Transparent mirrors: rays, waves and localization

Abstract

A stack of N 1 transparent plates with randomly varying thicknesses (e.g. viewgraphs) reflects light perfectly, as a result of the accumulation of reflections from interfaces at the air gaps separating the plates. Two theories of this effect are discordant. The naive ray theory assumes that the random phases associated with the thickness variations make all the reflections incoherent, and predicts that the transmitted intensity decays as 1/N. This theory is wrong because some distinct multiply reflected waves have identical path lengths and so superpose coherently. The true decay is exponential: exact averaging of the logarithm of the transmitted intensity over the random phases, assuming these are uniformly distributed modulo 2π, gives the transmitted intensity as τ 2N = exp{−2N log(1/τ)}, where τ is the intensity transmittance of a single interface. Transparent mirrors are naked-eye examples of the localization of light, for which the localization length (inverse decay exponent) can be calculated exactly. Experiments confirm the exponential decay. Zusammenfassung. Ein Stapel von N 1 transparenten Platten mit beliebig variierender Dicke (z.B.Overhead-Folien) reflektiert Licht perfekt. Diese perfekte Reflexion ist das Ergebnis der Akkumulation von Reflexionen an den Grenzflächen der Luftspalte zwischen den Platten. Die zwei Theorien dieses Effektes stimmen nichtüberein. Die naive Strahlen-Theorie nimmt an, daß die beliebigen Phasen, die mit den Variationen der Plattendicke verknüpft sind, zu absolut inkoherenten Reflexionen führen und sagt einen 1/N Verlauf für die durchgelassene Intensität voraus. Diese Theorie ist falsch, da einige der mehrfach reflektierten Wellen identische Weglängen haben und daher koherentüberlagern. Der wahre Verlauf ist exponential: exakte Mittelung des Logarithmus der durchgelassenen Intensitätüber die beliebigen Phasen, unter der Annahme sie seien gleichmässig verteilt modulo 2π , führt zur durchgelassenen Intensität τ 2N = exp{−2N log(1/τ)}, wo τ die durchgelassene Intensität einer einzelnen Grenzfläche ist. Durchsichtige Spiegel sind Beispiele der Lokalisation des Lichtes für das unbewaffenete Auge, für die die Lokalisationslänge (inverser Exponentialkoeffizient) exakt berechnet werden kann. Experimente bestätigen den exponentialen Verlauf.

Spectral Analysis of Wave Propagation through Rows of Scatterers via Random Sampling and a Coherent Potential Approximation

“…The modes are sought using two approaches. The first approach is referred to as a localization approach, in reference to the explanation of the phenomenon given by Berry and Klein [2]. It is closely related to the approaches adopted in McPhedran et al [5] and Peter and Meylan [6].…”

Colloquium: Unusual resonators: Plasmonics, metamaterials, and random media

“…A simple manifestation of this effect is the almost-total reflection of light from a thick stack of transparencies (Berry and Klein, 1997). However, there is a set of resonant frequencies, individual for each random sample, which correspond to a high transmission of the wave through the sample accompanied by a large concentration of energy in a finite region inside the sample (Azbel and Soven, 1983;Azbel, 1983) (see Fig.…”

Anderson localization in metamaterials and other complex media (Review Article)

“…56 and Section IV A). Come now to application of the transfer matrix approach to calculation of the Lyapunov exponent γ.…”

“…Ref. 56 ). Here wave propagation inside the j-th layer is described by diagonal transfer matrixŜ 20) where ϕ j = k j d j is the phase accumulated upon the wave propagating from left to right through the j-th layer, and k j = ω c √ ε j µ j .…”

“…Averaged localization decrement is associated with exponential decay of the incident wave deep into the infinite sample 5,39,49,56 . For a finite sample, this is so only for typical realizations.…”