Topology-imprinting in nonlinear metasurfaces

Abstract

Flat optical components, or metasurfaces, have transformed optical imaging, data storage, information processing, and biomedical applications by providing unprecedented control over light-matter interactions. These nano-engineered structures enable compact, multidimensional manipulation of light’s amplitude, phase, polarization, and wavefront, producing scalar and vector beams with unique properties such as orbital angular momentum and knotted topologies. This flexibility has potential applications in optical communication and imaging, particularly in complex environments such as atmospheric turbulence and undersea scattering. However, designing metasurfaces for shorter wavelengths, such as visible and ultraviolet light, remains challenging due to fabrication limitations and material absorption. Here, we introduce an innovative concept called topology imprinting using judiciously designed all-dielectric nonlinear optical metasurfaces to replicate desired waveforms at fundamental and harmonic frequencies, opening promising avenues for advanced photonic applications.

Nonlinear metasurfaces sculpt light and its harmonics into complex topologies.

INTRODUCTION

Light has long been described as an electromagnetic wave characterized by wavelength, amplitude, phase, and polarization. However, advances in optical physics have revealed that its structure extends well past this classical picture, encompassing multiple degrees of freedom (DoFs) that surpass traditional descriptions. These additional DoFs enable vast opportunities for encoding and manipulating optical information, giving rise to structured light—a class of light beams engineered to be tailored across all their DoFs (1–3). A key milestone in this field was the realization that photons can carry orbital angular momentum (OAM) in addition to spin angular momentum, enabling an infinite-dimensional alphabet of spatial modes (4–6). In this perspective, sculpting light beyond the transverse plane has recently led to more exotic field structures such as optical knots (7–9), Möbius strips (10), optical skyrmions (11–14), and spatiotemporal optical vortices (15–18), with applications extending beyond classical and quantum optics to areas such as acoustics (19) and neutron scattering (20, 21).

Historically, structured light has been realized using linear optical elements, leveraging interference and superposition to control spatial and polarization properties. More recently, structured light generation has been explored in nonlinear optics, where spatial structuring is influenced by processes such as second- and third-harmonic generation, exemplifying the concept of light shaping light to achieve more intricate and dynamic beam control (22). However, a long-standing challenge in nonlinear optics is the strict requirement for phase matching, which severly limits efficiency. Nonlinear frequency conversion relies on phase-matched light-matter interactions of multiple waves along the propagation distance, yet achieving phase-matching conditions often demands stringent constraints on wavelength, angle, and material dispersion. Traditional approaches to phase matching involve birefringent, quasi-phase matching, or modal phase-matching techniques, all of which impose limitations on flexibility and introduce additional fabrication complexity. Moreover, conventional nonlinear optics typically relies on bulk crystals and waveguides, making integration challenging due to the need for long interaction lengths and precise angular tuning. This bulkiness hinders miniaturization and scalability, particularly in applications requiring compact, high-performance optical systems. To overcome these limitations, several strategies have been explored, with flat optical devices emerging as a promising solution.

Flat optical devices transform how we manipulate light on the nanoscale (23–26). Optical metasurfaces (MSs) offer a compact, versatile means to shape light’s wavefront, facilitating advanced signal multiplexing, demultiplexing, and processing (27–30). This wavefront-based beam shaping expands modulation formats beyond traditional methods, including time, wavelength, and polarization multiplexing (3, 31, 32). Structured light beams that carry OAM and their three-dimensional (3D) counterparts—optical knots—show promise for applications in communication protocols leveraging space- or mode-division multiplexing and particle manipulation (1, 8, 9, 33–35). Key applications of these structured light beams—such as underwater probing, nanoscale optical lithography, and quantum information processing (36–39)—highlight the need for innovative generation strategies and advanced beam shaping at shorter wavelengths, particularly in visible and ultraviolet ranges. However, despite the rapid progress in nanotechnology and nanofabrication, tailoring light-matter interactions at shorter wavelengths within the visible spectrum and beyond remains challenging. First, optical MSs at these wavelengths demand extreme fabrication tolerances, high aspect ratios, and unit cell sizes that often exceed current nanofabrication capabilities. In addition, material absorption tends to compromise the efficiency of light manipulation at shorter wavelengths. Last, the simultaneous shaping of the amplitude and phase of light beams at shorter wavelengths is particularly complicated. The higher-harmonic generation is often used, but the resulting mode, or a superposition of modes, carries either higher-order topological charges (TCs) or forms scrambled optical modes (2).

All-dielectric nonlinear optical MSs provide a versatile platform for addressing these challenges by enabling new frequency generation and offering unprecedented control over the amplitude, phase, and polarization of optical fields through nonlinear interactions. Notably, their subwavelength thickness helps bypass the stringent phase-matching requirements typically associated with these processes (22, 40–47). However, existing nonlinear MS-based solutions face several limitations. For instance, designing a single meta-atom that efficiently modulates dual frequencies without compromising performance demands a delicate balance of material properties, geometric structuring, and electromagnetic interactions, all of which must be precisely tuned to optimize both efficiency and performance. To address these limitations, we propose an approach based on topology imprinting that enables the generation of identical copies of a wavefront at both fundamental and harmonic frequencies (more generally, a harmonic of any order) and overcomes traditional material absorption barriers that are believed to be inevitable at shorter wavelengths. We optimize the MS to produce the desired optical beam topology at the fundamental frequency (FF) and show that the same topology is imprinted onto the third harmonic (TH) beam due to the nonlinear Raman-Nath phenomenon (48–51). Moreover, the generation of the TH occurs despite the pronounced absorption of polycrystalline silicon in the visible spectrum, facilitated by the nonlinear phase-locking mechanism (52–57).

The phenomenon of light diffraction by thin periodic gratings when several diffracted waves are produced is referred to as the Raman-Nath regime (48). In analogy with the linear Raman-Nath diffraction, the propagation of a light beam in periodic nonlinear structure leads to multi-order diffraction of the harmonic waves (49–51), each carrying a phase that is defined by the diffraction order value and the distribution of the modulated nonlinear susceptibility. However, the material absorption and nonresonant behaviors of the nonlinear photonic crystals are limiting factors that have yet to be addressed to increase the efficiency of the generated harmonics in the higher energies. While material absorption appears inevitable, we demonstrate that generating any desired wavefront at higher harmonics, potentially in the opaque material region, is possible by leveraging phase-locking mechanisms (52–57). In particular, for the case of third harmonic generation (THG), when a pump pulse crosses an interface between linear and nonlinear media, three third harmonic components are generated. One component is reflected into the linear medium, while the other two components, homogeneous (HOM-TH) and inhomogeneous (INHOM-TH), are transmitted. When there is no absorption, the HOM-TH component travels at a group velocity corresponding to the material dispersion at the TH wavelength. In contrast, the INHOM-TH component, also known as the phase-locked (PL) component, is trapped by the pump pulse and copropagates with the same velocity as the FF with its wavenumber given by $k_{\text{INHOM}}=3k_0\left(\mathrm{\omega }\right)n\left(\mathrm{\omega }\right)$ . Therefore, while conventional wisdom suggests that the TH wave is absorbed in the absorption window of a material, the INHOM-TH persists and propagates inside the material. However, we note that only in scenarios where the material exhibits strong absorption at the TH wavelength and provides a sufficient interaction length, the HOM-TH component undergoes pronounced attenuation, walk-off from the pump, and permits the INHOM-TH component to be isolated and measured directly. On the other hand, in cases where the material has moderate absorption or the interaction length remains subwavelength, the effective nonlinear interaction length for TH generation is relatively short, making it impossible to completely suppress the HOM-TH component, leading the TH wave to be a combination of both HOM-TH and INHOM-TH components.

Following the above-mentioned points, here we demonstrate an all-dielectric optical MS that can simultaneously generate nonlinear TH signals within the opaque spectral range of the host material while also manipulating their respective wavefronts. In particular, we generate Laguerre Gaussian (LG) beams carrying OAM and optical knots that are identical at both fundamental and TH frequencies despite the tripled frequency being in the absorptive region of polycrystalline silicon. While the conservation of OAM dictates that the TC of the TH must be three times that of the FF (58–60), we illustrate that the TH produced by the optical MS resonating at the FF has a phase front replicating that of the FF, opening a unique opportunity of shaping light beams at shorter wavelengths and overcoming the limitations of materials’ absorption and nanofabrication.

RESULTS

Design principles

To illustrate the core concept of the proposed nonlinear wavefront manipulation technique, we implemented the method outlined in (61, 62) to design a phase-only MS that simultaneously modulates the amplitude and phase of the generated harmonics—incorporating a hologram that includes a diffraction grating and the desired phase $\left[\mathrm{\phi }(x,y)\right]$ (see text S1). In general, structuring material in the context of MSs can be understood as generating an effective optical susceptibility that considers the resonant field enhancement provided by the designed meta-atoms (63–66). Consequently, by incorporating multiple meta-atoms within an MS, their spatial distribution across the surface of the structure locally modulates the effective third-order nonlinear susceptibility corresponding to the region covered by each specific type of meta-atom. As a result, the effective nonlinear susceptibility, ${\mathrm{\chi }}_{\text{eff}}^{\left(3\right)}(x,y)$ , has the same pattern as that of the imprinted linear susceptibility distribution. Therefore, upon impinging a fundamental beam with an arbitrary wavefront of $E_{\text{FF}}(x,y)$ on the MS, the TH wave will be proportional to $E_{\text{TH}}\propto {\mathrm{\chi }}_{\text{eff}}^{\left(3\right)}(x,y)E_{\text{FF}}^3(x,y)$ , indicating that under the plane wave excitation, the far-field will be related to the Fourier transform of ${\mathrm{\chi }}_{\text{eff}}^{\left(3\right)}(x,y)$ , and thereby, the spatial distribution of the MS’s nonlinear susceptibility directly influences the far-field characteristics of the generated harmonic waves. Because the ${\mathrm{\chi }}_{\text{eff}}^{\left(3\right)}(x,y)$ contributes to the radiation source within the nonlinear Maxwell’s equations, homogeneous and inhomogeneous solutions will be affected by the same distribution of ${\mathrm{\chi }}_{\text{eff}}^{\left(3\right)}(x,y)$ . That is, both HOM-TH and INHOM-TH wavefronts undergo the same variations dictated by effective nonlinear susceptibility, whereas their propagation is affected by the material dispersion and absorption.

In this perspective, the spatial distribution of meta-atoms along the MS yields the local manipulation of effective optical susceptibility corresponding to each building block, such that the overall spatial effective susceptibility of the MS follows the imprinted distribution as $\begin{array}{c}{\mathrm{\chi }}_{\text{eff}}^{\left(3\right)}(x,y)={\sum }_m{a}_m\text{exp}\left\{im\left[\frac{2\mathrm{\pi }(x+y)}{\mathrm{\Lambda }}+\mathrm{\phi }(x,y)\right]\right\},\hfill \end{array}$ wherein $a_m$ represents the $m$ th Fourier component of the susceptibility, which takes into account the resonant field enhancement, the local geometrical variation of meta-atoms, and amplitude modulation term, and $\mathrm{\Lambda }$ denoting the grating pitch (61, 62). Therefore, the TH wave, $\begin{array}{c}{E}_{\text{TH}}\propto E_{\text{FF}}^3(x,y){\sum }_m{a}_m\text{exp}\left\{im\left[\frac{2\mathrm{\pi }(x+y)}{\mathrm{\Lambda }}+\mathrm{\phi }(x,y)\right]\right\}\hfill \end{array}$ , consists of a superposition of several diffraction orders each carrying a phase of $m\mathrm{\phi }(x,y)$ with their intensities proportional to ${\mid a_m\mid }^2$ (see text S1 for more details). We refer to this distinctive approach of generating simultaneous wavefront replicas at both the fundamental and nonlinear harmonics as the topology-imprinting method. We note that while this behavior resembles the nonlinear Raman-Nath phenomenon observed in periodic nonlinear structures, it differs in that the diffraction grating is achieved through the judicious arrangement of carefully designed meta-atoms. This configuration facilitates local field enhancement, leading to boosted third harmonic conversion efficiency, while simultaneously imprinting any desired wavefront (both phase and amplitude modulation) onto the desired diffracted orders.

MS design and characterization

Figure 1 illustrates the diagram of polycrystalline silicon MS that allows for the simultaneous transformation of a conventional Gaussian beam into a particular 2D or 3D structured light beam and the generation of a TH that replicates the shape of the beam at the fundamental wavelength. The design consists of discrete meta-atoms allowing for 2π phase control at the chosen fundamental wavelength of 1550 nm (see text S2). In polycrystalline silicon, the HOM-TH signal experiences absorption in the visible range (see fig. S7 in for the ellipsometry measurement of polycrystalline silicon). In contrast, the INHOM-TH component experiences a transparent medium. Therefore, although polycrystalline silicon is absorptive in the short wavelength range, the INHOM-TH is PL with the fundamental wave, and it is expected to carry an identical wavefront due to the topology-imprinting method outlined above.

Fig. 1. Design and characterization of topology imprinting concept.

(A) Diagram of polycrystalline silicon meta-atoms (refractive index n = 3.67 at 1550 nm with height H = 258 nm) on a fused silica substrate with the refractive index of n = 1.45. The INHOM-TH persists in the opaque region of the polycrystalline silicon while carrying the desired imprinted phase on the MS with high fidelity. (B) Designed intensity (left), phase distributions (middle), and the imprinted hologram, which is a combination of a grating intended to maximize the first diffraction order and the optical vortex phase with the TC l = 3 in the case of the fundamental beam being an OAM beam (top) and Hopf-link (bottom). The knotted solution was generated by superposing multiple LG modes carrying specific weightings and indices (9). The final phase masks of the phase-only hologram of OAM and Hopf-link were created through the inverse sinc-function encoding technique, including the amplitude profile onto the phase function (63, 64). (C) Scanning electron microscopy image of the meta-atoms (inset) and the MS structures. (D) Diagram of the experimental setup. The 1550-nm central wavelength laser was split into two paths: One beam was shaped to MS scale via lenses, creating structured first-order diffraction beams, and the other was serving as a reference, delayed for interference pattern generation. A 4f system with an iris filters higher orders. For 3D Hopf-link beam analysis, the camera tracks singularities along the beam’s path. The third harmonic’s interference pattern is achieved by focusing the reference beam onto an unpatterned polycrystalline silicon film, producing a Gaussian TH beam.

Figure 1B shows the designed intensity (left) and phase distributions (middle) of the vortex beams with a TC of l = 3 (top) and Hopf-link (bottom). Optical knots can be realized by superposing multiple LG modes carrying specific weights and indices, which can be analytically derived by projecting the Milnor polynomial associated with a specific knotted structure and projecting it onto the LG basis (8, 9, 34, 35). For the Hopf link, the corresponding field at z = 0 can be expressed as

$${\mathrm{\psi }}_{\text{link}}=-6.32\mathrm{L}{\mathrm{G}}_{01}+4.21\mathrm{L}{\mathrm{G}}_{02}+5.95\mathrm{L}{\mathrm{G}}_{20}+2.63\mathrm{L}{\mathrm{G}}_{00}$$ (1)

where LG^l_p is the Laguerre-Gaussian mode with radial and azimuthal indices p and l, respectively. The phase-only holograms for the optical vortex and Hopf-link are generated through the inverse sinc-function encoding technique (61, 62). In particular, the imprinted hologram is a combination of a grating to maximize the first diffraction order and the optical vortex phase with the TC l = 3 in the case of the fundamental beam being an OAM beam with the TC 3. The final phase masks are shown in the right panel of Fig. 1B. Following the discussion mentioned above, it is expected that the first diffracted order FF and TH waves carry the same phase of $m\mathrm{\phi }(x,y)$ , leading the TH wave to have a TC of $m\times l=3$ for the case of an optical vortex, and the same field distribution as that of the Hopf-link profile.

A foundational assumption in MS design is locality, indicating that the scattering response of each meta-atom is independent of its neighbors (67). This narrowness of optical response in real space (i.e., along the MS) carries an associated broadness of the response in reciprocal space, and as a result, “local MS” is well known for their broad angular functionalities, as opposed to their nonlocal counterparts. This assumption simplifies MS design by enabling a predefined library of meta-atom geometries, where each element’s response is presumed to remain consistent regardless of the surrounding configuration. In this perspective, one of the most common resonant modes considered local is the Mie-type resonance, whose low-order modes, such as electric and magnetic dipoles (ED and MD, respectively), are well established to exhibit locality features (67–69). The meta-atoms in this work, similar to our previous study where we comprehensively detailed the underlying physics (70), were designed based on the overlapping of ED and MD resonant modes, ensuring that all selected meta-atoms in our design exhibit local responses (see figs. S3 to S6 for more details on simulation results). The dimensions of the selected eight meta-atoms were optimized to allow for a phase variation from 0 to 2π and transmittance above 70%. Note that increasing the number of meta-atoms only serves to smooth the phase gradient, as the phase discretization follows 2π/N, where N is the number of meta-atoms. While a higher N may reduce phase quantization effects, it does not alter the fundamental physics of the concept or affect the underlying mechanism of topology imprinting. Furthermore, we have chosen N = 8 based on an effective balance between fabrication feasibility and maintaining high-quality MSs, as demonstrated in a previous study (70). The height and period of the meta-atoms were set to H = 258 nm and P = 900 nm on a SiO₂ substrate, respectively (see table S1 and fig. S3 for the near-field distribution within the meta-atoms). To fabricate the MSs, a layer of 258-nm polycrystalline silicon was deposited using low-pressure chemical vapor deposition on a fused quartz wafer, followed by standard electron beam lithography (EBL) and reactive ion etching (RIE) (See Materials and Methods). The resulting MSs contain 333 × 333 elements with a periodicity of 900 nm (Fig. 1C). While the fabricated sample closely matches theoretical predictions in both structure and dimensions, minor imperfections are unavoidable. These include deviations in the 3D meta-atom geometries, surface roughness, and slight variations in illumination conditions, such as the use of a Gaussian beam instead of an ideal plane wave, which introduces a range of k-vectors. These factors can influence both the amplitude and phase of the implemented meta-atoms, thereby affecting the resulting harmonic’s wavefront. A detailed analysis of these fabrication-related imperfections and their potential impact has been conducted and is provided in text S3.

Next, we performed experimental studies on the fundamental and TH frequencies of the structured optical beams. The experimental setup is illustrated in Fig. 1D. The laser beam—with a central wavelength of 1550 nm, repetition rate of 1 kHz, and pulse duration of 100 fs—was divided into two beams. The first beam was directed through two lenses to reduce its size to match the size of the MSs for an optimized illumination onto the MS and the generation of the structured beams at the first diffraction order. A 4f system with an iris at the focal point was used to select the first-order beam while filtering out all other orders. The second beam served as a reference and was directed through a mechanical delay line to generate the interference pattern with the first beam. For the 3D Hopf-link beams measurement, the camera was precisely moved along the propagation direction to capture the singularity positions. When the TH interference measurement was performed, an unpatterned polycrystalline silicon thin film was placed after the delay line, and a pair of lenses was used to focus the beam onto the reference thin film to generate a Gaussian TH beam. It is noteworthy that under pulsed laser illumination, thermal effects can influence the optical response due to the thermo-optic effect and potential localized heating. However, the repetition rate of the laser pulses plays a critical role in determining the extent of these effects and must be carefully considered. In our experiments, we used a Ti:sapphire laser system (Coherent Libra) with a 100-fs pulse width and a 1-kHz repetition rate (see Materials and Methods). Given this low repetition rate, thermal effects are minimal, as the time between pulses is sufficient for heat dissipation, preventing noticeable cumulative heating in the MS.Consequently, the contribution from the thermal nonlinearity mechanism is minimized, and no measurable impact on the nonlinear response or topology imprinting mechanism is expected.

Experimental characterization

Figure 2 (A and B) shows the intensity distribution and interference pattern of the fundamental vortex beams converted by the designed MS. The characteristic doughnut-shaped intensity profile and the forks-like pattern indicate that the generated vortex beam carries a TC of $l=3$ , as predicted by the theory. We then increased the power of the incident Gaussian beam to 53.1 GW/cm² to capture the TH images. We recall that in conventional bulk nonlinear optics, momentum and energy conservation principles must be satisfied to maximize the conversion efficiency from one frequency to another. In particular, momentum conservation includes orbital momentum that appears when the OAM beams are involved in the nonlinear interaction, where the high-harmonic wave inherits the OAM from the pump beam. That is, the FF beam with a TC of $l$ gives rise to an OAM of $3l$ for the THG process (22). However, as Fig. 2 (C and D) demonstrates, the intensity distribution and interference patterns of the TH vortex beams generated from the MS still show that the topology of the TH optical vortex is equal to 3 and is entirely preserved. We note that while a more detailed purity analysis becomes especially critical when the beam is propagated through complex environments such as atmospheric turbulence, where both phase and transmittance distortions can strongly affect the structured light’s integrity, the clear fork dislocation pattern and the well-preserved intensity distribution are good indications that the generated harmonics retain the desired mode characteristics. This preservation occurs because, in our case, the phase singularity with the desired TC ( $l=3$ ) is directly introduced into the nonlinear medium, and the vortex generation occurs within the MS. However, if the incoming pump beam also features a TC of $l_{\text{in}}$ , then the mth diffracted order of the TH wave, $E_{\text{TH}}\propto A(x,y,m)e^{i3l_{\text{in}}\mathrm{\phi }}{e}^{\mathit{im}{l}_{\text{MS}}\mathrm{\phi }}$ , acquires the TC of $l_{\text{TH}}=3l_{\text{in}}+m\times l_{\text{MS}}$ , with $l_{\text{MS}}$ denoting the TC of the imprinted hologram. It is evident that in this case, for the fundamental diffracted order ( $m=0$ ), the conventional relation for conservation of the angular momentum ( $l_{\text{TH}}=3l_{\text{in}}$ ), as the first-generation method suggests, is preserved. In contrast, for the other diffracted modes, an extra term of $m\times l_{\text{MS}}$ emerges. It should be noted that these considerations apply equally to both the homogeneous and inhomogeneous components, and the interplay between these components affects only the TH amplitude within the absorptive region of the material. In this context, the generation of TH structured light is not solely attributed to the INHOM-TH component; rather, it is the MS that governs the nonlinear wavefront shaping, ensuring that the generated TH vortex inherits the desired TC, regardless of whether the HOM-TH or INHOM-TH dominates. This observation is consistent with the findings of Liu et al. (71), where the authors demonstrated resonantly enhanced SHG below the nominal absorption edge of GaAs. Furthermore, unlike conventional nonlinear optical processes, where selection rules are typically dictated by crystal symmetry or multipole transitions, our MS-based approach does not enforce these selection rules in the traditional sense (22). However, despite this distinction, the observed harmonic wavefronts can still be understood by considering both the intrinsic OAM multiplication inherent to nonlinear processes and the engineered spatial modulation of both linear and nonlinear properties of the engineered optical material as discussed above.

Fig. 2. Linear and nonlinear interference measurement of the optical vortices.

(A) Characteristic doughnut-shaped intensity profile of the fundamental vortex beam. (B) Interference pattern of the fundamental vortex and Gaussian beams. The forks-like pattern converting a single line into the interference plot with three lines indicates that the topology of the TH optical vortex is equal to 3. (C) Characteristic doughnut-shaped intensity profile of the TH vortex beam. (D) Interference pattern of the TH vortex and Gaussian beams. The similar fork-like pattern proves that the topology of the TH optical vortex is equal to 3 and completely preserved. (E) Simultaneous generation of OAM beams with charge l = 3 at both fundamental and TH wavelengths. The difference in diffractive angles between the first-order OAM at FF and TH can be explained by the grating equation: The longer FF wavelength presents a larger diffraction angle than the shorter TH wavelength. The first-order FF OAM exhibited an elliptical profile, likely stemming from the camera’s tilted angle being aligned with the propagation angle of the first-order TH OAM.

We note that while different replicas of $m\mathrm{\phi }(x,y)$ of the fundamental beam are imprinted on both HOM-TH and INHOM-TH components of the generated waves, the former experiences material absorption and decays as it propagates inside the MS, whereas the latter is transparent in the opaque region. However, as mentioned earlier, only in cases where the material exhibits strong absorption at the TH wavelength and provides a sufficient interaction length does the HOM-TH component undergo pronounced attenuation and walk off from the pump, allowing the INHOM-TH component to be isolated and measured directly. In our experiments, two key factors make direct isolation of the INHOM-TH component challenging: the moderate absorption of poly-Si at the TH wavelength and the subwavelength thickness of the metasurface. Given these constraints, the observed vortices in our measurements are inherently a mixture of both HOM-TH and INHOM-TH components. We note that since the primary objective of this work is to demonstrate the topology-imprinting phenomenon, we have not attempted to optimize or separately quantify the power conversion efficiencies of the homogeneous and inhomogeneous components. However, in materials or configurations where absorption at the TH wavelength is considerably stronger—such as a thicker absorbing layer or at shorter wavelengths where polycrystalline silicon absorption increases—the HOMTH component could be substantially suppressed, yielding the potential isolation of the INHOM-TH contribution.

Therefore, the implemented method provides a platform to generate the desired wavefront with shorter wavelength corresponding to the high absorption of polycrystalline silicon, as shown in Fig. 2E. In particular, we realized the simultaneous generation of OAM beams with charge l = 3 at both fundamental and TH wavelengths in the first diffraction order, while the TC of $l=6$ corresponding to the second diffracted order is also visible in this plot with its amplitude modulated according to its corresponding $a_2$ coefficient (see fig. S13 for detailed visualization). The greater diffraction angle of the FF OAM beam compared to the TH OAM beam is attributed to its longer wavelength, aligning with the diffraction equation $d·\text{sin}\left(\mathrm{\theta }\right)=m·\mathrm{\lambda }$ , where $d$  is the grating line spacing, $\mathrm{\theta }$ is the diffraction angle, and $\mathrm{\lambda }$ is the wavelength. We note that while a diffraction grating can impose a phase profile on diffracted orders, our experiment demonstrates that the MS does more than simply diffract an unstructured fundamental beam. The key distinction is that the MS modulates effective nonlinear susceptibility, ensuring that the structured third harmonic is not merely a result of diffraction. Instead, the MS defines the transmittance and phase of the nonlinear response (desired TH structured beam), making it fundamentally different from a conventional diffraction grating, which only alters the propagation of an already-formed wavefront. Another key difference is the potential for the local field enhancement. While the current implementation prioritizes demonstrating the concept, it is entirely possible to incorporate resonant meta-atoms to considerably boost nonlinear efficiency. Unlike traditional diffraction gratings, which only affect the propagation of the incoming fields, our MS engineers the nonlinear response itself, meaning that higher-order Mie-type resonances could be used to amplify the nonlinear process while preserving full control over the wavefront. This is a crucial distinction, as it allows not only structuring the fundamental and harmonic fields but also enhancing the conversion efficiency in ways that a conventional diffraction grating simply cannot. Note that while the results shown in Fig. 2 share similarities with the Raman-Nath regime, they are not exactly the same. Our results resemble the nonlinear Raman-Nath phenomenon in terms of multi-order diffraction and structured harmonic generation. However, compared to the conventional nonlinear Raman-Nath diffraction, which arises from nonresonant phase gratings, our approach is different as the phase-only hologram is realized through a carefully engineered arrangement of meta-atoms, ensuring both localized field enhancement and tailored nonlinear wavefront shaping. This configuration enables local field enhancement—although relatively low in our case, it is still high enough to be observed under moderate pump intensity—leading to boosted third-harmonic conversion efficiency while simultaneously imprinting the desired wavefront onto the diffracted orders. We note that the primary focus of this work was to demonstrate the concept of topology imprinting in nonlinear MSs rather than to optimize nonlinear conversion efficiency. Therefore, we did not design a high-quality factor resonant MS that simultaneously provides 2π phase coverage while maximizing efficiency, as shown in fig. S3. As a result, the local field enhancement within each meta-atom remains relatively low. Consequently, the local field intensity inside the meta-atoms $(I_{\text{Loc}}\propto \mid E_{\text{Loc}}{\mid }^2)$ also remains low, meaning that a high nonlinear conversion efficiency was neither expected nor considered a key metric in our study. Nevertheless, to provide insight into the relative conversion efficiency, we have included an evaluation of the comparative efficiency between the zeroth-order and first-order TH signals and our analysis estimates that the ratio of first-order TH to zeroth-order TH is ~9%, suggesting that while the absolute experimental efficiency is expected to be low, the structured harmonic signals remain detectable and sufficient for validating the proposed concept (see fig. S15). However, while we acknowledge that the current efficiency of our MS is low, recent advancements in higher-order local resonances have demonstrated pathways for substantially improving nonlinear conversion efficiencies. For instance, some of the recent works have shown that using higher-order multipolar resonances, such as electric octupoles, can greatly enhance nonlinear efficiencies while allowing for nonlinear wavefront manipulation (72, 73). Incorporating these resonant modes into our MS design could provide a viable route toward increasing THG efficiency while maintaining structured beam generation capabilities.

The demonstrated capabilities of our topology-imprinted harmonic modulation in the generation of OAM beams suggest a promising avenue for transcending the conventional subwavelength constraints and material absorption limitations, paving the way for innovative holography applications, including the creation of complex 3D structured light. To demonstrate this, we extend our approach from the 2D topology imprinting to the generation of 3D Hopf links, representing not only a step up in the dimensionality but also a leap in functional complexity for advanced control of light topology. Figure 3 (A and B) displays the measured intensity distributions and the interferogram of the beam near the z = 0 plane, consistent with the theoretical target field and singularity positions shown in Fig. 1B. To accurately map the topology in 3D space near the z = 0 plane, we measured the interference distribution from 50 planes at different propagation distances and then connected the phase singularities to configure the structure of vorticity lines. Notably, the locations of the four-phase singularities evolve with beam propagation. From the sliced fields, we mapped the vorticity lines in 3D space (the scheme of this process is shown in Fig. 3C), and the corresponding configuration is shown in Fig. 3D, which presents a Hopf link. Figure 3E displays the top view of the Hopf-link structure.

Fig. 3. Linear and nonlinear characterization of 3D complex light beams.

(A) Measured intensity distributions of FF Hopf-link at z = 0 plane. The intensity profile matches the theoretical result in Fig. 1B very well. (B) Interferogram of the Hopf-linked field with Gaussian beams at z = 0 plane. The four red spots indicate the fork-like pattern positions where the singularities are. (C) Locating method via interference. Different transverse planes of the optical field along the beam propagation direction are imaged onto the camera by moving the position of the camera. We can map the vorticity lines in 3D space by connecting the singularity positions from each plane. (D) Isolated Hopf-link configured from interference patterns measured by the scanning method shown in (C). (E) Top view of the isolated Hopf-link structure. The cross between the singularities can be noticed. (F) Intensity distributions of TH Hopf Link beam at z = 0 plane. It shows a similar intensity profile as the FF Hopf link in (A). (G) Interferogram of the TH Hopf linked field with Gaussian beams. The four white spots indicate the fork-like pattern positions where the singularities are. (H) Isolated TH Hopf-link configured from interference patterns measured by the same scanning method used for FF. (I) Top view of the Isolated TH Hopf-link structure.

Figure 3 (F and G) displays the intensity distributions and interferograms of the TH Hopf-link beam near the z = 0 plane, respectively, which, due to phase locking, have very similar profiles to the results of the Hopf-link at the fundamental wavelength. The comparison of the experimental and theoretical results of the intensity distribution at z = 0.33z_R (Rayleigh length) along the propagation direction was shown in fig. S14. We then extracted the singularity positions from the multiplane interference images and reconstructed the Hopf-link structures at the TH wavelength, as shown in Fig. 3H, alongside their top view shown in Fig. 3I. The results in Fig. 3 affirm the imprinting of topology from the FF to the TH and validate the potential for advanced light manipulation techniques, with significant implications for holography and a spectrum of other photonic applications. In particular, conventional optical topology generation techniques create structured light by modulating phase, amplitude, or polarization. Two widely used methods are acousto-optic modulators (AOMs) and liquid crystal–based spatial light modulators (SLMs) (74, 75). AOMs excel in fast, dynamic frequency control but lack precision for complex wavefront shaping, especially at multiple harmonic frequencies. On the other hand, a major limitation of LC-based SLMs is their large pixel size, which restricts spatial resolution and field of view. For example, the commercial transmissive SLM HOLOEYE LC 2012 has a 36-μm pixel pitch, yielding a narrow 0.7° angular coverage of the first diffractive order (75). Further miniaturization is limited by pixel cross-talk, which creates a trade-off between the resolution and device performance. The proposed concept of nonlinear topology imprinting with MSs offers an alternative solution enabling simultaneously pronounced field enhancement and a smaller footprint compared to the conventional approaches. In particular, subwavelength meta-atoms not only improve resolution but also enable direct up-conversion to higher harmonics. In addition, MSs based on material platforms such as lithium niobate could extend topology imprinting to multiple wavelengths. While the current MS implementation has modest conversion efficiency, it can be markedly improved, making it a compact and scalable approach for all-optical information processing.

We note that, in principle, higher-order harmonic beams can be optimized to maintain their topological stability over long-distance transmission or in complex environments such as atmospheric turbulence (76). The stability of these structured beams is primarily dictated by their initial field distribution—comprising amplitude, phase, polarization, TC, and beam diameter—at the input plane of a turbulent medium. However, wavelength plays a crucial role in determining beam evolution in turbulent or scattering media. A key distinction between fundamental frequency and harmonic beams lies in their interaction with the environment, with shorter wavelengths exhibiting increased susceptibility to Rayleigh scattering, which scales as ${\mathrm{\lambda }}^{-4}$ (77). Consequently, TH beams experience stronger scattering than their fundamental counterparts, resulting in greater phase distortions and intensity fluctuations. Nevertheless, their topological properties, such as OAM, can remain robust over long-distance propagation, provided the initial wavefront quality is high. While the effects of turbulence and diffraction at shorter wavelengths cannot be eliminated, the generation method strongly influences propagation stability. Metasurfaces, with their subwavelength resolution, offer notable advantages over SLMs by producing higher-purity modes with reduced aberrations. A cleaner initial wavefront generated by MSs can minimize accumulated errors due to turbulence and diffraction, thereby enhancing beam resilience. This makes MSs a promising platform for structured harmonic beam generation aiming at propagation in complex environments, where preserving topological stability is critical.

DISCUSSION

In summary, topology-imprinting nonlinear MSs capable of simultaneously generating 2D OAM beams and 3D Hopf-links with identical topological properties at the fundamental wavelength and TH wavelengths have been demonstrated. Because the proposed concept relies on modulating the transmittance and phase of the effective nonlinear susceptibility via a phase-only MS, its underlying physical mechanism is not inherently limited to a specific nonlinear order, rather it is governed by the fundamental wavefront imprinting process, which applies to the nonlinear frequency conversion across all orders. It is noteworthy that although the third-harmonic structured light is generated at a wavelength corresponding to the absorption window of the host material, its generation is not solely attributed to the INHOM-TH component; rather, the MS governs the nonlinear wavefront shaping, ensuring that the generated TH vortex inherits the desired TC, irrespective of whether the HOM-TH or INHOM-TH contribution dominates. The found mechanism of topology imprinting applies to a broad range of material systems and is not fundamentally constrained by material properties, provided that the fundamental wavelength lies within a low-absorption region, as the topology imprinting process remains valid even under strong absorption. In this context, enhancing harmonic beam power and extending the concept to shorter wavelengths can be achieved by using materials with wider bandgaps that retain strong nonlinear susceptibilities—such as titanium dioxide, hafnium dioxide, or silicon nitride—thereby broadening the scope of potential applications in emerging optical and photonic technologies.

MATERIALS AND METHODS

Numerical simulations

The commercial software CST MICROWAVE STUDIO and COMSOL Multiphysics were used to perform linear simulations. The refractive index of Poly-Si measured using the spectroscopic ellipsometry (fig. S7) was used in the simulations. A dispersionless refractive index of 1.45 was assumed for the glass substrate. The transmittance and phase of the nano blocks at TH frequency were simulated, assuming that the Poly-Si is lossless and has a constant refractive index at that frequency. The numerical simulation for the calculation of nearfield distributions was carried out using the finite-element method implemented in the commercial software COMSOL Multiphysics. In particular, we used the Wave Optics Module to solve Maxwell’s equations in the frequency domain. We applied periodic boundary conditions in the x and y directions and the perfectly matched layers in the z direction to avoid undesired reflections. The tetrahedral mesh was also chosen to ensure the results’ accuracy of the results and facilitate numerical convergence. The meta-atoms were studied under the plane wave illumination along the z axis with an electric field pointing along the y axis. The near-field distribution inside the meta-atoms chosen for mapping the desired phase is shown in fig. S3.

Samples preparation

The Poly-Si film, with a thickness of 258 nm, was deposited on top of a glass wafer via an Low-Pressure Chemical Vapor Deposition (LPCVD) system. The linear refractive index of the deposited Poly-Si thin film was measured using a VASE J.A. Woollam spectroscopic ellipsometer. The thin film was cleaned by acetone, isopropanol, nitrogen, and O₂ plasma asher, followed by spin coating ZEP520A to form a 300-nm-thick layer and baking for 3 min at 180°C. A layer of EBL anticharging agent (discharge H₂O) with a thickness of 40 nm was spin-coated on top of ZEP520A. The pattern was written using an Elionix ELS-7500 EX EBL system and then was developed in ZED-N50 developer for 1 min after washing away the anticharging agent. The pattern was transferred to the Poly-Si layer by ICP-RIE etching (Oxford PlasmaPro ICP Etcher). Last, the ZEP520A mask was removed with 1165 Stripper (N-Methyl-2-pyrrolidone). The scanning electron microscopy images of the MS pattern were captured using an Apreo S system (Thermo Fisher Scientific).

Linear and nonlinear measurements

An 800-nm-wavelength beam was generated from a Ti:sapphire laser with a repetition rate of 1 kHz and 100-fs output pulse width (Libra system, Coherent) and went into an ultrafast optical parametric amplifier (TOPAS-C), which can covert the center wavelength to a range from 260 to 2600 nm. A manual filter wheel mount with neutral density filters was used to change the power of the laser beams. The beam was split into two beams: the first beam went through two lenses (300-mm and 50-nm N-BK7 lens, Thorlabs) to shrink the beam size to a comparable size to the sample size. Then, the beam incident onto the MS sample was passed through two lenses (35-mm and 250-nm N-BK7 lens, Thorlabs) and an iris to select the first diffraction order beam and magnify the beam size. The second (reference) beam passes through the mechanical delay line and then converges to the first beam to generate the interference pattern. An infrared camera (Bobcat 320, Xenics) was used to capture the images of the infrared beams. When we measured the 3D Hopf-link beams, the camera was moved along the propagation direction to scan the singularity positions. When the TH interference measurement was performed, an unpatterned Poly-Si thin film was placed after the delay line, and a lens (150-nm N-BK7 lens, Thorlabs) was used to focus the beam to generate a TH Gaussian beam with enough intensity. A near-infrared ND filter (NENIR40, Thorlabs) was used to filter the FF beam after the THG, and the infrared camera was placed by a visible camera (DCC1645, Thorlabs).

Supplementary Materials

This PDF file includes:

Text S1 to S4

Figs. S1 to S15

Table S1

References