Non-Diffracting Waves: A new introduction

TL;DR

This paper introduces and analyzes non-diffracting waves, providing exact solutions, methods for eliminating backward components, and exploring applications in optics, quantum mechanics, and related fields.

Contribution

It offers new analytic methods for describing finite-energy non-diffracting pulses and Frozen Waves, advancing understanding of their properties and applications.

Findings

Elimination of backward-traveling wave components

Analytic description of truncated non-diffracting beams

Application of NDWs to optical tweezers and quantum mechanics

Abstract

This work deals with exact solutions to the wave equations. We start by introducing the Non-Diffracting Waves (NDW), and by a definition of NDWs. Afterwards we recall -besides ordinary waves (gaussian beams, gaussian pulses)- the simplest non diffracting waves (Bessel beams, X-shaped pulses,...). In Sec.2 we show how to eliminate any backward-traveling components, first in the case of ideal NDW pulses, and then, in Sec.3, for realistic finite-energy NDW pulses. In particular, in subsec.3.1 we forward a general functional expression for any totally-forward non-diffracting pulses. Then, in Sec.4 an efficient method is set forth for the analytic description of truncated beams, a byproduct of its being the elimination of any need of lengthy numerical calculations. In Sec.5 we explore the question of the subluminal NDWs, or bullets, in terms of two different methods, the second one allowing the analytic description of non-diffracting waves with a static envelope ("Frozen Waves", FW), in terms of continuous Bessel beam superpositions. The production of such Frozen Waves (experimentally generated in recent time for Optics) is theoretically developed in Sec.6 also for the case of absorbing media. Sec.7 discusses the role of Special Relativity and of Lorentz transformations, relevant for the physical comprehension of the NDW issue. In Sec.8 we present further analytic solutions to the wave equations, with use of higher-order Bessel beams. Next, Sec.9 deals in detail with an application of NDWs to Optical Tweezers. In Sec.10 we show that "soliton-like" solutions can be found also in the different case of the ordinary linear Schroedinger equation within standard Quantum Mechanics. Some complementary issues are just mentioned in the last Section. This work also constitutes a part of a much longer Review in preparation.