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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

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.

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.

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.

Rydberg atoms have great potential in electric field measurement and have an advantage with a large frequency bandwidth from the kHz to the THz scale. However, the sensitivity for measuring a weak MHz electric field signal is limited by the spectroscopic resolution, because the weak electric field induces only a small perturbation of the population and energy level shift of the Rydberg atoms. Here, we report highly sensitive measurement of a weak MHz electric field using electromagnetically induced transparency with Rydberg atoms in a thermal atomic system. Using the heterodyne method on a 30-MHz electric field, we successfully measure the minimum electric field strength to be \textcolor{black}{37.3 $\mathrm{\mu V/cm}$} with a sensitivity up to $-65$ dBm/Hz and a linear dynamic range over 65 dB. Additionally, we measure an amplitude-modulated signal and demodulate the signal with a fidelity over 98\%. This work extends the sensitivity of atomic sensors for measuring MHz electric fields, which advances atomic electric field-sensing technology., Comment: 6 pages, 4 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.

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

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 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.

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

Cavity-free optical nonreciprocity components, which have an inherent strong asymmetric interaction between the forward- and backward-propagation direction of the probe field, are key to produce such as optical isolators and circulators. According to the proposal presented by Xia et al., [Phys. Rev. Lett. 121, 203602 (2018)], we experimentally build a device that uses cross-Kerr nonlinearity to achieve a cavity-free optical isolator and circulator. Its nonreciprocal behavior arises from the thermal motion of N-type configuration atoms, which induces a strong chiral cross-Kerr nonlinear response for the weak probe beam. We obtain a two-port optical isolator for up to 20 dB of isolation ratio in a specially designed Sagnac interferometer. The distinct propagation directions of the weak probe field determine its cross-phase shift and transmission, by which we demonstrate the accessibility of a four-port optical circulator., Comment: 5 pages, 3 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

A high-quality, compact, and narrow-bandwidth entangled photon source (EPS) is indispensable for realization of many quantum communication protocols. Usually, a free space cavity containing a nonlinear crystal is used to generate a narrow bandwidth EPS through spontaneous parametric down-conversion (SPDC). One major drawback is that this occupies a large space and requires complex optical and electrical control systems. Here we present a simple and compact method to generate a single-longitudinal-mode time-energy EPS via type II SPDC in a submillimeter Fabry-Perot cavity. We characterize the quality of the EPS by measuring the coincidence to accidental coincidence ratio, the two-photon time cross-correlation function and the two-photon interference fringes. All measured results clearly demonstrate that the developed source is of high quality when compared with EPSs generated using other configurations. We believe this source is very promising for applications in the quantum communication field., Comment: 7 pages, 5 figures,submitted to PRL, comments are welcome

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

Spin-orbit coupled (SOC) light fields with spatially inhomogeneous polarization have attracted increasing research interest within the optical community. In particular, owing to their spin-dependent phase and spatial structures, many nonlinear optical phenomena which we have been familiar with up to now are being re-examined, hence a revival of research in nonlinear optics. To fully investigate this topic, knowledge on how the topological structure of the light field evolves is necessary, but, as yet, there are few studies that address the structural evolution of the light field. Here, for the first time, we present a universal approach for theoretical tomographic treatment of the structural evolution of SOC light in nonlinear optics processes. Based on a Gedanken vector second harmonic generation, a fine-grained slice of evolving SOC light in a nonlinear interaction and the following diffraction propagation are studied theoretically and verified experimentally, and which at the same time reveal several interesting phenomena. The approach provides a useful tool for enhancing our capability to obtain a more nuanced understanding of vector nonlinear optics, and sets a foundation for further broad-based studies in nonlinear systems., Comment: 10 pages, 7 figures

Light with transverse polarization structure, such as radial and azimuthal polarization, enables and revives lots of applications based on light-matter interaction due to their unique focal properties. To date, studies referring to this topic mainly concentrate in weak-light domain, yet it should have gained more attention in strong-light domain. A main factor contributing to the current situation is that the generation devices cannot afford high power. Here, in this proof-of-principle work, we demonstrate the generation of strong single-longitudinal-mode (SLM) cylindrical vector (CV) short pulses via stimulated Brillouin amplification, specifically, the energy is transferred from a 700 ps pump to a 300 ps Stokes pulse via parametrically generated coherent phonons. After amplification, a 100 mJ-level SLM CV pulse light with 300 ps duration is obtained. Meanwhile, the phase and polarization structures are high fidelity maintained. This result provides a practicable way to generate strong CV light, and by extending this mechanism into multi-pump beam combing system, even an ultra-high intensity CV light can be expected.

Are quantum states real? This most fundamental question in quantum mechanics has not yet been satisfactorily resolved, although its realistic interpretation seems to have been rejected by various delayed-choice experiments. Here, to address this long-standing issue, we present a quantum twisted double-slit experiment. By exploiting the subluminal feature of twisted photons, the real nature of a photon during its time in flight is revealed for the first time. We found that photons' arrival times were inconsistent with the states obtained in measurements but agreed with the states during propagation. Our results demonstrate that wavefunctions describe the realistic existence and evolution of quantum entities rather than a pure mathematical abstraction providing a probability list of measurement outcomes. This finding clarifies the long-held misunderstanding of the role of wavefunctions and their collapse in the evolution of quantum entities., Comment: Comments are welcome.12 pages, 5 figures

A spatial light modulator (SLM) is one of the most useful and convenient device to generate structural light beams such as twisted light and complexed images used in modern optical science. The unbounded dimension of twisted light makes it promising in harnessing information carrying ability of a single photon, which greatly enhances the channel capacity in optical communications. We perform a detail theoretical study of the birth, evolution and reverse transformation of twisted light generated from a phase-only SLM based on diffraction theory, analytical expressions are obtained to show the special evolution behaviors of the light beam with the propagation distance. Beam intensity distributions calculated theoretically are in well agreement with experimental observations. Our findings clearly reveal how twisted light gradually emerges from a Gaussian spatial shape to a ring shape, and also the ring shape will gradually evolve to a quasi-Gaussian shape conversely, in the reverse transformation. These results will provide guidelines for using SLM in many optical experiments and OAM-mode multiplexing in optical communications., Comment: primary manscript, comments are welcome