
==== Front
Nat Commun
Nat Commun
Nature Communications
2041-1723
Nature Publishing Group UK London

39227596
52070
10.1038/s41467-024-52070-6
Article
Optical vortex ladder via Sisyphus pumping of Pseudospin
Lei Sihong 1
Xia Shiqi 1
http://orcid.org/0000-0001-9786-8209
Song Daohong songdaohong@nankai.edu.cn

12
http://orcid.org/0000-0002-2509-0175
Xu Jingjun 1
http://orcid.org/0000-0002-9808-6628
Buljan Hrvoje hbuljan@phy.hr

13
Chen Zhigang zgchen@nankai.edu.cn

12
1 https://ror.org/01y1kjr75 grid.216938.7 0000 0000 9878 7032 The MOE Key Laboratory of Weak-Light Nonlinear Photonics, TEDA Institute of Applied Physics and School of Physics, Nankai University, Tianjin, China
2 https://ror.org/03y3e3s17 grid.163032.5 0000 0004 1760 2008 Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi China
3 https://ror.org/00mv6sv71 grid.4808.4 0000 0001 0657 4636 Department of Physics, Faculty of Science, University of Zagreb, Bijenička c. 32, Zagreb, Croatia
3 9 2024
3 9 2024
2024
15 76934 4 2024
23 8 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
Robust high-order optical vortices are much in demand for applications in optical manipulation, optical communications, quantum entanglement and quantum computing. However, in numerous experimental settings, a controlled generation of optical vortices with arbitrary orbital angular momentum remains a challenge. Here, we present a concept of “optical vortex ladder” for the stepwise generation of optical vortices through Sisyphus pumping of pseudospin modes in photonic graphene. The ladder is applicable in various lattices with Dirac-like structures. Instead of conical diffraction and incomplete pseudospin conversion under conventional Gaussian beam excitations, the vortices produced in the ladder arise from non-trivial topology and feature diffraction-free Bessel profiles, thanks to the refined excitation of the ring spectrum around the Dirac cones. By employing a periodic “kick” to the photonic graphene, effectively inducing the Sisyphus pumping, the ladder enables tunable generation of optical vortices of any order even when the initial excitation does not involve any orbital angular momentum. The optical vortex ladder stands out as an intriguing non-Hermitian dynamical system, and, among other possibilities, opens a pathway for applications of topological singularities in beam shaping and wavefront engineering.

Controlled generation of OAM modes holds promise for applications in classical and quantum communication networks. Here the authors develop an optical vortex ladder in which an input beam transforms into vortices mediated by the Berry phase winding around topological singularities at the Dirac points.

Subject terms

Micro-optics
Photonic crystals
https://doi.org/10.13039/501100001809 National Natural Science Foundation of China (National Science Foundation of China) 12134006 Chen Zhigang issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

Optical vortices, featuring wavefront screw dislocations or phase singularities, have captivated great attention over the decades, particularly in connection with the orbital angular momentum (OAM) of light1–3. A host of fascinating phenomena has been demonstrated with optical vortex beams, from unusual diffraction patterns4 to toroidal vortices of light5, and from twisted photons6,7 to spin-orbit interactions8–10. Optical vortex generation has become a flourishing research area, driven by enormous applications including optical tweezers11–13, quantum information processing14–16, optical communications17,18, and vortex lasers19–21. Despite technological advancements in optical vortex generation such as spiral phase plates22,23, liquid crystal devices24,25, metasurfaces and nanophotonics19,26,27, the complex structural design and susceptibility to fabrication defects set unavoidable constraints and limitations in many OAM-based applications.

Over the past decade, topological photonics has opened a promising avenue for creating robust modes that are immune to impurities, disorders and defects28,29. The study of topological materials has sparked interdisciplinary interest30–34, leading to discoveries of new photonic topological edge modes35–37, higher-order topological phases38,39 and disclination states40–42, as well as Weyl and Dirac semimetals43–45. Photonic systems with momentum-space topological singularities have also been utilized for vortex generation through the spin-orbit interaction46,47. Due to the topological nature of the Dirac singularities, the vortex generation is immune to disorder and impurities in those systems and does not require precise alignment of the incident beam relative to the structures47, unlike in other cases with phase plates and metasurfaces. As such, topological systems are particularly attractive for the generation of optical vortices on demand. However, realization of higher-order vortices requires multiple Berry phase windings around topological singularities in momentum space, which are typically limited by the symmetry and design of microstructures46,47.

Here, we propose and demonstrate a scheme to construct an optical vortex ladder (OVL). In successive steps of propagation through periodically “kicked” photonic graphene, a probe beam transforms into vortices with the topological charge increasing by one in every step of the ladder (Fig. 1). The key ingredients of this process include a ring-spectrum excitation around the Dirac cone, involving (i) a complete conversion of pseudospin to OAM in every step of the ladder (where the pseudospin corresponds to sublattice degree of freedom), and (ii) Sisyphus pumping of the pseudospin achieved by a judicious “kick” applied on the photonic graphene at specific propagation lengths. This results in diffraction-free propagation of vortices (akin to the Bessel beams) during the conversion process, which benefits from the Berry phase winding around topological singularities. Our experimental results of vortex generation to any order with preserved OAM through the sequential ascension of the ladder stages are corroborated by theoretical analysis, with broad applications to other Dirac-like systems.Fig. 1 Illustration of an optical vortex ladder (OVL) and Sisyphus pumping in photonic graphene.

a Schematic of the pseudospin Sisyphus pumping process. The pseudospin modes are represented by the red and blue arrows within the “Sisyphus stone”, indicating whether pseudospin-up converts to pseudospin-down (during every stage), or in the opposite direction (between the stages). b Excitation of the ring spectra around Dirac cones generates a Bessel beam (l=0) in real space. At stage I of the propagation the beam undergoes a complete conversion from pseudospin-up to pseudospin-down with emerging OAM (l′=1). A judiciously chosen lattice “kick” in between stages I and II implements the Sisyphus pumping of pseudospin-down to pseudospin-up without altering the beam’s OAM. Stage II of the propagation is essentially the same as stage I, except that the topological charge after stage II is l′=2. The process repeats itself and forms an OVL that increases OAM by one unit at every stage of the ladder. The structure of photonic graphene is shown in the top-left inset, where the unit cell is highlighted by a shaded rhombus, and a1=3a/2x^+a/2y^,a2=3a/2x^−a/2y^ are the basis vectors with a being the lattice constant. In our experiments, the OVL is constructed and examined by taking the output at each step as the input for the next step with a cascade probing approach (see Methods).

Results

Theoretical analysis for a complete pseudospin conversion

We describe the pseudospin-orbit interaction in photonic graphene (inset in Fig. 1b), which consists of two sublattices (A and B) in one unit-cell. The propagation of the light is governed by the Schrödinger-type equation in the paraxial approximation. Furthermore, under the tight-binding approximation, the system is described by the two-band Hamiltonian1 Hk=0t1+eika2+eika1t1+e−ika2+e−ika10

where t is the nearest neighbor coupling, a1,a2 are the two basic vectors shown in Fig. 1b, and k=(kx,ky). Photonic graphene features a conical intersection at two inequivalent Dirac points (K and K′) and the topological features around these points are clearly visible in the effective Hamiltonian:2 H=κ2σxpx+σypy

where σi are Pauli matrices representing the components of the pseudospin angular momentum operator, px and py indicate the displacements of the transverse wavevectors with respect to the Dirac point K, κ=3at is related to the coupling coefficient t and lattice constant a. The eigenmodes around the Dirac cone are ψn,k=ϕA,ϕBT=n,expiθkT/2, where ϕA,ϕB represent the components on the two sublattices, kx+iky=kexpiθk and n=±1 denotes the band number. The Berry phase winding number around the Dirac point is w=1, which can be viewed as a momentum-space topological singularity46.

In graphene, pseudospin corresponds to the sublattice degree of freedom: pseudospin up (down) corresponds to the A (B) sublattice10. The eigenmodes of the pseudospin operator σz/2 are χ↑=1,0T and χ↓=0,1T, which relate to the Hamiltonian eigenmodes: χ↑=ψ+,k−ψ−,k, and χ↓expiθk=ψ+,k+ψ−,k. When we excite a single pseudospin component (say, in sublattice A), during propagation, a vortex is generated in the other sublattice (B) due to the mapping of topological singularity from momentum to real space46, see Supplementary Section 1. In the previous works, the Gaussian-type probe beams were commonly used for pseudospin mode excitations9,46. However, for such excitation methods, the probe beams occupy Bloch modes with different propagation constants which fail to achieve the complete conversion required by the OVL. Therefore, in this work, we redesign the probing conditions and, for the first time, employ a judicious excitation with a ring spectrum around the Dirac cone (see Supplementary Section 2 for more details). We initially excite all modes with eigenvalues Δβ/2 (ring at the upper cone) and −Δβ/2 (ring at the lower cone, see Fig. 2a1) and if such an excitation occurs solely on the A sublattice (pseudospin up), then it is straightforward to show that the wavefunction evolution can be written as Φz=∑k[χ↑cos(Δβz/2)exp(ikr)+iχ↓sin(Δβz/2)expikr+θk], where Φz excites both top and bottom bands around the Dirac cone.

This type of dynamics serves as the basis for constructing an OVL, in which a complete conversion of pseudospin up to pseudospin down (and vice versa) occurs at specific propagation distances set by the eigenvalue difference Δβ. The amount of power residing in pseudospin up is γ↑z=Φ(z)PAΦz∝cos2Δβz/2, whereas that in pseudospin down is γ↓z=⟨Φ(z)∣PB∣Φ(z)⟩∝sin2(Δβz/2), as illustrated in Fig. 2b1. Here, PA(B) is the projection operator on A(B) sublattice, i.e., the real-space spin-up (spin-down) projector (Supplementary Section 1). This serves as the first key ingredient for the OVL. The ring-spectrum excitations manifest as Bessel beam excitations in real space (Fig. 1b). The length L=π/Δβ of each stage in the OVL can be flexibly adjusted by tuning the diameters of the ring spectra, i.e., by tuning Δβ (Supplementary Section 3). The process of pseudospin conversion is topologically protected due to the nontrivial Berry phase winding around the Dirac cone46.

The second ingredient for the OVL is Sisyphus pumping. When the probe beam is exactly at the end of stage N, all the power is in the pseudospin-down mode. Then, all that power is pumped into the pseudospin-up mode (Fig. 2a2), initiating the Sisyphus pumping process. In the tight-binding approximation, the pumping is equivalent to multiplying Φ(z) by the ladder operator 0100 in the OVL, which corresponds to the pseudospin raising operator and raises up the eigenvalue of pseudospin components in Φ(z). Hence, the power residing in pseudospin down mode, which gradually increases throughout the stages, suddenly drops to zero at the end of the stage (blue line in Fig. 2b2), reminiscent of the Sisyphus process sketched in Fig. 1a. Since the topological charge carried by the pseudospin up modes in stage N+1 is inherited from that of the pseudospin down modes in stage N (Fig. 2a2), the topological charge of the probe beam increases by one at every stage (Fig. 2b2). The corresponding calculations in Fig. 2c demonstrate the topological charge carried by the probe beam at the end of each stage, further confirming the feasibility of the OVL generation.Fig. 2 Sisyphus pumping with the topological charge climbing up the ladder.

a1 Excitations of Bloch modes around the Dirac cone with a nontrivial Berry phase in k-space, as indicated by the green-shaded region. Δβ represents the well-defined eigenvalue difference. a2 Illustration of the total topological charges carried in pseudospin modes at adjacent ladder stages (N and N+1) due to Sisyphus pumping. The vertical arrows represent the pseudospin mode. b The evolution of the pseudospin mode in b1 the uniform (no Sisyphus pumping) and b2 the “kicked” (Sisyphus pumping) photonic graphene. The different shaded zones represent the different stages in b2. The total topological charges carried by the pseudospin down (up) are represented by dashed purple (orange) lines. c The associated pseudospin-down components at different stages under Sisyphus pumping, carrying a step increase of topological charges, as depicted by the phase diagram in the insets.

Experimental demonstration of periodic pseudospin conversion

To establish a photonic graphene lattice, we employ the multi-beam optical induction method48 (see more details in Method). A collimated laser beam (with a power of 100 mW and wavelength of 532 nm) is split into three broad beams (quasi-planewaves), which points towards three K points in momentum space and form a triangle intensity pattern in real space. Two sets of triangle patterns are sent in turn to a nonlinear crystal (with a refractive index of n0=2.35) for inducing the two sublattices of photonic graphene. An electric field (200kVm−1) applied across the crystal along the crystalline c-axis leads to the photorefractive effect48, which combines and translates the two triangle patterns into a coupled honeycomb waveguide array (photonic graphene). The nearest spacing between waveguides is approximately 9.6μm, as depicted in Fig. 3a. The probe beam is constructed with an overall Bessel distribution using a spatial light modulator (SLM) for excitation of the three K points in the first Brillouin zone of the lattice. This excitation approach is somewhat analogous to ring-shaped beam excitation with conical distribution in the momentum space for the generation of singular beams from spin-orbit interaction in uniaxial crystals49. However, it should be pointed out that, as opposed to circular polarization of light used in the spin-orbit interaction, here we have linear polarization, thus no “real” spin of light but rather the sublattice degree of freedom. In our graphene lattice, the pseudospin-orbit conversion is mediated by the phase winding around topological singularities. Since all components of the probe beam carry the same topological information46, we only need to extract the interferogram from one of the components at the lattice output.Fig. 3 Experimental demonstration of a complete pseudospin conversion by spectrum tuning.

a An experimentally established photonic graphene lattice with 9.6μm spacing between the nearest neighbor sites. b1 Probe pattern of pseudospin-up excitation at three K points. The inset is the phase distribution at one of the K points. The red circles denote the envelop of the lobes of the Bessel beam. b2 Output interferogram from one K point after 20 mm propagation. The position and helicity of the vortex are marked by the curved arrows. b3 The corresponding k-space spectrum with a ring diameter d1=0.16∣K∣. c1 The initial probe beam with a smaller overall envelope in real space. Interferograms of output patterns after c2 10 mm and c3 20 mm propagation. c4 The corresponding spectrum with a ring diameter d2=2d1. The vertical arrows represent different pseudospin modes as in Fig. 2.

When the probe beam (l=0) with diameter d1=0.16∣K∣ in momentum space (Fig. 3b3) is launched to cover sublattice A, it excites the pseudospin-up mode (zeroth-order Bessel envelope Fig. 3b1). After a 20 mm-long propagation through the lattice, the output exhibits a first-order Bessel envelope with topological charge l′=1, indicating a complete pseudospin conversion (Fig. 3b2). By tuning the diameters of the ring spectra, the probe beam is constructed with a larger ring in momentum space d2=0.32∣K∣ (Fig. 3c4) and smaller lobes in real space (Fig. 3c1). Now the propagation distance required to achieve a complete pseudospin conversion under this probe condition is exactly half of that for the d1=d2/2 beam. To verify this, we change to use a 10 mm-long crystal and capture the first-order Bessel envelope (l′=1) at the back facet of the crystal (Fig. 3c2). To show the periodicity, we use the same experimental parameters as those for the 20 mm-long experiment. The profile of the output beams returns to the initial mode distribution with l′=0 after 20 mm-long propagation (Fig. 3c3). Corresponding simulations and direct comparisons with Gaussian beam excitations are presented in the Supplementary Sections 4, which agree well with the experimental results and theoretical analysis. Thus, we demonstrate clearly the periodic pseudospin conversion under the Bessel beam excitations. These findings provide essential ingredients in the OVL construction.

Scheme for Sisyphus pumping in photonic graphene

To realize the Sisyphus pumping in experiment, we employ a periodically “kicked” structure, as shown in Fig. 4. If a pseudospin-up mode with a Bessel envelope is excited at sublattice A, as illustrated in Fig. 4a, it distributes completely on sublattice B after completing the pseudospin conversion in stage I. Before entering stage II, the lattice is shifted so that sublattice A maps exactly onto sublattice B of the previous stage, which is fully equivalent to pumping pseudospin down to pseudospin up. The lattice shifting follows the rule:3 Vn=V0r−n−1a1+a2/3,

where n is the number of steps in the OVL and V0(r) represents the refractive-index potential of photonic graphene, which is the same as that in stage I (Fig. 1a). It is evident that Hz=Hz+3L, where H(z) is the time-dependent Hamiltonian, the system returns to its initial position after 3L, i.e., VIV=VI. Such a sudden shifting (“kicking”) of the structures allows for the Sisyphus pumping and continuous mapping of topological singularity from momentum to real space48, which in turn leads to the OVL formation with output optical vortices of any order. Moreover, even when perturbations are applied on the nearest-neighbor couplings, the pseudospin conversion preserves (Supplementary Section 5), and the Bessel beam excitations enable self-healing property of the generated vortices against obstructions (Supplementary Section 2). These features give rise to the robustness of the OVLs.Fig. 4 Experimental demonstration of the OVL generation via Sisyphus pumping.

The center triangle illustrates the periodic “kicking” of the photonic graphene, while the spiral lines indicate vortices at different OVL stages. The orange dots in the schematic indicate the location of probe beam at the input and output facets. The relative position of the lattices in stage I is shaded for reference. a Stage I: The pseudospin-up modes with topological charge (l=0) on sublattice A (a1) transform into pseudospin-down modes (l′=1) after a propagation distance Z=20mm, where probe beams transfer entirely to sublattice B (a2). b Stage II: The probe beams excite sublattice A through lattice “kicking” and Sisyphus pumping (b1). After propagation, probe beams transfer to sublattice B again (top row in b2), and the topological charge increases to l′=2 (bottom row in b2). c Stage III: The probe beams with l=2 located on sublattice A (c1) increase to l′=3 (c2). The inset shows measured phase distribution of the main lobe of the Bessel beam. The position and helicity of the vortex are marked by the curved arrows. The green dashed arrows from input to output beams indicate the cascade probing method.

It should be noted that periodic “kicking” of the graphene lattice (Vn→Vn+1) implies that the dynamics in this Hamiltonian should generally be non-Hermitian as the power is not conserved. For our specific initial conditions and with a proper design of eigenvalue difference Δβ according to the length of the crystal, there is no loss of power as pseudospin-raising operator always acts on the system when it is in the lower mode. Moreover, it is worth highlighting that the OVL possesses a kind of topological protection owing to the nontrivial Berry phase winding around the Dirac point. Even when perturbations are introduced in the couplings that affect the system, causing a portion of energy to be trapped in the initially excited sublattices, the robustness of pseudospin conversion persists. The trapped energy is scattered into the higher bands due to non-Hermitian properties introduced by the Sisyphus pumping between the stages. These scattered modes can be interpreted as losses around the Dirac cones but will not impede the pseudospin conversion in the subsequent stage (Supplementary Section 5). Therefore, in spite of the mode sensitivity often encountered in non-Hermitian systems50, the combined effects of topological and non-Hermitian properties of periodically “kicked” photonic graphene contribute to the robust pseudospin conversion in the OVL.

Experimental realization of the OVL

In the experiment, the effectiveness of the OVL is limited by the length of the crystal and the lattice “kicking” between different stages. To overcome this limitation, we utilize a cascade-probing technique. Specifically, the amplitude and phase of the output beam from one waveguide section are captured using a camera, digitally replicated with a SLM, and then sent back into the next section of waveguide arrays. By transferring all amplitude and phase information from one section to another, this cascade-probing technique effectively connects all parts of the waveguide arrays, allowing us to observe the mode evolution through the otherwise length-limited lattice structure.

To initiate the OVL, we generate a probe beam with a zeroth-order Bessel envelope (l=0) using an SLM. This beam excites the pseudospin-up mode at the beginning of stage I (Fig. 4a1). At the end of stage I (z=20mm), the probe beam fulfills the complete pseudospin conversion (l′=1), transferring most energy to sublattice B (Fig. 4a2). In stage II, we apply the cascade-probing method to duplicate the output beam from stage I, resending it to cover sublattice A but aiming at the three K points in momentum space where the phase singularity (l=1) is inherited (Fig. 4b1). This process achieves the Sisyphus pumping, so the output of stage II has a topological charge l′=2 (Fig. 4b2). In stage III, the probe beam duplicated from the output of stage II carries the charge l=2 located on sublattice A (Fig. 4c1). After achieving one cycle of pseudospin conversion, the topological charge increases to l′=3 (Fig. 4c2). In the subsequent stages, the “kicked” lattice enters the next cycle, and the topological charge carried by the probe beam continues to increase. In principle, optical vortices of any order can be obtained through the OVL. However, in a realistic experiment, a generated high-order vortex undergoes splitting into multiple single-charge vortices due to a few reasons: (i) a truncated Bessel beam with a finite size is used for excitation, so the ring spectrum is not really a perfect ring, preventing a fully completed pseudospin conversion; (ii) a high-order vortex tends to split itself during free propagation; (iii) the non-uniformity of the writing beam and the crystal defects also introduce perturbation to the lattice potential (index changes), affecting the perfect conversion of topological charge as compared to simulations. The quality of the generated vortices and the limitation on the number of steps in a constructed OVL are influenced by excitation conditions. For instance, with a Bessel beam characterized by more lobes extended in real space or a narrower ring in momentum space (more close to an ideal Bessel beam with infinite energy), the order of OAM generation can be further increased (Supplementary Section 6). Overall, these experimental results confirm the effectiveness of the OVL, despite the impurity of the lattice structures (Fig. 3a) and the slight mismatch due to the cascade-probing method (Fig. 4).

Discussion

We have proposed and demonstrated a scheme for the generation of an optical vortex ladder that can produce vortices with arbitrary OAM. This is achieved by topological singularity mapping around the Dirac cones with ring-spectrum excitation and periodic “kicking” of the graphene lattice, which leads to Sisyphus pumping of the pseudospin modes. The concept of the “kicking” effect may also be relevant and applicable for guiding light via geometric phases51, offering a new approach for future studies in light manipulation. In contrast to conventional methods of vortex generation that rely on electro-optical elements arranged in series in real space which are sensitive to alignment and defects52, our generated vortices in the OVL show robust properties due to momentum-space nontrivial winding of the Berry phase, featuring diffraction-free Bessel profiles due to ring-spectrum excitation. In fact, the Sisyphus pumping process, along with relevant non-Hermitian properties associated with the OVL, heralds a new technique for OAM generation and light-field manipulation. This idea provides a framework adaptable to diverse Dirac-like systems, such as Lieb, super-honeycomb, T-graphene, and borophene lattices53–56, promising further advancements and applications of topological singularity mapping. We believe that the exploration of alternative setups based on this approach may unlock the full potential for generating high-order OAM in all-optical compact devices, particularly for engineering multi-charge and ultra-degree-of-freedom structured light for optical communications21,57–59. Furthermore, the OVLs may open exciting new avenues for application of topological singularities, with the potential to impact the fields of optical communications, quantum information computing and complex structured light control14,17,21,59–63. Thus, our results may pave the way for further exploration of topological systems for active and tunable generation and manipulation of optical vortices, as well as for the development of innovative technologies in the field of photonics and beyond.

Methods

Experimental setup

The scheme of the experimental setup for the generation of optically induced photonic lattices is depicted in Fig. 5a. To initiate the formation of photonic graphene, a uniformly modulated lattice intensity pattern is sent through a 20 mm-long photorefractive SBN crystal, which, due to nonlinearity, translates into a refractive index distribution. Utilizing a spatial light modulator (SLM), the lattice beam with a honeycomb pattern is generated, as shown in the inset of path 1. Upon applying a voltage along the crystalline c-axis, the lattice beam experiences a self-focusing nonlinearity, thereby converting the honeycomb intensity pattern into the index potential of photonic graphene. Employing the same SLM loaded with a predetermined phase pattern for three Bessel beams, the probe beam takes on a Bessel-shaped triangular lattice pattern before entering the crystal for probing the lattice. In momentum space, the orientations of the three Bessel beams align with the three Dirac K points of the first Brillouin zone (inset in Fig. 5a). In real space, the triangular intensity pattern is matched to either the A or B sublattice. The excitation condition facilitates the selective excitation of the two pseudospin states, starting from the initial propagation of probe beams at the beginning of the stages.Fig. 5 Experimental setup and phase extraction method.

a Experimental setup: SLM (spatial light modulator), L (lens), F (Fourier filter), BS (beam splitter), SBN (strontium barium niobite crystal). Path 1 is for the lattice beam (ordinarily polarized honeycomb pattern) and the probe beam (extraordinarily polarized Bessel beam) generated by the SLM. Path 2 serves as the reference beam (extraordinarily polarized) for phase measurement. The inset shows the Bessel beam excitation conditions in momentum space. b Example of phase extraction via the Fourier transform method, where b1 is the sample intensity distribution I(x,y) from the interferogram, and b2 is the Fourier transform of I(x,y). c Interferograms of an output beam obtained from experiments. Interferograms of a reference beam (Path 2) interfering with the output beam (c1) and a planewave (c2) generated in Path 1. c3 Extracted phase distribution of the output beam.

Cascade probing method

Due to the limitations imposed by the length of the crystal L=20mm, we need to construct and monitor the OVL in several stages using the cascade probing method. The length of each part is 20mm which matches the length of the crystal. At the end of stage I (z=20mm), both the intensity and phase distribution of the output beam are captured in the experiment, with the most energy transferred to the sublattice B (Fig. 4a2). In stage II, the intensity and phase distributions of the output beam captured in stage I are properly duplicated by the SLM and serve as the input beam for stage II (Fig. 4b1), resending it to cover the sublattice A but aiming at the three K points in momentum space. Consequently, although light in each stage propagates only 20 mm, the output of stage II has an effective 40 mm of propagation distance, thanks to the cascade probing method described above. The propagation distance of light can be further elongated to 60 mm within the allowed error range caused by the extraction-duplication process via the SLM.

Phase extraction via Fourier transform method

To analyze long-distance probe beam evolution, we need to extract both the intensity and phase distributions of the output beams in the experiment, and then use them as the input to the next section of the waveguide arrays. Therefore, we apply the Fourier transform of the interferogram of output beams: Let us assume we have the interferogram of the beams shown in Fig. 5b1, which can be described by Ix,y=I0x,y+I1x,y+I1*x,y, where I1(x,y)=Ax,yexp[iϕ(x,y)] and Ax,y and ϕ(x,y) are the amplitude and phase distribution, respectively. Then, the Fourier transform of Ix,y yields three parts FIx,y=G0k+G+k+G−k denoted in Fig. 5b2, where G+k contains the phase of the output beams. The inverse Fourier transform of g+x=F−1G+k uncovers the phase distribution of Ix,y. Moreover, we have4 g+x=g+0xexpiϕpx+ϕrx,

where g+0x is the intensity component, ϕpx is the phase carried by the probe beam, and ϕrx is the phase of the reference beam.

In the experiment, the interferogram between the output and the reference beam is recorded (Fig. 5c1), corresponding to ϕpx+ϕrx in Eq. 4. Since only the phase of the output beam ϕpx is necessary, the interferogram between the reference beam in Path 2 and plane-wave generated in Path 1 is captured (Fig. 5c2) to filter out the phase distribution ϕrx and showcase ϕpx in Fig. 5c3.

Supplementary information

Supplementary Information

Peer Review File

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-024-52070-6.

Acknowledgements

This work was supported by the National Key R&D Program of China (No. 2022YFA1404800); the National Nature Science Foundation of China (No. 12134006, 12274242, and 12204252); the Natural Science Foundation of Tianjin (No. 21JCJQJC00050) and the 111 Project (No. B23045) in China. H.B. acknowledges support from the QuantiXLie Center of Excellence.

Author contributions

S.L. and S.X. performed the experiments and simulation. J.X. and D.S. discussed the results. Z.C. and H.B. supervised the project. All authors contributed to this work.

Peer review

Peer review information

Nature Communications thanks Yijie Shen, and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

Data supporting key conclusions of this work are included within the article and Supplementary information. All other raw data that support the findings of this study are available from the corresponding authors upon request.

Competing interests

The authors declare no competing interests.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

These authors contributed equally: Sihong Lei, Shiqi Xia.
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