Mark R. Dennis

Singular Optics: Optical Vortices and Polarization Singularities

Publisher Summary The widespread availability of spatially and temporally coherent laser sources makes the production of optical vortices inevitable in any experiment involving scattered laser light. The realization of quantized vortices is not specific to optics: these objects occur in all spatial scalar fields. Although optical vortices are often referred to as “points of phase singularity within a cross section of the field,” physical optical fields extend over three dimensions, and the phase singularities are actually lines of perfect destructive interference that are embedded in the volume filled by the light. Optical vortices are examples of the singularity lines within all complicated scalar fields. By comparison, electromagnetic vector fields do not generally have nodes in all components simultaneously. However, vector fields possess singularities associated with the parameterization of elliptical and partial polarization rather than phase. Polarization singularities are present in many situations, ranging from sunlight to the light transmitted by birefringent materials. Their descriptors are more complicated than their scalar counterpart in that they have both handedness and additional categorization. The study of optical vortices and orbital angular momentum has led to a recognition that the energy flow—characterized by the Poynting vector—has features not immediately apparent from the intensity alone, nor from global properties of a beam.

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.

Phase singularities in isotropic random waves

The 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.

Polarization singularities in paraxial vector fields: morphology and statistics

Polarization 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.

Polarization singularities in isotropic random vector waves

Following 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.

Vortex knots in light

Optical 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.

Knotted and linked phase singularities in monochromatic waves

Exact solutions of the Helmholtz equation are constructed, possessing wavefront dislocation lines (phase singularities) in the form of knots or links where the wave function vanishes (‘knotted nothings’). The construction proceeds by making a nongeneric structure with a strength n dislocation loop threaded by a strength m dislocation line, and then perturbing this. In the resulting unfolded (stable) structure, the dislocation loop becomes an (m, n) torus knot if m and n are coprime, and N linked rings or knots if m and n have a common factor N; the loop or rings are threaded by an m-stranded helix. In our explicit implementation, the wave is a superposition of Bessel beams, accessible to experiment. Paraxially, the construction fails.

Topology of optical vortex lines formed by the interference of three, four, and five plane waves

When three or more plane waves overlap in space, complete destructive interference occurs on nodal lines, also called phase singularities or optical vortices. For super positions of three plane waves, the vortices are straight, parallel lines. For four plane waves the vortices form an array of closed or open loops. For five or more plane waves the loops are irregular. We illustrate these patterns numerically and experimentally and explain the three-, four- and five-wave topologies with a phasor argument.

Polarization singularities in the clear sky

Ideas from singularity theory provide a simple account of the pattern of polarization directions in daylight. The singularities (two near the Sun and two near the anti-Sun) are points in the sky where the polarization line pattern has index +1/2 and the intensity of polarization is zero. The singularities are caused by multiple scattering that splits into two each of the unstable index +1 singularities at the Sun and anti-Sun, which occur in the single-dipole scattering (Rayleigh) theory. The polarization lines are contours of an elliptic integral. For the intensity of polarization (unnormalized degree), it is necessary to incorporate the strong depolarizing effect of multiple scattering near the horizon. Singularity theory is compared with new digital images of sky polarization, and gives an excellent description of the pattern of polarization directions. For the intensity of polarization, the theory can reproduce not only the zeros but also subtle variations in the polarization maxima. It is not one of the least wonders of terrestrial physics, that the blue atmosphere which overhangs us, exhibits in the light which it polarises phenomena somewhat analogous to those of crystals with two axes of double refraction Brewster D 1863 Trans. R. Soc. Ed. 23 205-10

Super-oscillatory optical needle

Super-oscillatory optical lenses have recently been shown to achieve subwavelength focusing and have been used for super-resolution imaging. However, the subwavelength hotspots created by these lenses are always accompanied by sidebands containing a significant fraction of the optical energy and are highly localised in the axial direction. Here, we report a class of super-oscillatory lenses that form extended subwavelength optical needles on a 15λ field of view.