On the optimality of the Arf invariant formula for graph polynomials

We prove optimality of the Arf invariant formula for the generating function of even subgraphs, or, equivalently, the Ising partition function, of a graph.Comment: Advances in Mathematics, 201

Ultrafast and octave-spanning optical nonlinearities from strongly phase-mismatched cascaded interactions

Cascaded nonlinearities have attracted much interest, but ultrafast applications have been seriously hampered by the simultaneous requirements of being near phase-matching and having ultrafast femtosecond response times. Here we show that in strongly phase-mismatched nonlinear frequency conversion crystals the pump pulse can experience a large and extremely broadband self-defocusing cascaded Kerr-like nonlinearity. The large cascaded nonlinearity is ensured through interaction with the largest quadratic tensor element in the crystal, and the strong phase-mismatch ensures an ultrafast nonlinear response with an octave-spanning bandwidth. We verify this experimentally by showing few-cycle soliton compression with noncritical cascaded second-harmonic generation: Energetic 47 fs infrared pulses are compressed in a just 1-mm long bulk lithium niobate crystal to 17 fs (under 4 optical cycles) with 80% efficiency, and upon further propagation an octave-spanning supercontinuum is observed. Such ultrafast cascading is expected to occur for a broad range of pump wavelengths spanning the near- and mid-IR using standard nonlinear crystals.Comment: resubmitted, revised version, accepted for Phys. Rev. Let

Prototype laser-diode-pumped solid state laser transmitters

Monolithic, diode-pumped Nd:YAG ring lasers can provide diffraction-limited, single-frequency, narrow-linewidth, tunable output which is adequate for use as a local oscillator in a coherent communication system. A laser was built which had a linewidth of about 2 kHz, a power of 5 milliwatts, and which was tunable over a range of 30 MHz in a few microseconds. This laser was phase-locked to a second, similar laser. This demonstrates that the powerful technique of heterodyne detection is possible with a diode-pumped laser used as the local oscillator. Laser diode pumping of monolithic Nd:YAG rings can lead to output powers of hundreds of milliwatts from a single laser. A laser was built with a single-mode output of 310 mW. Several lasers can be chained together to sum their power, while maintaining diffraction-limited, single frequency operation. This technique was demonstrated with two lasers, with a total output of 340 mW, and is expected to be practical for up to about ten lasers. Thus with lasers of 310 mW, output of up to 3 W is possible. The chaining technique, if properly engineered, results in redundancy. The technique of resonant external modulation and doubling is designed to efficiently convert the continuous wave, infrared output of our lasers into low duty-cycle pulsed green output. This technique was verified through both computer modeling and experimentation. Further work would be necessary to develop a deliverable system using this technique

Thin presentation of knots and lens spaces

This paper concerns thin presentations of knots K in closed 3-manifolds M^3 which produce S^3 by Dehn surgery, for some slope gamma. If M does not have a lens space as a connected summand, we first prove that all such thin presentations, with respect to any spine of M have only local maxima. If M is a lens space and K has an essential thin presentation with respect to a given standard spine (of lens space M) with only local maxima, then we show that K is a 0-bridge or 1-bridge braid in M; furthermore, we prove the minimal intersection between K and such spines to be at least three, and finally, if the core of the surgery K_gamma yields S^3 by r-Dehn surgery, then we prove the following inequality: |r| <= 2g, where g is the genus of K_gamma.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-23.abs.htm