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Fibered Burnside rings appear as Grothendieck rings of fibered permutation representations of a finite group, generalizing Burnside rings and monomial representation rings. Their species, primitive idempotents and their conductors are of particular interest in representation theory as they encode information related to the structure of the group. In this note, we introduce fiber change maps between fibered Burnside rings, and we present results on their functoriality and naturality with respect to biset operations. We present some advances on the conductors for cyclic fiber groups, and fully determine them in particular cases, covering a wide range of interesting examples.

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

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

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

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

Let $G$ and $H$ be finite groups. We give a condition on $G$ and $H$ that implies that the $A$-fibered Burnside rings $B^A(G)$ and $B^A(H)$ are isomorphic. As a consequence, we show the existence of non-isomorphic groups $G$ and $H$ such that $B^A(G)$ and $B^A(H)$ are isomorphic rings. Here, the abelian fiber group $A$ can be chosen in a non-trivial way, that is, such that $B^A(G)$ and $B^A(H)$ are strictly bigger than the Burnside rings of $G$ and $H$, for which such counterexamples are already known.

In this article, we propose a concept of anti-involution for Green biset functors and we provide equivalent definitions. We present $\star$-Green biset functors and we study their orthogonal units and the orthogonal automorphisms in their associated categories.

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

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

In this note, we present a notion of species isomorphism for fibered Burnside rings. We prove that a species isomorphism can be restricted to an isomorphism of mark tables, that it can be extended to an isomorphism between the ghost rings and we provide three additional characterizations of this concept.

Quantum computation promises to execute certain computational tasks on time scales much faster than any known algorithm on an existing classical computer, for example calculating the prime factors of large integers. Recently a research team from Google claimed to have carried out such a task with a quantum computer, demonstrating in practice a case of this so-called quantum supremacy. Here we argue that this claim was not justified. Unlike other comments, our criticism is concerned with the missing verification of the output data of the quantum computation., Comment: 2 pages

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

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