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 Poincaré 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 Möbius strips.

Left arm structure and function late after subclavian flap repair of aortic coarctation in childhood.

OBJECTIVES: Concerns exist over the long-term consequences of subclavian artery ligation in subclavian flap repair for coarctation of the aorta. We sought to analyse upper limb structural and functional performance in adults who have had surgery in childhood for coarctation of the aorta, using either subclavian flap repair or end to end aortic anastomosis. METHODS: Two-group observational design using anatomical and upper limb functional performance measures. Purposive sampling from our specialist adult congenital heart disease database of patients who received subclavian flap repair or end to end anastomosis for coarctation of the aorta as children. Upper limb measurements were completed using MRI and blood flow velocity with ultrasound imaging. Bilateral standardised upper limb functional testing of assessment of strength, dexterity and a standardised self-report of upper limb disability was completed. RESULTS: Eighteen right-handed patients, 9 with subclavian repair, (38 ± 12 years, 78% males) were studied. Age at repair was 4.7 ± 5.9 years; mean time from initial repair 32 ± 9 years. The subclavian group had a larger difference between right and left when compared the end to end anastomosis group in: lower arm muscle mass (94.5 ± 42.3 mls versus 37.8 ± 94.5 mls, p = 0.008), lower arm maximal cross-sectional area, (5.9 ± 2.8 cm2 versus 2.9 ± 2.6 cm2, p = 0.038) and grip strength (14.7 ± 8.3 lbs versus 5.9 ± 5.3 lbs, p = 0.016) There were no significant functional differences between groups. CONCLUSIONS: In adults with repaired coarctation of the aorta, those with subclavian flap repair had a greater right to left arm muscle mass and grip strength differential when compared to those with end to end anastomosis repair.

Singular knot bundle in light.

As the size of an optical vortex knot, imprinted in a coherent light beam, is decreased, nonparaxial effects alter the structure of the knotted optical singularity. For knot structures approaching the scale of wavelength, longitudinal polarization effects become non-negligible, and the electric and magnetic fields differ, leading to intertwined knotted nodal structures in the transverse and longitudinal polarization components, which we call a knot bundle of polarization singularities. We analyze their structure using polynomial beam approximations and numerical diffraction theory. The analysis reveals features of spin-orbit effects and polarization topology in tightly focused geometry, and we propose an experiment to measure this phenomenon.

Weaving Knotted Vector Fields with Tunable Helicity.

We present a general construction of divergence-free knotted vector fields from complex scalar fields, whose closed field lines encode many kinds of knots and links, including torus knots, their cables, the figure-8 knot, and its generalizations. As finite-energy physical fields, they represent initial states for fields such as the magnetic field in a plasma, or the vorticity field in a fluid. We give a systematic procedure for calculating the vector potential, starting from complex scalar functions with knotted zero filaments, thus enabling an explicit computation of the helicity of these knotted fields. The construction can be used to generate isolated knotted flux tubes, filled by knots encoded in the lines of the vector field. Lastly, we give examples of manifestly knotted vector fields with vanishing helicity. Our results provide building blocks for analytical models and simulations alike.

Eigenpolarizations for giant transverse optical beam shifts.

We show how careful control of the incident polarization of a light beam close to the Brewster angle gives a giant transverse spatial shift on reflection. This resolves the long-standing puzzle of why such beam shifts transverse to the incident plane (Imbert-Fedorov shifts) tend to be an order of magnitude smaller than the related Goos-Hänchen shifts in the longitudinal direction, which are largest close to critical incidence. We demonstrate that with the proper initial polarization the transverse displacements can be equally large, which we confirm experimentally near Brewster incidence. In contrast to the established understanding, these polarizations are elliptical and angle dependent. We explain the magnitude of the Imbert-Fedorov shift by an analogous change of the symmetry properties for the reflection operators as compared to the Goos-Hänchen shift.

A super-oscillatory lens optical microscope for subwavelength imaging.

The past decade has seen an intensive effort to achieve optical imaging resolution beyond the diffraction limit. Apart from the Pendry-Veselago negative index superlens, implementation of which in optics faces challenges of losses and as yet unattainable fabrication finesse, other super-resolution approaches necessitate the lens either to be in the near proximity of the object or manufactured on it, or work only for a narrow class of samples, such as intensely luminescent or sparse objects. Here we report a new super-resolution microscope for optical imaging that beats the diffraction limit of conventional instruments and the recently demonstrated near-field optical superlens and hyperlens. This non-invasive subwavelength imaging paradigm uses a binary amplitude mask for direct focusing of laser light into a subwavelength spot in the post-evanescent field by precisely tailoring the interference of a large number of beams diffracted from a nanostructured mask. The new technology, which--in principle--has no physical limits on resolution, could be universally used for imaging at any wavelength and does not depend on the luminescence of the object, which can be tens of micrometres away from the mask. It has been implemented as a straightforward modification of a conventional microscope showing resolution better than λ/6.

Limits to superweak amplification of beam shifts.

The magnitudes of beam shifts (Goos-Hänchen 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 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.

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 wave packets (Gaussian beam-beams). Pendulum beams are the eigenfunctions of an operator that 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.

Incomplete Airy beams: finite energy from a sharp spectral cutoff.

We present a mathematical analysis of the finite-energy Airy beam with a sharply truncated spectrum, which can be generated by a uniformly illuminated, finite-sized spatial light modulator, or windowed cubic phase mask. The resulting "incomplete Airy beam" is tractable mathematically, and differs from an infinite-energy Airy beam by an additional oscillating modulation and the decay of its fringes. Its propagation can be described explicitly using an incomplete Airy function, from which we derive simple expressions for the beam's total power and mean position. Asymptotic analysis reveals a simple connection between the cutoff and the region of the beam with Airy-like behavior.

Auto-focusing and self-healing of Pearcey beams.

We present a new solution of the paraxial equation based on the Pearcey function, which is related to the Airy function and describes diffraction about a cusp caustic. The Pearcey beam displays properties similar not only to Airy beams but also Gaussian and Bessel beams. These properties include an inherent auto-focusing effect, as well as form-invariance on propagation and self-healing. We describe the theory of propagating Pearcey beams and present experimental verification of their auto-focusing and self-healing behaviour.

Topological aberration of optical vortex beams: determining dielectric interfaces by optical singularity shifts.

We predict the splitting of a high-order optical vortex into a constellation of unit vortices, upon total internal reflection of the carrier beam, and analyze the splitting. The reflected vortex constellation generalizes, in a local sense, the familiar longitudinal Goos-Hänchen and transverse Imbert-Fedorov shifts of the centroid of a reflected optical beam. The centroid shift is related to the center of the constellation, whose geometry otherwise depends on higher-order terms in an expansion of the reflection matrix. We derive an approximation of the amplitude around the constellation as a complex analytic polynomial, whose roots are the vortices. Increasing the order of the initial vortex gives an Appell sequence of complex polynomials, which we explain by an analogy with the theory of optical aberration.

Fermionic out-of-plane structure of polarization singularities.

A new classification of circular polarization C points in three-dimensional polarization ellipse fields is proposed. The classification type depends on the out-of-plane variation of the polarization ellipse axis, in particular, whether the ellipse axes are in the plane of circular polarization one or three times. A minimal set of parameters for this classification is derived and discussed in the context of the familiar in-plane C point classification into lemon, star, and monstar types. This new geometric classification is related to the Möbius index of polarization singularities recently introduced by Freund.

Paraxial and nonparaxial polynomial beams and the analytic approach to propagation.

We construct solutions of the paraxial and Helmholtz equations that are polynomials in their spatial variables. These are derived explicitly by using the angular spectrum method and generating functions. Paraxial polynomials have the form of homogeneous Hermite and Laguerre polynomials in Cartesian and cylindrical coordinates, respectively, analogous to heat polynomials for the diffusion equation. Nonparaxial polynomials are found by substituting monomials in the propagation variable z with reverse Bessel polynomials. These explicit analytic forms give insight into the mathematical structure of paraxially and nonparaxially propagating beams, especially in regard to the divergence of nonparaxial analogs to familiar paraxial beams.

Relativistic electron vortex beams: angular momentum and spin-orbit interaction.

Motivated by the recent discovery of electron vortex beams carrying orbital angular momentum (AM), we construct exact Bessel-beam solutions of the Dirac equation. They describe relativistic and nonparaxial corrections to the scalar electron beams. We describe the spin and orbital AM of the electron with Berry-phase corrections and predict the intrinsic spin-orbit coupling in free space. This can be observed as a spin-dependent probability distribution of the focused electron vortex beams. Moreover, the magnetic moment is calculated, which shows different g factors for spin and orbital AM and also contains the Berry-phase correction.

Particle-like topologies in light.

Three-dimensional (3D) topological states resemble truly localised, particle-like objects in physical space. Among the richest such structures are 3D skyrmions and hopfions, that realise integer topological numbers in their configuration via homotopic mappings from real space to the hypersphere (sphere in 4D space) or the 2D sphere. They have received tremendous attention as exotic textures in particle physics, cosmology, superfluids, and many other systems. Here we experimentally create and measure a topological 3D skyrmionic hopfion in fully structured light. By simultaneously tailoring the polarisation and phase profile, our beam establishes the skyrmionic mapping by realising every possible optical state in the propagation volume. The resulting light field's Stokes parameters and phase are synthesised into a Hopf fibration texture. We perform volumetric full-field reconstruction of the [Formula: see text] mapping, measuring a quantised topological charge, or Skyrme number, of 0.945. Such topological state control opens avenues for 3D optical data encoding and metrology. The Hopf characterisation of the optical hypersphere endows a fresh perspective to topological optics, offering experimentally-accessible photonic analogues to the gamut of particle-like 3D topological textures, from condensed matter to high-energy physics.

Suppression of collapse for spiraling elliptic solitons.

We reveal that orbital angular momentum can suppress catastrophic self-focusing in nonlinear Kerr media supporting stable spiraling solitons with an elliptic cross section. We discuss the necessary requirements for observation of this effect with coherent optical and matter waves.

Laser beams: knotted threads of darkness.

Destructive interference may lead to complete cancellation when light waves travelling in different directions cross, and in three-dimensional space this occurs along lines that are vortices of electromagnetic energy flow. Here we confirm theoretical predictions by experimentally creating combinations of optical laser beams in which these dark threads form stable loops that are linked and knotted.

Shaping caustics into propagation-invariant light.

Structured light has revolutionized optical particle manipulation, nano-scaled material processing, and high-resolution imaging. In particular, propagation-invariant light fields such as Bessel, Airy, or Mathieu beams show high robustness and have a self-healing nature. To generalize such beneficial features, these light fields can be understood in terms of caustics. However, only simple caustics have found applications in material processing, optical trapping, or cell microscopy. Thus, these technologies would greatly benefit from methods to engineer arbitrary intensity shapes well beyond the standard families of caustics. We introduce a general approach to arbitrarily shape propagation-invariant beams by smart beam design based on caustics. We develop two complementary methods, and demonstrate various propagation-invariant beams experimentally, ranging from simple geometric shapes to complex image configurations such as words. Our approach generalizes caustic light from the currently known small subset to a complete set of tailored propagation-invariant caustics with intensities concentrated around any desired curve.

Path ahead for 'low risk' adolescents living with a Fontan circulation.

OBJECTIVE: A high risk of morbidity and mortality is well documented in adults with a Fontan circulation. The difference in outcomes between those with and without significant morbidity at the time of transition to adult care has not been well characterised. METHODS: We analysed clinical outcomes in patients enrolled in the Australian and New Zealand Fontan Registry ≥16 years of age. Low risk (LR) Fontan patients were defined as those without history of sustained arrhythmia, thromboembolic event, transplantation, Fontan conversion, protein-losing enteropathy, plastic bronchitis, New York Heart Association class III/IV and/or moderate/severe atrioventricular valve regurgitation or ventricular dysfunction. Increased risk (IR) patients had one or more risk factor. RESULTS: Inclusion criteria were met in 822 patients; mean age 26±8 years, median follow-up from age 16 was 9 years, 203 had atriopulmonary connection (APC) and 619 had total cavopulmonary connection (TCPC). Survival at 30 years was higher in the LR versus IR; 94% versus 82% (p=0.005), 89% versus 77% (p=0.07) for APC and 96% versus 89% (p=0.05) for TCPC. LR patients experienced less Fontan failure (HR 0.34, 95% CI 0.23 to 0.49, p<0.001) and ventricular dysfunction (HR 0.46, 95% CI 0.29 to 0.71, p=0.001) compared with IR patients. For LR TCPC patients, modelled survival projections at 60 years were 49%-67%. CONCLUSIONS: Clinical outcomes for adolescents LR at transition to adult care are markedly superior to those who have established risk factors for Fontan failure, which is an important consideration when formulating individualised long-term risk estimates and counselling patients.

The plasmon Talbot effect.

The plasmon analog of the self-imaging Talbot effect is described and theoretically analyzed. Rich plasmon carpets containing hot spots are shown to be produced by a row of periodically-spaced surface features. A row of holes drilled in a metal film and illuminated from the back side is discussed as a realizable implementation of this concept. Self-images of the row are produced, separated from the original one by distances up to several hundreds of wavelengths in the examples under consideration. The size of the image focal spots is close to half a wavelength and the spot positions can be controlled by changing the incidence direction of external illumination, suggesting the possibility of using this effect (and its extension to non-periodic surface features) for far-field patterning and for long-distance plasmon-based interconnects in plasmonic circuits, energy transfer, and related phenomena.