The similarities and differences of spatial shifts to the centroids of reflected beams, and their (optical vortex) structure are discussed and reviewed. The differences between vortex-induced shifts to a beam centroid on reflection, and to the distribution of the vortices themselves is discussed. We conclude by discussing the shifts of a reflected beam containing a single anisotropic vortex.
On scattering, the high strength singularity of a vortex beam breaks into a configuration of single charge vortices. The precise geometry of such a vortex constellation depends on the angle of incidence and the material properties of the scatterer, but also on the optical spin-orbit coupling as choosing different input and output polarization results in a family of vortex constellation. Measuring the position of the individual vortices allows us to reconstruct elements in an systematic expansion of the scattering matrix, in an analogy to optical aberration theory. We discuss in detail the dependence of the constellation geometry on external parameters, which is the basis for an optical metrology based on vortices.
In optics, the Goos-Hänchen shift is a transverse displacement of a reflected light beam along a material interface.
It is usually associated with the presence of evanescent waves beyond the interface. We describe an analogous
displacement effect for scalar waves at a boundary satisfying Robin, or mixed, boundary conditions, although
the wave does not penetrate the boundary. We briefly discuss how the reflection of electromagnetic plane waves
differs from reflection due to Robin boundary conditions.
Optical vortices are points of zero intensity in a two dimensional, classical optical field. As first discussed by Berry
and Dennis [Berry, M. V. and Dennis, M. R., "Quantum cores of optical phase singularities" Journal of Optics
A 6, S178-S180 (2004)] these singularities are replaced by 'quantum cores' in a deeper level of description. In a
fully quantized theory of optical fields an excited atom trapped at the singularity can emit light spontaneously
and hence soften the perfect zero of an optical vortex. More recently Barnett [Barnett, S. M., "On the quantum
cores of a optical vortex," Journal of Modern Optics 55, 2279-2292 (2008)] presented a more realistic analysis of
quantum cores which accounts for the effect of the trapping potential on the transition dynamics inside a vortex
core. Here, we revisit the scenario of emission near an optical vortex in the realistic setting of Barnett.
Superoscillations are regions of band-limited waves where the local wavenumber, defined as the local phase
gradient, exceeds the global maximum wavenumber in the Fourier spectrum. In random functions, defined as
superpositions of plane waves with random complex amplitudes and directions, considerable regions are naturally
superoscillatory (M. R. Dennis, et al., Opt. Lett. 33, 2976-2978, 2008; M. V. Berry and M. R. Dennis, J. Phys. A:
Math. Theor. 42, 022003, 2009). We discuss this result by deriving the joint probability density function for
intensity and phase gradient of isotropic complex random wave in any dimension, with specific reference to the
Propagating three-dimensional speckle fields are threaded by random networks of nodal lines (optical vortices).
We review our recent numerical superpositions of simulations of random plane waves modelling speckle
(O'Holleran et al. Phys. Rev. Lett. in press), in which the nodal lines and loops were found to have the fractal
structure of brownian random walks. We discuss this result, and its comparison with the discrete vortices of the
Z<sub>3</sub> lattice model for cosmic strings. We argue that the scaling depends on the geometry of small vortex loops
and avoided crossings. The analytic statistics of these events, along with related singularities are discussed, and
the densities of vorticity-vanishing points and anisotropy C lines are found explicitly.
A new technique for fluid mechanics measurement is proposed that makes use of pseudophase singularities in an analytic signal representation of a speckle-like pattern generated by a Laguerre-Gauss filter operation. Based on the formal analogy between the polarization of the vector wave and the gradient field for the complex analytic signal, a set of Stokes-like parameters have been applied for the description of the anisotropic core structure of the pseudophase singularities, which serves as unique fingerprints attached to the seeding particles moving with the flow. Experimental results for flow velocity and acceleration measurement are presented that demonstrate the validity of the proposed optical vortex metrology for fluid mechanics measurement.
This paper is a review and extension of recent work by Berry and Dennis (Proc. Roy. Soc. Lond. A456, pp. 2059-2079, 2000; A457, pp. 141-155, 2001), where the geometric structure of phase singularities (wave dislocations) in waves is studied, particularly for singularities in isotropic random wavefields. The anisotropy ellipse of a generic dislocation is defined, and I derive an angular momentum rule for its phase. Random wavefields are discussed, and statistical results for density, anisotropy ellipse eccentricity, and planar correlation functions are stated. The properties of the correlation functions are compared to analogous features from ionic structure theory, and are discussed in those terms. The results are given explicitly for four particular spectra: monochromatic waves propagating in the plane, monochromatic waves propagating in space, a speckle pattern in the transverse plane of a paraxial beam, and the Planck spectrum for blackbody radiation.