We discuss the propagation of an electromagnetic field in an inhomogeneously anisotropic material where the optic axis is rotated in the transverse plane but is invariant along the propagation direction. In such a configuration, the evolution of an electromagnetic wavepacket is governed by the Pancharatnam-Berry phase (PBP), responsible for the appearance of an effective photonic potential. In a recent paper [A. Alberucci et al., "Electromagnetic confinement via spin-orbit interaction in anisotropic dielectrics", ACS Photonics \textbf{3}, 2249 (2016)] we demonstrated that the effective potential supports transverse confinement. Here we find the profile of the quasi-modes and show that the photonic potential arises from the Kapitza effect of light. The theoretical results are confirmed by numerical simulations, accounting for the medium birefringence. Finally, we analyze in detail a configuration able to support non-leaky guided modes.
In a recent letter, Milione et al. reported the Pancharatnam-Berry phase (PBP) on a higher-order Poincaré sphere (HOPS) [Phys. Rev. Lett, 108, 190401 (2012)]. They used two spin-orbit converters (SOCs) to make a trajectory along the HOPS. However, the SOCs actually do not make a path on the HOPS. We briefly discuss a way of acquiring a PBP made by SOCs.
Relying on the local orientation of nanostructures, Pancharatnam-Berry metasurfaces are currently enabling a new generation of polarization-sensitive optical devices. A systematical mesoscopic description of topological metasurfaces is developed, providing a deeper understanding of the physical mechanisms leading to the polarization-dependent breaking of translational symmetry in contrast with propagation phase effects. These theoretical results, along with interferometric experiments, contribute to the development of a solid theoretical framework for arbitrary polarization-dependent metasurfa…
We apply the Pancharatnam-Berry phase approach to plasmonic metasurfaces loaded by highly nonlinear multi-quantum well substrates, establishing a platform to control the nonlinear wavefront at will based on giant localized nonlinear effects. We apply this approach to design flat nonlinear metasurfaces for efficient second-harmonic radiation, including beam steering, focusing, and polarization manipulation. Our findings open a new direction for nonlinear optics, in which phase matching issues are relaxed, and an unprecedented level of local wavefront control is achieved over thin devices with g…
The Pancharatnam-Berry (PB) phase in metasurfaces obeys the symmetry restriction, according to which the PB phases of two orthogonal circularly polarized waves are the same but with opposite signs. Here, we reveal a general mechanism to break the axisymmetry of meta-atoms in order to break the PB-phase symmetry restriction. As a proof of concept, we designed a novel meta-atom with a QR-code structure and successfully demonstrated circular-polarization multiplexing metasurface holography. This study provides a fundamentally new understanding of the PB phase and opens a path for arbitrary wavefr…
Since its introduction by Sir Michael Berry in 1984, geometric phase became of fundamental importance in physics, with applications ranging from solid state physics to optics. In optics, Pancharatnam-Berry phase allows the tailoring of optical beams by a local control of their polarization. Here we discuss light propagation in the presence of an intensity-dependent local modulation of the Pancharatnam-Berry phase. The corresponding self-modulation of the wavefront counteracts the natural spreading due to diffraction, i.e., self-focusing takes place. No refractive index variation is associated …
In polarization optics unitary and non-unitary operations can be carried out by the Jones matrix. $\mathbf{Z}$ matrix is the $4\times 4$ analogue of the Jones matrix and the Mueller matrix of a nondepolarizing optical medium can be written as $\mathbf{M}=\mathbf{Z}\mathbf{Z}^*$. Jones matrix acts on the two component complex Jones vector, while the $\mathbf{Z}$ matrix acts on the four component real Stokes vector. Polarizer and retarder $\mathbf{Z}$ matrices can be written in compact forms in terms of the components of the position vector on the Poincaré sphere. In this note it is shown that t…
We derived the Berry connection of vector vortex states (VVSs) from the "true" Hamiltonian obtained through the Maxwell--Schrödinger equation for an inhomogeneous anisotropic (IA) medium, and we experimentally demonstrated measurement of the corresponding Pancharatnam--Berry (PB) geometrical phase of VVSs. The PB phase (PBP) of VVSs can be divided into two phases: homogeneous and inhomogeneous PBPs. Homogeneous and inhomogeneous PBPs are related to the conventional PBP and the spatially-dependent geometric phase given by an IA medium such as a polarization converter, respectively. We theoretic…
We theoretically and numerically investigate the properties of waveguides based on the Pancharatnam-Berry phase, obtained by a longitudinally periodic rotation of the optic axis in a transversely-twisted birefringent medium. In this paper we study the case where the period of the longitudinal modulation is chosen so that a net accumulation of geometric phase in propagation occurs. First, the interplay between different contributions to the optical potential is addressed. Second, a continuous evolution of the polarization structure of the quasi-modes is observed in the numerical simulations. We…
Perfect vortex beams are the orbital angular momentum (OAM)-carrying beams with fixed annular intensities, which provide a better source of OAM than traditional Laguerre- Gaussian beams. However, ordinary schemes to obtain the perfect vortex beams are usually bulky and unstable. We demonstrate here a novel generation scheme by designing planar Pancharatnam-Berry (PB) phase elements to replace all the elements required. Different from the conventional approaches based on reflective or refractive elements, PB phase elements can dramatically reduce the occupying volume of system. Moreover, the PB…
We tested the ability of human observers to discriminate distinct profiles of spatially dependant geometric phases when directly viewing stationary structured light beams. Participants viewed polarization coupled orbital angular momentum (OAM) states, or ``spin-orbit'' states, in which the OAM was induced through Pancharatnam-Berry phases. The coupling between polarization and OAM in these beams manifests as spatially dependant polarization. Regions of uniform polarization are perceived as specifically oriented Haidinger's brushes, and study participants discriminated between two spin-orbit st…
Recent developments in the field of photonic spin Hall effect (SHE) offer new opportunities for advantageous measurement of the optical parameters (refractive index, thickness, etc.) of nanostructures and enable spin-based photonics applications in the future. However, it remains a challenge to develop a tunable photonic SHE with any desired spin-dependent splitting for generation and manipulation of spin-polarized photons. Here, we demonstrate experimentally a scheme to realize the photonic SHE tunably by tailoring the space-variant Pancharatnam-Berry phase (PBP). It is shown that light beams…
Manipulating the polarisation of light is crucial for sensing and imaging applications. One such aspect in particular is selective transmission of one circular polarisation (spin) when light is transmitted through a medium or a device. However, most present methods of achieving this have relatively low efficiency and selectivity, whilst high selectivity examples rely on lossy and complex three-dimensional helical or multilayer structures. Here, we propose a dielectric metamaterial approach for achieving spin-selective transmission of electromagnetic waves, utilizing spin-controlled constructiv…
Pancharatnam Berry Phase as the Origin of Vector Nature Observed in Hermite Gaussian Superposition States
Graduate School of Engineering, Chiba University（千叶大学研究生院）；Molecular Chirality Research Centre, Chiba University（千叶大学分子手性研究中心）；Institute for Advanced Academic Research, Chiba University（千叶大学高级学术研究院）
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 oc…
In this paper, the implementation of optical elements in the form of Pancharatnam-Berry optics is considered. With respect to 3D bulk and diffractive optics, acting on the dynamic phase of light, Pancharatnam-Berry optical elements transfer a phase which is geometric in nature by locally manipulating the polarization state of the incident beam. They can be realized as space-variant sub-wavelengths gratings that behave like inhomogeneous form-birefringent materials. We present a comprehensive work of simulation, realization, and optical characterization at the telecom wavelength of 1310 nm of t…
In order to overcome the accommodation and convergence (A-C) conflict that commonly causes visual fatigue in AR display, we propose a Maxwellian-viewing-super-multi-view (MV-SMV) near-eye display system based on a Pancharatnam-Berry optical element (PBOE). The PBOE, which is constituted with an array of high-efficiency polarization gratings, is implemented to direct different views to different directions simultaneously, constructing the 3D light field. Meanwhile, each view is like a Maxwellian view display that possesses a small viewpoint and a large depth of field (DOF). Hence, the MV-SMV di…
We report the realization of a Pancharatnam-Berry phase optical element [Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, Opt. Lett. \textbf{27}, 1141 (2002)] for wavefront shaping working in the visible spectral domain, based on patterned liquid crystal technology. This device generates helical modes of visible light with the possibility of electro-optically switching between opposite helicities by controlling the handedness of the input circular polarization. By cascading this approach, fast switching among multiple wavefront helicities can be achieved, with potential applications to multi-s…
A simple variation of the traditional Young's double slit experiment can demonstrate several subtleties of interference with polarized light, including Berry and Pancharatnam's phase. Since the position of the fringes depends on the polarization state of the light at the input, the apparatus can also be used to measure the light's polarization without a quarter-wave plate or an accurate measurement of the light's intensity. In principle this technique can be used for any wavelength of photon as long as one can effectively polarize the incoming radiation.
The nonlinear processes of frequency conversion like second harmonic generation (SHG) usually obey certain selection rules, resulting from the preservation of different kind of physical quantities, e.g., the angular momentum. For SHG created by monolayer of transition-metal dichalcogenides (TMDCs) like WS2, the valley-exciton locked selection rule predicts an SHG signal in the cross-polarization state. By combining plasmonic nanostructures with a monolayer of TMDC, a hybrid metasurface is realized which affects this nonlinear process due to an additional polarization conversion process. Here, …
Parallel sorting of orbital angular momentum (OAM) and polarization has recently acquired paramount importance and interest in a wide range of fields ranging from telecommunications to high-dimensional quantum cryptography. Due to their inherently polarization-sensitive optical response, optical elements acting on the geometric phase prove to be useful for processing structured light beams with orthogonal polarization states by means of a single optical platform. In this work, we present the design, fabrication and test of a Pancharatnam-Berry optical element in silicon implementing a log-pol …
Whenever a quantum system undergoes a cycle governed by a slow change of parameters, it acquires a phase factor: the geometric phase. Its most common formulations are known as the Aharonov-Bohm, Pancharatnam and Berry phases, but both prior and later manifestations exist. Though traditionally attributed to the foundations of quantum mechanics, the geometric phase has been generalized and became increasingly influential in many areas from condensed-matter physics and optics to high energy and particle physics and from fluid mechanics to gravity and cosmology. Interestingly, the geometric phase …
Dielectric metasurfaces with spatially varying birefringence and high transmission efficiency can exhibit exceptional abilities for controlling the photonic spin states. We present here some of our works on spin photonics and spin-photonic devices with metasurfaces. We develop a hybrid-order Poincare sphere to describe the evolution of spin states of wave propagation in the metasurface. Both the Berry curvature and the Pancharatnam-Berry phase on the hybrid-order Poincare sphere are demonstrated to be proportional to the variation of total angular momentum. Based on the spin-dependent property…
This paper presents the design and experimental demonstration of an impedance-matched circular polarization selective surface which also offers spin-selective phase modulations at microwave frequencies. We achieve this by leveraging the theory of Pancharatnam-Berry phase and cascading four tensor impedance layers, each comprising an array of crossed meander lines. These meander lines are precisely tuned and rotated to implement particular tensor surface impedance values to satisfy the impedance-matching condition for the transmitted right-handed circularly-polarized field while inducing Pancha…
In this work, we develop a hybrid-order Poincaré sphere to describe the evolution of polarization states of wave propagation in inhomogeneous anisotropic media. We extend the orbital Poincaré sphere and high-order Poincaré sphere to a more general form. Polarization evolution in inhomogeneous anisotropic media with special geometry can be conveniently described by state evolution along the longitude line on the hybrid-order Poincaré sphere. Similar to that in previously proposed Poincaré spheres, the Berry curvature can be regarded as an effective magnetic field with monopole centered at the o…
We examine the spin-orbit coupling effects that appear when a wave carrying intrinsic angular momentum interacts with a medium. The Berry phase is shown to be a manifestation of the Coriolis effect in a non-inertial reference frame attached to the wave. In the most general case, when both the direction of propagation and the state of the wave are varied, the phase is given by a simple expression that unifies the spin redirection Berry phase and the Pancharatnam--Berry phase. The theory is supported by the experiment demonstrating the spin-orbit coupling of electromagnetic waves via a surface p…
Polarization vectors of light traveling in a coiled optical fiber rotate around its propagating axis even in the absence of birefringence. This rotation was usually explained due to the Pancharatnam-Berry phase of spin-1 photons. Here, we use a purely geometric method to understand this rotation. We show that similar geometric rotations also exist for twisted light carrying orbital angular momentum (OAM). The corresponding geometric phase can be applied in photonic OAM-state-based quantum computation and quantum sensing.
We propose and experimentally demonstrate that a metasurface consisting of Pancharatnam-Berry phase optical elements can enable the full control of optical spin-to-orbital angular momentum conversion. Our approach relies on the critical interference between the transmission and reflection upon the metasurfaceto create actively tunable and controllable conversion with a high output via coherent control of the two incident beams. The introduced control methodology is general and could be an important step toward the development of functional optical devices for practical applications.
We demonstrate experimentally an optical process in which the spin angular momentum carried by a circularly polarized light beam is converted into orbital angular momentum, leading to the generation of helical modes with a wavefront helicity controlled by the input polarization. This phenomenon requires the interaction of light with matter that is both optically inhomogeneous and anisotropic. The underlying physics is also associated with the so-called Pancharatnam-Berry geometrical phases involved in any inhomogeneous transformation of the optical polarization.
We report the realization of tunable spin-dependent splitting in intrinsic photonic spin Hall effect. By breaking the rotational symmetry of a cylindrical vector beam, the intrinsic vortex phases that the two spin components of the vector beam carries, which is similar to the geometric Pancharatnam-Berry phase, is no longer continuous in the azimuthal direction, and leads to observation of spin accumulation at the opposite edge of the beam. Due to the inherent nature of the phase and independency of light-matter interaction, the observed photonic spin Hall effect is intrinsic. Modulating the t…
We investigate electromagnetic propagation in uniaxial dielectrics with a transversely varying orientation of the optic axis, the latter staying orthogonal everywhere to the propagation direction. In such a geometry, the field experiences no refractive index gradients, yet it acquires a transversely-modulated Pancharatnam-Berry phase, that is, a geometric phase originating from a spin-orbit interaction. We show that the periodic evolution of the geometric phase versus propagation gives rise to a longitudinally-invariant effective potential. In certain configurations, this geometric phase can p…
We review the well-known polarization optics matrix methods, i.e., Jones and Stokes-Mueller calculus, and show how they can be formulated in terms of four-dimensional (4d) rotations of the four independent electromagnetic field quadratures. Since 4d rotations is a richer description than the conventional Jones and Stokes-Mueller calculi, having six rather than four degrees of freedom (DOF), we propose an extension of those calculi to handle all six DOF. For the Stokes-Mueller analysis, this leads to a novel and potentially useful extension that accounts for the absolute phase of the optical fi…
A remarkable property of quantum mechanics in two-dimensional (2D) space is its ability to support "anyons," particles that are neither fermions nor bosons. Theory predicts that these exotic excitations can be realized as bound states confined near topological defects, like Majorana zero modes trapped in vortices in topological superconductors. Intriguingly, in the simplest cases the nontrivial phase that arises when such defects are "braided" around one another is not intrinsically quantum mechanical; rather, it can be viewed as a manifestation of the geometric (Pancharatnam-Berry) phase in w…
Orbital angular momentum (OAM) is a promising degree of freedom for fundamental studies in electromagnetics and quantum mechanics. The unlimited state space of OAM shows a great potential to enhance channel capacities of classical and quantum communications. By exploring the Pancharatnam-Berry phase concept and engineering anisotropic scatterers in a metasurface with spatially varying orientations, a plane wave with zero OAM can be converted to a vortex beam carrying nonzero OAM. In this paper, we proposed two types of novel PEC (perfect electric conductor)-PMC (perfect magnetic conductor) ani…
