## Abstract

We recently reported the observation of transverse Anderson localization as the waveguiding mechanism in optical fibers with random transverse refractive index profiles [1]. Here, we explore the impact of the design parameters of the disordered fiber on the beam radius of the propagating transverse localized beam. We show that the optimum value of the fill-fraction of the disorder is 50% and a lower value results in a larger beam radius. We also explore the impact of the average size of the individual random features on the value of the localized beam radius and show how the boundary of the fiber can impact the observed localization radius. A larger refractive index contrast between the host medium and the disorder sites results in smaller value of the beam radius.

© 2012 Optical Society of America

## 1. Introduction

The possibility of micron-scale transverse confinement of light and its extremely low-loss propagation in optical fibers provides a robust platform to explore many new ideas in photonics as well as in other branches of science [2]. The 1996 introduction of photonic crystal fibers [3] was a revolutionary step towards a much greater control over the refractive index profile, resulting in the endlessly single-mode fibers [4], exotic dispersion properties [5], highly nonlinear fibers for supercontinuum generation [6], and bandgap fibers [7], among many others. The framework of the geometrical symmetries of fibers (or lack thereof) may guide its future progress as much as it has shaped its past and provide clues to what is yet to emerge in new scientific findings and applications of optical fibers. Traditional optical fibers are symmetric in both longitudinal and angular cylindrical coordinates with a broken radial symmetry required for light confinement. Further symmetry breaking has resulted in novel phenomena, such as polarization maintaining fibers, which break axial rotational symmetry, fiber Bragg gratings, which break longitudinal symmetry, or photonic crystal fibers where the angular symmetry is reduced to at most discrete rotations. It is of no coincidence that symmetry breaking plays a fundamental role at the heart of other scientific disciplines as well [8]. One can argue that besides aesthetics, symmetric structures are generally easier to study, model, and fabricate [3,9]. However, neither the guiding nor even the bandgap properties in photonic crystal fibers require periodicity at the fundamental level.

Disordered structures possess exotic universal physical characteristics, which sets them apart from deterministic structures [10]. For example, the propagation of light in highly disordered media can lead to strong confinement due to the Anderson localization effect [11–14]. Anderson localization has been the subject of intense investigation in various quantum and classical wave disordered systems over the years. There has been considerable progress during the past few years in research on the localization of light waves [14].

In order to observe Anderson localization, the disorder must be strong enough such that the mean free path for wave scattering *l*^{★} becomes of the order of the wavelength i.e. *kl*^{★} ≈ 1, where *k* is the effective wavevector in the medium. This is known as the Ioffe-Regel condition [15]. The Ioffe-Regel condition is very difficult to satisfy in three dimensional optical systems; however, the required conditions are considerably relaxed in two dimensional systems where strong localization effects for light can be readily observed. This interesting fact was pointed out in 1989 by De Raedt et al. [16] who proposed and numerically confirmed “transverse localization” of light in an optical system which is uniform in one (“longitudinal”) direction but contains disorder in the two “transverse” directions.

Two-dimensional disordered systems are always localized and the localization length *ξ* is given by *ξ* = *l*^{★} exp(*πk _{T}*

*l*

^{★}/2) (see [16–18]). In particular, if the randomness in the refractive index profile is only limited to the transverse plane of a propagating optical wave,

*k*is the effective transverse component of the wavevector and is 10–100 times smaller than

_{T}*k*; therefore, even small disorder (i.e. large

*l*

^{★}) can readily result in an effective localized beam width (

*ξ*), which is smaller than the transverse dimensions of the system. De Raedt et al. [16] observed that a narrow beam entering a medium with transverse disorder undergoes initial diffusive broadening until its width becomes of the order of the localization length, after which it propagates in a localized transverse profile. Transverse localization of light for the first time was observed experimentally by Segev’s group in a two-dimensional photonic lattice with random refractive index fluctuations induced on a photorefractive crystal using an optical interference pattern [18]. While this pioneering experiment clearly proves the transverse Anderson localization concept as introduced by De Raedt et al. [16], the photo-induced refractive index fluctuations are of the order of 10

^{−4}, resulting in a large localization length (effective beam radius or localization radius). Moreover, such a low index contrast is responsible for large fluctuations in the localization length among different realizations of the random refractive index profile. Transverse Anderson localization in 1D lattices is investigated both numerically in Ref. [19] and experimentally in Refs. [20–22], in great detail. In Ref. [20], the disordered structure is a 1D lattice of coupled waveguides on an AlGaAs substrate; and in Refs. [21,22], the waveguides are inscribed using ultrafast lasers inside transparent glass. In a recent work, we introduced an optical fiber with a binary compound with permanent refractive index fluctuations on the order of 0.1 and observed transverse localization of light with an effective propagating beam diameter, which was comparable to that of a typical index-guiding optical fiber [1]. We used a disordered polymer optical fiber to carry out our experiments and observed transverse localization of light for fibers as long as ≈60cm; however, longer propagation was affected by large longitudinal variations of the fiber due to the draw process. We made sure that all of the samples selected for our experiments are invariant along the length by inspecting their thickness under an optical microscope.

In order to manage and use transverse Anderson localized optical fibers for the device applications, it is of great importance to understand the effect of different optical and geometrical parameters of the fiber on the beam and propagation properties of the localized light, especially on its effective beam radius. In this paper, we investigate the impact of the disorder site size, fill-fraction, and refractive index contrast on the beam radius of the localized propagating beam in the Anderson localized optical fiber. In this paper, we mainly focus our studies on design variations of the polymer fiber we reported in Ref. [1]. In particular, we will compare our numerical results on the impact of different disorder site sizes with experimental measurements. However, we will show that our general observations are applicable to a wide range of parameters of interest for applications of transverse Anderson localized optical fibers.

In order to experimentally investigate the impact of the disorder site size, we have chosen the same polymer fibers that we used in Ref. [1], but now drawn to different side widths (fiber profiles are nearly rectangular). The fiber side widths studied in here are approximately 150*μm* and 250*μm*, resulting in approximate site sizes of 0.6*μm* and 0.9*μm*. For numerical investigations, we use the finite difference beam propagation method (FD-BPM) and calculate the beam radius of the localized light in the Anderson localized optical fibers. We will also investigate numerically the effect of different fill-fractions and different refractive index contrasts on the beam radius. Finally, we will compare some of our results with those obtained for a large index contrast created in a glass version of the disordered optical fiber where the random air holes create a much larger index contrast. Our observations for the glass version of the disordered fiber, especially the small localized beam radius and the reduced boundary effects, in addition to the intrinsically low material loss, may provide an avenue for potential applications in short-haul optical fiber communications.

## 2. Anderson localized optical fiber

Our disordered optical fiber is based on the structure proposed and numerically analyzed by De Raedt et al., where the refractive index is invariant in the longitudinal z-direction, but is randomly changing in the two “transverse” directions [16]. For the numerical simulations, we follow the same procedure that we used in our recent publication [1], the details of which are repeated below. We use an idealized structure, where we start with a square transverse geometry for the optical fiber of side dimension *L*. We then create a square grid of mesh size *d* covering the larger square; if *L* = *N* × *d*, the total number of cells covering the transverse profile of the fiber is *N*^{2}. The refractive index of each cell can be either *n*_{1} or *n*_{2}. The random numbers are drawn from a uniform distribution. We define *p* as the probability of each cell having the refractive index *n*_{1} and assume without loss of generality that *n*_{1} < *n*_{2}. Therefore, *p* can be regarded as the fill-fraction of the low-index material in the higher index host medium.

In order to fabricate the disordered fiber, we follow the procedure we reported recently in Ref. [1]. We randomly mixed 40, 000 pieces of Poly(methyl methacrylate) PMMA fibers with refractive index of 1.49 and 40, 000 pieces of Polystyrene (PS) fibers with refractive index of 1.59; each fiber was 8 inches long with an approximate diameter of 200*μm*. The mixture was fused together and redrawn to a fiber with a nearly square profile and different values of thickness, including fibers with approximate side width of 150*μm* and 250*μm*.

Our choice of the fill-fraction parameter *p* = 50% is optimal, in order to obtain the minimum localization length in these fibers. We note that *p* = 50% is below the percolation threshold 59.27% (of a square lattice); therefore, the host material with the higher refractive index *n*_{2} remains generally connected in the long range. Above the percolation threshold, disconnected clusters of the higher index material form, which can act as individual waveguides; in this case index-guiding by these higher-index localized clusters can mask any effects that can be attributed to Anderson localization and the optical field can be localized in a trivial sense.

In Fig. 1(a), we show an example cross section of a fiber with site size of 0.9*μm* used for our simulations, corresponding to a fiber of side width equal to 250*μm*. However, the square only shows a 24*μm*×24*μm* region of this fiber, in order to clearly capture the details of the structure used for our simulations. Similarly, SEM images of the fibers with side width of 150*μm* and 250*μm* over a 24*μm* × 24*μm* square are shown in Figs. 1(b) and 1(c), respectively. As it is apparent in the SEM images in Figs. 1(b) and 1(c), the fiber with the larger side width 250*μm* has larger feature sizes, (0.9*μm* versus 0.6*μm*), compared with those of the fiber with the side width of 150*μm*.

In order to carry out the experiment to observe the transverse Anderson localization, we coupled a He-Ne laser at the wavelength of 632.8 *nm* to a single mode fiber with the beam radius of 2*μm* and butt-coupled the single mode fiber to the Anderson localized optical fiber. The distance between the Anderson localized optical fiber and the single mode fiber is around 10–20*μm*, depending on how cleanly we can cleave and prepare the Anderson localized optical fiber. The polymer fibers are cleaved according to the recipe suggested in Ref. [23], where we modified their procedure to accommodate our set up. We used an X-ACTO curved carving blade heated to approximately 200 degrees Fahrenheit and the fiber is heated to approximately 100 degrees Fahrenheit. The fiber is directly cleaved by hand on a cutting surface and polished using 0.3*μm* aluminum oxide lapping sheet (LFG03P from Thorlabs), and inspected under a microscope for the final surface quality check.

In order to simulate the propagation of light in the fiber and observe transverse Anderson localization, a monochromatic Gaussian beam of radius *w* is launched into the center of the random fiber at *z* = 0 and propagated along the fiber by numerically solving the wave propagation equation Eq. 1, using the finite difference beam propagation method (FD-BPM) [18, 24].

*A*(

**r**) is the slowly-varying envelope of the primarily transverse electric field

*E*(

**r**,

*t*) = Re [

*A*(

**r**)

*e*

^{i(n0k0z−ωt)}] centered around frequency

*ω*and

*k*

_{0}= 2

*π*/

*λ*.

*n*(

*x*,

*y*) is the (random) refractive index of the optical fiber which is a function of the transverse coordinates, while

*n*

_{0}is average refractive index of the fiber. The forward propagation scheme is implemented using the fourth order Runge-Kutta method [25]. In order to obtain converged, stable, and non-dissipative simulations, we adopted

*δ*≤

*λ*and

*δ*≤

*d*, where

*δ*is size of the transverse grid in the finite-difference numerical scheme.

The size of the steps in the longitudinal direction are picked as *dz* = *αn*_{0}*k*_{0}*δ*^{2}.
$\alpha =1/\sqrt{2}$ is the standard stability condition for the Runge-Kutta method in a uniform medium. For our simulations we choose *α* = 0.02, in order to guarantee stability and also ensure no power dissipation for reliable long distance propagation. Transparent boundary condition [26] is implemented to absorb the wave that hits the lateral boundaries. In order to properly observe the localization effect, the size of the simulation domain should be selected to be considerably larger than the localization length in each case; therefore, the total power in the simulation region will remain unchanged along the fiber for Anderson localized beams. However, as we will discuss later, localization is still possible even if the tail of the optical field reaches the boundaries of the domain; we will show that the sharp index contrast at the fiber-jacket interface can assist the localization, though it can no longer be strictly viewed as Anderson localization.

We also note that we compared the results of the scalar wave equation with the full vectorial beam propagation method and the localization radius calculations were in excellent agreement. In order to increase the numerical efficiency of our large-scale simulations, we chose to use the scalar wave approximation for all the results obtained in this manuscript.

## 3. Impact of the site size on localization radius

We use *ξ* as a measure of the effective beam radius (localization length, when the beam reaches its localization length), given by the variance method [16] as

*x*−

*y*coordinates.

**R**= (

*x*,

*y*) is the transverse position vector and

**R̄**is the vector pointing to the center of the beam, defined as the mean intensity position by

**R̄**= 〈

*A*(

**r**)|

*R̄*|

*A*(

**r**)〉. The optical field is assumed to be normalized according to 〈

*A*(

**r**)|

*A*(

**r**)〉 = 1/2. We already noted that we carried out our measurements according to the same procedure described in our recent work [1]. We carried out our simulations with the physical parameters related to the experiment described above:

*p*= 50%,

*n*

_{1}= 1.49, and

*n*

_{2}= 1.59.

In Fig. 2, we plot *ξ*(*z*) as a function of the propagation distance along the fiber with different values of the site size: *d* = 0.9*μm* corresponding to a fiber side width of 250*μm*, and *d* = 0.6*μm* corresponding to a fiber side width of 150*μm*. The green band represents our experimental measurements of the effective beam radius over an ensemble of 100 separate measurements for 250*μm* wide fibers, where the green-highlighted region corresponds to *ξ*_{avg} ± *σ _{ξ}*.

*ξ*

_{avg}is the average of the measurements and

*σ*corresponds to the standard deviation. Similarly, the red band corresponds to 100 separate measurements for 150

_{ξ}*μm*wide fibers, where the overlap between red and green region is only ∼2–3

*μm*in Fig. 2.

In order to collect the 100 different measurements to carry out the required statistics, we used 20 different pieces of disordered fiber with the length of 5.5cm. We ensured that the fiber thickness remains invariant along each sample, by inspecting each fiber under an optical microscope. For each sample, we moved the incident beam to four different points around a best localized spot with the separation of 10*μm* on a square grid; therefore, 5 different launch positions were explored for each sample, resulting in a total of 100 separate measurements. We should point out that for each sample, the near-field intensity at the output clearly followed the shift in the location of the incident field. The near filed profile for each case was recorded using a 40x objective and a CCD camera (see Ref. [1] for further detail and also the movie (
Media 1) discussed in section 7).

The results of our simulations are averaged on an ensemble of 100 separate simulations, where the profile of each element of the ensemble is randomized differently, corresponding to a fixed fill-fraction of *p* = 50%. We use different seeds in our random generator to generate the 100 different random profiles, while keeping the incident beam fixed at the center of the disordered fiber. For each set of simulations, the region corresponding to *ξ*_{avg} ± *σ _{ξ}* is highlighted in black, where the average localization length

*ξ*

_{avg}and standard deviation

*σ*are calculated over the 100-element ensemble for the field profile after 5cm of propagation. Therefore, the error bars signify the expected variation in the effective beam width among the elements of the “statistically” identical ensemble. The input Gaussian beam initially undergoes diffusive broadening [16,18], until its effective width becomes of the order of the localization length, after which further expansion is halted and the beam remains localized. The mean effective beam radius for the 100 measured near-field beam intensity profiles for the 250

_{ξ}*μm*wide fibers was calculated to be

*ξ*

_{avg}= 30.1

*μm*, with a standard deviation

*σ*= 13.5

_{ξ}*μm*; the numerical simulation resulted in

*ξ*

_{avg}= 29.3

*μm*and

*σ*= 3.3

_{ξ}*μm*. For the 150

*μm*wide fibers, we obtained

*ξ*

_{avg}= 13.4

*μm*and

*σ*= 5.0

_{ξ}*μm*from the experimental measurements, and

*ξ*

_{avg}= 14.2

*μm*and

*σ*= 1.7

_{ξ}*μm*from the numerical simulations, respectively. The initial beam radius for our numerical simulations was 2.4

*μm*, and for our experimental measurements was 2 ± 0.25

*μm*; the experimental value was dictated by the mode field radius of the Thorlabs SMF 630HP fiber used to launch the beam into the disordered fibers.

If we estimate *k _{T}* using the initial FWHM width of the beam
${w}_{0}\approx \sqrt{2\text{ln}2}\times 2.4\mu m$ as

*k*= 2/

_{T}*w*

_{0}[18], we can calculate the mean free path for wave scattering

*l*

^{★}≈ 2.3

*μm*for the 250

*μm*wide fibers, which is more than 2.5 times larger than the site size of

*d*= 0.9

*μm*in these fibers and appears to be consistent with the design characteristics of the problem.

The simulations are in good agreement with the experimental measurements, yet the variation in the experimental measurements of the beam radius is larger than the variation in the theoretical results. The difference can be attributed to the variations in the thickness of the optical fiber in the draw process that result in approximately 20% sample-to-sample variation in the thickness of the 20 samples used in our measurements. We should note that even though we have ensured that the side width of our samples do not change along the 5.5cm length, the samples are cut from a 10-meter long segment of the fiber with an average side width of 250*μm* (or 150*μm* for the samples with the smaller side width), with an approximate variation of about 20%. For our simulations, we used the average value of the side width (250*μm* or 150*μm*). We expect to observe the same level of variation in the site sizes for each fiber (0.6±0.12*μm* for the 150*μm* samples and 0.9 ± 0.18*μm*) for the 250*μm* fiber samples, because of the conservation of mass in the draw process. This fact can also explain the ∼2–3*μm* overlap of the localization radius in the experimental data for the two different side widths in disordered fibers, as reported in Fig. 2.

In order to show that the refractive index profile remains invariant along the fiber, we have taken images of the index profile across the tip of a sample fiber with 1mm side width using an optical microscope at two different locations along the fiber, which are 5cm apart. The optical microscope is zoomed in at 4 different locations across the fiber at *z* =0cm where Figs. 3(a), 3(c), 3(e), and 3(g) are obtained; these images must be compared with Figs. 3(b), 3(d), 3(f), and 3(h), respectively, which are taken at *z* =5cm. Therefore, the pair of Figs. 3(a) and 3(b) correspond to the same (zoomed in) location across the fiber, but 5cm apart along the fiber; they clearly show that the refractive index profile remains invariant along the fiber over the 5cm long sample. Similarly, the Figs. 3(c) and 3(d) pair, Figs. 3(e) and 3(f) pair, and Figs. 3(g) and 3(h) pair correspond to the same (zoomed in) location across the fiber (different location among different pairs), but 5cm apart along the fiber. We note that finding the matching regions is a rather laborious task due to the difficulty in obtaining consistent high quality optical images of the tip of the fiber; the quality of the images are generally limited by the difficulties in the preparation quality of the fiber cleaves as explained in our recent publication [1]. We also note that our choice of the 1mm thickness fiber sample instead of the 250*μm* or 150*μm* fiber samples was both dictated by the higher quality of the cleave in the 1mm sample without having to polish the fiber (polishing lowers the quality of the optical images), as well as the higher image resolutions; we expect that the profiles do not change when the fiber is drawn further down to 250*μm* or 150*μm* thickness.

In Fig. 2, we might develop the impression that the localized beam radius is smaller for the optical fibers with smaller values of the site size (side width of 150*μm*), which is also well supported by the experimental measurements. However, our detailed numerical analysis revealed that the boundary of the fiber in our experiments is playing an important role in setting the observed localization radius. The tails of optical field in the simulation window for the fiber with the 150*μm* side width reach the boundary after a few millimeters of propagation. If a large step index contrast is used at the boundary to confine the field, similar to the experimental conditions where the fiber jacket is “air”, a confinement is observed which in every aspect resembles the standard Anderson localization, including the exponential decaying tails of the field. Moreover, the simulations agree closely with the localization radius measured over the 100 samples. Considering the fact that two-dimensional systems are always localized, we extended the simulation window to observe the true localization radius of this geometry (without the impact of the boundary), and observed that it is substantially larger than what we present in Fig. 2. In fact, in the absence of the boundary step index, when the fiber side width is taken to be much larger with the same small site size of the 150*μm* fiber, the localization radius is around 50*μm*, which is even larger than that of the fiber with the larger values of site size (side width of 250*μm*). The impact of the boundary on the localization radius is very important where the signatures (such as the exponential decay of the tails of the localized field) remain similar to those of the ordinary Anderson localization; such an impact has already been explored in a similar context in finite 1D and 2D disordered lattices in Refs. [27, 28].

Szameit et al. [27] have shown that the localization effect is reduced near the truncated boundary of a 1D lattice and in order to maintain the same degree of localization, a higher level of disorder is required near the boundary. Same way, Jović et al. [28] explore the boundary effect in 2D disordered lattices and draw similar conclusions to those of Ref. [27]. Our observations are consistent with and somewhat complement those reported by Refs. [27, 28]. We observe the impact of the boundary on the localization radius even when the beam is launched at the center of the fiber, away from the boundary. The key point is that the beam for the specific case of the 150*μm* fiber with the small approximate site size of 0.6*μm* reaches the boundary during the diffusive expansion stage (before reaching the localization stage) and the reflections from the boundary affect its final stabilized radius, yet its profile resembles that of the Anderson localized beam.

We emphasize again that for the case of the 250*μm* fiber with the approximate site size of 0.9*μm*, the choice of boundary condition (absorbing boundary condition versus an air jacket) makes virtually no difference in our results, because the beam that is launched in the center only excites the localized modes that are not impacted by the boundary. However, for the 150*μm* fiber with the small approximate site size of 0.6*μm*, if we do not consider an air jacket in our simulations (and use absorbing boundary condition instead), we observe the power absorption of about 40%, after 5cm of propagation; therefore, the presence of air jacket is required to get a consistent result that compares with the experimental measurements.

In Fig. 4, the localization radius in the polymer disordered fiber with the side width of 250*μm* (0.9*μm* site sizes) at the incident wavelength of 405nm is compared with the one at the wavelength of 632.8nm. Our calculations show the shorter incident wavelength results in a smaller localized beam radius. This is expected, considering the fact that Maxwell’s equations are scale invariant and a shorter incident wavelength is equivalent to larger site sizes, which also results in a smaller localized beam radius. For the wavelength of 1550nm which is interesting for telecommunication applications, the localized beam radius is so large that cannot properly localized within the boundaries of our fiber with the side width of 250*μm*. In order to observe localization at the wavelength of 1550nm, the side width of the fiber should be expanded to beyond 500*μm*.

In order to further confirm that our observations are in agreement with the standard signature of the Anderson localization, i.e. exponential decay of the tails of the field in the presence of disorder, we plot a cross section of the intensity profile of the localized beam, averaged over 100 samples in dB units. Fig. 5(a) shows a comparison between the experimental results for fibers with side width of 150*μm* and 250*μm*. Fig. 5(b) shows a comparison between the experimental and numerical results for fibers with side width of 150*μm*, where the difference between simulation and experiment is caused by the larger variation in the experimental results (see Fig. 2), and also the noise in the CCD beam profiler at low intensities. Similar results for the 250*μm* fiber were already reported in our recent publication [1].

## 4. Impact of the fill-fraction on the localization radius

Our choice of the fill-fraction parameter *p* = 50% is optimal, in order to obtain the minimum localization length in these fibers. In order to examine the effect of the fill-fraction *p* on the localization length, we numerically study the case of *p* = 40% and compare the results with *p* = 50% which was outlined in the previous section. In Fig. 6, we show the results of our simulations, where we plot *ξ*(*z*) as a function of the propagation distance along the fiber. Each curve, relating to a different value of the fill-fraction parameter *p*, represents the average of 100 independent simulations. The error bars indicate the standard-deviation of *ξ*(*z*) calculated over each 100-element ensemble. The localization radius for a given value of *p* always fluctuates around an average value because each realization of disordered fiber has a different refractive index profile. From Fig. 6, it is clear that the lower localization length is obtained for *p* = 50%. Any decrease in the value of the fill-fraction from the optimal choice of *p* = 50% results in an increase in the localization length. This observation agrees well with the physical intuition that *p* = 50% increases the overall probability of scattering and should result in a smaller localization length. We note that in order to obtain localization at smaller values of *p*, we needed to increase the side width of the simulation area in order to prevent the scattered light to reach the absorbing boundaries (250*μm* for *p* = 50% and 300*μm* for *p* = 40%), and for lower fill-fractions, the domain size needs to increase accordingly. While we verified this for a few samples, we decided against carrying out a full scale analysis for smaller values of *p*, due to the huge computational cost, which would have amounted on months of simulations on a large cluster. However, as shown in section 6, we carried out full simulations for *p* = 30%, as well, because the required simulation window was smaller due to the larger values of index contrast, resulting in smaller localization radius. As we discussed before, above the percolation threshold (*p* > 59.27%), index-guiding by the higher-index localized clusters can mask any effects that can be attributed to Anderson localization. In practice, we have observed that for the choice of parameters used for our simulations, the effective localized beam radius increases monotonically with increasing value of the fill-fraction *p* beyond the optimal value of *p* = 50%.

In the presence of large disorder, for example at *p* = 50%, little variation is expected among the individual elements of the ensemble due to a self-averaging behavior, which was also pointed out in our recent publication [1]. This “self-averaging” behavior is generally obtained from the theory of wave localization under appropriate conditions (see for example Refs. [29, 30]) and is responsible for the small size of the error bars at *p* = 50%. However, when the amount of disorder is low, such as for small values of *p* in our work or for small refractive index contrast in the case of Ref. [18], self-averaging may not hold and wave localization is only meaningful in a statistical averaging sense.

## 5. Impact of refractive index contrast

In this section, we show that lowering the index contrast Δ*n* = (*n*_{1} − *n*_{2}) increases the localization length. If the index contrast is too low, the beam of light will expand until it reaches the edges of the sample and transverse localization may never materialize in practice. This can be observed in Fig. 7, where the effective beam radius for *p* = 50% and different values of refractive index contrast Δ*n* are shown. Our simulations clearly show that for a waveguide with the side dimension of ≈ 400*λ* (250*μm* at *λ* =632.8 nm), the wave does not get localized for the refractive index contrast of 0.01, i.e. the localization length is larger than the side dimension of the structure, so the light is absorbed by the absorbing boundaries. We remind that two-dimensional random systems are always localized, however, the localization length can be larger than the domain of interest, such as discussed above. In order to observe the localization effect for the low refractive index contrast of 0.05, we need to use a larger side dimension for the waveguide of approximately 350-*μm* (555*λ*). We note that for all the simulations in this section, we only used absorbing boundary condition; therefore, the boundary did not impact the localization condition.

## 6. Glass Anderson localized optical fibers

In order to use Anderson localized optical fibers for possible applications in the telecommunication wavelengths, we need to have a disordered fiber with low loss materials at the telecommunication window of the spectrum: therefore, glass fibers are more desirable than polymer fibers. Here we numerically investigate the glass Anderson localized optical fibers with random air-holes. As far as our numerical simulations are concerned, everything remains the same, except the index contrast, which is now substantially larger, resulting in a smaller localization radius, which is more desirable. The refractive index profile is implemented based on the same procedure described for polymer optical fiber, yet the refractive index of sites are randomly picked as *n*_{2} = 1.5 and *n*_{1} = 1.0 for the glass host and random air-hole sites, respectively. The side width of each disordered fiber is 100*μm* and site sizes are 0.6*μm*. We note that for the glass-air structure with the refractive index contrast of 0.5, the fiber dimensions can be chosen to be smaller compared with that of the polymer fiber, because the localization radius is smaller.

In Fig. 8, the calculated beam radius *ξ*(*μm*) versus propagation distance for different values of fill-fraction are plotted, where each simulation is again performed for 100 different realization of randomness (100 different refractive index profiles). Similar to the case of the disordered polymer fiber, we observe that the localization radius is lowest for the optimal fill-fraction of *p* = 50%. We also observe that the beam radius of the localized beam for the glass Anderson localized optical fibers with random air-holes is smaller than the beam radius in the polymer fibers, because of the larger index contrast. These observations are consistent with our results in the previous section on the impact of the refractive index contrast on the localization radius. Another important advantage of the glass disordered optical fibers is that the larger index contrast results in a stronger self-averaging behavior, reducing the standard-deviation in the value of the beam radius; therefore, glass Anderson localized optical fibers provide a more reliable and predictable behavior for potential applications in optical fiber communications.

The exponential decay of the tail of averaged intensity for different values of fill-fraction are shown in Fig. 9, which presents a clear proof of Anderson localization in each case. As the exponential tails show, for lower values of fill-fraction, the decay coefficient is smaller and the wave expands farther. In the case of optimal localization for *p* = 50%, the decay coefficient is the largest and the wave gets localized much faster. It must be noted that the region with an appreciable intensity difference between the *p* = 50% and *p* = 30% structures has lower intensity than the peak value by at least 70dB; therefore, it will be extremely hard to distinguish between the beam profiles of the two structures, using conventional experimental techniques.

## 7. Conclusions

We explored the effect of site size, fill-fraction, and refractive index contrast for polymer optical fibers. We showed both numerically and experimentally that the large refractive index step at boundary of the fiber results in an anomalous reduction in the value of the localized beam radius, when the site sizes are decreased; this observation is consistent with and somewhat complement those reported by Refs. [27,28]. Our results show that the boundaries of the fiber assist the wave localization of the weakly localized modes, even though the incident beam is at the center of the fiber and away from boundaries. We also showed that *p* = 50% is the optimum fill-fraction to have the lowest value for the localized beam radius and also the minimal impact of boundaries on the wave localization. Lowering the refractive index contrast of the materials from which the fiber is drawn results in the increase of the localized beam radius. Using numerical simulations, we observed that a glass host with disordered airholes results in a substantially reduced value of the localized beam radius. The impact of other parameters in glass disordered fibers were similar to those of the polymer disordered fibers.

We would also like to point out that a typical reliable simulation of transverse Anderson localization for our optical fibers requires a transverse area in the range of ∼ 10^{5} − 10^{6} *λ*^{2}, and ∼ 10^{6} − 10^{7} steps in the longitudinal direction, which is computationally intensive. Therefore, an ensemble of 100 disordered fiber simulations to obtain proper statistics requires approximately 10^{5} CPU hours. The simulations were carried out on a local large HPC cluster consisting of 142 Nehalem 5550 nodes (1,136 cores), with 24 gigabytes of memory per node and a high-throughput, low-latency Infiniband network.

As yet another evidence of the strong transverse Anderson localization, we scan the in-coupling single-mode fiber across the input facet of the our disordered fiber, and image the near-field output from the disordered fiber on a CCD camera, using a 40x objective. The recorded movie clearly shows that the localized beam at the end facet of the fiber follows the location of the in-coupling beam ( Media 1).

## Acknowledgments

This research is supported by the grant number 1029547 from the National Science Foundation. The authors would like to acknowledge David J. Welker from Paradigm Optics Inc. for providing the initial fiber segments and the re-drawing of the final optical fiber. The authors would like also to acknowledge Jason Bacon for assisting in using the high throughput computing facilities at the University of Wisconsin-Milwaukee.

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