4 research outputs found
Aberration-like cusped focusing in the post-paraxial Talbot effect
We present an analysis of self-imaging in a regime beyond the paraxial, where
deviation from simple paraxial propagation causes apparent self-imaging
aberrations. The resulting structures are examples of aberration without rays
and are described analytically using post-paraxial theory. They are shown to
relate to, but surprisingly do not precisely replicate, a standard integral
representation of a diffraction cusp. Beyond the Talbot effect, this result is
significant as it illustrates that the effect of aberration -- as manifested in
the replacement of a perfect focus with a cusp-like pattern -- can occur as a
consequence of improving the paraxial approximation, rather than due to
imperfections in the optical system.Comment: 8 pages, 3 figures, IoP styl
Dispersive hyperasymptotics and the anharmonic oscillator
Hyperasymptotic summation of steepest-descent asymptotic expansions of integrals is extended to functions that satisfy a dispersion relation. We apply the method to energy eigenvalues of the anharmonic oscillator, for which there is no known integral representation, but for which there is a dispersion relation. Hyperasymptotic summation exploits the rich analytic structure underlying the asymptotics and is a practical alternative to Borel summation of the Rayleigh–Schrödinger perturbation series
Anharmonic oscillator discontinuity formulae up to second exponentially small order
The eigenvalues of the quartic anharmonic oscillator as functions of the anharmonicity constant satisfy a once-subtracted dispersion relation. In turn, this dispersion relation is driven by the purely imaginary discontinuity of the eigenvalues across the negative real axis. In this paper we calculate explicitly the asymptotic expansion of this discontinuity up to second-exponentially-small order
Incomplete Airy beams: finite-energy from a sharp spectral cutoff
We present a mathematical analysis of the finite-energy Airy beam with a sharply truncated spectrum, which can be generated by a uniformly illuminated, finite-sized spatial light modulator, or windowed cubic phase mask. The resulting “incomplete Airy beam” is tractable mathematically, and differs from an infinite-energy Airy beam by an additional oscillating modulation and the decay of its fringes. Its propagation can be described explicitly using an incomplete Airy function, from which we derive simple expressions for the beam’s total power and mean position. Asymptotic analysis reveals a simple connection between the cutoff and the region of the beam with Airy-like behavior