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The Higher-Order Poincar\'e Sphere (HOPS) provides a powerful geometrical tool for representing vector beams as points on the surface of a unitary sphere. Since a particular position on the surface represents any spatial mode regardless of its shape, this representation cannot be used to discern between the spatial modes geometries of vector modes. For instance, Laguerre- and Ince-Gauss vector beams are ambiguously represented using the same unitary sphere, even though their spatial profiles are circular and elliptical, respectively. As such, in this manuscript, we propose a generalisation of the HOPS that we call the Higher-Order Poincar\'e Ellipsoid (HOPE). Our approach allows an unambiguous representation of helical Ince-Gauss vector modes of ellipticity $\varepsilon$ onto the surface of an ellipsoid of eccentricity $\bf e$, providing a unique way to visualise elliptically-shaped vector modes. We provide a transformation that links the ellipticity $\varepsilon$ of helical Ince-Gauss vector modes to the eccentricity $\bf e$ of an ellipsoid, such that the HOPS is recovered for $\varepsilon=0$. Since this representation preserves the Stokes parameters formalism, the transition from the HOPS to the HOPE is straightforward, thus making its implementation appealing for the structured light community. We anticipate the concepts outlined here will pave the path toward the representation of structured light beams' properties using other geometrical objects., Comment: 16 pages, 6 figures

The strong coupling between the spatial and polarisation degrees of freedom (DoF) in vector modes enables a diverse array of exotic, inhomogeneous polarisation distributions through a non-separable superposition, which are conventionally generated in circular-cylindrical symmetry. Here, we theoretically and experimentally demonstrate a generalized class of vector modes specified in elliptical spatial coordinates and elliptical polarisation. This generalisation gives rise to an even larger set of vector beams with more intricate polarisation distributions. Crucially, controlling the beam parameters allows engineering of vector beams with predefined polarisation trajectories on the Poincar\'e sphere. This capability offers potential applications, for example in optical communications, where precise polarisation control can significantly enhance data transmission and security.

Maxwell's demon (MD) has proven an instructive vehicle by which to explore the relationship between information theory and thermodynamics, fueling the possibility of information driven machines. A long standing debate has been the concern of entropy violation, now resolved by the introduction of a quantum MD, but this theoretical suggestion has proven experimentally challenging. Here, we use classical vectorially structured light that is non-separable in spin and orbital angular momentum to emulate a quantum MD experiment. Our classically entangled light fields have all the salient properties necessary of their quantum counterparts but without the experimental complexity of controlling quantum entangled states. We use our experiment to show that the demon's entropy increases during the process while the system's entropy decreases, so that the total entropy is conserved through an exchange of information, confirming the theoretical prediction. We show that our MD is able to extract useful work from the system in the form of orbital angular momentum, opening a path to information driven optical spanners for the mechanical rotation of objects with light. Our synthetic dimensions of angular momentum can easily be extrapolated to other degrees of freedom, for scalable and robust implementations of MDs at both the classical and quantum realms, enlightening the role of a structured light MD and its capability to control and measure information., Comment: 14 pages, 6 Figures

For the past few decades, structured light has been gaining popularity across various research fields. Its fascinating properties have been exploited for both previously unforeseen and established applications from new perspectives. Crucial to this is the several techniques that have been proposed for both their generation and characterisation. On one hand, the former has been boosted by the invention of computer-controlled devices, which combined with a few optical components allow flexible and complete control of the spatial and polarisation degrees of freedom on light, thus enabling a plethora of proof-of-principle experiments for novel and old applications. On the other hand, characterising light beams is important not only for gaining better insights into light's properties but also for potentially being used as metrics. In this perspective, we thus offer our take on a few key applied research fields where structured light is particularly promising, as well as some pivotal generation and characterisation techniques. In addition, we share our vision of where we believe structured light's applications are moving towards., Comment: The following article has been submitted to Applied Physics Letters. After it is published, it will be found at https://publishing.aip.org/resources/librarians/products/journals/

In this manuscript, we put forward two new types of structured light beams, the vortex Pearcey-Gauss (VPeG) beam, with a homogeneous polarisation distribution, and the vector vortex Pearcey-Gauss (VVPeG) beam, with a non-homogeneous polarisation distribution. The later generated as a non-separable superposition of the spatial and polarisation degrees of freedom. We also achieve their experimental realization through the combination of a spatial light modulator, which creates a scalar Pearcey-Gauss beam, and a q-plate which transforms it into a vortex or a vortex vector beam, depending on its input polarisation state. Their intensity and polarisation distribution was performed through Stokes polarimetry, along the propagation direction, which was compared with numerical simulations. As demonstrated, the VVPeG beam evolves from a pure vector beam into a vector mode of quasi-homogeneous polarisation distribution. The proposed vector beams add to the already extensive family of non-separable states of light. We anticipate that both types of beams will find applications in fields as diverse as optical metrology, optical communications, and optical tweezers, amongst others., Comment: 10 pages, 4 figures

Stellar convection plays an important role in atmospheric dynamics, wind formation and the mass-loss processes in Asymptotic Giant Branch (AGB) stars. However, a direct characterization of convective surface structures in terms of size, contrast, and life-span is quite challenging. Spatially resolving these features requires the highest angular resolution. In this work, we aim at characterizing the size of convective structures on the surface of the O-rich AGB star R Car to test different theoretical predictions, based on mixing-length theory from solar models. We used infrared low-spectral resolution (R~35) interferometric data in the H-band (~1.76 $\mu$m) with the instrument PIONIER at the Very Large Telescope Interferometer (VLTI) to image the star's surface at two epochs separated by ~6 years. Using a power spectrum analysis, we estimate the horizontal size of the structures on the surface of R Car. The sizes of the stellar disk, at different phases of a pulsation cycle, were obtained using parametric model-fitting in the Fourier domain. Our analysis supports that the sizes of the structures in R Car are correlated with variations of the pressure scale height in the atmosphere of the target, as predicted by theoretical models based on solar convective processes. We observe that these structures grow in size when the star expands within a pulsation cycle. While the information is still scarce, this observational finding highlights the role of convection in the dynamics of those objects. New interferometric imaging campaigns with the renewed capabilities of the VLTI are envisioned to expand our analysis to a larger sample of objects., Comment: Accepted by A&A on 15/05/2024, 11 pages, 7 figures

Our purpose is to study the effect of binary companions located within the first 10 stellar radii from the primary AGB star. In this work, we target the mass-losing carbon star V Hydrae (V Hya), looking for signatures of its companion in the dust forming region of the atmosphere. The star was observed in the L- and N-bands with the VLTI/MATISSE instrument at low spectral resolution. We reconstructed images of V Hya's photosphere and surroundings using the two bands and compared our interferometric observables with VLTI/MIDI and VISIR archival data. To constrain the dust properties, we used DUSTY to model the spectral energy distribution. The star is dominated by dust emission in the L- and N- bands. The VISIR image confirms the presence of a large-scale dusty circumstellar envelope surrounding V Hya. The MATISSE reconstructed images show asymmetric and elongated structures in both infrared bands. In the L-band, we detected an elongated shape of approximately 15 mas, likely to be of photospheric origin. In the N-band, we found a 20 mas extension North-East from the star, and perpendicular to the L-band elongated axis. The position angle and the size of the N-band extension match the prediction of the companion position at MATISSE epoch. By comparing MATISSE N-band with MIDI data, we deduce that the elongation axis in the N-band has rotated since the previous interferometric measurements 13 years ago, supporting the idea that the particle enhancement is related to the dusty clump moving along with the companion. The MATISSE images unveil the presence of a dust enhancement at the companion position, opening new doors for further analysis on the binary interaction with an AGB component., Comment: Accepted for publication in Astronomy & Astrophysics

Vector modes are well-defined field distributions with spatially varying polarisation states, rendering them irreducible to the product of a single spatial mode and a single polarisation state. Traditionally, the spatial degree of freedom of vector modes is constructed using two orthogonal modes from the same family. In this letter, we introduce a novel class of vector modes whose spatial degree of freedom is encoded by combining modes from both the Hermite- and Laguerre-Gaussian families. This particular superposition is not arbitrary, and we provide a detailed explanation of the methodology employed to achieve it. Notably, this new class of vector modes, which we term Hybrid Hermite-Laguerre-Gaussian (HHLG) vector modes, gives rise to subsets of modes exhibiting polarisation dependence on propagation due to the difference in mode orders between the constituent Hermite- and Laguerre-Gaussian modes. To the best of our knowledge, this is the first demonstration of vector modes composed of two scalar modes originating from different families. We anticipate diverse applications for HHLG vector modes in fields such as free-space communications, information encryption, optical metrology, and beyond., Comment: 4 pages, 5 figures

Astigmatic unitary transformations allow for the adiabatic connections of all feasible states of paraxial Gaussian beams on the same modal sphere, i.e., Hermite-Laguerre-Gaussian (HLG) modes. Here, we present a comprehensive investigation into the unitary modal evolution of complex structured Gaussian beams, comprised by HLG modes from disparate modal spheres, via astigmatic transformation. The non-synchronized higher-order geometric phases in cyclic transformations originates a Talbot-effect-like modal evolution in the superposition state of these HLG modes, resulting in pattern variations and revivals in transformations with specific geodesic loops. Using Ince-Gaussian modes as an illustrative example, we systematically analyze and experimentally corroborate the beamforming mechanism behind the pattern evolution. Our results outline a generic modal conversion theory of structured Gaussian beams via astigmatic unitary transformation, offering a new approach for shaping spatial modal structure. These findings may inspire a wide variety of applications based on structured light., Comment: 7 pages, 5 figures

In this work, we propose and demonstrate experimentally a compact technique for the generation of cylindrical vector beams, based on a Michelson interferometer and a $\pi$-astigmatic mode converter, capable of inverting the topological charge of higher-order Laguerre-Gauss beams. Compared to previously demonstrated methods, this is relatively easy to align, and very compact. In addition, it generalises the concept of astigmatic mode conversion, commonly associated with scalar beams, to vector beams with non-homogeneous polarisation distribution. We anticipate that many of the applications based on Michelson interferometers will benefit from the unique properties of vector beams., Comment: 5 pages, 4 figures

Recent advances in planar optics with geometric-phase superstructures have brought a new paradigm in the control of structured light and, in particular, has substantially enhanced the capabilities of generating and detecting orbital angular momentum (OAM) states of light and associated spatial modes. However, the structured modal interface that can reciprocally link OAM states via adiabatic control and access-associated higher-order geometric phase remains absent in planar optics. In this work, we propose and experimentally demonstrate a planar optical astigmatic retarder fabricated with liquid-crystal (LC) geometric phase. The LC superstructure was designed with the principle of fractional Fourier transformation and is capable of reciprocal conversion between all possible OAM states on the same modal sphere. Such a planar device paves the way towards an easily deployed modal interface of paraxial OAM states, unlocks the resource of higher-order geometric phase, and has promising applications in high-dimensional classical/quantum information.

In this manuscript, we propose the generation of complex vector beams with high quality and stability based on a novel approach that relies on the combination of two techniques that seem incompatible at first glance. The first is Complex Amplitude Modulation (CAM), which produces scalar structured light fields in phase and amplitude with high accuracy. The second is on-axis modulation for the generation of vector beams, a method that requires phase-only holograms, therefore yielding beams of reduced quality. More precisely, the idea behind our technique is to send the shaped light produced by CAM co-axially to the zeroth order, rather than to the first order, as commonly done. We describe our technique, explaining the generation of the hologram and experimental setup to isolate the desired vector mode, and then present experimental results that corroborate our approach. We first address the quality of the generated beams using Stokes polarimetry to reconstruct their transverse polarisation distribution, and then compare their stability against the same mode produced using a popular interferometric method. Our vector beams are of good quality and remarkably stable, two qualities that we expect will appeal to the community working with vector modes., Comment: 9 pages, 4 figures

Since the diffraction behavior of a light field is fully determined by its spatial wavefunction, i.e., its spatial complex amplitude (SCA), full characterization of spatial wavefunction, plays a vital role in modern optics from both the fundamental and applied aspects. In this work, we present a novel complex-amplitude profiler based on spatial Stokes tomography with the capability to fully determine the SCA of a light field in a single shot with high precision and resolution. The SCA slice observed at any propagation plane provides complete information about the light field, thus allowing us to further retrieve the complete beam structure in 3 dimensions space, as well as the exact modal constitution in terms of spatial degrees of freedom. The principle demonstrated here provides an important advancement for the full characterization of light beams with a broad spectrum of potential applications in various areas of optics, especially for the growing field of structured light.

Vector beams display varying polarisation over planes transversal to their direction of propagation. The variation of polarisation implies that the electric field cannot be expressed as a product of a spatial mode and its polarisation. This non-separability has been analysed for particular vector beams in terms of non--quantum entanglement between the spatial and the polarisation-degrees of freedom, and equivalently, with respect to the degree of polarisation of light. Here we demonstrate theoretically and experimentally that Mathieu-Gauss vector modes violate a Bell-like inequality known as the Clauser-Horn-Shimony-Holt-Bell (CHSH-Bell) inequality. This demonstration provides new insights into that fact that a more general class of vector modes with elliptical symmetry also violate Bell inequalities., Comment: 8 pages, 6 figures

In this work, we propose and demonstrate experimentally a new family of vector beams, the Helico-Conical Vector Beams (HCVB), whose spatial degree of freedom is encoded in the Helico-Conical Optical Beams. We use Stokes polarimetry to study their properties and find that upon propagation their transverse polarisation distribution evolves from non-homogeneous to quasi-homogeneous, such that even though their global degree of nonseparability remains constant, locally it decreases to a minimum value as z tends to infinit. We corroborated this quantitatively using the Hellinger distance, a novel metric for vectorness that applies to spatially-disjoint vector modes. To the best of our knowledge, HCVBs are the second family of vector beams in which this behaviour has been observed, paving the way for applications in optical tweezing or information encryption., Comment: 4 pages, 4 figures

The 'Great Dimming' of the prototypical red supergiant Betelgeuse, which occurred between December 2019 and April 2020, gives us unprecedented insight into the processes occurring on the stellar surface and in the inner wind of this type of star. In particular it may bring further understanding of their dust nucleation and mass loss processes. Here, we present and analyse VLTI/MATISSE observations in the N-band (8 - 13 $\mu$m) taken near the brightness minimum in order to assess the status of the dusty circumstellar environment. We explore the compatibility of a dust clump obscuring the star with our mid-infrared interferometric observations using continuum 3D radiative transfer modelling, and probe the effect of adding multiple clumps close to the star on the observables. We also test the viability of a large cool spot on the stellar surface without dust present in the ambient medium. Using the visibility data, we derive a uniform disk diameter of 59.02 $\pm$ 0.64 mas in the spectral range 8 to 8.75 $\mu$m. We find that both the dust clump and the cool spot models are compatible with the data. Further to this, we note that the extinction and emission of our localised dust clump in the line of sight of the star, directly compensate each other making the clump undetectable in the spectral energy distribution and visibilities. The lack of infrared brightening during the 'Great Dimming' therefore does not exclude extinction due to a dust clump as one of the possible mechanisms. The visibilities can be reproduced by a spherical wind with dust condensing at 13 stellar radii and a dust mass-loss rate of (2.1 - 4.9) $\times$ 10$^{-10}$ $\mathit{M}_{\odot} {\rm yr}^{-1}$, however, in order to reproduce the complexity of the observed closure phases, additional surface features or dust clumps would be needed., Comment: 13 Pages, accepted for publication in A&A

The mass-loss mechanisms in M-type AGB stars are not well understood, in particular, the formation of dust-driven winds from the innermost gaseous layers around these stars. One way to understand the gas-dust interaction in these regions and its impact on the mass-loss mechanisms is through the analysis of high-resolution observations of the stellar surface and its closest environment. We aim at characterizing the inner circumstellar environment (~3 R*) of the M-type Mira star R Car in the near-infrared at different phases of a pulsation period. We used GRAVITY interferometric observations in the K-band obtained at two different epochs over 2018. Those data were analyzed using parametric models and image reconstruction of both the pseudo-continuum and the CO band-heads observed. The reported data are the highest angular resolution observations on the source in the K-band. We determine sizes of R Car's stellar disk of 16.67 +- 0.05 mas (3.03 au) in January 2018 and 14.84+-0.06 mas (2.70 au) in February 2018, respectively. From our physical model, we determined temperatures and size ranges for the innermost CO layer detected around R Car. We find that magnesium composites, Mg2SiO4 and MgSiO3, have temperatures and condensation distances consistent with the ones obtained for the CO layer model and pure-line reconstructed images, being them the most plausible dust types responsible of wind formation. Our reconstructed images show evidence of asymmetrical and inhomogeneous structures, which might trace a complex and perhaps clumpy structure of the CO molecule distribution. Our work demonstrates that the conditions for dust nucleation and thus for initialising dust-driven winds in M-type AGB stars are met in R Car and we identify Magnesium composites as the most probable candidates. This observational evidence is crucial to constrain the role of convection and pulsation in M-type stars., Comment: Accepted to be published by A&A, 20 pages, 21 figures

Complex vector modes, entangled in spin and orbital angular momentum, are opening burgeoning opportunities for a wide variety of applications. Importantly, the flexible manipulation the various properties of such beams will pave the way to novel applications. As such, in this manuscript, we demonstrate a longitudinal spin-orbit separation of complex vector modes propagating in free space. To achieve this we employed the recently demonstrated circular Airy Gaussian vortex vector (CAGVV) modes, which feature a self-focusing property. More precisely, by properly manipulating the intrinsic parameters of CAGVV modes, the strong coupling between the two constituting orthogonal components of CAGVV mode undergo a spin-orbit separation along the propagation direction namely, while one polarisation component, focuses at a specific plane, the other focuses at a different plane. Such spin-orbit separation, which we demonstrated by numerical simulations and corroborated experimentally, can be adjusted on-demand by simply changing the initial parameters of CAGVV modes. Our findings will be of great relevance, for example in optical tweezers, to manipulate micro- or nano-particles at two different parallel planes.

Reciprocal spin-orbit coupling (SOC) via geometric phase with flat optics provides a promising platform for shaping and controlling paraxial structured light. Current devices, from the pioneering q-plates to the recent J-plates, provide only spin-dependent wavefront modulation without amplitude control. However, achieving control over all the spatial dimensions of paraxial SOC states requires spin-dependent control of corresponding complex amplitude, which remains challenging for flat optics. Here, to address this issue, we present a new type of flat-optics elements termed structured geometric phase gratings that is capable of conjugated complex-amplitude control for orthogonal input circular polarizations. By using a microstructured liquid crystal photoalignment technique, we engineered a series of flat-optics elements and experimentally showed their excellent precision in arbitrary SOC control. This principle unlocks the full-field control of paraxial structured light via flat optics, providing a promising way to develop an information exchange and processing units for general photonic SOC states, as well as extra-/intracavity mode convertors for high-precision laser beam shaping.

Orbital angular momentum (OAM) conservation plays an important role in shaping and controlling structured light with nonlinear optics. The OAM of a beam originating from three-wave mixing should be the sum or difference of the other two inputs because no light-matter OAM exchange occurs in parametric nonlinear interactions. Here, we report anomalous OAM conservation during parametric upconversion, in which a Hermite-Gauss mode signal interacts with a specially engineered pump capable of astigmatic transformation in a crystal, resulting in Laguerre-Gaussian mode sum-frequency generation (SFG). The anomaly here refers to the fact that the pump and signal carry no net OAM, while their SFG does. We show that the lost OAM with the opposite sign that maintains OAM conservation in the system is hidden in the residual pump. This unexpected OAM selection rule improves our understanding of OAM conservation in parametric nonlinear systems and may inspire new ideas for controlling OAM states via nonlinear optics, especially in quantum applications.

Optical interferometers are pillars of modern precision metrology, but their resolution is limited by the wavelength of the light source, which cannot be infinitely reduced. Magically, this limitation can be circumvented by using an entangled multiphoton source because interference produced by an N-photon amplitude features a reduced de Broglie wavelength {\lambda}/N. However, the extremely low efficiency in multiphoton state generation and coincidence counts actually negates the potential of using multiphoton states in practical measurements. Here, we demonstrate a novel interferometric technique based on structured nonlinear optics, i.e., parametric upconversion of a structured beam, capable of superresolution measurement in real time. The main principle relies in that the orbital angular momentum (OAM) state and associated intramodal phase within the structured beam are both continuously multiplied in cascading upconversion to mimic the superresolved phase evolution of a multiphoton amplitude. Owing to the use of bright sensing beams and OAM mode projection, up to a 12-photon de Broglie wavelength with almost perfect visibility is observed in real time and, importantly, by using only a low-cost detector. Our results open the door to real-time superresolution interferometric metrology and provide a promising way toward multiphoton superiority in practical applications.

Complex vector modes of light, non-separable in their spatial and polarisation degrees of freedom, are revolutionising a wide variety of research fields. It is therefore not surprising that the generation techniques have evolved quite dramatically since their inception. At present it is common to use computer-controlled devices, among which digital micromirror devices have become popular. Some of the reason for this are their low-cost, their polarisation-insensitive and their high-refresh rates. As such, in this manuscript we put forward a novel technique characterised by its high stability, which is achieved through a common-path interferometer. We demonstrate the capabilities of this technique experimentally, first by generating arbitrary vector modes on a higher-order Poincar\'e sphere, secondly, by generating vector modes in different coordinates systems and finally, by generating various vector modes simultaneously. Our technique will find applications in fields such as optical manipulations, optical communications, optical metrology, among others., Comment: 7 pages, 6 figures

Autocorrelation measurement based on second-harmonic generation (SHG), the best-known technique for measuring the temporal duration of ultrashort pulses, could date back to the birth of ultrafast lasers. Here, we propose and experimentally demonstrate that such well-established technique can also be used to measure the orbital angular momentum of ultrashort Laguerre-Gauss (LG) pulses. By analysing the far-field pattern of the SHG signal, the full spatial structure of ultrashort LG pulses, including both azimuthal and radial indices, are unambiguously determined. Our results provide an important advancement for the well-established autocorrelation technique by extending it to reach its full potential in laser characterization, especially for structured ultrashort pulses.

Controlling the various degrees-of-freedom (DoFs) of structured light at both quantum and classical levels is of paramount importance in optics. It is a conventional paradigm to treat diverse DoFs separately in light shaping. While, the more general case of nonseparable states of light, in which two or more DoFs are coupled in a nonseparable way, has become topical recently. Importantly, classical nonseparable states of light are mathematically analogue to quantum entangled states. Such similarity has hatched attractive studies in structured light, e.g. the spin-orbit coupling in vector beams. However, nonseparable classical states of light are still treated in a fragmented fashion, while its forms are not limited by vector beams and its potential is certainly not fully exploited. For instance, exotic space-time coupled pulses nontrivial light shaping towards ultrafast time scales, and ray-wave geometric beams provide new dimensions in optical manipulations. Here we provide a bird's eye view on the rapidly growing but incoherent body of work on general nonseparable states involving various DoFs of light and introduce a unified framework for their classification and tailoring, providing a perspective on new opportunities for both fundamental science and applications., Comment: Under review in Laser & Photonics Review

Vectorial forms of structured light that are non-separable in their spatial and polarisation degrees of freedom have become topical of late, with an extensive toolkit for their creation and control. In contrast, the toolkit for quantifying their non-separability, the inhomogeneity of the polarisation structure, is far less developed, and in some cases fails altogether. To overcome this, here we introduce a new measure for vectorial light, which we demonstrate both theoretically and experimentally. We consider the general case where the local polarisation homogeneity can vary spatially across the field, from scalar to vector, a condition that can arise naturally if the composite scalar fields are path separable during propagation, leading to spatially disjoint vectorial light. We show how the new measure correctly accounts for the local path-like separability of the individual scalar beams, which can have varying degrees of disjointness, even though the global vectorial field remains intact. Our work attempts to address a pressing issue in the analysis of such complex light fields, and raises important questions on spatial coherence in the context of vectorially polarised light., Comment: 16 pages, 5 figures

Photonic states encoded in spatial modes of paraxial light fields provide a promising platform for high-dimensional quantum information protocols and related studies, where several pioneering theoretical and experimental demonstrations have paved the path for future technologies. Crucially, critical issues encountered in free-space propagation still represent a major challenge. This is the case of asynchronous diffraction between spatial modes with different modal orders, which experience variations in their transverse structure upon free-space propagation. Here we address this issue by proposing an encoding method based on the use of Laguerre-Gaussian (LG) modes of the same modal order N to define a N + 1 dimensional space. Noteworthy, such modes endowed with orbital angular momentum (OAM) experience the same propagation aberrations featuring an identical Gouy phase and wavefront curvature. We demonstrate our proposal experimentally by using time-correlated-single-photon imaging combined with a digital propagation technique. Importantly, our technique allows to eliminate, without the use of imaging systems, all issues related to asynchronous diffraction, providing an accessible way to generate propagation-invariant OAM qudits for quantum optical protocols.

Abruptly autofocusing waves and associated ring-Airy (RA) beams are attracting increasing interest owing to their fascinating properties such as their ability of abruptly autofocusing to small F-number. Optical frequency conversion via nonlinear interactions can further expand their applications to new area, yet are rarely studied. In this work, we report the frequency conversion of RA beams via sum-frequency generation using perfect flattop and common Gauss beams as the pump beams. The nonlinear transformation of the spatial complex amplitude of the signal and associated influences on autofocusing behavior, under different conditions of interaction location (i.e., original, autofocusing, and Fourier planes) and pump structure, were systematically studied and experimentally investigated. This proof-of principle demonstration provides a general guideline to build the frequency interface for abruptly autofocusing waves and a reference for relevant studies involving nonlinear transformation of abruptly autofocusing waves., Comment: 12 pages, 8 figures

Vectorial structured light with spatially varying amplitude, phase, and polarization is reshaping many areas of modern optics, including nonlinear optics, as diverse parametric processes can be used to explore interactions between such complex vector fields, extending the frontiers of optics to new physical phenomena. However, the most basic nonlinear application, i.e., frequency conversion, still remains challenging for vectorial structured light since parametric processes are polarization dependent, leading to a change in the spatial topological structure of signals. In this work, to break this fundamental limit, we propose a novel conformal frequency conversion scheme that allows to maintain the full spatial structure of vectorial structured light in the conversion; and systematically examine its spatial polarization independence based on non-degenerate sum-frequency generation with type-0 phase matching. This proof-of-principle demonstration paves the way for a wide range of applications requiring conformal frequency conversion, and, particularly, to implement frequency interfaces with multimodal communication channels, high-dimensional quantum states, and polarization-resolved upconversion imaging., Comment: 15 pages with 10 figures

Optical aberrations have been studied for centuries, placing fundamental limits on the achievable resolution in focusing and imaging. In the context of structured light, the spatial pattern is distorted in amplitude and phase, often arising from optical imperfections, element misalignment, or even from dynamic processes due to propagation through perturbing media such as living tissue, free-space, underwater and optical fibre. Here we show that the polarisation inhomogeneity that defines vectorial structured light is immune to all such perturbations, provided they are unitary. By way of example, we study the robustness of vector vortex beams to tilted lenses and atmospheric turbulence, both highly asymmetric aberrations, demonstrating that the inhomogeneous nature of the polarisation remains unaltered from the near-field to far-field, even as the structure itself changes. The unitary nature of the channel allows us to undo this change through a simple lossless operation, tailoring light that appears robust in all its spatial structure regardless of the medium. Our insight highlights the overlooked role of measurement in describing classical vectorial light fields, in doing so resolving prior contradictory reports on the robustness of vector beams in complex media. This paves the way to the versatile application of vectorial structured light, even through non-ideal optical systems, crucial in applications such as imaging deep into tissue and optical communication across noisy channels.

In recent time, the optical-analogous skyrmions, topological quasiparticles with sophisticated vectorial structures, have received an increasing amount of interest. Here we propose theortically and experimentally a generalized family of these, the tunable optical skyrmion, unveiling a new mechanism to transform between various skyrmionic topologies, including N\'eel-, Bloch-, and antiskyrmion types, via simple parametric tuning. In addition, Poincar\'e-like geometric representation is proposed to visualize the topological evolution of tunable skyrmions, which we termed Skyrme-Poincar\'e sphere, akin to the spin-orbit representation of complex vector modes. To generate experimentally the tunable optical skyrmions we implemented a digital hologram system based on a spatial light modulator, showing great agreement with our theoretical prediction.

Complex vector light fields have become a topic of late due to their exotic features, such as their non--homogeneous transverse polarisation distributions and the non-separable coupling between their spatial and polarisation degrees of freedom. In general, vector beams propagate in free space along straight lines, being the Airy-vector vortex beams the only known exception. Here, we introduce a new family of vector beams that exhibit novel properties that have not been observed before, such as their ability to freely accelerate along parabolic trajectories. In addition, their transverse polarisation distribution only contains polarisation states oriented at exactly the same angle but of different ellipticity. We anticipate that these novel vector beams might not only find applications in fields such as optical manipulation, microscopy or laser material processing, but extend to others., Comment: 7 pages, 5 images

Optical trapping describes the interaction between light and matter to manipulate micro-objects through momentum transfer. In the case of 3D trapping with a single beam, this is termed optical tweezers. Optical tweezers are a powerful and non-invasive tool for manipulating small objects, which have become indispensable in many fields, including physics, biology, soft condensed matter, amongst others. In the early days, optical trapping were typically used with a single Gaussian beam. In recent years, we have witnessed the rapid progress in the use of structured light beams with customized phase, amplitude and polarization in optical trapping. Unusual beam properties, such as phase singularities on-axis, propagation invariant nature, have opened up novel capabilities to the study of micromanipulation in liquid, air and vacuum. In this review, we summarize the recent advances in the field of optical trapping using structured light beams., Comment: 69 pages, 46 figures

Exploring the physics and potential applications of vectorially structured light with propagation-invariant transverse structures has benefited many areas of modern optics and photonics. In this paper, we investigate the non-eigen vector modes of paraxial light fields, focusing on the propagation variations and revivals of their transverse structures, including both spatial and polarization structures. We show that the physical mechanism behind the variations and revivals of their transverse structure is linked to the evolution of the intramodal phases between the constituting spatial modes. Such evolution originates from fractional Gouy phases, or rather, Geometric-phase difference between spatial modes with different orders under a same unitary transformation. This underlying principle, provides a general guideline for shaping vectorially structured light with custom propagation-evolution properties, and may also inspire a wide variety of new applications based on structured light.

Engineering vector spatial modes of photons is an important approach for manipulating high-dimension photonic states in various quantum optical experiments. In this work, we demonstrate generation of heralded single photons with well-defined vector spatial modes by using a self-locking polarizing interferometer comprising a spatial light modulator. Specifically, it is shown that, by carefully tailoring and compensating spatial and temporal amplitudes of manipulated photons, one can exactly convert ultrafast single photons into desired spin-orbit states with extremely high purity. This compact and robust device provides a versatile way for not only generation, but also manipulation and characterization of arbitrary photonic spin-orbit states.

Complex vector modes have become topical of late due to their fascinating properties and the many applications they have found across a broad variety of research fields. Even though such modes can be generated in a wide variety of ways, digital holography stands out as one of the most flexible and versatile. Along this line, Digital Micromirror Devices (DMDs) have gain popularity in recent time due to their high refresh rates, which allows the generation of vector modes at kHz rates. Nonetheless, most techniques are limited either by the diversity of vector modes that can be generated or by the speed at which they can be switched. Here we propose a technique based on the concept of random encoding, which allows the generation of arbitrary vector beams at speeds limited only by the refresh rate of the DMD. Our technique will be of great relevance in research fields such as optical communications, laser material processing and optical manipulation, amongst others., Comment: 9 pages, 9 figures

Vector modes represent the most general state of light in which, the spatial and polarisation degrees of freedom are coupled in a non-separable way. Crucially, while polarisation is limited to a bi-dimensional space, the spatial degree of freedom can take any spatial profile. However, most generation and application techniques are mainly limited to spatial modes with polar cylindrical symmetry, such as Laguerre- and Bessel-Gauss modes. In this manuscript we put forward a novel class of vector modes with its spatial degree of freedom encoded in the set of helical Mathieu-Gauss beams of the elliptical cylindrical coordinates. We first introduce these modes theoretically and outline their geometric representation on the higher-order Poincar\'e sphere. Later on, we demonstrate their experimental generation using a polarisation-insensitive technique comprising the use of a digital micromirror device. Finally, we provide with a qualitative and a quantitative characterisation of the same using modern approaches based on quantum mechanics tools. It is worth mentioning that non-polar vector beams are highly desired in various applications, such as optical trapping and optical communications., Comment: 9 pages, 9 figure

One of the most prominent features of quantum entanglement is its invariability under local unitary transformations, which implies the degree of entanglement remains constant during free-space propagation. While this is true for quantum and classically--entangled modes, here we demonstrate a novel type of classically-entangled modes that experience an entanglement decay upon free-space propagation. We show this by numerical simulations and corroborate experimentally. Our results evinces novel properties of classically-entangled modes, which pave the way to novel applications., Comment: 6 pages, 5 figures

Complex vector light modes, classically-entangled in their spatial and polarisation degrees of freedom (DoF), havebecome ubiquitous in a vast diversity of research fields. Crucially, while polarisation is limited to a bi-dimensionalspace, the spatial mode is unbounded, it can be specified by any of the sets of solutions the wave equation can supportin the different coordinate systems. Here we report on a class of vector beams with elliptical symmetry where thespatial DoF is encoded in the Ince-Gaussian modes of the cylindrical elliptical coordinates. We outline their geometricrepresentation on the Higher-Order Poincar\'e Sphere, demonstrate their experimental generation and analyse the qualityof the generated modes via Stokes polarimetry. We anticipate that such vector modes will be of great relevance inapplications, such as, optical manipulations, laser material processing and optical communications amongst others., Comment: 5 Pages, 6 figures

Spatial polarization independent (SPI) parametric conversion is the basis of many optical applications, such as SPI frequency interface for communication channels carried by vector modes and upconversion detection for polarization-resolved imaging. However, realizing such conversion remains a challenge. In this proof-of-principle work, we demonstrated SPI parametric upconversion using a polarization Sagnac nonlinear interferometer based on type-II second-harmonic generation (SHG). Our results show that the vector (including both polarization and intensity) profile and associated SOC state of the vector signal beam could be transferred to the SHG beam with a high fidelity. The principle lays a foundation of SPI frequency interface for quantum/classical channels based on vector modes and also paves the way for upconversion detection of polarization-resolved imaging in Mid-/far-infrared region.

Ince-Gaussian (IG) mode, a recently discovered type of structured Gaussian beam, corresponds to eigenfunctions of the paraxial wave equation in elliptical coordinates. This propagation-invariant mode is of significance in various domains, and in particular, its nonlinear transformation; however, there have been few relevant studies to date. In this work, we report the parametric upconversion of IG modes and associated full-field selection rule for the first time. We demonstrate that IG signals can be perfectly upconverted by a flattop-beam pump; in contrast, significant mode distortion occurred when using the most common Gaussian pump. Particular attention was given to the origin of the distortion, i.e., radial-mode degeneration induced by the sum-frequency generation excited by Gaussian pump. This proof-of-principle demonstration has great significance in relevant areas, such as high-dimension quantum frequency interfacing and upconversion imaging.

In recent time there has been an increasing amount of interest in developing novel techniques for the generation of complex vector light beams. Amongst these, digital holography stands out as one of the most flexible and versatile with almost unlimited freedom to generate scalar and vector light beams with arbitrary polarisation distributions and spatial transverse profile. Recently, we put forward a novel method to quantify the non-separability of vector modes in which we reported first measurements of a compact and robust device to generate such vector modes that fully exploits the polarisation-independence of Digital Micromirror Devices (DMDs). In this manuscript we fully characterise this device and provide qualitative and quantitative analysis of the generated modes. First by reconstructing their transverse polarisation distribution, using stokes polarimetry, followed by a measure of their degree of non-separability, determined through the concurrence., Comment: 7 pages, 5 figures

Stokes polarimetry is widely used to extract the polarisation structure of optical fields, typically from six measurements, although it can be extracted from only four. To measure the required intensities, most approaches are based on optical polarisation components. In this work, we present an all-digital approach that enables a rapid measure of all four intensities without any moving components. Our method employs a Polarisation Grating (PG) to simultaneously project the incoming mode into left- and right-circular polarised states, followed by a polarisation-insensitive Digital Micromirror Device (DMD), which digitally introduces a phase retardance for the acquisition of the remaining two polarisation states. We demonstrate how this technique can be applied to measuring the SoP, vectorness and intra-modal phase of optical fields, without any moving components and shows excellent agreement with theory, illustrating fast, real-time polarimetry.

The non-separability between the spatial and polarisation Degrees of Freedom (DoFs) of complex vector light fields has drawn significant attention in recent time. Key to this are its remarkable similarities with quantum entanglement, with quantum-like effects observed at the classical level. Crucially, this parallelism enables the use of quantum tools to quantify the coupling between the spatial and polarisation DoFs, usually implemented with polarisation-dependent spatial light modulators, which requires the splitting of the vector mode into two orthogonal polarisation components. Here we put forward a novel approach that relies on the use of Digital Micromirror Devices (DMDs) for fast, cheap and robust measurement, while the polarisation-independent nature of DMDs enables a reduction in the number of required measurements by 25\%. We tested our approach experimentally on cylindrical vector modes with arbitrary degrees of non-separability, of great relevance in a wide variety of applications. Our technique provides a reliable way to measure in real time the purity of vector modes, paving the way to novel applications where the degree of non-separability can be used as an optical sensor., Comment: Five pages, 5 figure

Optical orbital angular momentum transformation and corresponding azimuthal-mode selection rules have been studied exhaustively for various nonlinear optical interactions. However, nonlinear transformation of radial mode has not been systematically studied since the pioneering work [Phys. Rev. A 56, 4193, 1997]. In this paper, we theoretically study and experimentally verify the radial modal transitions of Laguerre-Gauss (LG) modes in parametric upconversion. Specifically, we provide a general solution that describes the sum-frequency generation (SFG) field excited by two arbitrary LG modes. Based on the solution, one can predict the full spatial complex amplitude of SFG fields upon propagation precisely and readily obtain the associated full-field selection rule including both azimuthal and radial modes. This work provides a theoretical basis for quantum and nonlinear optical research involving parametric upconversion of complex structured light, and paves the way for future work on full-field transformation of spatial modes in other nonlinear interactions., Comment: 10 pages, 6 figures

Stokes polarimetry (SP) is a powerful technique that enables spatial reconstruction of the state of polarization (SoP) of a light beam using only intensity measurements. A given SoP is reconstructed from a set of four Stokes parameters, which are computed through four intensity measurements. Since all intensities must be performed on the same beam, it is common to record each intensity individually, one after the other, limiting its performance to light beams with static SoP. Here, we put forward a novel technique to extend SP to a broader set of light beams with dynamic SoP. This technique relies on the superposition principle, which enables the splitting of the input beam into identical copies, allowing the simultaneous measurement of all intensities. For this, the input beam is passed through a multiplexed digital hologram displayed on a polarization-insensitive Digital Micromirror Device (DMD) that grants independent and rapid (20 kHz) manipulation of each beam. We are able to reliably reconstruct the SoP with high fidelity and at speeds of up to 27 Hz, paving the way for real-time polarimetry of structured light, Comment: 6 pages, 5 figure

Vectorial nonlinear optics refers to the investigation of optical processes whose nonlinear polarization (NP) undergoes spin-orbit coupling (SOC) interactions where, in general, the driving light field or the new field generated by the interaction containing the SOC property. To contribute to fundamental knowledge in this domain, we examine the type-II second harmonic generation (SHG) induced by vectorial laser modes. First, we provide a general theory to analyze the vectorial SHG process. Second, by using two typical vector modes as examples, we show how the SOC of the pump field dictates nonlinear interaction. Finally, we corroborate our theoretical predictions through experiments to confirm the crucial role of the SOC in nonlinear interactions. These results enhance our fundamental understanding of SOC-mediated nonlinear optics and lay the foundation for further fundamental studies as well as possible applications., Comment: 15 pages, 10 figures

Characterization and analysis of spin-orbit coupled (SOC) states, as a measurement problem, play a vital role in research on the modern optics and photonics based on structured light. Here, we demonstrate determination of photonic SOC states via spatial-Stokes measurement, in which two spatial complex probability amplitudes of spin-dependent spatial modes within SOC states and their relative (intramode) phase can be measured directly. Compared with the standard quantum-state tomography, by avoiding wavefront-flattening operations, the apparatus can completely record photons' realistic SOC structure, leading to a more accurate and precise determination of wavefunction. This simple and general approach for SOC state determination can provide a powerful toolkit for in-situ measuring photonic SOC state, characterizing the quality of SOC light source and associated geometric-phase devices., Comment: 5 pages, 3 figures

It is well known that when a laser is reflected from a rough surface or transmitted through a diffusive medium, a speckle pattern will be formed at a given observation plane. Speckle is commonly produced by laser beams with a homogeneous intensity, for which, well-known relations have been derived, relating the speckle size to the area of illumination. Here we investigate the speckle generated by higher-order Laguerre-Gaussian (LG) modes, characterized by a non-uniform intensity distribution of concentric rings.We show that the ring-structure of the LG modes does not play any role in the speckle size, which happens to be the same as that obtained for a homogeneous intensity distribution. This allow us to provide with a simple expression that relates the speckle size to the spot size of the LG modes. Our findings will be of great relevance in many speckle-based applications., Comment: 5 pages, 5 figures

Laser remote sensing represents a powerful tool that enables the accurate measurement of the speed of moving targets. Crucially, most sensing techniques are 2-Dimensional and only enable direct determination of the speed along the line of sight. A disadvantage that is very often overcome using two-dimensional techniques that in many cases are hard to implement and expensive. Here we put forward a novel 3-Dimensional technique that enables the direct and simultaneous measurement of both, the speed and the spin rate of cooperative targets. This technique is based on the use of complex vector light beams, whose polarization and spatial degree of freedom are coupled in a non-separable way. We present experimental evidence of our technique by performing a laboratory proof-of-principle experiment., Comment: 4 pages, 6 figures

Complex vectorial light fields, non-separable in their polarization and spatial degree of freedom, are of relevance in a wide variety of fields encompassing microscopy, metrology, communication and topological studies. Controversially, they have been suggested as analogues to quantum entanglement, raising fundamental questions on the relation between non-separability in classical systems, and entanglement in quantum systems. Here we propose and demonstrate basis-independent tomography of arbitrary vectorial light fields by relating their concurrence to spatially resolved Stokes projections. We generate vector fields with controllable non-separability using a novel compact interferometer that incorporates a digital micro-mirror device (DMD), thus offering a holistic toolbox for the generation and quantitative analysis of arbitrary vectorial light fields.