The following paper is an attempt to connect what, at first glance, appears to be an unrelated phenomenon in glass to an example of Anderson localization using the common themes involving waves and disorder. This common behavior of localization in all cases results from wave interference. The unique insight of Anderson [1] was utilizing the Schrodinger equation to treat the diffusion of defects in crystals. What was the natural approach for treating the behavior of photons and phonons was now used for electrons to explain how localized states could occur as a consequence of structural disorder. The physical description of disorder is different for electrons with dopants in an otherwise perfect lattice than for photons where the disorder would correspond to fluctuations of the refractive index; for acoustic phonons it would be regions of different velocities of sound. (Equation (3)) Clearly, the specific wave equations governing the different wave phenomena are not the same as well as the resulting incorporation of the disorder. For these two examples the disorder is incorporated from the refractive index and the sound velocity, respectively. For the electron case the Hamiltonian is altered either by letting the defect have a broad distribution of energies or letting the potential function V(r) vary spatially. As we will see in the case of electrons for the Anderson treatment to be operative requires that there is ignorable interaction between sites so that the wavefunction of the defect can be uniquely defined. For photons and phonons this is always the case. A simple statement can be made; Waves + Disorder = Localization.

Localized states for photons become localized propagating modes. It appears that one could have arrived at this directly from Maxwell’s equations because interference phenomena is inherent in light propagation in a medium with a spatially varying refractive index as we will see below, but here we interpret as an example of Anderson localization.

∇ × ∇ × E + ε c 2 ∂ 2 E ∂ t 2 = 4 π c 2 ∂ 2 P N L ∂ t 2 (1)

For the 2-D case one can write the nonlinear wave equation (Equation (1)) with an intensity dependent refractive index but now one replaces this term with a static spatial refractive index variation shown here in Equation (2).

i ∂ E ∂ z = ( 1 2 k ) ∇ T 2 E + ( ∫ 2 n 0 2 ) k | E 2 | E = 0 (2)

i ∂ A ∂ z + ( 1 2 k ) ∇ T 2 A + ( k n 0 ) Δ n ( x , y ) A = 0 (3)

Equation (3) appears similar to the Schrodinger equation [2] [3] shown below in Equation (4); where the disorder is represented in the potential wells, V(r) of the electrons on the various sites.

i h ∂ φ ∂ t + ( i h 2 2 m ) ∇ 2 φ + Δ V ( r ) φ ( r , t ) = 0 (4)

From the similarity one can work it backward; viz. localized optical modes from disorder by analogy implies localized electronic states as “a wave is a wave”.

1) Mixed polymers

Below is shown an example of redrawing a mix of two polymer optical fibers of differing refractive index [4]. This can be interpreted as propagating modes via the Anderson localization phenomenon as long the light is not confined to the higher index fiber. The first picture in Figure 1 shows an SEM image resulting from the co-drawing of the 50 - 50 mix of two 0.25-mm polymer fibers with different refractive indices, thus resulting in a refractive index pattern through which light guiding occurs from the Anderson localization phenomenon.

2) Glass

The next example of a fiber made from the elongation of phase separated glass [5] [6]. This glass undergoes a liquid-liquid phase separation on heating, (see Figure 2). The elongation of the droplets seen in Figure 2 is produced in the present case by what is termed a down-draw method. Here, by gravity, the glass flows out of a hole in the bottom of the crucible at the appropriate viscosity (temperature) and is formed into a fiber. The observation of image transport in these fibers is indicative of Anderson localization. Light launched into an arbitrary point at the input face of the fiber, emerges in the same transverse location on the output face. The processing method used to create these fibers is more adaptable to a manufacturing process than one based on stacking filaments of different compositions.

The elongated regions are continuous thoughout the length of the fiber as shown in Figure 2(a) and a cross sectional view in shown in Figure 2(b) showing the random refractive index pattern.

One can obtain an estimate of the relationship between the initial phase droplet size and the ultimate length of the continuous fiber that can result. We utilize the volume conservation of the particles in the blank with radius, R as it relates to the filament radius, r over a length, L. Then, by conservation of volume, we have the following.

4 3 π R 3 = π r 2 L . (5)

Solving for the initial particulate radius we have

R = ( 3 r 2 L 4 ) 1 / 3 ≅ 0.909 ( r 2 L ) 1 / 3 (6)

One can see from Figure 3 that for a 10-µm droplet that the fiber radius would be ~0.1 µm for the fiber length of 30 cm and ~0.5 µm for a 2-cm length.

It is useful at this point to be more specific about the proposed light guiding mechanism. The light is not guided by the elongated particles (strands) per se, this would not represent Anderson localization. The light guiding is by regions of the index pattern where regions of high index are surrounded by regions low index. The actual delta n is not that of the refractive of respective phases, rather by the difference of the effective indices of the two respective regions. Therefore, this results in a distribution of the values of delta n and effective light guiding radii. This is consistent with why we see the guiding behavior irrespective of whether the droplets are the high (borosilicate glass case) or low index phase (borate glass). So truly the light propagation is determined by Equation (2), namely Anderson localization.

In this section a calculation of the possible propagating modes is accomplished by using the SEM result shown in Figure 4(b) to make a map of the refractive and then numerically solving Equation (2) for the modes, realizing that the spatial inhomogeneity remains substantially axially invariant over large distances. This was done for the two cases where the elongated droplet phase was the higher index phase and then the lower index phase. From Figure 4(a) one observes the multiplicity of modes that can propagate. This represents a true example of Anderson localization.

In the picture below in Figure 5 is shown the image transfer property in a sample that is in a larger diameter cane form.

In the following two Figures, pictures are shown of images from 0.5-mm fiber samples; Figure 6 indicating the resolution and Figure 7 indicating the longest length where an image can be transmitted.

Compositions of the two fibers of Figure 6, Figure 7 (wt%).

Imaging via an Anderson localization mechanism in an optical fiber can be produced from a particle-elongated, phase-separated glass medium. The approach suggests a more manufacturable approach to producing such fibers.

We wish to thank Joseph Schroeder, Derek Webb, and Heath Filkins for technical support in the processing of the glasses reported in this paper.

The authors declare no conflicts of interest regarding the publication of this paper.

Borrelli, N.F., Seward, T.P., Koch, K.W. and Lamberson, L.A. (2022) Anderson Localization Light Guiding in a Two-Phase Glass. Journal of Modern Physics, 13, 768-775. https://doi.org/10.4236/jmp.2022.135045