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Journal ArticleDOI

Parabolic nondiffracting optical wave fields

TL;DR: The existence of parabolic beams that constitute the last member of the family of fundamental nondiffracting wave fields and their associated angular spectrum is demonstrated and their eigenvalue spectrum is continuous.
Abstract: We demonstrate the existence of parabolic beams that constitute the last member of the family of fundamental nondiffracting wave fields and determine their associated angular spectrum. Their transverse structure is described by parabolic cylinder functions, and contrary to Bessel or Mathieu beams their eigenvalue spectrum is continuous. Any nondiffracting beam can be constructed as a superposition of parabolic beams, since they form a complete orthogonal set of solutions of the Helmholtz equation. A novel class of traveling parabolic waves is also introduced for the first time.
Citations
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Journal ArticleDOI
TL;DR: This work investigates the acceleration dynamics of quasi-diffraction-free Airy beams in both one- and two-dimensional configurations and shows that this class of finite energy waves can retain their intensity features over several diffraction lengths.
Abstract: We investigate the acceleration dynamics of quasi-diffraction-free Airy beams in both one- and two-dimensional configurations. We show that this class of finite energy waves can retain their intensity features over several diffraction lengths. The possibility of other physical realizations involving spatiotemporal Airy wave packets is also considered.

1,522 citations

Journal ArticleDOI
TL;DR: The authors survey the steady refinement of techniques used to create optical vortices, and explore their applications, which include sophisticated optical computing processes, novel microscopy and imaging techniques, the creation of ‘optical tweezers’ to trap particles of matter, and optical machining using light to pattern structures on the nanoscale.
Abstract: Thirty years ago, Coullet et al. proposed that a special optical field exists in laser cavities bearing some analogy with the superfluid vortex. Since then, optical vortices have been widely studied, inspired by the hydrodynamics sharing similar mathematics. Akin to a fluid vortex with a central flow singularity, an optical vortex beam has a phase singularity with a certain topological charge, giving rise to a hollow intensity distribution. Such a beam with helical phase fronts and orbital angular momentum reveals a subtle connection between macroscopic physical optics and microscopic quantum optics. These amazing properties provide a new understanding of a wide range of optical and physical phenomena, including twisting photons, spin-orbital interactions, Bose-Einstein condensates, etc., while the associated technologies for manipulating optical vortices have become increasingly tunable and flexible. Hitherto, owing to these salient properties and optical manipulation technologies, tunable vortex beams have engendered tremendous advanced applications such as optical tweezers, high-order quantum entanglement, and nonlinear optics. This article reviews the recent progress in tunable vortex technologies along with their advanced applications.

1,016 citations

Journal ArticleDOI
TL;DR: In this paper, two promising adjacent approaches tackle fundamental limita- tions by utilizing non-optical forces which are, however, induced by optical light fields, namely, dielectrophoretic and photophoretic forces.
Abstract: Optical tweezers, a simple and robust implementa- tion of optical micromanipulation technologies, have become a standard tool in biological, medical and physics research labo- ratories. Recently, with the utilization of holographic beam shap- ing techniques, more sophisticated trapping configurations have been realized to overcome current challenges in applications. Holographically generated higher-order light modes, for exam- ple, can induce highly structured and ordered three-dimensional optical potential landscapes with promising applications in op- tically guided assembly, transfer of orbital angular momentum, or acceleration of particles along defined trajectories. The non- diffracting property of particular light modes enables the op- tical manipulation in multiple planes or the creation of axially extended particle structures. Alongside with these concepts which rely on direct interaction of the light field with particles, two promising adjacent approaches tackle fundamental limita- tions by utilizing non-optical forces which are, however, induced by optical light fields. Optoelectronic tweezers take advantage of dielectrophoretic forces for adaptive and flexible, massively parallel trapping. Photophoretic trapping makes use of thermal forces and by this means is perfectly suited for trapping ab- sorbing particles. Hence the possibility to tailor light fields holo- graphically, combined with the complementary dielectrophoretic and photophoretic trapping provides a holistic approach to the majority of optical micromanipulation scenarios.

338 citations

Journal ArticleDOI
TL;DR: In this article, a detailed study of the propagation of an arbitrary nondiffracting beam whose disturbance in the plane z = 0 is modulated by a Gaussian envelope is presented.
Abstract: A detailed study of the propagation of an arbitrary nondiffracting beam whose disturbance in the plane z=0 is modulated by a Gaussian envelope is presented. We call such a field a Helmholtz–Gauss (HzG) beam. A simple closed-form expression for the paraxial propagation of the HzG beams is written as the product of three factors: a complex amplitude depending on the z coordinate only, a Gaussian beam, and a complex scaled version of the transverse shape of the nondiffracting beam. The general expression for the angular spectrum of the HzG beams is also derived. We introduce for the first time closed-form expressions for the Mathieu–Gauss beams in elliptic coordinates and for the parabolic Gauss beams in parabolic coordinates. The properties of the considered beams are studied both analytically and numerically.

238 citations

Journal ArticleDOI
TL;DR: In this paper, the temporal degree of freedom can be exploited to efficiently synthesize one-dimensional pulsed light sheets that propagate self-similarly in free space, with no need for nonlinearity or dispersion.
Abstract: Diffraction-free optical beams propagate freely without change in shape and scale. Monochromatic beams that avoid diffractive spreading require two-dimensional transverse profiles and there are no corresponding solutions for profiles restricted to one transverse dimension. Here, we demonstrate that the temporal degree of freedom can be exploited to efficiently synthesize one-dimensional pulsed light sheets that propagate self-similarly in free space, with no need for nonlinearity or dispersion. By introducing programmable conical (hyperbolic, parabolic or elliptical) spectral correlations between the beam’s spatiotemporal degrees of freedom, a continuum of families of propagation-invariant light sheets is generated. The spectral loci of such beams are the reduced-dimensionality trajectories at the intersection of the light-cone with spatiotemporal spectral planes. Far from being exceptional, self-similar axial-propagation in free space is a generic feature of fields whose spatial and temporal degrees of freedom are tightly correlated. These ‘space–time’ light sheets can be useful in microscopy, nonlinear spectroscopy, and non-contact measurements. One-dimensional non-diffracting sheets of light are achieved without exploiting nonlinearity. Such light sheets may be exploited in microscopy and sensing applications.

224 citations

References
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Book
01 Jan 1937

11,054 citations

Book
01 Jan 1902
TL;DR: The volume now gives a somewhat exhaustive account of the various ramifications of the subject, which are set out in an attractive manner and should become indispensable, not only as a textbook for advanced students, but as a work of reference to those whose aim is to extend the knowledge of analysis.
Abstract: This classic work has been a unique resource for thousands of mathematicians, scientists and engineers since its first appearance in 1902 Never out of print, its continuing value lies in its thorough and exhaustive treatment of special functions of mathematical physics and the analysis of differential equations from which they emerge The book also is of historical value as it was the first book in English to introduce the then modern methods of complex analysis This fifth edition preserves the style and content of the original, but it has been supplemented with more recent results and references where appropriate All the formulas have been checked and many corrections made A complete bibliographical search has been conducted to present the references in modern form for ease of use A new foreword by Professor SJ Patterson sketches the circumstances of the book's genesis and explains the reasons for its longevity A welcome addition to any mathematician's bookshelf, this will allow a whole new generation to experience the beauty contained in this text

8,965 citations


"Parabolic nondiffracting optical wa..." refers methods in this paper

  • ...We emphasize that our approach to expressing the solutions of the PCDE differs from the common methods found in the literature.(3,10,12) In the typical approach the PCDE is transformed into a Weber differential equation whose solutions are the parabolic cylinder functions Dn m , where n and m are in general complex parameters....

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Journal ArticleDOI
TL;DR: In this paper, exact nonsingular solutions of the scalar-wave equation for beams that are non-diffracting were presented, which means that the intensity pattern in a transverse plane is unaltered by propagating in free space.
Abstract: We present exact, nonsingular solutions of the scalar-wave equation for beams that are nondiffracting. This means that the intensity pattern in a transverse plane is unaltered by propagating in free space. These beams can have extremely narrow intensity profiles with effective widths as small as several wavelengths and yet possess an infinite depth of field. We further show (by using numerical simulations based on scalar diffraction theory) that physically realizable finite-aperture approximations to the exact solutions can also possess an extremely large depth of field.

2,283 citations