Geometry of rank 2 distributions with nonzero Wilczynski invariants

In the famous 1910 “cinq variables” paper Cartan showed in particular that for maximally nonholonomic rank 2 distributions in ?⁵ with non-zero covariant binary biquadratic form the dimension of the pseudo-group of local symmetries does not exceed 7 and among such distributions he described the one-parametric family of distributions for which this pseudo-group is exactly 7-dimensional. Using the novel interpretation of the Cartan covariant binary biquadratic form via the classical Wilczynski invariant of curves in projective spaces associated with abnormal extremals of the distributions [4, 27, 28] one can generalize this Cartan result to rank 2 distributions in ?ⁿ satisfying certain genericity assumption, called maximality of class, for arbitrary n ≥ 5.

In the present paper for any rank 2 distribution of maximal class with at least one nonvanishing generalized Wilczynski invariants we construct the canonical frame on a (2n — 3)-dimensional bundle and describe explicitly the moduli spaces of the most symmetric models. The relation of our results to the divergence equivalence of Lagrangians of higher order is given as well.     

Stable bright and vortex solitons in photonic crystal fibers with inhomogeneous defocusing nonlinearity

We predict that a photonic crystal fiber whose strands are filled with a defocusing nonlinear medium can support stable bright solitons and also vortex solitons if the strength of the defocusing nonlinearity grows toward the periphery of the fiber. The domains of soliton existence depend on the transverse growth rate of the filling nonlinearity and nonlinearity of the core. Remarkably, solitons exist even when the core material is linear.     

Optical vortex detector as a basis for a data transfer system: Operational principle,model, and simulation of the influence of turbulence and noise

The principle of determining the topological charge of an optical vortex is suggested based on measuring the light field intensity and designing the corresponding detector. A mathematical model of the performance of the detector of topological vortex charge is presented. Results of numerical experiments imitating the vortex recognition in the presence of turbulence or (amplitude or phase) noise in registered radiation as well as of the displacement of the optical beam source and detector axes are presented. Principles are formulated of designing the position finder for an optical vortex (that is, the detector of vortex coordinates) that allows us to consider its realization in the form of mathematical and numerical model. Conditions of reliable operation of the vortex detector and singular optical communication line constructed on its basis are estimated. Dependencies of the probability of error in data transfer on the turbulence intensity, photodetector noise amplitude, and displacement of the optical axes are investigated for different coding algorithms (absolute and differential with fixed or adaptive threshold). The data of modeling confirm the results of analytical calculations.     

On the Structure of Eigenfunctions Corresponding to Embedded Eigenvalues of Locally Perturbed Periodic Graph Operators

The article is devoted to the following question. Consider a periodic self-adjoint difference (differential) operator on a graph (quantum graph) G with a co- compact free action of the integer lattice $\mathbb{Z}^{n}The article is devoted to the following question. Consider a periodic self-adjoint difference (differential) operator on a graph (quantum graph) G with a co- compact free action of the integer lattice . It is known that a local perturbation of the operator might embed an eigenvalue into the continuous spectrum (a feature uncommon for periodic elliptic operators of second order). In all known constructions of such examples, the corresponding eigenfunction is compactly supported. One wonders whether this must always be the case. The paper answers this question affirmatively. What is more surprising, one can estimate that the eigenmode must be localized not far away from the perturbation (in a neighborhood of the perturbation’s support, the width of the neighborhood dependent upon the unperturbed operator only). The validity of this result requires the condition of irreducibility of the Fermi (Floquet) surface of the periodic operator, which is known in some cases and is expected to be satisfied for periodic Schr?dinger operators.     

Polarization-controlled switching between diffraction orders in transverse-periodically aligned nematic liquid crystals

Transverse-periodic-oriented nematic liquid crystals (LCs) are a special type of optical axis grating that are capable of very high efficiency diffraction (theoretically, 100%) in thin layers of materials with thickness comparable to the radiation wavelength. In particular, they fully diffract linearly polarized input beam into circularly polarized +1st and -1st diffraction orders. We experimentally demonstrate switching between diffraction orders of such gratings when the polarization of the incident beam changes from right-circular to left-circular and vice versa with the aid of an electrically controlled LC phase retarder. Such a setup in which the diffraction efficiency and direction are controlled externally, without application of an electric field to the transverse-periodic grating, provides additional control opportunities and does not compromise the quality of the grating. The grating used in the experiment was 1.5 microm thick and had a period of 4 microm. The contrast ratio of switching between the +1st and -1st orders was as high as 267:1 for a He-Ne laser beam with a switching time of 6.6 ms.     

Effects of coupled vibrations on the parameters of tangentially polarized stripe-electroded piezoelectric cylinders (L)

The effects of coupled vibrations on the piezoelectric properties and electroacoustic transducer performance of radially polarized hollow cylindrical elements as a function of the choice of height-to-diameter ratio have been well documented [J. Acoust. Soc. Am. 120(3), 1374-1380 (2006); J. Acoust. Soc. Am. 122(6), 3419-3427 (2007)]. This letter presents experimental work on the effects of coupled (circumferential, axial, and flexural) vibrations on the resonance frequencies, effective electromechanical coupling coefficient, and acoustical performance of tangentially polarized piezoceramic cylindrical elements. Comparisons are made with analytical predictions of the properties of uniformly circumferentially polarized cylinders.     

Stability of solitons in parity-time-symmetric couplers

Families of analytical solutions are found for symmetric and antisymmetric solitons in a dual-core system with Kerr nonlinearity and parity-time (PT)-balanced gain and loss. The crucial issue is stability of the solitons. A stability region is obtained in an analytical form, and verified by simulations, for the PT-symmetric solitons. For the antisymmetric ones, the stability border is found in a numerical form. Moving solitons of both types collide elastically. The two soliton species merge into one in the "supersymmetric" case, with equal coefficients of gain, loss, and intercore coupling. These solitons feature a subexponential instability, which may be suppressed by periodic switching ("management").     

Unveiling the correlation between non‐diffracting tractor beam and its singularity in Poynting vector

This paper investigates the singular optics of nonparaxial light beams in the near field when the light behaves as a tractor beam. New insights into the optical pulling force, which is usually represented by integrating the stress tensor at a black box enclosing the object, are interpreted by the optical singularity of the Poynting vector. The negative nonconservative pulling force originates from the transfer of the azimuthal Poynting vector to the longitudinal component partly owing to the presence of a scatterer. The separatrice pattern and singularity shifts of the Poynting vector unanimously exhibit a differentiable near‐field distribution in the presence of optical pulling force. A new method is established to calculate the near‐field optical force using the differential Poynting vector in the far field. The results obtained provide a clear physical interpretation of the light–matter interaction and manifest the significance of singular optics in manipulating objects.

     

Unitary representations of superconformal algebra

superconformal algebra is algebra with two Virasoro operators. The Kac determinant is calculated and the complete list of unitary representations is determined. Two types of extensions of algebra are discussed. A new approach to construction of algebras from rational conformal field theories is proposed.     

Gap soliton dynamics in a nonuniform resonant structure

A model for the propagation of coherent pulses along a one-dimensional, resonantly absorbing Bragg grating that includes localized inhomogeneous population inversion at its center is presented. The long-range coupling between the optical field and resonant atoms allows for controllable trapping of a gap soliton by the local inversion, thus opening new opportunities for control of signal transmission and localization of light.