## Abstract

In physics, wave propagation is traditionally analyzed by means of regular solutions of wave equations. However, solutions of wave equations in two and three dimensions often possess singularities, that is, points or lines in space at which mathematical quantities that describe physical properties of waves become infinite or change abruptly (Berry [2000]). For example, at the point of phase singularity, the phase of the wave is undefined and wave intensity vanishes. Phase singularities are recognized as important features common to all waves. They were first discussed in depth in a seminal paper by Nye and Berry [1974]. However, the earliest known scientific description of phase singularity was made in the 1830s by Whewell, as discussed by Berry [2000]. While Whewell studied the ocean tides, he came to the extraordinary conclusion that rotary systems of tidal waves possess a singular point at which all cotidal lines meet and at which tide height vanishes. Waves that possess a phase singularity and a rotational flow around the singular point are called vortices. They can be found in physical systems of different nature and scale, ranging from water whirlpools and atmospheric tornadoes to quantized vortices in superfluids and quantized lines of magnetic flux in superconductors (Pismen [1999]). In a light wave, the phase singularity is known to form an optical vortex: The energy flow rotates around the vortex core in a given direction; at the center, the velocity of this rotation would be infinite and thus the light intensity must vanish. The study of optical vortices and associated localized objects is important from the viewpoint of both fundamental and applied physics. The unique nature of vortex fields is expected to lead to applications in many areas that include optical data storage, distribution, and processing. Optical vortices propagating, e.g., in air, have been suggested also for the establishment of optical interconnects between electronic chips and boards (Scheuer and Orenstein [1999]), as well as free-space communication links (Gibson, Courtial, Padgett, Vasnetsov, Pas'ko, Barnett and Franke-Arnold [2004], Bouchal and Celechovsky [2004]), based on the multidimensional alphabets afforded by the corresponding angular momentum states (Molina-Terriza, Torres and Torner [2002]). The ability to use light vortices to create reconfigurable patterns of complex intensity in an optical medium could aid optical trapping of particles in a vortex field (Gahagan and Swartzlander [1999]), and could enable light to be guided by the light itself, or in other words by the waveguides created by optical vortices (Truscott, Friese, Heckenberg and Rubinsztein-Dunlop [1999], Law, Zhang and Swartzlander [2000], Carlsson, Malmberg, Anderson, Lisak, Ostrovskaya, Alexander and Kivshar [2000], Salgueiro, Carlsson, Ostrovskaya and Kivshar [2004]). Thus, singular optics, the study of wave singularities in optics (Nye and Berry [1974], Vasnetsov and Staliunas [1999], Soskin and Vasnetsov [2001]), is now emerging as a new discipline (for an extended list of references, see http://www.u.arizona.edu/~grovers/SO/so.html). In a broad perspective, the study of optical vortices brings inspiring similarities between different and seemingly disparate fields of physics; the comparison of singularities of optical and other origins leads to theories that transcend the confines of specific fields. Vortices play an important role in many branches of physics, even those not directly related to wave propagation. An example is the Kosterlitz-Thouless phase transition (Kosterlitz and Thouless [1973]) in solid-state physics models, characterized by creation of tightly bound pairs of point-like vortices that restore the quasi-long-range order of a two-dimensional model at low temperatures. Such vortex-induced phase transitions can be observed in superfluid helium films, thin superconducting films, and surfaces of solids, as well as in models of interest to particle physicists and cosmologists. The Bose-Einstein condensate (BEC), a state of matter in which a macroscopic number of particles share the same quantum state, constitutes a well-researched example of a superfluid in which topological defects with a circulating persistent current are observed. Nearly 75 years ago, Bose and Einstein introduced the idea of condensate of a dilute gas at temperatures close to absolute zero. The BEC was experimentally created in 1995 by the JILA group (Anderson, Ensher, Matthews, Wieman and Cornell [1995]), who trapped thousands (later, millions) of alkali ^{87}Rb atoms in a 10-μm cloud and then cooled them to a millionth of a degree above absolute zero. The extensive study of vortices in BEC (Williams and Holland [1999], Matthews, Anderson, Haljan, Hall, Wieman and Cornell [1999], Madison, Chevy, Wohlleben and Dalibard [2000], Raman, Abo-Shaeer, Vogels, Xu and Ketterle [2001]) promises a better understanding of deep links between the physics of superfluidity, condensation, and nonlinear singular optics. To introduce the notion of optical vortices, we recall that a light wave can be represented by a complex scalar function ψ (e.g., an envelope of an electric field), which varies smoothly in space and/or time. Phase singularities of the wave function ψ appear at the points (or lines in space) at which its modulus vanishes, i.e., when Re ψ = Im ψ = 0. Such points are referred to as wave-front screw dislocations or optical vortices, because the surface of constant phase structurally resembles a screw dislocation in a crystal lattice, and because the phase gradient direction swirls around the singular line much like fluid in a whirlpool. Optical vortices are associated with zeros in light intensity (black spots) and can be recognized by a specific helical wave front. If the complex wave function is presented as ψ ( r, t ) = ρ ( r, t ) exp { i θ ( r, t ) }, in terms of its real modulus ρ ( r, t ) and phase θ ( r, t ), the dislocation strength (sometimes referred to as the vortex topological charge) is defined by the circulation of the phase gradient around the singularity, {A formula is presented} here dl is the element of an arbitrary counter-clockwise path closed around the dislocation. The result is an integer because the phase changes by a multiple of 2π. Under appropriate conditions, it also measures an orbital angular momentum of the vortex associated with the helical wave-front structure. If a light wave is characterized by an extra parameter, e.g., the wave polarization, its mathematical representation is no longer a scalar but a vector field. In vector fields, several types of line singularity exist; for example, those analogous to disclinations in liquid crystals, which could be edge type, screw type, or mixed edge-screw type, that could move relative to background wave fronts and could interact in several different ways (Nye and Berry [1974], Soskin and Vasnetsov [2001]). In the linear theory of waves, each wave dislocation could be understood as a simple consequence of destructive wave interference. In this review we mostly address screw phase singularities existing in scalar wave fields and thus we concentrate our analysis on the corresponding vortices. However, other types of singularities whose analysis falls beyond the scope of this review, such us polarization singularities (Freund [2004a, 2004b]), do exist and exhibit fascinating properties. A laser beam with a phase singularity generally has a doughnut-like shape and diffracts when it propagates in free space. However, when the vortex-bearing beam propagates in a nonlinear medium, a variety of interesting effects can be observed. Nonlinear optical media are characterized by an electromagnetic response that depends on the strength of the propagating light. The polarization of such a medium can be described as {Mathematical expression}, where E is the amplitude of the light wave's electric field, and the coefficients characterize both the linear and the nonlinear response of the medium (Shen [1984], Butcher and Cotter [1992], Boyd [1992]). The χ^{( 1 )} coefficient describes the linear refractive index of the medium. When χ^{( 2 )} vanishes (as happens in the case of centro-symmetric media), the main nonlinear effect is produced by the third term that can be presented as an intensity-induced change of the refractive index proportional to χ^{( 3 )} E^{3}. An important consequence of such intensity-dependent nonlinearity is the spontaneous focusing of a beam that is due to the lensing property of a self-focusing medium (i.e. when χ^{( 3 )} > 0). This focusing action of a nonlinear medium can precisely balance the diffraction of a laser beam, resulting in the creation of optical solitons, which are self-trapped light beams that do not change shape during propagation (Kivshar and Luther-Davies [1998]). A stable bright spatial soliton is radially symmetric, and it has no nodes in its intensity profile. If, however, a beam with elaborate geometry carries a topological charge and propagates in a self-focusing nonlinear medium, it has a doughnut like structure. However, such a doughnut beam is unstable, and it decays into a number of more fundamental bright spatial solitons, such an example is shown in fig. 1. The resulting field distribution does not preserve the radial symmetry, and the vortex beam decays into several solitons that repel and twist around one another as they propagate. This rotation is due to the angular momentum of the vortex beam transferred to the splinters. Remarkably, the behavior of a laser beam in a self-defocusing nonlinear medium (i.e. when χ^{( 3 )} < 0) is distinctly different, see an example in fig. 2. Such a medium cannot produce a lensing effect and therefore cannot support bright solitons. ...

Original language | English |
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Title of host publication | Progress in Optics |

Publisher | Elsevier |

Pages | 291-391 |

Number of pages | 101 |

ISBN (Print) | 0444515984, 9780444515988 |

DOIs | |

Publication status | Published - 2005 |

### Publication series

Name | Progress in Optics |
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Volume | 47 |

ISSN (Print) | 0079-6638 |