Abstract
Optical vector vortex beams provide additional degrees of freedom for spatially distinguishable channels in data transmission. Although several coherent light sources carrying a topological singularity have been reported, it remains challenging to develop a general strategy for designing ultrasmall, highquality photonic nanocavities that generate and support optical vortex modes. Here we demonstrate wavelengthscale, lowthreshold, vortex and antivortex nanolasers in a C_{5} symmetric optical cavity formed by a topological disclination. Various photonic disclination cavities are designed and analysed using the similarities between tightbinding models and optical simulations. Unique resonant modes are strongly confined in these cavities, which exhibit wavelengthscale mode volumes and retain topological charges in the disclination geometries. In the experiment, the optical vortices of the lasing modes are clearly identified by measuring polarizationresolved images, Stokes parameters and selfinterference patterns. Demonstration of vortex nanolasers using our facile design procedure will pave the way towards nextgeneration optical communication systems.
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Main
Control of the angular momentum of light has attracted a great deal of attention in photonics. The optical vector vortex is particularly useful for generating different degrees of freedom for spatially distinguishable channels in data transmission^{1,2,3,4,5,6,7,8,9,10,11}. Several vortex microlasers have been successfully demonstrated using microring resonators with asymmetric scatterers^{3,4}, planar spiral nanostructures^{5}, micropillar chains^{6} and symmetric photoniccrystal slabs^{7,8}. Although the directional output and generation efficiency of these vortex beams are noteworthy, substantial scattering loss and large energy consumption are unavoidable when constructing ultrasmall optical devices due to a lack of mechanisms for highquality (highQ) light confinement while maintaining the optical vortices^{3,4,5,6,7,8}. It thus remains challenging to realize lowthreshold ultracompact laser devices capable of the selfconfiguration of optical vortex modes and robust localization of resonant modes.
However, photonic topological insulators (PTIs) have recently been proposed as attractive tools for the robust manipulation of light^{12,13,14,15,16,17,18}. Studies on PTIs have been extended to topological defects in topological crystalline insulators (TCIs), such as dislocations and disclinations, which commonly disrupt the symmetric geometries of periodic structures^{19,20,21,22,23}. Notably, the fractional disclination charge, which belongs to the class of higherorder topological insulators (HOTIs)^{24,25,26,27}, can be trapped at the boundary of disclination defects as a topological bound state^{28,29,30}. Disclination defects have been experimentally demonstrated using artificial TCI metamaterials in massive domains such as microwave circuits and millimetrescale photonic systems^{31,32}. However, the concept of disclination has not yet been applied to nanophotonics for the implementation of ultrasmall light sources of quantized topological charges.
In this Article we demonstrate wavelengthscale optical vortex and antivortex nanolasers with topological charges of ±1. Using the correspondence between tightbinding (TB) models and optical simulations, various disclination geometries are converted into unprecedented optical nanocavities. Optimized optical feedback is achieved in the cores of photonic disclination cavities transformed from C_{4} to C_{n} symmetries (n = 3, 5 and 6). In the experiment, the optical vortices of all lasing modes in the C_{5} symmetric photonic disclination cavity were identified by measuring polarizationresolved images, Stokes parameters and selfinterference patterns. This unique design procedure for photonic disclination cavities can be utilized to develop lowthreshold vortex nanolasers for nextgeneration optical communication systems.
Results
Procedure for cavity design
Figure 1a shows our laser cavity, which has a disclination structure that can generate optical vortex and antivortex modes. Localized bound states form at the disclinations, supporting eigenstates with different angular momenta due to the C_{n} rotational symmetry^{20,32}. However, it is not straightforward to design highQ photonic disclination nanocavities that not only allow for strong light confinement at the wavelength scale, but also carry eigenmodes with topological charges. To generate the desired optical vortices in a cavity, a rigorous analysis and classification of the eigenstates based on the type of disclination are required.
The TB calculation is generally used to determine the interface states of HOTIs in the twodimensional (2D) Su–Schrieffer–Heeger (SSH) model, under the assumption of a discrete lattice structure with onsite atomic elements^{15,33,34}. For example, in a C_{4} symmetric bulk lattice, a quadrumer composed of four discrete sites constructs a 2D rectangular grid (Fig. 1b). The energy band is governed by intracell coupling (t_{1}), intercell coupling (t_{2}) and nextnearest neighbour (NNN) coupling (t_{nn}). For a comparison of the TB model with an optical simulation, we established a squarelattice photoniccrystal slab consisting of four air holes in C_{4} symmetry. The fundamental transverseelectriclike (TElike) bandstructure and field profile were calculated (Fig. 1c). The H_{z}field profile at the M point presents quadrupolar antinodes confined in the four air holes at the corner (Extended Data Fig. 1).
We took advantage of the similarities between TB calculations and optical simulations—specifically, the charge density confinement in atomic sites of the TB model and the H_{z}field confinement in the air holes of the optical cavity—to develop a design strategy for a photonic disclination cavity. In particular, we asked whether optical modes with angular momenta can be confined in a photonic disclination cavity, in a similar manner to fractional charges in a disclination core^{20,30,32}. Therefore, we started by solving the TB model to determine all possible localized resonant modes and the angular momentum of each mode, and then designed a corresponding optical cavity with the same properties as the TB model.
Our design procedure was used to create C_{3}, C_{5} and C_{6} symmetric photonic disclination cavities, starting with a C_{4} symmetric bulk lattice that can be transformed into various structures through structural modification (Fig. 1d–g). By applying the Frank angle Ω (=π/2) to the C_{4} lattice with four atomic sites (Fig. 1d), disclination geometries with C_{n} symmetry can be constructed via the Volterra process (Fig. 1e,f). For example, a C_{3} symmetric disclination structure is created by removing one sector with −Ω from a C_{4} symmetric bulk lattice and combining three identical HOTI sectors. Similarly, the C_{5} and C_{6} symmetric disclination structures are created by adding one sector (+Ω) and two sectors (+2Ω) to the C_{4} lattice, respectively. Then, each atomic site is converted into an air hole to form a photonic disclination cavity (Fig. 1g).
To validate this design strategy, we compared the angular momenta of disclination structures in the TB model with those of the corresponding photonic disclination cavities from an optical simulation (Fig. 2). Here we show how disclination traps localized bound states while clarifying angular momentum characteristics in the TB calculations, and then we examine whether optical simulations of the designed photonic disclination cavities agree with the TB calculation results. To this end, the three disclination geometries depicted in Fig. 1d–g were investigated using TB calculations. Detailed structural parameters and simulation conditions for each symmetry are shown in Extended Data Fig. 2. The calculation results show that localized bound states of C_{n} symmetric disclination structures appear in the bandgap and are classified by probability density distributions with angular momenta l associated to the relative phase between the interior corners (Fig. 2, left column).
In the case of the C_{3} symmetric disclination, one nondegenerate energy state and two degenerate states are observed at a nearzero energy level (Fig. 2a). These three energy states are classified by index l, and are related to the point group symmetry between the three interior corners, as calculated in the probability density distributions of eigenvectors (Fig. 2b). Similarly, five and six eigenstates are observed at nearzero energy levels for the C_{5} and C_{6} symmetric disclinations, respectively (Fig. 2d,e,g,h). All these states in the probability density distributions with l are localized to the interior corner sites. Taken together, the C_{n} point group symmetry is satisfied by the localization of probability charge density at the C_{n} symmetric interior corners (n = 3, 5 and 6), and the energy states of l exhibit the relation \({\psi\left(r,\,\varphi +\frac{2\uppi }{n}\right)=\psi\left(r,\,\varphi \right){\rm{e}}^{i2\pi l/n}}\), where ψ is the eigenvector or the H_{z}field profile in the polar coordinate.
Next, we created photonic disclination cavities using the above TB calculations to derive eigenmodes with l (Fig. 2, right column). As described in the design strategy in Fig. 1g, the atomic sites are replaced by air holes. The airhole positions and spacings are determined by the coupling strengths t_{1}, t_{2} and t_{nn} in the TB model. We then performed 2D finiteelement method (FEM) simulations to calculate the H_{z} fields and angular momenta of the resonant modes excited in the C_{3}, C_{5} and C_{6} symmetric photonic disclination cavities (Methods and Extended Data Fig. 3). The FEM simulations reveal resonant modes confined to the disclination region that are almost identical to the TB results (Fig. 2c,f,i). This onetoone correspondence between the topological and photonic (H_{z}) ingap states, with the same angular momenta, originates from the similarities between TB and optical simulations. We note that no resonant mode is localized in the core of a photonic disclination cavity made using a trivial unit cell (Extended Data Fig. 4).
Vector vortex analysis and optical simulations
The modes with l = 0 and l = ±2 in the C_{5} symmetric photonic disclination cavity are of particular interest because they have central singularities in the electric fields. To excite these modes in a 3D cavity, the C_{5} symmetric photonic disclination cavity of Fig. 2f was formed in a photoniccrystal slab structure (Fig. 3). We calculated their topological charges, defined as \({q=1/2{{\uppi }}{\oint }_{{{C}}}{\rm{d}}\varphi {\partial }_{\varphi }{{\arg }}{E}(r,\,\varphi )}\) (ref. ^{35}), to examine the optical vortices. First, when considering the TE modes, the electric field of the C_{5} disclination mode with l = 0 is given by
where u_{0}(r) is a radial part of the electric field, and \({\hat{{\bf{e}}}}_{\rm{R}}\) and \({\hat{{\bf{e}}}}_{\rm{L}}\) are the right and leftcircular polarization bases, respectively (Supplementary Note 1). q is then calculated as +1, which is the vortex mode, also known as the azimuthal polarization state^{36,37}. In addition, the electricfield profiles of antivortex modes are produced by disclination modes with l = ±2 as follows:
which denote even and odd hybrid polarized states with radial electricfield components of u_{+}(r) and u_{−}(r), respectively^{36,37}. These are the antivortex modes, with q values calculated as −1. As a result of the superposition of ±(l − 1) (Supplementary Note 1), polarization vortices are formed^{7,37,38,39}.
As in the 2D case, we designed a 3D photonic disclination cavity by merging four, two and five air holes in the bulk, disclination boundary and core, respectively (Fig. 3a,d and Extended Data Fig. 5). The periodic air holes in the bulk open a photonic bandgap into which the defect resonant modes localized in the disclination region are introduced (Fig. 3b,e). In addition, the air holes at the disclination boundary are modified to excite different types of resonant mode with higher Q factors (Fig. 3a,d, red dashed pentagons). Without this modification in the disclination boundary, only the vortex mode is excited, even if the shape of the core changes (Extended Data Fig. 6).
We then performed systematic 3D FEM simulations in the photonic disclination slab cavities with expanded and shrunken cores to calculate the frequencies and Q factors of the resonant modes. The disclination structures are made of an InGaAsP slab with a thickness of 275 nm and lattice constant of 500 nm (Extended Data Fig. 5). In the photonic disclination cavity with expanded core (Fig. 3a), a single mode with a Q factor of 2.0 × 10^{4} at 193.9 THz was calculated as an ingap resonant state (Fig. 3b). The calculated mode volume was 0.58 (λ/n)^{3}, where λ and n are the resonant wavelength and refractive index of the slab, respectively. We also obtained nearfield and farfield profiles for this mode, including the H_{z} field, polarization vector field and phase distribution. The H_{z} field was strongly confined in the five interior air holes and disclination core (Fig. 3c), which corresponds to the eigenstate with l = 0 in Fig. 2f. The polarization vector distribution exhibits an azimuthal polarization state with a singularity at the centre. Also, the phase of (E_{x} + iE_{y}), the winding number of the polarization, represents a counterclockwise closed loop around the singularity^{35}, which indicates a vortex mode, q = +1.
We also modified a core design by shrinking the core to excite a different type of mode (Fig. 3d). The structural parameters are the same as in Fig. 3a, except for the disclination core. Doubly degenerated modes were then excited in this shrunkencore photonic disclination cavity, showing Q factors of 4.3 × 10^{3} at 201.0 THz (Fig. 3e). Their mode volumes were calculated to be 0.61 (λ/n)^{3}, indicating light confinement at the wavelength scale. In addition, the distinguishable features from those in Fig. 3c are displayed in the calculated field profiles (Fig. 3f,g). Strong confinements in the H_{z} profiles are observed only at the disclination boundaries, not at the cores, which correspond to the eigenstates with l = ±2 in Fig. 2f. In Fig. 3f,g, the polarization vector distributions are referred to as odd and even hybrid polarized states, respectively. Also, because the phase of (E_{x} + iE_{y}) is given by a clockwise closed loop around the central singularity, both modes exhibit the antivortex feature, q = −1.
Furthermore, the FEM simulation showed the mode transition from optical vortex to antivortex mode by shrinking the core. We calculated the frequencies (Fig. 3h) and Q factors (Fig. 3i) of the resonant modes as a function of Δ, the centretocentre distance between air holes in the core. The frequency of the vortex mode decreased as Δ increased from the expanded core, eventually moving out of the bandgap with a substantial decrease in Q. In contrast, the antivortex mode moved from the higherfrequency region to the bandgap with a reasonably high Q factor. This local deformation of the disclination core can increase the Q factor of a specific mode, allowing mode transition with desired polarization distributions and topological charges.
Measurements of vortex/antivortex nanolasers
For the experimental demonstration of vortex and antivortex nanolasers, we fabricated two types of photonic disclination cavity in a 287nmthick InGaAsP slab with three quantum wells (Methods). As designed using the FEM simulation in Fig. 3, the disclination core consisting of five air holes moved radially to form expanded or shrunken ones (Fig. 4a,b). The photonic disclination cavities were optically pumped at room temperature with a 980nm laser diode at a repetition rate of 1 MHz and a 2% duty cycle (Methods). We then measured photoluminescence (PL) spectra and light in–light out (L–L) curves (Fig. 4c–e). Each cavity exhibited clear lasing features, including a sharp singlemode peak and a superlinear growth above the threshold. The divergence angle of the emitted beams was measured to be 34.5°. The slightly different wavelengths and threshold values between degenerate modes (Fig. 4d,e) originated from fabrication imperfections.
We examined the polarizationresolved mode images (Fig. 4f–h), which show the following unique properties. First, without a polarizer, all lasing modes displayed doughnutshaped intensity profiles (top left). The intensity node in the centre is one of the distinguishable features of vortex/antivortex modes. In addition, the polarizationresolved mode images in each mode are fully distinguishable. For example, images with an intensity minimum along the polarizer axis were observed in the photonic disclination cavity with an expanded core, indicating the azimuthal polarization state (Fig. 4f). On the other hand, in the cavity with shrunken core, the two degenerate modes exhibited more complicated images that are completely opposite to each other (Fig. 4g,h). These images correspond to the odd and even hybrid polarization states. The measured images agree well with the simulated ones (Extended Data Fig. 7), demonstrating the effectiveness of our design strategy.
Next, to quantitatively assess the purity of the vectorial states in the lasing modes, we analysed spatially resolved Stokes parameters^{40}. To this end, we measured complete sets of polarizationresolved mode images from the vortex and antivortex lasers, including 0°, ±45° and 90° linearly polarized images and right/left circularly polarized images (Extended Data Fig. 8). Figure 5a–c shows the Stokes parameters (S_{0}, S_{1}, S_{2} and S_{3}) obtained from these polarizationresolved images (Methods). In all three cases, S_{1} and S_{2} consist of four intensity lobes that are rotated 45° from each other, whereas S_{3} has a weak intensity profile, indicating that the vector beam is of high quality^{41}. Specifically, the Stokes parameters in Fig. 5a display the azimuthal polarization distribution of the vortex beam. The colour distributions in S_{2} in Fig. 5b and S_{1} in Fig. 5c are opposite to those in Fig. 5a, demonstrating the antivortex feature. We observe a nonperfectly zero S_{3} mainly due to the small laser size and fabrication imperfections (Supplementary Fig. 1).
Furthermore, selfinterference patterns were measured in the vortex and antivortex lasing modes^{3,7} (Methods and Supplementary Fig. 2). The measured patterns using an offcentre selfinterferometry setup show that all modes have a pair of correct and inverted fork fringes^{7} (Fig. 5d–f). We observe that the order of the fork fringes differs between the two disclination cavities. These interference results, as well as the polarizationresolved mode images and Stokes parameters, show that the expanded core cavity has q = +1 (vortex), whereas the shrunkencore cavity has q = −1 (antivortex). Therefore, we have successfully demonstrated vortex and antivortex nanolasers that were strongly confined in the photonic disclination cavities, as designed in the TB models and FEM simulations of Figs. 2 and 3. We note that the mode transition between these vortex and antivortex lasing modes can occur simply by changing the core, whereas there is no transition in a photonic disclination cavity without modifying the disclination boundary (Extended Data Fig. 9).
Discussion
We have introduced and experimentally realized novel nanolasers generating optical vortices from wavelengthscale photonic disclination cavities with C_{n} symmetry. The facile design approach based on the exceptional correlation between TB models and optical simulations has been developed to transform various topological disclinations into unprecedented optical nanocavities. The vortex and antivortex modes with q = ±1 were obtained in a C_{5} symmetric photonic disclination cavity (Supplementary Note 2). Our strategy was also used to design resonant modes with q = 2 or a radial polarization state in C_{6} symmetric holetype and C_{5} symmetric rodtype photonic disclination cavities, respectively (Extended Data Fig. 10). Furthermore, our disclination nanolaser outperforms other types of vortex laser in terms of lasing threshold and cavity size (Supplementary Table 1).
This demonstration will be useful for the development of the ultimate laser sources with superior optical properties, such as small footprint, low threshold and selfconfiguration of vortex modes. Through vector mode multiplexing, various vector vortex beams can be generated that are suitable for increasing the capacity of optical communications^{39,42,43}. Multiple data streams can be transmitted simultaneously by encoding information into different vector beams. Furthermore, vortex nanolasers can have a fast modulation speed due to their small mode volume, and thus will be a promising technology for providing a new platform for highbandwidth optical communication.
Methods
TB calculations
The TB model can be used to calculate the topological energy states and angular momenta when the structural lattice undergoes complex transformations such as disclination^{30}. We performed TB calculations (Python) for the 2D SSH model including NNN hopping. The C_{4} symmetric SSH lattice with four discrete sites (Fig. 1b) was connected by weak intracell coupling (t_{1} = −0.2) and strong intercell coupling (t_{2} = −1.0), with additionally incorporated NNN coupling (t_{nn} = t_{2}/\({\sqrt{2}}\)). The lattice constant was set as 1. The C_{n} symmetric disclination geometries (Fig. 1f) consisted of a bulk lattice (grey) and a disclination region (magenta and purple) with 13 unit cells per side. The bulk lattice obeys the coupling of the SSH lattice, and the modification of coupling occurs only in the disclination region (Extended Data Fig. 2). Volterra’s process in our TB model does not violate the C_{4} symmetry of the lattice, and disclination boundary effects only alter the energy levels of the disclination states. In addition, the spatial probability density P(r) is given by P(r) = ψ(r)^{2}, where ψ(r) is an eigenvector (Fig. 2). The angular momentum l is represented by the phase shift at neighbouring interior corners of the eigenvector.
FEM simulations
The 2D (Fig. 2) and 3D (Fig. 3) FEM simulations (COMSOL Multiphysics) were performed to calculate optical resonant modes and bandstructures. Floquet periodic boundaries and perfectly matched layers were used in the lateral and outofplane directions, respectively. The geometries of the TB model were used to derive the structural parameters of the photonic disclination cavities (Extended Data Figs. 3 and 5). The refractive index of the dielectric material was 3.33. The angular momentum l was determined by the resonant frequency and farfield profile of each eigenmode. In addition, farfield images of the polarization vector fields and phase distributions were calculated at 1.5 μm above the slab surface.
Device fabrication
The samples were fabricated using a 287nmthick InGaAsP/1μmthick InP/100nmthick InGaAs/InP substrate wafer. The InGaAsP layer included three quantum wells in the middle, whose central emission wavelength was ~1.55 μm. The InP and InGaAs layers served as sacrificial and etch stop layers, respectively. The designed patterns using the COMSOL simulation were directly converted to CAD to define air holes in a photonic disclination cavity, and electronbeam lithography was performed at 30 keV on a poly(methyl methacrylate) (PMMA) layer coated on the wafer. Chemically assisted ionbeam etching was performed to drill air holes into the InGaAsP layer while using the PMMA layer as an etch mask. Finally, the sacrificial InP layer was selectively wet etched using a diluted HCl:H_{2}O (4:1) solution at room temperature, and the remaining PMMA layer on top of the slab was removed by O_{2} plasma.
Optical measurements
A 980nm pulsed laser diode (2.0% duty cycle, 1MHz period) with a spot size of ~3 μm was used to optically pump the fabricated samples at room temperature. The light emitted from the photonic disclination cavities was collected by a ×100 objective lens with a numerical aperture of 0.85 (LCPLN100XIR, Olympus) and focused onto a spectrometer equipped with an infrared array detector (SP 2300i and PyLoN, Princeton Instruments) or an InGaAs infrared camera (PA1280F70NCL, OZRAY, 1,280 × 1,024 pixels). The resolution of the spectrometer was ~0.5 nm. The measured L–L curves were plotted as a function of the peak pump power (Fig. 4c–e insets and Extended Data Fig. 9b). The polarizationresolved mode images were captured after placing a halfwave plate and polarizing beamsplitter (for linearly polarized images) or a quarterwave plate, halfwave plate and polarizing beamsplitter (for circularly polarized images) in front of the infrared camera (Fig. 4f–h and Extended Data Figs. 8 and 9c).
Stokes parameters
We obtained spatially resolved Stokes parameters by measuring the intensity of the laser beam in six polarizations: horizontal (I_{H}), vertical (I_{V}), diagonal (I_{D}), antidiagonal (I_{A}), right circular (I_{R}) and left circular (I_{L}). The Stokes parameters were as follows: S_{0} = I_{H} + I_{V}, S_{1} = I_{H} − I_{V}, S_{2} = I_{D} − I_{A} and S_{3} = I_{R} − I_{L}. All parameters were normalized by the total intensity, S_{0}.
Offcentre selfinterference measurements
The optical vortex of the lasing mode was measured using the offcentre selfinterferometry setup. In this setup, the lasing mode was split into two paths by a beamsplitter and reconverged in image space, forming the selfinterference pattern. To clearly observe this interference pattern, the central singularities of the vortex and antivortex lasing modes were slightly offcentre and magnified by a pair of the concave and convex lenses in front of the infrared camera (Supplementary Fig. 2). The selfinterference pattern obtained with such an offcentred overlap revealed a pair of fork fringes.
Data availability
All the data supporting the findings of this study are available within this Article and its Supplementary Information. Any additional information can be obtained from the corresponding authors on reasonable request. Source data are provided with this paper.
Code availability
The codes used in this work are available from the corresponding authors upon reasonable request.
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Acknowledgements
We thank D. Leykam for helpful discussions. This work was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean Government (MSIT) (nos. 2021R1A2C3006781, 2022R1I1A1A01073189 and 2022R1I1A1A01072807). H.G.P. was supported by the Institute of Applied Physics, Seoul National University. Y.K. was supported by the Australian Research Council (grant nos. DP200101168 and DP210101292) and the International Technology Center IndoPacific (ITC IPAC) via Army Research Office (contract FA520923C0023). B.J.Y. was supported by the Institute for Basic Science in Korea (grant no. IBSR009D1), the Samsung Science and Technology Foundation (project no. SSTFBA200206) and NRF grants (nos. 2021R1A2C4002773 and NRF2021R1A5A1032996).
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M.S.H., Y.K. and H.G.P. conceived the idea. M.S.H. designed the cavities and conducted the simulations. M.S.H. and H.R.K. fabricated the samples and performed the optical measurements. M.S.H., H.R.K., J.K., B.J.Y., Y.K. and H.G.P. analysed the experimental and theoretical data. All authors contributed to writing and editing the manuscript.
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Extended data
Extended Data Fig. 1 Topological phase transition in photonic bandstructures.
ac, Photonic bandstructures of topologically trivial (a), symmetric (b), and nontrivial (c) unit cells. Insets, corresponding unitcell structures. The SSHlike lattice is created with the following parameters: a = 459 nm, r_{0} = 0.20 a, and h = 275 nm, where a, r_{0}, and h are the lattice constant, radius of air hole, and slab thickness, respectively. The bandstructures are calculated as a function of a centertohole distance, d_{0}. d_{0} is 0.17 a (a), 0.35 a (b), and 0.54 a (c). The gray and orange regions in the bandstructures indicate the light cone and bandgap, respectively. d, Calculated frequency at the M point of a reciprocal lattice as a function of d_{0}/a. Topological phase transition occurs with varying d_{0}. ef, Normalized H_{z} field (left column) and phase profiles (right column) in the topologically trivial (e) and nontrivial (f) unit cells at the M point. The H_{z} field profiles for the fundamental (▲) and second bands (▼) show a band inversion between the trivial and nontrivial phases.
Extended Data Fig. 2 Detailed parameters of C_{n} disclination structures in TB model.
The black dots represent atomic sites, and the white lines represent unitcell boundaries. The coupling strengths of all atomic sites in the bulk lattice (gray) are the same as those in Fig. 1b, whereas the coupling strengths in the disclination region (magenta and purple) vary with atomic site positions. Strong coupling connects atomic sites in the disclination core (purple), whereas inverted coupling strengths to the bulk lattice define atomic sites on the disclination boundary (magenta). In our TB calculation, the weak coupling of t_{w} = –0.2 (black dashed lines) and the strong coupling of t_{s} = –1.0 (red solid lines) are set in the C_{n} disclination boundary.
Extended Data Fig. 3 Photonic disclination cavities for 2D FEM simulation.
a, C_{4} symmetric unitcell structure with a lattice constant a = 500 nm. r_{0} is the radius of air hole and d_{0} is the centertohole distance. Three different sets of (r_{0}, d_{0}) parameters are used to convert to the C_{3}, C_{5}, and C_{6} symmetric disclination geometries: (r_{0}, d_{0}) = (0.25 a, 0.51 a), (0.20 a, 0.45 a), and (0.19 a, 0.48 a), respectively. b, Designed C_{3} (top), C_{5} (middle), and C_{6} (bottom) symmetric photonic disclination cavities from the C_{4} bulk lattice. These cavities are identical to the ones in Fig. 2c,f, and i. c, Magnified structures inside the orange dashed regions in b. For each cavity, the shifting parameters of the core (d_{c}; red circle), boundary (d_{b}; green circle), and interior corner (d_{i}; blue circle) are as follows: (d_{c}, d_{b}, d_{i}) = (0.42 a, 0.34 a, 0), (0.25 a, 0.23 a, 0), and (0.23 a, 0.20 a, 0.14 a) for C_{3}, C_{5}, and C_{6} symmetric photonic disclination cavities, respectively. We note that the radius of the air hole at the disclination boundary is 0.14 a, and d_{c} = 0.40 a, for the mode with l = 0 in the C_{6} symmetric photonic disclination cavity (Fig. 2i).
Extended Data Fig. 4 Topologically trivial C_{5} disclination structure.
a, Calculated energy spectrum in a C_{5} symmetric disclination structure based on the trivial unit cell (inset). The orange box indicates the bandgap at t_{1} = –1.0 and t_{2} = –0.2. Because a trivial lattice has no corner states, fractional charges are not trapped in a disclination structure^{32}. Thus, there are no ingap modes observed. b, Calculated frequency of eigenmodes in a photonic disclination cavity based on the trivial unit cell (inset), which was constructed using the TB structure in a. Similar to a, no resonant mode is localized in the core of the cavity. The scale bar (inset) is 1 μm.
Extended Data Fig. 5 Detailed structural parameters of 3D C_{5} symmetric photonic disclination cavities.
a, Top and tilted views of the squarelattice slab structure with the following parameters: a = 500 nm, r_{0} = 0.18 a, d_{0} = 0.54 a, and h = 275 nm, where a, r_{0}, d_{0}, and h are the lattice constant, radius of air hole, centertohole distance, and slab thickness, respectively. bc, C_{5} symmetric photonic disclination cavities with nonmodified (b) and modified (c) disclination boundaries. The shifting parameters of the core (d_{c}; red circle) and boundary (d_{b}; green circle) are as follows: (d_{c}, d_{b}) = (0.28 a, 0), (0.25 a, 0.31 a), and (0.40 a, 0.31 a) for the cavity with nonmodified disclination boundary, the expandedcore cavity with modified disclination boundary, and shrunkencore cavity with modified disclination boundary, respectively.
Extended Data Fig. 6 FEM simulation in a photonic disclination cavity with a nonmodified disclination boundary.
3D FEM simulation results in the C_{5} symmetric photonic disclination cavity with nonmodified disclination boundary that is designed as shown in Extended Data Fig. 5b. a, Photonic disclination cavity. The red dashed pentagon denotes the disclination boundary. b, Calculated eigenmode frequencies (top) and Q factors (bottom) for the cavity of a. The red dot represents the resonant mode within the photonic bandgap (orange box). c, Optical vortex mode excited in the cavity of a. The H_{z} field (top), polarization vector field (bottom left), and phase distribution (bottom right) represent the vortex feature of q = +1. d, Calculated frequencies (top) and Q factors (bottom) of the vortex mode as a function of Δ, the centertocenter distance between air holes in the expanded and shrunken cores (inset). No other resonant mode appears in the bandgap. The black circles indicate the value of Δ for the cavity of a.
Extended Data Fig. 8 Measured polarizationresolved images.
Complete sets of measured polarizationresolved mode images from the vortex (a) and antivortex lasers (b and c), including 0°, ±45°, 90° linearly polarized images (middle column) and right/left circularly polarized images (right column). The images captured without a linear polarizer are shown in the top left panels. The schematics of polarization directions are shown in the bottom left panels.
Extended Data Fig. 9 Experimental vortex nanolaser in a photonic disclination cavity with a nonmodified disclination boundary.
a, SEM image of a fabricated C_{5} symmetric photonic disclination cavity with a nonmodified disclination boundary. The scale bar is 500 nm. b, Measured abovethreshold PL spectrum and L–L curve (inset) from the cavity of a. Peak wavelength and lasing threshold are 1513.2 nm and 90 μW, respectively. c, Measured polarizationresolved lasing images from the cavity of a. The mode profiles were captured after placing a linear polarizer in front of the IR camera. White arrows indicate the directions of the linear polarizer: 0°, 45°, 90°. The image captured without a linear polarizer is shown in the top left panel. The scale bar is 3 μm. d, Simulated polarizationresolved images corresponding to c.
Extended Data Fig. 10 Photonic disclination cavities for various topological charges and polarization states.
Our design strategy is used to generate resonant modes with q = 2 or radial polarization. a, C_{6} symmetric holetype photonic disclination cavity (Fig. 2i). The colored region represents the dielectric material with a refractive index of 3.33. b, Optical vortex mode excited in the cavity of a. The H_{z} field (left), polarization vector field (upper right), and phase distribution (lower right) are calculated, showing the vortex feature of q = 2. c, C_{5} symmetric rodtype photonic disclination cavity. The colored region represents the dielectric material with a refractive index of 3.33. d, Radial polarization mode excited in the cavity of c. The E_{z} field (left), polarization vector field (upper right), and phase distribution (lower right) are calculated.
Supplementary information
Supplementary Information
Supplementary Figs. 1 and 2, Table 1, Notes 1 and 2 and references.
Source data
Source Data Figs. 2, 3 and 4 and Extended Data Figs. 1, 4, 6 and 9
Statistical source data.
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Hwang, MS., Kim, HR., Kim, J. et al. Vortex nanolaser based on a photonic disclination cavity. Nat. Photon. 18, 286–293 (2024). https://doi.org/10.1038/s41566023013382
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DOI: https://doi.org/10.1038/s41566023013382
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