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Natural and artificially created light fields in three-dimensional space contain lines of zero intensity, known as optical vortices 1-3. Here, we describe a scheme to create optical beams with isolated optical vortex loops in the forms of knots and links using algebraic topology. The required complex fields with fibred knots and links 4 are constructed from abstract functions with braided zeros and the knot function is then embedded in a propagating light beam. We apply a numerical optimization algorithm to increase the contrast in light intensity, enabling us to observe several optical vortex knots. These knotted nodal lines, as singularities of the wave's phase, determine the topology of the wave field in space, and should have analogues in other three-dimensional wave systems such as superfluids 5 and Bose-Einstein condensates 6,7. In nature, one never finds the plane waves described in textbooks; real physical waves, such as light beams, are superpositions of many plane waves propagating in different directions. Generically, interference of three or more plane waves results in optical vortex lines, where the intensity is zero, and around which the phase increases by 2π (refs 1, 2). This contrasts with a single plane wave, for which the intensity is uniform, and the wavefronts-the surfaces of constant phase-are nested planes perpendicular to the wavevector k. The presence of vortices disrupts this regular arrangement of wavefronts 3 : a straight vortex line parallel to k has helicoidal wavefronts; a circular vortex loop perpendicular to k is the edge of a punctured wavefront. The situation is more complicated when a single, isolated vortex loop is knotted 8. Here, we establish theoretically and demonstrate experimentally that waves may contain a single isolated knotted vortex loop. The global topology of such a knotted vortex field is nontrivial. Wavefront surfaces with all phase labels must intersect on the vortex lines in any optical field, and fill all space. If the vortex line is knotted, all wavefront surfaces have a knotted boundary curve, and are thus multiply connected. Although the vortices occupy a finite volume of space, this topology affects the entire wave field. Such non-trivial topology of physical fields arises elsewhere in physics, such as flows in fluid dynamics 9 (including Lord Kelvin's vortex atom hypothesis 10), quantum condensates 6,7 and field theory 11,12. Time-dependent solutions of Maxwell's equations in which all electric field lines have the form of torus knots (knots that can be drawn, without crossing, on a torus) were found in ref. 13. This knotting of light complements our present approach, in which the topology resides in the optical complex amplitude, giving knots that are directly observable in the intensity distribution of the beam. Knotted vortex loops are hypothesized to be generic in turbulent and chaotic wave fields, including superfluids 5 , optical 14 and biological waves 15 , although only unknotted linked vortex rings have been identified in large-scale simulations ...

SummaryA key feature of Notch signaling is that it directs immediate changes in transcription via the DNA-binding factor CSL, switching it from repression to activation. How Notch generates both a sensitive and accurate response—in the absence of any amplification step—remains to be elucidated. To address this question, we developed real-time analysis of CSL dynamics including single-molecule tracking in vivo. In Notch-OFF nuclei, a small proportion of CSL molecules transiently binds DNA, while in Notch-ON conditions CSL recruitment increases dramatically at target loci, where complexes have longer dwell times conferred by the Notch co-activator Mastermind. Surprisingly, recruitment of CSL-related corepressors also increases in Notch-ON conditions, revealing that Notch induces cooperative or “assisted” loading by promoting local increase in chromatin accessibility. Thus, in vivo Notch activity triggers changes in CSL dwell times and chromatin accessibility, which we propose confer sensitivity to small input changes and facilitate timely shut-down.

Light emerging from a spiral phase plate with a non-integer phase step has a complicated vortex structure and is unstable on propagation. We generate light carrying fractional orbital angular momentum (OAM) not with a phase step but by a synthesis of Laguerre-Gaussian modes. By limiting the number of different Gouy phases in the superposition we produce a light beam which is well characterised in terms of its propagation. We believe that their structural stability makes these beams ideal for quantum information processes utilising fractional OAM states. 3427-3435 (1990). http: //link.aps.org/abstract/PRA/v41/p3427. 17. J. Courtial, "Self-imaging beams and the Guoy [sic] effect," Opt. Commun.

Natural light fields are threaded by lines of darkness. For monochromatic light, the phenomenon is familiar in laser speckle, i.e., the black points that appear in the scattered light. These black points are optical vortices that extend as lines throughout the volume of the field. We establish by numerical simulations, supported by experiments, that these vortex lines have the fractal properties of a Brownian random walk. Approximately 73% of the lines percolate through the optical beam, the remainder forming closed loops. Our statistical results are similar to those of vortices in random discrete lattice models of cosmic strings, implying that the statistics of singularities in random optical fields exhibit universal behavior.