A fast-Fourier-transform method of topography and interferometry is proposed. By computer processing of a noncontour type of fringe pattern, automatic discrimination is achieved between elevation and depression of the object or wave-front form, which has not been possible by the fringe-contour-generation techniques. The method has advantages over moire topography and conventional fringe-contour interferometry in both accuracy and sensitivity. Unlike fringe-scanning techniques, the method is easy to apply because it uses no moving components.

Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry

Topological photonics is a rapidly emerging field of research in which geometrical and topological ideas are exploited to design and control the behavior of light. Drawing inspiration from the discovery of the quantum Hall effects and topological insulators in condensed matter, recent advances have shown how to engineer analogous effects also for photons, leading to remarkable phenomena such as the robust unidirectional propagation of light, which hold great promise for applications. Thanks to the flexibility and diversity of photonics systems, this field is also opening up new opportunities to realize exotic topological models and to probe and exploit topological effects in new ways. This article reviews experimental and theoretical developments in topological photonics across a wide range of experimental platforms, including photonic crystals, waveguides, metamaterials, cavities, optomechanics, silicon photonics, and circuit QED. A discussion of how changing the dimensionality and symmetries of photonics systems has allowed for the realization of different topological phases is offered, and progress in understanding the interplay of topology with non-Hermitian effects, such as dissipation, is reviewed. As an exciting perspective, topological photonics can be combined with optical nonlinearities, leading toward new collective phenomena and novel strongly correlated states of light, such as an analog of the fractional quantum Hall effect.

Topological Photonics

Thirty years ago, Coullet et al. proposed that a special optical field exists in laser cavities bearing some analogy with the superfluid vortex. Since then, optical vortices have been widely studied, inspired by the hydrodynamics sharing similar mathematics. Akin to a fluid vortex with a central flow singularity, an optical vortex beam has a phase singularity with a certain topological charge, giving rise to a hollow intensity distribution. Such a beam with helical phase fronts and orbital angular momentum reveals a subtle connection between macroscopic physical optics and microscopic quantum optics. These amazing properties provide a new understanding of a wide range of optical and physical phenomena, including twisting photons, spin-orbital interactions, Bose-Einstein condensates, etc., while the associated technologies for manipulating optical vortices have become increasingly tunable and flexible. Hitherto, owing to these salient properties and optical manipulation technologies, tunable vortex beams have engendered tremendous advanced applications such as optical tweezers, high-order quantum entanglement, and nonlinear optics. This article reviews the recent progress in tunable vortex technologies along with their advanced applications.

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Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities

The current understanding of the role of topology in non-Hermitian (NH) systems and its far-reaching physical consequences observable in a range of dissipative settings are reviewed. In particular, how the paramount and genuinely NH concept of exceptional degeneracies, at which both eigenvalues and eigenvectors coalesce, leads to phenomena drastically distinct from the familiar Hermitian realm is discussed. An immediate consequence is the ubiquitous occurrence of nodal NH topological phases with concomitant open Fermi-Seifert surfaces, where conventional band-touching points are replaced by the aforementioned exceptional degeneracies. Furthermore, new notions of gapped phases including topological phases in single-band systems are detailed, and the manner in which a given physical context may affect the symmetry-based topological classification is clarified. A unique property of NH systems with relevance beyond the field of topological phases consists of the anomalous relation between bulk and boundary physics, stemming from the striking sensitivity of NH matrices to boundary conditions. Unifying several complementary insights recently reported in this context, a picture of intriguing phenomena such as the NH bulk-boundary correspondence and the NH skin effect is put together. Finally, applications of NH topology in both classical systems including optical setups with gain and loss, electric circuits, and mechanical systems and genuine quantum systems such as electronic transport settings at material junctions and dissipative cold-atom setups are reviewed.

Exceptional topology of non-Hermitian systems

Visible, circularly polarised light can be transformed into light-carrying orbital angular momentum by a plasmonic metasurface. That is the finding of Ebrahim Karimi and co-workers at the University of Ottawa in Canada and the University of Rochester in the United States. Light with orbital angular momentum (owing to a twisted phase front) is traditionally generated using specially designed optical elements such as spatial light modulator, cylindrical lens mode converters and q-plate. The researchers have now shown that a plasmonic metasurface comprising an array of nano-antennas can couple spin-to-orbital angular momentum at thickness much smaller than the wavelength of the light with an efficiency of around 3%. The conversion takes place due to the birefringence present in the nanostructure array. This approach could yield ultrathin generators of visible light with orbital angular momentum, for potential applications in spectroscopy, imaging, sensing and quantum information.

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Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface

We present an investigation on ultrashort laser surface structuring with structured light fields generated by various q-plates. In particular, q-plates with topological charges q = 1, 3/2, 2, 5/2 are used to generate femtosecond (fs) vector vortex beams, and form complex periodic surface structures through multi-pulse ablation of a solid crystalline silicon target. We show how optical retardation tuning of the q-plate offers a feasible way to vary the fluence transverse distribution of the beam, thus allowing the production of structures with peculiar shapes, which depend on the value of q. The features of the generated surface structures are compared with the vector vortex beam characteristics at the focal plane, by rationalizing their relationship with the local state of the laser light. Our experimental findings demonstrate how irradiation with fs complex light beams can offer a valuable route to design unconventional surface structures.

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Surface structures with unconventional patterns and shapes generated by femtosecond structured light fields.

In quantum information, complementarity of quantum mechanical observables plays a key role. Theeigenstates of two complementary observables form a pair of mutually unbiased bases (MUBs). Moregenerally,asetofMUBsconsistsofbasesthatareallpairwiseunbiased.ExceptforspecificdimensionsoftheHilbert space, the maximal sets of MUBs are unknown in general. Even for a dimension as low as six, theidentification of a maximal set of MUBs remains an open problem, although there is strong numericalevidence that no more than three simultaneous MUBs do exist. Here, by exploiting a newly developedholographic technique, we implement and test different sets of three MUBs for a single photonsix-dimensionalquantumstate(a‘‘qusix’’),encodedexploitingpolarizationandorbitalangularmomentumof photons. A close agreement is observed between theory and experiments. Our results can findapplications in state tomography, quantitative wave-particle duality, quantum key distribution.

Test of mutually unbiased bases for six-dimensional photonic quantum

We report on a method for the characterization of intense, structured optical fields through the analysis of the size and surface structures formed inside the annular ablation crater created on the target surface. In particular, we apply the technique to laser ablation of crystalline silicon induced by femtosecond vector vortex beams. We show that a rapid direct estimate of the beam waist parameter is obtained through a measure of the crater radii. The variation of the internal and external radii of the annular crater as a function of the laser pulse energy, at fixed number of pulses, provides another way to evaluate the beam spot size through numerical fitting of the obtained experimental data points. A reliable estimate of the spot size is of paramount importance to investigate pulsed laser-induced effects on the target material. Our experimental findings offer a facile way to characterize focused, high intensity complex optical vector beams which are more and more applied in laser-matter interaction experiments.

Simple method for the characterization of intense Laguerre-Gauss vector vortex beams

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.

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Topological features of vector vortex beams perturbed with uniformly polarized light

Using spatial modes for quantum key distribution (QKD) has become highly topical due to their infinite dimensionality, promising high information capacity per photon. However, spatial distortions reduce the feasible secret key rates and compromise the security of a quantum channel. In an extreme form such a distortion might be a physical obstacle, impeding line-of-sight for free-space channels. Here, by controlling the radial degree of freedom of a photon's spatial mode, we are able to demonstrate hybrid high-dimensional QKD through obstacles with self-reconstructing single photons. We construct high-dimensional mutually unbiased bases using spin-orbit hybrid states that are radially modulated with a non-diffracting Bessel-Gaussian (BG) profile, and show secure transmission through partially obstructed quantum links. Using a prepare-measure protocol we report higher quantum state self-reconstruction and information retention for the non-diffracting BG modes as compared to Laguerre-Gaussian modes, obtaining a quantum bit error rate (QBER) that is up to 3 times lower. This work highlights the importance of controlling the radial mode of single photons in quantum information processing and communication as well as the advantages of QKD with hybrid states.

Filippo Cardano

Papers

Surface structures with unconventional patterns and shapes generated by femtosecond structured light fields.

Test of mutually unbiased bases for six-dimensional photonic quantum

Simple method for the characterization of intense Laguerre-Gauss vector vortex beams

Topological features of vector vortex beams perturbed with uniformly polarized light

Self-healing high-dimensional quantum key distribution using hybrid spin-orbit Bessel states