## 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 [4–6]. 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) [7–11]. 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 [15–17], 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 $S(\mathbf{q})$ vanishing in the neighborhood of $|\mathbf{q}|=0$. The geometrical properties of hyperuniform point patterns have been extensively studied, in particular in terms of packing properties [21–25]. Regarding wave propagation, it has been shown that bandgaps could be observed for electromagnetic waves in two-dimensional (2D) disordered hyperuniform materials [26–30]. 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 $S(\mathbf{q})=0$ in a finite domain around $|\mathbf{q}|=0$, 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]

vanishes in the neighborhood of $|\mathbf{q}|=0$ [23], with $N$ as the number of points and ${\mathbf{r}}_{j}$ 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 $V={L}^{2}$, with periodic boundary conditions to avoid finite-size effects. We consider hyperuniform point distributions, such that $S(\mathbf{q})=0$ in a square domain $\mathrm{\Omega}$ of reciprocal space with size $K$, centered at the origin. The square shape of $\mathrm{\Omega}$ 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 $S(\mathbf{q})$ is controlled on a finite number of points located on a square lattice inside $\mathrm{\Omega}$ and separated by $2\pi /L$. The number of points $M(K)$ depends on the size $K$ of the domain $\mathrm{\Omega}$ and satisfies $2M(K)+1=({K}^{2}{L}^{2})/(4{\pi}^{2})$, where the factor of 2 results from the symmetry property $S(\mathbf{q})=S(-\mathbf{q})$ and the “$+1$” correction accounts for the fact that the value of the structure factor at the exact point $\mathbf{q}=0$ cannot be controlled [$S(\mathbf{q}=0)=N$]. 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 $\chi =M(K)/(2N)$. For $\chi =0$, the system is uncorrelated (fully disordered), while for $\chi =1$, the system is a perfect crystal [21]. From the expression of $M(K)$, one immediately gets $K=(2\pi ){L}^{-1}\sqrt{4N\chi +1}$, 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 $\mathrm{\Omega}$, the structure factor vanishes [except at the origin, where $S(\mathbf{q}=0)=N$]. 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.

## 3. SCATTERING FROM STEALTH HYPERUNIFORM MATERIALS

A hyperuniform scattering material is built by dressing each point with an electric polarizability $\alpha (\omega )$ describing the electrodynamics’ response of the individual scatterers at frequency $\omega $ (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 $\alpha (\omega )=-4\gamma {c}^{2}/[{\omega}_{0}({\omega}^{2}-{\omega}_{0}^{2}+i\gamma {\omega}^{2}/{\omega}_{0})]$, where ${\omega}_{0}$ is the resonant frequency and $\gamma $ 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 ${\mathbf{k}}_{i}$) 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 ${E}_{\text{sca}}(\mathbf{r})$ 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 ${\mathbf{k}}_{s}$). When ${k}_{0}r\gg 1$, $r$ being the observation distance and ${k}_{0}=\omega /c=2\pi /\lambda $, with $c$ as the speed of light and $\lambda $ as the wavelength in vacuum, the scattered field takes the asymptotic form

We first consider the single scattering regime, defined by ${b}_{B}(\omega )<1$, where ${b}_{B}(\omega )=L/{\ell}_{B}(\omega )$ is the optical thickness in the independent scattering (or Boltzmann) approximation, for which the scattering mean free path is ${\ell}_{B}(\omega )={[\rho {\sigma}_{s}(\omega )]}^{-1}$ and the condition ${k}_{0}{\ell}_{B}(\omega )\gg 1$ is satisfied. Here, $\rho =N/V$ is the number density of scatterers, and ${\sigma}_{s}(\omega )=({k}_{0}^{3}/4){|\alpha (\omega )|}^{2}$ is their scattering cross section. Under plane-wave illumination with amplitude ${E}_{0}$, 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):

In the single scattering regime, the vanishing of $S(\mathbf{q})$ in a finite domain $\mathrm{\Omega}$ 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 ${b}_{B}=0.5$, and we change the incident wavelength, or equivalently, ${k}_{0}$, in order to tune the range of scattered wavevector $\mathbf{q}={\mathbf{k}}_{s}-{\mathbf{k}}_{i}={k}_{0}{\widehat{\mathbf{k}}}_{s}-{k}_{0}{\widehat{\mathbf{k}}}_{i}$ that is involved in the scattering process (here, $^$ denotes a unit vector). The first situation is chosen with ${k}_{0}=K/8$, so that the circle described by the scattered wavevector $\mathbf{q}$ in Fig. 2(b) lies entirely within the domain $\mathrm{\Omega}$. In this case, we expect a substantial reduction of scattering for a hyperuniform material [with $S(\mathbf{q})\simeq 0$ in $\mathrm{\Omega}$], 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 $\theta $ 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 $\mathbf{q}$ lies entirely within the domain $\mathrm{\Omega}$, independently of the exact shape of $\mathrm{\Omega}$. 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 ${k}_{0}$), another behavior can be observed. Choosing ${k}_{0}=K/3.5$ as an illustrative example, the circle described by the scattered wavevector $\mathbf{q}$ in Fig. 2(b) is now only partially included inside the domain $\mathrm{\Omega}$. Therefore, a reduced level of scattering, due to a vanishing structure factor, is expected only for the observation angles $\theta \in [-{\theta}_{\text{lim}},+{\theta}_{\text{lim}}]$ such that $\mathbf{q}$ falls within $\mathrm{\Omega}$. Theoretically, we find ${\theta}_{\text{lim}}=\mathrm{arccos}[1-K/(2{k}_{0})]=138\xb0$. 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 ${\theta}_{\text{lim}}=134\xb0$ in good agreement with the estimate from single scattering theory (for ${b}_{B}=0.5$, the higher-order scattering is not fully negligible). Finally, let us note that in terms of frequency bandwidth for a predefined angular range $[-{\theta}_{\text{lim}},+{\theta}_{\text{lim}}]$, scattering would be suppressed for all wavelengths satisfying $\lambda >{\lambda}_{\text{lim}}$, with ${\lambda}_{\text{lim}}=(4\pi /K)(1-\mathrm{cos}\text{\hspace{0.17em}}{\theta}_{\text{lim}})$.

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 ${k}_{0}{\ell}_{B}\gg 1$ [2]. In this framework, the angular dependence of the diffuse intensity is driven by the phase function $p(\mathbf{u},{\mathbf{u}}^{\prime})$, where ${\mathbf{u}}^{\prime}$ and $\mathbf{u}$ 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)

## 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]

where the circles denote scatterers, the horizontal solid lines represent free-space Green functions ${G}_{0}$, 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 ${k}_{0}<K/4$. The value of the effective scattering mean free path is obtained by an angular integration (over direction $\mathbf{u}$) 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 $\alpha (\omega )$, this leads toThe integral term in Eq. (9) is on the order of ${({k}_{0}{\ell}_{B})}^{-1}$, which can be used to estimate the right-hand side in Eq. (7), leading to $\ell \sim {\ell}_{B}\times {k}_{0}{\ell}_{B}$. Transparency is observed provided that $\ell \gg L$, which leads to the criterion

Under this condition, a stealth hyperuniform material becomes transparent for frequencies satisfying ${k}_{0}<K/4$. In particular, this means that even for ${b}_{B}\gg 1$ (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.

## REFERENCES

**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]

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

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

**13. **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]

**14. **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]

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

**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). [CrossRef]

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

**18. **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]

**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). [CrossRef]

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

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

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

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

**24. **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]

**25. **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]

**26. **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]

**27. **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]

**28. **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]

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

**30. **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]

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

**32. **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).

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

**34. **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]

**35. **J. M. Ziman, *Models of Disorder: The Theoretical Physics of Homogeneously Disordered Systems* (Cambridge University, 1979).

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

**37. **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]

**38. **R. A. Farrell and R. L. McCally, “On corneal transparency and its loss with swelling,” J. Opt. Soc. Am. **66**, 342 (1976). [CrossRef]

**39. **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]

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

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

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

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

**44. **S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, *Principles of Statistical Radiophysics* (Springer-Verlag, 1989), Vol. 4.