Level-spacing distributions beyond the Wigner surmise

We introduce a dynamical system, for which it is possible to get such a large number of eigenvalues that deviations from Wigner's surmise are visible. The obtained level-spacing distribution agrees much better with the distribution derived from random matrix theory.     

Level spacing distributions and the Bessel kernel

Scaling models of randomN×N hermitian matrices and passing to the limitN→∞ leads to integral operators whose Fredholm determinants describe the statistics of the spacing of the eigenvalues of hermitian matrices of large order. For the Gaussian Unitary Ensemble, and for many others'as well, the kernel one obtains by scaling in the “bulk” of the spectrum is the “sine kernel” $\frac{{\sin \pi (x - y)}}{{\pi (x - y)}}$ . Rescaling the GUE at the “edge” of the spectrum leads to the kernel $\frac{{Ai(x)Ai'(y) - Ai'(x)Ai(y)}}{{x - y}}$ , where Ai is the Airy function. In previous work we found several analogies between properties of this “Airy kernel” and known properties of the sine kernel: a system of partial differential equations associated with the logarithmic differential of the Fredholm determinant when the underlying domain is a union of intervals; a representation of the Fredholm determinant in terms of a Painlevé transcendent in the case of a single interval; and, also in this case, asymptotic expansions for these determinants and related quantities, achieved with the help of a differential operator which commutes with the integral operator. In this paper we show that there are completely analogous properties for a class of kernels which arise when one rescales the Laguerre or Jacobi ensembles at the edge of the spectrum, namely $$\frac{{J_\alpha (\sqrt x )\sqrt y J'_\alpha (\sqrt y ) - \sqrt x J'_\alpha (\sqrt x )J_\alpha (\sqrt y )}}{{2(x - y)}},$$ , whereJ _α(z) is the Bessel function of order α. In the cases α=?1/2 these become, after a variable change, the kernels which arise when taking scaling limits in the bulk of the spectrum for the Gaussian orthogonal and symplectic ensembles. In particular, an asymptotic expansion we derive will generalize ones found by Dyson for the Fredholm determinants of these kernels.     

Level-spacing statistics for the Anderson model in one and two dimensions

We investigate the energy level-spacing distribution of the discrete one and two-dimensional Anderson model. The numerical results suggest that, for any finite disorder, the level spacing is Poisson distributed.     

The optical Airy transform and its application in generating and controlling the Airy beam

We propose an optical Airy transform in this paper, and obtain the analytical expressions for the Airy transform of fundamental Gaussian beams and finite energy Airy beams. The setup for performing the optical Airy transform is presented. The Airy transform for Gaussian beams and finite energy Airy beams are theoretically calculated and analyzed. Our results show that the Airy beam can be conveniently generated and controlled through the optical Airy transform of the Gaussian beam. The optical Airy transform also can be used to directly modulate the beam parameters of the incident Airy beam, and it can transform the incident Airy beam into the Gaussian beam.     

How to get Airy disc from Airy pattern

Airy pattern is formed by diffraction of a plane wave through a circular aperture in the far field with certain side lobes. It is shown that Airy pattern can be truncated at the first dark ring by a circular aperture to pick up the center spot (called Airy disc) without causing additional diffraction fringes, even in the very far field. This may be called zero field truncation technique. And is very useful in practice, for example, as a good quality optical source.     

Airy beam laser

A method to design lasers that emit an arbitrary beam profile is studied. In these lasers, output-coupling is performed by a diffraction grating that imposes a phase and amplitude distribution onto the diffracted light. A solid-state laser emitting beams with a two-dimensional Airy intensity profile is demonstrated both theoretically and experimentally. In this case, the diffraction grating adds a transverse cubic phase to the diffracted light. An Airy beam is obtained by performing optical Fourier transform of the out-coupled light. The laser beam profile and power characteristics are shown to agree with theory.     

Airy beams beyond the paraxial approximation

The behavior of the vectorial Airy beams beyond the paraxial approximation is investigated. Indeed, closed-form (even though non exact) expressions for the electric components of the fields generated by the same boundary conditions, which should pertain to the scalar Airy beams, are obtained on the basis of the vectorial Rayleigh-Sommerfeld diffraction integrals under suitable approximations. Such expressions may accompany more complete approaches, like that in Opt. Expr. 17, 22432 (2009), where a fully numerical analysis of the propagation of exponentially smoothed Airy beams has been presented, faithfully reproducing the conditions of their experimental demonstration as reported in Phys. Rev. Lett. 99, 213907 (2007). Comments on other well known approaches to the investigation of the nonparaxial propagation of definite paraxial beams are also given.     

Specular and vortical Airy beams

A new kind of truncated Airy beams is investigated and discussed. These beams are a superposition of shifted and truncated Airy functions and its specular counterparts, where zeroes or extremal points of the Airy function are chosen as a truncation point. The specular Airy beams are smooth at the truncation point and produce a diffraction pattern similar to Hermite-Gaussian modes. Under propagation in Fresnel zone, specular Airy beams demonstrate a symmetrical acceleration in opposite sides and the beam divergence is proportional to the traveled distance squared. The astigmatic mode converter transforms a two-dimensional specular Airy beam into a quasi-annular field with a nonzero orbital angular momentum. Vortical Airy beams based on truncated Airy functions are also discussed. These beams are similar to Laguerre-Gaussian modes, while their annular structure is changed during propagation.     

On Asymptotics for the Airy Process

The Airy process t→A(t), introduced by Prähofer and Spohn, is the limiting stationary process for a polynuclear growth model. Adler and van Moerbeke found a PDE in the variables s ₁,s ₂, and t for the probability Pr(A(0)≤s ₁, A(t)≤s ₂). Using this they were able, assuming the truth of a certain conjecture and appropriate uniformity, to obtain the first few terms of an asymptotic expansion for this probability as t→∞, with fixed s ₁ and s ₂. We shall show that the expansion can be obtained by using the Fredholm determinant representation for the probability. The main ingredients are formulas obtained by the author and C. A. Tracy in the derivation of the Painlevé II representation for the distribution function F ₂ plus a few others obtained in the same way.     

The Airy transform and associated polynomials

The Airy transform is an ideally suited tool to treat problems in classical and quantum optics. Even though the relevant mathematical aspects have been thoroughly investigated, the possibilities it offers are wide and some features, such as the link with special functions and polynomials, still contain unexplored aspects. In this note we will show that the so called Airy polynomials are essentially the third order Hermite polynomials. We will also prove that this identification opens the possibility of developing new conjectures on the properties of this family of polynomials.     

Airy Functions in the Thermodynamic Bethe Ansatz

Thermodynamic Bethe ansatz equations are coupled nonlinear integral equations which appear frequently when solving integrable models. Those associated with models with N=2 supersymmetry can be related to differential equations, among them Painlevé III and the Toda hierarchy. In the simplest such case, the massless limit of these nonlinear integral equations can be solved in terms of the Airy function. This is the only known closed-form solution of thermodynamic Bethe ansatz equations, outside of free or classical models. This turns out to give the spectral determinant of the Schrödinger equation in a linear potential.     

Reshaping the trajectory and spectrum of nonlinear Airy beams

We demonstrate theoretically and experimentally that a finite Airy beam changes its trajectory while maintaining its acceleration in nonlinear photorefractive media. During this process, the spatial spectrum reshapes dramatically, leading to negative (or positive) spectral defects on the initial spectral distribution under a self-focusing (or defocusing) nonlinearity.     

Reduced side-lobe Airy beams

The primary interest in the finite-energy Airy beam derives from the special properties possessed by the maximum intensity at its central lobe. However, the defining spatial dependence, the Airy function, is an oscillatory function that consists of decaying side lobes. For some applications these side lobes may create deleterious effects. Fortunately, a nonsymmetric apodization of the beam in Fourier space is shown to enhance the central lobe as the side lobes are reduced and clipped. The properties of the central lobe are unaffected over a large proportion of the propagation path.     

Scale Invariance of the PNG Droplet and the Airy Process

We establish that the static height fluctuations of a particular growth model, the PNG droplet, converges upon proper rescaling to a limit process, which we call the Airy process A(y). The Airy process is stationary, it has continuous sample paths, its single time (fixed y) distribution is the Tracy–Widom distribution of the largest eigenvalue of a GUE random matrix, and the Airy process has a slow decay of correlations as y ^–2. Roughly the Airy process describes the last line of Dyson's Brownian motion model for random matrices. Our construction uses a multi-layer version of the PNG model, which can be analyzed through fermionic techniques. Specializing our result to a fixed value of y, one reobtains the celebrated result of Baik, Deift, and Johansson on the length of the longest increasing subsequence of a random permutation.     

Ballistic dynamics of Airy beams

We demonstrate both theoretically and experimentally that optical Airy beams propagating in free space can perform ballistic dynamics akin to those of projectiles moving under the action of gravity. The parabolic trajectories of these beams as well as the motion of their center of gravity were observed in good agreement with theory. The possibility of circumventing an obstacle placed in the path of the Airy beam is discussed.     

Low-loss Airy surface plasmon polaritons

We propose a scheme to obtain a low-loss propagation of Airy surface plasmon polaritons(SPPs) along the interface between a dielectric and a negative-index metamaterial(NIMM). We show that by using the transverse-magnetic mode and the related destructive interference effect between electric and magnetic absorption responses, the propagation loss of the Airy SPPs can be largely suppressed when the optical frequency is close to the lossless point of the NIMM. As a result, the Airy SPPs obtained in our scheme can propagate more than a 6times longer distance than that in conventional dielectric–metal interfaces.     

Product of three Airy beams

A two-dimensional field that is a product of three Airy beams is proposed and investigated. It is shown that the Fourier image of this field has a cubic phase and a radially symmetric intensity with a super-Gaussian decrease. Propagation of the product of three Airy beams in a Fresnel zone is investigated numerically.