On-chip vectorial structured light field manipulation by inverse design

Abstract

Structured light, with its multidimensional control over amplitude, phase, space and frequency, is a key enabler for advanced technologies such as high-capacity communications, quantum information, and super-resolution imaging. Here, we propose a unified inverse-design methodology for arbitrary on-chip vectorial structured-light. Inspired by quantum-state representations, we describe complex vector fields as finite-dimensional vectors in a Hilbert space and introduce a transmission-matrix formalism that links input waveguide modes to target topological edge states. By combining this mapping with adjoint-based topology optimization, we obtain the permittivity distribution within a compact design window that realizes the desired vector transformation while preserving topological transport. We experimentally demonstrate two representative domain-wall configurations on a valley photonic crystal (VPC) platform, termed Type-I and Type-II topological couplers, which efficiently couple the fundamental {\mathrm{T}\mathrm{E}}_0 mode into valley pseudospin edge states. Simulations of the ideally designed device show insertion losses of 0.04 dB and 0.09 dB at 1550 nm with 3-dB bandwidths of 132 nm and 65 nm, respectively. Experimentally, the fabricated device, which was designed accounting for fabrication tolerances, maintains a broadband low-loss performance, with measured losses of < 0.6 dB at 1550 nm with 3-dB bandwidth over > 60 nm and < 0.8 dB at 1550 nm with 3-dB bandwidth over 87 nm. Mirror-symmetric designs further validate selective excitation of orthogonal pseudospin states. Our results establish this inverse-design methodology as a powerful tool for strictly controlling on-chip vectorial light, paving the way toward compact, broadband, and multifunctional photonic integrated circuits for optical computing, communications, and beyond.

Graphical abstract

Keywords

Inverse design / On-chip structured light / Valley photonic crystal / Chalcogenide photonics

1 Introduction

Structured light [1,2], with its programmability in multiple degrees of freedom including amplitude, phase, space, frequency and polarization [3–5], represents a frontier field in modern optics. Unlike conventional Gaussian beams or plane waves, structured light offers extended control over spatial and polarization states, such as orbital-angular-momentum (OAM) beams [6], vector beams with spatially varying polarization [7–10], and self-accelerating Airy beams [11]. This diversity of field structures opens new avenues for advanced applications, including high-capacity optical communications [12,13], quantum information processing [13,14], super-resolution imaging [15–17], and optical micromanipulation [18]. However, the generation and manipulation of these fields in free space rely on bulky, alignment-sensitive setups [19], which severely limit their practicality. This fundamental limitation makes the transfer of these capabilities to on-chip integration [20–23] a necessary step toward compact, stable, and scalable optical systems. For this purpose, photonic integrated circuits (PICs) [24–26] offer an ideal platform for bringing on chip structured light into real-world applications [27–29], with the ultimate goal of achieving precise generation and manipulation of complex optical fields.

Valley photonic crystals (VPCs) [30], a key PIC platform, provide an ideal physical platform for on-chip structured-light manipulation [31]. These structures support pseudospin states characterized by spatially varying field and polarization distributions [32,33]. Their modal field profiles inherently exhibit the essential features of structured light, effectively serving as an on-chip “reservoir” of structured light states. Furthermore, VPCs host two inequivalent valleys (|K⟩ and |K^{\prime }⟩) as additional degrees of freedom [34] and support topological edge states with distinct chiral responses [35], thereby offering a robust foundation for multidimensional optical control. However, despite this potential, a unified design paradigm for precise on-chip structured-light manipulation within such topological platforms is still lacking.

Inverse design [36,37] offers a new approach. It allows for coordinated control of multidimensional vectorial properties within a compact footprint, while preserving the robust transport characteristics of topological edge states. By directly defining the desired optical functionality and using the adjoint method to optimize in a high-dimensional parameter space [38], inverse design can discover optimal structures that are often inaccessible through traditional empirical approaches. This goal-oriented design paradigm overcomes the limitations of analytic modeling and manual parameter tuning, offering far greater flexibility and performance potential. Adjoint-based inverse optimization has been successfully applied to various photonic devices, such as wavelength demultiplexers [39,40] and logic components [41,42], demonstrating its generality [43,44]. When incorporated into the design of on-chip structured light manipulation, this methodology enables highly flexible, multi-parameter control. However, its comprehensive application to vectorial structured light control in topological photonic systems has not yet been systematically explored.

In this work, we propose a quantum-state–inspired [45,46] inverse design methodology that enables arbitrary on-chip manipulation of vectorial structured light on a VPC platform. This methodology models the evolution of the optical field from input to output as a linear transformation in a finite-dimensional Hilbert space. Within this space, a transmission matrix acts as the core operator for the coordinated optimization of spatial, phase, and polarization properties. We designed and realized two compact topological couplers corresponding to distinct domain-wall configurations (Type-I and Type-II). At the 1550 nm telecommunication band, simulations of the ideal design achieved ultra-low insertion losses of 0.04 dB and 0.09 dB, with broad 3-dB bandwidths of 132 nm and 65 nm, respectively. Experimentally, the fabricated devices, which were designed accounting for fabrication tolerances, exhibited measured insertion losses below 0.6 dB and 0.8 dB, with 3-dB bandwidths exceeding 60 nm and 80 nm. This study not only verifies the feasibility of achieving precise vector-field control within topological photonic crystals via inverse design but also establishes a unified design paradigm for structured light on-chip, offering a new pathway for developing multifunctional photonic integrated circuits and polarization-multiplexed communication systems.

2 On-chip structure light

The concept of structured light has greatly expanded our ability to manipulate optical fields in multiple dimensions, making its migration from free space to on-chip applications highly attractive. In the context of conventional free-space structured light, engineering amplitude, phase, polarization, and frequency degrees of freedom (DoFs) [3–5] enables the construction of a wide variety of complex light-field distributions (Fig. 1a). Representative free-space structured-light modes shown in Fig. 1 include Gaussian-like OAM modes with zero topological charge ℓ = 0 (Fig. 1c), as well as OAM beams with a central phase singularity, such as ℓ = 1 (Fig. 1e), ℓ = 2 (Fig. 1g), and even higher orders like ℓ = 3 (Fig. 1i). On chip, the goal is to generate analogous structured textures within guided vectorial fields (see Section S1 in the Supporting Information for definitions). Realizing this capability, however, remains an open problem. Conventional integrated photonic devices predominantly rely on standard transverse electromagnetic modes (TEM, Fig. 1b), which do not explicitly encode rich spatial phase/polarization textures, making it difficult to realize comparable structured-field patterns at the chip scale. It is therefore of significant interest that recent advances in topological photonics provide new physical DoFs and design paradigms for constructing structured light on-chip [24–26], particularly on VPC platforms.

In VPCs, the transverse phase distributions of the interface states often exhibit localized phase-winding patterns. For example, the valley pseudospin mode shown in Fig. 1d displays clear local phase vortices within each unit cell, visually resembling the vortex-like local phase structure found in free-space OAM beams. Similarly, higher-order topological interface states (Fig. 1f) can exhibit double or multiple phase windings at the unit-cell center, with qualitative analogous to free-space OAM modes with ℓ = 2 (Fig. 1g) or even higher orders (Fig. 1i) of increasing complexity. Meanwhile, the trivial state shown in Fig. 1h presents yet another distinct on-chip field configuration, offering a broader comparison between free-space and chip-based modal structures. Unlike the global topological charge of free-space OAM beams, these localized phase vortices in VPCs originate from the superposition of the valley-degenerate states |K⟩ and |K^{\prime }⟩, reflecting the valley Chern-number contrast (± 1/2) across the interface. Therefore, Fig. 1 as a whole illustrates the correspondence between free-space structured-light modes (Fig. 1c, e, g, i) and on-chip topological structured-light modes (Fig. 1b, d, f, h) in terms of phase, amplitude, and spatial patterns, highlighting that topological photonic platforms can naturally support high-dimensional vectorial light fields analogous to structured light, thus providing a physical foundation for realizing structured light on-chip. (See Section S1 in the Supporting Information for a rigorous discussion).

Looking beyond the devices demonstrated in this work, the inverse-design method introduced here is inherently extensible and offers a unified pathway toward multidimensional structured-light manipulation on integrated photonic platforms. While our study focuses on vectorial transformations between conventional strip-waveguide modes and valley pseudospin states, the same transmission-matrix formalism can be generalized to accommodate additional optical degrees of freedom. In the phase domain, the method may enable broadband, topology-compatible phase modulation elements, forming the basis for reconfigurable interferometric units or phase-controlled logic within topological photonic circuits. In the wavelength domain, extending the mapping matrix toward joint space–wavelength–valley control could support novel wavelength-division functionalities and selective spectral routing, which are essential for scaling future quantum and classical communication architectures. Furthermore, in the spatial domain, the inverse-design strategy may facilitate direct conversion between free-space structured beams and topological edge states, establishing on-chip interfaces capable of bridging free-space channels, optical fibers, and topological photonic crystals. More broadly, since the framework can naturally evolve from a matrix representation into a high-dimensional tensor formulation, it may eventually enable simultaneous control over phase, wavelength, polarization, orbital angular momentum, and temporal properties within a single design paradigm. These prospects highlight the versatility and scalability of inverse-designed topological photonics and point toward a promising route for realizing compact, broadband, and multifunctional integrated platforms for advanced optical signal processing and quantum information technologies.

3 Structure light manipulation by inverse design

To achieve arbitrary on-chip manipulation of vectorial structured light within a VPC platform, we employ a quantum-state–inspired representation to guide the inverse design framework centered on the transmission matrix, as illustrated in Fig. 2. It provides a compact Hilbert-space notation, which arises as the vector space spanned by orthogonal spatial and polarization modes naturally and provides a rigorous framework for describing field superposition and linear optical transformations (see Section S2 in the Supporting Information for a detailed discussion). We could represent any on-chip vectorial structured light field as a quantum-state superposition of basis vectors [46]:

|E⟩=\sum _{i=1}^N\sum _{j\in \{x,y,z\}}{C}_{ij}|r_i⟩\otimes |p_j⟩.

Here, |r_i⟩ denotes the basis state associated with the i-th spatial sampling point on the observation plane, N is the total number of spatial sampling points, and |p_j⟩denotes the orthonormal polarization basis state corresponding to the Cartesian unit vector {\hat{e}}_j. The complex coefficient C_{ij} corresponds to the complex electric-field amplitude of the j-th polarization component evaluated at the spatial location r_i, thereby encoding the amplitude and phase information of the structured light field. Directly operating in a continuous, infinite-dimensional space is neither necessary nor efficient for numerical implementation. Therefore, we sample the vectorial fields on selected transverse planes, restricting the spatial and polarization degrees of freedom to a finite set. Consequently, the input and output fields are represented as finite-dimensional complex matrices:

E_{\mathrm{i}\mathrm{n}}\in C^{N_{\mathrm{i}\mathrm{n}}\times d},E_{\mathrm{o}\mathrm{u}\mathrm{t}}\in C^{N_{\mathrm{o}\mathrm{u}\mathrm{t}}\times d},

where N_{\mathrm{i}\mathrm{n}}/N_{\mathrm{o}\mathrm{u}\mathrm{t}} denote the number of sampling points or modes, d denotes the dimensionality of the polarization subspace, and C denotes the complex number field. When the spatial sampling grid is sufficiently dense and the discretization satisfies the constraints of Maxwell’s equations, this representation is information-equivalent to the original continuous vector field. It thus serves as the computational workspace for subsequent modeling and optimization, as illustrated in Fig. 2a (for more information regarding the selection criteria for the discretized-observation plane and clarifications, see Section S4 in the Supporting Information). Under this representation, the device functionality can be abstracted as a linear transformation from E_{\mathrm{i}\mathrm{n}} to E_{\mathrm{o}\mathrm{u}\mathrm{t}}:

E_{\mathrm{o}\mathrm{u}\mathrm{t}}=T\cdot E_{\mathrm{i}\mathrm{n}},

where T is the transmission matrix, which characterizes how the input spatial-polarization states are transformed into the target structured-light states within the chosen finite-dimensional basis. Physically, T encodes the scattering and propagation behavior of electromagnetic waves under the given dielectric distribution. For the operator-level derivation relating the continuous Maxwell operator to the finite-dimensional transmission matrix (via projection/synthesis operators), see Section S3 in the Supporting Information.

In this work, we first specify the desired target field E_{\mathrm{o}\mathrm{u}\mathrm{t}}, such as achieving efficient, phase-preserving, or polarization-selective transformation from the single-mode waveguide {\mathrm{T}\mathrm{E}}_0 mode to a specific topological pseudospin edge state. Subsequently, by numerically solving Maxwell’s equations, we establish the mapping from the dielectric permittivity distribution \epsilon (r) inside the device to the transmission matrix T(\epsilon ) evaluated on the discretized observation plane. Figure 2a conceptually illustrates the relationship among the input optical field, the transmission matrix, and the target vectorial structured-light state. It depicts how, within the chosen finite-dimensional basis, the input spatial–polarization states are mapped to the desired output state. The complete three-dimensional evolution of the optical field can be reconstructed by solving Maxwell’s equations numerically on a discretized grid, ensuring the accuracy of this mapping within its finite-dimensional representation.

Using the inverse-design optimization algorithm, we design a nonuniform dielectric permittivity distribution within a predefined compact region to realize the desired transmission matrix. The overall design workflow is illustrated in Fig. 2b. After defining the target mapping and monitors, the design is parameterized by a pixelated density field within the optimization window. In each iteration, a forward simulation evaluates the figure of merit (FOM) associated with the desired transmission-matrix mapping, and an adjoint simulation provides the gradient with respect to the design variables. A gradient-based optimizer then updates the design field. To obtain physically realizable layouts, the updated design is processed by spatial filtering (e.g., top-hat convolution) and a projection step (e.g., a Heaviside-type mapping), which progressively drives the permittivity distribution toward a near-binary two-material pattern. The projection strength is increased through a continuation scheme until the design becomes sufficiently binarized. To improve the manufacturability of the optimized designs, we incorporate a design-for-manufacturing (DFM) stage into the topology-optimization workflow. By introducing a minimum-feature-size penalty term, this stage suppresses ultra-fine structures that are sensitive to fabrication errors, thereby enhancing process robustness, albeit at the cost of some ideal simulated performance. Representative permittivity patterns in different stages are shown in Fig. 2c. Further workflow details are provided in Section S5 of the Supporting Information.

Traditional optical-field control approaches often rely on analytical formulations, which become cumbersome for complex vectorial fields—especially on-chip structured light in VPCs. In contrast, our framework provides a unified, implementation-friendly route by constructing a transmission matrix that maps between input and target vectorial field states within a chosen finite-dimensional basis, while being physically realized by a compact dielectric structure governed by Maxwell’s equations. Importantly, this transmission-matrix–based inverse-design paradigm is not limited to the pseudospin edge states demonstrated here: once the desired target field is specified on the observation plane(s), the same workflow can be directly extended to synthesize and manipulate other representative vectorial structured-light states (e.g., cylindrical vector beams [8], etc.) on chip, by appropriately defining the target amplitude–phase–polarization distribution and the corresponding figure of merit.

4 Inverse designed coupler for topological photonics

The efficient coupling between standard single-mode strip waveguides and topological photonic crystals is fundamentally limited by modal field mismatch. Here, we systematically overcome this limitation using our inverse-design framework, designing two compact couplers for distinct valley-topological domain-wall configurations.

The first configuration, referred to as the Type-I domain wall, consists of two interleaved honeycomb valley photonic crystal lattices (VPC1 and VPC2), as shown in Fig. 3a, forming a complete valley photonic crystal structure illustrated in Fig. 3b and 3c. By breaking spatial inversion symmetry via a structure with a = 480 nm, R = 223 nm, and r = 78 nm, a photonic bandgap is opened in the 1460–1630 nm wavelength range (Fig. 3d). Due to the valley Hall effect, light propagates along the interface between the two oppositely polarized VPC regions in the form of pseudospin modes. Previous studies have attempted coupling through line-defect waveguides, multimode waveguides, or asymmetric junctions; however, these approaches often suffer from high insertion loss (IL), large footprint, or the inability to achieve single-mode coupling.

Using the aforementioned inverse-design approach, we designed a compact 5 μm × 5 μm structure, which is referred to as the Type-I coupler, that satisfies the transmission-matrix mapping between the {\mathrm{T}\mathrm{E}}_0 mode and the valley pseudospin field. The optimization objective was defined over the target wavelength band of 1540−1560 nm and the resulting ideal design is shown in Fig. 3e. The device connects a single-mode strip waveguide (500 nm × 200 nm) to the VPC, and its optical-intensity distribution was simulated at a wavelength of 1550 nm. Under ideal fabrication conditions, the Type-I coupler achieves an insertion loss of only 0.04 dB at 1550 nm (Fig. 3g, red dashed line, simulated result) and exhibits a 3-dB bandwidth of 132 nm (1484−1616 nm), enabling efficient conversion of the wavevector from the {\mathrm{T}\mathrm{E}}_0 mode into the pseudospin mode within the VPC. Figure 3g and 3i comprehensively demonstrate the superiority of our design. At the wavelength of 1550 nm, optical energy injected from the single-mode waveguide into the topological coupler enters the valley photonic crystal with near-zero reflection, and then continues seamlessly into the output waveguide (Fig. 3i, left).

For comparison, when the VPC is directly connected to the single-mode waveguide (Fig. 3i, center), the field mismatch causes significant insertion loss (IL = 2.54 dB @ 1550 nm, green dashed line in Fig. 3g) and oscillations along the transmission path. Simulated power-flow distributions clearly show standing-wave patterns both inside the waveguide and within the photonic crystal, indicating strong back-reflection. It is also noteworthy that the two pseudospin modes possess spatial chiral symmetry. Aligning the Type-I coupler in a mirror-symmetric configuration enables excitation of the orthogonal pseudospin mode. As shown in Fig. 3i (right), the injected pseudospin light corresponds to the opposite local handedness (RCP instead of LCP) due to selective excitation of the opposite pseudospin branch. Under ideal fabrication conditions, the coupler yields a peak extinction ratio (ER) of 11.79 dB at 1550 nm (the difference between the insertion loss of the red curve and that of the blue dashed curve in Fig. 3g). This behavior arises because the two pseudospin modes in the VPC are orthogonal and counter-propagating, causing optical energy to remain largely confined within the input waveguide and suppressing transmission into the photonic-crystal region when the opposite pseudospin is excited.

In addition to the ideal design in Fig. 3e, we carried out a fabrication-aware re-optimization to obtain the fabrication-aware design shown in Fig. 3f. Fabrication tolerance is incorporated at the optimization level by imposing manufacturability constraints via spatial filtering and a minimum-feature-size control. Concretely, a larger filtering strength is applied and a minimum-feature-size penalty is enabled to eliminate ultra-fine features (e.g., small islands, sharp corners, and narrow bridges) that are highly sensitive to lithography/etching variations. As a result, the fabrication-aware design is obtained by re-optimizing the structure under stricter constraints rather than by simply correcting or smoothing the ideal design. Since the feasible design space is reduced by these constraints, the fabrication-aware design can exhibit a different field pattern and a slightly degraded ideal simulated performance, but it is expected to be more robust and closer to experimental behavior under the available fabrication process. Taking into account the constraints of the fabrication platform, we fabricated the fabricated device based on this fabrication-aware design, and the measured results are shown in Fig. 3h. The measured results, indicated by the red dotted line, show an insertion loss of 0.58 dB at 1550 nm, with a 3-dB bandwidth exceeding 60 nm (ranging from 1526 nm to 1586 nm).

The Type-II domain wall is obtained by symmetrically transforming the VPC configuration shown in Fig. 3a, resulting in two new VPCs illustrated in Fig. 4a. When arranged together, they form a distinct topological boundary state, as shown in Fig. 4b, the Type-II domain wall opens a photonic bandgap spanning 1440–1620 nm (Fig. 4d). Using the same inverse-design procedure, we constructed the Type-II coupler (Fig. 4c) to verify that the proposed framework can adaptively optimize different domain-wall configurations with the optimization objective defined over the target wavelength band of 1540−1560 nm. Under ideal fabrication conditions, the ideal design achieved an insertion loss of 0.09 dB at 1550 nm (Fig. 4g, red dashed line, simulated result) and a 3 dB bandwidth of 65 nm (1504−1569 nm), realizing efficient wavevector conversion from the TE₀ mode to the pseudospin mode in the VPC. Figure 4i presents the optical-intensity distributions resulting from excitation through the ideal design of Type-II coupler, a direct connection, and its mirror-symmetric counterpart. Similar to the Type-I coupler, the Type-II coupler enables near-zero-loss mode conversion. In contrast, direct coupling between the single-mode waveguide and VPC (Fig. 4i, middle) causes significant field mismatch, resulting in an insertion loss of 3.17 dB at 1550 nm (Fig. 4g, green dashed line, simulation). With the mirror-symmetric configuration, the device exhibits a peak extinction ratio (ER) of 10.55 dB at 1554 nm, defined as the transmission contrast between the two opposite pseudospin excitations (Fig. 4g, simulation). Finally, the fabricated device based on the fabrication-aware design was fabricated on the GSSe chalcogenide glass platform, as shown in Fig. 4c. The fabricated device (Fig. 4f) exhibits an insertion loss of 0.77 dB at 1550 nm and a 3 dB bandwidth of 87 nm (1486−1573 nm), as indicated by the red dotted line in Fig. 4h, in good agreement with simulation results of the fabrication-aware design.

5 Conclusion

In conclusion, we have proposed and experimentally demonstrated an inverse-design-based framework for on-chip vectorial structured-light manipulation, achieving efficient control of complex optical fields on a VPC platform, achieving efficient and low-loss coupling from conventional waveguide modes to topological pseudospin states. By introducing a quantum-inspired mapping matrix, the optical-field conversion problem in an infinite-dimensional Hilbert space is transformed into a finite-dimensional matrix optimization, enabling precise control of complex vector fields. Simulations of the ideally designed device show insertion losses of 0.04 dB and 0.09 dB at 1550 nm with 3-dB bandwidths of 132 nm and 65 nm, respectively. Measurements of the fabricated device, which was designed to account for fabrication tolerances, confirm a broadband low-loss performance, with losses below 0.6 dB at 1550 nm with 3-dB bandwidth over > 60 nm and below 0.8 dB at 1550 nm with 3-dB bandwidth over 87 nm. These results validate the effectiveness and practicality of inverse design for topological photonics, advancing the field from passive transport toward active control, and paving the way for high-performance, multifunctional integrated photonic circuits such as optical neural networks and quantum photonic chips.

6 Appendixes

6.1 Methods

1) Fabrication [47]

The chalcogenide glass (ChG) film was composed of Ge₂₈Sb₁₂Se₆₀. The thin films (500 nm thickness) were deposited on top of a standard silicon wafer with 2 µm thick silicon oxide by thermal evaporation. The film was then spin-coated by the positive-tone e-beam resist (ARP 6200.13). The photonic crystals and the inverse design structure were patterned by using electron beam lithography (Raith VOYAGER). The ChG of the electron beam exposed area was then etched by inductively coupled plasma etching (Oxford Plasmapro100 Cobra 180) and the patterns were finally transferred to the film by the removal of the e-beam resist.

2) Optical measurements [48]

The C band test system for the measurement was mainly composed of a broadband tuning laser (Santec TSL-570), a vertical fiber coupling platform, and a photodetector (MPM-210/210-H). The polarization direction of the light was controlled using a manual polarization controller. The transmission spectrum was measured by scanning the wavelength of the laser, and the output signal power was detected by the photodetector.

6.2 Inverse design method

Except for the proposed initialization strategy, the inverse-design procedure follows the standard adjoint-based topology optimization implemented in the Lumerical lumopt framework. In each iteration, a forward simulation evaluates the objective function F defined by the target mapping, and an adjoint simulation provides the gradient \mathrm{\partial }F/\mathrm{\partial }ϵ(r). The design is then updated using the L-BFGS optimizer. To enforce manufacturable features and promote a near-binary permittivity distribution, spatial filtering and a Heaviside projection are applied, with the projection sharpness parameter \beta following the default \beta-continuation scheme in lumopt. The optimization terminates when the objective change satisfies the convergence criterion . After convergence, an optional binarization/DFM step is performed to obtain the final fabricable structure.

For the ideal design we used filter_R = 60 nm and min_feature_size = 60 nm, whereas for the fabrication-aware design we used filter_R = 180 nm and min_feature_size = 90 nm.

References

[1] Forbes , A. , De Oliveira, M. , Dennis , M.R. : Structured light. Nat. Photonics 15(4), 253–262(2021)

[2] Nape , I. , Singh , K. , Klug , A. , Buono , W. , Rosales-Guzman , C. , McWilliam , A. , Franke-Arnold , S. , Kritzinger , A. , Forbes , P. , Dudley , A. , Forbes , A. : Revealing the invariance of vectorial structured light in complex media. Nat. Photonics 16(7), 538–546(2022)

[3] Zang , W. , Yuan , Q. , Chen , R. , Li , L. , Li , T. , Zou , X. , Zheng , G. , Chen , Z. , Wang , S. , Wang , Z. , Zhu , S. : Chromatic dispersion manipulation based on metalenses. Adv. Mater 32(27), e1904935(2020)

[4] Yu , N. , Capasso , F. : Flat optics with designer metasurfaces. Nat. Mater 13(2), 139–150(2014)

[5] Zou , X.J. , Zheng , G.G. , Yuan , Q. , Zang , W. , Chen , R. , Li , T. , Li , L. , Wang , S. , Wang , Z. , Zhu , S. : Imaging based on metalenses. PhotoniX 1(1), 2(2020)

[6] Berry , M. : Making waves in physics. Nature 403(6765), 21(2000)

[7] Bauer , T. , Banzer , P. , Karimi , E. , Orlov , S. , Rubano , A. , Marrucci , L. , Santamato , E. , Boyd , R.W. , Leuchs , G. : Observation of optical polarization Möbius strips. Science 347(6225), 964–966(2015)

[8] Mao , D. , Zheng , Y. , Zeng , C. , Lu , H. , Zhao , J. : Generation of polarization and phase singular beams in fibers and fiber lasers. Adv. Photon 3(1), 014002(2021)

[9] Liu , M. , Huo , P. , Zhu , W. , Zhang , C. , Zhang , S. , Song , M. , Zhang , S. , Zhou , Q. , Chen , L. , Lezec , H.J. , Agrawal , A. , Lu , Y. , Xu , T. : Broadband generation of perfect Poincaré beams via dielectric spin-multiplexed metasurface. Nat. Commun 12(1), 2230(2021)

[10] Ndagano , B. , Nape , I. , Cox , M.A. , Rosales-Guzman , C. , Forbes , A. : Creation and detection of vector vortex modes for classical and quantum communication. J. Lightwave Technol 36(2), 292–301(2018)

[11] Efremidis , N.K. , Chen , Z. , Segev , M. , Christodoulides , D.N. : Airy beams and accelerating waves: an overview of recent advances. Optica 6(5), 686(2019)

[12] Wang , J. , Yang , J.Y. , Fazal , I.M. , Ahmed , N. , Yan , Y. , Huang , H. , Ren , Y. , Yue , Y. , Dolinar , S. , Tur , M. , Willner , A.E. : Terabit free-space data transmission employing orbital angular momentum multiplexing. Nat. Photonics 6(7), 488–496(2012)

[13] Willner , A.E. , Huang , H. , Yan , Y. , Ren , Y. , Ahmed , N. , Xie , G. , Bao , C. , Li , L. , Cao , Y. , Zhao , Z. , Wang , J. , Lavery , M.P.J. , Tur , M. , Ramachandran , S. , Molisch , A.F. , Ashrafi , N. , Ashrafi , S. : Optical communications using orbital angular momentum beams. Adv. Opt. Photonics 7(1), 66(2015)

[14] Arrazola , J.M. , Bergholm , V. , Brádler , K. , Bromley , T.R. , Collins , M.J. , Dhand , I. , Fumagalli , A. , Gerrits , T. , Goussev , A. , Helt , L.G. , Hundal , J. , Isacsson , T. , Israel , R.B. , Izaac , J. , Jahangiri , S. , Janik , R. , Killoran , N. , Kumar , S.P. , Lavoie , J. , Lita , A.E. , Mahler , D.H. , Menotti , M. , Morrison , B. , Nam , S.W. , Neuhaus , L. , Qi , H.Y. , Quesada , N. , Repingon , A. , Sabapathy , K.K. , Schuld , M. , Su , D. , Swinarton , J. , Száva , A. , Tan , K. , Tan , P. , Vaidya , V.D. , Vernon , Z. , Zabaneh , Z. , Zhang , Y. : Quantum circuits with many photons on a programmable nanophotonic chip. Nature 591(7848), 54–60(2021)

[15] Devlin , R.C. , Ambrosio , A. , Rubin , N.A. , Mueller , J.P.B. , Capasso , F. : Arbitrary spin-to-orbital angular momentum conversion of light. Science 358(6365), 896–901(2017)

[16] Curd , A. , Cleasby , A. , Makowska , K. , York , A. , Shroff , H. , Peckham , M. : Construction of an instant structured illumination microscope. Methods 88, 37–47(2015)

[17] York , A.G. , Chandris , P. , Nogare , D.D. , Head , J. , Wawrzusin , P. , Fischer , R.S. , Chitnis , A. , Shroff , H. : Instant super-resolution imaging in live cells and embryos via analog image processing. Nat. Methods 10(11), 1122–1126(2013)

[18] Yang , Y. , Ren , Y.X. , Chen , M. , Arita , Y. , Rosales-Guzmán , C. : Optical trapping with structured light: a review. Adv. Photon 3(3), 034001(2021)

[19] Rubinsztein-Dunlop , H. , Forbes , A. , Berry , M.V. , Dennis , M.R. , Andrews , D.L. , Mansuripur , M. , Denz , C. , Alpmann , C. , Banzer , P. , Bauer , T. , Karimi , E. , Marrucci , L. , Padgett , M. , Ritsch-Marte , M. , Litchinitser , N.M. , Bigelow , N.P. , Rosales-Guzmán , C. , Belmonte , A. , Torres , J.P. , Neely , T.W. , Baker , M. , Gordon , R. , Stilgoe , A.B. , Romero , J. , White , A.G. , Fickler , R. , Willner , A.E. , Xie , G. , McMorran , B. , Weiner , A.M. : Roadmap on structured light. J. Opt 19(1), 013001(2017)

[20] Bozinovic , N. , Yue , Y. , Ren , Y. , Tur , M. , Kristensen , P. , Huang , H. , Willner , A.E. , Ramachandran , S. : Terabit-scale orbital angular momentum mode division multiplexing in fibers. Science 340(6140), 1545–1548(2013)

[21] Toyoda , K. , Takahashi , F. , Takizawa , S. , Tokizane , Y. , Miyamoto , K. , Morita , R. , Omatsu , T. : Transfer of light helicity to nanostructures. Phys. Rev. Lett 110(14), 143603(2013)

[22] Kozawa , Y. , Matsunaga , D. , Sato , S. : Superresolution imaging via superoscillation focusing of a radially polarized beam. Optica 5(2), 86(2018)

[23] Deng W., Chen X., Zhao F., Dong J. : Transverse angular momentum in topological photonic crystals. J. Optics 20, 014006(2018)

[24] Bao , J. , Fu , Z. , Pramanik , T. , Mao , J. , Chi , Y. , Cao , Y. , Zhai , C. , Mao , Y. , Dai , T. , Chen , X. , Jia , X. , Zhao , L. , Zheng , Y. , Tang , B. , Li , Z. , Luo , J. , Wang , W. , Yang , Y. , Peng , Y. , Liu , D. , Dai , D. , He , Q. , Muthali , A.L. , Oxenløwe , L.K. , Vigliar , C. , Paesani , S. , Hou , H. , Santagati , R. , Silverstone , J.W. , Laing , A. , Thompson , M.G. , O’Brien , J.L. , Ding , Y. , Gong , Q. , Wang , J. : Very-large-scale integrated quantum graph photonics. Nat. Photonics 17(7), 573–581(2023)

[25] Elshaari , A.W. , Pernice , W. , Srinivasan , K. , Benson , O. , Zwiller , V. : Hybrid integrated quantum photonic circuits. Nat. Photonics 14(5), 285–298(2020)

[26] Zelaya , K. , Honari-Latifpour , M. , Mandal , K.K. , Friedman , J. , Madamopoulos , N. , Miri , M.A. : On-chip unitary generation of arbitrary complex spatial photonic states. Optica 12(9), 1492(2025)

[27] Yablonovitch , E. : Photonic band-gap structures. J. Opt. Soc. Am. B 10(2), 283(1993)

[28] Yablonovitch , E. : Inhibited spontaneous emission in solid-state physics and electronics. Phys. Rev. Lett 58(20), 2059–2062(1987)

[29] John , S. : Strong localization of photons in certain disordered dielectric superlattices. Phys. Rev. Lett 58(23), 2486–2489(1987)

[30] Wang , Y. , Wang , H.X. , Liang , L. , Zhu , W. , Fan , L. , Lin , Z.K. , Li , F. , Zhang , X. , Luan , P.G. , Poo , Y. , Jiang , J.H. , Guo , G.Y. : Hybrid topological photonic crystals. Nat. Commun 14(1), 4457(2023)

[31] Chen , Y. , He , X.T. , Cheng , Y.J. , Qiu , H.Y. , Feng , L.T. , Zhang , M. , Dai , D.X. , Guo , G.C. , Dong , J.W. , Ren , X.F. : Topologically protected valley-dependent quantum photonic circuits. Phys. Rev. Lett 126(23), 230503(2021)

[32] He , X.T. , Liang , E.T. , Yuan , J.J. , Qiu , H.Y. , Chen , X.D. , Zhao , F.L. , Dong , J.W. : A silicon-on-insulator slab for topological valley transport. Nat. Commun 10(1), 872(2019)

[33] Shalaev , M.I. , Walasik , W. , Tsukernik , A. , Xu , Y. , Litchinitser , N.M. : Robust topologically protected transport in photonic crystals at telecommunication wavelengths. Nat. Nanotechnol 14(1), 31–34(2019)

[34] Gao , F. , Xue , H. , Yang , Z. , Lai , K. , Yu , Y. , Lin , X. , Chong , Y. , Shvets , G. , Zhang , B. : Topologically protected refraction of robust kink states in valley photonic crystals. Nat. Phys 14(2), 140–144(2018)

[35] Dong , J.W. , Chen , X.D. , Zhu , H. , Wang , Y. , Zhang , X. : Valley photonic crystals for control of spin and topology. Nat. Mater 16(3), 298–302(2017)

[36] Lalau-Keraly , C.M. , Bhargava , S. , Miller , O.D. , Yablonovitch , E. : Adjoint shape optimization applied to electromagnetic design. Opt. Express 21(18), 21693–21701(2013)

[37] Lu , J. , Vučković , J. : Nanophotonic computational design. Opt. Express 21(11), 13351–13367(2013)

[38] Yang , K.Y. , Shirpurkar , C. , White , A.D. , Zang , J. , Chang , L. , Ashtiani , F. , Guidry , M.A. , Lukin , D.M. , Pericherla , S.V. , Yang , J. , Kwon , H. , Lu , J. , Ahn , G.H. , Van Gasse, K. , Jin , Y. , Yu , S.P. , Briles , T.C. , Stone , J.R. , Carlson , D.R. , Song , H. , Zou , K. , Zhou , H. , Pang , K. , Hao , H. , Trask , L. , Li , M. , Netherton , A. , Rechtman , L. , Stone , J.S. , Skarda , J.L. , Su , L. , Vercruysse , D. , MacLean , J.W. , Aghaeimeibodi , S. , Li , M.J. , Miller , D.A.B. , Marom , D.M. , Willner , A.E. , Bowers , J.E. , Papp , S.B. , Delfyett , P.J. , Aflatouni , F. , Vučković , J. : Multi-dimensional data transmission using inverse-designed silicon photonics and microcombs. Nat. Commun 13(1), 7862(2022)

[39] Piggott , A.Y. , Lu , J. , Lagoudakis , K.G. , Petykiewicz , J. , Babinec , T.M. , Vučković , J. : Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer. Nat. Photonics 9(6), 374–377(2015)

[40] Su , L. , Piggott , A.Y. , Sapra , N.V. , Petykiewicz , J. , Vučković , J. : Inverse design and demonstration of a compact on-chip narrowband three-channel wavelength demultiplexer. ACS Photonics 5(2), 301–305(2018)

[41] Mohammadi Estakhri, N. , Edwards , B. , Engheta , N. : Inverse-designed metastructures that solve equations. Science 363(6433), 1333–1338(2019)

[42] Camacho , M. , Edwards , B. , Engheta , N. : A single inverse-designed photonic structure that performs parallel computing. Nat. Commun 12(1), 1466(2021)

[43] Sapra , N.V. , Yang , K.Y. , Vercruysse , D. , Leedle , K.J. , Black , D.S. , England , R.J. , Su , L. , Trivedi , R. , Miao , Y. , Solgaard , O. , Byer , R.L. , Vučkovicć , J. : On-chip integrated laser-driven particle accelerator. Science 367(6473), 79–83(2020)

[44] White , A.D. , Su , L. , Shahar , D.I. , Yang , K.Y. , Ahn , G.H. , Skarda , J.L. , Ramachandran , S. , Vučković , J. : Inverse design of optical vortex beam emitters. ACS Photonics 10(4), 803–807(2023)

[45] Goodman , J.W.: Introduction to Fourier Optics. 3rd ed. Roberts & Company Publishers (2005)

[46] Nape , I. , Singh , K. , Klug , A. , Buono , W. , Rosales-Guzman , C. , McWilliam , A. , Franke-Arnold , S. , Kritzinger , A. , Forbes , P. , Dudley , A. , Forbes , A. : Revealing the invariance of vectorial structured light in complex media. Nat. Photonics 16(7), 538–546(2022)

[47] Lin , X. , Wei , M. , Lei , K. , Yang , S. , Ma , H. , Zhong , C. , Luo , Y. , Li , D. , Li , J. , Lin , C. , Zhang , W. , Dai , S. , Hu , X. , Li , L. , Li , E. , Lin , H. : Compact mid‐infrared chalcogenide glass photonic devices based on robust‐inverse design. Laser Photonics Rev 17(2), 2200445(2023)

[48] Wei , M. , Lin , X. , Xu , K. , Wu , Y. , Wang , C. , Wang , Z. , Lei , K. , Bao , K. , Li , J. , Li , L. , Li , E. , Lin , H. : Inverse design of compact nonvolatile reconfigurable silicon photonic devices with phase-change materials. Nanophotonics 13(12), 2183–2192(2024)

RIGHTS & PERMISSIONS

The author(s)