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Much ado about knotting
The flow of light beams flow through space is analogous to the flow of water in a river. Although it mainly flows in a straight line along the beam, light also flows in whirls and eddies, forming lines in space called 'optical vortices' on which the intensity of the light is zero. called optical vortices, along which the light is completely zero. Optical vortices can be created with holograms which direct the flow of light. My student Robert King and I found a class of solutions of the wave equations for propagating laser light which had knotted optical vortices, using the abstract mathematical theory of fibred knots. These solutions led to designs of holograms, which Miles Padgett's team in Glasgow used to realise isolated, fibred optical vortex knots in the lab. (See publication "Isolated optical vortex knots" for more information). This work received much media attention, including an article in Physics Today.
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More ado about knotting
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Previous work in knotted vortices was based on another means of generating knotted optical vortices, with Michael Berry, based on perturbing cylindrically symmetric vortex configurations in light beams (see "Knotted and linked phase singularities..." and "Knotting and unknotting..." for more information). These solutions were also generated in the Glasgow lab, and the vortex knots and links were threaded by other vortex lines. (See publications "Knotted threads of darkness" and "Vortex knots in light" for more information.)
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Fractality in darkness
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The phenomenon of optical speckle can be seen as the fine bright and dark pattern in a laser spot. This random interference structure extends in three-dimensional space, within which are entwined lines of darkness (optical vortices). Recent work with Kevin O'Holleran, Florian Flossmann and Miles Padgett of the University of Glasgow revealed that the large-scale structure of these lines is fractal, with the same random shape as brownian motion (i.e. fractal dimension 2). Similar random fractal patterns are studied in other areas of physics, such as vortices in superfluid helium and cosmic strings. Some of these lines are loops, which sometimes are linked together; the probability that a loop is part of a link scales with the loop length. (See publications "Topology of light's darkness", "Fractality of light's darkness" and "The fractal shape of speckled darkness" for more information.) The work on topological scaling was featured by the APS Physics spotlight.
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The sound of drums with position-varying boundary conditions
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The allowed frequencies of resonators, such as drums, violin strings and organ pipes, depend on the form of the wave on the boundary (perimeter of drum, ends of violin string, etc.). For instance, violin strings do not vibrate at their ends, whereas the vibration gradient is zero at the top of an organ pipe; mathematically, every point along the boundary has to be a mixture (mixed boundary conditions). Michael Berry and I have recently been exploring the properties of circular drums (circular quantum billiards) whose mixed boundary conditions are a continuous function of position, which can have very surprising mathematical properties. (See "Boundary-condition-varying circle billiards..." for more information).
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Self-imaging optical carpets on metal surfaces
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Javier Garcia de Abajo (Madrid), Nikolay Zheludev (Southampton) and I recently worked out the theory of optical surface waves (plasmons) evolving from a row of nano-holes on the surface of a metal. The resulting plasmon interference pattern self-images the original row of hotspots at various lengthscales (including apparently sub-wavelength focusing), in the surface plasmon analogue of the Talbot effect of propagating light beams. (See "The plasmon Talbot effect" for more information). Our figure was used in the editorial of Nature Materials, June 2008.
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The geometry of twisted ribbons
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The twisting geometry of a ribbon loop in 3D space is characterized by three numbers: Link, the integer number of turns of the ribbon around its axis; Writhe, a measure of non-planarity of the ribbon's axis curve; and Twist, determined by the rate of rotation of the ribbon around its axis. Work of Calugareanu, White and Fuller led to the remarkable theorem that Link equals Writhe plus Twist, which plays an important role in understanding ribbon-type physical objects, such as DNA loops and optical vortices. John Hannay and I found an extremely simple proof of this result, which led to deeper insight into the geometry of three-dimensional curve and ribbon loops. (See "Geometry of Calugareanu's theorem" for more information.) 
Polarization fingerprints in the sky revealed by simple mathematics
| With Michael Berry and Raymond Lee, I found a simple representation of the polarization pattern of skylight. The pattern is governed by neutral points, which are simple polarization singularities. The predicted quartic polynomial structure, which agrees excellently both with (Chandrasekhar's) scattering theory and new observations, is revealed elegantly by elliptic integrals in the sky. (See publication "Polarization singularities in the clear sky" for more information.) 
Maxwell's multipoles, Majorana's sphere and the Cosmic Microwave Background?
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A real mode on the sphere can be defined democratically using a unique set of directions, known as Maxwell's multipole vectors. It turns out that this representation is a special case of the more general Majorana representation of quantum spin. (See publication "Canonical representation ..." for more information.) The probability distribution between directions of these vectors for gaussian random spherical functions can be calculated using random polynomials. (See "Correlations between Maxwell's multipoles..." for more information.) The cosmic microwave background (CMB) is apparently an example of such a random spherical function, and with Kate Land, I have explored how well the gaussian prediction compares with WMAP observations. (See "Probability distribution of the multipole vectors..." for more information.) |
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