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The resonance wavelengths of optical Möbius strip microcavities can be continuously tuned via geometric phase manipulation by changing the thickness-to-width ratio of the strip.

We present a theoretical study and experimental measurements of a broadened orbital angular momentum power spectrum due to the phase modulation in the generation of spatially varying polarization beams and monstar singularities.

Asymmetric polarization disclinations, such as monstars, can be generated in two distinct ways: (a) by an inseparable superposition of three spatial modes bearing optical vortices with circular polarization states; (b) by using a modulated Poincaré beam, consisting of an inseparable superposition of a circularly-polarized fundamental Gaussian beam TEM00 and a second beam exhibiting an azimuthally-modulated vortex with an m-fold rotational symmetry and the opposite circular polarization. Based on the analysis of the spatial modes indirectly involved into the superposition through the latter method, we investigate its capability of spanning as many disclinations as possible, as well as its capability of enabling effective predictions about the generated patterns, such as relevant geometric features, already at the design stage.

We present a device based on liquid crystal and via Pancharatnam-Berry phase to generate Poincare beams by the coherent collinear superposition of two Free-Form Dark Hollow (FFDH) beams. We generate beams with spatially-variable polarization encoded on their cross section showing disclinations in the azimuth orientation and mappings of the Poincare sphere onto the transverse mode. We report generated beams characterized by nonuniform rotation rate of the local polarization azimuth in different polarization configurations, radial and azimuthal, lemon and star disclinations, and other richer and complex higher-order disclinations, by using tailored space-varying-axis plates based on liquid crystals.

Entanglement in high-dimensional quantum systems, where one or more degrees of freedom of light are involved, offers increased information capacities and enables new quantum protocols. Here, we demonstrate a functional source of high-dimensional, noise-resilient hyperentangled states encoded in time-frequency and vector-vortex structured modes, which in turn carry single-particle entanglement between polarisation and orbital angular momentum. Pairing nonlinearity-engineered parametric downconversion in an interferometric scheme with spin-to-orbital-angular-momentum conversion, we generate highly entangled photon pairs at telecom wavelength that we characterise via two-photon interference and quantum state tomography, achieving near-unity visibilities and fidelities. While hyperentanglement has been demonstrated before in photonic qubits, this is the first instance of such a rich entanglement structure involving spectrally and spatially structured light, where three different forms of entanglement coexist in the same biphoton state., 5 pages, 4 figures, Supplement: 6 pages, 2 figures

Quantum walks are powerful tools for quantum applications and for designing topological systems. Although they are simulated in a variety of platforms, genuine two-dimensional realizations are still challenging. Here we present an innovative approach to the photonic simulation of a quantum walk in two dimensions, where walker positions are encoded in the transverse wavevector components of a single light beam. The desired dynamics is obtained by means of a sequence of liquid-crystal devices, which apply polarization-dependent transverse "kicks" to the photons in the beam. We engineer our quantum walk so that it realizes a periodically-driven Chern insulator, and we probe its topological features by detecting the anomalous displacement of the photonic wavepacket under the effect of a constant force. Our compact, versatile platform offers exciting prospects for the photonic simulation of two-dimensional quantum dynamics and topological systems., Published version of the manuscript

A new photonic platform allows implementing 2D Quantum Walks in the space of transverse wavevector components of a single light beam. Detection of an anomalous velocity demonstrates that this system simulates a Quantum Hall Insulator.

We present a novel mechanism for performing common-path shearing interferometry using geometric phase, enabling almost any directional derivative while securing compactness and robustness. Important applications in freeform optics testing may result.

We mimic one dimensional forced quantum walks by using the photonic implementation obtained by means of a sequence of liquid-crystal devices (”g-plates”), which apply polarization-dependent transverse kicks to the photons in the beam. We observed refocusing phenomena for localized initial states.

CV entangled beams carrying OAM are successfully generated by converting entangled polarization modes into vortex beams. Entanglement is detected by homodyning in the OAM space with the homodyne local oscillator switching between different OAM states.

Space-variant polarization patterns present in the transverse mode of optical beams highlight disclination patterns of polarization about a singularity, often a C-point. These patterns are important for understanding rotational dislocations and for characterizing complex polarization patterns. Liquid-crystal devices known as q-plates have been used to produce two of the three types of disclination patterns in optical beams: lemons and stars. Here we report the production of the third type of disclination, which is asymmetric, known as the monstar. We do so with elliptically-symmetric q-plates. We present theory and measurements, and find excellent agreement between the two.

The working principle of ordinary refractive lenses can be explained in terms of the space-variant optical phase retardations they introduce, which reshape the optical wavefront curvature and hence affect the subsequent light propagation. These phases, in turn, are due to the varying optical path length seen by light at different transverse positions relative to the lens centre. A similar lensing behavior can however be obtained when the optical phases are introduced by an entirely different mechanism. Here, we consider the "geometric phases" that arise from the polarization transformations occurring in anisotropic optical media, named after Pancharatnam and Berry. The medium anisotropy axis is taken to be space-variant in the transverse plane and the resulting varying geometric phases give rise to the wavefront reshaping and lensing effect, which however depends also on the input polarization. We describe the realization and characterization of a cylindrical geometric-phase lens that is converging for a given input circular polarization state and diverging for the orthogonal one, which provides one of the simplest possible examples of optical element based on geometric phases. The demonstrated lens is flat and only few microns thick (not including the supporting substrates); moreover, its working wavelength can be tuned and the lensing can be switched on and off by the action of an external control electric field. Other kinds of lenses or more general phase elements inducing different wavefront distortions can be obtained by a similar approach. Besides their potential for optoelectronic technology, these devices offer good opportunities for introducing college-level students to an advanced topic of modern physics, such as the Berry phase, with the help of interesting optical demonstrations., Comment: 15 pages, 8 figures

Light beams carrying orbital angular momentum are key resources in modern photonics. In many applications, the ability of measuring the complex spectrum of structured light beams in terms of these fundamental modes is crucial. Here we propose and experimentally validate a simple method that achieves this goal by digital analysis of the interference pattern formed by the light beam and a reference field. Our approach allows one to characterize the beam radial distribution also, hence retrieving the entire information contained in the optical field. Setup simplicity and reduced number of measurements could make this approach practical and convenient for the characterization of structured light fields., 8 pages (including Methods and References), 6 figures

Topological insulators are fascinating states of matter exhibiting protected edge states and robust quantized features in their bulk. Here we propose and validate experimentally a method to detect topological properties in the bulk of one-dimensional chiral systems. We first introduce the mean chiral displacement, an observable that rapidly approaches a value proportional to the Zak phase during the free evolution of the system. Then we measure the Zak phase in a photonic quantum walk of twisted photons, by observing the mean chiral displacement in its bulk. Next, we measure the Zak phase in an alternative, inequivalent timeframe and combine the two windings to characterize the full phase diagram of this Floquet system. Finally, we prove the robustness of the measure by introducing dynamical disorder in the system. This detection method is extremely general and readily applicable to all present one-dimensional platforms simulating static or Floquet chiral systems., The detection of topological invariants in the bulk remains challenging even in state-of-the-art experiments. Here, Cardano et al. propose a method to read-out the Zak phases and topological invariants in one-dimensional chiral systems and detect those in a photonic quantum walk of twisted photons.

Optical singularities manifesting at the center of vector vortex beams are unstable, since their topological charge is higher than the lowest value permitted by Maxwell’s equations. Inspired by conceptually similar phenomena occurring in the polarization pattern characterizing the skylight, we show how perturbations that break the symmetry of radially symmetric vector beams lead to the formation of a pair of fundamental and stable singularities, i.e. points of circular polarization. We prepare a superposition of a radial (or azimuthal) vector beam and a uniformly linearly polarized Gaussian beam; by varying the amplitudes of the two fields, we control the formation of pairs of these singular points and their spatial separation. We complete this study by applying the same analysis to vector vortex beams with higher topological charges, and by investigating the features that arise when increasing the intensity of the Gaussian term. Our results can find application in the context of singularimetry, where weak fields are measured by considering them as perturbations of unstable optical beams.

Known methods for transverse confinement and guidance of light can be grouped into a few basic mechanisms, the most common being metallic reflection, total internal reflection and photonic-bandgap (or Bragg) reflection. All of them essentially rely on changes of the refractive index, that is on scalar properties of light. Recently, processes based on "geometric Berry phases", such as manipulation of polarization states or deflection of spinning-light rays, have attracted considerable interest in the contexts of singular optics and structured light. Here, we disclose a new approach to light waveguiding, using geometric Berry phases and exploiting polarization states and their handling. This can be realized in structured three-dimensional anisotropic media, in which the optic axis lies orthogonal to the propagation direction and is modulated along it and across the transverse plane, so that the refractive index remains constant but a phase distortion can be imposed on a beam. In addition to a complete theoretical analysis with numerical simulations, we present a proof-of-principle experimental demonstration of this effect in a discrete element implementation of a geometric phase waveguide. The mechanism we introduce shows that spin-orbit optical interactions can play an important role in integrated optics and paves the way to an entire new class of photonic systems that exploit the vectorial nature of light., Comment: Publication supported by European Union (EU) within Horizon 2020 - ERC-Advanced Grant PHOSPhOR, grant no. 694683. This is the final peer-reviewed manuscript as accepted for publication (including methods and supplementary information)

We demonstrate light guiding in a locally twisted anisotropic medium in the absence of a refractive index gradient. The transverse phase modulation required to compensate diffraction is provided by the Pancharatnam-Berry phase.

Many phenomena in solid-state physics can be understood in terms of their topological properties. Recently, controlled protocols of quantum walk (QW) are proving to be effective simulators of such phenomena. Here we report the realization of a photonic QW showing both the trivial and the non-trivial topologies associated with chiral symmetry in one-dimensional (1D) periodic systems. We find that the probability distribution moments of the walker position after many steps can be used as direct indicators of the topological quantum transition: while varying a control parameter that defines the system phase, these moments exhibit a slope discontinuity at the transition point. Numerical simulations strongly support the conjecture that these features are general of 1D topological systems. Extending this approach to higher dimensions, different topological classes, and other typologies of quantum phases may offer general instruments for investigating and experimentally detecting quantum transitions in such complex systems., Topological phases play a fundamental role in a variety of physical systems, yet there is a lack of efficient tools for revealing the occurrence of associated quantum transitions. Here, Cardano et al. report that such transitions can be identified in the statistics of quantum-walk dynamics and validate this idea in a photonic experimental implementation.

We introduce a laser for generating all states on the Higher-order Poincare sphere by coupling spin angular momentum to orbital angular momentum and we demonstrate that the OAM degeneracy of a laser may be broken.

The standard method for experimentally determining the probability distribution of an observable in quantum mechanics is the measurement of the observable spectrum. However, for infinite-dimensional degrees of freedom, this approach would require ideally infinite or, more realistically, a very large number of measurements. Here we consider an alternative method which can yield the mean and variance of an observable of an infinite-dimensional system by measuring only a two-dimensional pointer weakly coupled with the system. In our demonstrative implementation, we determine both the mean and the variance of the orbital angular momentum of a light beam without acquiring the entire spectrum, but measuring the Stokes parameters of the optical polarization (acting as pointer), after the beam has suffered a suitable spin–orbit weak interaction. This example can provide a paradigm for a new class of useful weak quantum measurements., The more degrees of freedom a quantum observable has, the more complicated it is to measure its probability distribution. Here, the authors deduce the mean and variance of an infinite-dimensional variable, the orbital angular momentum of light, from a two-dimensional one: spin angular momentum.

The angular momentum state of light can be described by positions on a higher-order Poincar\'e (HOP) sphere, where superpositions of spin and orbital angular momentum states give rise to laser beams that have found many applications, including optical communication, quantum information processing, microscopy, optical trapping and tweezing and materials processing. Many techniques exist to create such beams but none to date allow their creation at the source. Here we report on a new class of laser that is able to generate all states on the HOP sphere. We exploit geometric phase control with a non-homogenous polarization optic and a wave-plate inside a laser cavity to map spin angular momentum (SAM) to orbital angular momentum (OAM). Rotation of these two elements provides the necessary degrees of freedom to traverse the entire HOP sphere. As a result, we are able to demonstrate that the OAM degeneracy of a standard laser cavity may be broken, producing pure OAM modes as the output, and that generalized vector vortex beams may be created from the same laser, for example, radially and azimuthally polarized laser beams. It is noteworthy that all other aspects of the laser cavity follow a standard design, facilitating easy implementation.

We present newly conceived liquid-crystal-based retardation waveplates in which the optic axis distribution has asuperellipticallyŽsymmetricazimuthal structure with a topologicalcharge q superimposed. Such devices, namedsuperelliptical q-plates, act as polarization-to-spatialmodes converters that can be used to produce optical beamshaving peculiar spiral spectra. These spectra re”ect the topological charge of the optic axis distribution as well asthe symmetry properties of the underlying superellipse. The peculiar capability of q-plates of producing opticalmodes entangled with respect to spin and orbital angular momentum is here extended to superelliptical q-platesin order to create more complex optical modes structurally inseparable with respect to polarization and spatialdegrees of freedom. Such superelliptical modes can play a crucial role in studying polarization singularities or todevelop polarization metrology.Keywords: Optical angular momentum, topological charge, wavefront and polarization singularities

Two-dimensional fluid of polaritons sheds light on quantum vortex dynamics., Vortices are archetypal objects that recur in the universe across the scale of complexity, from subatomic particles to galaxies and black holes. Their appearance is connected with spontaneous symmetry breaking and phase transitions. In Bose-Einstein condensates and superfluids, vortices are both point-like and quantized quasiparticles. We use a two-dimensional (2D) fluid of polaritons, bosonic particles constituted by hybrid photonic and electronic oscillations, to study quantum vortex dynamics. Polaritons benefit from easiness of wave function phase detection, a spinor nature sustaining half-integer vorticity, strong nonlinearity, and tuning of the background disorder. We can directly generate by resonant pulsed excitations a polariton condensate carrying either a full or half-integer vortex as initial condition and follow their coherent evolution using ultrafast imaging on the picosecond scale. The observations highlight a rich phenomenology, such as the spiraling of the half-vortex and the joint path of the twin charges of a full vortex, until the moment of their splitting. Furthermore, we observe the ordered branching into newly generated secondary couples, associated with the breaking of radial and azimuthal symmetries. This allows us to devise the interplay of nonlinearity and sample disorder in shaping the fluid and driving the vortex dynamics. In addition, our observations suggest that phase singularities may be seen as fundamental particles whose quantized events span from pair creation and recombination to 2D+t topological vortex strings.

In this paper the role played by the angular momentum of light in the optical reorientation of a nematic liquid crystal was studied in the case of a normal incident elliptically shaped laser beam with circular polarization. Above the threshold power for the OFT, the molecular director suffered a stationary reorientation due to the balance between the orbital and spin angular momentum of light transferred to the sample. The director started to rotate only above a second threshold power. The experimental results qualitatively matched theoretical predictions.

In this work we studied experimentally the optical effects produced by the orbital angular momentum of light on the Optical Freedericksz Transition (OFT) in a nematic liquid crystal film. The measurements were carried out using a linearly polarized astigmatic laser beam with elliptical cross-section. The laser-induced reorientation was found to be strongly dependent on the angle between the polarization and the major axis of the beam profile. A threshold increase, out-of-polarization-plane reorientation and persistent oscillations were found to be the newest features observed in our experimental conditions. We ascribed all these features to the presence of an additional torque due to the orbital angular momentum of light.

We here demonstrate on both theoretical and experimental bases a method to recover the topological structure of a monochromatic optical field that has suffered diffuse transmission. This method consists of two steps: first, a linearly polarized sample beam is mixed with a coaxial Gaussian beam in orthogonal polarization states resulting in a Poincaré beam; second, a polarization-related spatial correlation function is considered and measured for the overall speckle field arising by optical diffusion. The singularities of the sample beam turn out to be imaged into the correlation function of the vector speckle field.

Standard multiple-beam holography has been largely used to produce gratings in polymer-liquid crystal composites, like POLICRYPS, H-PDLC gratings and POLIPHEM [1]. In this work we present a different approach to liquid crystalpolymeric grating production, based on the Computer-Generated Holography (CGH). The great advantage of using CGH is that interferometer-based schemes are no longer necessary, avoiding problems related to long term stability of the interference pattern and multi-beam complex optical setup. Moreover, the CGH technique allows a wider choice of pattern designs. In this preliminary work, we obtained promising results, as for instance the patterning of a square-wave refractive index modulation of a LCpolymeric composite, a pattern which is not achievable with standard two-beam holography., 6 pages, 3 figures

Multiple-beam holography has been widely used for the realization of photonic quasicrystals with high rotational symmetries not achievable by the conventional periodic crystals. Accurate control of the properties of the interfering beams is necessary to provide photonic band-gap structures. Here we show, by FDTD simulations of the transmission spectra of 8-fold quasiperiodic structures, how the geometric tiling of the structure affects the presence and properties of the photonic band-gap for low refractive index contrasts. Hence, we show an interesting approach to the fabrication of photonic quasicrystals based on the use of a programmable Spatial Light Modulator encoding Computer-Generated Holograms, that permits an accurate control of the writing pattern with almost no limitations in the pattern design. Using this single-beam technique we fabricated quasiperiodic structures with high rotational symmetries and different geometries of the tiling, demonstrating the great versatility of our technique., Comment: 8 pages, 6 figures

The angular momentum of light can be split into a spin and an orbital component (SAM and OAM). A few years ago, an optical process involving a conversion of angular momentum from one form to the other was conceived and experimentally demonstrated in a singular patterned liquid crystal cell, also known as “q-plate”. In this paper, after reviewing the q-plate concept and technology, we will survey some of the most significant results that have originated from it, with particular attention to the possibility of realizing a physical one-to-one mapping between the polarization Poincare sphere and an OAM subspace of an optical beam or of a single photon.

We present a tunable liquid crystal device that converts pure orbital angular momentum eigenmodes of a light beam into equal-weight superpositions of opposite-handed eigenmodes and vice versa. For specific input states, the device may thus simulate the behavior of a {\pi}/2 phase retarder in a given two-dimensional orbital angular momentum subspace, analogous to a quarter-wave plate for optical polarization. A variant of the same device generates the same final modes starting from a Gaussian input., Comment: 6 pages, 6 figures

The polarization and intensity patterns of the outgoing beam at the exit face of the q plate reported in Fig. 2(d) of the original paper does not match with its caption. Figure 1 shows the associate polarization and intensity patterns of the emerging optical beam at the exit face of the q plate when the input beam is horizontally polarized. However, this correction bears no consequence on the conclusions in our original paper.

The mechanism of photon spin and orbital angular momentum transfer to nonlinear non-absorbing material is described on both theoretical and experimental grounds.

A few years ago the possibility of coupling and inter-converting the spin and orbital angular momentum (SAM and OAM) of paraxial light beams in inhomogeneous anisotropic media was demonstrated. An important case is provided by waveplates having a singular transverse pattern of the birefringent optical axis, with a topological singularity of charge q at the plate center, hence named 'q-plates'. The introduction of q-plates has given rise in recent years to a number of new results and to significant progress in the field of orbital angular momentum of light. Particularly promising are the quantum photonic applications, because the polarization control of OAM allows the transfer of quantum information from the SAM qubit space to an OAM subspace of a photon and vice versa. In this paper, we review the development of the q-plate idea and some of the most significant results that have originated from it, and we will briefly touch on many other related findings concerning the interaction of the SAM and OAM of light.

We demonstrate electromagnetic quantum states of single photons and of correlated photon pairs exhibiting "hybrid" entanglement between spin and orbital angular momentum. These states are obtained from entangled photon pairs emitted by spontaneous parametric down conversion, by employing a $q$-plate for coupling the spin and orbital degrees of freedom of a photon. Entanglement and contextual quantum behavior (that is also non-local, in the case of photon pairs) is demonstrated by the reported violation of the Clauser-Horne-Shimony-Holt inequality. In addition a classical analog of the hybrid spin-orbit photonic entanglement is reported and discussed., Comment: 5 pages, 3 figures

The orbital angular momentum (OAM) of light, associated with a helical structure of the wavefunction, has a great potential for quantum photonics, as it allows attaching a higher dimensional quantum space to each photon. Hitherto, however, the use of OAM has been hindered by its difficult manipulation. Here, exploiting the recently demonstrated spin-OAM information transfer tools, we report the first observation of the Hong-Ou-Mandel coalescence of two incoming photons having nonzero OAM into the same outgoing mode of a beam-splitter. The coalescence can be switched on and off by varying the input OAM state of the photons. Such effect has been then exploited to carry out the 1 \rightarrow 2 universal optimal quantum cloning of OAM-encoded qubits, using the symmetrization technique already developed for polarization. These results are finally shown to be scalable to quantum spaces of arbitrary dimension, even combining different degrees of freedom of the photons., 5 pages, 3 figures

We present a novel optical device to encode and decode two bits of information into different Orbital Angular Momentum (OAM) states of a paraxial optical beam. Our device generates the four angular momentum states of order $\pm 2$ and $\pm4$ by Spin-To-Orbital angular momentum Conversion (STOC) in a triangular optical loop arrangement. The switching among the four OAM states is obtained by changing the polarization state of the circulating beam by two quarter wave plates and the two-bit information is transferred to the beam OAM exploiting a single $q$-plate. The polarization of the exit beam is left free for additional one bit of information. The transmission bandwidth of the device may be as large as several megahertz if electro-optical switches are used to change the beam polarization. This may be particularly useful in communication system based on light OAM., 5 pages, 5 figures, 1 table. Submitted

We present an easy, efficient and fast method to generate arbitrary linear combinations of light orbital angular momentum eigenstates $\ell=\pm 2$ starting from a linearly polarized TEM$_{00}$ laser beam. The method exploits the spin-to-orbital angular momentum conversion capability of a liquid-crystal-based $q$-plate and a Dove prism inserted in a Sagnac polarizing interferometer. The nominal generation efficiency is 100\%, being limited only by reflection and scattering losses in the optical components. When closed paths are followed on the polarization Poincar\'{e} sphere of the input beam, the associated Pancharatnam geometric phase is transferred unchanged to the orbital angular momentum state of the output beam., Comment: 5 pages and 5 figures

Exploiting electro-optic effects in liquid crystals, we achieved real-time control of the retardation of liquid- crystal-based $q$-plates through an externally applied voltage. The newly conceived electro-optic $q$-plates can be operated as electrically driven converters of photon spin into orbital angular momentum, enabling a variation of the orbital angular momentum probabilities of the output photons over a time scale of milliseconds., Comment: 4 pages, 5 figures, submitted

The recently demonstrated possibility of entangling opposite values of the orbital angular momentum (OAM) of a photon with its spin enables the realization of nontrivial one-photon spinorbit four-dimensional states for quantum information purposes. Hitherto, however, an optical device able to perform arbitrary unitary transformations on such spinorbit photon states has not been proposed yet. In this work we show how to realize such a ``universal unitary gate'' device, based only on existing optical technology, and describe its operation. Besides the quantum information field, the proposed device may find applications wherever an efficient and convenient manipulation of the combined OAM and spin of light is required., 7 pages, 2 figures

We calculated the Fresnel paraxial propagator in a birefringent plate having topological charge $q$ at its center, named "$q$-plate". We studied the change of the beam transverse profile when it traverses the plate. An analytical closed form of the beam profile propagating in the "$q$-plate" can be found for many important specific input beam profiles. We paid particular attention to the plate having a topological unit charge and we found that if small losses due to reflection, absorption and scattering are neglected, the plate can convert the photon spin into orbital angular momentum with up to 100% efficiency, provided the thickness of the plate is less than the Rayleigh range of the incident beam., 3 pages and 3 figures. Optics Letters style

Liquid crystals interacting with light represent a unique class of soft-matter systems that exhibit various generic nonlinear behaviors, including chaotic rotational dynamics. Despite several experimental observations, complex nematic liquid crystal director rotations in presence of spin and orbital angular momentum transfer processes were left unexplained. We present a self-consistent three-dimensional model able to describe the previous experimental observations, accounting for the dependence on the incident beam intensity, polarization, finite size and shape. More generally, our model is able to describe quantitatively the dynamics of, and beyond, the optical Fréedericksz transition under realistic experimental conditions almost three decades after its experimental discovery.