Airy Functions Demystified — I

Resonance ◽  
2021 ◽  
Vol 26 (5) ◽  
pp. 629-647
Author(s):  
M. S. Ramkarthik ◽  
Elizabeth Louis Pereira
Keyword(s):  
Airy Functions

Derivation of tunneling probabilities for arbitrarily graded potential barriers using modified Airy functions

2010 ◽  
Vol 42 (2) ◽  
pp. 129-141
Author(s):  
Kyu-Tae Lee ◽  
Eun Joo Jung ◽  
Chul Han Kim ◽  
Chang-Min Kim
Keyword(s):  
Airy Functions ◽  
Potential Barriers ◽  
Modified Airy Functions

Analytical account for off-axis effects in X-ray radiation of free electron lasers

2021 ◽  
Vol 64 (1) ◽  
pp. 21-28
Author(s):  
K.V. Zhukovsky ◽  
Keyword(s):  
Free Electron ◽  
Finite Size ◽  
Photon Radiation ◽  
Beam Parameter ◽  
Beam Size ◽  
Free Electron Lasers ◽  
Airy Functions ◽  
X Ray ◽  
Harmonic Radiation ◽  
Gain Length

We give analytical description of generation of harmonics of the undulator radiation (UR) with account for the finite electron beam size, emittance, off-axis beam deviation and electron energy spread, as well as for the constant magnetic components and field harmonics effects. We give exact analytical expressions for the generalized Bessel and Airy functions, which describe the spectrum line shape and intensities in the two-frequency bi-harmonic undulator with account for the above factors. The obtained analytical formulae distinguish contributions of each field component and every undulator and beam parameter on the harmonic radiation in free electron lasers (FEL). The effect of the field on the harmonic radiation is analyzed with account for the beam finite size and its off-axis deviation. The phenomenological model is employed for the FEL modeling; with its help we study the harmonic generation, including even ones, in the experiments LCLS and LEUTL. We demonstrate analytically that strong second FEL harmonic in X-ray FEL at the wavelengths λ = 0.75nm in the LCLS experiment is caused by the deviation of the electron trajectories off the axis in 15 μm on the gain length 1.6 m, which is comparable with the beam size; the strong second FEL harmonic in the LEUTL experiment at the wavelength λ = 192nm can be attributed to interaction of the electrons in wide, ~ 0.2 mm, beam with the photon radiation at the gain length 0.87 m. The modeling results fully agree with the measurements. The developed formalism allows the analysis of projected and built FELs and their radiation, helps minimizing losses and correcting magnetic fields; it also shows physical background and reasons for each harmonic radiated power in the FEL.

scholarly journals Calculating the Energy Spectrum of Complex Low-Dimensional Heterostructures in the Electric Field

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Svetlana N. Khonina ◽  
Sergey G. Volotovsky ◽  
Sergey I. Kharitonov ◽  
Nikolay L. Kazanskiy
Keyword(s):  
Steady State ◽  
Electric Field ◽  
Ground State ◽  
State Energy ◽  
Airy Functions ◽  
Piecewise Constant ◽  
Matrix Rank ◽  
The Matrix ◽  
Constant Potential ◽  
Low Dimensional

An algorithm for solving the steady-state Schrödinger equation for a complex piecewise-constant potential in the presence of theE-field is developed and implemented. The algorithm is based on the consecutive matching of solutions given by the Airy functions at the band boundaries with the matrix rank increasing by no more than two orders, which enables the characteristic solution to be obtained in the convenient form for search of the roots. The algorithm developed allows valid solutions to be obtained for the electric field magnitudes larger than the ground-state energy level, that is, when the perturbation method is not suitable.

scholarly journals Uniform asymptotic expansions for the Whittaker functions M κ , μ ( z ) and W κ , μ ( z ) with μ large

2021 ◽  
Vol 477 (2252) ◽  
pp. 20210360
Author(s):  
T. M. Dunster
Keyword(s):  
Analytic Continuation ◽  
Error Bounds ◽  
Asymptotic Expansions ◽  
Hypergeometric Functions ◽  
Whittaker Functions ◽  
Airy Functions ◽  
Elementary Functions ◽  
Confluent Hypergeometric Functions ◽  
Uniform Asymptotic Expansions

Uniform asymptotic expansions are derived for Whittaker’s confluent hypergeometric functions M κ , μ ( z ) and W κ , μ ( z ) , as well as the numerically satisfactory companion function W − κ , μ ( z   e − π i ) . The expansions are uniformly valid for μ → ∞ , 0 ≤ κ / μ ≤ 1 − δ < 1 and 0 ≤ arg ⁡ ( z ) ≤ π . By using appropriate connection and analytic continuation formulae, these expansions can be extended to all unbounded non-zero complex z . The approximations come from recent asymptotic expansions involving elementary functions and Airy functions, and explicit error bounds are either provided or available.

Back Where We Started

2017 ◽  
Author(s):  
John A. Adam
Keyword(s):  
Integral Equation ◽  
Black Hole ◽  
Angular Momentum ◽  
Gravitational Lensing ◽  
Airy Functions ◽  
Ray Optics ◽  
Gravitational Fields ◽  
The Subject ◽  
Luneberg Lens

This chapter returns to the subject of rainbows, offering some reflections based on the author's review of the book The Rainbow Bridge: Rainbows in Art, Myth, and Science by Raymond L. Lee, Jr. and Alistair B. Frase. In particular, it discusses various topics related to the rainbow, including historical descriptions of the rainbow, some common misperceptions about rainbows, theories of the rainbow, angular momentum, rainbow ray, and Airy functions. The chapter also considers ray optics, with emphasis on Luneberg inversion and gravitational lensing, Abel's integral equation, and the Luneberg lens. Finally, it explains the rainbow's connection with classical scattering and gravitational lensing, focusing on weak gravitational fields and the black hole lens.

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