Lommel Functions Research Articles

Diffraction-free lommel beams

We consider a new family of nonparaxial diffraction-free laser beams with their complex amplitude being proportional to the n-th order Lommel function of two variables. Therefore, these beams are called Lommel beams (L-beams). We obtained explicit analytical expressions for the angular spectrum of plane waves and for the orbital angular momentum of the L-beams. Transverse intensity of the L-beams has a reflective symmetry with respect to both Cartesian coordinate axes. Since transverse intensity distribution of L-beams does not change upon propagation, L-beams are modes of free space (L-modes). Functions of complex amplitudes of even (n = 2p) and odd (n = 2p + 1) L-modes are mutually orthogonal. For certain parameter, L-modes become traditional Bessel modes.

Magnetoelastic Field of a Multilayered and Functionally Graded Cylinder With a Dynamic Polynomial Eigenstrain

In this paper, an analytical solution is obtained for the magnetoelastic response of a multilayered and functionally graded cylinder with an embedded dynamic polynomial eigenstrain. The internal core of the cylinder endures a harmonic eigenstrain of cubic polynomial distribution along the radial direction. Both plane strain and plane stress conditions are assumed for the axisymmetric cylinder. The composite cylinder is placed in a constant magnetic field parallel to its axis. The magnetoelastic governing equations are solved exactly and the displacement and stress components are obtained in terms of Bessel, Struve, and Lommel functions. Using the analytical solution for the multilayered, composite cylinder, the magnetoelastic response of a functionally graded cylinder with exponential and power law distribution of material properties is investigated. Finally, the numerical results reveal the effects of external magnetic field, eigenstrain, and nonhomogeneity indices on the magnetoelastic response of the heterogeneous cylinders.

The inverse Laplace transform of the modified Lommel functions

Particular solutions of the inhomogeneous Bessel differential equation are the Lommel functions, usually denoted as s μ, ν(x), S μ, ν(x), and S μ, ν(x). The inhomogeneous modified Bessel differential equation has the particular solutions t μ, ν(x)=−i1−μ s μ, ν(ix), T μ, ν(x)=−i1−μ S μ, ν(ix), and T μ, ν(x)=−i1−μ S μ, ν(ix). For the modified Lommel functions T μ, ν(x) and T μ, ν(x) as well as T μ, ν(√x) and T μ, ν(√x), we give the inverse Laplace transform for μ=−1 and μ=0. For μ<−1, the inverse Laplace transform can be obtained from a recurrence relation.

On a Unification of Generalized Mittag-Leffler Function and Family of Bessel Functions

In the present work, a unification of certain functions of mathematical physics is proposed and its properties are studied. The proposed function unifies Lommel function, Struve function, the Bessel-Maitland function and its generalization, Dotsenko function, generalized Mittag-Leffler function etc. The properties include absolute and uniform convergence, differential recurrence relation, integral representations in the form of Euler-Beta transform, Mellin-Barnes transform, Laplace transform and Whittaker transform. The special cases namely the generalized hypergeometric function, generalized Laguerre polynomial, Fox H-function etc. are also obtained.

Coherence properties of Stokes beams

A Stokes beam is a light beam which is expressible as a superposition of a completely polarized and a completely unpolarized beam. In this note we study coherence properties of a wide class of beams of this kind. The behaviors of the degree of coherence of a Stokes beam produced by a circular planar source are expressed in terms of Lommel functions and theoretical results are illustrated by computed examples.

On complex oscillation, function-theoretic quantization of non-homogeneous periodic ordinary differential equations and special functions

New necessary and sufficient conditions are given for the quantization of a class of periodic second-order non-homogeneous ordinary differential equations in the complex plane. The problem is studied from the viewpoint of complex oscillation theory first developed in works by Bank and Laine and Gundersen and Steinbart. We show that, when a solution is complex non-oscillatory (finite exponent of convergence of zeros), then the solution, which can be written as special functions, must degenerate. This gives a necessary and sufficient condition when the Lommel function has finitely many zeros in every branch, and this is a type of quantization for the non-homogeneous differential equation. The degenerate solutions are polynomial/rational-type functions, which are of independent interest. In particular, this shows that complex non-oscillatory solutions of this class of differential equations are equivalent to the subnormal solutions considered in a recent paper by Chiang and Yu. In addition to the asymptotics of special functions, the other main idea that we apply in our proof is a classical result by Wright that gives precise asymptotic locations of large zeros of a functional equation.

Diffraction and order conversion of Bessel beams in uniaxial crystals

The order of a Bessel light beam propagating along the optical axes of a uniaxial crystal can be changed by two due to the influence of crystal anisotropy on beam propagation. This effect occurs because of spin-orbit coupling in crystal-traveling optical field and it is interesting for generation of high-order Bessel beams possessing optical vortices. If the input Bessel beam is apertured, the order-conversion may be distorted by diffraction spreading. A description of polarization and spatial structure evolution in crystal can be accomplished using propagation equations, which are obtained in the paper by using integral Hankel transform and solutions of Maxwell's equations for Bessel beams. The simple form of propagation equations are also derived using the Lommel functions. As an example, the order-conversion of strongly apertured zero-order Bessel beam is studied.

Exact solutions for dynamic response of a periodic spring and mass structure

This paper derives exact analytical solutions for the dynamic response of a periodic structure for which the unit cell consists of one mass and one spring. The solutions cover arbitrary initial conditions and both polynomial and harmonic external forces. They involve Bessel, Lommel, Weber and hypergeometric functions.

A generalized hypergeometric function method for axisymmetric vibration analysis of a piezoelectric actuator

Axisymmetrical piezoelectric vibration is a classical problem. Generally, the Lommel function method is used to solve this problem, so, the conventional electro-mechanical solutions take the form of infinite series, which converge relatively slowly and are a bit unsuitable for engineering application. For the first time, the present work constructs the exact solutions to axisymmetric electro-mechanical fields of transversely isotropic piezoceramics in terms of generalized hypergeometric functions instead, which can help to avoid the calculation of infinite series to a large extent. Though the mathematical essence of the present solutions is the same as that of the infinite series solutions, the present expression form and numerical computation are more straightforward and thus more suitable for engineering applications. As an example, the current solutions are used to analyze the axisymmetric vibration of a radially polarized cylindrical piezoelectric actuator installed on a metal shaft. Numerical results of the electrical and mechanical resonant fundamental frequencies are verified in the special case of a single hollow piezoelectric cylinder. In addition, the effects of interfacial imperfection and the portion of piezoelectric layer are also discussed.

Tight Bounds for Ergodic Capacity of Dual-Hop Fixed-Gain Relay Networks under Rayleigh Fading

Since a closed-form expression for the exact ergodic capacity of dual-hop fixed-gain relay networks is not mathematically tractable, thus we will present tight closed-form bounds for the ergodic capacity in Rayleigh fading channels. First we express the exact ergodic capacity in terms of a Lommel function and a single integral and then we apply integral inequalities to present one upper bound and two lower bounds on the ergodic capacity. We also show that our closed-form bounds match perfectly with Monte-Carlo simulations, particularly for moderate and high average signal-to-noise ratio (SNR) of the first hop.

Inviscid critical and near-critical reflection of internal waves in the time domain

A solution that describes the inviscid reflection of internal waves off a sloping bottom in time is derived under conditions of linearity and uniform stratification. The solution is well behaved even under critical conditions. In the region ky &lt; Nt, where k is the along-slope wavenumber of the incoming wave, y is the slope-normal direction, N is the Brünt–Väisälä frequency and t is time, an approximation can be written in terms of Lommel's function of two variables. The analysis can easily be extended to the case of a beam of finite width. In the non-critical case, the streamfunction relaxes to the classical Phillips steady-state solution in the region y &lt; cgt, where cg is the slope-normal component of the group velocity for waves at the forcing frequency. However, it is found that the region where the along-boundary component of the velocity relaxes to the Phillips solution is also bounded from below, leaving a region very close to the wall where the classical solution misses important elements of the reflectionprocess. This leads to interesting properties near the boundary, especially relatively to the formation of shear-driven unstable conditions.

Integral representations for the exceptional univariate Lommel functions

An elementary integral representation for the Lommel function S−1, 0(z) is given and extended to other exceptional cases.

An efficient algorithm of the electromagnetic dyadic Green’s function in a horizontal-layered anisotropic medium

In this paper, we advance an efficient algorithm of electromagnetic spatial dyadic Greens function in a horizontal-layered anisotropic medium through high order window function and so on. First, we use the high order window function to transform the Sommerfeld integrals of dyadic Greens function into integrals along a generalized steep descent path. And we give a new and higher order of power series of expansion expression of Hankel transforms of the window function and an accurate formula of Lommel function so that the dyadic Greens function can be precisely computed either near or far from the transmitter. Then, we divide the integral path into a series of subintervals based on the zeros of Bessel function and compute integral per subinterval using an improved adaptive Gauss quadrature. Furthermore, an algorithm of the finite continued fraction expansion is used to sum up the integrals on each subinterval to greatly accelerate the convergence velocity of the numerical integration along the descent path. Finally, our numerical results validated the efficiency of the algorithm.

Analytical benchmarks for topological optimization IV: square-shaped line support

Michell’s problem of optimizing truss topology for stress or compliance constraints under a single load condition is solved analytically for plane trusses having a square-shaped line support. Geometrical characteristics of the Hencky nets giving the truss layout are expressed in terms of Lommel functions. Analytically derived truss volumes for the above problem are compared with those of trusses supported along circles of equivalent area. Some general implications of the results are also discussed.

Exact Closed-Form Expression of a Sommerfeld Integral for the Impedance Plane Problem

A Sommerfeld integral for an impedance half-plane is one of the classical problems in electromagnetic theory. In this paper, the integral is evaluated into two series representations, which are expressed in terms of the exponential integral and the Lommel function, respectively. It is shown that the series expansions absolutely converge for any parameters, such as distance or surface impedance. Then based on the Lommel function expansion, an exact closed-form expression of the integral is formulated. This expression is written in terms of incomplete Weber integrals, which are directly related to incomplete cylindrical functions such as incomplete Lipschitz-Hankel integrals. Additionally, a complete asymptotic series of the integral is obtained based on the exponential integral expansion. With conventional methods such as steepest descent method, it is cumbersome to derive the divergent series. The validity of all the formulations derived in this paper is demonstrated through a comparison with a numerical integration of the integral for various situations

Transient excitation of linear space-charge waves by a punctual source in a drifting cold plasma

We study the effects of a uniform drift on the excitation emanating from an external antenna immersed in a cold modelled plasma. Instead of the well-known approach based on the steady-state regime approximation, the causal and transient evolution of the excited plasma waves is investigated. A comprehensive description of the computational method is proposed. We show that a technique based on the Heaviside direct operational method turns out to be very effective in order to handle the integrodifferential expression for the solution. Making use of a pure analytical calculation, an exact algebraic expression for the plasma response is inferred. The solution takes the form of a combination of Bessel's functions and series of Lommel's functions. Then some specific analysis of the result allow us to gain more understanding on the dynamics of the fast and slow space-charge waves. A special attention is paid to the analysis of the extended singularity of the wave field and the secular behaviour of the fast wave mode at the resonant excitation.

Intensity distributions near focus of strongly converging spherical waves diffracted at a circular aperture

Based on the Rayleigh–Sommerfeld diffraction integral, the diffraction of converging spherical waves at a circular aperture is studied in a general case. The expression for the intensity near focus of strongly converging spherical diffracted waves is derived, which reduces to the well known result expressed in terms of Lommel functions for the case of weakly converging spherical diffracted waves. The intensity distributions at the geometrical focal plane and along the axis are given. Numerical comparative examples are presented to illustrate the more general applicability of our results.

Focusing of truncated Gaussian beams

It is shown that the focusing of truncated Gaussian beams can be treated by the same manner as uniform spherical waves, i.e., the diffraction integral can be expressed by the Lommel functions, which offers a very efficient way for the calculation of the three-dimensional light distribution near focus. All the expressions for the uniform spherical waves hold good for Gaussian beams if the first variable in the Lommel functions is extended to the complex domain. The intensity distribution depending on the Fresnel number and the truncation coefficient is calculated. The location of the first few minima and maxima of the intensity in focal plane is given for different values of the truncation coefficient. The phase behavior depending on the truncation coefficient is studied.

Accurate efficient evaluation of Lommel functions for arbitrarily large arguments

It can be difficult to evaluate Lommel functions Un(u,v) and Vn(u,v) when u and v are both large. Here, we present an accurate means to evaluate such functions using contour integration. The approach suggests the potential for developing analogous techniques in diffraction calculations for many optical systems.

INTEGRAL TRANSFORMATIONS BY THE INDEX OF LOMMEL'S FUNCTION

In this paper we study a class of integral transformations with kernels consisting of Lommel's functions of a certain index. Boundedness properties, Parseval type formula, inversion theorem are proved in Lebesgue spaces of integrable functions.