Abstract

We show that materials made of scatterers distributed on a stealth hyperuniform point pattern can be transparent at densities for which an uncorrelated disordered material would be opaque due to multiple scattering. The conditions for transparency are analyzed using numerical simulations, and an explicit criterion is found based on a perturbative theory. The broad applicability of the concept offers perspectives for various applications in photonics and more generally in wave physics.

© 2016 Optical Society of America

1. INTRODUCTION

The study of light propagation in scattering media has been a very active field in the past decades, stimulated by fundamental questions in mesoscopic physics [1,2] and by the development of innovative imaging techniques [3]. Recently, a new trend has emerged, with the possibility to control electromagnetic wave propagation in disordered media up to the optical frequency range. On the one hand, wavefront shaping techniques offer the possibility to overcome the distortions induced by a scattering material, even in the multiple scattering regime [46]. On the other hand, the possibility to engineer the disorder itself by controlling the degree of structural correlation opens new perspectives for the design of materials with specific properties (e.g., absorbers or filters for photonics) [711]. These materials combine the advantages of disordered materials, in terms of process scalability and robustness to fabrication errors, with the possibility of developing a real engineering of their scattering and transport properties through the control of the degree of correlation in the disorder. For example, it has been shown that correlations can substantially change basic transport properties, such as the mean free path [12], the density of states [13,14], including the appearance of bandgaps [1517], or the Anderson localization length [18].

A specific class of correlated materials has appeared recently, initially referred to as “superhomogeneous materials” [19], and now called “hyperuniform materials” [20]. These materials are made of discrete scatterers distributed on a hyperuniform point pattern, a correlated pattern with a structure factor S(q) vanishing in the neighborhood of |q|=0. The geometrical properties of hyperuniform point patterns have been extensively studied, in particular in terms of packing properties [2125]. Regarding wave propagation, it has been shown that bandgaps could be observed for electromagnetic waves in two-dimensional (2D) disordered hyperuniform materials [2630]. Although understanding the origin of the bandgaps is still a matter of study [31,32], these results have stimulated the design and fabrication of three-dimensional (3D) hyperuniform structures for wave control at optical frequencies [33,34].

In this Letter, we demonstrate that stealth hyperuniform point patterns, a special class of hyperuniform structures for which S(q)=0 in a finite domain around |q|=0, offer the possibility to design disordered materials that can be both dense and transparent in a specific and broad range of frequencies and directions of incidence. The analysis is based on full numerical simulations and theoretical modeling. In the single scattering regime, transparency can be explained in simple terms, as a direct consequence of the vanishing of the structure factor. Interestingly, transparency can survive in the multiple scattering regime under a general condition that we establish using a theoretical analysis that applies to a broad range of materials.

2. HYPERUNIFORM POINT PATTERNS

A distribution of points in space is hyperuniform when the normalized variance of the number of points within a sphere of radius R vanishes when R tends to infinity [20]. In other words, the fluctuations in the number of points in a volume increase slower with the volume than the average number of points, a feature at the origin of the name “hyperuniform.” In Fourier space, this is equivalent to stating that the structure factor defined as [35,36]

S(q)=1N|j=1Nexp[iq·rj]|2,
vanishes in the neighborhood of |q|=0 [23], with N as the number of points and rj as their positions. In this work, we study the scattering of light or other electromagnetic waves by disordered materials made of discrete scatterers whose positions coincide with a hyperuniform point pattern. In the present work, we restrict the numerical study to 2D materials, which is convenient for the sake of computer time, while the theory developed in the last section covers both 2D and 3D geometries. Although the design of 3D materials is a major goal, the interest of 2D architectures should not be underestimated. Artificial 2D structures can produce efficient microwave reflectors or filters [27], and promising 2D materials for photonics and optoelectronics can be fabricated using bottom-up approaches [37]. Even natural materials, such as the cornea, can exhibit photonic properties resulting from an underlying 2D microstructure [38].

The first step consists of generating 2D stealth hyperuniform point patterns. We have adapted the algorithm described in Refs. [21,39,40] and based on the minimization of an interaction potential (see Section 1 of Supplement 1). The 2D medium used in the numerical simulations fills a square region with size L and volume V=L2, with periodic boundary conditions to avoid finite-size effects. We consider hyperuniform point distributions, such that S(q)=0 in a square domain Ω of reciprocal space with size K, centered at the origin. The square shape of Ω has been chosen for convenience. Although it induces an anisotropy in the structure factor, this choice does not affect the generality of the transparency property discussed in this work. Due to the periodic boundary conditions, the structure factor S(q) is controlled on a finite number of points located on a square lattice inside Ω and separated by 2π/L. The number of points M(K) depends on the size K of the domain Ω and satisfies 2M(K)+1=(K2L2)/(4π2), where the factor of 2 results from the symmetry property S(q)=S(q) and the “+1” correction accounts for the fact that the value of the structure factor at the exact point q=0 cannot be controlled [S(q=0)=N]. For large K, the system is very constrained, creating stronger correlations in the point pattern. The degree of order in hyperuniform structures is usually measured by the parameter χ=M(K)/(2N). For χ=0, the system is uncorrelated (fully disordered), while for χ=1, the system is a perfect crystal [21]. From the expression of M(K), one immediately gets K=(2π)L14Nχ+1, showing that by increasing K, one increases the degree of order in the point pattern. An example of point distribution generated numerically is shown in Fig. 1(a), together with the corresponding structure factor averaged over 20 configurations in Fig. 1(b). In the square area Ω, the structure factor vanishes [except at the origin, where S(q=0)=N]. For the sake of comparison, an uncorrelated pattern with the same number of points is shown in Fig. 1(c), with the corresponding structure factor in Fig. 1(d). The structure factor of the uncorrelated disordered medium is almost everywhere close to unity.

 figure: Fig. 1.

Fig. 1. (a) Hyperuniform medium generated with N=90000 and χ=0.222. (b) Structure factor of the hyperuniform medium averaged over 20 configurations. (c) Uncorrelated medium generated with N=90000 and χ=0. (d) Structure factor of the uncorrelated medium.

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3. SCATTERING FROM STEALTH HYPERUNIFORM MATERIALS

A hyperuniform scattering material is built by dressing each point with an electric polarizability α(ω) describing the electrodynamics’ response of the individual scatterers at frequency ω (in this study, the scatterers are assumed to be much smaller than the wavelength and are treated in the electric-dipole limit). For 2D waves with an electric field parallel to the invariance axis of the scatterers, the polarizability is α(ω)=4γc2/[ω0(ω2ω02+iγω2/ω0)], where ω0 is the resonant frequency and γ is the linewidth, this expression being consistent with the optical theorem (energy conservation). The scattering medium is illuminated by a Gaussian beam at normal incidence (direction defined by wavevector ki) and focused in the middle of the medium, as shown in Fig. 2(a). The beam waist w is chosen large compared to the wavelength to mimic a weakly focused beam similar to a plane wave. The electric field is calculated numerically by solving Maxwell’s equations using the coupled dipoles method [41]. A description of the method in a similar geometry can be found in Ref. [42]. It allows us to compute the scattered electric field Esca(r) at any position inside or outside the medium. In wave scattering by disordered media, a measurable quantity is the average diffuse intensity, proportional to the average power radiated in a given direction in the far field (defined by wavevector ks). When k0r1, r being the observation distance and k0=ω/c=2π/λ, with c as the speed of light and λ as the wavelength in vacuum, the scattered field takes the asymptotic form

Esca(r,ω)=i(1i)4πk0rexp[ik0r]Esca(θ,ω),
where θ is the observation angle (angle between ki and ks) indicated in Fig. 2(a). By definition, the far-field average diffuse intensity is
Idiff(θ,ω)=|Esca(θ,ω)|2|Esca(θ,ω)|2,
where denotes an average over the configurations of disorder (positions of scatterers).

 figure: Fig. 2.

Fig. 2. (a) Scattering geometry. The 2D medium is illuminated by a Gaussian beam at normal incidence (beam waist w=60/k0), and the diffuse intensity is calculated in direction θ. (b) Schematic view of the domain described by the scattered wavevector q=kski in Fourier space. For a hyperuniform material, the structure factor vanishes in the square domain Ω. (c) Angular pattern of the average diffuse intensity in the single scattering regime (bB=0.5) for a hyperuniform medium with k0=K/8 (blue line) and for uncorrelated disorder (red dashed line). The same scatterers and the same density are used in both cases, the intensity is averaged over 20 configurations, and k0B=444. (d) Same as (c) with k0=K/3.5 and k0B=1015.

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We first consider the single scattering regime, defined by bB(ω)<1, where bB(ω)=L/B(ω) is the optical thickness in the independent scattering (or Boltzmann) approximation, for which the scattering mean free path is B(ω)=[ρσs(ω)]1 and the condition k0B(ω)1 is satisfied. Here, ρ=N/V is the number density of scatterers, and σs(ω)=(k03/4)|α(ω)|2 is their scattering cross section. Under plane-wave illumination with amplitude E0, it is well known that the diffuse intensity can be directly written in terms of the average structure factor. The full expression takes the following form (see Section 2 of Supplement 1):

Idiff(θ,ω)NS(q)|E0|2N2V2|Θ(q)|2|E0|2,
where q=kski is the scattered wavevector. On the right-hand side, Θ(q) is the Fourier transform of a window function equal to unity inside the volume V occupied by the medium and vanishing outside. This function describes the diffraction pattern of the finite size volume V, and in practice, it is sharply peaked around q=0. In particular, for q=0 corresponding to forward scattering, the two terms on the right-hand side in Eq. (4) exactly compensate each other.

In the single scattering regime, the vanishing of S(q) in a finite domain Ω suppresses scattering for a given frequency and/or angular range, a property at the origin of the denomination of stealth hyperuniform materials [39]. To illustrate this interesting behavior, we choose the parameters such that bB=0.5, and we change the incident wavelength, or equivalently, k0, in order to tune the range of scattered wavevector q=kski=k0k^sk0k^i that is involved in the scattering process (here, ^ denotes a unit vector). The first situation is chosen with k0=K/8, so that the circle described by the scattered wavevector q in Fig. 2(b) lies entirely within the domain Ω. In this case, we expect a substantial reduction of scattering for a hyperuniform material [with S(q)0 in Ω], compared to a fully disordered material with the same density of scatterers. This is observed in Fig. 2(c), where we plot the average diffuse intensity versus the observation angle θ calculated from the full numerical simulation without approximation (the averaging is performed over 20 configurations). Also note that this regime is observed as long as the circle described by the scattered wavevector q lies entirely within the domain Ω, independently of the exact shape of Ω. For the uncorrelated disordered medium (red dashed line), a diffuse intensity pattern is observed, while for the hyperuniform structure made with the same scatterers and the same density, the diffuse intensity (blue line) is negligible [reduced by a factor of about 800, so that it is hardly visible in Fig. 2(c)].

For smaller wavelengths (larger k0), another behavior can be observed. Choosing k0=K/3.5 as an illustrative example, the circle described by the scattered wavevector q in Fig. 2(b) is now only partially included inside the domain Ω. Therefore, a reduced level of scattering, due to a vanishing structure factor, is expected only for the observation angles θ[θlim,+θlim] such that q falls within Ω. Theoretically, we find θlim=arccos[1K/(2k0)]=138°. The result of the full numerical simulation of the diffuse intensity pattern is shown in Fig. 2(d). The change between the pattern produced by an uncorrelated disordered medium (red dashed line) and the hyperuniform material (blue line) is clearly visible. For the latter, scattering is suppressed for a wide angular range, with the value θlim=134° in good agreement with the estimate from single scattering theory (for bB=0.5, the higher-order scattering is not fully negligible). Finally, let us note that in terms of frequency bandwidth for a predefined angular range [θlim,+θlim], scattering would be suppressed for all wavelengths satisfying λ>λlim, with λlim=(4π/K)(1cosθlim).

The examples above have illustrated the interest of hyperuniform point patterns in the design of materials with a tunable level of scattering. An interesting question is to see under which conditions these properties could survive in the multiple scattering regime. The transport of the average intensity in this case is well described by the radiative transfer equation [43], which is derived from the Bethe–Salpeter equation in the limit k0B1 [2]. In this framework, the angular dependence of the diffuse intensity is driven by the phase function p(u,u), where u and u are unit vectors denoting the incoming and outgoing directions. For subwavelength scatterers, the phase function is connected to the average structure factor through the relation (see Section 3 of Supplement 1)

p(u,u)S[k0(uu)]N1V2|Θ[k0(uu)]|2,
where the term involving Θ appears as in Eq. (4), but with a different weighting factor. This difference results from the connection between the phase function and the vertex of the Bethe–Salpeter equation that contains irreducible terms only. Note that for an uncorrelated disordered medium, the phase function is constant [i.e., p(u,u)=1]. In the case of a hyperuniform structure, under the condition k0<K/4, the scattered wavevector k0(uu) falls inside the domain Ω [see the construction in Fig. 2(b)], and the phase function vanishes for all directions, except in the forward direction u=u. Indeed, the second term involving Θ in Eq. (5) does not exactly compensate the value of the structure factor at the origin. This means that scattering occurs, but only in the forward direction. For an infinite medium, this would lead to an effective scattering mean free path tending to infinity. In practice, for a medium with size L, the width of Θ(k) is on the order of 2π/L, keeping finite but permitting it in principle to reach arbitrary large values. This is particularly interesting, since getting L leads to transparency. To check this idea, we have used numerical simulations in the multiple scattering regime (bB=5), with k0=K/8 and k0B=44.4. The result is displayed in Fig. 3, where we clearly observe that the average diffuse intensity for the hyperuniform medium (blue line) is extremely small compared to that obtained for a fully disordered medium (red dashed line), except in the forward direction, as expected. Our analysis demonstrates that, in conditions (in terms of type and density of scatterers) under which a fully disordered material would induce strong multiple scattering, a hyperuniform material can chiefly generate forward scattering, leading to an effective scattering mean free path L, which is the condition for transparency.

 figure: Fig. 3.

Fig. 3. (a) Angular pattern of the average diffuse intensity in the multiple-scattering regime (bB=5) for a hyperuniform medium with k0=K/8 (blue line) and for uncorrelated disorder (red dashed line). The same scatterers and the same density are used in both cases, the intensity is averaged over 20 configurations, and k0B=44.4. (b) Zoom on the central part of the polar plot.

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4. EXPLICIT CRITERION FOR TRANSPARENCY

It is possible to establish a criterion for the existence of transparency in a dense (stealth) hyperuniform disordered material, based on a perturbative theory valid for both 2D and 3D materials. To proceed, we use a diagrammatic expansion of the phase function in the form [44]

optica-3-7-763-e001
where the circles denote scatterers, the horizontal solid lines represent free-space Green functions G0, the vertical solid lines stand for identical scatterers, and the dashed lines connect different but correlated scatterers. The first two terms in Eq. (6) correspond to Eq. (5), and according to the analysis above, they compensate for a stealth hyperuniform medium, provided that k0<K/4. The value of the effective scattering mean free path is obtained by an angular integration (over direction u) of the phase function [44]. For a “standard” short-range ordered material, the first-order correction is obtained by integrating the first two terms, leading to an expression equivalent to that introduced initially in Ref. [12] to describe multiple scattering in interacting (but non-hyperuniform) suspensions. In the specific case of stealth hyperuniform disorder, the first two terms do not contribute, and the effective scattering mean free path is obtained by an angular integration of the next four diagrams. For subwavelength scatterers with polarizability α(ω), this leads to
1=8πρk02B0R[α(ω)G0(r)]h(r)F(k0r)rd1dr,
where h is the pair correlation function [20] and d is the dimension of space. Note that for large scatterers with sizes comparable with the wavelength (such as Mie spheres), the dependence of the scattering amplitude of an individual scatterer on the scattered wavevector q would change the analytical expressions. We can expect, in this case, the results to change qualitatively. For scalar waves in 2D, F(x)=J0(x) and G0(r)=(i/4)H0(1)(k0r), with J0 (H0(1)) as the zero-order Bessel (Hankel) function of the first kind. For scalar waves in 3D, F(x)=2sinc(x) and G0(r)=exp(ik0r)/(4πr). An estimate of the right-hand side can be obtained by computing the self-energy Σ. To first order in (k0B)1, one has [2]
optica-3-7-763-e002
which in Fourier space with k=k0u reads
Σ=ρk02α(ω)[1+2πρk02α(ω)0G0(r)h(r)F(k0r)rd1dr].

The integral term in Eq. (9) is on the order of (k0B)1, which can be used to estimate the right-hand side in Eq. (7), leading to B×k0B. Transparency is observed provided that L, which leads to the criterion

bBk0B.

Under this condition, a stealth hyperuniform material becomes transparent for frequencies satisfying k0<K/4. In particular, this means that even for bB1 (corresponding to the multiple scattering regime for a fully disordered material), transparency can be reached, provided that Eq. (10) is satisfied. Therefore, our analysis establishes a criterion for transparency of a hyperuniform material even at high density, i.e., far beyond the single scattering regime.

5. CONCLUSION

In summary, we have shown that materials made of non-absorbing subwavelength scatterers distributed on a stealth hyperuniform point pattern can be transparent, even in conditions under which, for the same density of scatterers, an uncorrelated disordered material would be opaque due to multiple scattering. This occurs under very general conditions that we have established theoretically for 2D and 3D materials. In this first study, the numerical examples have been restricted to 2D materials (which by themselves are of practical interest) for the sake of computer time, but there is no fundamental limitation for the extension of the numerical simulations to 3D. The transparency property of dense hyperuniform media opens new perspectives in the engineering of disordered materials combining specific photonic properties with a high robustness to fabrication errors. More generally, the design of dense and transparent hyperuniform materials is in principle achievable for any kind of wave, and the applications of the concept cover many fields of wave physics.

Funding

LABEX WIFI, Agence Nationale de la Recherche (ANR) (ANR-10-IDEX-0001-02 PSL*, ANR-10-LABX-24); French Centre National de la Recherche Scientifique (CNRS).

Acknowledgment

We thank Kevin Vynck for the helpful discussions.

 

See Supplement 1 for supporting content.

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References

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  1. P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena (Academic, 1995).
  2. E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University, 2007).
  3. P. Sebbah, Waves and Imaging through Complex Media (Kluwer Academic, 2001).
  4. I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32, 2309 (2007).
    [Crossref]
  5. S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
    [Crossref]
  6. A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6, 283–292 (2012).
    [Crossref]
  7. L. F. Rojas-Ochoa, J. M. Mendez-Alcaraz, J. J. Sáenz, P. Schurtenberger, and F. Scheffold, “Photonic properties of strongly correlated colloidal liquids,” Phys. Rev. Lett. 93, 073903 (2004).
    [Crossref]
  8. M. Reufer, L. F. Rojas-Ochoa, S. Eiden, J. J. Sáenz, and F. Scheffold, “Transport of light in amorphous photonic materials,” Appl. Phys. Lett. 91, 171904 (2007).
    [Crossref]
  9. L. Corté, P. M. Chaikin, J. P. Gollub, and D. J. Pine, “Random organization in periodically driven systems,” Nat. Phys. 4, 420–424 (2008).
    [Crossref]
  10. P. D. García, R. Sapienza, A. Blanco, and C. López, “Photonic glass: a novel random material for light,” Adv. Mater. 19, 2597–2602 (2007).
    [Crossref]
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  16. M. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and M. Segev, “Amorphous photonic lattices: band gaps, effective mass, and suppressed transport,” Phys. Rev. Lett. 106, 193904 (2011).
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  19. A. Gabrielli, M. Joyce, and F. Sylos Labini, “Glass-like universe: real-space correlation properties of standard cosmological models,” Phys. Rev. D 65, 083523 (2002).
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  40. O. U. Uche, S. Torquato, and F. H. Stillinger, “Collective coordinate control of density distributions,” Phys. Rev. E 74, 031104 (2006).
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2015 (2)

T. Amoah and M. Florescu, “High-Q optical cavities in hyperuniform disordered materials,” Phys. Rev. B 91, 020201(R) (2015).
[Crossref]

R. Dreyfus, Y. Xu, T. Still, L. A. Hough, A. G. Yodh, and S. Torquato, “Diagnosing hyperuniformity in two-dimensional, disordered, jammed packings of soft spheres,” Phys. Rev. E 91, 012302 (2015).
[Crossref]

2014 (2)

G. M. Conley, M. Burresi, F. Pratesi, K. Vynck, and D. S. Wiersma, “Light transport and localization in two-dimensional correlated disorder,” Phys. Rev. Lett. 112, 143901 (2014).
[Crossref]

O. Leseur, R. Pierrat, J. J. Sáenz, and R. Carminati, “Probing two-dimensional Anderson localization without statistics,” Phys. Rev. A 90, 053827 (2014).
[Crossref]

2013 (6)

J. Haberko and F. Scheffold, “Fabrication of mesoscale polymeric templates for three-dimensional disordered photonic materials,” Opt. Express 21, 1057 (2013).
[Crossref]

J. Haberko, N. Muller, and F. Scheffold, “Direct laser writing of three-dimensional network structures as templates for disordered photonic materials,” Phys. Rev. A 88, 043822 (2013).
[Crossref]

W. Man, M. Florescu, K. Matsuyama, P. Yadak, G. Nahal, S. Hashemizad, E. Williamson, P. Steinhardt, S. Torquato, and P. Chaikin, “Photonic band gap in isotropic hyperuniform disordered solids with low dielectric contrast,” Opt. Express 21, 19972 (2013).
[Crossref]

N. Muller, J. Haberko, C. Marichy, and F. Scheffold, “Silicon hyperuniform disordered photonic materials with a pronounced gap in the shortwave infrared,” Adv. Opt. Mater. 2, 115–119 (2013).
[Crossref]

M. Florescu, P. J. Steinhardt, and S. Torquato, “Optical cavities and waveguides in hyperuniform disordered photonic solids,” Phys. Rev. B 87, 165116 (2013).
[Crossref]

W. Man, M. Florescu, E. P. Williamson, Y. He, S. R. Hashemizad, B. Y. C. Leung, D. R. Liner, S. Torquato, P. M. Chaikin, and P. J. Steinhardt, “Isotropic band gaps and freeform waveguides observed in hyperuniform disordered photonic solids,” Proc. Natl. Acad. Sci. USA 110, 15886–15891 (2013).
[Crossref]

2012 (1)

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6, 283–292 (2012).
[Crossref]

2011 (3)

M. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and M. Segev, “Amorphous photonic lattices: band gaps, effective mass, and suppressed transport,” Phys. Rev. Lett. 106, 193904 (2011).
[Crossref]

J.-H. Tian, J. Hu, S.-S. Li, F. Zhang, J. Liu, J. Shi, X. Li, Z.-Q. Tian, and Y. Chen, “Improved seedless hydrothermal synthesis of dense and ultralong ZnO nanowires,” Nanotechnology 22, 245601 (2011).
[Crossref]

C. E. Zachary, Y. Jiao, and S. Torquato, “Hyperuniform long-range correlations are a signature of disordered jammed hard-particle packings,” Phys. Rev. Lett. 106, 178001 (2011).
[Crossref]

2010 (3)

J.-K. Yang, C. Schreck, H. Noh, S.-F. Liew, M. I. Guy, C. S. O’Hern, and H. Cao, “Photonic-band-gap effects in two-dimensional polycrystalline and amorphous structures,” Phys. Rev. A 82, 053838 (2010).
[Crossref]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref]

P. D. García, R. Sapienza, and C. López, “Photonic glasses: a step beyond white paint,” Adv. Mater. 22, 12–19 (2010).
[Crossref]

2009 (1)

M. Florescu, S. Torquato, and P. J. Steinhardt, “Designer disordered materials with large, complete photonic band gaps,” Proc. Natl. Acad. Sci. USA 106, 20658–20663 (2009).
[Crossref]

2008 (4)

R. D. Batten, F. H. Stillinger, and S. Torquato, “Classical disordered ground states: super-ideal gases and stealth and equi-luminous materials,” J. Appl. Phys. 104, 033504 (2008).
[Crossref]

A. Gabrielli, M. Joyce, and S. Torquato, “Tilings of space and superhomogeneous point processes,” Phys. Rev. E 77, 031125 (2008).
[Crossref]

L. Corté, P. M. Chaikin, J. P. Gollub, and D. J. Pine, “Random organization in periodically driven systems,” Nat. Phys. 4, 420–424 (2008).
[Crossref]

K. Edagawa, S. Kanoko, and M. Notomi, “Photonic amorphous diamond structure with a 3D photonic band gap,” Phys. Rev. Lett. 100, 013901 (2008).
[Crossref]

2007 (3)

M. Reufer, L. F. Rojas-Ochoa, S. Eiden, J. J. Sáenz, and F. Scheffold, “Transport of light in amorphous photonic materials,” Appl. Phys. Lett. 91, 171904 (2007).
[Crossref]

P. D. García, R. Sapienza, A. Blanco, and C. López, “Photonic glass: a novel random material for light,” Adv. Mater. 19, 2597–2602 (2007).
[Crossref]

I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32, 2309 (2007).
[Crossref]

2006 (1)

O. U. Uche, S. Torquato, and F. H. Stillinger, “Collective coordinate control of density distributions,” Phys. Rev. E 74, 031104 (2006).
[Crossref]

2005 (1)

A. Donev, F. Stillinger, and S. Torquato, “Unexpected density fluctuations in jammed disordered sphere packings,” Phys. Rev. Lett. 95, 090604 (2005).
[Crossref]

2004 (2)

O. U. Uche, F. H. Stillinger, and S. Torquato, “Constraints on collective density variables: two dimensions,” Phys. Rev. E 70, 046122 (2004).
[Crossref]

L. F. Rojas-Ochoa, J. M. Mendez-Alcaraz, J. J. Sáenz, P. Schurtenberger, and F. Scheffold, “Photonic properties of strongly correlated colloidal liquids,” Phys. Rev. Lett. 93, 073903 (2004).
[Crossref]

2003 (2)

A. Yamilov and H. Cao, “Density of resonant states and a manifestation of photonic band structure in small clusters of spherical particles,” Phys. Rev. B 68, 085111 (2003).
[Crossref]

S. Torquato and F. H. Stillinger, “Local density fluctuations, hyperuniformity, and order metrics,” Phys. Rev. E 68, 041113 (2003).
[Crossref]

2002 (1)

A. Gabrielli, M. Joyce, and F. Sylos Labini, “Glass-like universe: real-space correlation properties of standard cosmological models,” Phys. Rev. D 65, 083523 (2002).
[Crossref]

2001 (1)

C. Jin, X. Meng, B. Cheng, Z. Li, and D. Zhang, “Photonic gap in amorphous photonic materials,” Phys. Rev. B 63, 195107 (2001).
[Crossref]

1990 (1)

S. Fraden and G. Maret, “Multiple light scattering from concentrated, interacting suspensions,” Phys. Rev. Lett. 65, 512–515 (1990).
[Crossref]

1976 (1)

1952 (1)

M. Lax, “Multiple scattering of waves. II. The effective field in dense systems,” Phys. Rev. 85, 621–629 (1952).
[Crossref]

Akkermans, E.

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University, 2007).

Amoah, T.

T. Amoah and M. Florescu, “High-Q optical cavities in hyperuniform disordered materials,” Phys. Rev. B 91, 020201(R) (2015).
[Crossref]

Batten, R. D.

R. D. Batten, F. H. Stillinger, and S. Torquato, “Classical disordered ground states: super-ideal gases and stealth and equi-luminous materials,” J. Appl. Phys. 104, 033504 (2008).
[Crossref]

Blanco, A.

P. D. García, R. Sapienza, A. Blanco, and C. López, “Photonic glass: a novel random material for light,” Adv. Mater. 19, 2597–2602 (2007).
[Crossref]

Boccara, A. C.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref]

Burresi, M.

G. M. Conley, M. Burresi, F. Pratesi, K. Vynck, and D. S. Wiersma, “Light transport and localization in two-dimensional correlated disorder,” Phys. Rev. Lett. 112, 143901 (2014).
[Crossref]

Cao, H.

J.-K. Yang, C. Schreck, H. Noh, S.-F. Liew, M. I. Guy, C. S. O’Hern, and H. Cao, “Photonic-band-gap effects in two-dimensional polycrystalline and amorphous structures,” Phys. Rev. A 82, 053838 (2010).
[Crossref]

A. Yamilov and H. Cao, “Density of resonant states and a manifestation of photonic band structure in small clusters of spherical particles,” Phys. Rev. B 68, 085111 (2003).
[Crossref]

Carminati, R.

O. Leseur, R. Pierrat, J. J. Sáenz, and R. Carminati, “Probing two-dimensional Anderson localization without statistics,” Phys. Rev. A 90, 053827 (2014).
[Crossref]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref]

Chaikin, P.

Chaikin, P. M.

W. Man, M. Florescu, E. P. Williamson, Y. He, S. R. Hashemizad, B. Y. C. Leung, D. R. Liner, S. Torquato, P. M. Chaikin, and P. J. Steinhardt, “Isotropic band gaps and freeform waveguides observed in hyperuniform disordered photonic solids,” Proc. Natl. Acad. Sci. USA 110, 15886–15891 (2013).
[Crossref]

L. Corté, P. M. Chaikin, J. P. Gollub, and D. J. Pine, “Random organization in periodically driven systems,” Nat. Phys. 4, 420–424 (2008).
[Crossref]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, 1950).

Chen, Y.

J.-H. Tian, J. Hu, S.-S. Li, F. Zhang, J. Liu, J. Shi, X. Li, Z.-Q. Tian, and Y. Chen, “Improved seedless hydrothermal synthesis of dense and ultralong ZnO nanowires,” Nanotechnology 22, 245601 (2011).
[Crossref]

Cheng, B.

C. Jin, X. Meng, B. Cheng, Z. Li, and D. Zhang, “Photonic gap in amorphous photonic materials,” Phys. Rev. B 63, 195107 (2001).
[Crossref]

Conley, G. M.

G. M. Conley, M. Burresi, F. Pratesi, K. Vynck, and D. S. Wiersma, “Light transport and localization in two-dimensional correlated disorder,” Phys. Rev. Lett. 112, 143901 (2014).
[Crossref]

Corté, L.

L. Corté, P. M. Chaikin, J. P. Gollub, and D. J. Pine, “Random organization in periodically driven systems,” Nat. Phys. 4, 420–424 (2008).
[Crossref]

Damasceno, P. F.

L. S. Froufe-Pérez, M. Engel, P. F. Damasceno, N. Muller, J. Haberko, S. C. Glotzer, and F. Scheffold, “The role of short-range order and hyperuniformity in the formation of band gaps in disordered photonic materials,” arXiv:1602.01002 (2016).

Donev, A.

A. Donev, F. Stillinger, and S. Torquato, “Unexpected density fluctuations in jammed disordered sphere packings,” Phys. Rev. Lett. 95, 090604 (2005).
[Crossref]

Dreisow, F.

M. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and M. Segev, “Amorphous photonic lattices: band gaps, effective mass, and suppressed transport,” Phys. Rev. Lett. 106, 193904 (2011).
[Crossref]

Dreyfus, R.

R. Dreyfus, Y. Xu, T. Still, L. A. Hough, A. G. Yodh, and S. Torquato, “Diagnosing hyperuniformity in two-dimensional, disordered, jammed packings of soft spheres,” Phys. Rev. E 91, 012302 (2015).
[Crossref]

Edagawa, K.

K. Edagawa, S. Kanoko, and M. Notomi, “Photonic amorphous diamond structure with a 3D photonic band gap,” Phys. Rev. Lett. 100, 013901 (2008).
[Crossref]

Eiden, S.

M. Reufer, L. F. Rojas-Ochoa, S. Eiden, J. J. Sáenz, and F. Scheffold, “Transport of light in amorphous photonic materials,” Appl. Phys. Lett. 91, 171904 (2007).
[Crossref]

Engel, M.

L. S. Froufe-Pérez, M. Engel, P. F. Damasceno, N. Muller, J. Haberko, S. C. Glotzer, and F. Scheffold, “The role of short-range order and hyperuniformity in the formation of band gaps in disordered photonic materials,” arXiv:1602.01002 (2016).

Farrell, R. A.

Fink, M.

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6, 283–292 (2012).
[Crossref]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref]

Florescu, M.

T. Amoah and M. Florescu, “High-Q optical cavities in hyperuniform disordered materials,” Phys. Rev. B 91, 020201(R) (2015).
[Crossref]

M. Florescu, P. J. Steinhardt, and S. Torquato, “Optical cavities and waveguides in hyperuniform disordered photonic solids,” Phys. Rev. B 87, 165116 (2013).
[Crossref]

W. Man, M. Florescu, E. P. Williamson, Y. He, S. R. Hashemizad, B. Y. C. Leung, D. R. Liner, S. Torquato, P. M. Chaikin, and P. J. Steinhardt, “Isotropic band gaps and freeform waveguides observed in hyperuniform disordered photonic solids,” Proc. Natl. Acad. Sci. USA 110, 15886–15891 (2013).
[Crossref]

W. Man, M. Florescu, K. Matsuyama, P. Yadak, G. Nahal, S. Hashemizad, E. Williamson, P. Steinhardt, S. Torquato, and P. Chaikin, “Photonic band gap in isotropic hyperuniform disordered solids with low dielectric contrast,” Opt. Express 21, 19972 (2013).
[Crossref]

M. Florescu, S. Torquato, and P. J. Steinhardt, “Designer disordered materials with large, complete photonic band gaps,” Proc. Natl. Acad. Sci. USA 106, 20658–20663 (2009).
[Crossref]

Fraden, S.

S. Fraden and G. Maret, “Multiple light scattering from concentrated, interacting suspensions,” Phys. Rev. Lett. 65, 512–515 (1990).
[Crossref]

Froufe-Pérez, L. S.

L. S. Froufe-Pérez, M. Engel, P. F. Damasceno, N. Muller, J. Haberko, S. C. Glotzer, and F. Scheffold, “The role of short-range order and hyperuniformity in the formation of band gaps in disordered photonic materials,” arXiv:1602.01002 (2016).

Gabrielli, A.

A. Gabrielli, M. Joyce, and S. Torquato, “Tilings of space and superhomogeneous point processes,” Phys. Rev. E 77, 031125 (2008).
[Crossref]

A. Gabrielli, M. Joyce, and F. Sylos Labini, “Glass-like universe: real-space correlation properties of standard cosmological models,” Phys. Rev. D 65, 083523 (2002).
[Crossref]

García, P. D.

P. D. García, R. Sapienza, and C. López, “Photonic glasses: a step beyond white paint,” Adv. Mater. 22, 12–19 (2010).
[Crossref]

P. D. García, R. Sapienza, A. Blanco, and C. López, “Photonic glass: a novel random material for light,” Adv. Mater. 19, 2597–2602 (2007).
[Crossref]

Gigan, S.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref]

Glotzer, S. C.

L. S. Froufe-Pérez, M. Engel, P. F. Damasceno, N. Muller, J. Haberko, S. C. Glotzer, and F. Scheffold, “The role of short-range order and hyperuniformity in the formation of band gaps in disordered photonic materials,” arXiv:1602.01002 (2016).

Gollub, J. P.

L. Corté, P. M. Chaikin, J. P. Gollub, and D. J. Pine, “Random organization in periodically driven systems,” Nat. Phys. 4, 420–424 (2008).
[Crossref]

Guy, M. I.

J.-K. Yang, C. Schreck, H. Noh, S.-F. Liew, M. I. Guy, C. S. O’Hern, and H. Cao, “Photonic-band-gap effects in two-dimensional polycrystalline and amorphous structures,” Phys. Rev. A 82, 053838 (2010).
[Crossref]

Haberko, J.

J. Haberko, N. Muller, and F. Scheffold, “Direct laser writing of three-dimensional network structures as templates for disordered photonic materials,” Phys. Rev. A 88, 043822 (2013).
[Crossref]

J. Haberko and F. Scheffold, “Fabrication of mesoscale polymeric templates for three-dimensional disordered photonic materials,” Opt. Express 21, 1057 (2013).
[Crossref]

N. Muller, J. Haberko, C. Marichy, and F. Scheffold, “Silicon hyperuniform disordered photonic materials with a pronounced gap in the shortwave infrared,” Adv. Opt. Mater. 2, 115–119 (2013).
[Crossref]

L. S. Froufe-Pérez, M. Engel, P. F. Damasceno, N. Muller, J. Haberko, S. C. Glotzer, and F. Scheffold, “The role of short-range order and hyperuniformity in the formation of band gaps in disordered photonic materials,” arXiv:1602.01002 (2016).

Hansen, J.-P.

J.-P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic, 2005).

Hashemizad, S.

Hashemizad, S. R.

W. Man, M. Florescu, E. P. Williamson, Y. He, S. R. Hashemizad, B. Y. C. Leung, D. R. Liner, S. Torquato, P. M. Chaikin, and P. J. Steinhardt, “Isotropic band gaps and freeform waveguides observed in hyperuniform disordered photonic solids,” Proc. Natl. Acad. Sci. USA 110, 15886–15891 (2013).
[Crossref]

He, Y.

W. Man, M. Florescu, E. P. Williamson, Y. He, S. R. Hashemizad, B. Y. C. Leung, D. R. Liner, S. Torquato, P. M. Chaikin, and P. J. Steinhardt, “Isotropic band gaps and freeform waveguides observed in hyperuniform disordered photonic solids,” Proc. Natl. Acad. Sci. USA 110, 15886–15891 (2013).
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Heinrich, M.

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O. Leseur, R. Pierrat, J. J. Sáenz, and R. Carminati, “Probing two-dimensional Anderson localization without statistics,” Phys. Rev. A 90, 053827 (2014).
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M. Reufer, L. F. Rojas-Ochoa, S. Eiden, J. J. Sáenz, and F. Scheffold, “Transport of light in amorphous photonic materials,” Appl. Phys. Lett. 91, 171904 (2007).
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N. Muller, J. Haberko, C. Marichy, and F. Scheffold, “Silicon hyperuniform disordered photonic materials with a pronounced gap in the shortwave infrared,” Adv. Opt. Mater. 2, 115–119 (2013).
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L. F. Rojas-Ochoa, J. M. Mendez-Alcaraz, J. J. Sáenz, P. Schurtenberger, and F. Scheffold, “Photonic properties of strongly correlated colloidal liquids,” Phys. Rev. Lett. 93, 073903 (2004).
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L. S. Froufe-Pérez, M. Engel, P. F. Damasceno, N. Muller, J. Haberko, S. C. Glotzer, and F. Scheffold, “The role of short-range order and hyperuniformity in the formation of band gaps in disordered photonic materials,” arXiv:1602.01002 (2016).

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J.-K. Yang, C. Schreck, H. Noh, S.-F. Liew, M. I. Guy, C. S. O’Hern, and H. Cao, “Photonic-band-gap effects in two-dimensional polycrystalline and amorphous structures,” Phys. Rev. A 82, 053838 (2010).
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Steinhardt, P. J.

W. Man, M. Florescu, E. P. Williamson, Y. He, S. R. Hashemizad, B. Y. C. Leung, D. R. Liner, S. Torquato, P. M. Chaikin, and P. J. Steinhardt, “Isotropic band gaps and freeform waveguides observed in hyperuniform disordered photonic solids,” Proc. Natl. Acad. Sci. USA 110, 15886–15891 (2013).
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M. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and M. Segev, “Amorphous photonic lattices: band gaps, effective mass, and suppressed transport,” Phys. Rev. Lett. 106, 193904 (2011).
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J.-H. Tian, J. Hu, S.-S. Li, F. Zhang, J. Liu, J. Shi, X. Li, Z.-Q. Tian, and Y. Chen, “Improved seedless hydrothermal synthesis of dense and ultralong ZnO nanowires,” Nanotechnology 22, 245601 (2011).
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J.-H. Tian, J. Hu, S.-S. Li, F. Zhang, J. Liu, J. Shi, X. Li, Z.-Q. Tian, and Y. Chen, “Improved seedless hydrothermal synthesis of dense and ultralong ZnO nanowires,” Nanotechnology 22, 245601 (2011).
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R. Dreyfus, Y. Xu, T. Still, L. A. Hough, A. G. Yodh, and S. Torquato, “Diagnosing hyperuniformity in two-dimensional, disordered, jammed packings of soft spheres,” Phys. Rev. E 91, 012302 (2015).
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W. Man, M. Florescu, K. Matsuyama, P. Yadak, G. Nahal, S. Hashemizad, E. Williamson, P. Steinhardt, S. Torquato, and P. Chaikin, “Photonic band gap in isotropic hyperuniform disordered solids with low dielectric contrast,” Opt. Express 21, 19972 (2013).
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M. Florescu, P. J. Steinhardt, and S. Torquato, “Optical cavities and waveguides in hyperuniform disordered photonic solids,” Phys. Rev. B 87, 165116 (2013).
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W. Man, M. Florescu, E. P. Williamson, Y. He, S. R. Hashemizad, B. Y. C. Leung, D. R. Liner, S. Torquato, P. M. Chaikin, and P. J. Steinhardt, “Isotropic band gaps and freeform waveguides observed in hyperuniform disordered photonic solids,” Proc. Natl. Acad. Sci. USA 110, 15886–15891 (2013).
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C. E. Zachary, Y. Jiao, and S. Torquato, “Hyperuniform long-range correlations are a signature of disordered jammed hard-particle packings,” Phys. Rev. Lett. 106, 178001 (2011).
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M. Florescu, S. Torquato, and P. J. Steinhardt, “Designer disordered materials with large, complete photonic band gaps,” Proc. Natl. Acad. Sci. USA 106, 20658–20663 (2009).
[Crossref]

A. Gabrielli, M. Joyce, and S. Torquato, “Tilings of space and superhomogeneous point processes,” Phys. Rev. E 77, 031125 (2008).
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O. U. Uche, S. Torquato, and F. H. Stillinger, “Collective coordinate control of density distributions,” Phys. Rev. E 74, 031104 (2006).
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A. Donev, F. Stillinger, and S. Torquato, “Unexpected density fluctuations in jammed disordered sphere packings,” Phys. Rev. Lett. 95, 090604 (2005).
[Crossref]

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O. U. Uche, S. Torquato, and F. H. Stillinger, “Collective coordinate control of density distributions,” Phys. Rev. E 74, 031104 (2006).
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R. Dreyfus, Y. Xu, T. Still, L. A. Hough, A. G. Yodh, and S. Torquato, “Diagnosing hyperuniformity in two-dimensional, disordered, jammed packings of soft spheres,” Phys. Rev. E 91, 012302 (2015).
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C. E. Zachary, Y. Jiao, and S. Torquato, “Hyperuniform long-range correlations are a signature of disordered jammed hard-particle packings,” Phys. Rev. Lett. 106, 178001 (2011).
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J.-H. Tian, J. Hu, S.-S. Li, F. Zhang, J. Liu, J. Shi, X. Li, Z.-Q. Tian, and Y. Chen, “Improved seedless hydrothermal synthesis of dense and ultralong ZnO nanowires,” Nanotechnology 22, 245601 (2011).
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Adv. Mater. (2)

P. D. García, R. Sapienza, A. Blanco, and C. López, “Photonic glass: a novel random material for light,” Adv. Mater. 19, 2597–2602 (2007).
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Adv. Opt. Mater. (1)

N. Muller, J. Haberko, C. Marichy, and F. Scheffold, “Silicon hyperuniform disordered photonic materials with a pronounced gap in the shortwave infrared,” Adv. Opt. Mater. 2, 115–119 (2013).
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Appl. Phys. Lett. (1)

M. Reufer, L. F. Rojas-Ochoa, S. Eiden, J. J. Sáenz, and F. Scheffold, “Transport of light in amorphous photonic materials,” Appl. Phys. Lett. 91, 171904 (2007).
[Crossref]

J. Appl. Phys. (1)

R. D. Batten, F. H. Stillinger, and S. Torquato, “Classical disordered ground states: super-ideal gases and stealth and equi-luminous materials,” J. Appl. Phys. 104, 033504 (2008).
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Nanotechnology (1)

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Supplementary Material (1)

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Figures (3)

Fig. 1.
Fig. 1. (a) Hyperuniform medium generated with N = 90000 and χ = 0.222 . (b) Structure factor of the hyperuniform medium averaged over 20 configurations. (c) Uncorrelated medium generated with N = 90000 and χ = 0 . (d) Structure factor of the uncorrelated medium.
Fig. 2.
Fig. 2. (a) Scattering geometry. The 2D medium is illuminated by a Gaussian beam at normal incidence (beam waist w = 60 / k 0 ), and the diffuse intensity is calculated in direction θ . (b) Schematic view of the domain described by the scattered wavevector q = k s k i in Fourier space. For a hyperuniform material, the structure factor vanishes in the square domain Ω . (c) Angular pattern of the average diffuse intensity in the single scattering regime ( b B = 0.5 ) for a hyperuniform medium with k 0 = K / 8 (blue line) and for uncorrelated disorder (red dashed line). The same scatterers and the same density are used in both cases, the intensity is averaged over 20 configurations, and k 0 B = 444 . (d) Same as (c) with k 0 = K / 3.5 and k 0 B = 1015 .
Fig. 3.
Fig. 3. (a) Angular pattern of the average diffuse intensity in the multiple-scattering regime ( b B = 5 ) for a hyperuniform medium with k 0 = K / 8 (blue line) and for uncorrelated disorder (red dashed line). The same scatterers and the same density are used in both cases, the intensity is averaged over 20 configurations, and k 0 B = 44.4 . (b) Zoom on the central part of the polar plot.

Equations (8)

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S ( q ) = 1 N | j = 1 N exp [ i q · r j ] | 2 ,
E sca ( r , ω ) = i ( 1 i ) 4 π k 0 r exp [ i k 0 r ] E sca ( θ , ω ) ,
I diff ( θ , ω ) = | E sca ( θ , ω ) | 2 | E sca ( θ , ω ) | 2 ,
I diff ( θ , ω ) N S ( q ) | E 0 | 2 N 2 V 2 | Θ ( q ) | 2 | E 0 | 2 ,
p ( u , u ) S [ k 0 ( u u ) ] N 1 V 2 | Θ [ k 0 ( u u ) ] | 2 ,
1 = 8 π ρ k 0 2 B 0 R [ α ( ω ) G 0 ( r ) ] h ( r ) F ( k 0 r ) r d 1 d r ,
Σ = ρ k 0 2 α ( ω ) [ 1 + 2 π ρ k 0 2 α ( ω ) 0 G 0 ( r ) h ( r ) F ( k 0 r ) r d 1 d r ] .
b B k 0 B .

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