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Stokes surfaces of diffraction catastrophes with codimension three
No full textThe Stokes set, where exponentially small complex (i.e. evanescent) rays appear and disappear, is the locus of wavefield positions where stationary points of a diffraction integral have equal phase. In three dimensions, it is a surface. Stokes surfaces are calculated and displayed for the diffraction patterns decorating the swallowtail, elliptic and hyperbolic umbilic singularities. The surfaces are smooth apart from cusped edges where they meet the cusp lines of the real bifurcations set (caustic), and finite-angled creases at the complex whiskers of the singularity
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
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
Phase singularities in isotropic random waves
Get PDFThe singularities of complex scalar waves are their zeros; these are dislocation lines in space, or points in the plane. For waves in space, and waves in the plane (propagating in two dimensions, or sections of waves propagating in three), we calculate some statistics associated with dislocations for isotropically random Gaussian ensembles, that is, superpositions of plane waves equidistributed in direction but with random phases. The statistics are: mean length of dislocation line per unit volume, and the associated mean density of dislocation points in the plane; eccentricity of the ellipse describing the anisotropic squeezing of phase lines close to dislocation cores; distribution of curvature of dislocation lines in space; distribution of transverse speeds of moving dislocations; and position correlations of pairs of dislocations in the plane, with and without their strength (topological charge) -1. The statistics depend on the frequency spectrum of the waves. We derive results for general spectra, and specialize to monochromatic waves in space and the plane, and black-body radiation
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
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
Polarization singularities in isotropic random vector waves
Get PDFFollowing Nye & Hajnal, we explore the geometry of complex vector waves by regarding them as a field of polarization ellipses. Singularities of this field are the C lines and L lines, where the polarization is purely circular and purely linear, respectively. The singularities can be reinterpreted as loci of photon spin 1 (C lines) and 0 (L lines). For Gaussian random superpositions of plane waves equidistributed in direction but with an arbitrary frequency spectrum, we calculate the density (length per unit volume) of C and L lines
The optical singularities of birefringent dichroic chiral crystals
No full textUsing a new formalism involving projection from the sphere of directions to the stereographic plane, and associated complex variables, explicit formulae are obtained for the two refractive indices and polarizations in optically anisotropic crystals that are both dichroic (absorbing) and chiral (optically active). This enables three types of polarization singularity to be classified and explored: singular axes, which are degeneracies where the two refractive indices are equal, and which for a transparent non-chiral crystal condense pairwise onto the optic axes; C points, where the polarization is purely circular (right- or left-handed), with topological index +1, +12 or +14 and whose positions are independent of the chirality; and L lines, where the polarization is purely linear, dividing direction space into regions with right- and left-handedness. A local model captures essential features of the general theory. Interference figures generated by slabs of crystal viewed directly or through a polarizer and/or analyser enable the singularities to be displayed directly
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