An experimentum crucis on transversely disordered optical fibers (Schirmacher et al., Phys. Rev. Lett. 120, 067401 (2018)) established that Anderson localization of light is described by the modulus-type (MT) stochastic wave equation, in which disorder enters through the dielectric modulus inside the spatial derivative, rather than by the traditional potential-type (PT) equation in which disorder multiplies the frequency term. This reopens the three-dimensional problem: the PT-based canon of 3D light localization, including the photon mobility edge and its critical exponents predicted since 1984, has never been re-derived within the MT description; the MT comparison has been done only for scalar wave equations (continuum self-consistent treatments in the 2D transverse geometry, discretized scalar lattices for 3D comparisons), so the full vector nature of the electromagnetic field is unaddressed beyond mean-field treatments; and no three-dimensional experiment tests MT-specific predictions. The open problem is to complete the 3D MT theory of light localization, including mobility-edge existence and position, critical behavior, and vector Maxwell effects, and to determine which PT-based predictions survive; experimental tests in 3D samples and consequences for applications such as photonic devices, solar cells, endoscopy, and transparency of disordered media are motivating prospects downstream of this theory. Three-dimensional Anderson localization of electromagnetic waves has been established only in high-contrast metallic systems, via exact numerics and a microwave experiment, while MT continuations have been built at the coherent-potential-approximation and scalar-lattice levels; the mobility edge, critical exponents, and experimental test of MT for 3D dielectric light localization have not yet been determined.

Background

Anderson localization is the complete arrest of wave diffusion in a strongly disordered medium by interference between multiply scattered partial waves, originally discovered for electrons and later pursued for classical waves such as sound, microwaves, and light. In optics, one speaks of transverse Anderson localization when a medium is disordered only in the plane transverse to the propagation direction and translationally invariant along the axis, as in an optical fiber whose refractive index is random across the cross-section: a launched beam then spreads in the disordered plane only up to a finite localization length, so such Anderson localizing optical fibers behave like a bundle of disorder-defined single-mode waveguides and have become a platform for image transport.

The theoretical starting point is the Helmholtz equation for a medium with spatially fluctuating permittivity, written in terms of the normalized relative permittivity ϵ~(r)\tilde\epsilon(\mathbf{r}) and the vacuum wavenumber k0=2π/λk_0=2\pi/\lambda. Two competing stochastic wave equations exist. In the traditional potential-type (PT) description, written for the electric field E\mathbf{E}, the disorder multiplies the frequency term, ϵ~(r)k02E+∇2E=0(PT),\tilde\epsilon(\mathbf{r})k_0^2\mathbf{E}+\nabla^2\mathbf{E}=0\qquad(\text{PT}), which is mathematically analogous to a Schrödinger equation with a random potential whose strength scales with k02k_0^2; this analogy allowed the entire electronic theory of localization to be transferred to light. In the modulus-type (MT) description the exact wave equation, written for the divergence-free magnetic field B\mathbf{B} (∇⋅B=0\nabla\cdot\mathbf{B}=0), is retained by placing the disorder inside the spatial derivative, k02B+∇⋅(ϵ~(r)−1∇B)=0(MT),k_0^2\mathbf{B}+\nabla\cdot\big(\tilde\epsilon(\mathbf{r})^{-1}\nabla\mathbf{B}\big)=0\qquad(\text{MT}), in analogy with sound waves in a material whose elastic moduli fluctuate in space; here the fluctuating quantity is the dielectric modulus 1/ϵ1/\epsilon. Schirmacher, Abaie, Mafi, Ruocco, and Leonetti (Phys. Rev. Lett. 120, 067401 (2018)) showed that the PT equation is obtained only by tacitly neglecting a term of the exact equation, namely the nonzero divergence of E\mathbf{E} in a medium with fluctuating permittivity, and performed a decisive experiment on transversely disordered fibers: PT predicts a strong dependence of the average transverse localization length on the wavelength λ\lambda, whereas MT predicts essentially none, and the measured localization radii were wavelength-independent. This established MT as the correct description for two-dimensional (2D) transverse disorder.

The 2018 result closed the 2D transverse question but opened the three-dimensional (3D) one. In 3D the MT eigenvalue problem has ω2\omega^2 as its spectral parameter, and the localization characteristics are expected to differ from the 2D case. The 3D theoretical canon for light localization was built on PT, beginning with John's prediction of a photon mobility edge, i.e. a boundary in frequency separating extended from localized states (Phys. Rev. Lett. 53, 2169 (1984)), so its predictions of a mobility edge and its critical behavior have never been re-examined within MT. The empirical side is equally unsettled: claims of 3D light localization in strongly scattering TiO2 samples were re-analyzed as artifacts of absorption and finite size (Sperling et al., New J. Phys. 18, 013039 (2016); Skipetrov and Page, New J. Phys. 18, 021001 (2016)), and 3D Anderson localization of light in ordinary dielectric media has remained unobserved. A further caveat recorded by the source is that the PT-versus-MT comparison has thus far been carried out only for scalar wave equations, disregarding the vector character of the electromagnetic field: the 2018 treatment uses continuum self-consistent field-theoretic equations for scalar fields, while the existing 3D comparisons use discretized scalar lattices, whereas the physical electromagnetic field is vectorial.

Answering the 3D question therefore requires: a localization theory (analytic and/or numerical) for the 3D modulus-type operator, in which disorder enters through the dielectric modulus and the spectral parameter is ω2\omega^2; a determination of whether and where a mobility edge exists and of its critical behavior; a re-evaluation of the body of PT-based 3D predictions; and an extension to the full vector Maxwell problem. Experimental verification in 3D samples, where the dielectric-modulus correlation length is expected to control the spectral window in which localization can occur, is a motivating prospect downstream of the theory.

Problem Statement

Establish the theory of Anderson localization of light in three-dimensionally disordered media on the modulus-type footing that the 2018 transverse-fiber experiment established for 2D. Within the MT description, i.e. the exact wave equation, written for the divergence-free magnetic field B\mathbf{B}, in which the disorder enters through the dielectric modulus 1/ϵ(r)1/\epsilon(\mathbf{r}) inside the spatial derivative and the spectral parameter is ω2\omega^2: (1) determine the localization characteristics of light in 3D, in particular whether a mobility edge exists, where in the spectrum and parameter space (disorder strength, index contrast, dielectric-modulus correlation length) it lies, and what its critical behavior is, where establishing the absence of a mobility edge over the spectrum region examined resolves the existence sub-question on its own, so that the location and critical-behavior demands apply only if a mobility edge is present, in which case existence, location, and critical behavior are joint requirements; (2) re-evaluate the body of PT-based 3D theoretical work, such as the predicted photon mobility edge and its exponents, to determine which predictions survive the correction from PT to MT; (3) extend the comparison to the full vector Maxwell field, beyond the scalar wave equations used so far, including the role of longitudinal-field transport channels (the evanescent, non-transverse components of the vector field, which provide additional energy-transport pathways absent from scalar treatments). All components serve one overarching objective: replacing the potential-type description of 3D light localization with the correct modulus-type one and establishing what it actually predicts for the long-sought three-dimensional localization of light. Experimental verification of MT-specific predictions in 3D samples, for instance in media with wavelength-scale correlated dielectric-modulus disorder or combined electric and magnetic disorder, and the consequences for applications such as photonic devices, solar cells, endoscopy, and transparency of disordered media, are motivating prospects of this problem rather than conjunctive completion requirements.

References

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