Light beams carrying a point singularity with a screw-type phase distribution are associated with an optical vortex. The corresponding momentum flow leads to an orbital angular momentum of the photons 1-3 . The study of optical vortices has led to applications such as particle micro-manipulation 4,5 , imaging 6 , interferometry 7 , quantum information 8 and highresolution microscopy and lithography 9 . Recent analyses showed that transitions forbidden by selection rules seem to be allowed when using optical vortex beams 10 . To exploit these intriguing new applications, it is often necessary to shorten the wavelength by nonlinear frequency conversion. However, during the conversion the optical vortices tend to break up 11-13 . Here we show that optical vortices can be generated in the extreme ultraviolet (XUV) region using high-harmonic generation 14,15 . The singularity impressed on the fundamental beam survives the highly nonlinear process. Vortices in the XUV region have the same phase distribution as the driving field, which is in contradiction to previous findings 16 , where multiplication of the momentum by the harmonic order is expected. This approach opens the way for several applications based on vortex beams in the XUV region.Places where physical quantities become infinite or change abruptly are called singularities. The presence of phase dislocations (singularities) in the wavefront of a light beam determines both the phase and intensity structure around them. As the phase becomes indeterminate at singularities, both the real and the imaginary parts of the field amplitude (that is, also the field intensity) vanish. The characteristic helical phase profiles of optical vortices are described by exp(imθ) multipliers, where θ is the azimuthal coordinate and the integer number m is their topological charge, also called dislocation strength or winding number.Recalling the fact that in free space the Poynting vector gives the momentum flow, for helical phase fronts the Poynting vector has an azimuthal component that produces an orbital angular momentum parallel to the axis of the beam. The momentum circulates around the beam axis, so such beams are said to contain an optical vortex. As has been shown 1-3 , an m-fold charged optical vortex beam carries an orbital angular momentum of mh per photon independent of the spin angular momentum (that is, the polarization state). It was shown that transitions that are forbidden by known selection rules in the electric and magnetic dipole approximation seem to be allowed when using optical vortex beams 10 . This provides a new degree of freedom in the spectroscopy of forbidden transitions. Multi-coloured optical vortices can be generated through a nonlinear frequency-conversion process such as second-harmonic generation 16 or four-wave mixing, which is an important process in the white-light vortex generation. However, as predicted in ref. 11 and observed in ref. 12, vortex breakup in self-focusing nonlinear media is an important issue for supercontinuum vortex genera...

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In 2009, Briane and Milton proved mathematically the existence of three-dimensional isotropic metamaterials with a classical Hall coefficient which is negative with respect to that of all of the metamaterial constituents. Here, we significantly simplify their blueprint towards an architecture composed of only a single constituent material in vacuum/air, which can be seen as a special type of porosity. We show that the sign of the Hall voltage is determined by a separation parameter between adjacent tori. This qualitative behavior is robust even for only a small number of metamaterial unit cells. The combination of simplification and robustness brings experimental verifications of this striking sign-inversion into reach.

Effectively inverting the sign of material parameters is a striking possibility arising from the concept of metamaterials. Here, we show that the electrical properties of a p-type semiconductor can be mimicked by a metamaterial solely made of an n-type semiconductor. By fabricating and characterizing threedimensional simple-cubic microlattices composed of interlocked hollow semiconducting tori, we demonstrate that sign and magnitude of the effective metamaterial Hall coefficient can be adjusted via a tori separation parameter-in agreement with previous theoretical and numerical predictions.