A stack of NN transparent plates with randomly varying thicknesses (viewgraphs, microscope slides) reflects light almost perfectly: Berry and Klein (1997) showed that the transmitted intensity decays exponentially, with the exact effective transmission τeff=exp⁡{−2Nlog⁡(1/τ)}\tau_{\rm eff}=\exp\{-2N\log(1/\tau)\} for identical plates, where τ\tau is the intensity transmittance of a single face, and identified this as Anderson localization of light observable with the naked eye. The exact exponent holds only for strong random-phase disorder (rms thickness variation Δd≫λ/2n\Delta d\gg\lambda/2n, making interface phases independent and uniform modulo 2π2\pi), normal incidence, and strictly one-dimensional, lossless propagation. The open problem is to compute the localization exponent and transmission statistics of randomly layered media outside this idealization: for oblique incidence and both polarizations on discrete random-thickness multilayers, for weak or correlated phase disorder, for the finite-NN fluctuation distribution and the crossover from the ray-like regime (Nρ≪1N\rho\ll 1) to localization, and for the transverse inhomogeneities of real stacks. Later literature has produced exact critical-incidence results for continuous white-noise models (localization length proportional to λ4/3\lambda^{4/3}) that conflict with the linear-in-λ\lambda scaling reported for deep-subwavelength binary multilayers, leaving the exponent for the discrete random-thickness stack at general incidence angles and general phase statistics unresolved.

Background

A stack of many transparent plates separated by air gaps reflects visible light strongly even though each interface reflects only a few per cent of the incident intensity. Berry and Klein (1997) analyzed this 'transparent mirror' and exposed a discordance between two theories. The naive ray theory treats the multiple reflections between interfaces as incoherent and adds intensities; it predicts that the transmitted intensity decays only as 1/N1/N. The wave theory keeps amplitudes and averages the logarithm of the transmitted intensity over the random phases of the layers; because distinct multiply reflected paths of equal length interfere coherently and destructively, the result is an exact exponential decay. If the rms variation of the layer thicknesses is Δd\Delta d and the condition Δd≫λ/(2n)\Delta d\gg\lambda/(2n) holds (λ\lambda the wavelength, nn the refractive index), the phases nkdnkd (with k=2π/λk=2\pi/\lambda the wavenumber) can be treated as independent and uniformly distributed modulo 2π2\pi ('strong phase disorder'), and for a stack of NN identical plates (2N2N faces, each transmitting intensity τ\tau) the logarithmically averaged transmission is the product formula

⟨log⁡∣T1…2N∣2⟩=∑s=12Nlog⁡τs,τ2Neff=exp⁡{−2Nlog⁡(1/τ)},\langle\log|T_{1\ldots 2N}|^2\rangle = \sum_{s=1}^{2N} \log\tau_s, \qquad \tau^{\rm eff}_{2N} = \exp\{-2N\log(1/\tau)\},

with decay exponent E=2log⁡(1/τ)E=2\log(1/\tau) per plate; their experiments on plastic-film staircases confirmed the exponential law. The exponential decay is Anderson localization — the suppression of wave transport by disorder-induced interference — in an effectively one-dimensional geometry, and the localization length is the inverse decay exponent.

Mathematically, the decay exponent of a random multilayer is the Lyapunov exponent (the exponential growth rate of the norm of the product) of a product of random 2×22\times 2 transfer matrices (each matrix relating forward- and backward-propagating wave amplitudes across one layer). Furstenberg's theorem on products of random matrices guarantees that almost every individual stack shows the same exponential decay for many layers (self-averaging, i.e., sample-to-sample independence of the asymptotic rate), but its explicit formula for the exponent requires the invariant measure of the matrix product, which is known only in special cases. Berry and Klein note that 'usually, it is difficult to calculate decay exponents' for this reason. Their exact result is the random-phase-approximation special case (introduced for one-dimensional localization by Anderson, Thouless, Abrahams and Fisher and obtained for layered acoustics by Baluni and Willemsen): normal incidence, strong phase disorder, identical lossless layers. Other cases were treated earlier only numerically (Kondilis and Tzanetakis, random-thickness multilayers), at critical incidence (Bouchaud and Le Doussal: intermittency of random optical layers at total reflection), or within the general one-dimensional theory (Pendry's review, which also covers transmission fluctuations).

The exact regime fails in several directions, and each has a precise physical meaning. At oblique incidence the wave enters with an angle θ\theta, and the two polarizations differ: s-polarized (TE) waves have the electric field perpendicular to the plane of incidence, p-polarized (TM) waves have it in that plane. When light travels from a higher to a lower average index there is a critical angle above which total internal reflection occurs, and near it the localization length competes with the evanescent tunneling decay length. At a single interface the p-polarized reflectance vanishes at the Brewster angle; in random stacks an analogous 'Brewster anomaly' can suppress localization at a special angle. Weak phase disorder (Δd\Delta d smaller than or comparable to λ/2n\lambda/2n) leaves interface phases correlated rather than uniform, so the logarithmic average no longer factorizes. Finally, real stacks are not strictly one-dimensional: thickness varies across the beam footprint, light scatters transversely out of the stack, interfaces are contaminated, and for finite NN the transmitted intensity fluctuates strongly from sample to sample; the ray picture is valid only while Nρ≪1N\rho\ll 1, with ρ=1−τ\rho=1-\tau the per-face reflectance, and the full distribution of transmitted intensity at finite NN (not only its logarithmic average) is not fixed by Furstenberg asymptotics alone.

Problem Statement

For a stack of NN transparent plates with randomly varying thicknesses — and more generally for a randomly stratified dielectric medium — determine the localization exponent (the asymptotic decay rate of the typical and log-averaged transmitted intensity with NN, equivalently the Lyapunov exponent of the transfer-matrix product) and the statistics of transmission, outside the regime in which the exact Berry-Klein result τeff=exp⁡{−2Nlog⁡(1/τ)}\tau_{\rm eff}=\exp\{-2N\log(1/\tau)\} holds. The four parts below are independent sub-problems contributing to one objective: a quantitative theory of localization in randomly layered media beyond the strong-random-phase, normal-incidence, strictly one-dimensional idealization.

  1. Angle and polarization dependence: closed-form or controlled-asymptotic localization exponents for both s- and p-polarized waves incident at an arbitrary angle on discrete random-thickness multilayers, including angles near and above the critical angle for total internal reflection. The existing strong-disorder oblique-incidence theory (Bliokh and Freilikher) is approximate, and exact results exist only at normal incidence and, for continuous white-noise models, at the critical angle.

  2. Model dependence at critical incidence: resolve the discrepancy between the localization length proportional to λ4/3\lambda^{4/3} found for a continuous Gaussian delta-correlated (spatially white) random stratification and the localization length proportional to λ\lambda found for deep-subwavelength binary multilayers (two alternating refractive indices with random layer thicknesses much smaller than λ\lambda), establishing which scaling — or what interpolation between them — applies to the discrete random-thickness plate stack and why (the role of disorder correlations and of the continuum limit).

  3. Weak and correlated phase disorder: the localization exponent when the rms thickness variation is smaller than or comparable to λ/(2n)\lambda/(2n), so that interface phases are no longer independent and uniform modulo 2π2\pi, and for stacks with short-range correlated disorder — in particular how the exponent crosses over from the strong-disorder value 2log⁡(1/τ)2\log(1/\tau) to its weak-disorder limit.

  4. Finite samples and non-ideal geometry: the distribution of the transmitted intensity at finite NN (not only its logarithmic average), the crossover scale and functional form of the passage from the ray-like regime Nρ≪1N\rho\ll 1 (ρ=1−τ\rho=1-\tau the per-face reflectance) to the localized exponential regime, and the quantitative effect of transverse thickness inhomogeneity, transverse scattering loss, and interface contamination — the effects behind the measured single-film transmittance of about 0.94 versus the ideal 16n2/(n+1)416n^2/(n+1)^4 of about 0.90 at n=1.58n=1.58 — on the exponent and on the sample-to-sample fluctuations.

References

  1. M. V. Berry and S. Klein, 'Transparent mirrors: rays, waves and localization', European Journal of Physics 18 (1997) 222-228. DOI: 10.1088/0143-0807/18/3/017, https://doi.org/10.1088/0143-0807/18/3/017
  2. H. Furstenberg, 'Noncommuting random products', Transactions of the American Mathematical Society 108 (1963) 377-428. DOI: 10.1090/S0002-9947-1963-0163345-0, https://doi.org/10.1090/S0002-9947-1963-0163345-0
  3. P. W. Anderson, D. J. Thouless, E. Abrahams and D. S. Fisher, 'New method for a scaling theory of localization', Physical Review B 22 (1980) 3519-3526. DOI: 10.1103/PhysRevB.22.3519, https://doi.org/10.1103/PhysRevB.22.3519
  4. V. Baluni and J. Willemsen, 'Transmission of acoustic waves in a random layered medium', Physical Review A 31 (1985) 3358-3363. DOI: 10.1103/PhysRevA.31.3358, https://doi.org/10.1103/PhysRevA.31.3358
  5. J. P. Bouchaud and P. Le Doussal, 'Intermittency in random optical layers at total reflection', Journal of Physics A: Mathematical and General 19 (1986) 797-810. DOI: 10.1088/0305-4470/19/5/033, https://doi.org/10.1088/0305-4470/19/5/033
  6. A. Kondilis and P. Tzanetakis, 'Numerical calculations on optical localization in multilayer structures with random-thickness layers', Physical Review B 46 (1992) 15426-15431. DOI: 10.1103/PhysRevB.46.15426, https://doi.org/10.1103/PhysRevB.46.15426
  7. J. B. Pendry, 'Symmetry and transport of waves in one-dimensional disordered systems', Advances in Physics 43 (1994) 461-542. DOI: 10.1080/00018739400101515, https://doi.org/10.1080/00018739400101515
  8. K. Yu. Bliokh and V. D. Freilikher, 'Localization of transverse waves in randomly layered media at oblique incidence', Physical Review B 70 (2004) 245121. DOI: 10.1103/PhysRevB.70.245121, arXiv: cond-mat/0407074, https://arxiv.org/abs/cond-mat/0407074
  9. Kihong Kim, 'Exact localization length for s-polarized electromagnetic waves incident at the critical angle on a randomly-stratified dielectric medium', Optics Express 25 (2017) 28752-28763. DOI: 10.1364/OE.25.028752, arXiv: 1708.02021, https://arxiv.org/abs/1708.02021
  10. H. Herzig Sheinfux, I. Kaminer, A. Z. Genack and M. Segev, 'Interplay between evanescence and disorder in deep subwavelength photonic structures', Nature Communications 7 (2016) 12927. DOI: 10.1038/ncomms12927, https://doi.org/10.1038/ncomms12927
  11. Kihong Kim and Seulong Kim, 'Anderson localization and Brewster anomaly of electromagnetic waves in randomly-stratified anisotropic media', Materials Research Express 6 (2019) 085803. DOI: 10.1088/2053-1591/ab1aa9, arXiv: 1710.10354, https://arxiv.org/abs/1710.10354
  12. Seulong Kim and Kihong Kim, 'Scaling behavior of the localization length for TE waves at critical incidence on short-range correlated stratified random media', Results in Physics 62 (2024) 107820. DOI: 10.1016/j.rinp.2024.107820, arXiv: 2406.08156, https://arxiv.org/abs/2406.08156
  13. V. D. Freilikher and Yu. V. Tarasov, 'Propagation of wave packets in randomly stratified media', Physical Review E 64 (2001) 056620. DOI: 10.1103/PhysRevE.64.056620, arXiv: cond-mat/0107180, https://arxiv.org/abs/cond-mat/0107180
  14. Y. Bliokh, E. I. Chaikina, N. Lizarraga, E. R. Mendez, V. Freilikher and F. Nori, 'Disorder-induced cavities, resonances, and lasing in randomly layered media', Physical Review B 86 (2012) 054204. DOI: 10.1103/PhysRevB.86.054204, arXiv: 1205.5212, https://arxiv.org/abs/1205.5212
  15. K. J. Kemp, S. Barker, J. Guthrie, B. Hagood and M. D. Havey, 'One-dimensional light localization with classical scatterers: An advanced undergraduate laboratory experiment', American Journal of Physics 84 (2016) 746-751. DOI: 10.1119/1.4958615, arXiv: 1603.07913, https://arxiv.org/abs/1603.07913