4,785 research outputs found
Constructing a polynomial whose nodal set is the three-twist knot
We describe a procedure that creates an explicit complex-valued polynomial
function of three-dimensional space, whose nodal lines are the three-twist knot
. The construction generalizes a similar approach for lemniscate knots: a
braid representation is engineered from finite Fourier series and then
considered as the nodal set of a certain complex polynomial which depends on an
additional parameter. For sufficiently small values of this parameter, the
nodal lines form the three-twist knot. Further mathematical properties of this
map are explored, including the relationship of the phase critical points with
the Morse-Novikov number, which is nonzero as this knot is not fibred. We also
find analogous functions for other knots with six crossings. The particular
function we find, and the general procedure, should be useful for designing
knotted fields of particular knot types in various physical systems.Comment: 19 pages, 6 figure
Limits to superweak amplification of beam shifts
The magnitudes of beam shifts (Goos-H\"anchen and Imbert-Fedorov, spatial and
angular) are greatly enhanced when a reflected light beam is postselected by an
analyzer, by analogy with superweak measurements in quantum theory.
Particularly strong enhancements can be expected close to angles at which no
light is transmitted for a fixed initial and final polarizations. We derive a
formula for the angular and spatial shifts at such angles (which includes the
Brewster angle), and we show that their maximum size is limited by higher-order
terms from the reflection coefficients occurring in the Artmann shift formula.Comment: 3 pages, 2 figures, Optics Letters styl
Topological aberration of optical vortex beams and singularimetry of dielectric interfaces
The splitting of a high-order optical vortex into a constellation of unit
vortices, upon total reflection, is described and analyzed. The vortex
constellation generalizes, in a local sense, the familiar longitudinal
Goos-H\"anchen and transverse Imbert-Federov shifts of the centroid of a
reflected optical beam. The centroid shift is related to the centre of the
constellation, whose geometry otherwise depends on higher-order terms in an
expansion of the reflection matrix. We present an approximation of the field
around the constellation of increasing order as an Appell sequence of complex
polynomials whose roots are the vortices, and explain the results by an analogy
with the theory of optical aberration.Comment: 5 pages, 3 figures, REVTeX 4.
Propagation-invariant beams with quantum pendulum spectra: from Bessel beams to Gaussian beam-beams
We describe a new class of propagation-invariant light beams with Fourier
transform given by an eigenfunction of the quantum mechanical pendulum. These
beams, whose spectra (restricted to a circle) are doubly-periodic Mathieu
functions in azimuth, depend on a field strength parameter. When the parameter
is zero, pendulum beams are Bessel beams, and as the parameter approaches
infinity, they resemble transversely propagating one-dimensional Gaussian
wavepackets (Gaussian beam-beams). Pendulum beams are the eigenfunctions of an
operator which interpolates between the squared angular momentum operator and
the linear momentum operator. The analysis reveals connections with Mathieu
beams, and insight into the paraxial approximation.Comment: 4 pages, 3 figures, Optics Letters styl
Position, spin and orbital angular momentum of a relativistic electron
Motivated by recent interest in relativistic electron vortex states, we
revisit the spin and orbital angular momentum properties of Dirac electrons.
These are uniquely determined by the choice of the position operator for a
relativistic electron. We overview two main approaches discussed in the
literature: (i) the projection of operators onto the positive-energy subspace,
which removes the zitterbewegung effects and correctly describes spin-orbit
interaction effects, and (ii) the use of Newton-Wigner-Foldy-Wouthuysen
operators based on the inverse Foldy-Wouthuysen transformation. We argue that
the first approach [previously described in application to Dirac vortex beams
in K.Y. Bliokh et al., Phys. Rev. Lett. 107, 174802 (2011)] has a more natural
physical interpretation, including spin-orbit interactions and a nonsingular
zero-mass limit, than the second one [S.M. Barnett, Phys. Rev. Lett. 118,
114802 (2017)].Comment: 10 pages, 1 table, to appear in Phys. Rev.
A random wave model for the Aharonov-Bohm effect
We study an ensemble of random waves subject to the Aharonov-Bohm effect. The
introduction of a point with a magnetic flux of arbitrary strength into a
random wave ensemble gives a family of wavefunctions whose distribution of
vortices (complex zeros) are responsible for the topological phase associated
with the Aharonov-Bohm effect. Analytical expressions are found for the vortex
number and topological charge densities as functions of distance from the flux
point. Comparison is made with the distribution of vortices in the isotropic
random wave model. The results indicate that as the flux approaches
half-integer values, a vortex with the same sign as the fractional part of the
flux is attracted to the flux point, merging with it at half-integer flux.
Other features of the Aharonov-Bohm vortex distribution are also explored.Comment: 16 pages, 5 figure
On the Burgers vector of a wave dislocation
Following Nye and Berry's analogy with crystal dislocations, an approach to
the Burgers vector of a wave dislocation (phase singularity, optical vortex) is
proposed. It is defined to be a regularized phase gradient evaluated at the
phase singularity, and is computed explicitly. The screw component of this
vector is naturally related to the helicoidal twisting of wavefronts along a
vortex line, and is related to the helicity of the phase gradient. The edge
component is related to the nearby current flow (defined by the phase gradient)
perpendicular to the vortex, and the distribution of this component is found
numerically for random two-dimensional monochromatic waves.Comment: 15 pages, 4 figures, IoP styl
Geometric phases in 2D and 3D polarized fields: geometrical, dynamical, and topological aspects
Geometric phases are a universal concept that underpins numerous phenomena
involving multi-component wave fields. These polarization-dependent phases are
inherent in interference effects, spin-orbit interaction phenomena, and
topological properties of vector wave fields. Geometric phases have been
thoroughly studied in two-component fields, such as two-level quantum systems
or paraxial optical waves. However, their description for fields with three or
more components, such as generic nonparaxial optical fields routinely used in
modern nano-optics, constitutes a nontrivial problem. Here we describe
geometric, dynamical, and total phases calculated along a closed spatial
contour in a multi-component complex field, with particular emphasis on 2D
(paraxial) and 3D (nonparaxial) optical fields. We present several equivalent
approaches: (i) an algebraic formalism, universal for any multi-component
field; (ii) a dynamical approach using the Coriolis coupling between the spin
angular momentum and reference-frame rotations; and (iii) a geometric
representation, which unifies the Pancharatnam-Berry phase for the 2D
polarization on the Poincar\'e sphere and the Majorana-sphere representation
for the 3D polarized fields. Most importantly, we reveal close connections
between geometric phases, angular-momentum properties of the field, and
topological properties of polarization singularities in 2D and 3D fields, such
as C-points and polarization M\"obius strips.Comment: 21 pages, 11 figures, to appear in Rep. Prog. Phy
Singular Values, Nematic Disclinations, and Emergent Biaxiality
Both uniaxial and biaxial nematic liquid crystals are defined by
orientational ordering of their building blocks. While uniaxial nematics only
orient the long molecular axis, biaxial order implies local order along three
axes. As the natural degree of biaxiality and the associated frame, that can be
extracted from the tensorial description of the nematic order, vanishes in the
uniaxial phase, we extend the nematic director to a full biaxial frame by
making use of a singular value decomposition of the gradient of the director
field instead. New defects and degrees of freedom are unveiled and the
similarities and differences between the uniaxial and biaxial phase are
analyzed by applying the algebraic rules of the quaternion group to the
uniaxial phase.Comment: 5 pages, 1 figure, submitted to PR
A three-dimensional degree of polarization based on Rayleigh scattering
A measure of the degree of polarization for the three-dimensional
polarization matrix (coherence matrix) of an electromagnetic field is proposed,
based on Rayleigh scattering. The degree of polarization, due to dipole
scattering of the three-dimensional state of polarization, is averaged over all
scattering directions. This gives a well-defined purity measure, which, unlike
other proposed measures of the three-dimensional degree of polarization, is not
a unitary invariant of the matrix. This is demonstrated and discussed for
several examples, including a partially polarized transverse beam.Comment: 17 pages, 3 figures. OSA styl
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