Poincaré-sphere equivalent for light beams containing orbital angular momentum

The polarization state of a light beam is related to its spin angular momentum and can be represented on the Poincaré sphere. We propose a sphere for light beams in analogous orbital angular momentum states. Using the Poincaré-sphere equivalent, we interpret the rotational frequency shift for light beams with orbital angular momentum [Phys. Rev. Lett. 80, 3217 (1998)] as a dynamically evolving geometric phase.     

Generation and application of the twisted beam with orbital angular momentum

The twisted Laguerre-Gaussian beam was generated by transforming of Hermite-Gaussian beams through an optical system consisting of three rotated cylindrical lenses. The intensity distribution and phase structure of the twisted hollow beam were theoretically analyzed by using Collins diffraction integral. By utilizing the method of mode decomposition, the theory of transformation was analyzed. In the experiment, micro particles were trapped and rotated by this twisted beam.     

Synchronization of Huygens' clocks and the Poincaré method

We study two models of connected pendulum clocks synchronizing their oscillations, a phenomenon originally observed by Huygens. The oscillation angles are assumed to be small so that the pendulums are modeled by harmonic oscillators, clock escapements are modeled by the van der Pol terms. The mass ratio of the pendulum bobs to their casings is taken as a small parameter. Analytic conditions for existence and stability of synchronization regimes, and analytic expressions for their stable amplitudes and period corrections are derived using the Poincaré theorem on existence of periodic solutions in autonomous quasi-linear systems. The anti-phase regime always exists and is stable under variation of the system parameters. The in-phase regime may exist and be stable, exist and be unstable, or not exist at all depending on parameter values. As the damping in the frame connecting the clocks is increased the in-phase stable amplitude and period are decreasing until the regime first destabilizes and then disappears. The results are most complete for the traditional three degrees of freedom model, where the clock casings and the frame are consolidated into a single mass.     

Flexible and high-efficiency generation of arbitrary vector vortex beams on hybrid-order Poincaré sphere

We propose theoretically and verify experimentally a compact optical configuration to directly generate arbitrary vector vortex beams on a hybrid-order Poincaré sphere with good flexibility and high efficiency based on a reflective phase-only liquid crystal spatial light modulator(LC-SLM). The conversion system, consisting of an LC-SLM and a quarter-wave plate, can be considered a flexible dielectric metasurface to simultaneously modulate inhomogeneous polarization and helical phase-front. This approach has some advantages, including a simple experimental setup, good flexibility, and high efficiency. Orthogonally polarized modes alignment and an explicit superposition existing in the conventional method are not necessary in the proposed method, which exhibits potential applications in many advanced domains.     

Measuring the fractional orbital angular momentum of a vortex light beam by cascaded Mach–Zehnder interferometers

We design an interferometric method for measuring the fractional orbital angular momentum of a vortex light beam by cascading Mach–Zehnder interferometers. The validity of this method is verified by simulation and theoretical analysis. We demonstrate the method experimentally for two stages of cascaded Mach–Zehnder interferometers, which can measure the fractional topological charge up to two. The experimental results agree with the theoretical results well. Since fractional orbital angular momentum may have a potential application in the field of quantum information, one can utilize the method to detect them easily and precisely.     

Direct gauging of the Poincaré group

Realization of the Poincaré group as a subgroup ofGL(5,R) that maps an affine set into itself is shown to lead to a well-defined minimal replacement operator when the Poincaré group is allowed to act locally. The minimal replacement operator is obtained by direct application of the Yang-Mills procedure without the explicit introduction of fiber bundle techniques. Its application gives rise to compensating 1-formsW , 1 6, for the local action of the Lorentz groupL(4,R), and to compensating 1-forms ^k , 1k4, for the translation groupT(4). When applied to the basis 1-formsdx ⁱ of Minkowski space, distortion 1-formsB ^k result that define a canonical anholonomic coframe that contains both theT(4) and theL(4,R) compensating fields. When the canonical coframe is considered as a differential system onM ₄, it gives rise to gauge curvature expressions and Cartan torsion, but the latter has important differences from that usually encountered in the associated literature in view of the inclusion of the compensating fields forL(4,R). The standard Yang-Mills minimal coupling construct is used to obtain a total Lagrangian. This leads to a system of field equations for the matter fields, theT(4) compensating fields, and theL(4,R) compensating fields. Part of the current that drives theT(4) compensating fields is the 3-form of gauge momentum energy that obtains directly from the momentum-energy tensor of the matter fields onM ₄ under minimal replacement. Introduction of the Cartan torsion in the free-field Lagrangian is shown to lead to a direct spin decoupling in the sense that the gauge momentum energy (orbital) contribution of the matter fields to the spin current is eliminated. Explicit conservation laws for total momentum energy current and total spin current are obtained.     

Nucleon internal structure: a new set of quark, gluon momentum, angular momentum operators and parton distribution functions

It is unavoidable to deal with the quark and gluon momentum and angular momentum contributions to the nucleon momentum and spin in the study of nucleon internal structure. However we never have the quark and gluon momentum, orbital angular momentum and gluon spin operators which satisfy both the gauge invariance and the canonical momentum and angular momentum commutation relation. The conflicts between the gauge invariance and canonical quantization requirement of these operators are discussed. A new set of quark and gluon momentum, orbital angular momentum and spin operators, which satisfy both the gauge invariance and canonical momentum and angular momentum commutation relation, are proposed. The key point to achieve such a proper decomposition is to separate the gauge field into the pure gauge and the gauge covariant parts. The same conflicts also exist in QED and quantum mechanics and have been solved in the same manner. The impacts of this new decomposition to the nucleon internal structure are discussed.     

Do gravitational waves carry energy‐momentum and angular momentum?

We show that gravitational waves which possess a non‐vanishing Riemann tensor R_iklm ≠ 0 always carry energy‐momentum and angular momentum. Our proof uses canonical superenergy and supermomentum tensors for the gravitational field.     

Tuning the polarization states of optical spots at the nanoscale on the Poincaré sphere using a plasmonic nanoantenna

It is shown that the polarization states of optical spots at the nanoscale can be manipulated to various points on the Poincaré sphere using a plasmonic nanoantenna. Linearly, circularly, and elliptically polarized near-field optical spots at the nanoscale are achieved with various polarization states on the Poincaré sphere using a plasmonic nanoantenna. A novel plasmonic nanoantenna is illuminated with diffraction-limited linearly polarized light. It is demonstrated that the plasmonic resonances of perpendicular and longitudinal components of the nanoantenna and the angle of incident polarization can be tuned to obtain optical spots beyond the diffraction limit with a desired polarization and handedness.     

Parameter optimization for a reflective bistable twisted nematic display by use of the Poincar sphereé method

Although reflective bistable twisted nematic (RBTN) displays have potential in low-power-consumption applications, to achieve the optimum conditions for both bistable states simultaneously remains a challenge. We use a geometrical method based on the Poincaré sphere representation to obtain the optimum conditions that can simultaneously satisfy both bistable states for a RBTN structure. With this method, the optimum conditions can be obtained analytically and the operation modes can be clearly visualized and better understood.     

On the Poincaré Gauge Theory of Gravitation

We present a compact, self-contained review of the conventional gauge theoretical approach to gravitation based on the local Poincaré group of symmetry transformations. The covariant field equations, Bianchi identities and conservation laws for angular momentum and energy-momentum are obtained.     

On the difference between Poincaré and Lorentz gravity

The Poincaré invariance of GR is usually interpreted as Lorentz invariance plus diffeomorphism invariance. In this paper, by introducing the local inertial coordinates (LIC), it is shown that a theory with Lorentz and diffeomorphism invariance is not necessarily Poincaré invariant. Actually, the energy–momentum conservation is violated there. On the other hand, with the help of the LIC, the Poincaré invariance is reinterpreted as an internal symmetry. In this formalism, the conservation law is derived, which has not been sufficiently explored before.     

The K-Orbits, Identities, and Classification of Four-Dimensional Homogeneous Spaces with the Group of Poincaré and de Sitter Transforms

The method of orbits traditionally applied to geometric quantization problems is used to study homogeneous spaces. Based on the proposed classification of the orbits of co-adjoint representation (K-orbits), a classification of homogeneous spaces is constructed. This classification allows one, in particular, to point out the explicit form of identities – functional relations between the transform-group generators – which are of great importance in applied problems (e.g., in the theory of separation of variables). All four-dimensional homogeneous spaces with the group of Poincaré and de Sitter transforms are classified and all independent identities on these spaces are given in explicit form.     

Geometrical meaning of the Poincaré group gauge theory

We find a condition (6) under which a gauge theory of the Poincaré group is equivalent to the Einstein-Cartan theory of gravitation.     

Direct gauging of the Poincaré group. II

This paper continues the study of direct gauge theory of the Poincaré groupP ₁₀. The meanings and implications of transformations induced by the local action ofP ₁₀ are studied, and transformation rules for all field quantities are derived for the local action ofP ₁₀ in a sufficiently small neighborhood of the identity. These results lead directly to a system of fundamental partial differential equations that are both necessary and sufficient for invariance of the free field Lagrangian density. Homogeneity arguments and the classical theory of invariants are used to obtain the most general free field Lagrangian density. Gauge conditions are shown to imply coordinate conditions, and an algebraic system of antiexact gauge conditions is implemented. The underlying Minkowski space,M ₄, and the resulting Riemann-Cartan space,U ₄, become attached at their centers, as do their respective frame and coframe bundles. Weak constraints of vanishing torsion are studied. All field quantities are shown to be determined in terms of the compensating l-forms for the Lorentz sector alone provided an explicit system of integrability conditions is satisfied. Field equations of the Einstein type are shown to result.     

Poincaré recurrence and measure of hyperbolic and nonhyperbolic chaotic attractors

We study Poincaré recurrence of chaotic attractors for regions of finite size. Contrary to the standard case, where the size of the recurrent regions tends to zero, the measure is no longer supported solely by unstable periodic orbits of finite length inside it, but also by other special recurrent trajectories, located outside that region. The presence of the latter leads to a deviation of the distribution of the Poincaré first return times from a Poissonian. Consequently, by taking into account the contribution of these special recurrent trajectories, a corrected estimate of the measure is obtained. This has wide experimental implications, as in the laboratory all returns can exclusively be observed for regions of finite size, and only unstable periodic orbits of finite length can be detected.