Search results for "Bessel function"
showing 10 items of 61 documents
Stealth dicing with ultrafast Bessel beams with engineered transverse profiles
2017
International audience; We investigate high-speed glass cleaving with ultrafast laser beams with engineered transverse intensity profile. We achieve accuracy of ~ 1 µm at 25 mm/s and drastically enhance cleavability compared to standard Bessel beams.
Crack formation and cleaving of sapphire with ultrafast bessel beams
2017
Sapphire is a transparent crystalline dielectric of high hardness with many important applications, specifically to the next-generation touchscreens and to the LED growth, as substrates. However, sapphire cutting by ablative techniques is rather slow therefore fast material separation techniques are needed. Material separation by “stealth dicing” has been recently developed, it is based on material cleaving along a plane weakened by multiple ultrafast laser illuminations. This allows usually generating taper-free cutting and avoids material loss. However, the illuminated plane needs small spacing between the shot to shot (typically a few μm) and long damages inside the bulk. This requires l…
Exact Fourier expansion in cylindrical coordinates for the three-dimensional Helmholtz Green function
2009
A new method is presented for Fourier decomposition of the Helmholtz Green Function in cylindrical coordinates, which is equivalent to obtaining the solution of the Helmholtz equation for a general ring source. The Fourier coefficients of the Helmholtz Green function are split into their half advanced+half retarded and half advanced-half retarded components. Closed form solutions are given for these components in terms of a Horn function and a Kampe de Feriet function, respectively. The systems of partial differential equations associated with these two-dimensional hypergeometric functions are used to construct a fourth-order ordinary differential equation which both components satisfy. A s…
Frames and weak frames for unbounded operators
2020
In 2012 G\u{a}vru\c{t}a introduced the notions of $K$-frame and of atomic system for a linear bounded operator $K$ in a Hilbert space $\mathcal{H}$, in order to decompose its range $\mathcal{R}(K)$ with a frame-like expansion. In this article we revisit these concepts for an unbounded and densely defined operator $A:\mathcal{D}(A)\to\mathcal{H}$ in two different ways. In one case we consider a non-Bessel sequence where the coefficient sequence depends continuously on $f\in\mathcal{D}(A)$ with respect to the norm of $\mathcal{H}$. In the other case we consider a Bessel sequence and the coefficient sequence depends continuously on $f\in\mathcal{D}(A)$ with respect to the graph norm of $A$.
New indefinite integrals from a method using Riccati equations
2018
ABSTRACTAn earlier method for obtaining indefinite integrals of special function from the second-order linear equations which define them has been reformulated in terms of Riccati equations, which ...
Further monotonicity and convexity properties of the zeros of cylinder functions
1992
AbstractLet cvk be the kth positive zero of the cylinder function Cv(x,α)=Jv(x) cos α−Yv sin α, 0⩽α<π, where Jv(x) and Yv(x) are the Bessel functions of the first and the second kind, respectively. We prove that the function v(d2cvkddv2+δ)cvk increases with v⩾0 for suitable values of δ and k−απ⩾ 0.7070… . From this result under the same conditions we deduce, among other things, that cvk+12δv2 is convex as a function of v⩾0. Moreover, we show some monotonicity properties of the function c2vkv. Our results improve known results.
Jaynes-Cummings model with atomic position distribution
1995
The position as well as the width of the atomic wave packet passing through an optical cavity may affect the matter-field interaction. As a result, the internal dynamics of a two-level atom, such as the Rabi oscillations or the collapse and revival phenomenon, may be strongly modified with respect to the standard Jaynes-Cummings model. In particular, for a sufficiently large spread of the atomic position, the atomic population inversion displays the characteristic ringing behavior of the Bessel function ${\mathit{J}}_{0}$ differently from the usual full Rabi oscillation.
Indefinite integrals of special functions from hybrid equations
2019
Elementary linear first and second order differential equations can always be constructed for twice differentiable functions by explicitly including the function's derivatives in the definition of ...
Indefinite integrals of Lommel functions from an inhomogeneous Euler–Lagrange method
2015
ABSTRACTA method given recently for deriving indefinite integrals of special functions which satisfy homogeneous second-order linear differential equations has been extended to include functions which obey inhomogeneous equations. The extended method has been applied to derive indefinite integrals for the Lommel functions, which obey an inhomogeneous Bessel equation. The method allows integrals to be derived for the inhomogeneous equation in a manner which closely parallels the homogeneous case, and a number of new Lommel integrals are derived which have well-known Bessel analogues. Results will be presented separately for other special functions which obey inhomogeneous second-order linear…
Subwavelength Bessel beams in wire media
2013
Recent progress is emerging on nondiffracting subwavelength fields propagating in complex plasmonic nanostructures. In this paper, we present a thorough discussion on diffraction-free localized solutions of Maxwell’s equations in a periodic structure composed of nanowires. This self-focusing mechanism differs from others previously reported, which lie on regimes with ultraflat spatial dispersion. By means of the Maxwell–Garnett model, we provide a general analytical expression of the electromagnetic fields that can propagate along the direction of the cylinder’s axis, keeping its transverse waveform unaltered. Numerical simulations based on the finite element method support our analytical a…