Diffraction-Attenuation Resistant Beams: their Higher Order Versions and Finite-Aperture Generations

Recently, a method for obtaining diffraction-attenuation resistant beams in absorbing media was developed through suitable superposition of ideal zero-order Bessel beams. In this work, we will show that such beams maintain their resistance to diffraction and absorption even when generated by finite apertures. Also, we shall extend the original method to allow a higher control over the transverse intensity profile of the beams. Although the method has been developed for scalar fields, it can be applied to paraxial vector wave fields as well. These new beams can possess potential applications, such as free space optics, medical apparatuses, remote sensing, optical tweezers, etc..Comment: 24 pages, 6 figure

Analytical expressions for the longitudinal evolution of nondiffracting pulses truncated by finite apertures

In this paper, starting from some general and plausible assumptions based on geometrical optics and on a common feature of the truncated Bessel beams, a heuristic derivation is presented of very simple analytical expressions, capable of describing the longitudinal (on-axis) evolution of axially-symmetric nondiffracting pulses when truncated by finite apertures. We apply our analytical formulation to several situations involving subluminal, luminal or superluminal localized pulses and compare the results with those obtained by numerical simulations of the Rayleigh-Sommerfeld diffraction integrals. The results are in excellent agreement. The present approach can be very useful, because it can yield, in general, closed analytical expressions, avoiding the need of time-consuming numerical simulations, and also because such expressions provide a powerful tool for exploring several important properties of the truncated localized pulses, as their depth of fields, the longitudinal pulse behavior, the decaying rates, and so on.Comment: 27 pages, 7 figure

Chirped optical X-shaped pulses in material media

In this paper we analyze the properties of chirped optical X-shaped pulses propagating in material media without boundaries. We show that such ("superluminal") pulses may recover their transverse and longitudinal shape after some propagation distance, while the ordinary chirped gaussian-pulses can recover their longitudinal shape only (since gaussian pulses suffer a progressive spreading during their propagation). We therefore propose the use of chirped optical X-type pulses to overcome the problems of both dispersion and diffraction during the pulse propagation.Comment: Replaced with a much larger and deepened version (the number of pages going on from 4 to 24; plus 4 Figures added

Frozen Waves: Stationary optical wavefields with arbitrary longitudinal shape, by superposing equal-frequency Bessel beams

In this paper it is shown how one can use Bessel beams to obtain a stationary localized wavefield with high transverse localization, and whose longitudinal intensity pattern can assume any desired shape within a chosen interval 0 < z < L of the propagation axis. This intensity envelope remains static, i.e., with velocity v=0; and because of this we call "Frozen Waves" such news solutions to the wave equations (and, in particular, to the Maxwell equations). These solutions can be used in many different and interesting applications, as optical tweezers, atom guides, optical or acoustic bistouries, various important medical purposes, etc.Comment: LaTeX file (10 pages, including 2 sets of two Figures

Diffraction-Attenuation Resistant Beams in Absorbing Media

In this work, in terms of suitable superpositions of equal-frequency Bessel beams, we develop a theoretical method to obtain nondiffractive beams in absorbing media (weakly conductive) capable to resist the loss effects for long distances.Comment: 12 pages, 3 figure

Further results for the Dunkl Transform and the generalized Ces\`aro operator

In this paper, we consider Dunkl theory on R^d associated to a finite reflection group. This theory generalizes classical Fourier anal- ysis. First, we give for 1 < p <= 2, sufficient conditions for weighted Lp-estimates of the Dunkl transform of a function f using respectively the modulus of continuity of f in the radial case and the convolution for f in the general case. In particular, we obtain as application, the integrability of this transform on Besov-Lipschitz spaces. Second, we provide necessary and sufficient conditions on nonnegative functions phi defined on [0; 1] to ensure the boundedness of the generalized Ces\`aro operator C_phi on Herz spaces and we obtain the corresponding operator norm inequalities.Comment: 19 page