Gradient Marker: A Universal Wave Pattern in an Inhomogeneous Continuum

Abstract

Wave transport in a media with slow spatial gradient of its characteristics is found to exhibit a universal wave pattern ("gradient marker") in a vicinity of the maxima/minima of the gradient. The pattern is common for optics, quantum mechanics and any other propagation governed by the same wave equation. Derived analytically, it has an elegantly simple yet nontrivial profile found in perfect agreement with numerical simulations for specific examples. We also found resonant states in continuum in the case of quantum wells, and formulated criterium for their existence.PACS numbers: 03.50. De, 72.15 [7], dark-soliton grids [8], "scars" in "quantum billiard" [9], "quantum carpets" in QM potentials [10], nanostratification of local field in finite lattices [11], etc. In all of those, the presence of multi-modes or a broad-bend spectrum is pre-requisite for interference and pattern formation in inhomogeneous or confining structures.In this Letter we show, however, that a localized wave pattern -an immobile single-cycle intensity profile -can emerge in a single-mode wave in a vicinity of a min/max of the gradient of QM potential or optical refractive index. The phenomenon is universal for both optics and quantum mechanics, and for any other propagation described by a wave equation (1) below. What makes it unusual is that it emerges in media with no potential wells and only a smooth inhomogeneity yielding no reflection, -and is originated by a purely traveling wave with apparently no other modes to interfere with. We found, however, that this wave here generates a co-traveling but localized "satellite" of slightly different phase and amplitude resulting in "self-interference". The wave ideally is not trapped and carries its momentum and energy flux unchanged through the area. To a degree, the pattern mimics a 2-nd order spatial derivative of the refractive index (or potential function); it would be natural to call it a "gradient (G) marker". In QM it may be most pronounced for an above-barrier propagation of electron in continuum over smoothly-varying potential; in solid state it might emerge above the critical temperature for the Anderson localization to vanish. Even for a potential well, when the energy of electrons exceeds the ionization potential and there is no trapping, the G-markers emerge as the main non-resonant localized feature.To demonstrate the effect and elucidate analytical results (to be compared with numerical simulations) we consider 1D-case written, for the sake of compactness, in "optical" terms, using space-varying refractive index n(x); yet we consistently "translate" all the effects and approaches into QM-terms. A 1D spatial dynamics of an ω-monochromatic plane wave with linearly polarized electrical field E =ê p E(x) exp(−iωt) + c.c., propagating in the x-axis (hereê p ⊥ê x is a polarization unity vector), is governed by wave equationwhere k 0 = ω/c = 2π/λ 0 , and "prime" stands for d/dξ.(For H field,ê x ⊥ H ⊥ê p , one has H = −iE ′ in nonmagnetic materials; for a traveling wave, |H| = n|E|, if...