Nye and Berry's 1974 paper 'Dislocations in wave trains' founded the physics of phase singularities: in any generic monochromatic or quasi-monochromatic scalar wave field the complex amplitude vanishes along curving, intersecting, annihilating and reconnecting lines (wavefront dislocations, today also called optical vortices or nodal lines). Their closing discussion left a bundle of problems, most of which the ensuing field of singular optics has since settled: the caustic-dislocation synthesis, the crystal-dislocation analogy, and the paraxial matter-wave (electron vortex beam) realization. The open problems that the paper pointed to are: (i) a complete analytic and rigorous characterization of the three-dimensional tangle of dislocation lines (the 'tangled cobweb' of nodal lines) in generic wave fields - random waves, speckle, and chaotic-cavity or dispersive matter-wave eigenfunctions - beyond the local density and curvature laws already derived, including the torsion distribution, loop statistics, knotting and linking, and large-scale self-similar scaling; (ii) an effective dynamical theory of interactions between dislocation lines in three-dimensional fields, generalizing the two-dimensional pair-interaction results; and (iii) the paper's still-unsettled conjecture that edge and mixed dislocations of strength s>1 cannot exist as solutions of the wave equation.

Background

A scalar wave field can be written in complex form as ψ=ρ eiχ\psi=\rho\,e^{i\chi}, with real amplitude ρ\rho and phase χ\chi. Nye and Berry (1974) studied what happens where ρ=0\rho=0: there the phase is indeterminate, so wavefronts (surfaces of constant χ\chi) can end on the zero lines, which they called wavefront dislocations by analogy with the line defects of imperfect crystals. The analogy is exact for topology and loose for mechanics. A closed circuit (a Burgers circuit) encircling the line accumulates a phase change 2πN2\pi N with integer NN (the Burgers vector, or topological charge); when the charge circulates around the line the dislocation is screw-like (helical wavefronts), and when the Burgers vector is transverse to the line it is edge-like (extra or missing wavefront sheets). Unlike crystal dislocations, the wave equation stays linear right through the core - there is no 'bad crystal' region - so dislocated wave fields can be superposed freely, and dislocations can glide and climb (move in the directions of and perpendicular to their Burgers vector, the two line-motion modes inherited from crystal terminology), collide, annihilate, and be created as loops, all as exact solutions. Nye and Berry were led to these objects by ultrasonic pulses reflected from rough surfaces, and noted that they must occur generically wherever wave trains from a common oscillator interfere, for example in scattering; their discussion then raised the problems considered here, including the extension to dispersive systems such as the time-dependent Schrödinger equation, where the dislocation lines are lines on which the probability density vanishes. The subject grew into singular optics, reviewed by Dennis, O'Holleran and Padgett (2009). Three strands of the later literature define what is now known. First, the connection to caustics: Berry, Nye and Wright (1979) analyzed the dislocation lines threading the diffraction pattern of the elliptic umbilic diffraction catastrophe (the wave decoration of a caustic singularity of geometrical optics), Nye (2006, 2009) extended this to the hyperbolic umbilic and to higher-order catastrophes, and Berry and Upstill (1980) gave the general catastrophe-optics framework; the caustics-focal-relationships thread of the 1974 discussion is thereby substantially developed. Second, the matter-wave strand: free-electron vortex beams - electron beams whose wavefunction carries a designed screw dislocation and quantized orbital angular momentum - were realized by Uchida and Tonomura (2010), Verbeeck, Tian and Schattschneider (2010), and McMorran et al. (2011, charges up to 100), which settles the existence question for paraxial (slowly spreading, small-angle beam-like) matter-wave dislocations but says nothing about the statistics of a generic three-dimensional tangle. Third, the statistical strand, which descends directly from the 'tangled cobweb' remark: for isotropic Gaussian random waves (the random-wave model of speckle - the granular interference pattern produced by waves arriving from many uncorrelated directions - and of quantum chaos, in which a monochromatic field is a superposition of plane waves with random directions and phases), Berry and Dennis (2000) computed exact laws for the local structure of the dislocation tangle - line density and the probability distribution of curvature - and the density and correlations of dislocation points have been measured in microwave billiards (flat microwave resonators whose shapes make the ray trajectories chaotic, used as laboratory analogues of quantum-chaotic eigenfunctions) by Höhmann et al. (2009). Beyond the local regime almost everything is open: Taylor and Dennis (2014) tracked the tangle in large numerical random-wave fields, found individual lines to become Brownian-like beyond the coherence length and reported that even the torsion distribution 'seems analytically intractable'; Taylor (2018) found numerical evidence for universal power-law scaling of the tangle, describing it as a fractal loop soup; Lin (2022) obtained partial rigorous asymptotic laws for the topology of Gaussian random zero sets, a program tied to questions posed by Berry. On the dynamical side, Nye (2004) constructed local exact solutions describing the interaction of wave dislocations, and De Angelis and Kuipers (2021) extracted an effective pair-interaction for the dislocation points of two-dimensional random waves from their liquid-like pair correlations, but no three-dimensional line-level dynamical theory with anything playing the role of the forces and line tensions of crystal dislocations exists. Finally, Nye and Berry's sharpest single question remains untouched as stated: multiple screw dislocations of strength s>1s>1 exist exactly (the paper itself gives ψ=(Ars+B/rs)exp⁡[i(kz−ωt±sϕ)]\psi=(Ar^{s}+B/r^{s})\exp[i(kz-\omega t\pm s\phi)]), but for multiple edge and mixed dislocations the paper stated the conjecture 'we conjecture that these cannot exist, although we are unable to give a general proof'; adjacent results - the dynamical splitting of a doubly charged vortex in a Bose-Einstein condensate (Shin et al. 2004) and the splitting of higher-order optical vortices (Upadhaya et al. 2016) - show that multiply charged vortices are structurally unstable in matter-wave and optical settings, but no dedicated proof or construction settles the wave-equation statement, and Nye (2011) has returned to the clustering of dislocations in a carrier wave.

Problem Statement

Provide a complete characterization of the three-dimensional tangle of wavefront-dislocation lines (the 'tangled cobweb' of nodal lines) in generic scalar wave fields, of which isotropic random waves and speckle, eigenfunctions of chaotic three-dimensional cavities, and dispersive matter-wave fields obeying the time-dependent Schrödinger equation are representative instances, together with the local structural question of multiple edge dislocations raised in the source paper. Concretely: (1) determine, analytically or rigorously, the statistics and topology of the dislocation-line network beyond the local density and curvature laws already known, including the distribution of torsion, the statistics of closed loops, the incidence and distribution of knotting and linking of the lines, and the large-scale self-similar scaling (exponents and universality classes) of the tangle, currently established only numerically; (2) construct an effective dynamical theory for the dislocation lines themselves in three-dimensional fields - their interactions, motion, and any analogue of the forces and line tension of crystal dislocations - generalizing the two-dimensional pair-interaction results and the exact local interaction solutions, and extend it to dispersive (Schrödinger) wave trains, where the carrier-envelope structure (the separation between the rapidly oscillating carrier and a slowly varying envelope that defines a narrow-band, quasi-monochromatic, wave train and thus a well-defined dislocation) must be re-derived; and (3) settle the conjecture of Nye and Berry (1974) that edge and mixed dislocations of strength s>1 (Burgers vector of transverse magnitude s wavelengths, with s an integer greater than 1) cannot exist as solutions of the wave equation, either by a general proof of impossibility or by an explicit construction. No restriction on wave model, geometry, or field class is imposed beyond genericity; the individual designed vortex (electron vortex beam) case and the caustic-diffraction relation are already treated in the literature and are not part of the target. The properties enumerated in (1) are jointly the content of that part - each is a component of the characterisation to be determined - and (1)-(3) are posed together.

References

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