Configuration spaces and Vassiliev classes in any dimension

The real cohomology of the space of imbeddings of S^1 into R^n, n>3, is studied by using configuration space integrals. Nontrivial classes are explicitly constructed. As a by-product, we prove the nontriviality of certain cycles of imbeddings obtained by blowing up transversal double points in immersions. These cohomology classes generalize in a nontrivial way the Vassiliev knot invariants. Other nontrivial classes are constructed by considering the restriction of classes defined on the corresponding spaces of immersions.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-39.abs.htm

Whispering gallery mode resonator based ultra-narrow linewidth external cavity semiconductor laser

We demonstrate a miniature self-injection locked DFB laser using resonant optical feedback from a high-Q crystalline whispering gallery mode resonator. The linewidth reduction factor is greater than 10,000, with resultant instantaneous linewidth less than 200 Hz. The minimal value of the Allan deviation for the laser frequency stability is 3x10^(-12) at the integration time of 20 us. The laser possesses excellent spectral purity and good long term stability.Comment: To be published in Optics Letter

Reducing the linewidth of a diode laser below 30 Hz by stabilization to a reference cavity with finesse above 10^5

An extended cavity diode laser operating in the Littrow configuration emitting near 657 nm is stabilized via its injection current to a reference cavity with a finesse of more than 10^5 and a corresponding resonance linewidth of 14 kHz. The laser linewidth is reduced from a few MHz to a value below 30 Hz. The compact and robust setup appears ideal for a portable optical frequency standard using the Calcium intercombination line.Comment: 8 pages, 4 figures on 3 additional pages, corrected version, submitted to Optics Letter

On polynomially integrable domains in Euclidean spaces

Let $D$ be a bounded domain in $\mathbb R^n,$ with smooth boundary. Denote $V_D(\omega,t), \ \omega \in S^{n-1}, t \in \mathbb R,$ the Radon transform of the characteristic function $\chi_{D}$ of the domain $D,$ i.e., the $(n-1)-$ dimensional volume of the intersection $D$ with the hyperplane $\{x \in \mathbb R^n: =t \}.$ If the domain $D$ is an ellipsoid, then the function $V_D$ is algebraic and if, in addition, the dimension $n$ is odd, then $V(\omega,t)$ is a polynomial with respect to $t.$ Whether odd-dimensional ellipsoids are the only bounded smooth domains with such a property? The article is devoted to partial verification and discussion of this question