Pseudo-ovals in even characteristic and ovoidal Laguerre planes

Pseudo-arcs are the higher dimensional analogues of arcs in a projective plane: a pseudo-arc is a set $\mathcal{A}$ of $(n-1)$-spaces in $\mathrm{PG}(3n-1,q)$ such that any three span the whole space. Pseudo-arcs of size $q^n+1$ are called pseudo-ovals, while pseudo-arcs of size $q^n+2$ are called pseudo-hyperovals. A pseudo-arc is called elementary if it arises from applying field reduction to an arc in $\mathrm{PG}(2,q^n)$. We explain the connection between dual pseudo-ovals and elation Laguerre planes and show that an elation Laguerre plane is ovoidal if and only if it arises from an elementary dual pseudo-oval. The main theorem of this paper shows that a pseudo-(hyper)oval in $\mathrm{PG}(3n-1,q)$, where $q$ is even and $n$ is prime, such that every element induces a Desarguesian spread, is elementary. As a corollary, we give a characterisation of certain ovoidal Laguerre planes in terms of the derived affine planes

On organizing principles of Discrete Differential Geometry. Geometry of spheres

Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. In this survey we discuss the following two fundamental Discretization Principles: the transformation group principle (smooth geometric objects and their discretizations are invariant with respect to the same transformation group) and the consistency principle (discretizations of smooth parametrized geometries can be extended to multidimensional consistent nets). The main concrete geometric problem discussed in this survey is a discretization of curvature line parametrized surfaces in Lie geometry. We find a discretization of curvature line parametrization which unifies the circular and conical nets by systematically applying the Discretization Principles.Comment: 57 pages, 18 figures; In the second version the terminology is slightly changed and umbilic points are discusse

Optimal Light Beams and Mirror Shapes for Future LIGO Interferometers

We report the results of a recent search for the lowest value of thermal noise that can be achieved in LIGO by changing the shape of mirrors, while fixing the mirror radius and maintaining a low diffractional loss. The result of this minimization is a beam with thermal noise a factor of 2.32 (in power) lower than previously considered Mesa Beams and a factor of 5.45 (in power) lower than the Gaussian beams employed in the current baseline design. Mirrors that confine these beams have been found to be roughly conical in shape, with an average slope approximately equal to the mirror radius divided by arm length, and with mild corrections varying at the Fresnel scale. Such a mirror system, if built, would impact the sensitivity of LIGO, increasing the event rate of observing gravitational waves in the frequency range of maximum sensitivity roughly by a factor of three compared to an Advanced LIGO using Mesa beams (assuming all other noises remain unchanged). We discuss the resulting beam and mirror properties and study requirements on mirror tilt, displacement and figure error, in order for this beam to be used in LIGO detectors.Comment: 9 pages, 11 figure

A Step-by-step Guide to the Realisation of Advanced Optical Tweezers

Since the pioneering work of Arthur Ashkin, optical tweezers have become an indispensable tool for contactless manipulation of micro- and nanoparticles. Nowadays optical tweezers are employed in a myriad of applications demonstrating the importance of these tools. While the basic principle of optical tweezers is the use of a strongly focused laser beam to trap and manipulate particles, ever more complex experimental set-ups are required in order to perform novel and challenging experiments. With this article, we provide a detailed step- by-step guide for the construction of advanced optical manipulation systems. First, we explain how to build a single-beam optical tweezers on a home-made microscope and how to calibrate it. Improving on this design, we realize a holographic optical tweezers, which can manipulate independently multiple particles and generate more sophisticated wavefronts such as Laguerre-Gaussian beams. Finally, we explain how to implement a speckle optical tweezers, which permit one to employ random speckle light fields for deterministic optical manipulation.Comment: 29 pages, 7 figure

Elliptical beams

A very general beam solution of the paraxial wave equation in elliptic cylindrical coordinates is presented. We call such a field an elliptic beam (EB). The complex amplitude of the EB is described by either the generalized Ince functions or the Whittaker-Hill functions and is characterized by four parameters that are complex in the most general situation. The propagation through complex ABCD optical systems and the conditions for square integrability are studied in detail. Special cases of the EB are the standard, elegant, and generalized Ince-Gauss beams, Mathieu-Gauss beams, among others