2,867 research outputs found
Polarization singularities in paraxial vector fields: morphology and statistics
No full textPolarization patterns in the transverse plane generically contain singularities: points of circular polarization (C points), lines of linear polarization (L lines), instantaneous zeros (disclinations) and component zeros. We investigate the geometry of ellipse fields at these singularities, using the Stokes parameters and others to characterize the singular geometry and morphology. Comparison is made with analogous structures on random surfaces, namely umbilic points and parabolic lines. The densities and correlations of the different types of polarization singularities are calculated in random polarization fields, and compared to the statistics of phase singularities and random surfaces
Geometric interpretation of the three-dimensional coherence matrix for nonparaxial polarization
Get PDFThe three-dimensional coherence matrix is interpreted by emphasizing its invariance with respect to spatial rotations. Under these transformations, it naturally decomposes into a real symmetric positive definite matrix, interpreted as the moment of inertia of the ensemble (and the corresponding ellipsoid), and a real axial vector, corresponding to the mean angular momentum of the ensemble. This vector and tensor are related by several inequalities, and the interpretation is compared to those in which unitary invariants of the coherence matrix are studied
Correlations and screening of topological charges in Gaussian random fields
Get PDFTwo-point topological charge correlation functions of several types of geometric singularity in Gaussian random fields are calculated explicitly, using a general scheme: zeros of n-dimensional random vectors, signed by the sign of their Jacobian determinant; critical points (gradient zeros) of real scalars in two dimensions signed by the Hessian; and umbilic points of real scalars in two dimensions, signed by their index. The functions in each case depend on the underlying spatial correlation function of the field. These topological charge correlation functions are found to obey the first Stillinger-Lovett sum rule for ionic fluids
Canonical representation of spherical functions: Sylvester's theorem, Maxwell's multipoles and Majorana's sphere
No full textAny eigenfunction of the Laplacian on a sphere is given in terms of a unique set of directions: these are Maxwell's multipoles, their existence and uniqueness being known as Sylvester's theorem. Here, the theorem is proved by realizing the multipoles are pairs of opposite vectors in Majorana's sphere representation of quantum spins. The proof involves a physicist's standard tools of quantum angular momentum algebra, integral kernels and Gaussian integration. Various other proofs are compared, including an alternative using the calculus of spacetime spinors
Geometry of Călugăreanu's theorem
No full textA central result in the space geometry of closed twisted ribbons is Călugăreanu's theorem (also known as White's formula, or the Călugăreanu–White–Fuller theorem). This enables the integer linking number of the two edges of the ribbon to be written as the sum of the ribbon twist (the rate of rotation of the ribbon about its axis) and its writhe. We show that twice the twist is the average, over all projection directions, of the number of places where the ribbon appears edge-on (signed appropriately)—the ‘local’ crossing number of the ribbon edges. This complements the common interpretation of writhe as the average number of signed self-crossings of the ribbon axis curve. Using the formalism we develop, we also construct a geometrically natural ribbon on any closed space curve—the ‘writhe framing’ ribbon. By definition, the twist of this ribbon compensates its writhe, so its linking number is always zero
Topological events on wave dislocation lines: birth and death of small loops, and reconnection
No full textIn three-dimensional space, a wave dislocation, that is, a quantized (optical) vortex or phase singularity, is a line zero of a complex scalar wavefunction. As a 'time' parameter varies, the topology of the vortex can change by encounter with a line of vanishing vorticity (curl of the current associated with the wavefunction). An isolated critical point of the field intensity, sliding along the zero-vorticity line like a bead on a wire, meets the vortex as it encounters the line, and so participates in the singular event. Local expansio n and gauge and coordinates transformations show that the vortex topology can change generically by the appearance or disappearance of a loop, or by the reconnection of branches of a pair of hyperbolas
Vortex knots in light
Get PDFOptical vortices generically arise when optical beams are combined. Recently, we reported how several laser beams containing optical vortices could be combined to form optical vortex loops, links and knots embedded in a light beam (Leach et al 2004 Nature 432 165). Here, we describe in detail the experiments in which vortex loops form these structures. The experimental construction follows a theoretical model originally proposed by Berry and Dennis, and the beams are synthesized using a programmable spatial light modulator and imaged using a CCD camera
Quantum cores of optical phase singularities
Get PDFThe nodal line singularities (optical vortices) of classical scalar optics are smoothed in quantum optics, because of spontaneous emission into unoccupied modes. The radius of the 'quantum core' surrounding each classical singularity is proportional to ??. A trapped excited atom, steered into a nodal line of the classical field, is a possible detector for the effect. Analogous phenomena are anticipated for other waves, for example sound, where the silence at a nodal line is disturbed by pressure fluctuations of the fluid molecules
Black polarization sandwiches are square roots of zero
Get PDFIn the 2 x 2 matrices representing retarders and ideal polarizers, the eigenvectors are orthogonal. An example of the opposite case, where eigenvectors collapse onto one, is matrices M representing crystal plates sandwiched between a crossed polarizer and analyser. For these familiar combinations, M^2 = 0, so black sandwiches can be regarded as square roots of zero. Black sandwiches illustrate physics associated with degeneracies of non-Hermitian matrices
Knotting and unknotting of phase singularities: Helmholtz waves, paraxial waves and waves in 2+1 spacetime
No full textAs a parameter a is varied, the topology of nodal lines of complex scalar waves in space (i.e. their dislocations, phase singularities or vortices) can change according to a structurally stable reconnection process involving local hyperbolas whose branches switch. We exhibit families of exact solutions of the Helmholtz equation, representing knots and links that are destroyed by encounter with dislocation lines threading them when a is increased. In the analogous paraxial waves, the paraxial prohibition against dislocations with strength greater than unity introduces additional creation events. We carry out the analysis with polynomial waves, obtained by long-wavelength expansions of the wave equations. The paraxial events can alternatively be interpreted as knotting and linking of worldlines of dislocation points moving in the plane
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