Abstract
We present an integral of diffraction based on particular eigenfunctions of the Laplacian in two dimensions. We show how to propagate some fields, in particular a Bessel field, a superposition of Airy beams, both over the square root of the radial coordinate, and show how to construct a field that reproduces itself periodically in propagation, i.e., a field that renders the Talbot effect. Additionally, it is shown that the superposition of Airy beams produces self-focusing.
Highlights
We present an integral of diffraction based on particular eigenfunctions of the Laplacian in two dimensions
beams[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15], which present weak diffraction, i.e. they remain propagation invariant for distances that are much longer than the usual diffraction length of Gaussian beams with the same beamwidth[16], self-healing, i.e. they regenerate themselves when a part of the beam is o bstructed[17], and abrupt a utofocusing[18,19], i.e. their maximum intensity remains constant while propagating and close to a particular point they autofocus increasing its maximum intensity by orders of magnitude
By using eigenfunctions of the perpendicular Laplacian in polar coordinates, we propose a novel diffraction integral which we use to propagate some fields, namely, B essel[29,30] functions and superposition of Airy functions, both divided by the square root of the radial coordinate
Summary
We begin our analysis by recalling the paraxial equation, usually written as:. ∂2 ∂y2. In order to obtain the commonly used diffraction integral from (2), we first define the = ∂∂y , we may write the propagated field as operators Dx. we may write E(x, y, 0) in terms of its two-dimensional Fourier transform, i.e.,. Where we have used the fact that eiux is an eigenfunction of the operator Dx , with eigenvalue given by iu (similar expressions are obtained for the y coordinate). We employ the concepts of eigenfunctions and eigenvaules to produce an integral of diffraction that may be used when the field to be propagated is divided by the square root of the radial coordinate. Follow the procedure employed to obtain Eq (6), a diffraction integral can be readily written as (we set k = 1 ) (In all calculations, by replacing z → kz , arbitrary k’s may be considered). We have applied the property that a function of the operator ∇⊥2 applied to the eigenfunction is the function of the eigenvalue times the eigenfunction, i.e., F(∇⊥2 )
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