Scaling analysis of transverse Anderson localization in a disordered optical waveguide
SScaling analysis of transverse Anderson localization in a disordered optical waveguide
Behnam Abaie and Arash Mafi ∗ Department of Physics & Astronomy, University of New Mexico, Albuquerque, NM 87131, USACenter for High Technology Materials, University of New Mexico, Albuquerque, NM 87106, USA (Dated: November 7, 2018)The intention of this manuscript is twofold. First, the mode-width probability density function(PDF) is introduced as a powerful statistical tool to study and compare the transverse Andersonlocalization properties of a disordered one dimensional optical waveguide. Second, by analyzingthe scaling properties of the mode-width PDF with the transverse size of the waveguide, it isshown that the mode-width PDF gradually converges to a terminal configuration. Therefore, itmay not be necessary to study a real-sized disordered structure in order to obtain its statisticallocalization properties and the same PDF can be obtained for a substantially smaller structure.This observation is important because it can reduce the often demanding computational effort thatis required to study the statistical properties of Anderson localization in disordered waveguides.Using the mode-width PDF, substantial information about the impact of the waveguide parameterson its localization properties is extracted. This information is generally obscured when disorderedwaveguides are analyzed using other techniques such as the beam propagation method. As anexample of the utility of the mode-width PDF, it is shown that the cladding refractive index canbe used to quench the number of extended modes, hence improving the contrast in image transportproperties of disordered waveguides.
PACS numbers: 42.25.Dd, 42.82.Et, 72.15.Rn
I. INTRODUCTION
Anderson localization has been a topic of great sci-entific interest for over five decades . It has beensuccessfully demonstrated in highly scattering classicalwave systems including acoustics , electromagnetics ,optics , as well as quantum optical systems, such asatomic lattices and propagating photons . Trans-verse Anderson localization of light was first suggestedby Abdullaev, et al. and De Raedt, et al. and wasconfirmed experimentally by Schwartz, et al. . In par-ticular, De Raedt, et al. analyzed an optical waveguidewith a transversely random and longitudinally invariantrefractive index profile. They showed in this quasi-two-dimensional (quasi-2D) system that an optical beam canpropagate freely in the longitudinal direction while be-ing trapped (Anderson localized) in the transverse direc-tion. Transverse Anderson localization of light has sincebeen observed in various quasi-one-dimensional (quasi-1D) and quasi-2D optical systems .Most recently, Karbasi, et al. reported the first obser-vation of transverse Anderson localization in disorderedoptical fibers . The disordered optical fibers wereused for image transport and it was shown that the highquality image transport was achieved because of, not inspite of, the high level of disorder and randomness in thefiber . A high-quality imaging optical fiber based ontransverse Anderson localization requires a narrow and uniform point spread function (PSF) across the tip ofthe fiber. The width of the PSF is determined by thelocalization radius; it has been argued that a large re-fractive index contrast is essential in ensuring that thelocalization radius is sufficiently small and does not varyappreciably across the fiber profile . The large re- fractive index fluctuations of aorund 0.1 in the disorderedpolymer fiber by Karbasi, et al. ensured a strong and uni-form transverse localization across the tip of the fiber forhigh quality image transport.It has been previously shown that coherent waves inone-dimensional (1D) and and two-dimensional (2D) un-bounded disordered systems are always localized . Forbounded 1D and 2D systems, if the sample size is consid-erably larger than the localization radius, the boundaryeffects are minimal and can often be ignored . Adisordered waveguide can support both localized and ex-tended modes simultaneously because it is transverselybounded. By the extended, we refer to those modeswhich span the transverse dimension(s) of the waveg-uide whose transverse size also increases with increasingthe transverse dimension(s) of the waveguide. However,as we shall see later, the occurrence of these modes be-comes less probable as the transverse size of the waveg-uide becomes larger; and becomes improbable when thedisordered waveguide becomes sufficiently wide to be ap-proximated by an unbounded disordered system.Because the refractive index of a disordered waveg-uide is random, the properties of the localized and ex-tended modes should be studied using statistical tech-niques. The most relevant physical quantity that char-acterizes the localization properties of disordered waveg-uides is the mode width, which is defined rigorously inEq. 3 and characterizes the transverse size of a guidedmode. Therefore, detailed understanding of the mode-width statistics is the gateway to uncovering the local-ization properties. In this manuscript, we report on the probability density function (PDF) of the mode width asa powerful tool to study Anderson localization in disor-dered waveguides. Using the mode-width PDF : we canobtain the average width of the localized modes which a r X i v : . [ phy s i c s . op ti c s ] J un determines the size of the PSF for a disordered imagingfiber; we can obtain the standard-deviation of the mode-width distribution which determines the uniformity ofthe transported image across the tip of the fiber; we canobtain the distribution of the extended modes which af-fect the image contrast; and we can study the impactof the total size of the structure and the cladding indexcontrast on the localized and extended modes.The mode-width PDF contains all the relevant in-formation on the mode-width statistics of a disorderedwaveguide ensemble. However, computing the mode-width PDF is a challenging numerical problem. In or-der to compute the PDF, a large number of differentwaveguide samples are generated in a given ensemble forproper statistics. The guided modes for each waveguideare calculated and the corresponding mode-width val-ues are extracted to generate the PDF. Calculating theguided modes of even a single fiber structure may be-come highly challenging. For example, the V-number ofthe disordered polymer fiber in Ref. with air claddingis approximately 2,200 at 405 nm wavelength resulting inmore than 2.3 million guided modes . Recall that theV-number is given by V = πtλ (cid:113) n − n , (1)where λ is the optical wavelength, t is the core diameterof the fiber (or the core width for the case of a 1D slabwaveguide), and n co ( n cl ) is the effective refractive indexof the core (cladding). The total number of the boundguided modes in a step-index optical fiber is ≈ V / we employ the power of scalinganalysis and study the mode-width PDF as a function ofthe total size of the waveguide. We show that the PDFconverges to a terminal form as the waveguide dimen-sions are increased. As such, the mode-width PDF of areal-sized disordered waveguide may be obtained by sim-ulating a waveguide ensemble of a considerably smallerdimensions. Moreover, we will show that the region ofthe PDF corresponding to the Anderson localized modesconverges to its terminal form considerably faster thanthe entire PDF as the size of the structure grows larger,while the region of the PDF corresponding to the ex-tended modes is rather generic looking. Therefore, toobtain the most useful information corresponding to theAnderson-localized region of the PDF, it is often possibleto even further reduce the size of the structures resultingin substantial reduction in computational effort. In cer-tain systems, this may actually turn the computationalproblem from nearly impossible to one that can be han-dled by moderate sized computer clusters. The focus of this manuscript is to establish a frame-work for a comprehensive analysis of the mode-widthstatistics for transverse Anderson localization in opticalfibers in the future. A mode-width distribution that ismore strongly peaked at narrower mode-width values isfavored because it can result in a smaller PSF for imag-ing applications. Moreover, a narrower distribution of themode-widths indicates PSF uniformity across the fiber.In order to lay the groundwork for understanding thescaling behavior of the statistical distribution for boththe localized and extended modes in a 2D Anderson lo-calizing optical fiber, we have decided to present a com-prehensive characterization of a 1D Anderson localizedoptical waveguide in this manuscript. This exercise isquite illuminating as it sheds light on the scaling behav-ior of the statistical distribution of the mode widths andshows the extent of information that can be extractedfrom such an exercise. The detailed analysis of a 2Ddisordered fiber structure will be presented in a futurepublication.Here, we have chosen to calculate the transverse elec-tric (TE) guided modes of the disordered waveguide us-ing finite element method (FEM) presented in Refs. .Similar observations can be drawn for transverse mag-netic (TM) guided modes, but we limit our analysis to TEin this paper for simplicity. The appropriate partial dif-ferential equation that will be solved in this manuscriptis the paraxial approximation to the Helmholtz equationfor electromagnetic wave propagation in dielectrics ∇ A ( x T ) + n ( x T ) k A ( x T ) = β A ( x T ) (2)where A ( x T ) is the transverse profile of the (TE) electricfield E ( x T , z, t ) = A ( x T ) exp( iβz − iω t ), β is the prop-agation constant, n ( x T ) is the (random) refractive indexof the waveguide, x T is the one (two) transverse dimen-sion(s) in 1D (2D), ω = ck , and k = 2 π/λ where c is the speed of light in vacuum. Equation 2 is an eigen-value problem in β and guided modes are those solutions(eigenfunctions) with β > n k . For each guided mode,the mode width is defined as the standard deviation σ ofthe (1D) normalized intensity distribution I ( x ) ∝ | A ( x ) | of the mode according to σ = (cid:18)(cid:90) + ∞−∞ ( x − ¯ x ) I ( x ) dx (cid:19) / , (3)where the mode center is defined as¯ x = (cid:90) + ∞−∞ x I ( x ) dx. (4) x is the spatial coordinate across the width of the waveg-uide and the mode intensity profile is normalized suchthat (cid:82) + ∞−∞ I ( x ) dx = 1. σ is a measure of width of themodes i.e. a larger σ signifies a wider mode intensityprofile distribution.Finally, we would like to contrast the power of the sta-tistical simulation in the modal method with that of the FIG. 1. Sample refractive index profiles of (a) ordered and(b) disordered slab waveguides are shown. finite difference beam propagation method (FD-BPM) employed earlier in Refs. . When using the FD-BPMto analyze the Anderson localization in optical waveg-uide, one is always worried about the extent to which theresults are dependent on the shape and size of the inputbeam. The modal description is superior because it reliessolely on the physics of the disordered system and is in-dependent of the properties of the external excitation .As it will be shown in this manuscript, the mode-widthPDF can reveal the subtle interplay between Andersonlocalization, step-index waveguiding, and support of non-localized extended modes, all of which can be present si-multaneously in a disordered waveguide–something thatcannot be achieved using the FD-BPM analysis. II. 1D DISORDERED LATTICE INDEXPROFILE
A 1D ordered optical lattice can be realized by periodi-cally stacking dielectric layers with different refractive in-dexes on top of each other. Fig. 1(a) shows the refractiveindex profile of a periodic 1D optical waveguide where n , n , and n c correspond to the lower index layers, higherindex layers, and cladding, respectively. In order to makea disordered waveguide, randomness can be introduced indifferent ways in the geometry or refractive index profileof a waveguide structure. For example, in Refs. thethickness of the layers is randomized around an averagevalue. In this manuscript, we adopt a different random-ization method: we keep the thickness of all dielectriclayers identical but assign a refractive index value of n or n to each layer with a 50% probability. This is thesame as the method prescribed by De Raedt, et al. inrandomizing a disordered 2D waveguide which was alsoadopted by Karbasi, et al. to fabricate an Andersonlocalizing fiber. As we explained in the Introduction,our intention is to extend our current analysis to 2D dis-ordered Anderson localizing fibers in the future and wewould like to stay as close as possible to the practicaldisordered 2D structure for proper comparison.Fig. 1(b) shows the refractive index profile of a disor-dered 1D optical waveguide. In Fig. 2(a), we plot twoguided modes of a 1D periodic waveguide with the high-est propagation constant, where we have assumed that n = 1 . n = 1 .
50, and n c = 1 .
49. These two modesbelong to a large group of standard extended
Bloch pe-
FIG. 2. Typical mode profiles for (a) an ordered slab waveg-uide where each mode extends over the entire waveguide, and(b) a disordered slab waveguide, where the modes are local-ized. riodic guided modes supported by the ordered opticalwaveguide, which are modulated by the overall mode pro-file of the 1D waveguide . The total number of guidedmodes depends on the total thickness and the refractiveindex values of the slabs and cladding. The key point isthat each mode of the periodic structure extends over theentire width of the waveguide structure. A similar exer-cise can be done with a 1D disordered waveguide, wheretwo arbitrarily selected modes are plotted in Fig. 2(b)using the same refractive index parameters as that of theperiodic waveguide. It is clear that the modes becomelocalized in the disordered 1D waveguide. While thereare variations in the shape and width of the modes, themode profiles shown in Fig. 2(b) are typical.It is important to note that the disordered core of thelattice is sandwiched between a cladding with a refractiveindex of n c that can also be adjusted to resemble exper-imental situations where a waveguide is surrounded byair or a dielectric with a refractive index higher than airor even n (but always less than n to ensure waveg-uiding). As we will see later, the value of n c influencesthe mode-width distribution of the extended modes in a1D Anderson localized waveguide and should be carefullystudied in practical implementations of such structures,e.g. for image transport . III. ANALYSIS OF THE MODE-WIDTH PDF
In the absence of localization, guided modes are Blochperiodic and extend over the entire width of a waveguideas shown in Fig. 2(a). In this case, the confinement ismerely due to the total internal reflection at the effec-tive index step between the waveguide and the cladding.In Fig. 3, we plot the mode-width PDF for the periodicwaveguide with N slabs, for N = 20, 40, 60, 80, and100. The width of each slab is equal to the wavelength d = λ , n c = n = 1 .
49, and n = 1 .
50 so the index step∆ n core = 0 .
01 where ∆ n core = n − n . The horizon-tal axis is in units of λ and the vertical axis is in units of1 /λ such that the PDF integrates to one (unit area underthe PDF curve). The width of the cladding is assumedto be 25 λ on each side of the waveguide. The guidedmodes decay exponentially (in the transverse direction)in the cladding. We have verified that the 25 λ claddingis sufficiently wide so that the exponential decay in the FIG. 3. Mode-width PDF of an ordered waveguide defined as∆ n core = 0 .
01 and ∆ n clad = 0, for N = 20, 40, 60, 80, and 100slabs. Mode width increases as the number of slabs increases,so the average mode width scales proportional to the size ofthe structure.FIG. 4. Mode-width PDF of a disordered waveguide definedby ∆ n core = 0 .
01, ∆ n clad = 0, for N = 20, 40, 60, 80, and100 slabs. The inset is the magnified version of the localizedpeaks of the PDFs . Saturation of the PDF beyond N sat ≈ cladding combined with the Dirichlet boundary conditionof vanishing mode profile imposed at the outer edge ofeach cladding properly approximate an infinite cladding.Figure 3 shows that for the periodic slab waveguide, themode widths are determined by the width of the waveg-uide as can be seen clearly in Fig. 2(a). Therefore, themode widths, on the average, scale linearly with the sizeof the waveguide structure and the peak of the PDF shiftsto larger values of mode width as the waveguide becomeswider.For a disordered waveguide, the scaling behavior ofthe PDF with the size of the waveguide is completelydifferent from that of the periodic waveguide shown inFig. 3. When Anderson localization comes into playdue to the disorder in the structure of waveguide, mostguided modes become transversely localized as shown in FIG. 5. ∆ n core = 0 .
02 is increased in comparison to Fig. 4;The small-mode-width peak of the PDF shifts towards smallermode width values indicating a stronger localization for alarger index contrast in the disordered waveguide. A mag-nified version is shown as an inset. N sat ≈
40 is smaller for astronger transverse scattering.
Fig. 2(b), while a few extended guided modes may stillbe supported depending on the waveguide configuration.As the number of slabs is increased, the PDF saturatesto a terminal form. In Fig. 4, we show the PDF for anensemble of disordered waveguides with N slabs definedby n c = n = 1 .
49, and n = 1 .
50 (∆ n core = 0 . d = λ . The PDFs are plotted for N = 20, 40, 60, 80, and100. The PDF shows two localized peaks at width val-ues less than 4 λ with a long tail signifying the extendedmodes. The shape of the PDF changes with the numberof slabs; however, it remains nearly unchanged beyond N sat ≈
60. The near saturation of the PDF beyond acritical number of slabs N sat is of utmost importance fortwo reasons: 1) N sat can be viewed as the effective trans-verse scale (waveguide width) beyond which the averagelocalization dynamic is no longer dictated by the bound-ary; and 2) if we need to calculate the PDF for a widedisordered waveguide, it is sufficient to simulate a waveg-uide with only N sat slabs because it gives the same PDF;therefore, the computational effort can be significantlyreduced. In order to see the saturation behavior of thePDF more clearly, the inset shows a magnified versionof the PDFs, which is zoomed in at smaller mode widthvalues. The transition to the terminal form of the PDFis clearly observed in the Anderson localized region ofthe PDFs where mode width is approximately less than ≈ λ . A. Impact of the index difference in the disorderedwaveguide
The results shown in Fig. 4 are for the waveguide indexdifference of ∆ n core = 0 .
01 ( n = 1 .
49 and n = 1 . n core = 0 .
02 ( n = 1 .
48 and n = 1 .
50) in Fig. 5and its magnified inset. The small mode width peakof the PDF relating to the Anderson localized modes inFig. 5 has shifted to lower mode width values comparedwith Fig. 4 because of the larger ∆ n and stronger trans-verse scattering. Also, the convergence of the PDF hap-pens with a smaller number of slabs, i.e., N sat is smallerwhen ∆ n is larger. Otherwise, the qualitative behav-ior of the PDFs are similar in the sense that the bothwaveguides support localized and extended modes simul-taneously. B. Impact of the boundary index difference
In the previous figures (Figs. 4, 5), the refractive in-dex of the cladding n c is assumed to be the same asthe refractive index n of the lower index layers. Forthe practical 2D disordered optical fiber of Ref. , thecladding of the structure is air with a refractive index of n c = 1, which is considerably smaller than the lower in-dex n = 1 .
49 of the fiber. The cladding index of the fibercan be controlled by an additional cladding layer or an in-dex matching gel. As such, understanding the impact ofthe refractive index of the boundary on the guided modestructure of the disordered waveguide is of practical im-portance. A lower cladding index increases the effectiveV-number of the whole disordered waveguide, resulting inan increase in the total number of modes. In this section,we will investigate the impact of the cladding refractiveindex on the distribution of the localized and extendedmodes, as well as on the scaling and eventual convergenceof the mode-width PDF with the transverse size of thewaveguide. Moreover, we will show that the impact of achange in the cladding index is primarily on the extendedmodes, while the localized modes are hardly affected bychanges in the cladding index step.In Fig. 6, we consider a disordered waveguide with∆ n core = 0 .
01 ( n = 1 .
49 and n = 1 .
50) and ∆ n clad =0 .
01 where ∆ n clad = n − n c . This waveguide is identicalin structure to that of Fig. 4 except for ∆ n clad . The maindifference between Fig. 6 and Fig. 4 is in the distributionof the extended modes. The presence of a larger claddingindex difference in Fig. 6 results in a greater number ofextended modes which appears as a large bump in thePDF for N = 40 slabs and smooths down when the PDFsaturates to the terminal shape for large N . Another im-portant difference between Fig. 6 and Fig. 4 is that theconvergence of the PDF in Fig. 6 (larger ∆ n clad ) happensat a larger value of N. The inset in Fig. 6 (∆ n clad = 0 . n clad = 0). Whilethe two figures are visually similar, the localized peakof Fig. 4 is observed to be clearly higher when comparing FIG. 6. Mode-width PDF of a disordered waveguide definedas ∆ n core = 0 .
01, ∆ n clad = 0 .
01, and N = 40, 80, 120, 160,200 slabs. A lower cladding index significantly changes thedistribution of the extended modes and PDF saturates at amuch larger value of N ( N sat ≈ the vertical scales of the PDF plots. This is due to thefact that the total area under PDF is normalized to unityand the larger number of extended modes in Fig. 6 resultsin a reduction in the overall amplitude of the PDF overthe entire domain. As such, the PDF in its present formcannot provide a fair comparison between the localizedmode structure of Fig. 6 and Fig. 4. We will get back tothis important point later in this section.In Fig. 7 and Fig. 8 we investigate the effect of furtherlowering n c to have ∆ n clad = 0 .
02 and ∆ n clad = 0 . n clad results inan increase in the number of extended modes, empha-sizing that we have yet to show in this section that thelocalized modes are not affected by the change in thecladding index. Moreover, an increase in the claddingindex difference results in a delayed convergence of thePDF to its terminal form resulting in a larger value of N sat . In fact, it can be seen that N sat ≈
200 for Fig. 7and N sat (cid:29)
200 for Fig. 8.Our discussion will not be complete without discussingthe reverse effect of raising n c above n , hence a nega-tive value of ∆ n clad = − . n c < n ; otherwise, no guiding mode would exist.Figure 9 (∆ n clad = − . n clad = 0. It is clear that raising n c above n (negative ∆ n clad ) removes a considerable number ofextended modes from the system. Recall that the PDFin Fig. 4 showed two distinct peaks in the region nearthe localized modes and raising n c above n appears toremove the second localized peak (with a larger mode FIG. 7. The same as Fig. 6 except ∆ n clad = 0 .
02. The PDFsaturates to its terminal shape at a larger number of slabs( N sat ≈ n clad = 0 .
04. Furtherdecreasing the cladding index delays the saturation of thePDF to larger values of N ( N sat (cid:29) width). Therefore, we conclude that a negative ∆ n clad not only removes many of the extended modes, it alsoremoves those more weakly localized modes associatedwith the second peak in Fig. 4.Previously in this Section, we mentioned that it is hardto judge the impact of the cladding refractive index onthe width distribution of the localized modes by com-paring the PDFs from two different waveguides. Thereason is that the total area under PDF is normalized tounity and different waveguide parameters result in differ-ent number of modes. As such, we need to come up witha method to clearly differentiate between the impact ofthe cladding index on the extended modes versus local-ized modes across different lattice parameters. In orderto do this, it is best to use the normalized PDF which isthe PDF multiplied by a constant factor such that totalarea under the normalized PDF curve equals the averagenumber of modes in each class of random waveguide.In Fig. 10, we plot the normalized PDF for disor- FIG. 9. The same as Fig. 4 except ∆ n clad = − . n not only reduces the probabilitydensity of the extended modes, but also diminishes the secondpeak of the localized regime. The inset represents a magnifiedversion. dered waveguides with ∆ n core = 0 .
01 ( n = 1 .
49 and n = 1 . d = λ , and N = 200 slabs. Different curvesin Fig. 10 correspond to different values of ∆ n clad rang-ing from -0.005 to 0.04. The curves belonging to thelargest three values of ∆ n clad are not fully saturated tothe terminal PDF because N = 200 is smaller than N sat in these cases, hence resulting in a bump in the extendmode region. The inset shows the magnified version ofthe same figure in the region of the localized modes. Fig-ure 10 clearly shows that increasing the cladding indexstep merely introduces new extended modes and the lo-calized modes are hardly affected. We re-emphasize theutility of the normalized PDF in revealing this importantbehavior. The case of ∆ n clad = − .
005 is quite interest-ing, as it can be seen that raising n c above n strongly de-couples extended modes and trims the large-mode-widthedge of the localized mode region of the PDF. Therefore,if having more localized modes versus extended modes isa desired outcome of a design, a small or even negative∆ n clad is preferable. C. Impact of the unit slab thickness
In the previous sections, we learned much about thebehavior of the mode-width PDF for various refractiveindex configurations in the core and cladding. In all pre-vious simulations, we assumed that the width of eachslab is equal to the wavelength d = λ . However, themode-width PDF depends on the value of d as well. Un-derstanding the behavior of the mode-width PDF as afunction of d is quite important because d is a parameterthat can be used to optimize the disordered lattice givenan objective function. For example, our objective can beto obtain the smallest possible mean value of the modewidth calculated using the PDF, where d in addition tothe refractive indexes can be used as an optimization FIG. 10. Normalized PDF for disordered waveguides definedby ∆ n core = 0 . N = 200, and ∆ n clad = -0.005, 0.01, 0.02,0.03, and 0.04. The magnified inset clearly shows that thestatistics of the localized modes is independent of the claddingindex unless for a negative ∆ n clad . parameter. In Figs. 11 and 12 we plot the normalizedPDFs for disordered lattices defined by n c = n = 1 . n = 1 .
50. The value of the unit slab thicknessis different in each case, taking the values ranging over d = 0 . λ − . λ , while keeping the total waveguide widthequal to 200 λ . Therefore, the case with d = 0 . λ cor-responds to N = 400, while the case with d = 2 λ cor-responds to N = 100. As we discussed before, the nor-malized PDF integrates to the total number of modes,which varies from an average of 49 modes for the caseof d = 0 . λ to an average of 39 modes for the case of d = 2 . λ . In Fig. 11, it is clear that d = 0 . λ corre-sponds to a normalized mode-width PDF with a longtail in the extended mode region. When d is increased to d = λ and further to d = 1 . λ , the extended tail is grad-ually lowered contributing more to the localized region.It seems as if that the extended modes trade off theirrole with the localized modes of the second PDF hump.Another important observation is that the localized peakshifts slightly towards the smaller mode width values asthe unit slab thickness increases.In Fig. 12, the normalized PDFs for disordered lat-tices with d = 2 λ and d = 2 . λ are shown. The normal-ized PDF in these figures exhibit sharp peaks, which aremarkedly different from the PDFs we have observed inprevious figures. Below, we will argue that these sharppeaks are mainly due to step-index waveguiding behaviorof individual discrete waveguides accidentally formed inthe random structure. In order to understand this, con-sider the case of d = 2 λ , where discrete local waveguidesof widths t = 2 λ , t = 4 λ , t = 6 λ , etc appear, respec-tively, with decreasing probability. The V-number of theslab waveguide from Eq. 1 ( n co = n and n cl = n ) isequal to 1.09 for t = 2 λ and is proportionally larger for t = 4 λ , t = 6 λ , etc. We recall that the single-mode cut-off condition for the TE modes of a slab waveguide is V = π/
2. Therefore, t = 2 λ is near cut-off and t = 4 λ FIG. 11. Normalized mode-width PDF of disordered waveg-uides defined by ∆ n core = 0 .
01, ∆ n clad = 0, N = 200, andunit slab thickness of d = 0 . , . , and 1 . λ . Overall, thewaveguides show stronger localization for a thicker unit slab.FIG. 12. The same as Fig. 11, except that the thickness ofthe unit slabs in the structure of the disordered waveguidesis larger. The formation of local waveguides leads to sharpdiscrete peaks in the normalized PDF. or larger are multimode. The large V-number in thesewaveguides results in highly confined modes that cannotinteract with the modes of the neighboring waveguidesto allow for randomized interaction to form Anderson lo-calized modes. Therefore, in addition to the extendedmodes and the Anderson localized modes that stem fromthe more-loosely-bound modes, we encounter the regu-lar step-index waveguiding modes in the form of sharpdiscrete peaks. The peaks are centered at mode-widthvalues of the corresponding waveguides of discrete thick-ness values of t = 2 λ , t = 4 λ , t = 6 λ , etc. The decreasingvalues of the discrete peaks in the PDF are indicativeof the decreasing probability of having local waveguideswith t = 2 λ , t = 4 λ , t = 6 λ , etc, respectively. This sit-uation is even more prominent in the case of d = 2 . λ ,where discrete local waveguides have widths of t = 2 . λ , t = 5 λ , t = 7 . λ , etc.Our argument in the previous paragraph was based onthe value of the V-number created in the locally formed FIG. 13. The unit slab thickness is fixed at d = λ but∆ n core = 0 . , . , and 0 .
05. As ∆ n core increases, more localwaveguides form in the structure and the PDF becomes morediscrete. waveguides. Therefore, if the refractive index contrast∆ n core is increased, we should observe a similar behavior,where narrow peaks related to regular step-index waveg-uiding modes should appear alongside with the extendedmodes and the Anderson localized modes. The discretepeaks in the PDF observed in Fig. 5 are in fact of this na-ture. In order to see this more clearly, in Fig. 13 we studythe impact of increasing the value of the waveguide indexdifference ∆ n core by comparing the mode-width PDFs for∆ n core = 0 .
01, ∆ n core = 0 .
03, and ∆ n core = 0 .
05, all for d = λ . Sharp peaks clearly appear when the refractiveindex contrast is increased.The results presented in this section so far give a thor-ough overview on the statistical behavior of Andersonlocalized modes, extended modes, and regular step-indexwaveguiding modes, all of which can be present in a dis-ordered waveguide at the same time.We conclude this section by visualizing the interplaybetween the impact of the localized and extended modesin a disordered waveguide. In Fig. 14, we numericallysimulate the propagation of light in a disordered waveg-uide and plot the intensity distribution of the guidedbeam as it propagates along the waveguide. The dis-ordered waveguide is defined with N = 200 slabs, whereeach slab’s thickness is d = λ , and the refractive indexesare given by n = 1.5 and n = 1.49. The cladding indexin Fig. 14 is n c = 1.49, so ∆ n clad = 0, while the claddingindex in Fig. 15 is n c = 1.45 resulting in ∆ n clad = 0 . n clad in Fig. 15 results ina larger number of extended modes.In Figs. 14 and 15, the injected beam is a Gaussiancharacterized by the electric field distribution of the form E ( x ) ∝ exp( − x /ω ) with ω = 3 λ at the entrance, where x is the coordinate across of the waveguide. The center ofthe Gaussian beam is assumed to be in the middle of the FIG. 14. Propagation of a Gaussian beam ( ω = 3 λ ) along adisordered waveguide defined by ∆ n core = 0 .
01, ∆ n clad = 0 , and N = 200. The beam eventually localizes to a relativelystable width after an initial expansion.FIG. 15. The refractive index profile of the disordered waveg-uide is exactly the same as Fig. 14, except ∆ n clad = 0 .
04. For-mation of the extended modes generates a background noise. disordered lattice. In the single realization of the disor-dered waveguide shown in Fig. 14 with very few extendedmodes, there is virtually no background noise and the ini-tial excitation is clearly Anderson localized after a shortpropagation distance. However, in the presence of a largenumber of extended modes in Fig. 15, a background noisedue to extended modes is evident throughout the prop-agation, while the Anderson localized modes still play aprominent role in the center that is similar to the oneobserved in Fig. 14.
IV. DISCUSSION
There is a vast literature over the past five decades onAnderson localization, especially in 1D, which is the mainfocus of this paper. In this section, we will establish aconnection between the key aspects of the work presentedhere and the existing literature. In particular, scalingproperties of electron transport and conductance havereceived considerable attention over the years. There is aone-to-one relationship between the Schr¨odinger equationfor electron in a disordered potential V ( x ) − − (cid:126) m ∂ ∂x A ( x ) + V ( x ) A ( x ) = EA ( x ) , (5)and the paraxial Helmholtz equation 2 for optical wavepropagation–along the z-direction–in a longitudinally (z-direction) invariant and transversally (x-direction) dis-ordered waveguide. The analogy can be established bymaking the following identifications: V ( x ) = (cid:126) k m (cid:16) n − n ( x ) (cid:17) , (6a) E = (cid:126) k m (cid:16) n − n (cid:17) , (6b)where n eff = β/k . Our study in this manuscript hasfocused on guided waves with n eff > n cl , which is equiva-lent to the problem of electronic bound-states with E < V ( x ∈ boundary) = 0). In Ref. , we established a rela-tionship between the mode-width of the localized statesand the localization length. Briefly, for an exponentiallylocalized state of the form A ( x ) ∼ exp( −| x | /L c ), themode-width ( σ defined by Eq. 3) is given by σ = √ L c .The localization length ¯ L c , is defined through logarith-mic averaging of the localized beam profile intensity(modulus-squared) .Scaling theories of localization have been discussed inmultiple publications especially in late 1970s and early1980s . These and similar work have primarily fo-cused on the scaling properties of conductivity. Thereare similarities between the scaling analyses of these pa-pers and our work especially at the formal level of thegoverning differential equations 2 and 5. However, thereare subtle and important differences which arise primar-ily due the physical nature of the problem here whichonly deals with the transversely localized guided opti-cal modes propagating in the longitudinal direction. Thedifferences are mathematically manifested in the differentboundary conditions used in these problems. For exam-ple, consider the work of Pichard that studies the 1Dscaling of the Anderson model and resembles our workbecause of the 1D nature of both problems and the un-derlying Eqs. 5 and 6. Pichard studies the scaling be-havior of the eigenvalue λ N of the unimodular matrix τ † N τ N with the length of a disordered chain N , where τ N is the 2 × E > V ( x ∈ boundary) = 0) and explores the scaling behaviorof λ N where both λ N and resistance depend on the valueof E . In this manuscript, we formally study the samedifferential equation 5 as that of Pichard, with the mi-nor difference that Ref. only considers diagonal disor-der and our disorder is mixed due to the practical natureof the studied problem. However, the main differencearises in the boundary condition, where we treat Eq. 5 asan eigenvalue problem and only study bound-states with E <
0, hence exponentially decaying tails of the eigen-states because V ( x ∈ boundary) = 0 (or equivalently animaginary wave-vector in the boundary). Moreover, wefocus our studies on the mode-widths of these eigenstates which are roughly related to their near exponential de-cay (on each side) through σ = √ L c that was derivedabove (and there is no E dependence because E here is aneigenvalue which we solve for). Therefore, unlike Ref. that analyses the eigenvalues of the modulus squared ofthe transmission matrix, our focus is on the eigenstates of Eq. 5 with E < L c (and localization properties) of an eigenstate and the cor-responding eigenvalue, which is discussed in Refs. .Similarly, the PDF has been studied extensively in thepast (see Refs. and references therein), again in thecontext of the localization length determined through thescattering problem discussed above. Our analysis is fo-cused on the PDF of the eigenvalue problem and theemphasis has been placed on the statistics and scalingof the PDF of the mode-width directly calculated fromthe eigenstates, which is the relevant quantity for theexperiments presented on disordered optical fibers in ref-erences such as Ref. and Ref. . We would also like toacknowledge an interesting body of work on the scalingproperties of the scattering problems in optical systems,e.g. in Ref. which resemble more the work of electrontransport than the bound-state problem studiedhere. V. CONCLUSION
In this manuscript, we have introduced the mode-width PDF as a powerful tool to study the transverseAnderson localization properties of guided modes of a dis-ordered one dimensional optical waveguide. The mode-width PDF has been used for detailed statistical analysisof the impact of various structural and optical parame-ters of the disordered waveguide. A disordered waveg-uide supports both Anderson localized modes as well asextended modes. The mode-width PDF sheds light intothe distribution of these modes and provides a powerfulframework to manipulate such distributions, for example0to quench the number of extended modes while mini-mally affecting the localized ones. An important obser-vation in this manuscript is the convergence of the mode-width PDF to a terminal configuration as a function ofthe transverse dimension of the disordered waveguide.This has been shown by performing a scaling analysis ofthe mode-width PDF and can be quite helpful in turning a formidable computational problem from nearly impos-sible to a tractable one. The results presented in themanuscript are intended to establish the framework for acomprehensive analysis of the mode-width statistics for2D transverse Anderson localization in optical fibers inthe future. ∗ mafi@unm.edu P. W. Anderson, Phys. Rev. , 1492 (1958). E. Abrahams,
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