Topological orbital angular momentum extraction and twofold protection of vortex transport

Abstract

Vortex phenomena are ubiquitous in nature. In optics, despite the availability of numerous techniques for vortex generation and detection, topological protection of vortex transport with desired orbital angular momentum (OAM) remains a challenge. Here, by use of topological disclination, we demonstrate a scheme to confine and guide vortices featuring arbitrary high-order charges. Such a scheme relies on twofold topological protection: a non-trivial winding in momentum space due to chiral symmetry, and a non-trivial winding in real space due to the complex coupling of OAM modes across the disclination structure. We unveil a vorticity-coordinated rotational symmetry, which sets up a universal relation between the vortex topological charge and the rotational symmetry order of the system. As an example, we construct photonic disclination lattices with a single core but different C_n symmetries and achieve robust transport of an optical vortex with preserved OAM solely corresponding to one selected zero-energy vortex mode at the mid-gap. Furthermore, we show that such topological structures can be used for vortex filtering to extract a chosen OAM mode from mixed excitations. Our results illustrate the fundamental interplay of vorticity, disclination and higher-order topology, which may open a new pathway for the development of OAM-based photonic devices such as vortex guides, fibres and lasers.

Main

Vortices are observed in a wide range of natural systems, from vortices of quantum particles and living cells to tornados and black holes^{1,2,3,4,5,6,7,8,9}. In optics, vortices are typically characterized by a circulating flux that gives rise to orbital angular momentum (OAM)¹⁰, playing a crucial role in numerous optical phenomena and applications^{10,11,12,13,14,15}. Apart from classical waves, vortex beams carrying OAM have been experimentally realized with photons³, electrons⁴ and even non-elementary particles such as neutrons⁵, atoms and molecules⁶. The ability to generate and manipulate vortex beams has sparked substantial scientific interest, leading to research in fundamental phenomena and enabling unconventional implementations across different fields. Like all localized waves, however, vortices of any field tend to spread out during evolution.

Confining vortex flows is important across diverse areas of science and technology. For instance, in free space, localized vortex beams naturally diffract due to their wave nature, making them susceptible to atmospheric turbulence and other environmental factors. High-order vortices, characterized by topological charges $\left|l\right| > 1$ , are particularly prone to disintegration during propagation, often breaking up into several ‘pieces’ of singly charged vortices¹⁶ and further complicating their control and manipulation over long distances. To effectively confine an optical vortex without distortion in shape and preserve its OAM during transport is crucial, particularly for OAM-based optical communications¹⁷. However, unlike vortex generation and detection^10,11, vortex transport with preserved OAM is a non-trivial task. Addressing this challenge requires innovative approaches in both theoretical design and experimental implementation to create robust and efficient vortex transport systems^17,18.

Recently, there has been a surge of interest in the study of topological disclinations^{19,20,21,22,23,24,25}, unveiling non-trivial topological phases, including higher-order topological phases^26,27,28. Topological disclinations, as a representative type of topological defects of a point-group rotational symmetry, can support localized topological states within the bulk^20,21,29 rather than at the boundaries³⁰. However, lattice defects and disclinations typically break chiral symmetry, a key element for topological protection in a large family of topological insulators³¹. If disclinations are designed to preserve chiral symmetry, their bounded states can lie exactly in the middle of the bandgap and be pinned at zero energy²⁴—reminiscent of Majorana-bound states¹⁹. Such chiral-symmetric disclinations are ideal for the realization of a topological vortex guide (TVG), where spectral isolation, OAM-mode spatial confinement and topological protection can be guaranteed. Nevertheless, most topological disclination structures established so far, including the recent work on disclination-based vortex nanolasers³² and solitons³³, consist of a multi-site rather than a single-site core for confining the vortex mode. In curved or engineered three-dimensional (3D) synthetic materials^34,35,36,37, localized higher-orbital and vortex states can also emerge, but they rely on complex design of topological defects, and the localized states are not zero-energy OAM modes. In fact, a single-channel TVG has never been realized. In the so-called Dirac-vortex topological structure, the ‘vortex’ refers to a Kekulé modulation of a Dirac lattice with a vortex phase (as in the Jackiw–Rossi model)^38,39,40, but the bounded mode itself is not an OAM mode.

Here, we demonstrate a universal principle for the realization of robust vortex transport by using specially designed topological disclination structures. As illustrated in Fig. 1a, a vortex beam preserves its circular shape and phase singularity along the central waveguide—the single-site disclination core in a C₃-rotational and chiral-symmetric lattice. It is ‘doubly’ protected by both non-trivial momentum-space (k-space) band topology and real-space topology, with the latter characterized by the non-trivial winding of complex vortex-mode coupling (Fig. 1b). The k-space band topology gives rise to the localized mid-gap (zero-energy) vortex states protected by chiral symmetry, while the real-space non-trivial winding ensures the selection of just one OAM mode along the disclination core without the interference from all other possible modes. The principle applies to any high-order OAM modes with topological charge l for any rotational order n of the ${C}_{n}$ lattice symmetry, as long as $2l/n$ is not an integer. We unravel this condition as the vorticity-coordinated rotational symmetry (VRS)—essential for the real-space protection of vortex transport in topological disclinations. Experimentally, we employ a laser-writing technique to establish different photonic disclination lattices as a test bench and observe that optical vortices with topological charges $l=1,\,2,\,3$ can be protected by non-trivial ${C}_{3}$ and ${C}_{5}$ structures, but an $l=2$ vortex breaks up into a quadrupole-like pattern in the ${C}_{4}$ structure owing to the lack of real-space protection. Numerical simulations (Fig. 1c) further corroborate the conditions needed for protected vortex transport. Moreover, we show that, under a mixed-mode excitation, our topological approach leads to an effective vortex filter, which extracts and protects the transport of a selected OAM mode while filtering out other unwanted modes. Our work represents a demonstration of single-channel TVGs for robust transport of vortices in any system⁴¹.

**Fig. 1: Illustration of doubly protected vortex transport via topological disclinations.**

Results

Our disclination lattices are constructed by ‘cutting and gluing’^20,21 a two-dimensional Su–Schrieffer–Heeger (SSH) lattice^42,43 that results in a single-site core at the centre of the structure (Methods and Extended Data Fig. 1). Three disclination lattices featuring C₃-, C₄- and C₅-rotational symmetry are established by laser-writing waveguides in an otherwise uniform crystal (Supplementary Note 1). A typical example of the ${C}_{3}$ disclination and corresponding results are shown in Fig. 2, where Fig. 2a is the disclination lattice. The 3D intensity plots in Fig. 2b(1)–b(3) clearly illustrate the difference between a guided and an unguided vortex. Figure 2c(1) is the input vortex beam with $l=1$ used to probe the central disclination core. The input beam size is 36 μm, and it expands to about 240 μm after 20 mm propagation through the crystal (without any written waveguide), as seen from the corresponding output intensity pattern (Fig. 2c(2)) and interferogram (Fig. 2c(3)). In contrast, when the same vortex is launched into the disclination core, its intensity is well confined in the core, preserving both the vortex ring pattern and the topological charge (Fig. 2c(4)). Results from numerical simulations (Fig. 2c(5)) agree well with such observations, showing the robustness of the vortex transport even at much longer distances through the vortex guide (Methods, Extended Data Fig. 2 and Supplementary Note 2). Likewise, a high-order vortex with $l=2$ (Fig. 2d(1)–d(5)) or $l=3$ (Extended Data Fig. 3) is also well guided in the disclination core.

**Fig. 2: Experimental demonstration of vortex transport along a single channel in topological disclination.**

Protection by momentum-space topology

To understand the essence of the ‘double protection’ needed for the TVGs, let us first consider the k-space band topology, characterized by the theory of topological invariants in momentum space^31,44. The band structure of the disclination Hamiltonian considering complex vortex-mode coupling for a C_n-symmetric lattice is calculated using the tight-binding model (Methods and Supplementary Note 4). In our model, the k-space band topology lies in a chiral-symmetric topological phase, which ensures that the topological defect states appear right at zero-energy mid-gap (see Fig. 3a(1),a(2) for $l=1$ and $l=2$ cases in the ${C}_{3}$ lattice) and occupy only one sublattice (Fig. 3b(1),b(2),c(1),c(2)). In contrast to photonic crystals or photonic crystal fibres, here the vortex modes are spatially localized modes in the bandgap of propagation constants, as opposed to time-domain frequencies. The disclination states here cannot be characterized by the topological invariants conventionally used for higher-order topology, for example, the fractional charge density^20,23,27. To solve this issue, we employ the concept of the multipole chiral number (MCN)²⁸. The MCN is a bulk integer topological invariant $N$ recently developed for predicting the number of degenerate zero-energy corner states in higher-order topological systems enriched by chiral symmetry. It is essentially a real-space representation of the winding number generalized from one- to higher-dimensional systems. Since the number of lattice sites belonging to different sublattices in our C_n-symmetric disclination structures are not equal (Fig. 1a), we use $N$ to evaluate the overall difference between the multipole moments of two sublattice wave functions (Fig. 3d,e and Supplementary Note 5). Physically, it describes the winding of the wave function for the B sublattice with respect to the A sublattice. To present the whole structure of a ${C}_{n}$ lattice, we use different coordinates for every sector of the ${C}_{n}$ lattice to define multipole operators with respect to the central defect and thus get $N$ . The system is topologically non-trivial when ${d}_{2} < {d}_{1}$ , where the dimerization parameters ${d}_{1}$ and ${d}_{2}$ are the waveguide distances associated with the intra-cell and inter-cell coupling in the SSH model, respectively^28,31,43. In this case, a non-zero $N$ corresponds to a non-trivial winding in the momentum space, as depicted in Fig. 1b. For example, in Fig. 3d, $N=2$ indicates that there are two degenerate zero-energy vortex states (Fig. 3a(1),a(2)) with opposite vortex-phase circulation. In contrast, if ${d}_{2} > {d}_{1}$ , we have $N=0$ (Fig. 3d), implying a topologically trivial winding and the absence of topological disclination states. Results for other examples of ${C}_{n}$ lattices are shown in Extended Data Fig. 4. This generalized MCN can be applied to characterize higher-order topological phases in other non-periodic C_n-symmetric structures that exhibit chiral symmetry. Results in Fig. 3 highlight one key ingredient needed for protecting the vortex transport: momentum-space topology featured by chiral symmetry of the disclination structure. The role of chiral symmetry is to guarantee that the guided vortex mode is at zero energy. In practice, even if the mode is not exactly at the middle of the bandgap, it can still enjoy topological protection if it resides close to the mid-gap.

**Fig. 3: Momentum-space protection of zero-energy vortex modes via topological disclination.**

Protection by real-space topology

Notwithstanding that there are two degenerate zero-energy vortex modes, robust transport of a vortex requires that only a single vortex state be present at the disclination core during propagation. Every waveguide can support both clockwise ( $l < 0$ ) and anti-clockwise ( $l > 0$ ) phase circulations; thus, there are two types of vortex mode coupling between waveguides (Fig. 4a(1)): the same-vorticity mode coupling (SVMC), which is always real regardless of the coupling direction, and the direction-dependent opposite-vorticity mode coupling (OVMC). The OVMC can be illustrated by a coupling vector ${t}_{{\rm{OV}}}$ (Supplementary Note 4) in the complex plane (Fig. 4a(2)). To ensure that a TVG supports only a single clockwise (or anti-clockwise) vortex mode at any propagation distance, no anti-clockwise (or clockwise) components should arise at the disclination core during the transport, as analysed in Supplementary Note 6. This indicates that the collective contribution of the OVMC from all waveguides across the entire lattice to the disclination core must be zero.

**Fig. 4: Real-space protection of vortex transport and universal rule for VRS-mediated non-trivial winding.**

In a ${C}_{n}$ disclination lattice, we evaluate the OVMC between the core and all other waveguides belonging to each distinct sector and then examine the winding of the complex coupling ${T}_{j}$ from each of the $n$ rotational sectors (see Fig. 4b(1)–b(3) for the ${C}_{3}$ lattice, Fig. 4c(1)–c(3) for the ${C}_{4}$ lattice and Fig. 4d(1)–d(3) for the ${C}_{5}$ lattice). Real-space topology can be examined by defining a coupling winding number as

w = \frac{1}{2 π i} \sum_{j = 1}^{n} l n (\frac{T_{j + 1}}{T_{j}}) .

$w=\frac{1}{2\uppi i}{\mathop{\sum}\limits_{j=1}^{n}}{\mathrm{ln}}\left(\frac{{T}_{j+1}}{{T}_{j}}\right).$

(1)

We find that $w$ is non-zero only when $2l/n$ is not an integer number, indicating the existence of a topologically non-trivial phase. In this case, we have the total complex coupling ${\sum }_{j=1}^{n}{T}_{j}=0$ . On the contrary, when $2l/n$ is an integer, $w$ becomes zero and also ${\sum }_{j=1}^{n}{T}_{j}\ne 0$ (Supplementary Note 7). In the examples of ${C}_{3}$ and ${C}_{5}$ lattices, the coupling winding is found to be non-zero for both $l=1$ and $l=2$ , thus resulting in protected vortex transport (Figs. 2 and 4f(1)–f(3)). However, in the ${C}_{4}$ lattice (Fig. 4e(1)), the winding is non-zero for $l=1$ but vanishes for $l=2$ . As such, the ${C}_{4}$ lattice can stably guide a single-charge vortex (Fig. 4e(2)) but not a double-charge vortex. In the latter case, the $l=2$ vortex breaks up into a quadrupole-like pattern (Fig. 4e(3)), in agreement with simulation results (Fig. 1c and Extended Data Fig. 2). This winding picture (see also Fig. 1b) resembles the skyrmion-like spin texture in a magnetic structure⁴⁵. Here, the non-trivial real-space winding of the complex coupling vectors depends on the lattice rotational symmetry order $n$ with respect to the vortex topological charge $l$ for a chosen OAM mode. For this reason, we name it vorticity-coordinated rotational symmetry (VRS). Although not easily visualized in a simple picture, we can consider the VRS intuitively as follows. For a given clockwise (or counter-clockwise) vortex mode excitation at the disclination core, coupling along an arbitrary closed path in the ${C}_{n}$ disclination structure will not induce counter-clockwise (or clockwise) vortex modes when coming back to the disclination core. This is guaranteed by real-space topology: one can always find other corresponding paths in the structure that are equivalent up to a rotation by $2\uppi /n$ , such that the interference of all these OVMC paths is zero, which in turn protects the excited vortex mode. Hence, the VRS can be thought of as a rotational symmetry that coordinates with the vorticity of the optical field, and it sets up a universal rule for real-space protection of a single high-order vortex (with either clockwise or anti-clockwise phase circulation but without mixing) propagating along the ${C}_{n}$ disclination core.

We further explore the general cases featuring arbitrary $n$ and $l$ and theoretically prove that the total complex couplings accounting for the OVMC between the central defect waveguide and all contributions originating from $n$ sectors vanish if and only if the winding number is non-zero, which requires a non-integer $2l/n$ (Supplementary Notes 6 and 7). Such a condition (summarized in Fig. 4a(3)) sets up a universal rule for protecting the transport of a single high-order vortex (with either clockwise or anti-clockwise phase circulation but not both) along the ${C}_{n}$ disclination core (Supplementary Note 8). By considering the alternative example of a ${C}_{5}$ disclination for even higher topological charges, we show that a vortex beam with $l=5$ or $l=10$ cannot maintain its shape during propagation, but other high-order OAM modes are well guided as long as $2l/5$ is non-integer (Extended Data Fig. 5), further validating the established condition. Results in Fig. 4 highlight another key ingredient needed for protecting the vortex transport: real-space topology mediated by the VRS of the disclination structure.

Topological extraction of an OAM mode

The presented features of real and momentum-space topology can be readily employed to single out a selected OAM mode (non-trivial) from a mixed-mode excitation, while other modes (trivial) dissipate into the bulk. In Fig. 5, we show a proof-of-concept demonstration. For a mixed excitation of $l=1$ and $l=2$ modes, by judiciously choosing the lattice parameters (Supplementary Note 3), we achieve different transport dynamics of the OAM modes through a single waveguide versus a disclination structure: in a single waveguide, beating between $l=1$ and $l=2$ modes leads to a strongly deformed and unstable vortex pattern (Fig. 5b(1)–b(4)), whereas, in the TVG, the $l=2$ mode is ‘filtered’ out but the $l=1$ mode is preserved during transport (Fig. 5c(1)–c(4)). In Fig. 5a, we also plot the modal weighting into the OAM basis (OAM mode distribution) before (blue) and after (red) filtering takes place, which clearly shows that the $l=2$ mode is suppressed after propagating through the TVG, especially when compared with a single waveguide that does not have any filtering (Methods). These experimental results along with numerical simulations clearly demonstrate that topological disclinations can be implemented to extract and transport a desired OAM mode, promising for structure-based vortex filters.

**Fig. 5: OAM extraction (filtering) from mixed-mode excitation via topological disclination.**

Discussion

We have demonstrated the fundamental principle behind OAM mode extraction and twofold protection of vortex transport via chiral-symmetric topological disclinations. The principle allows for robust guidance of a single zero-energy vortex mode, or for selection of one of the degenerate OAM modes from mixed-mode excitations, which cannot be achieved in topologically trivial waveguide structures including specially designed optical fibres^12,17. Even if the structure has non-trivial topology in momentum space, that is, it supports zero-energy mode due to chiral symmetry inherent to the SSH-like system, it can support and protect an OAM mode only when the VRS is also satisfied. This reflects the interplay of vorticity, symmetry and topological phases. Detailed stability analyses under different perturbations (respecting chiral symmetry, subsymmetry⁴⁶ and rotational symmetry) are presented in Supplementary Notes 9 and 10, confirming the advantage of twofold protection from the TVG approach. In Supplementary Note 11, we show how a high-order vortex ( $l=5$ ) remains guided (no diffraction, no splitting) in the TVG, however it dramatically diffracts and breaks up into multiple $l=1$ vortices under the same perturbation in the absence of a waveguide (Extended Data Fig. 6 and Supplementary Media File (3)).

Over the past decades, guiding light has been based on the paradigm of either total internal reflection or photonic bandgaps, but recent exploration has heralded new mechanisms for unconventional transport, including, for example, guiding light by geometric phases⁴⁷, by centrifugal barriers from the OAM of light itself⁴⁸ and by optical Coriolis forces around the Lagrange points⁴⁹. Our topological approach certainly opens a new avenue for guiding light, particularly for protected vortex transport.

While this work focuses mainly on the fundamental principle, it may bring about a solution to the long-standing challenge of controlling vortex transport applicable to different fields, since the underlying physics for topological protection of vortices is broadly valid. For instance, it may be applied to acoustics and topolectrical circuitry where chiral symmetry has already been realized^24,50,51. In technologically important structures like photonic crystals, recent work has demonstrated that photonic crystal fibres can be designed to host topological supermodes across multiple cores⁵². Thus, we envision that our scheme may be adopted for the design of microstructured optical fibres enabling protection of OAM modes in future communication networks¹⁷. With the rapid advancement of integrated vortex generation^11,32,53,54, the topological approach may open a pathway for routing and protecting vortices, particularly classical and quantum OAM modes, from one place to another with unprecedented transport properties.

Methods

Construction of single-site disclination lattices

The C_n-symmetric disclination used in this work is constructed by a modified cutting and gluing procedure^20,21. Compared with previous lattice structures derived from the standard two-dimensional SSH model⁴², our disclination structure is uniquely designed to have a single-site core, yet featuring chiral symmetry.

As illustrated in Extended Data Fig. 1a,b, the conventional disclination structures belong to either the type-I or type-II categories^20,21. The type of disclination is identified by the amount of translation and rotation of a vector around a chosen path (depending on the translation value $\left[{\mathbf{a}}\right]$ and the Frank angle $\varOmega$ )²³. Type-I disclination lattices with Frank angle $\varOmega =-90^\circ$ and the holonomy value of a closed path around the core ${\left[{\mathbf{a}}\right]}^{\left(4\right)}=0$ are terminated by weak bonds at the centre location (Extended Data Fig. 1a). Complementarily, type-II lattices with $\varOmega =-90^\circ$ and ${\left[{\mathbf{a}}\right]}^{\left(4\right)}=1$ have strong bonds around the defect core (Extended Data Fig. 1b). In the non-trivial phase, the Wannier centres (quadrangular yellow stars) are positioned at the intersection among four-unit cells. We note that chiral symmetry is not present in both types of disclination²⁰.

The ${C}_{n}$ disclination with a single-site core used in our work cannot be simply categorized as one of the above classes, and its formation requires the removal of some lattice sites instead of just cutting and gluing. To guarantee the existence of zero-energy bound states, we appropriately modify an initial type-II disclination structure. The resulting lattice displays three-unit cells composed of four sites intersecting at the centre, each of which belongs to one of the C₃-symmetric sectors (Extended Data Fig. 1c). We first shift every lattice sector with respect to the core until the three nearest waveguides perfectly overlap. The white arrows in the inset indicate the directions to shift the lattice sites. Then, any overlapped (extra) lattice sites that break chiral symmetry in the traditional disclination structure are removed, so that the array index is still uniform. In a similar way, other C_n-symmetric lattices can be readily constructed. A characteristic difference between our scheme and those previously shown type-I and type-II disclinations is that our single-core ${C}_{n}$ disclinations possess chiral symmetry and can thus support topologically protected zero-energy bound states.

Discrete vortex Hamiltonian

In the OAM domain, we express the real-space Hamiltonian of a C_n-symmetric disclination lattice with topological charge $l$ under the tight-binding approximation as

H = \sum_{R, R^{'}, l} ξ (R - R^{'}) [κ_{S V} c_{R, l}^{†} c_{R^{'}, l} + κ_{O V} e^{2 i l θ (R - R^{'})} c_{R, l}^{†} c_{R^{'}, - l}],

$H=\mathop{\sum }\limits_{{\bf{R}},{{\bf{R}}}^{{{{\prime} }}},l}\xi \left({\bf{R}}{{-}}{{\bf{R}}}^{{{{\prime} }}}\right)\left[{\kappa }_{{\rm{SV}}}{c}_{{\bf{R}}{{,}}l}^{\dagger }{c}_{{{\bf{R}}}^{{{{\prime} }}},l}+{\kappa }_{{\rm{OV}}}{{\mathrm{e}}}^{2{il}\theta \left({\bf{R}}{{-}}{{\bf{R}}}^{{{{\prime} }}}\right)}{c}_{{\bf{R}}{{,}}l}^{\dagger }{c}_{{{\bf{R}}}^{{{{\prime} }}},-l}\right],$

(M1)

where $\xi \left({\bf{R}}{{-}}{{\bf{R}}}^{{{{\prime} }}}\right)={{\mathrm{e}}}^{-\rho \left|{\bf{R}}{{-}}{{\bf{R}}}^{{{{\prime} }}}\right|}$ is the hopping amplitude between two nearest-neighbour waveguides of the C_n-disclination lattice located at the positions ${\bf{R}}$ and ${\bf{R}}^{\prime}$ , and $\rho$ is a scale factor. The hopping amplitudes are approximated as an exponential decay function of the difference $\left|{\bf{R}}{\boldsymbol{-}}{{\bf{R}}}^{{\boldsymbol{{\prime} }}}\right|$ (ref. ⁵⁵). The parameter ${\kappa }_{{\rm{SV}}}$ describes the SVMC, while ${\kappa }_{{\rm{OV}}}$ describes the OVMC, with $\theta \left({\bf{R}}{{-}}{{\bf{R}}}^{{{{\prime}}}}\right)$ being the azimuth angle of the vector ${\bf{R}}{{-}}{{\bf{R}}}^{{{{\prime} }}}$ . ${c}_{{\bf{R}}{{,}}l}^{\dagger }$ is the creation operator at the lattice site with position ${\bf{R}}$ , corresponding to a vortex mode with a topological charge $l$ . An analogous definition is given for the annihilation operators. The vortex band structures reported in Fig. 3 are calculated by diagonalizing $H$ for the same C₃-disclination structure, but distinct $l$ values. Related vortex-mode distributions are found by retrieving both clockwise ( $l < 0$ ) and anti-clockwise ( $l > 0$ ) components from the calculated eigenvectors of $H$ .

Experimental methods

We create three disclination photonic lattices with C₃-, C₄- and C₅-rotational symmetries by employing a site-to-site CW-laser-writing technique in a 20-mm-long photorefractive crystal^43,56. For the ${C}_{3}$ lattice shown in Fig. 2a, the intra-cell and inter-cell waveguide distances (corresponding to ${d}_{1}$ and ${d}_{2}$ in Fig. 3b(1)) are, respectively, 57.5 μm and 42.5 μm for guiding the $l=1$ vortex, and 69 μm and 51 μm for guiding the $l=2$ vortex. In the writing process, an ordinary-polarized laser beam with a 532 nm wavelength and a low power of about 70 mW is phase-modulated in the Fourier domain by a spatial light modulator, to create a quasi-non-diffracting beam at variable writing positions. Every waveguide remains intact during each set of measurements owing to the photorefractive ‘memory effect’^43,56. The probing process is performed by launching into the disclination core an extraordinary-polarized vortex beam at the same wavelength with different topological charges. The probe vortex is generated by imposing a helical phase together with an amplitude modulation on a Gaussian-like beam assisted with the spatial light modulator, which can be described as $\varPsi (x,y,0)=A{r}^{|{l}|}\exp (-\frac{{r}^{2}}{{w}^{2}})\exp (il\varphi )$ . This indicates that the probe beam features a high-order vortex with a topological charge $l$ (here, $A$ is an amplitude parameter, $r=\sqrt{{x}^{2}+{y}^{2}}$ and $\varphi ={\tan }^{-1}\left(y/x\right)$ are the radial and azimuthal coordinates, while w is a normalization width). Such generated vortices resemble the Laguerre–Gauss beams that carry OAM. Experimentally, interferograms are obtained by setting the interference between the vortex beam and a reference quasi-plane wave or spherical wave to identify the vorticity (see Supplementary Note 1 for more details). Numerically, we calculate the guided modes (eigenmodes) of individual waveguides in the disclination structure using experimental parameters and find that the mode profiles well fit those of Laguerre–Gauss modes. Thus, since the excitation beam is modulated to have radial modes approximately matching the eigenmodes, it will evolve into a disclination vortex mode during propagation through the structure with preserved OAM.

To demonstrate OAM filtering from a mixed-mode excitation, in Fig. 5, we plot the modal weighting into the OAM basis (OAM mode distribution) before and after the filtering from our experimental results. The OAM spectrum is obtained by calculating ${\left|\left\langle {\varPsi }_{\exp }|{\varPsi }_{l}\right\rangle \right|}^{2}$ , where ${\varPsi }_{\exp }$ is the amplitude of the light field from the experimental output beam in the disclination core, and ${\Psi }_{l}$ is the eigenmode of our photonic waveguide obtained from numerical simulations (with an induced index change $\Delta n/n=1.11\times {10}^{-4}$ and a waveguide width 36 μm; similar to our experimental condition). Both ${\varPsi }_{\exp }$ and ${\varPsi }_{l}$ are normalized so that ${\left|\left\langle {\varPsi }_{\exp }|{\varPsi }_{\exp }\right\rangle \right|}^{2}={\left|\left\langle {\varPsi }_{l}|{\varPsi }_{l}\right\rangle \right|}^{2}=1$ . The possible OAM modes involved under this condition are $l=-3,-2,\,-1,\,+1,+2,+3$ . The amplitude of the experimental output is directly acquired from the charged-coupled device (CCD) image, with its corresponding phase measured through the plane-wave interferogram using the method previously established⁵⁷. Due to the limitation of the method (projecting experimental data onto the eigenmodes calculated for an ideal single waveguide) to attain the OAM spectrum, small portions of initially unexcited modes (for example, $l=0$ ) appear to be present at output, but in reality, these modes are not involved.

Numerical methods

The propagation dynamics of an optical vortex beam are simulated using a continuum model of the nonlinear Schrödinger-like equation (NLSE)⁴⁶

i \frac{\partial Ψ}{\partial z} = - \frac{1}{2 k} \nabla_{⊥}^{2} Ψ - \frac{k Δ n_{0}}{n_{0}} \frac{Ψ}{1 + I_{L} (x, y) + I_{P} (x, y)},

$i\frac{\partial \varPsi }{\partial z}=-\frac{1}{2k}{\nabla }_{\perp }^{2}\varPsi -\frac{k\Delta {n}_{0}}{{n}_{0}}\frac{\varPsi }{1+{I}_{{\mathrm{L}}}\left(x,y\right)+{I}_{{\mathrm{P}}}\left(x,y\right)},\,$

(M2)

where $\varPsi (x,y,z)$ is the electric field envelope, x and y denote the transverse coordinates, $z$ is the longitudinal propagation distance and ${\nabla }_{\perp }^{2}={\partial }^{2}/\partial {x}^{2}+{\partial }^{2}/\partial {y}^{2}$ is the transversal Laplacian operator. Here, $k$ is the wavenumber in the medium, ${n}_{0}=2.35$ is the refractive index for our specific photorefractive crystal and $\Delta {n}_{0}=-{n}_{0}^{3}{r}_{33}{E}_{0}/2$ is the refractive-index change, where r₃₃ = 280 pm V⁻¹ is the electro-optic coefficient along the crystalline $c$ axis, and ${E}_{0}$ is the bias electric field. The two terms ${I}_{{\mathrm{L}}}(x,y)$ and ${I}_{{\mathrm{P}}}\left(x,y\right)$ denote the intensity patterns of the lattice-writing and lattice-probing beams, respectively. To confirm the theoretical prediction of TVG formation in the proposed C_n-disclination lattice, experimental measurements are also corroborated by numerical simulations using the NLSE in equation (M2) (Supplementary Note 2). For a linear vortex-beam excitation, the NLSE solutions are found via a split-step Fourier transform method under the condition that ${I}_{{\mathrm{P}}}\left(x,y,z\right)$ is weak, so the probe beam itself does not undergo nonlinear self-action during propagation.