Disordered Potential Landscapes for Anomalous Delocalization and

Jan 30, 2018 - The prominent distinction between order and disorder in optics has been understood in terms of the spatial spreading of waves...
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Disordered Potential Landscapes for Anomalous Delocalization and Superdiffusion of Light Sunkyu Yu, Xianji Piao, and Namkyoo Park ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.7b01532 • Publication Date (Web): 30 Jan 2018 Downloaded from http://pubs.acs.org on February 3, 2018

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Disordered Potential Landscapes for Anomalous Delocalization and Superdiffusion of Light Sunkyu Yu, Xianji Piao, and Namkyoo Park* Photonic Systems Laboratory, Department of Electrical and Computer Engineering, Seoul National University, Seoul 08826, Korea KEYWORDS Anderson localization; Delocalization; Disordered media; Superdiffusion

ABSTRACT: The prominent distinction between order and disorder in optics has been understood in terms of the spatial spreading of waves. In the Anderson picture of optical disorder, light localization has been elucidated by the interference of multiple scatterings from disorders, thus implying a natural correspondence between the localization and disordered potentials. Here, we focus on the disorder of a wave itself to achieve a new class of disordered optical potentials with continuous landscapes, distinguished from conventional Anderson disorder or abnormal disorders in discrete systems. Starting from the disordered but extended ground state for the Schrödinger-like wave equation, we inversely develop the landscape of an optical potential, the disorder pattern of which is similar to Brownian random-walk motion. We then demonstrate that the modes in such a structure can extend over an anomalously large region of space, and also exhibit superdiffusive wave transport. Such behaviors are in contrast to the wavelength-scale localization commonly referred to as Anderson localization in conventional disordered potentials. Our results enable wave delocalization and signal transport in generalized disordered potentials with anomalous modal properties, without the aid of interactions between on-site and hopping energies.

The concept of order in physics is definite, as shown in the crystal periodicity,1 the cut-and-project construction of quasicrystals,2 and the fractal self-similarity.3 By contrast, disorder itself contains an inherent ambiguity4 because the randomness for breaking the order can be introduced in various ways. With the simplest random perturbation of potentials, the famous Anderson disorder5 with the wavelength-scale localization and transport blockade6-9 has been intensively studied in solid-state physics,10 phononics,11 graph modeling,12 and optics.7, 13, 14 Due to the evident contrast between Bloch extended modes in crystals and Anderson localized modes in disordered materials, the direct correspondence between localized modes and highly disordered materials has been taken for granted.6-9 However, by controlling the degree and pattern of the randomness in potentials, the traditional aspect of Anderson localization can be controlled significantly. For example, by changing the strength of the disorder, a continuous transition from ballistic to diffusive transport and to Anderson localization has been achieved.9 Especially, from the distinct distributions of on-site and hopping energies, the existence of delocalized eigenmodes in disordered discrete systems has been demonstrated,15-23 by searching and engineering the spectral region,15-17 by partially introducing the correlation,18, 22, 23 by inversely designing the on-site energy distribution for randomly distributed hopping energy,21 or by constructing the building

block of designed on-site and hopping energy distributions.19, 20 Anomalous transports in unconventional regimes between order and disorder have also been investigated for superdiffusion,24-26 non-diffracting transport,27 and functional wave transport.21, 28 The concept of hyperuniformity for short-range order29-32 enables the understanding of crystal-like scattering or large bandgap in disordered materials. In most previous approaches, the discrete systems having two degrees of freedom of on-site and hopping energies have been considered by simplifying full wave phenomena to the interactions of bound modes: e.g., random dimer models or tight-binding lattices15-23 or the packing density of elemental structures29-32. Due to the restricted degree of freedom on hopping energy in ‘continuous’ potential landscapes, the delocalization and anomalous transport in a continuous disordered potential have not been intensively studied. Therefore, the inverse and deterministic design method using the rigorous wave equation is strongly desired to realize abnormal disorders of continuous landscapes. In this paper, from the perspective of “disordered waves”, we derive disordered potential landscapes with anomalous trends in localization and wave excitations at the given frequency. From the inverse application of the Schrödinger-like wave equation for obtaining randomly perturbed but extended waves, we reveal the existence of a potential that has a tunable degree of disorder but simultaneously maintains the spatial extension of eigenmodes

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over an anomalously large region. From a designed set of delocalized eigenmodes and their collective modal excitations, we demonstrate the wave transport superior to diffusion even in the extreme level of disorder analogous to Brownian random-walk motions. We also show the fundamental difference between conventional Anderson disorders and the proposed disorders in terms of the trends of modal localizations and excitations, realizing the disorder-robustness of the ground state and its nearby states. Our approach, allowing the deterministically tunable delocalization in highly disordered potentials, will provide a new viewpoint on the relation between wave phenomena and disordered potentials.

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is the gauge potential, εorder and korder are the amplitude and spatial frequency of the periodic potential, respectively, and b(x) is the bound function for smoothly varying potentials (see Supplementary Note S1). The Andersondisordered potential εA(x) is then constructed by imposing random perturbations in amplitude and phase on εo(x):

ε A ( x) = ε o ( x) + ε ∆ b( x)

∫K u (0,1) cos[kx + u(−π , π )]dk , (1) max( ∫K u(0,1) cos[kx + u(−π , π )]dk )

where εΔ is the perturbation amplitude, K = [0,k0] is the perturbation spectrum, and u(p,q) is the uniform density function between p and q.

RESULTS AND DISCUSSION Concept of non-Anderson disorders. To clarify the effects of disordered potentials on light-matter interactions, we investigate the 1-dimensional (1D) optical platform defined by the inhomogeneous relative permittivity landscape ε = ε(x), which allows the transverse electric (TE) mode propagation. By assigning the electric field form E(x,y) = ψ(x)∙e–iβy, the Helmholtz wave equation ∇2E + k02ε(x,y)Ez = 0 is then reduced to the 1D Schrödinger-like eigenvalue equation33, 34 as Hψ = γψ, where k0 is the freespace wavenumber, the Hamiltonian operator H is H = – (1/k02)∙∂x2 – ε(x) and the eigenvalue is γ = –(β/k0)2. For an ordered structure having a periodic potential εo(x), Bloch-type eigenmodes are extended over the entire space of the potential (ψ0-2 in Fig. 1a). The conventional Anderson disorder εA(x) is readily obtained by imposing a random perturbation on the ordered potential (Fig. 1b). The eigenmodes (ψA0-2) are localized with exponential decay along the transverse direction, with a decay rate that varies with the disorder strength.7, 9, 11 Due to the spatially isolated eigenmodes, the localization also restricts the collective excitation of eigenmodes, thus prohibiting wave transport in disordered materials. By contrast, our approach to the disordered material with delocalization, which we call “non-Anderson disorder”, is inspired by the fact that the wave equation intrinsically describes wave-matter interactions, linking the spatial information of waves and matter. We conceive a set of eigenmodes that have extended but randomly perturbed shapes (ψN0-2 in Fig. 1c), in contrast to the shapes of Bloch-type eigenmodes in the ordered potential (ψ0-2 in Fig. 1a) and Anderson-type eigenmodes in the disordered potential (ψA0-2 in Fig. 1b). To support disordered profiles of ψN0-2, the corresponding potential εN(x) will then have a disordered landscape, but will simultaneously allow the delocalization feature of ψN0-2. Deterministic design methodology. For the comparison, we generate ordered, Anderson-disordered, and nonAnderson-disordered potentials. The ordered potential is defined by εo(x) = εg + εorder∙b(x)∙[1+cos(korder∙x)]/2, where εg

Figure 1. Non-Anderson disorder from ‘disordered’ waves. (a) The periodic potential εo(x) and its first three eigenmodes ψ0-2. (b) The Anderson disorder with a perturbed ‘potential’ εA(x) = εo(x) + Δε(x) and its first three eigenmodes ψA0-2. (c) The non-Anderson-disorder with the potential εN(x) designed from a perturbed ‘ground state’ ψN0(x) = ψ0(x) + Δψ(x) and its first three eigenmodes ψN0-2. For the implementation of a non-Anderson material, we instead start from the ground state ψ0(x) of the ordered potential εo(x). We impose the random perturbation on the ground state ψ0(x), not on the potential εo(x), while keeping the same functional form of the perturbation as in the case of Anderson disorder in Eq. (1). The resulting ground state of non-Anderson disorder ψN0(x) then becomes

ψ N0 ( x) = ψ 0 ( x) +ψ ∆ ( x)b( x)

∫K u(0,1) cos[kx + u(−π , π )]dk ,(2) max( ∫K u(0,1) cos[kx + u(−π , π )]dk )

where ψΔ(x) is the perturbation amplitude defined by the scalar ρN = ψΔ(x) / ψ0(x), to guarantee the nodeless ψN0(x). Because the disordered potential is uniquely determined from ψN0(x) as εN(x) = [–(1/k02)∙∂x2ψN0(x) – γN0∙ψN0(x)]/ψN0(x) where γN0 is the target eigenvalue, Eq. (2) derives the disordered potential landscape ‘deterministically’ with the designed ground state (see Supplementary Note S2 for detailed structures of ordered, Andersondisordered, and non-Anderson-disordered potentials).

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Note that a real function ψN0(x) guarantees the design of a real-valued optical potential εN(x). To examine the relation between disordered potentials and wave profiles, we utilize characteristic quantities for the disorder of the potential landscape and the strength of the wave localization. First, the modal extension9 wi = (∫|ψi(x)|2dx)2 / ∫|ψi(x)|4dx is utilized to quantify the localization property of each eigenmode. For the degree of disorders, the Hurst exponent H that measures the longrange correlation of the potential landscape35, 36 is applied to classify the type of randomness,34, 37 where H = 0.5 is for the uncorrelated, Brownian-motion-like randomness, 0 ≤ H < 0.5 for negatively correlated potentials, and 0.5 < H ≤ 1 for positively correlated potentials.

Figure 2. Delocalization in Anderson and non-Anderson disorders. (a,c) Hurst exponents H and (b,d) modal sizes w in (a,b) Anderson and (c,d) non-Anderson disorders, as functions of ρA and ρN, respectively. εg = 2, εorder = 0.2, korder = 0.5∙k0, and b(x) is defined for x = [–100∙λ0,100∙λ0]. The target ground-state eigenvalue is γN0 = –2.12 for all cases. Each dot (200 dots for each ‘single’ value of ρA or ρN) denotes an ensemble obtained by the random design process

using Eqs (1) and (2), and solid lines represent the average of 200 ensembles. The signs in (a,c) depict positive and negative correlations. Black dotted lines in (a,c) denote the Brownian random-walk. The grid size for ρA or ρN is 0.01. When the random perturbation ρA = εΔ / εg increases in Anderson disorders, the Hurst exponent of the potential landscape εA(x) varies from H ~ 0.06 to H ~ 0.57 (Fig. 2a), representing the transition from the negative correlation to the Brownian random-walk. Also, a decrease in the average modal size of all eigenmodes w = wi is evident with increasing disorder (Fig. 2b), including wavelengthscale Anderson localization (w < 2λ0 at ρA > 0.25 for the wavelength λ0). In non-Anderson disorders, when the wave perturbation ρN increases, H alters from H ~ 0.06 to H ~ 0.48 (Fig. 2c), showing that the disorder in waves leads to the disorder in materials as similar to the case of Anderson disorder. In contrast, the reduction of modal sizes is significantly moderated, keeping w >> λ0 (Fig. 2d, w > 10λ0), contrary to the λ0-scale Anderson localization. Anomalous localization trend in non-Anderson disorders. To inspect the trend of the localization in more detail, parts of eigenmodes for the potentials in Fig. 2 are presented in Fig. 3. In comparison with the extended eigenmodes in the ordered potential1 (Fig. 3a), the eigenmodes in conventional Anderson disorders are strongly localized by increasing the degree of disorder5, 11 (Fig. 3b to 3c). In contrast, the localization of eigenmodes in the non-Anderson disorders (Figs 3d,e) is significantly moderated when compared with the Anderson disorders with the same degree of disorder. Although the delocalization in non-Anderson disorder is realized from the perturbed ground state (ψ0), other excited states (ψ1, ψ2, ψ49, ψ50, ψ51) are also relatively delocalized, due to the globally suppressed scattering. Although the delocalization is achieved for all eigenmodes (Fig. 3d,e versus Fig. 3b,c), the degree of delocalization is much greater for the modes close to the ground state (ψ0, ψ1, ψ2 versus ψ49, ψ50, ψ51).

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Figure 3. The corresponding eigenmodes of the ordered, Anderson-disordered, and non-Anderson-disordered potentials. The ground state (ψ0) and some excited states (ψ1, ψ2, ψ49, ψ50, ψ51) are depicted for the (a) periodic potential (H = 0.06), (b,c) Anderson-disordered potentials of (b) H = 0.25 (ρA = 0.02) and (c) H = 0.50 (ρA = 0.10), and (d,e) non-Andersondisordered potentials of (d) H = 0.25 (ρN = 0.30) and (e) H = 0.50 (ρN = 0.90). All other parameters are the same as in Fig. 2. The exotic feature of non-Anderson disorders, distinct from the cases of Anderson disorders5 or other correlated disorders,15-23 is evident in the localization trend of excited states. In detail, non-Anderson disorder possesses abnormal trend in the modal localization, in terms of the resistance to random perturbations. In Anderson disorder (Fig. 3b), the eigenmodes near the ground state (ψ0, ψ1, ψ2) are more sensitively localized to the disorder than higherorder eigenmodes (ψ49, ψ50, ψ51). However, in nonAnderson disorder (Fig. 3e), the delocalization of the designed ground state and nearby eigenmodes (ψ0, ψ1, ψ2) is more resistant to the disorder than that of higher-order eigenmodes (ψ49, ψ50, ψ51). In consequence, the trends of the modal localization are opposite between Anderson and non-Anderson disorders, emphasizing the distinctive nature of non-Anderson disorders.

tial (Fig. 4c and Supplementary Note S4). Remarkably, for all cases, the separation of the regimes of Anderson and non-Anderson disorders in the H-w spaces is apparent (red dashed line in Fig. 4b), implying their naturally distinct relations between long-range order and localization properties. Also can be seen in the opposite trends of modal localizations (Fig. 3 for Anderson disorder with lower-order mode localization and non-Anderson disorder with higher-order mode localization), these fundamental differences prove the intrinsic distinction between matter-to-wave (Anderson) and wave-to-matter (nonAnderson) perspectives on disordered materials.

Phase diagrams for Anderson and non-Anderson disorders. To gain an in-depth understanding, we analyze the H-w relations in Anderson and non-Anderson disorders (Fig. 4a), demonstrating the unique delocalization in non-Anderson disorder. For the same degree of the disorder quantified by H, the modal size w in non-Anderson disorder is orders of magnitude larger than that of Anderson one. Although non-Anderson disorder is derived from the extended but perturbed ground state, the contribution of ‘extended’ excited states (Fig. 3) is also apparent as shown in the increase of the average modal size w (Fig. 4a), signifying the global suppression of coherent backscattering in non-Anderson disorder. The regime of non-Anderson disorders can be extended significantly to the entire correlation space H (from negative correlation to Brownian uncorrelation, and to positive correlation) by controlling the spectral range K for the random perturbation (Fig. 4b and Supplementary Note S3) or tuning the initial correlation of the ordered poten-

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Figure 4. The H-w relations for the modal size (w) with respect to the potential correlation (H): (a) the result from Fig. 2, (b) the effect of perturbation bandwidth ΔK: K = [k0/2 – ΔK/2, k0/2 + ΔK/2], and (c) the effect of modulated εo(x) with korder = 0. Arrows in (b) depict evolutions from the increase of ΔK. The red dashed line in (b) presents the boundary between Anderson and non-Anderson disorders. In all figures, each dot denotes an ensemble obtained by the random design process (200 dots for each ρA or ρN), and the ensembles for each K range are plotted together in (b). Wave transport in non-Anderson disorder. To examine the signal transport in non-Anderson disorder, we compare the transport feature in the order, Andersonand non-Anderson-disorders (Fig. 5). The transport wave profile can be calculated by exerting the point-source-like excitation9, 38 around x ~ 0, which enables the collective excitation of eigenstates and the following energy transport from their superposition. For the set of eigenmodes ψq(x) and their effective permittivities εeq (q = 1, 2, …), the point-source-like excitation φi(x) can be approximated to the linear combination of eigenmodes as φi(x) = Σaq∙ψq. The propagating field then becomes φ(x,y) = Σaq∙ψq∙exp(–ik0∙εeq1/2∙y). As a quantity for transport efficiency, we utilize the mean-square displacement39 (MSD) M(y) from the field profile φ(x,y), as ∞

M ( y) = x 2 =

2

2 ∫−∞ ( x − xm ) ⋅ ϕ ( x, y ) . 2 ∞ ∫−∞ ϕ ( x, y )

(3)

The MSD M(y) is fitted for y exponentially as M(y) ~ cα∙yα, for the diffusion exponent α: α = 2 for ballistic transport, α = 1 for diffusive transport, and α = 0 for localization.39 Whereas scattering-free ballistic wave transport is achieved in the ordered structure (Fig. 5a, α = 2), the increased random perturbation of the ‘material’ drastically suppresses wave transport because of localized eigenmodes, eventually resulting in a complete blockade (Fig. 5b to 5c). As indicated by the variation of α in Fig. 5f, the transport feature is fragile in Anderson-type disorder. In detail, a subdiffusive feature is observed after the weak perturbation (α < 1 when ρA ≥ 0.02), in agreement with the trend of the modal localization (Fig. 2b). However, wave transport is highly resistant to nonAnderson disorder because of delocalized eigenmodes, even exhibiting the transport in the Brownian state (Fig. 5d to 5e). The decrease of α is highly moderated in nonAnderson disorder (Fig. 5g), inheriting the trend of the modal delocalization (Fig. 2d). For most of ensembles in a whole range of disorder ρN, the superdiffusive transport is preserved (α > 1, Fig. 5g), in contrast to the subdiffusive transport or complete blockade in Anderson disorder. As demonstrated by the H-α relation in Fig. 5h, the proposed non-Anderson disorder (regime III, H ~ 0.5 with α > 1) again establishes the distinctive regime between the order and disorder of potential landscapes, in sharp contrast to the cases of ordered (regime I, H ~ 0 with α ~ 2) and Anderson-disordered potentials (regime II, H ~ 0.5 with α ~ 0).

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Figure 5. Wave transport in the order, Anderson- and non-Anderson-disorders. (a-e) x-axis wave transport in the (a) ordered potential, (b,c) Anderson-disorders with (b) H = 0.25 (ρA = 0.02) and (c) H = 0.50 (ρA = 0.10), and (d,e) nonAnderson-disorders with (d) H = 0.25 (ρN = 0.30) and (e) H = 0.50 (ρN = 0.90). Diffusion exponent α in (f) Anderson and (g) non-Anderson disorders, as a function of ρA and ρN, respectively. (h) The H-α relation representing the diffusion exponent α with respect to the correlation H in Anderson and non-Anderson disorders. Each dot again denotes an ensemble obtained by the random design process (200 dots for each ρA or ρN). The approximated point source (Gaussian envelope with the spatial width of λ0/100) is excited at x = 0. Black dotted lines in (f-h) denote the diffusive transport of α = 1. Anomalous modal excitations. The distinct transport feature of non-Anderson disorder is clarified by looking into the modal overlaps between eigenmodes and incident waves. Figure 6 shows an example of the amplitude of modal excitations for the point source incidence to each structure. The ordered potential, with the paritysymmetric excitation (x = 0), leads to collective excitations of even-numbered eigenmodes (0th, 2nd, 4th, …, Fig. 6a), from the modal overlap between extended eigenmodes and incident waves. The Anderson disorder,

however, allows for the excitation of only few eigenmodes in cases of strong randomness, owing to spatially isolated eigenmodes (Fig. 6b,c). By contrast, collective modal excitations are achieved in non-Anderson disorder from delocalized eigenmodes (Fig. 6d,e), comparable to those of the ordered potential. Note that superdiffusive transport in non-Anderson disorder originates from the spatial extension of eigenmodes near the ground state (Mode numbers between 0 and 20 in Fig. 6d,e), leading to collective modal excitations from incident waves.

Figure 6. The excitation of eigenmodes in ordered, Anderson-disordered, and non-Anderson-disordered potentials: (a) the ordered potential, (b,c) Anderson-disordered potentials with (b) ρA = 0.02 and (c) ρA = 0.10, and (d,e) nonAnderson-disordered potentials with (d) ρN = 0.30 and (e) ρN = 0.90. All other parameters are the same as in Fig. 2. The statistical analysis of modal excitations (Fig. 7, including 200 random ensembles for each case) clarifies the origin of distinctive transport in non-Anderson disorder. The increase in the random perturbations in Anderson disorder imposes strong randomness on modal excitations (Fig. 7a,b) supported by few dominant modes. However, in the case of non-Anderson disorder, the distribution and amplitudes of collective modal excitations still

preserve the well-arranged pattern near the ground state (red dashed marks in Fig. 7c,d). As demonstrated by the delocalization of excited states (Fig. 3), superdiffusive wave transport in the case of non-Anderson disorder thus mainly originates from the collective spatial extension of eigenmodes near the perturbed but extended ground state.

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Note that the abnormal localization in non-Anderson disorders is also observed in modal excitations. In Anderson disorder, eigenmodes near the ground state are weakly excited (Fig. 7a) due to their sensitivity to the disorder (Fig. 3b). In contrast, in non-Anderson disorders (Fig. 7d),

eigenmodes near the ground state have strong and stable excitation conditions. Following the trends of the modal localization, the trends of the modal excitations are opposite between Anderson and non-Anderson disorders.

Figure 7. Random ensembles of the excitation of eigenmodes in Anderson and non-Anderson disorders: (a,b) Andersondisorders with (a) ρA = 0.02 and (b) ρA = 0.10, and (c,d) non-Anderson-disorders with (d) ρN = 0.30 and (e) ρN = 0.90. Each point denotes a statistical ensemble (200 random ensembles, each with over 106 modal information) of each modal excitation. All other parameters are the same as in Fig. 2. CONCLUSION In conclusion, by imposing the random perturbation on the wave itself, we revealed and designed a set of disordered materials possessing wave delocalization at the given frequency, even in the Brownian random-walk realization. In terms of signal transport, the proposed realization of non-Anderson disorder also supports superdiffusive wave transport at the extreme level of disordered materials. We demonstrated that non-Anderson disorders possess fundamentally different natures from Anderson disorders. In terms of modal localization and excitation, the randomly ‘molded’ ground state and lower-order excited modes are more robust to random perturbations than higher-order modes, which is the opposite trend to that in Anderson disorders. In the H-w phase diagram for the relation between the degree of disorders and localizations, the regimes of Anderson and Non-Anderson disorders are evidently separated. One of the critical differences of non-Anderson disorders from other abnormal disorders15-20, 22, 23 is the applicability to continuous potentials. In most previous works, ‘discrete’ models such as random dimer models or tightbinding lattices have been assumed. The competition between on-site (or diagonal) and hopping (or off-diagonal) energies in these discrete disordered models cannot be realized in ‘continuous’ potential landscapes which have uniform interactions between spatial positions. The inverse design technique based on the fundamental wave equation allows the deterministic control of the delocalization in random continuous potentials.

Because the scale of the randomness strongly relates to the wavelength of light, the delocalization and anomalous transport in disordered optical potentials are maintained only at the given frequency. This frequency-selective nature can be applied to sensing applications. Also note that, the inverse application of the wave equation for nonAnderson disorder represents the “wave-to-matter perspective”, distinct from the matter-to-wave viewpoint of the Anderson picture. In a broad sense, the investigation of disorder in a wave itself is analogous to the relation between coordinate-transformation optics40 and fieldtransformation optics.41

ASSOCIATED CONTENT The Supporting Information is available free of charge on the ACS Publications website at DOI: xx.xxxx/acsphotonics. xxxxxxx. Supporting Information S1−S4 (PDF).

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]

ACKNOWLEDGMENT We acknowledge financial support from the National Research Foundation of Korea (NRF) through the Global Frontier Program (2014M3A6B3063708). S. Yu was supported by the Basic Science Research Program (2016R1A6A3A04009723), and X. Piao and N. Park were supported by the Korea Research Fellowship Program (2016H1D3A1938069) through the NRF. S.Y. conceived the presented idea. S.Y. and X.P. developed the theory and performed the computations. N.P. encouraged S.Y. to investi-

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gate the relation between the disorder and Anderson localization while supervising the findings of this work. All authors discussed the results and contributed to the final manuscript.

REFERENCES 1. Zhang, Z.; Satpathy, S., Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell’s equations. Phys. Rev. Lett. 1990, 65, 2650. 2. Levine, D.; Steinhardt, P. J., Quasicrystals: a new class of ordered structures. Phys. Rev. Lett. 1984, 53, 2477. 3. Lavrinenko, A.; Zhukovsky, S.; Sandomirski, K.; Gaponenko, S., Propagation of classical waves in nonperiodic media: Scaling properties of an optical Cantor filter. Phys. Rev. E 2002, 65, 036621. 4. Goodrich, C. P.; Liu, A. J.; Nagel, S. R., Solids between the mechanical extremes of order and disorder. Nat. Phys. 2014, 10, 578-581. 5. Anderson, P. W., Absence of diffusion in certain random lattices. Phys. Rev. 1958, 109, 1492. 6. Lahini, Y.; Avidan, A.; Pozzi, F.; Sorel, M.; Morandotti, R.; Christodoulides, D. N.; Silberberg, Y., Anderson localization and nonlinearity in one-dimensional disordered photonic lattices. Phys. Rev. Lett. 2008, 100, 013906. 7. Segev, M.; Silberberg, Y.; Christodoulides, D. N., Anderson localization of light. Nat. Photon. 2013, 7, 197-204. 8. Wiersma, D. S., Disordered photonics. Nat. Photon. 2013, 7, 188-196. 9. Schwartz, T.; Bartal, G.; Fishman, S.; Segev, M., Transport and Anderson localization in disordered twodimensional photonic lattices. Nature 2007, 446, 52-55. 10. Semeghini, G.; Landini, M.; Castilho, P.; Roy, S.; Spagnolli, G.; Trenkwalder, A.; Fattori, M.; Inguscio, M.; Modugno, G., Measurement of the mobility edge for 3D Anderson localization. Nat. Phys. 2015, 11, 554-559. 11. Sheng, P., Introduction to wave scattering, localization and mesoscopic phenomena. Springer Science & Business Media: 2006; Vol. 88. 12. Tikhonov, K.; Mirlin, A.; Skvortsov, M., Anderson localization and ergodicity on random regular graphs. Phys. Rev. B 2016, 94, 220203. 13. Crespi, A.; Osellame, R.; Ramponi, R.; Giovannetti, V.; Fazio, R.; Sansoni, L.; De Nicola, F.; Sciarrino, F.; Mataloni, P., Anderson localization of entangled photons in an integrated quantum walk. Nat. Photon. 2013, 7, 322-328. 14. Sperling, T.; Buehrer, W.; Aegerter, C. M.; Maret, G., Direct determination of the transition to localization of light in three dimensions. Nat. Photon. 2013, 7, 48-52. 15. Dunlap, D. H.; Wu, H.; Phillips, P. W., Absence of localization in a random-dimer model. Phys. Rev. Lett. 1990, 65, 88. 16. Zhang, W.; Ulloa, S. E., Extended states in disordered systems: role of off-diagonal correlations. Phys. Rev. B 2004, 69, 153203. 17. Rodriguez, A.; Chakrabarti, A.; Römer, R. A., Controlled engineering of extended states in disordered systems. Phys. Rev. B 2012, 86, 085119. 18. Sedrakyan, T.; Kestner, J.; Sarma, S. D., Proposed signature of Anderson localization and correlation-induced delocalization in an N-leg optical lattice. Phys. Rev. A 2011, 84, 053621. 19. Pal, B.; Maiti, S. K.; Chakrabarti, A., Complete absence of localization in a family of disordered lattices. Europhys. Lett. 2013, 102, 17004.

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20. Pal, B.; Chakrabarti, A., Engineering bands of extended electronic states in a class of topologically disordered and quasiperiodic lattices. Phys. Lett. A 2014, 378, 2782-2789. 21. Yu, S.; Piao, X.; Hong, J.; Park, N., Metadisorder for designer light in random systems. Sci. Adv. 2016, 2, e1501851. 22. Sánchez, A.; Maciá, E.; Domínguez-Adame, F., Suppression of localization in Kronig-Penney models with correlated disorder. Phys. Rev. B 1994, 49, 147. 23. de Moura, F. A.; Lyra, M. L., Delocalization in the 1D Anderson model with long-range correlated disorder. Phys. Rev. Lett. 1998, 81, 3735. 24. Bertolotti, J.; Vynck, K.; Wiersma, D. S., Multiple scattering of light in superdiffusive media. Phys. Rev. Lett. 2010, 105, 163902. 25. Martínez, A. J.; Kevrekidis, P. G.; Porter, M. A., Superdiffusive transport and energy localization in disordered granular crystals. Phys. Rev. E 2016, 93, 022902. 26. Bertolotti, J.; Vynck, K.; Pattelli, L.; Barthelemy, P.; Lepri, S.; Wiersma, D. S., Engineering disorder in superdiffusive Levy glasses. Adv. Funct. Mater. 2010, 20, 965-968. 27. Hsieh, P.; Chung, C.; McMillan, J.; Tsai, M.; Lu, M.; Panoiu, N.; Wong, C. W., Photon transport enhanced by transverse Anderson localization in disordered superlattices. Nat. Phys. 2015, 11, 268-274. 28. Yu, S.; Piao, X.; Park, N., Target decoupling in coupled systems resistant to random perturbation. Sci. Rep. 2017, 7, 2139. 29. Torquato, S.; Zhang, G.; Stillinger, F., Ensemble Theory for Stealthy Hyperuniform Disordered Ground States. Phys. Rev. X 2015, 5, 021020. 30. Froufe-Pérez, L. S.; Engel, M.; Damasceno, P. F.; Muller, N.; Haberko, J.; Glotzer, S. C.; Scheffold, F., Role of shortrange order and hyperuniformity in the formation of band gaps in disordered photonic materials. Phys. Rev. Lett. 2016, 117, 053902. 31. Florescu, M.; Torquato, S.; Steinhardt, P. J., Designer disordered materials with large, complete photonic band gaps. Proc. Natl. Acad. Sci. 2009, 106, 20658-20663. 32. Torquato, S.; Stillinger, F. H., Local density fluctuations, hyperuniformity, and order metrics. Phys. Rev. E 2003, 68, 041113. 33. Longhi, S., Quantum-optical analogies using photonic structures. Laser Photon. Rev. 2009, 3, 243-261. 34. Yu, S.; Piao, X.; Hong, J.; Park, N., Bloch-like waves in random-walk potentials based on supersymmetry. Nat. Commun. 2015, 6, 8269. 35. Hurst, H. E., Long-term storage capacity of reservoirs. Trans. Amer. Soc. Civil Eng. 1951, 116, 770-808. 36. Roche, S.; Bicout, D.; Maciá, E.; Kats, E., Long range correlations in DNA: scaling properties and charge transfer efficiency. Phys. Rev. Lett. 2003, 91, 228101. 37. King, C.; Horsley, S.; Philbin, T., Perfect Transmission through Disordered Media. Phys. Rev. Lett. 2017, 118, 163201. 38. Christodoulides, D. N.; Lederer, F.; Silberberg, Y., Discretizing light behaviour in linear and nonlinear waveguide lattices. Nature 2003, 424, 817-823. 39. Hahn, K.; Kärger, J.; Kukla, V., Single-file diffusion observation. Phys. Rev. Lett. 1996, 76, 2762. 40. Pendry, J. B.; Schurig, D.; Smith, D. R., Controlling electromagnetic fields. Science 2006, 312, 1780-1782. 41. Liu, F.; Liang, Z.; Li, J., Manipulating polarization and impedance signature: a reciprocal field transformation approach. Phys. Rev. Lett. 2013, 111, 033901.

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Disordered Potential Landscapes for Anomalous Delocalization and Superdiffusion of Light Sunkyu Yu, Xianji Piao, and Namkyoo Park*

Synopsis: Compared with Anderson disorder which derives wave localization with a randomly perturbed ‘potential’, non-Anderson disorder for anomalous delocalization and superdiffusion can be inversely designed from a randomly perturbed but extended ‘waves’.

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