Local Wave Vector Research Articles

Canonical and Poynting currents in propagation and diffraction of structured light: tutorial

The local propagation and the energy flux in structured optical fields are often associated with the Poynting vector. However, the local phase gradient (i.e., local wavevector) in monochromatic fields in free space is described by another fundamental quantity: the canonical momentum density. Distributions of the Poynting and canonical momentum densities can differ significantly from each other in structured fields. We examine the role of these quantities in the propagation and diffraction of structured optical fields, exemplified by various circularly polarized vortex beams carrying orbital angular momentum. We describe the canonical and Poynting momentum distributions in such beams, experimentally measure the local transverse momentum density by a Shack-Hartmann wavefront sensor, and investigate fine features of the diffraction of various vortex beams on a knife-edge aperture. In all cases, the measured local momentum density and local beam evolution are consistent with the canonical momentum distribution rather than the Poynting vector. Furthermore, we introduce the local angular velocity in vortex beams and determine the universal integral π angle of azimuthal rotation in an arbitrary (yet circularly symmetric) propagating and diffracting vortex beam. Finally, we discuss the “supermomentum” and “backflow” effects; both of these phenomena are examples of superoscillations and are related to the properties of the canonical momentum. Our results reveal the profound role of the canonical momentum in the evolution of light and demonstrate the importance of distinguishing between it and the Poynting vector in structured light.

Probing Spacetime Foam with Extragalactic Sources of High-Energy Photons

Quantum fluctuations can endow spacetime with a foamy structure. In this review article, we discuss our various proposals to observationally constrain models of spacetime foam. One way is to examine if the light wave-front from a distant quasar or GRB can be noticeably distorted by spacetime-foam-induced phase incoherence. As the phase fluctuations are proportional to the distance to the source but inversely proportional to the wavelength, ultra-high energy photons (>1 TeV) from distant sources are particularly useful. We elaborate on several proposals, including the possibility of detecting spacetime foam by observing “seeing disks” in the images of distant quasars and active galactic nuclei. We also discuss the appropriate distance measure for calculating the expected angular broadening. In addition, we discuss our more recent work in which we investigate whether wave-front distortions on small scales (due to spacetime foam) can cause distant objects become undetectable because the phase fluctuations have accumulated to the point at which image formation is impossible. Another possibility that has recently become accessible is to use interferometers to observe cosmologically distant sources, thereby giving a large baseline perpendicular to the local wave vector over which the wave front could become corrugated and thus distorted, reducing or eliminating its fringe visibility. We argue that all these methods ultimately depend on the availability of ways (if any) to carry out proper averaging of contributions from different light paths from the source to the telescope.

Hole probability for noninteracting fermions in a d-dimensional trap

The hole probability, i.e., the probability that a region is void of particles, is a benchmark of correlations in many-body systems. We compute analytically this probability P(R) for a sphere of radius R in the case of N noninteracting fermions in their ground state in a d-dimensional trapping potential. Using a connection to the Laguerre-Wishart ensembles of random matrices, we show that, for large N and in the bulk of the Fermi gas, P(R) is described by a universal scaling function of k F R, for which we obtain an exact formula (k F being the local Fermi wave vector). It exhibits a super-exponential tail where is a universal amplitude, in good agreement with existing numerical simulations. When R is of the order of the radius of the Fermi gas, the hole probability is described by a large deviation form which is not universal and which we compute exactly for the harmonic potential. Similar results also hold in momentum space.

Semiclassical superoscillations: interference, evanescence, post-WKB

The concept of superoscillations is extended beyond bandlimited functions, to include monochromatic waves in space-varying media, such as wavefunctions representing quantum particles in non-constant potentials. ‘Semiclassical superoscillations’ are defined as places where the phase gradient (local wavevector, or weak momentum) exceeds the classically allowed momentum. The phenomenon is essentially quantum (or, in optics, essentially beyond geometrical optics), and can arise in three different ways: from interference, evanescence (e.g. barrier penetration), and WKB corrections. Illustrations are given, including superpositions of few and many states in a linear potential, angular-momentum circular harmonic-oscillator states, eigenstates of degenerate harmonic-oscillator modes, and above-barrier scattering.

Impurities in systems of noninteracting trapped fermions

We study the properties of spin-less non-interacting fermions trapped in a confining potential in one dimension but in the presence of one or more impurities which are modelled by delta function potentials. We use a method based on the single particle Green's function. For a single impurity placed in the bulk, we compute the density of the Fermi gas near the impurity. Our results, in addition to recovering the Friedel oscillations at large distance from the impurity, allow the exact computation of the density at short distances. We also show how the density of the Fermi gas is modified when the impurity is placed near the edge of the trap in the region where the unperturbed system is described by the Airy gas. Our method also allows us to compute the effective potential felt by the impurity both in the bulk and at the edge. In the bulk this effective potential is shown to be a universal function only of the local Fermi wave vector, or equivalently of the local fermion density. When the impurity is placed near the edge of the Fermi gas, the effective potential can be expressed in terms of Airy functions. For an attractive impurity placed far outside the support of the fermion density, we show that an interesting transition occurs where a single fermion is pulled out of the Fermi sea and forms a bound state with the impurity. This is a quantum analogue of the well-known Baik-Ben Arous-Péché (BBP) transition, known in the theory of spiked random matrices. The density at the location of the impurity plays the role of an order parameter. We also consider the case of two impurities in the bulk and compute exactly the effective force between them mediated by the background Fermi gas.

Wave-Vector-Varying Pancharatnam-Berry Phase Photonic Spin Hall Effect.

The geometric Pancharatnam-Berry (PB) phase not only is of physical interest but also has wide applications ranging from condensed-matter physics to photonics. Space-varying PB phases based on inhomogeneously anisotropic media have previously been used effectively for spin photon manipulation. Here we demonstrate a novel wave-vector-varying PB phase that arises naturally in the transmission and reflection processes in homogeneous media for paraxial beams with small incident angles. The eigenpolarization states of the transmission and reflection processes are determined by the local wave vectors of the incident beam. The small incident angle breaks the rotational symmetry and induces a PB phase that varies linearly with the transverse wave vector, resulting in the photonic spin Hall effect (PSHE). This new PSHE can address the contradiction between spin separation and energy efficiency in the conventional PSHE associated with the Rytov-Vladimirskii-Berry phase, allowing spin photons to be separated completely with a spin separation up to 2.2 times beam waist and a highest energy efficiency of 86%. The spin separation dynamics is visualized by wave coupling equations in a uniaxial crystal, where the centroid positions of the spin photons can be doubled due to the conservation of the angular momentum. Our findings can greatly deepen the understanding in the geometric phase and spin-orbit coupling, paving the way for practical applications of the PSHE.

Superoscillations for monochromatic standing waves

For complex scalar waves, a convenient measure of the local oscillations and (‘faster than Fourier’) superoscillations is the phase gradient vector: the local wavevector, or weak value of the momentum operator. This vanishes for standing waves, described by real functions ψ( r ); for such waves, an alternative descriptor of oscillations is the local weak value of the square of one of the momentum components, i.e. , here called the ‘weak curvature’. Superoscillations correspond to places where K 2 lies outside the interval 0 ⩽ K 2 ⩽ 1. Two illustrations are given. First is an explicit family of real waves in dimension d = 2, with arbitrarily strong superoscillations; this could represent Neumann standing modes in a strip waveguide. Second is an exact calculation of the probability distribution of K 2 for Gaussian random real waves in d dimensions. This decays as , as a consequence of the codimension 1 of nodes (e.g. nodal lines for d = 2). The superoscillation probability varies from 0.3918 (d = 2) to 0.3041 (d = ∞).

Vector wave simulation of active imaging through random media.

When a target is embedded in random media, the quality of optical imaging can be improved by actively controlling the illumination and exploiting vector wave properties. A rigorous description, however, requires expensive computational resources to fully account for the electromagnetic boundary conditions. Here, we introduce a statistically equivalent scaling model that allows for reducing the complexity of the problem. The new scheme describes the entanglement between the local wave vector and the polarization state in random media and also accounts for cumulative properties such as geometric phase. The approach is validated for different scenarios where the coherent background noise alters substantially the performance of active imaging.

Geometry of 3D monochromatic light: local wavevectors, phases, curl forces, and superoscillations*

This is an exploration of some geometric properties, and their interconnections, for nonparaxial electromagnetic waves. The properties involve the electric field alone, and also in combination with the magnetic field (‘electric-magnetic democracy’). The orbital part of the Poynting vector, represented in terms of the electric field, is proportional to the nonconservative ‘curl force’ exerted on small absorbing polarisable particles; its circuit integral represents the work done transporting a particle. Normalised versions of the electric, magnetic, and electric-magnetic orbital Poynting vectors are natural candidates for wavevectors representing the direction of the wave at each point, generalising the phase gradient for scalar waves; there are associated total, dynamical and geometric phases. For isotropic ensembles of random waves, the probability distributions of the different wavevectors are estimated by codimension arguments and calculated exactly for statistically isotropic superpositions of plane waves. The superoscillation probability, that the magnitude of the local wavevector exceeds that of the waves in the superposition, is 0.031 40 for the electric wavevector and 0.000 109 for the electric-magnetic wavevector. Both values are much smaller than the known superoscillation probability 0.350 48 for scalar waves in three-dimensions, illustrating a general phenomenon: interference detail is weaker for electric properties than for scalar, and weaker still for electric-magnetic properties.

\u201cPlasmonics\u201d in free space: observation of giant wavevectors, vortices, and energy backflow in superoscillatory optical fields

Evanescent light can be localized at the nanoscale by resonant absorption in a plasmonic nanoparticle or taper or by transmission through a nanohole. However, a conventional lens cannot focus free-space light beyond half of the wavelength λ. Nevertheless, precisely tailored interference of multiple waves can form a hotspot in free space of an arbitrarily small size, which is known as superoscillation. Here, we report a new type of integrated metasurface interferometry that allows for the first time mapping of fields with a deep subwavelength resolution ~λ/100. The findings reveal that an electromagnetic field near the superoscillatory hotspot has many features similar to those found near resonant plasmonic nanoparticles or nanoholes: the hotspots are surrounded by nanoscale phase singularities and zones where the phase of the superoscillatory field changes more than tenfold faster than a free-propagating plane wave. Areas with high local wavevectors are pinned to phase vortices and zones of energy backflow (~λ/20 in size) that contribute to tightening of the main focal spot size beyond the Abbe–Rayleigh limit. Our observations reveal some analogy between plasmonic nanofocusing of evanescent waves and superoscillatory nanofocusing of free-space waves and prove the fundamental link between superoscillations and superfocusing, offering new opportunities for nanoscale metrology and imaging.

Ultra-dispersive anomalous diffraction from Pancharatnam-Berry metasurfaces

Achieving ultra-dispersive diffractions is fundamentally important to improve the chromatic resolution of spectrometers for numerous applications, such as Raman measurements, atom and molecule identification, and so on. Gratings, as traditionally widely used diffraction elements, disperse chromatic light into different angles according to the phase matching condition, which resorts to the transverse reciprocal vectors of the grating lattice. Mathematically, gratings show higher diffraction dispersion for larger diffraction angles. Either increasing grating line frequencies or steepening the groove angles is adopted to enlarge the deflection angles. However, all of them cause problems of near-zero diffraction efficiency and complicated fabrication. Here, we realize an ultra-dispersive diffraction in the framework of metasurfaces using an alternative phase matching strategy, in which the contributions of both the reciprocal vectors of the lattice and the local wave vectors arisen by the phase gradient are considered simultaneously. The diffraction angle of more than 80° is achieved with the resulting dispersion 4 times larger than the Littrow grating counterparts.

Ground roll attenuation by synchrosqueezed curvelet transform

Ground roll is a type of coherent noise in land seismic data that has low frequency, low velocity and high amplitude. It damages reflection events that contain important information about subsurface structures, hence the removal of ground roll is a crucial step in seismic data processing. A suitable transform is needed for removal of ground roll. Curvelet transform is an effective sparse transform that optimally represents seismic events. In addition, the curvelets can provide a multiscale and multidirectional decomposition of the input data in time-frequency and angular domain, which can help distinguish between ground roll and useful signals. In this paper, we apply synchrosqueezed curvelet transform (SSCT) for ground roll attenuation. The synchrosqueezing technique in SSCT is used to precisely reallocate the energy of local wave vectors in order to separate ground roll from the original data with higher resolution and higher fidelity. Examples of synthetic and field seismic data reveal that SSCT performs well in the suppression of aliased and non-aliased ground roll while preserving reflection waves, in comparison with high-pass filtering, wavelet and curvelet methods.

Hyperbolic accelerating beams and their relation with Hermite-Gaussian beams.

We derive the initial distributions of phase and complex amplitude of accelerating beams with arbitrary predesigned hyperbolic trajectories using the caustic-design method and explore the relation between these beams and Hermite-Gaussian beams. The results show the hyperbolic accelerating beams are a larger class of beams than Hermite-Gaussian beams. When the bending parameter is an integer, the hyperbolic accelerating beams have a similar initial complex amplitude distribution and almost the same propagating characteristics as Hermite-Gaussian beams. Through the analysis of the ray-based method, we also derive an approximate expression for the initial complex amplitude of Hermite-Gaussian beams after introducing an amplitude distribution function. Although the proposed approximate expressions of complex amplitude are more complex than the usually used Hermite-Gaussian function, they explicitly indicate the information on local amplitude, wave vector, and internal ray structure (including caustics) of these beams and thus provide us clearer geometrical insights into these beams.

Extraordinary spin momenta in birefringent structures

Mechanical action caused by the optical forces connected with the canonical momentum density associated with the local wave vector or classical spin angular momentum, the helicity dependent and the helicity independent forces determined by spin momenta of different nature open attractive prospects to use optical structures for manipulating with nanoobjects of different nature. The main finding of our study consists in demonstration of mechanical action of extraordinary transverse component of the spin angular momentum arising in an evanescent light wave due to the total internal reflection of linearly polarized probing beam with azimuth 45° at the interface ‘birefringent plate–air’.

Influence of evanescent wave on birefringent microplates.

Mechanical action caused by the optical forces connected with the canonical momentum density associated with the local wavevector or Belinfante's spin angular momentum is experimentally verified. The helicity-dependent and the helicity-independent forces determined by spin momenta of different nature open attractive prospects for the use of optical structures for manipulating minute quantities of matter of importance in nanophysics, nanooptics and nanotechnologies, precision chemistry and pharmacology and in numerous other areas. Investigations in this area reveal new, extraordinary manifestations of optical forces, including the helicity-independent force caused by the transverse helicity-independent spin or vertical spin of a diagonally polarized wave, which was not observed and exploited up to recently. The main finding of our study consists in a direct experimental demonstration of the physical existence and mechanical action of this recently discovered extraordinary transverse component of the spin here arising in an evanescent light wave due to the total internal reflection of a linearly polarized probing beam with azimuthal angle 45° at the interface between the birefringent plate and air, which is oriented perpendicularly to the wavevector of an evanescent wave and localized over the boundary of the transparent media with polarization-dependent refraction indices.

Commensurate 4a0-period charge density modulations throughout the Bi2Sr2CaCu2O8+x pseudogap regime

Theories based upon strong real space (r-space) electron-electron interactions have long predicted that unidirectional charge density modulations (CDMs) with four-unit-cell (4a0) periodicity should occur in the hole-doped cuprate Mott insulator (MI). Experimentally, however, increasing the hole density p is reported to cause the conventionally defined wavevector QA of the CDM to evolve continuously as if driven primarily by momentum-space (k-space) effects. Here we introduce phase-resolved electronic structure visualization for determination of the cuprate CDM wavevector. Remarkably, this technique reveals a virtually doping-independent locking of the local CDM wavevector at [Formula: see text] throughout the underdoped phase diagram of the canonical cuprate Bi2Sr2CaCu2O8 These observations have significant fundamental consequences because they are orthogonal to a k-space (Fermi-surface)-based picture of the cuprate CDMs but are consistent with strong-coupling r-space-based theories. Our findings imply that it is the latter that provides the intrinsic organizational principle for the cuprate CDM state.

Theoretical and Numerical Modeling of Acoustic Metamaterials for Aeroacoustic Applications

The advent, during the first decade of the 21st century, of the concept of acoustic metamaterial has disclosed an incredible potential of development for breakthrough technologies. Unfortunately, the extension of the same concepts to aeroacoustics has turned out to be not a trivial task, because of the different structure of the governing equations, characterized by the presence of the background aerodynamic convection. Some of the approaches recently introduced to circumvent the problem are biased by a fundamental assumption that makes the actual realization of devices extremely unlikely: the metamaterial should guarantee an adapted background aerodynamic convection in order to modify suitably the acoustic field and obtain the desired effect, thus implying the porosity of the cloaking device. In the present paper, we propose an interpretation of the metamaterial design that removes this unlikely assumption, focusing on the identification of an aerodynamically-impermeable metamaterial capable of reproducing the surface impedance profile required to achieve the desired scattering abatement. The attention is focused on a moving obstacle impinged by an acoustic perturbation induced by a co-moving source. The problem is written in a frame of reference rigidly connected to the moving object to couple the convective wave equation in the hosting medium with the inertially-anisotropic wave operator within the cloak. The problem is recast in an integral form and numerically solved through a boundary-field element method. The matching of the local wave vector is used to derive a convective design of the metamaterial applicable to the specific problem analyzed. Preliminary numerical results obtained under the simplifying assumption of a uniform aerodynamic flow reveal a considerable enhancement of the masking capability of the convected design. The numerical method developed shows a remarkable computational efficiency, completing a simulation of the entire field in a few minutes on mid-end workstations. The results are re-interpreted in term of boundary impedance, assuming a locally-reacting behavior of the outer boundary of the cloaking layer. The formulation is currently being extended to the analysis of arbitrarily complex external flows in order to remove the limitation of the background uniform stream in the host.

Metasurface transformation for surface wave control.

Metasurfaces (MTSs) constitute a class of thin metamaterials used for controlling plane waves and surface waves (SWs). At microwave frequencies, they are constituted by a metallic texture with elements of sub-wavelength size printed on thin grounded dielectric substrates. These structures support the propagation of SWs. By averaging the tangential fields, the MTSs can be characterized through homogenized isotropic or anisotropic boundary conditions, which can be described through a homogeneous equivalent impedance. This impedance can be spatially modulated by locally changing the size/orientation of the texture elements. This allows for a deformation of the SW wavefront which addresses the local wavevector along not-rectilinear paths. The effect of the MTS modulation can be analysed in the framework of transformation optics. This article reviews theory and implementation of this MTS transformation and shows some examples at microwave frequencies.

Analytic derivation of electrostrictive tensors and their application to optical force density calculations

Using multiple scattering theory, we derived for the first time analytical formulas for electrostrictive tensors for two dimensional metamaterial systems. The electrostrictive tensor terms are found to depend explicitly on the symmetry of the underlying lattice of the metamaterial and they also depend explicitly on the direction of a local effective wave vector. These analytical results enable us to calculate light induced body forces inside a composite system (metamaterial) using the Helmholtz stress tensor within the effective medium formalism in the sense that the fields used in the stress tensor are those obtained by solving the macroscopic Maxwell equation with the microstructure of the metamaterial replaced by an effective medium. Our results point to some fundamental questions of using an effective medium theory to determine optical force density. In particular, the fact that Helmholtz tensor carries electrostrictive terms that are explicitly symmetry dependent means that the standard effective medium parameters cannot give sufficient information to determine body force density, even though they can give the correct total force. A more challenging issue is that the electrostrictive terms are related to a local effective wave vector, and it is not always obtainable in systems with boundary reflections within the context of a standard effective medium approach.

Crystal Image Analysis Using 2D Synchrosqueezed Transforms

We propose efficient algorithms based on a band-limited version of 2D synchrosqueezed transforms to extract mesoscopic and microscopic information from atomic crystal images. The methods analyze atomic crystal images as an assemblage of nonoverlapping segments of 2D general intrinsic mode type functions, which are superpositions of nonlinear wave-like components. In particular, crystal defects are interpreted as the irregularity of local energy; crystal rotations are described as the angle deviation of local wave vectors from their references; the gradient of a crystal elastic deformation can be obtained by a linear system generated by local wave vectors. Several numerical examples of synthetic and real crystal images are provided to illustrate the efficiency, robustness, and reliability of our methods.