not only in photonics Wang  _et al._ (2020); Xie  _et al._
(2022), but also in acoustics Wang  _et al._ (2021b); Deng  _et al._ (2022);
Liang  _et al._ (2022). Different from aforementioned achievements reported
only in linear media, the impact of nonlinearity on photonic disclination
states was addressed theoretically only recently Ren  _et al._ (2023), while
experimental observation of nonlinear disclination states has never been
performed so far.

At the same time, nonlinear effects, such as self-action of light, attract
more and more attention in topological photonics Smirnova  _et al._ (2020)
because they enable all-optical control of the properties of the topological
states. New effects of topological origin emerging due to self-action include
topological phase transitions Maczewsky  _et al._ (2020), nonlinear Thouless
pumping Jürgensen  _et al._ (2021); Fu  _et al._ (2022a, b); Jürgensen  _et
al._ (2023), formation of the topological solitons Ablowitz  _et al._ (2014);
Leykam and Chong (2016); Ablowitz and Cole (2017); Lumer  _et al._ (2013);
Zhang  _et al._ (2020a); Mukherjee and Rechtsman (2020); Ivanov  _et al._
(2020, 2021); Zhong  _et al._ (2021); Ren  _et al._ (2021), development of the
modulational instabilities of the edge states Kartashov and Skryabin (2016);
Zhang  _et al._ (2019a) and rich bistability effects Kartashov and Skryabin
(2017); Zhang  _et al._ (2019b), to name only a few. Nonlinear higher-order
topological insulators supporting corner solitons have been also reported
theoretically Zhang  _et al._ (2020b) and in experiment Kirsch  _et al._
(2021); Hu  _et al._ (2021), while the Floquet version of higher-order
nonlinear topological insulator was proposed just recently in Zhong  _et al._
(2023).

Disclination states appearing in aperiodic structures obtained by specific
deformations of periodic arrays, formally belong to a class of higher-order
topological states. However, in contrast to the previously considered higher-
order insulator geometries with periodic bulk, disclination systems may
feature other types of discrete rotational symmetries, not compatible with
crystallographic symmetries and not attainable in usual higher-order
insulators. One can thus expect that such symmetry properties of the
disclination systems should find their manifestation in a completely different
structure of their linear eigenmodes, properties of nonlinear self-sustained
states bifurcating from them, and in their excitation dynamics. Our work is
thus aimed at the exploration of the interplay of nonlinear self-action
effects and topology in the disclination structures with different discrete
rotational symmetries.

Here we report on the first experimental observation of the nonlinear
topological states in disclination arrays with both pentagonal and heptagonal
cores, obtained by removing or adding sectors into periodic honeycomb
structures, where topological phase arises due to the Kekulé distortion Wu and
Hu (2015); Noh  _et al._ (2018b); Liu  _et al._ (2021); Wu  _et al._ (2021);
Wang  _et al._ (2021a) introduced into positions of six sites in each unit
cell of the structure. Our disclination arrays are inscribed in nonlinear
fused silica, using the fs-laser direct writing technique Rechtsman  _et al._
(2013); Kirsch  _et al._ (2021); Kartashov  _et al._ (2022); Tan  _et al._
(2021); Lin and Hong (2021); Li  _et al._ (2022). We observe that when
disclination arrays are in topological phase, one can excite thresholdless
disclination solitons existing in a broad range of input powers by Gaussian
beam focused into one of the waveguides on the disclination core. The
excitation of the same waveguides in nontopological regime yields strong
diffraction at low powers, while formation of nontopological self-sustained
states occurs only above considerable power threshold. We thus compare
behaviour of nonlinear excitations for different values of the distortion
coefficient $r$. The results obtained here are relevant for a broad class of
nonlinear physical systems, including matter waves, polariton condensates,
photonic crystals, atomic vapors, and many others, where potentials with
disclinations can be created. They also highlight the potential of these
topological structures for realisation of higher-harmonic generation and
lasing that may benefit from strong topological state confinement and its
resilience to disorder.

Figure 1: Disclination arrays and their linear spectra. (a) Microphotographs
of the fs-laser written waveguide arrays with a pentagonal disclination core
for different values of the distortion coefficient $r$. The orange, blue, and
green dotted circles indicate the nodes 1, 2 and 3, that will be used below
for probing of excitation dynamics. (b) Propagation constants $b$ of the
eigenmodes of pentagonal disclination array vs distortion coefficient $r$. Red
curves are associated with states residing at the disclination core. (c)
Spectrum at $r=1.68$. The bands corresponding to the bulk states are shown in
gray, while propagation constants of corner, edge, zero-energy and
disclination states are represented by dots of different colors. (d-f)
Microphotographs and spectrum for the array with heptagonal disclination core.
The arrangement of panels is the same as in (a)-(c).

##  II Results and discussions

###  II.1 Spectra of the arrays with disclinations

We consider paraxial propagation of a light beam along the $z$ axis of a
medium with the focusing cubic nonlinearity and shallow transverse refractive
index modulation that can be described by the nonlinear Schrödinger-like
equation for the dimensionless light field amplitude $\psi$:

$i\frac{\partial\psi}{\partial
z}=-\frac{1}{2}\left(\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial
y^{2}}\right)\psi-\mathcal{R}(x,y)\psi-|\psi|^{2}\psi,$ (1)

where $x,y$ are the scaled transverse coordinates, $z$ is the propagation
distance that plays the same role as time in the Schrödinger equation
describing a quantum particle in a potential, and the function
${\mathcal{R}}(x,y)$ describes disclination array with the straight
waveguides:

${\mathcal{R}}=p\sum_{m,n}e^{[-(x-x_{m,n})^{2}/a_{x}^{2}-(y-y_{m,n})^{2}/a_{y}^{2}]},$

where $a_{x}$ and $a_{y}$ are the widths of waveguides that are elliptical due
to the writing process, $x_{m,n}$ and $y_{m,n}$ are the positions of the
waveguide centers (depending on the particular type of introduced
disclination), and $p$ is the array depth proportional to the refractive index
contrast $\delta n$ in the structure (see Methods for adopted normalizations).
To create topologically nontrivial arrays with disclinations we use two-step
process. We start from usual periodic honeycomb waveguide array with identical
waveguide spacing $d$ in the entire structure and first introduce shift of the
waveguides in the direction perpendicular to the borders of the unit cell,
whose magnitude can be characterized by the Kekulé distortion coefficient
$r=\ell_{\rm intra}/\ell_{\rm inter}$, with $\ell_{\rm intra}$ and $\ell_{\rm
inter}$ being intra-cell and inter-cell spacing between waveguides after shift
Wu and Hu (2015); Noh  _et al._ (2018b); Wu  _et al._ (2021); Liu  _et al._
(2021). Clearly, $r=1$ corresponds to non-deformed structure with $\ell_{\rm
intra}=\ell_{\rm inter}=d$. As it will be shown below, by changing the value
of $r$ one can achieve the transition between topologically trivial and
nontrivial geometries. On the second step, to create the arrays with
disclination, we remove or add sectors into honeycomb structure with shifted
waveguides. At this step, after removing of the sector we deform the unit
cells in the remaining structure such that they fill the entire $2\pi$ polar
angle, while to add the sector we compress unit cells accordingly (see Methods
for the description of the deformation process).

In Fig. 1(a), we display the microphotographs of the arrays with pentagonal
disclination core inscribed with fs-laser in $10~{}\textrm{cm}$ long fused
silica sample (total number of waveguides in this structure is 90), obtained
by removing a sector from honeycomb array, with coordinates of the waveguides
obtained using the above two-step process for three different values of the
distortion coefficient $r$. Black hexagons superimposed on the
microphotographs are guides for the eye indicating different unit cells of the
structure. Similar microphotographs, but for the structure with the heptagonal
disclination core (total number of waveguides is 126) that was obtained by
adding the sector into honeycomb array are presented in Fig. 1(d), also for
three different $r$ values. Topological properties of these structures are
controlled by the distortion coefficient $r$. One can see that for $r<1$ the
inter-cell coupling becomes weaker than the intra-cell one, while for $r>1$
the situation is reversed, and the inter-cell coupling becomes stronger than
the intra-cell one, indicating on the possible transition of the disclination
array into higher-order topological insulator phase. This transition is
manifested in qualitative modification of the linear spectrum of eigenmodes
supported by these structures. To obtain such modes, we first use the ansatz
$\psi=u(x,y)e^{ibz}$, where $b$ is the propagation constant and $u(x,y)$ is
the real function, for Eq. (1) to get the equation

$bu=\frac{1}{2}\left(\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\right)u+\mathcal{R}u+u^{3}.$ (2)

We then omit last nonlinear term in Eq. (2) and calculate all linear
eigenmodes of the system using the plane-wave expansion method.

Figure 2: The families of nonlinear disclination states. (a) Peak intensity
$I_{p}$ (blue solid curve) and power $U$ (orange solid curve) of the nonlinear
disclination states vs propagation constant $b$ in the array with the
pentagonal core at $r=1.68$. Gray regions represent the bulk band, while the
vertical dotted color lines show propagation constants of linear corner, edge,
zero-energy, and disclination states. (b) Intensity distributions of selected
nonlinear disclination states with different propagation constants that
correspond to circles in (a). (c,d) The families of nonlinear disclination
states and examples of their profiles in the array with heptagonal core.
Intensity distributions in (b) are shown within the window $-40\leq x,y\leq
40$, while those in (d) are shown within the window $-46\leq x,y\leq 46$.

The transformation of linear spectrum of the array with the pentagonal
disclination core with increase of the distortion coefficient $r$ is
illustrated in Fig. 1(b). While at $r<1$ no localized states are present in
the gap between two bulk bands, at $r>1$ the spectrum changes qualitatively,
with several different types of localized states emerging in the gap. The
appearance of such states can be explained by the bulk-disclination
correspondence principle which gives the link between the fractional charge
and topologically nontrivial states (see Methods for discussion of the
topological invariants). Among these states, five states marked with red color
(their number is dictated by the symmetry of the structure) are disclination
states residing at the disclination core in the center of the array. Some of
these states can be degenerate depending on the value of $r$, but in general
they have different eigenvalues. These states have different phase structure,
their localization at the disclination core increases with the increase of
$r$. To describe the structure of the spectrum in more details, we chose
$r=1.68$ [cyan dashed line in Fig. 1(b)] and show eigenvalues of all modes in
the interval $0.36\leq b\leq 0.50$ in Fig. 1(c). Besides disclination states,
in the same gap there appear corner (cyan dots), edge (purple dots), and zero-
energy states (green dots), but all of them emerge at the outer edge/corners
of the structure due to its finite size and on this reason they do not
hybridize for sufficiently large $r$ with red disclination states localized on
the central disclination core. Calculated intensity distributions of all
eigenmodes forming in the gap at $r=1.68$ for the arrays with pentagonal and
heptagonal disclination cores are presented in Supplemental Materials — to
stress generality of these results, we present them for even larger structures
with 300 (420) waveguides for pentagonal (heptagonal) cases — while in
experiments we use sufficiently large and most compatible with writing
technology structures from Fig. 1.

Similar transformation of linear spectrum with increase of $r$ is observed
also in the array with heptagonal disclination core, see Fig. 1(d). In this
structure seven disclination states with different phase structures emerge in
the spectrum (some of them are nearly degenerate so that there are seemingly
five red curves) shown in Fig. 1(e). Detailed structure of spectrum for this
case is presented in Fig. 1(f) for $r=1.68$, where one can again see that
disclination states at the disclination core may coexist with spatially
separated from them corner (cyan dots), edge (purple dots), and zero-energy
states (green dots) at the outer edge.

The emergence of disclination states of topological origin at the inner
disclination core is consistent with the bulk-disclination correspondence
principle Peterson  _et al._ (2021); Liu  _et al._ (2021); Chen  _et al._
(2022) that establishes the link between the fractional disclination charge
$\mathcal{Q}$ (see Methods for details of topological characterization) and
the localized states emerging at the disclination core. For our arrays,
$\mathcal{Q}=1/2$ in topologically nontrivial phase at $r>1$ signalizing on
the appearance of disclination states, while $\mathcal{Q}=0$ in nontopological
regime, when $r<1$ and disclination states are absent.

Figure 3: Excitation of nonlinear modes in disclination arrays without
distortion, at $r=1.0$. Comparison of experimentally measured (maroon
background) and theoretically calculated (white background) output intensity
distributions in pentagonal (a)-(c) and heptagonal (d)-(f) disclination arrays
for different input powers (pulse energies) of single-site Gaussian
excitation. Results for beams focused into different nodes 1, 2 and 3 of the
structure [indicated by colored circles and corresponding to the circles in
Figs. 1(a) and 1(d)] are additionally highlighted with the orange, blue, and
green backgrounds. Pulse energies $E$ for the experimental outputs and
dimensionless input powers $U$ for theoretical outputs are indicated on each
panel. White and black lines are guides for the eye illustrating unit cells of
the array. All theoretical panels are shown within the window $-30\leq x,y\leq
30$ and are obtained for the array depth $p=5.0$. Figure 4: Excitation of
nonlinear modes in nontopological disclination array with $r=0.8$. Comparison
of the output theoretical and experimental output intensity distributions for
different input powers in pentagonal (a)-(c) and heptagonal (d)-(f) arrays in
topologically trivial phase. The arrangement of panels is similar to Fig. 3.
The array depth is $p=4.9$ in (a)-(c) and $p=5.0$ in (d)-(f). Figure 5:
Excitations of nonlinear modes in topological disclination arrays with
$r=1.68$. Comparison of experimental and theoretical outputs in pentagonal
(a)-(c) and heptagonal (d)-(f) arrays in the topological phase illustrating
the formation of disclination solitons (a),(d) and corner solitons (c),(f)
existing for a broad range of powers, and considerable diffraction at all
power levels for excitation of node 2. In all cases the array depth $p=5.0$.

###  II.2 Properties of nonlinear disclination states

We now address the properties of stationary nonlinear disclination states,
whose shapes are governed by the Eq. (2), where we keep the last nonlinear
term. Such states can be found using the Newton relaxation method. By analogy
with corner solitons encountered in higher-order topological insulators with
periodic bulk Kirsch  _et al._ (2021); Hu  _et al._ (2021), such nonlinear
disclination states can be called disclination solitons. For their theoretical
description we adopt the large-scale disclination arrays schematically
depicted in Fig. 2. In both pentagonal and heptagonal arrays the families of
the nonlinear disclination states bifurcate from linear modes localized at the
disclination core. We consider bifurcation from the disclination state with
the largest propagation constant (see Supplemental Materials). With the
increase of the propagation constant $b$ the power $U=\iint|\psi|^{2}dxdy$ of
nonlinear disclination state monotonically grows [see Figs. 2(a) and 2(c) for
pentagonal and heptagonal cases, respectively], while the state first rapidly
localizes and already at $U\sim 0.1$ concentrates practically on one side of
the disclination core [see representative intensity distributions in Figs.
2(b) and 2(d)]. This clearly stresses solitonic nature of such state, since
without nonlinearity its power would be redistributed between different sites
at the disclination core due to beating between several disclination states
(notice that this process may be slow because eigenvalues of linear
disclination states are close). The peak intensity
$I_{p}=\max\\{|\psi|^{2}\\}$ of nonlinear disclination state [blue curve in
Figs. 2(a) and 2(c)] practically linearly increases with $b$. Even though
propagation constant of the nonlinear disclination state crosses eigenvalues
of linear edge (magenta vertical dashed lines) and corner (cyan vertical
dashed lines) states, no coupling with them occurs because they are located at
the outer edge of the array. However, when the propagation constant of the
nonlinear disclination state penetrates into the bulk band, shown with gray
color in Figs. 2(a) and 2(c), the coupling with bulk states occurs that leads
to strong expansion over entire array [see right panels in Figs. 2(b) and
2(d)]. As a result, the power $U$ in the band rapidly increases with increase
of $b$. These results illustrate that nonlinear disclination states are the
modes of topological origin, whose position inside the gap and localization
degree strongly depend on their power. Linear stability analysis performed for
these nonlinear families shows that they are stable in the entire gap, in both
pentagonal and heptagonal structures, thus they can be readily excited in the
experiment.

###  II.3 Observation of nonlinear disclination states

For observation of nonlinear disclination states we inscribed (see Methods for
details of fabrication) the arrays with pentagonal and heptagonal disclination
cores with different values of the distortion coefficient $r=0.8$, $1.0$ and
$1.68$, to be able to compare dynamics in topologically trivial and nontrivial
structures. In experiments, we employed single-waveguide excitations using
$280\,\textrm{fs}$ pulses of variable energy $E$ from $1\,\textrm{kHz}$ fs
Ti:sapphire laser at $800\,\textrm{nm}$ central wavelength. The input peak
power in the waveguide defined as a ratio of the pulse energy $E$ to the pulse
duration $\tau$ taking into account the losses for matching with the focusing
lens is evaluated as $2.5\,\textrm{kW}$ for each $1\,\textrm{nJ}$. We compare
excitations of three different waveguides (nodes) numbered 1 (at the
disclination core), 2 (in the bulk), and 3 (in the outer corner) indicated by
colored circles in Figs. 1(a) and 1(d).

In Fig. 3 we present comparison of the output intensity distributions for
these three types of excitations calculated theoretically (images with white
background) using Eq. (1) and measured experimentally (images with maroon
background) for both pentagonal and heptagonal disclination arrays without
distortion, i.e. with $r=1$. In this “borderline” case between topological and
nontopological phases, no localized states are present in spectra of the
arrays. On this reason the excitation of any of the nodes 1, 2, or 3 is
accompanied by strong diffraction in the linear regime for $E=10\,\rm nJ$
pulses (in Fig. 3 the position of excitation in each case is marked by colored
ring). Increasing pulse energy (power in theoretical simulations) results in
gradual contraction of light towards the excited waveguide. For excitation of
node 1 at disclination core one can observe the formation of well-localized
soliton at highest shown pulse energy $E\sim 600\,\rm nJ$, i.e. above
considerable threshold [Figs. 3(a) and 3(d)]. The same pulse energy level in
general is not sufficient for soliton formation for excitation in the bulk
[Figs. 3(b) and 3(e)] and at the outer corner [Figs. 3(c) and 3(f)], since at
this energy level the tendency for light contraction to the excited waveguide
only begins. To achieve good localization in these cases one has to increase
the pulse energy even further, approximately to $E\sim 900\,\rm nJ$ level.

In Fig. 4 we consider the same three types of excitations in the trivial
insulator phase, when distortion coefficient $r=0.8$. According to the Wannier
center analysis in each unit cell, the filling anomaly does not occur in this
case, and, consequently, no localized corner, edge or disclination states can
appear in the linear spectrum of the system, despite the fact that forbidden
gap opens for this value of $r$ [see Figs. 1(b) and 1(e)], i.e. all linear
eigenmodes are delocalized bulk modes. Thus, one again observes diffraction in
the linear regime for $E=10\,\rm nJ$ pulses, for both pentagonal Figs.
4(a)-4(c) and heptagonal Figs. 4(d)-4(f) disclination arrays for all three
types of excitations. Moreover, now localization does not occur for excitation
at the disclination core even for pulse energies $E\sim 600\,\rm nJ$ that was
sufficient for nonlinear localization at $r=1.0$. Thus, there exists the
tendency for increase of the pulse energy required for localization at the
disclination core with decrease of $r$. For depicted pulse energies
localization was not observed neither for bulk nor for corner excitations (it
occurs only around $E\sim 900\,\rm nJ$).

The picture changes qualitatively at $r=1.68$ in topologically nontrivial
phase. In this case, the disclination core supports topologically nontrivial
localized disclination states, thus the input beam focused into node 1 excites
localized states even at the lowest pulse energies $E\sim 10\,\rm nJ$ in both
pentagonal [Fig. 5(a)] and heptagonal [Fig. 5(d)] arrays. Notice that even
though in this quasi-linear regime single-site excitation leads to
simultaneous population of several localized disclination eigenmodes, the
beating between them occurs on the scale much larger than sample length (due
to small difference of propagation constants $b$ of such eigenmodes, see
Supplementary Materials) and is therefore not visible in experiment at
$10~{}\textrm{cm}$ of propagation. Even weak nonlinearity suppresses this
beating leading to the formation of well-localized disclination solitons that
exist over broad range of input pulse energies (powers), as long as
propagation constants of such states remain in the forbidden gap of the
spectrum [Figs. 5(a) and 5(d)]. Notice that because for $r=1.68$ the gap is
already wide, in experiment we do not reach power levels (below optical damage
threshold), at which strong coupling with bulk states occurs. In contrast,
when node 2 in the bulk is excited, one observes diffraction, and nonlinear
localization does occur even for pulse energies $E\sim 600\,\rm nJ$ [see Figs.
5(b) and 5(e)]. An interesting situation is encountered for excitation of the
node 3 [Figs. 5(c) and 5(f)]. This excitation has the largest overlap with the
corner states that are also well-localized for this value of $r$ at the outer
edge of the array, and it does not excite zero-energy states (because the
latter have different symmetry, see Supplementary Materials). As a result, in
this case one observes the formation of nonlinear corner states in both
pentagonal and heptagonal disclination arrays, whose localization degree only
weakly changes in the considered range of input powers. Theoretical
simulations fully support these observations.

##  III Conclusions

In conclusion, we have reported on the experimental observation of nonlinear
disclination states in disclination lattices inscribed in transparent
nonlinear optical medium. Such states form when Kekulé distortion of waveguide
positions drives the array into the topological phase, where several different
types of localized states appear: disclination states residing at the
disclination core and not overlapping with them spatially corner, edge, and
zero-energy states at the outer edge of the structure. The nonlinearity
enables strong light localization on one side of the disclination core. Our
findings are reported for the pentagonal and heptagonal structures, with
symmetry different from previously considered higher-order insulators with
periodic bulk (such as $\mathcal{C}_{3}$, $\mathcal{C}_{4}$, $\mathcal{C}_{6}$
ones). They pave the way for the development of new types of topological
lasers on disclination states, efficient harmonic generation in topologically
protected states, and observation of new interesting topological objects, such
as topological disclination bound states in the continuum Qin  _et al._
(2023).

##  IV Materials and methods

###  IV.1 Fs-laser inscription of the waveguide arrays

The waveguide arrays shown in Figs. 1(a,d) were inscribed in
$10\,\textrm{cm}$-long fused silica glass samples (JGS1) using focused (with
an aspheric lens with $\textrm{NA}=0.3$) under the surface of sample at the
depth range of $550\sim 1050\,\mu\textrm{m}$ fs laser pulses at the wavelength
of $515\,\textrm{nm}$ with the duration $280\,\textrm{fs}$, repetition rate
$1\,\textrm{MHz}$, and energy $320\,\textrm{nJ}$. Translation of the sample
during the writing process of each waveguide was performed by the high-
precision air-bearing positioner (Aerotech) with identical for all waveguides
velocity of $1\,\textrm{mm/s}$. All such waveguides are elliptical and single-
mode, and they exhibit propagation losses not exceeding $0.3\,\textrm{dB/cm}$
at $\lambda=800\,\textrm{nm}$. After the waveguide arrays were inscribed, the
input/output facets of the sample were optically polished, so that the sample
length was shortened to $99\,\rm mm$.

###  IV.2 Numerical simulations and normalizations

For numerical simulations of evolution and excitation of the nonlinear
disclination states, we used dimensionless continuous nonlinear Schrödinger-
like equation (1), in which the transverse coordinates $x,y$ are normalized to
the characteristic scale $r_{0}=10\,\mu\textrm{m}$, the propagation distance
$z$ is normalized to the diffraction length $kr_{0}^{2}\approx
1.14\,\textrm{mm}$, $k=2\pi n/\lambda$ is the wavenumber in the medium with
the background refractive index $n$ (for fused silica $n\approx 1.45$), and
$\lambda=800~{}\textrm{nm}$ is the working wavelength. Our waveguides are
single-mode and elliptical due to the writing process, the dimensionless
widths of the waveguides are $a_{x}=0.25$ and $a_{y}=0.75$ (corresponding to
$2.5$ and $7.5\,\mu\textrm{m}$, respectively), but the eigenmode of such
waveguides is only slightly elliptical. The waveguide spacing in structure
without distortion is $d=3.2$ (corresponding to $32\,\mu\rm m$). The array
depth $p=k^{2}r_{0}^{2}\delta n/n$ is proportional to the refractive index
contrast $\delta n$ in the structure. For instance, $p=1.0$ corresponds to
$\delta n\sim 1.1\times 10^{-4}$. In the majority of presented results (unless
specifically stated in the caption), we use the depth $p=5.0$ that provides
the best agreement between experiments and theory.

Figure 6: Construction of the disclination array from honeycomb structure. (a)
Original honeycomb array with intra-cell separation $\ell_{\rm intra}$ and
inter-cell separation $\ell_{\rm inter}$. The Kekulé distortion coefficient
$r=\ell_{\rm intra}/\ell_{\rm inter}$. Each unit cell is indicated by white
hexagons. $\sqrt{3}d$ denotes the length of one side of the hexagonal unit
cell. (b) Disclination array with the pentagonal core obtained by removing the
sector with the Frank angle $2\pi/6$ from array in (a) and gluing the cutting
edges. (c) Disclination array with the heptagonal core obtained by inserting
the sector with the Frank angle $2\pi/6$ into array (a) and subsequent
compression of the unit cells. All arrays are shown within the window $-30\leq
x,y\leq 30$.

Pentagonal and heptagonal disclination arrays were obtained from regular
honeycomb arrays using two-step process described in the main text, when at
the first stage by shifting the waveguides one introduces controllable Kekulé
distortion, quantified by the distortion coefficient $r=\ell_{\rm
intra}/\ell_{\rm inter}$, where $\ell_{\rm intra}$ and $\ell_{\rm inter}$ is
the intra-cell and inter-cell spacing between waveguides after shift [see
notations in Fig. 6(a)], while at the second stage one removes or inserts
Frank sector into the array, and then expands or contracts unit cells such as
to obtain the resulting disclination structure [Fig. 6(b) and 6(c)]. While
deforming the structure we keep the longer axes of all elliptical waveguides
parallel to the $y$ axis.

###  IV.3 Topological indices

The topological properties of disclination arrays can be discussed by
analyzing the fractional “charge” that is carried by each unit cell, that is
employed in established bulk-disclination correspondence principle Liu  _et
al._ (2021). Note that the “charge” here is spectral charge that can be
defined through the local density of states. It is an analog of the real
charge in electric systems, and it can be used to evaluate the number of
states in the unit cell with states considered below the topological band gap
Wang  _et al._ (2021a). The spectral charge $\mathcal{Q}$ bound to a
disclination with a Frank angle $\Omega$ is defined by Noh  _et al._ (2018b);
Benalcazar  _et al._ (2019); Li  _et al._ (2020); Peterson  _et al._ (2021);
Liu  _et al._ (2021); Wu  _et al._ (2021)

$\mathcal{Q}=\frac{\Omega}{2\pi}\left(\frac{3}{2}\chi_{\rm M}-\chi_{\rm
K}\right)\,{\rm modulo}\,1,$ (3)

where the high symmetry indicators are $\chi_{\rm M}=\\#{\rm
M}_{1}^{(2)}-\\#\Gamma_{1}^{(2)}$ and $\chi_{\rm K}=\\#{\rm
K}_{1}^{(3)}-\\#\Gamma_{1}^{(3)}$ that should be calculated directly in the
honeycomb array before removing or inserting the Frank sector. Here
$\\#\Pi_{q}^{(n)}$ is the number of bands below the forbidden gap at a high-
symmetry point $\Pi=\Gamma,\,\rm M,\,K$ with the eigenvalue of the $C_{n}$
rotation matrix $e^{i2\pi(q-1)/n}~{}(q=1,\cdots,n)$ Benalcazar  _et al._
(2019), and $\Omega=2\pi/6$ for the disclination arrays adopted in this work.
For the topological nontrivial case with $r>1$, one can find that $(\chi_{\rm
M},\chi_{\rm K})=(2,\,0)$ for both the pentagonal disclination array and the
heptagonal disclination array. While for the topological trivial case with
$r<1$, $(\chi_{\rm M},\chi_{\rm K})=(0,\,0)$. Thus, the fractional charge is
$\mathcal{Q}=1/2$ for $r>1$ and $\mathcal{Q}=0$ for $r<1$. The fractional
charge can be also obtained by counting the number of the Wannier centers
occupied by each unit cell. The Wannier centers are located at the edges of
the unit cell if $r>1$ and at its center if $r<1$. The unit cell around the
pentagonal disclination core has five bulk Wannier centers if $r>1$, which
give a $5/2$ charge per unit cell. If $r<1$, the charge per unit cell around
the disclination core is $3$. See also the Supplementary Materials.

## Acknowledgements

This research was funded by the National Natural Science Foundation of China
(Grant No.: 12074308), the research project FFUU-2021-0003 of the Institute of
Spectroscopy of the Russian Academy of Sciences and partially by the RSF grant
21-12-00096, the Foundation for the Advancement of Theoretical Physics and
Mathematics “BASIS” (Grant No.: 22-2-2-26-1), and the Fundamental Research
Funds for the Central Universities (Grant No.: xzy022022058).

## Author contributions

All authors made significant contribution to this work.

## Conflict of interest

The authors declare no competing interests.

## Supplementary information

The online version contains supplementary material available at
https://doi.org/.

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