Light in correlated disordered media Kevin Vynck∗ Univ. Bordeaux, Institut d’Optique Graduate School, CNRS, Laboratoire Photonique Numérique et Nanosciences (LP2N), F-33400 Talence, France Univ. Claude Bernard Lyon 1, CNRS, Institut Lumière Matière (iLM), F-69622 Villeurbanne, France Romain Pierrat and Rémi Carminati† ESPCI Paris, PSL University, CNRS, Institut Langevin, F-75005 Paris, France Luis S. Froufe-Pérez and Frank Scheffold‡ Physics Department, University of Fribourg, CH-1700 Fribourg, Switzerland Riccardo Sapienza Imperial College London, Blackett Laboratory, London, SW7 2AZ, United Kingdom Silvia Vignolini Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge, CB2 1EW, United Kingdom Juan José Sáenz Donostia International Physics Center (DIPC), 20018 Donostia-San Sebastian, Spain IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Spain (Dated: May 2, 2023) The optics of correlated disordered media is a fascinating research topic emerging at the interface between the physics of waves in complex media and nanophotonics. Inspired by photonic structures in nature and enabled by advances in nanofabrication processes, recent investigations have unveiled how the design of structural correlations down to the subwavelength scale could be exploited to control the scattering, transport and local- ization of light in matter. From optical transparency to superdiffusive light transport to photonic gaps, the optics of correlated disordered media challenges our physical in- tuition and offers new perspectives for applications. This article reviews the theoretical foundations, state-of-the-art experimental techniques and major achievements in the study of light interaction with correlated disorder, covering a wide range of systems – from short-range correlated photonic liquids, to Lévy glasses containing fractal hetero- geneities, to hyperuniform disordered photonic materials. The mechanisms underlying light scattering and transport phenomena are elucidated on the basis of rigorous theoret- ical arguments. We overview the exciting ongoing research on mesoscopic phenomena, such as transport phase transitions and speckle statistics, and the current development of disorder engineering for applications such as light-energy management and visual ap- pearance design. Special efforts are finally made to identify the main theoretical and experimental challenges to address in the near future. 2 CONTENTS I. Introduction 2 II. Theory of multiple light scattering by correlated disordered media 5 A. General framework 5 1. Average field and self-energy 5 2. Refractive index and extinction mean free path 6 3. Multiple-scattering expansion 6 4. Average intensity and four-point irreducible vertex 7 5. Radiative transfer limit and scattering mean free path 9 6. Transport mean free path and diffusion approximation 10 B. Media with fluctuating continuous permittivity 10 1. Weak permittivity fluctuations 10 2. Lorentz local fields: strong fluctuations 11 3. Average exciting field 12 4. Long-wavelength solutions: Bruggeman versus Maxwell-Garnett models 12 5. Expressions for the scattering and transport mean free paths from the average intensity 13 C. Particulate media 14 1. Expansion for identical scatterers 14 2. Extinction mean free path and effective medium theories 15 3. Scattering and transport mean free paths for resonant scatterers 16 D. Summary and further remarks 17 III. Structural properties of correlated disordered media 19 A. Continuous permittivity versus particulate models in practice 19 B. Fluctuation-correlation relation 20 C. Classes of correlated disordered media 20 1. Short-range correlated disordered structures 21 2. Polycrystalline structures 21 3. Imperfect ordered structures 22 4. Disordered hyperuniform structures 23 5. Disordered fractal structures 23 D. Numerical simulation of correlated disordered media 24 1. Structure generation 24 2. Electromagnetic simulations 25 E. Fabrication of correlated disordered media 26 1. Jammed colloidal packing 26 2. Thermodynamically-driven self-assembly 27 3. Optical and e-beam lithography 28 F. Measuring structural correlations 29 IV. Modified transport parameters 29 A. Light scattering and transport in colloids and photonic materials 29 1. Impact of short-range correlations: first insights 29 2. Enhanced optical transparency 31 3. Tunable light transport in photonic liquids 31 4. Resonant effects in photonic glasses 32 5. Modified diffusion in imperfect photonic crystals 32 B. Anomalous transport in media with large-scale heterogeneity 33 1. Radiative transfer with non-exponential extinction 33 ∗ kevin.vynck@univ-lyon1.fr † remi.carminati@espci.psl.eu ‡ frank.scheffold@unifr.ch 2. From normal to super-diffusion 34 V. Mesoscopic and near-field effects 35 A. Photonic gaps in disordered media 35 1. Definition and identification of photonic gaps in disordered media 36 2. Competing viewpoints on the origin of photonic gaps 37 3. Reports of photonic gaps in the litterature 38 B. Mesoscopic transport and light localization 39 C. Near-field speckles on correlated materials 41 1. Intensity and field correlations in bulk speckle patterns 41 2. Near-field speckles on dielectrics 41 D. Local density of states fluctuations 42 VI. Photonics applications 44 A. Light trapping for enhanced absorption 44 B. Random lasing 45 C. Visual appearance 46 1. Photonic structures in nature 46 2. Synthetic structural colors 47 VII. Summary and perspectives 48 A. Near-field-mediated mesoscopic transport in 3D high-index correlated media 48 B. Mesoscopic optics in fractal and long-range correlated media 49 C. Towards novel applications 50 Acknowledgments 50 A. Green functions in Fourier space 51 1. Dyadic Green tensor 51 2. Dressed Green tensor 51 B. Derivation of Eq. (28) 51 C. Configurational average for statistically homogeneous systems 52 1. Particle correlation functions 52 2. Fluctuations of the number of particles in a volume 53 D. Local density of states and quasinormal modes 53 References 54 I. INTRODUCTION Correlated disordered media are non-crystalline het- erogeneous materials exhibiting pronounced spatial cor- relations in their structure and morphology. The topic has bloomed in the context of optics and photonics, grad- ually unveiling the considerable impact of structural cor- relations on the scattering, transport, and localization of light in matter. In essence, correlations engender con- structive and destructive interferences that survive con- figurational average. This leads not only to more pro- nounced spectral and angular features at the single scat- tering level, but also to profound modifications of the radiation properties of quantum emitters and the macro- scopic diffusion of photons by intricate near-field and multiple wave scattering phenomena. Recent findings let us envision novel types of materials with unprecedented optical functionalities and raise a number of challenges mailto:kevin.vynck@univ-lyon1.fr mailto:remi.carminati@espci.psl.eu mailto:frank.scheffold@unifr.ch 3 in theoretical modelling, material fabrication and optical spectroscopy. This article aims to provide an overview of this emerging research field, starting from the basic principles of light interaction with heterogeneous media to the most recent and still actively debated topics. The scattering of light by heterogeneous media has a long and venerable history, which started more than a century ago with pioneering studies on the refractive index of fluids of atoms or molecules (Lorentz, 1880; Lorenz, 1880), the electromagnetic scattering by parti- cles (Maxwell Garnett, 1904; Mie, 1908; Rayleigh, 1899), and the phenomenon of critical opalescence in binary fluid mixtures (Einstein, 1910; Ornstein and Zernike, 1914; Smoluchowski, 1908). The foundations of a rig- orous theoretical treatment of multiple light scattering were built in the 1930s (Kirkwood, 1936; Yvon, 1937) to take its full dimension a few decades later with various important contributions (Foldy, 1945; Keller, 1964; Lax, 1951, 1952; Twersky, 1964). These early works already pointed out the key role played by structural correla- tions on light scattering, a nice illustration of this being the transparency of the cornea resulting from short-range correlations in ensembles of discrete scatterers (Benedek, 1971; Hart and Farrell, 1969; Maurice, 1957; Twersky, 1975). A new branch of research exploiting light waves to study mesoscopic phenomena in disordered systems emerged in the 1980s, prompted by experimental demon- strations of weak localization (Tsang and Ishimaru, 1984; Van Albada and Lagendijk, 1985; Wolf and Maret, 1985) and theoretical predictions for the three-dimensional Anderson localization of light (Anderson, 1985; John, 1984). The advent of photonic crystals (John, 1987; Yablonovitch, 1987), wherein photonic band gaps are created by a periodic modulation of the refractive in- dex in two or three dimensions, gave additional mo- mentum to research by greatly stimulating the devel- opment of nanofabrication techniques for high-index di- electrics (López, 2003). The following decade witnessed a flourishment of studies on periodic dielectric nanos- tructures (Joannopoulos et al., 2011) and disordered media made of resonant (Mie) scatterers (van Albada et al., 1991; Busch and Soukoulis, 1995; Lagendijk and van Tiggelen, 1996) from two overlapping communi- ties (Soukoulis, 2012). The importance of short-range structural correla- tions on light transport in disordered systems (Fraden and Maret, 1990; Saulnier et al., 1990) and of ran- dom imperfections on light propagation in periodic sys- tems (Asatryan et al., 1999; Sigalas et al., 1996; Vlasov et al., 2000) was recognized quite early. Research on cor- related disordered media in optics however really took off in the mid-2000s with experimental studies show- ing that disorder could be engineered to harness light transport (Barthelemy et al., 2008; Garćıa et al., 2007; Rojas-Ochoa et al., 2004). The surprising observation of photonic gaps in disordered structures with short-range correlations (Edagawa et al., 2008; Liew et al., 2011), reports of mesoscopic phenomena in imperfect photonic crystals (Conti and Fratalocchi, 2008; Garcia et al., 2012; Toninelli et al., 2008) and the prospects of new genera- tions of photonic devices like random lasers (Gottardo et al., 2008), thin-film solar cells (Martins et al., 2013; Oskooi et al., 2012; Vynck et al., 2012) and integrated spectrometers (Redding et al., 2013), greatly contributed to the emergence of the research field. Figure 1 presents some of the early achievements and applications of cor- related disordered media in optics and photonics. Important efforts have been made in recent years to elucidate the role of structural correlations on the emer- gence of photonic gaps and Anderson localization of light in two-dimensional (Conley et al., 2014; Froufe- Pérez et al., 2016, 2017; Monsarrat et al., 2022) and three-dimensional disordered systems (Aubry et al., 2020; Haberko et al., 2020; Klatt et al., 2019; Ricouvier et al., 2019; Scheffold et al., 2022). Near-field interaction and light polarization considerably complicate theoret- ical modelling (Cherroret et al., 2016; van Tiggelen and Skipetrov, 2021; Vynck et al., 2016), explaining the widespread use of full-wave numerical methods to address this issue, alongside phenomenological models (Naraghi et al., 2015). The so-called hyperuniform disordered structures (Torquato and Stillinger, 2003), introduced in photonics by Florescu et al. (2009), have received consid- erable attention in this context, leading to a wider explo- ration of their optical properties (Bigourdan et al., 2019; Froufe-Pérez et al., 2017; Gorsky et al., 2019; Leseur et al., 2016; Piechulla et al., 2021a; Rohfritsch et al., 2020; Sheremet et al., 2020; Torquato and Kim, 2021) and advances on top-down and bottom-up fabrication techniques (Chehadi et al., 2021; Maimouni et al., 2020; Man et al., 2013; Muller et al., 2017; Piechulla et al., 2022; Ricouvier et al., 2017; Weijs et al., 2015). In a different context, the interplay of order and dis- order appeared quite early as an essential ingredient to explain the colored appearance of certain plants and an- imals (Kinoshita and Yoshioka, 2005). Research on nat- ural photonic structures continued at a fast pace with important findings, such as the ubiquity of correlated disorder in animals exhibiting vivid diffuse blue col- ors (Johansen et al., 2017; Magkiriadou et al., 2012; Moy- roud et al., 2017; Noh et al., 2010; Yin et al., 2012), the use of short-range correlations to reduce light re- flectance (Deparis et al., 2009; Pomerantz et al., 2021; Siddique et al., 2015) or structural anisotropy to en- hance whiteness (Burresi et al., 2014). Efforts are nowa- days made to realize artificial materials exhibiting cor- related disorder to create materials with versatile visual appearances (Chan et al., 2019; Forster et al., 2010; Go- erlitzer et al., 2018; Jacucci et al., 2021; Park et al., 2014; Salameh et al., 2020; Schertel et al., 2019a; Shang et al., 2018; Takeoka, 2012). 4 FIG. 1 Early achievements and applications of correlated disordered media in optics and photonics. (a) Male mandrill (Mandrillus sphinx) blue facial skin and cross-section of its dermis in the structurally colored area (blue) that reveals parallel collagen fibres organised in a correlated array. Adapted with permission from (Prum and Torres, 2004). (b) Modified light transport (described by the inverse transport mean free path ℓt) by engineering short-range structural correlations thanks to Coulomb repulsion between charged particles (filled symbols and solid line). Hard sphere systems (open symbols and dotted line) exhibit weaker correlations. The dashed line is a model neglecting completely structural correlations. Adapted with permission from (Rojas-Ochoa et al., 2004). (c) Anomalous light transport in Lévy glasses. A fractal heterogeneity is engineered by adding transparent spheres with sizes varying over orders of magnitude in a host matrix. Transport can be modelled by a truncated Lévy walk. On finite size samples, this leads to an anomalous scaling of the total transmittance T ∼ Lα/2 (experiments shown as symbols, lines are fits). Adapted with permission from (Barthelemy et al., 2008). (d) Light localization in randomly perturbed inverse opal photonic crystals (upper left inset). Simulations reveal spatially-localized modes near the photonic band edge (lower left inset). Their typical spatial extent (the localization length ξ) depends strongly on the degree of disorder. Adapted with permission from (Conti and Fratalocchi, 2008). (e) Existence of photonic gaps in amorphous photonic materials. Simulations of the spectral density - a quantity proportional to the density of states - in a connected amorphous diamond structure exhibiting short-range order shows a photonic gap near d/λ ≃ 0.23, where d is the average bond length. Adapted with permission from (Edagawa et al., 2008). (f) Random lasing in two-dimensional photonic structures with correlated disorder. Short-range correlations are shown to increase the lasing efficiency at certain frequencies due to enhanced optical confinement. Adapted with permission from (Noh et al., 2011). In this review, we will introduce the key concepts and techniques in the study of light in correlated dis- ordered media, assess the current state of knowledge on the topic, and define the main challenges that lie ahead of us. Compared to existing reviews on correlated dis- order and disorder engineering in optics and photon- ics (Cao and Eliezer, 2022; Shi et al., 2013; Wang and Zhao, 2020; Wiersma, 2013; Yu et al., 2021), we provide here a broader view on the field and sufficient technical details for the readers who wish to dive into it, be it from the theoretical or experimental side. This article is also an attempt to bridge the gap between different research fields for which excellent textbooks already exist, namely on random heterogeneous materials (Torquato, 2013), multiple light scattering in complex media (Akkermans and Montambaux, 2007; Carminati and Schotland, 2021; Sheng, 2006; Tsang and Kong, 2001) and periodic pho- tonic crystals (Joannopoulos et al., 2011), and which may serve as complementary literature. We focus on two and three-dimensional dielectric materials, intentionally leav- ing aside one-dimensional dielectric structures (i.e., lay- ered media) (Izrailev et al., 2012) and metallic nanostruc- tures (Shalaev, 2002). Quasicrystalline media, which are non-periodic, yet deterministic structures, are not dis- cussed explicitly here, despite many conceptual overlaps discussed at length, for instance, by Dal Negro (2022). We also do not discuss the very fertile fields of meta- materials and metasurfaces, which show some apparent similarities with the present topic in terms of theoreti- cal models and concepts (Mackay and Lakhtakia, 2020), yet with different scopes of application. Finally, certain concepts discussed here relate to transport theory in cor- related, stochastic media, where spatial correlations take place on scales larger than the wavelength and do not give rise to interferences. This article only covers a tiny portion of the vast literature on the topic, which has been 5 instead meticulously reviewed by d’Eon (2022). The remainder of the article is structured as follows. Section II introduces the basic concepts and important quantities for light scattering and transport in corre- lated disordered media, namely the extinction, scatter- ing and transport mean free paths. We derive math- ematically explicit results, as a function of the degree of structural correlations, from rigorous multiple scatter- ing theories for both continuous permittivity media and discrete particulate media, emphasizing conceptual sim- ilarities between these two viewpoints. Section III ad- dresses the statistical description of the structural prop- erties of correlated disordered media. Different classes of correlated systems are discussed, together with numer- ical and experimental techniques to realize and charac- terize them. Section IV reviews experimental and theo- retical studies wherein structural correlations yield sub- stantial variations of light transport parameters, includ- ing enhanced scattering in colloidal suspensions of par- ticles, optical transparency in hyperuniform media and anomalous diffusion in materials with large-scale (frac- tal) heterogeneities. Section V is concerned with emer- gent mesoscopic phenomena relying on an interplay of order and disorder, most of which are not yet fully un- derstood. This includes the formation of photonic gaps and localized states in disordered systems, and the statis- tical properties of near-field speckles and local density of states. Section VI describes various applications of corre- lated disordered media in optics and photonics, namely light trapping for enhanced absorption, random lasing and visual appearance design. Section VII concludes the review with a discussion on some open challenges in the field. II. THEORY OF MULTIPLE LIGHT SCATTERING BY CORRELATED DISORDERED MEDIA The theoretical study of light propagation in disor- dered media is a notoriously difficult problem that has experienced many developments for more than a century. In this section, we introduce the basic concepts of mul- tiple light scattering by heterogeneities with the aim to give a solid theoretical ground to the role of structural correlations in light scattering and transport. In Sec. II.A, we first focus on the “constitutive” lin- ear relation between the average electric field and the average polarization density in disordered media, which allows us to introduce the concepts of effective permit- tivity tensor and extinction mean free path. Many of the derived results have been used in the study of the effective optical response and homogeneization processes of peri- odic and amorphous dielectrics (Mackay and Lakhtakia, 2020; Van Kranendonk and Sipe, 1977). We derive the main equations that govern the propagation of the aver- age intensity and introduce the scattering and transport mean free paths, two experimentally measurable quan- tities that form the backbone of radiative transfer the- ory (Chandrasekhar, 1960). Within this unique theoretical framework, we then ad- dress the light scattering problem for non-absorbing me- dia described by either a continuous permittivity that fluctuates in space [Sec. II.B] or discrete particles cor- related in their position [Sec. II.C], see Fig. 2. We de- rive analytical expressions for the characteristic lengths, allowing us to show, on rigorous grounds, how struc- tural correlations impact scattering and transport. In- terestingly, we find that the choice of a specific effective medium model does not affect the form of the expres- sions, thereby demonstrating their generality. The main outcomes of the theoretical analysis are sum- marized in Sec. II.D for the readers who prefer to skip the mathematical details. Table I in this section provides the final expressions for the scattering and transport lengths (mean free paths), to be used in practical situations. FIG. 2 Disordered media may be described by a continuous permittivity model (left) in the most general case, or by a particulate model (right) in the case where the permittivity variation is compact. A. General framework 1. Average field and self-energy We consider a region of space filled with a non- magnetic, isotropic material (relative permeability µ(r) = 1), described by a scalar spatially-varying rel- ative permittivity ϵ(r) in a uniform host medium with relative permittivity ϵh. Throughout this review, we con- sider harmonic fields at frequency ω with the e−iωt con- vention and drop the explicit dependence on ω in the permittivities, fields, etc. In absence of charges and cur- rents, the electric field E(r) at frequency ω satisfies the propagation equation ∇×∇×E(r)− k20ϵhE(r) = k20P(r)/ϵ0, (1) where k0 = ω/c is the vacuum wave number and P(r) = ϵ0(ϵ(r) − ϵh)E(r) is the polarization density (electric dipole moment per unit volume). The permittivity vari- ation ∆ϵ(r) = ϵ(r) − ϵh readily appears as the source of light scattering in the system. Upon statistical average over an ensemble of realiza- tions of disorder, denoted here as ⟨. . . ⟩, Eq. (1) de- scribes the propagation of the average field ⟨E(r)⟩ with 6 the average polarization density ⟨P(r)⟩ as a source term. The difficulty to solve this general problem essentially comes from the fact that the permittivity variation and the electric field are not statistically-independent, i.e., ⟨∆ϵ(r)E(r)⟩ ≠ ⟨∆ϵ(r)⟩⟨E(r)⟩. The standard approach to solve this problem is to make an ansatz about the effec- tive permittivity ϵeff of the medium, thereby establishing a constitutive linear relation between the average field and the average polarization density, and then calculate it perturbatively from the spatial permittivity fluctua- tions. Let us then rewrite Eq. (1) as (Ryzhov et al., 1965) ∇×∇×E(r)− k20ϵbE(r) = X (r)E(r), (2) where ϵb is a constant, auxiliary, background permittivity that can differ from the permittivity of the host medium ϵh, and X is an effective scattering potential (or electric susceptibility) defined as X (r) = k20(ϵ(r)− ϵb)1, (3) with 1 the unit tensor. The average field ⟨E⟩ then fulfills the vector wave equation ∇×∇× ⟨E(r)⟩ − k2b⟨E(r)⟩ = ⟨X (r)E(r)⟩, (4) where k2b = k20ϵb. We now introduce the susceptibility tensor Σ, commonly known as the “self-energy” or “mass operator” following the language of many-body scatter- ing theory (Dyson, 1949a), as ⟨X (r)E(r)⟩ ≡ ∫ Σ(r, r′)⟨E(r′)⟩dr′, (5) with Σ(r, r′) = k20 (ϵeff(r, r ′)− ϵb1δ(r− r′)) . (6) The self-energy depends on the non-local effective per- mittivity tensor that results from multiple scattering in the disordered medium. Hereafter, we assume that the system has a proper thermodynamic limit in which it becomes spatially homogeneous and translationally in- variant on average (i.e., ϵeff(r, r ′) = ϵeff(r− r′)). Aspects related to the effective medium description in large but finite systems and the deep connection with the Ewald- Oseen extinction theorem (Hynne and Bullough, 1987; Van Kranendonk and Sipe, 1977) will thus not be dis- cussed here. In Fourier space, the self-energy is given by Σ(k,k′) = ∫∫ e−ik·rΣ(r− r′)eik ′·r′drdr′ (7) = (2π)3Σ(k)δ(k− k′). (8) It is further convenient to decompose Σ(k) into its trans- verse (⊥) and longitudinal (∥) components as Σ(k) = Σ⊥(k) (1− u⊗ u) + Σ∥(k)u⊗ u, (9) where u = k/|k| and u ⊗ u is the outer tensor product between u and itself. Defining e as the unit polarization vector with e · u = 0, the transverse component of the self-energy reads Σ⊥(k) = e ·Σ(k)e. (10) 2. Refractive index and extinction mean free path To understand the role played by the self-energy in wave propagation and scattering, we can seek for trans- verse solutions of the vector wave propagation equation of the form ⟨E(r)⟩ = E0ee ikeff·r, (11) with keff = k0neffu the wavevector describing propa- gation in a homogeneous medium with effective refrac- tive index neff. Assuming a statistically isotropic sys- tem, substituting Eq. (11) in Eq. (4) and making use of Eqs. (5),(9),(10) leads to a transcendental equation for the effective wave number keff = k0neff = √ k2b +Σ⊥(keff), ≡ kr + i 1 2ℓe . (12) The real part of the effective index Re[neff] = kr/k0 describes the phase velocity of the average field (often called “coherent” or “ballistic” component) in the ma- terial, while the imaginary part Im[neff] = (2k0ℓe) −1 de- scribes its exponential decay with propagation due to ab- sorption and/or scattering on a characteristic length scale that is the extinction mean free path, ℓe. In the weak ex- tinction regime (i.e., ImΣ⊥(keff) ≪ k2b + ReΣ⊥(keff) and krℓe ≫ 1), Eq. (12) leads to 1 ℓe ≃ ImΣ⊥(kr) kr . (13) In non-absorbing dielectric materials, extinction is purely driven by scattering (ℓe = ℓs with ℓs the scattering mean free path). The problem of light scattering by correlated disordered media can therefore be apprehended by deter- mining the self-energy of the system. 3. Multiple-scattering expansion A key ingredient in solving multiple light scattering problems is the electromagnetic Green tensor Gb(r, r ′), which is the solution of the wave equation in a homoge- neous medium with permittivity ϵb [Eq. (2)] with a point 7 source, ∇×∇×Gb(r, r ′)− k2bGb(r, r ′) = δ(r− r′)1. (14) Physically, it corresponds to the electric field produced at a point r by a radiating point electric dipole at r′, and is given by Gb(r, r ′) = − 1 3k2b δ(r− r′) + lim a→0 Θ(|r− r′| − a) {( 1+ ∇⊗∇ k2b ) eikb|r−r′| 4π|r− r′| } , (15) where Θ is the Heaviside step function. The Dirac delta function in the right hand side gives the well- known singularity in the source region while the second term corresponds to the non-singular, principal value of the Green function (Van Bladel and Van Bladel, 1991; Yaghjian, 1980). The exclusion volume defining the source region is chosen here to be spherical, but non- spherical (e.g., spheroidal, cubic, etc.) regions may also be used (Torquato and Kim, 2021; Tsang and Kong, 1981; Yaghjian, 1980). The choice of a nonspherical geome- try can be particularly adapted to certain microstruc- tures, for instance, with anisotropic correlation functions. Mathematically, the geometry of the source region affects the values of integrals involving the individual singular or non-singular contributions of the Green function. The Green function enables writing a general solution of the wave equation in the form E(r) = Eb(r) + ∫ Gb(r, r ′)X (r′)E(r′)dr′, (16) where Eb(r) is the solution of the homogeneous prob- lem, which can be seen as a background (incident) field with wave number kb. Equation (16), known as the Lippmann-Schwinger equation, can conveniently be writ- ten in operator form as E = Eb + GbXE, (17) where Gb is an integral operator. Equation (17) can be formally solved by successive iterations, leading to a multiple-scattering expansion on orders of X (i.e., single scattering, double scattering, etc.). Eventually, all multiple-scattering orders are taken into account by defining the transition operator T relating the polariza- tion induced in the medium to the background field, as E = Eb + GbT Eb, (18) with T = X +XGbX + · · · = [1−XGb] −1 X . (19) Keeping only the lowest order in the expansion, T = X , is known as the Born approximation, which corresponds to single scattering. In the general case of multiple scattering, the transition operator is spatially non-local, T Eb ≡ ∫ T(r, r′)Eb(r ′)dr′. Upon statistical average of Eqs. (17) and (18), and having Eq. (5), we finally reach a general expression for the average field as a function of the self-energy operator Σ , known as the Dyson equation (Dyson, 1949a,b; Rytov et al., 1989; Yvon, 1937) ⟨E⟩ = Eb + GbΣ⟨E⟩, (20) with Σ = ⟨T ⟩ [1+ Gb⟨T ⟩]−1 . (21) In summary, the disordered medium is described as a permittivity that fluctuates around an auxiliary back- ground permittivity ϵb via the scattering potential X [Eq. (3)]. The field propagates from fluctuation to fluctu- ation via the Green tensor Gb in the homogeneous back- ground with wave number kb [Eq. (15)]. The multiple scattering process on the scattering potential X is de- scribed (to infinite order) via the transition operator T [Eq. (19)]. The average transition operator ⟨T ⟩ finally defines the self-energy Σ [Eq. (21)], which describes the propagation of the average field ⟨E⟩ in the disordered medium, and leads to the extinction mean free path in the medium [Eq. (13)]. 4. Average intensity and four-point irreducible vertex Light transport, that is the propagation of the energy, is formally described by the average intensity 〈 |E(r)|2 〉 , which we will now consider. First and foremost, let us re- mark that the average intensity can be decomposed into two components with distinct physical meaning. Indeed, writing the field as the sum of its average value and a fluctuating part, E = ⟨E⟩ +∆E with ⟨∆E⟩ = 0 by defi- nition, straighforwardly leads to〈 |E(r)|2 〉 = | ⟨E(r)⟩ |2 + 〈 |∆E(r)|2 〉 . (22) The first term, | ⟨E⟩ |2, corresponds to the so-called bal- listic (or coherent) intensity, which describes the part of the intensity that propagates in the direction of the incident light and is attenuated (exponentially) by scat- tering and absorption. Its behavior is fully determined by the theory for the average field presented in the pre- vious section. Our attention here should be given in- 8 stead to the second term, 〈 |∆E|2 〉 , which corresponds to the so-called diffuse (or incoherent) intensity, describ- ing the part of the intensity that spreads throughout the volume of the medium by successive scattering events. The diffuse intensity will lead to the definition of the scattering and transport mean free paths, two additional length scales at the heart of light propagation in disor- dered media (Apresyan and Kravtsov, 1996; van Rossum and Nieuwenhuizen, 1999; Rytov et al., 1989). Let us then consider the spatial correlation func- tion of the electric field, or “coherence matrix” (Man- del and Wolf, 1995), C(r, r′) ≡ ⟨E(r)⊗E∗(r′)⟩, with ∗ denoting complex conjugate. Starting from the Lippmann-Schwinger equation [Eq. (17)], one can eas- ily show that C depends on the correlator of the po- larization density in the effective scattering potential ⟨(X (r)E(r))⊗ (X ∗(r′)E∗(r′))⟩. Similarly to the self- energy that allowed us to relate the average polariza- tion to the average field [Eq. (5)] eventually leading to the Dyson equation [Eq. (20)], we can introduce here an operator Γ known as the “four-point irreducible vertex” – or intensity vertex – that relates the effective polar- ization density correlation to the electric field correla- tion. This leads to a closed-form equation, known as the Bethe-Salpeter equation (Salpeter and Bethe, 1951), which reads C(r, r′) = ⟨E(r)⟩⊗⟨E∗(r′)⟩+ ∫ ⟨G(r, r1)⟩⊗⟨G∗(r′, r′1)⟩ · Γ(r1, r2, r′1, r′2) ·C(r2, r ′ 2)dr1dr ′ 1dr2dr ′ 2, (23) where the average Green function ⟨G⟩ is given by the Dyson equation [Eq. (20)], ⟨G⟩ = Gb + GbΣ ⟨G⟩ . (24) The symbol · denotes here a tensor contraction, defined such that (A ⊗ B) · (e ⊗ f) = (Ae) ⊗ (Bf) and (A ⊗ B) · (C⊗D) = (AC)⊗ (BD), where e, f are vectors and A,B,C,D second-rank tensors. The first term in Eq. (23) is the correlation function on the average field that leads to the coherent intensity. The second term expresses the field correlation as a multiple scattering process, wherein the propagation is described by the average Green tensors and scattering by the ver- tex Γ that connects two pairs of points (for the field and the complex conjugate). Following similar steps as those leading to Eq. (21), we obtain the following general ex- pression for Γ (Carminati and Schotland, 2021) Γ = [GbG∗ b] −1 [ (1+ Gb ⟨T ⟩+ G∗ b ⟨T ∗⟩+ ⟨T ⟩GbG∗ b ⟨T ∗⟩)−1 − (1+ Gb ⟨T ⟩+ G∗ b ⟨T ∗⟩+ ⟨T GbG∗ bT ∗⟩)−1 ] . (25) To proceed further, it is convenient to rewrite the Bethe-Salpeter equation in Fourier space. Assuming that the scattering events take place on distances larger than the wavelength, the average Green tensor can be approx- imated by its transverse component, ⟨G(k)⟩ = [ k2P(u)− k2b1−Σ(k) ]−1 , (26) ≃ ⟨G⊥(k)⟩P(u), (27) where P(u) = 1−u⊗u is the transverse projection oper- ator and ⟨G⊥(k)⟩ = [ k2 − k2b − Σ⊥(k) ]−1 is the (scalar) transverse component. After some algebra provided in details in Appendix B, we find that [( k− q 2 )2 − ( k+ q 2 )2 − Σ∗ ⊥ ( k− q 2 ) +Σ⊥ ( k+ q 2 )] L⊥(k,q) = [〈 G⊥ ( k+ q 2 )〉 − 〈 G∗ ⊥ ( k− q 2 )〉] ∫ Γ̄⊥ ( k+ q 2 ,k′ + q 2 ,k− q 2 ,k′ − q 2 ) · L⊥ (k′,q) dk′ (2π)3 , (28) where we have assumed statistical homogeneity and translational invariance of the medium and neglected the exponentially small coherent intensity. The field correla- tion, described by a new function L⊥(k,q) ≡ C⊥ ( k+ q 2 ,k− q 2 ) , (29) with C⊥ ( k,k′) = P(u)⊗P(u′) ·C ( k,k′) , (30) depends only on the transverse part of the intensity ver- tex, which is given by Γ⊥(k,κ,k ′,κ′) = P(u)⊗P(u′) · Γ(k,κ,k′,κ′) = (2π)3δ(k− κ− k′ − κ′)Γ̄⊥(k,κ,k ′,κ′). (31) Equation (28) is very general, as it considers all multi- ple scattering events within the medium and does not make any explicit assumption on the kind of disorder. Note however that neglecting the longitudinal compo- nent of the Green tensor implicitly excludes near-field 9 interactions between scattering centers, that might be important, for example, in dense packings of high-index resonant particles. 5. Radiative transfer limit and scattering mean free path Further approximations are required to obtain an ex- plicit transport equation for the average intensity. First, we take the large-scale approximation |q| ≪ {|k|, |k′|}, also known as the radiative transfer limit (Barabanenkov and Finkel’berg, 1968; Ryzhik et al., 1996), which as- sumes that the average intensity varies on length scales 2π/|q| much larger than the wavelength in the medium 2π/kr. This amounts to assuming krℓe ≫ 1, which cor- responds to the weak extinction regime. Equation (28) becomes [−2k · q+ 2i ImΣ⊥(k)]L⊥(k,q) = 2i Im ⟨G⊥(k)⟩ × ∫ Γ̄⊥ (k,k′,k,k′) · L⊥(k ′,q) dk′ (2π)3 . (32) The weak extinction regime also corresponds to |Σ⊥| ≪ k2b, see Eqs. (12),(13). Using the relation lim ϵ→0+ 1 x− x0 − iϵ = PV [ 1 x− x0 ] + iπδ(x− x0), (33) where PV stands for the Cauchy principal value operator, the imaginary part of the average Green function reduces to Im ⟨G⊥(k)⟩ = πδ [ k2 − k2b − ReΣ⊥(k) ] . (34) This relation fixes the real part of the effective wavevector kr = Re keff to kr = √ k2b +ReΣ⊥(kr), (35) which is the so-called “on-shell approximation”. Sec- ond, we assume that the field is fully depolarized, which is valid when the observation point is at a large dis- tance from the source compared to the average distance between scattering events (Bicout and Brosseau, 1992; Gorodnichev et al., 2014; Vynck et al., 2016). This means that C ( k,k′) = C ( k,k′)1, (36) leading to L⊥(k,q) = L(k,q)P(u)⊗P(u′) · 1. (37) An inverse Fourier transform of the trace of Eq. (32) to- gether with Eqs. (34) and (37) eventually leads to the well-known Radiative Transfer Equation (RTE) (Chan- drasekhar, 1960)[ u ·∇r + 1 ℓe ] I(r,u) = 1 ℓs ∫ p(u,u′)I (r,u′) du′, (38) where du means an integration over the unit sphere or equivalently over the solid angle, and I is the specific intensity, defined as δ(k − kr)I(r,u) = L(r,k). (39) The specific intensity can be interpreted as a local (at po- sition r) and directional (on direction u) radiative flux. In the RTE, ℓs and p(u,u′) are the scattering mean free path and the phase function, describing respectively the average distance between two scattering events and the angular diagram for an incident planewave along u′ scat- tered along direction u. Both quantities are related to the intensity vertex via the relation 1 ℓs p(u,u′) = 1 32π2 Tr [P(u)⊗P(u) ·Γ̄(kru, kru′, kru, kru ′) ·P(u′)⊗P(u′) · 1 ] , (40) and the phase function is normalized as∫ p(u,u′)du′ = 1. (41) This normalization immediately shows that 1/ℓs is given by the integral of the right hand side of Eq. (40) over u′. The trace appearing in Eq. (40) is a consequence of the assumption of a depolarized field. The RTE [Eq. (38)] can be seen as an energy balance (Chandrasekhar, 1960). The spatial variation of the specific intensity (term in- volving the derivative) is due to the loss induced by ex- tinction along the direction u [term involving ℓe] and the gain from scattering from direction u′ to direction u [term involving ℓs and p(u,u′)]. Equations (40) and (41) show that ℓs, the key quantity to describe the scat- tering strength of a medium, is obtained in the radiative transfer limit from the angular integral of the intensity vertex. Previously, we showed that the extinction mean free path ℓe could be obtained from the self-energy Σ and noted that, in absence of absorption, we should have ℓe = ℓs, the latter being defined from the intensity vertex Γ. It is worth emphasizing at this point that the two operators are indeed formally linked by the Ward identity (Bara- banenkov and Ozrin, 1995; Cherroret et al., 2016), which may be seen as a generalization of the extinction (opti- cal) theorem and ensures energy conservation (Apresyan and Kravtsov, 1996; Carminati and Schotland, 2021; La- gendijk and van Tiggelen, 1996; Sheng, 2006; Tsang and Kong, 2001). 10 6. Transport mean free path and diffusion approximation Many experiments on light in disordered media are per- formed in situations where light experiences not just a few but many scattering events on average. In the deep multiple scattering regime, the RTE can be simplified into a diffusion equation. In this limit, light transport is driven by a new length scale, known as the transport mean free path ℓt, which we will introduce here. We start by taking the first moment of Eq. (38) (i.e., multiplying both sides by u and integrating over u) which directly leads to∫ [u ·∇rI(r,u)]udu+ 1 ℓt j(r) = 0, (42) where j(r) = ∫ I(r,u)udu is the radiative flux vector and ℓt ≡ ℓs 1− g , (43) is the transport mean free path. g = ∫ p(u,u′)u · u′du (44) is the average cosine of the scattering angle, or scatter- ing anisotropy factor. Structural correlations impact the transport mean free path via both the scattering mean free path ℓs and the scattering anisotropy described by g. After a large number of scattering events, we can as- sume that the specific intensity becomes quasi-isotropic. Expanding the specific intensity in the RTE [Eq. (38)] on Legendre polynomials to first order in u, which is known as the P1-approximation (Ishimaru, 1978), leads to the diffusion equation. In the steady-state regime, it reads −D∆u(r) = s(r) (45) where u(r) = v−1 E ∫ I(r,u)du is the energy density with vE the energy velocity (Lagendijk and van Tiggelen, 1996), D = vEℓt/3 is the diffusion constant and s is a source term. An analysis of the diffusion equation shows that it is valid on length scales large compared to ℓt. This allows us to reinterpret ℓt as the distance after which the intensity distribution is quasi-isotropic (Carminati and Schotland, 2021; Ishimaru, 1978). Resolving this equation in a slab geometry of thickness L under planewave illumination at normal incidence gives the following asymptotic behavior for the total transmit- tance T ∼ 5 3 ℓt L , (46) which is Ohm’s law for light (van Rossum and Nieuwen- huizen, 1999). Many transport observations in the dif- fusive limit depend directly on the transport mean free path, including the linewidth of the coherent backscat- tering cone (Akkermans et al., 1988, 1986), the time- resolved transmittance and reflectance (Contini et al., 1997), and long-range speckle correlations (Fayard et al., 2015; Scheffold and Maret, 1998; Shapiro, 1999). In summary, the average transition operator of the medium ⟨T ⟩ defines the four-point irreducible vertex Γ [Eq. (25)]. Neglecting near-field interaction between scat- tering elements, taking the radiative transfer limit and assuming fully depolarized light allow us to relate the transverse component of Γ to the scattering mean free path ℓs and phase function p(u,u′) [Eq. (40)]. In the diffusion approximation, the transport mean free path ℓt [Eq. (43)] drives the energy flux. It is related to the intensity vertex via Eqs. (40) and (41). The theoretical framework described here will now be used to get closed-form expressions for the different opti- cal length scales in the cases of random media described by a continuous permittivity [Sec. II.B] or as an assembly of discrete particles [Sec. II.C]. B. Media with fluctuating continuous permittivity 1. Weak permittivity fluctuations We consider a statistically homogeneous and isotropic disordered medium, described by a spatially-dependent permittivity ϵ(r) = ⟨ϵ⟩ + ∆ϵ(r), where ∆ϵ is the fluctu- ating part with statistics ⟨∆ϵ(r)⟩ = 0, (47) ⟨∆ϵ(r)∆ϵ(r′)⟩ = ⟨ϵ⟩2δ2ϵhϵ(|r− r′|). (48) Here, δ2ϵ = ⟨∆ϵ2⟩/⟨ϵ⟩2 is the normalized variance of ϵ and hϵ(|r− r′|) = ⟨∆ϵ(r)∆ϵ(r′)⟩/⟨∆ϵ2⟩ is the normalized permittivity-permittivity correlation function (hϵ(0) = 1). Hereafter, we assume ergodicity, such that the en- semble average is equivalent to a volume average in the infinite-volume limit, and isotropic permittivity fluctua- tions, keeping in mind, however, that anisotropic fluctua- tions may take place even in isotropic materials (Landau et al., 2013). The statistical properties of ϵ(r) straightforwardly translate into statistical properties of X (r) via Eq. (3). The self-energy Σ can be expressed in terms of X by in- serting the expression for the transition operator T given by Eq. (19) into Eq. (21), leading to Σ = 〈 X [1− GbX ] −1 〉〈 [1− GbX ] −1 〉−1 . (49) In the simplest approach, we proceed by assuming that the scattering potential X weakly fluctuates around its average value ⟨X ⟩. Expanding the last expression near 11 ⟨X ⟩ leads to Σ ∼ ⟨X ⟩+ ⟨(X − ⟨X ⟩)Gb (X − ⟨X ⟩)⟩+ · · · (50) At this stage, we need to define explicitly the constant auxiliary background permittivity ϵb, which describes the reference value around which the permittivity fluctuates. A reasonable choice is to set it to the average permittiv- ity, ϵb = ⟨ϵ⟩ ≡ ϵav, which, for a two-component medium with permittivities ϵp and ϵh at filling fractions f and 1−f , respectively, would simply be ϵav = fϵp+(1−f)ϵh. Having then ⟨X ⟩ = 0, the leading term for the self-energy becomes ⟨XGbX ⟩, such that Σ(r− r′) = k4avδ 2 ϵhϵ(|r− r′|)Gav(r− r′), (51) where Gav is the Green tensor in a homogeneous medium with permittivity ϵav. Correlated permittivity fluctua- tions mutually interacting viaGav are readily responsible for the non-local character of the self-energy [Eq. (6)]. In Fourier space, Eq. (51) becomes Σ(k) = k4avδ 2 ϵ ∫ hϵ(|k− k′|)Gav(k ′) dk′ (2π)3 . (52) The extinction mean free path ℓe can finally be deter- mined using Eq. (13) with kr = kav. For non-absorbing media [ϵ(r) real], the imaginary part of the Green tensor is given by ImGav(k) = πδ ( k2 − k2av ) P(u), (53) with kav = k0 √ ϵav. Using Eq. (10) for the transverse component with e the unit vector defining the polariza- tion direction such that e · u = 0, we find that 1 ℓe ≃ ImΣ⊥(kav) kav , = k4av 16π2 δ2ϵ ∫ hϵ(kav|u− u′|) (e ·P(u′)e) du′. (54) Introducing the scattering wavenumber q = kav|u − u′|, we eventually reach 1 ℓe = k40 8πk2av ϵ2avδ 2 ϵ ∫ 2kav 0 P ( q 2kav ) hϵ(q)qdq, (55) where P (k) ≡ 1− 2k2 + 2k4, (56) is specific to the vector nature of light. Equation (55), obtained in non-absorbing disordered media with weak permittivity fluctuations, constitute the first analytical expression for the extinction mean free path in correlated media. Very importantly, it shows that besides the amplitude of the permittivity fluctuations, ϵ2avδ 2 ϵ = 〈 ∆ϵ2 〉 , spatial correlations, described here by hϵ, also play a crucial role in light scattering. 2. Lorentz local fields: strong fluctuations The approximation of weak fluctuations is prohibitive in many realistic cases. Fortunately, this constraint can be eliminated by properly handling the singularity of the dyadic Green function at the origin [Eq. (15)], which constitutes the basis of a strong fluctuation the- ory (Finkel’berg, 1964; Ryzhov et al., 1965; Tsang and Kong, 1981). Related approaches were introduced by Bedeaux and Mazur (1973) and Felderhof (1974) in the description of the optical response of non-polar fluids. Extensions to chiral and anisotropic media have also been proposed (Mackay and Lakhtakia, 2020; Michel and Lakhtakia, 1995; Ryzhov and Tamoikin, 1970), but will not be discussed here. Noteworthy is the recent work by Torquato and Kim (2021), which relies on the same theoretical grounds and proposes an analytical expression for the effective permittivity of two-phase composite me- dia that takes into account structural correlations up to an arbitrary order n (whereas we restrict the discussion to correlations of order n = 2 in the present review). The singularity of the Green function is handled by considering the scattering medium as being made of in- finitesimal volume elements within which the polariza- tion density P(r) is constant. This physical viewpoint is the basis of the renowned discrete dipole approxima- tion (Draine and Flatau, 1994; Lakhtakia, 1992). Let us then write the Green function as a sum of two contribu- tions Gb(r, r ′) = Θ(|r− r′| − a)G̃b(r, r ′) + Θ(a− |r− r′|)gb(r, r ′), (57) where gb contains the singular part of the Green func- tion and G̃b is the so-called Lorentz propagator, which is purely non-local. The contributions are distinguished as belonging or not to a spherical region with radius a and volume v = 4πa3/3 around r′. Choosing a such that kba ≪ 1, the singular part reads gb(r, r ′)|kba≪1 = − 1 3k2b δ(r− r′)1+ i kb 6π 1+ · · ·(58) Keeping the lowest-order terms in the real and imaginary parts, the Lippmann-Schwinger equation [Eq. (16)] can be rewritten as E(r) = Eb(r) + ( − 1 3k2b + i kbv 6π ) X (r)E(r) + ∫ G̃b(r, r ′)X (r′)E(r′)dr′. (59) The actual field at r is then given by the sum of the 12 external field Eb(r), the local contributions, and the non- local contributions coming from neighboring permittivity fluctuations (integral term). In this framework, the field exciting a small volume element around r, Eexc(r), is the sum of the incident (background) field and the field scattered by other per- mittivity fluctuations. Using again operator notations, we thus reach an important set of equalities Eexc = Eb + G̃bXE = [1− gbX ]E = Eb + G̃bT̃ Eexc = [ 1+ G̃bT ] Eb. (60) We have introduced here a new quantity, T̃ , that plays the role of a local transition operator. From Eqs. (60), we straightforwardly obtain T = T̃ [ 1− G̃bT̃ ]−1 . (61) The transition operator of the medium can be seen as a multiple-scattering expansion on independent scattering elements, connected via the (non-local) Lorentz propa- gator. Consistently with this picture, from Eqs. (60), we also obtain T̃ = X [1− gbX ] −1 . (62) For kba ≪ 1, the (local) transition operator T̃(r, r′) is directly proportional to the polarizability of the volume element, T̃(r, r′) = k2b α(r) v δ(r− r′)1, (63) with α(r) = α0(r) 1− i k3 b 6πα0(r) , α0(r) = 3v ϵ(r)− ϵb ϵ(r) + 2ϵb , (64) where α0 is the quasi-static polarizability. Note that α(r) is a space-dependent local polarizability defined in a con- tinuous medium. 3. Average exciting field To determine the self-energy Σ of the system, it is convenient to introduce a self-energy Σ̃ for the exciting field defined from a Dyson equation ⟨Eexc⟩ = Eb + G̃bΣ̃⟨Eexc⟩, (65) thereby leading to Σ̃ = 〈 T̃ [ 1− G̃bT̃ ]−1 〉〈[ 1− G̃bT̃ ]−1 〉−1 . (66) Note the similarity of this expression with Eq. (49), where the self-energy Σ was expressed directly in terms of the scattering potential X . From Eqs. (21), (61) and (66), one shows that the two self-energies are related as Σ = Σ̃ [ 1+ gbΣ̃ ]−1 . (67) Let us then expand Σ̃ in Eq. (66) near ⟨T̃ ⟩ = T̃ −∆T̃ , Σ̃ ∼ 〈 T̃ 〉 + 〈 ∆T̃ Ĝb∆T̃ 〉 + · · · ≡ Σ̃1 + Σ̃2 + · · · .(68) We have introduced here a new “dressed” propagator, Ĝb = [ 1− G̃b 〈 T̃ 〉]−1 G̃b. (69) Noting that ⟨T̃ ⟩ corresponds to an average polarizability of the medium (for kba ≪ 1, see Eqs. (63)), one under- stands that Ĝb describes the fact that the fields propa- gate from fluctuation to fluctuation via a medium with a permittivity that can differ from the background permit- tivity ϵb (Bedeaux and Mazur, 1973; Felderhof, 1974). The self-energy for the average exciting field, Σ̃ in Eq. (68), now explicitly depends on the spatial corre- lations of the fluctuations of T̃ (i.e., of the polarizability of small volume elements). In most practical cases, the expansion is limited to second order, corresponding to the so-called “bilocal” approximation (Tsang and Kong, 2001), due to the lack of information on higher-order cor- relation functions in real systems. Expanding Σ in Eq. (67) near Σ̃1, we obtain Σ ∼ Σ̃1 [ 1+ gbΣ̃1 ]−1 + Σ̃2 [ 1+ gbΣ̃1 ]−2 + · · · ≡ Σ1 +Σ2 + · · · (70) The self-energy is now expressed in terms of the scatter- ing properties of vanishingly small, individual scattering elements. 4. Long-wavelength solutions: Bruggeman versus Maxwell-Garnett models As in the case of weakly fluctuating media discussed previously, we now need to give an explicit definition of the constant auxiliary background permittivity ϵb, that describes the homogeneous effective medium in which the permittivity fluctuations scatter light. We will see that this sole parameter constitutes the essential difference be- tween the two celebrated “mixing rules” due to Brugge- man (1935) and Maxwell Garnett (1904), presented here in a unique theoretical framework. Despite the arbitrari- ness in the choice of ϵb, it is important to realize that all models would eventually be strictly equivalent when carried out to infinite order. The applicability of a model 13 is thus mainly a question of accuracy at low orders and convergence. A first possibility for ϵb is to set it such that ⟨T̃ ⟩ = 0. In the limit of small volume elements, this corresponds to having a zero average polarizability, see Eqs. (63)-(64). Considering a two-component medium with relative per- mittivities ϵp (at filling fraction f) and ϵh (at filling frac- tion 1− f) in the quasi-static limit (α = α0 in Eq. (64)), one obtains ϵp − ϵBG ϵp + 2ϵBG f + ϵh − ϵBG ϵh + 2ϵBG (1− f) = 0 (71) which is the Bruggeman mixing rule (Bruggeman, 1935) with ϵb ≡ ϵBG. The generalization to N -component media is straightforward. Having k2b = k2BG = k20ϵBG, Ĝb = G̃BG and Σ ∼ Σ̃ since Σ̃1 = 0, we eventually find that Σ(k) = k4BGδ 2 α ∫ hα(|k− k′|)G̃BG(k ′) dk′ (2π)3 , (72) with δ2α = ⟨∆α2⟩/v2 a normalized variance of the polar- izability and hα(|k − k′|) the Fourier transform of the normalized polarizability-polarizability correlation func- tion hα(|r − r′|) = ⟨∆α(r)∆α(r′)⟩/⟨∆α2⟩. The function hα plays the same role as hϵ in the weakly fluctuating permittivity model to describe structural correlations. Assuming non-absorbing media and following the same steps as those leading to Eq. (55) with ImG̃BG(k) = πδ ( k2 − k2BG ) P(u), we obtain 1 ℓe = k40 8πk2BG ϵ2BGδ 2 α ∫ 2kBG 0 P ( q 2kBG ) hα(q)qdq, (73) with q = kBG|u− u′|. Equation (73) is strikingly similar to Eq. (55), the essential differences being (i) the per- mittivity of the homogeneous effective medium and (ii) the description of the medium via a local polarizability instead of a local permittivity. A second possibility for the choice of ϵb is to set it to the permittivity of the host medium (i.e., ϵb = ϵh), in which case ⟨T̃ ⟩ ≠ 0. Considering again a two-component system in the quasi-static limit, we obtain Σ̃1(r− r′) = k2hρα0δ(r− r′)1, (74) with ρ = f/v the average number density of the small volume elements with permittivity ϵp, and Σ̃2(r− r′) = k4hδ 2 αhα(|r− r′|)Ĝh(r− r′). (75) Using Eq. (70), we find the following expression for the self-energy in reciprocal space Σ(k) = k20 (ϵMG − ϵh)1 + k4hδ 2 αfL ∫ hα(|k− k′|)Ĝh(k ′) dk′ (2π)3 , (76) where fL = (∂ϵMG/∂ρ)/(ϵhα0) and ϵMG = ϵh + ϵh ρα0 1− ρα0/3 , (77) is the Maxwell-Garnett mixing rule (Markel, 2016; Maxwell Garnett, 1904). By contrast with the previ- ous case, the lowest-order term now provides the renor- malization of the wave number in the effective medium, structural correlations appearing at the next order. The factor fL is a local-field correction coming from the fact that the fluctuation of polarizability (∆α in δ2α) is eval- uated with respect to the host medium. Following again the same steps as those leading to Eq. (55), noting that ϵMG is real in the quasi-static limit for non-absorbing media, and using ImĜh(k) = πfLδ ( k2 − k2MG ) P(u), (78) which is derived in Appendix A.2, we eventually obtain 1 ℓe = k40 8πk2MG f2 Lϵ 2 hδ 2 α ∫ 2kMG 0 P ( q 2kMG ) hα(q)qdq, (79) with q = kMG|u − u′|. This expression for ℓe takes the same form as Eq. (73) with differences in the definition of the effective medium and a prefactor that accounts for local-field corrections. All in all, the similarity between Eqs. (55), (73) and (79) demonstrates the deep physical implication of struc- tural correlations for light scattering. The same func- tional structure is kept, whatever the approach used to define the effective medium. 5. Expressions for the scattering and transport mean free paths from the average intensity We conclude this part on continuous permittivity me- dia by deriving the expressions for ℓs and ℓt from the theory for the average intensity. A second-order expan- sion of the intensity vertex Γ in Eq. (25), with T given by Eq. (19), leads to Γ ∼ ⟨T T ∗⟩ − ⟨T ⟩ ⟨T ∗⟩ ∼ ⟨XX ∗⟩ − ⟨X ⟩ ⟨X ∗⟩ . (80) This expansion is valid in the weak extinction limit krℓe ≫ 1. Similarly to the case of weakly fluctuating media, we set the auxiliary background permittivity as ϵb = ⟨ϵ⟩ ≡ ϵav, such that ⟨X ⟩ = 0, and assume a non- 14 absorbing material. This leads to Γ̄(kavu, kavu ′, kavu, kavu ′) = k40ϵ 2 avδ 2 ϵhϵ(kav|u− u′|) × 1⊗ 1. (81) The scattering mean free path is then obtained by inte- grating Eq. (40) over u′, and making use of the equation above, we find that 1 ℓs = k40 8πk2av ϵ2avδ 2 ϵ ∫ 2kav 0 P ( q 2kav ) hϵ(q)qdq, (82) with q = kav|u − u′|. Similarly, the transport mean free is obtained by calculating the average cosine of Eq. (40) and using Eqs. (43)-(44), leading to 1 ℓt = k40 16πk4av ϵ2avδ 2 ϵ ∫ 2kav 0 P ( q 2kav ) hϵ(q)q 3dq. (83) In absence of absorption, we expect ℓs = ℓe, which is actually found by comparing Eqs. (82) and (55). C. Particulate media 1. Expansion for identical scatterers Let us now consider a system whose morphology con- sists in localized (i.e., compact) permittivity variations in a uniform background. We take the most natural choice for the background permittivity ϵb = ϵh from the start but the theory can also be developed with an ar- bitrary ϵb. We also restrict the discussion to composite media made of identical inclusions with relative permit- tivity ϵp confined to a volume v, centered at positions R = [R1,R2, . . .RN ]. The medium permittivity then reads ϵ(r) = ∑ j ϵp(r−Rj)Θ(a− |r−Rj |). (84) A configuration of the medium is described statistically by the probability distribution function p(R). Implic- itly, we neglect here the possibility to have orientational correlations between particles (otherwise, the distribu- tion should include orientational variables). Under the ergodic hypothesis, defining the statistical average as an average over all possible particle positions as ⟨f(R)⟩ =∫ f(R)p(R)dR, where f(R) is an arbitrary tensor, the statistical properties of the medium can be described by n-particle probability density functions (Lebowitz and Percus, 1963; Tsang et al., 2004) ρn(r1, · · · rn) = 〈 ∑ j1 ̸=j2···̸=jn δ(r1 −Rj1) · · · δ(rn −Rjn) 〉 , (85) or equivalently by n-particle correlation functions gn(r1, · · · rn) = 1 ρn ρn(r1, · · · rn), (86) where ρ is the constant particle number density reached in the limit of infinite system size (ρ = limN,V→∞ N/V ). In statistically homogeneous ensembles of impenetra- ble spheres, the particle correlation functions gn are formally related to the probability functions of finding n points separated by given distances in the particle phase (Torquato and Stell, 1982; Torquato, 2013). Similar to the case of random media described by a continuous permittivity, the first step is to derive an ex- pression for the transition operator T of the medium. Having a discrete set of identical particles allows us to express the multiple-scattering problem in such a way as to separate the effects associated to (local) particle res- onances and (non-local) structural correlations on light scattering. We start by rewritting the integral equation for the total field, Eq. (17), for particulate media, E = Eh + ∑ j GhX jE, (87) with the effective scattering potential X j(r − Rj) ≡ k20[ϵp(r − Rj)Θ(a − |r − Rj |) − ϵh]1. We can then ex- press the polarization induced in particle j in terms of the polarization induced in all particles as X jE = X jEh +X j ∑ k GhX kE, (88) = X jEh +X jGhX jE+X j ∑ k ̸=j GhX kE, (89) = T jEh + T j ∑ k ̸=j GhX kE. (90) In the second expression, we separated the self- contribution of particle j from the contribution of all other particles. The last expression was obtained by introducing the transition operator T j of an individual particle centered at Rj as T j = X j [1− GhX j ] −1 . (91) Very importantly, T j can be determined for particles of virtually any size, shape and composition, either analyt- ically using Mie theory for simple geometries like spheri- cal particles (Bohren and Huffman, 2008) or numerically by any method for solving Maxwell’s equations other- wise (Mishchenko et al., 1999). This allows us to consider resonant particles exhibiting high-order multipolar reso- nances as the building blocks of the disordered medium. Inserting Eq. (90) into Eq. (87) and iterating over scat- tering sequences, we reach an expression for the transi- tion operator T of the entire system [Eq. (18)] in terms 15 of the transition operator of the individual particle, as T = ∑ j T j + ∑ j T j ∑ k ̸=j GhT k + ∑ j T j ∑ k ̸=j GhT k ∑ l ̸=k GhT l + · · · (92) This series expansion is the root of multiple scattering theory for particulate media and was introduced in the pioneering works of Kirkwood (1936) and Yvon (1937) to determine the permittivity of molecular liquids. Simi- lar multiple scattering equations were later discussed by Foldy (1945) and Lax (1951). It is further interesting to note the occurrence of so-called “recurrent scattering”, that is, scattering sequences that involve the same par- ticle multiple times (for instance, l can be equal to j in the last displayed term). The Green function in Eq. (91) for the transition oper- ator T j of a specific particle j always connects two points that belong to the same particle, whereas the Green func- tion in Eq. (92) for the transition operator T of the entire medium always connects two points that belong to differ- ent particles. This is conceptually analogous to Eq. (57) for continuous permittivity media where the Green func- tion was split into local and non-local terms. To deter- mine the self-energy Σ in the Dyson equation [Eq. (20)], we may then follow the strategy used for continuous per- mittivity media and consider the exciting field Eexc. Let us then write the Green function Gh as gh when connect- ing two points in the same particle and G̃h otherwise. Removing the local contribution on the induced polar- ization in Eq. (89) and rewriting X kE in terms of the exciting field leads to X jEexc = X jEh +X j ∑ k G̃hT̃ kEexc, (93) with T̃ j = X j [1− ghX j ] −1 = T j , see Eq. (91). Note that the sum now runs over all particles k. Further defin- ing T̃ ≡ ∑ j T̃ j , we find T = T̃ [ 1− G̃hT̃ ]−1 , (94) which agrees with Eq. (61) derived for continuous media. Expanding the self-energy Σ̃ for the average exciting field near ⟨T̃ ⟩ = T̃ −∆T̃ then leads to Eq. (68), and expand- ing Σ near Σ̃1 to Eq. (70). The problem of scattering by particulate media is thus described in a strictly similar manner to that of scatter- ing by strongly fluctuating continuous permittivity me- dia (i.e., including local-field corrections). The essen- tial difference is that the volume elements composing the medium are no longer vanishingly small but actual finite- size scattering particles. Finally, let us remined that Eq. (68) is obtained by neglecting particle correlations beyond second order. Higher-order correlations may yet be taken into account by treating them as sequences of two-particle correlations (which is formally exact for crystalline media). This ap- proach corresponds to the so-called quasicrystalline ap- proximation (QCA) originally introduced by Lax (1952), further developed by Fikioris and Waterman (1964) and Tsang and Kong (1980, 1982) among others, and used nowadays in various contexts (Gower et al., 2018; Kristensson, 2015; Tsang et al., 2000; Wang and Zhao, 2018a). 2. Extinction mean free path and effective medium theories To get an expression for ℓe, we need an expression for the self-energy Σ̃ . Using Eq. (68) to the lowest order, we obtain Σ̃1 ≡ 〈 T̃ 〉 = 〈∑ j T̃ j 〉 . (95) Writing the transition operator of an individual particle as T j ≡ T0(r−Rj , r ′ −Rj), this can be rewritten as Σ̃1(r− r′) = ρ ∫ T0(r− rp, r ′ − rp)drp. (96) Similarly, the second-order contribution is Σ̃2 ≡ 〈 ∆T̃ Ĝh∆T̃ 〉 = 〈 T̃ ĜhT̃ 〉 − 〈 T̃ 〉 Ĝh 〈 T̃ 〉 , (97) which leads to Σ̃2(r− r′) = ρ ∫ [δ(|rp − rq|) + ρh2(|rp − rq|)] × T0(r− rp, r ′′ − rp)Ĝh(r ′′ − r′′′) × T0(r ′′′ − rq, r ′ − rq)dr ′′dr′′′drpdrq.(98) In this expression, we have defined the total pair corre- lation function h2(r) ≡ g2(r)− 1, (99) and g2 is defined from Eq. (86). To reach a general expression for the self-energy Σ via Eq. (70), we need to define the operator gh that describes the local propagation of radiation within each particle. As first pointed out by Sullivan and Deutch (Sullivan and Deutch, 1976), gh may be chosen to obtain different lowest order results for the effective permittivities and refractive index, such as the Maxwell-Garnett (Bedeaux and Mazur, 1973; Felderhof, 1974), Onsager-Bütcher (Böttcher et al., 1978; Hynne and Bullough, 1987; On- sager, 1936) or Wertheim (Wertheim, 1973) models. Dif- 16 ferences vanish when all orders of the expansion are taken into account, but the rate of convergence of the differ- ent formulations is influenced by the particular choice of the Green function (Bedeaux and Mazur, 1973; Bedeaux et al., 1987; Geigenmüller and Mazur, 1986; Sullivan and Deutch, 1976). For pedagogical reasons, we restrict ourselves here to the Maxwell-Garnett result obtained in the long- wavelength limit. For some applications dealing with particles with complex shapes and relatively low scat- tering contrast, one can simplify the problem using the well-known Rayleigh-Gans approximation (Bohren and Huffman, 2008) or subsequent generalizations (Acquista, 1976). In this framework, the transition operator can be written as T0(r−Rj , r ′ −Rj) = k2h αp v Θ [a− |r−Rj |] δ(r− r′)1 (100) with αp the particle polarizability. Inserting this ex- pression in Eqs. (96) and (98) straightforwardly leads to an expression for the self-energy that reads in reciprocal space as Σ(k) = k20 (ϵMG − ϵh)1+ ρ k4hα 2 p ϵhαp ∂ϵMG ∂ρ × ∫ S(|k− k′|) ∣∣∣∣3j1(|k− k′|a) |k− k′|a ∣∣∣∣2 Ĝh(k ′) dk′ (2π)3 . (101) In this expression, jn is the spherical Bessel function of the first kind and order n, ϵMG is the Maxwell-Garnett permittivity given by Eq. (77) with α0 ≡ αp, and S(k) ≡ 1 + ρh2(k), (102) is the static structure factor, a key quantity for light scat- tering studies in correlated disordered media. Following the same steps as those for continuous per- mittivity random media, assuming non-absorbing mate- rials, we eventually reach a simple expression for the ex- tinction mean free path, 1 ℓe = 2πρ k4MG ∫ 2kMG 0 F (q)S(q)qdq, (103) with q = kMG|u − u′|. We have defined here the form factor F (q) = k2MG dσ dΩ ( q 2kMG ) f2 L ∣∣∣∣3j1(qa)qa ∣∣∣∣2 , (104) and the Rayleigh differential scattering cross-section dσ dΩ (k) = k4h α2 p (4π)2 P (k), (105) where P (k) is given by Eq. (56). Remarkably, the respective contributions of the in- dividual scattering elements, via the form factor F (q), and of their spatial arrangement, via the structure fac- tor S(q), on the extinction (or scattering) strength of the medium are treated independently. Structural correla- tions act as a weighting function to the optical response of a random assembly of identical scatterers. Physically, the structure factor describes far-field interferences be- tween fields scattered by pairs of particles. Equation (103) was obtained here in the long- wavelength limit, for small (non-resonant) particles. The more general situation of resonant particles is signifi- cantly more difficult to address within the theory for the average field. A heuristic extension of the Maxwell- Garnett approximation to resonant dipolar particles has been proposed by Doyle (1989) and further analyzed by Grimes and Grimes (1991); Ruppin (2000). The approach provides some understanding to the spectral resonances observed in certain light scattering experi- ments, but it fails to fulfill a fundamental scaling law between the material and effective permittivities, as ob- served by Bohren (2009). A more rigorous framework is given by the so-called Coherent Potential Approximation (CPA) (Tsang and Kong, 2001; Tsang and Kong, 1980), which may be seen as a generalization of the approach leading to the Bruggeman mixing rule for continuous per- mittivity media, as presented above, for particulate me- dia. Considering scattering elements that are not only the particles but also the host medium, one looks for an auxiliary background permittivity ϵb such that the aver- age transition operator vanishes ⟨T ⟩ = 0, or equivalently that the background Green function Gb equals the actual averaged Green function ⟨G⟩. Different CPA-like mod- els have been developed based on different self-consistent conditions (Busch and Soukoulis, 1995; Soukoulis et al., 1994). The so-called Energy-density CPA (ECPA) in- troduced by Busch and Soukoulis (1995), in particular, stands out from classical effective medium approaches as it focuses on energy transport (described by the average intensity) rather than on wave propagation and attenua- tion (described by the average field). 3. Scattering and transport mean free paths for resonant scatterers In this last part, we show that, in the presence of res- onant particles, it is more convenient to use the theory for the average intensity. To this aim, we assume that an effective index can be defined, such that Re[neff] = kr/k0 (but an explicit model for neff is not needed). In the weak extinction limit (i.e., krℓe ≫ 1) and using the expansion of the transition operator T in Eq. (92), the intensity vertex given by Eq. (25) can be reduced to its lowest 17 order terms as Γ ∼ ⟨T T ∗⟩ − ⟨T ⟩ ⟨T ∗⟩ ∼ 〈∑ i,j T iT ∗ j 〉 − 〈∑ i T i 〉〈∑ j T ∗ j 〉 . (106) In Fourier space, this leads to Γ̄(kru, kru ′, kru, kru ′) = ρT0(kru, kru ′)⊗T∗ 0(kru, kru ′) × S(kr(u− u′)). (107) In the radiative transfer limit, taking the on-shell approx- imation and using Eqs. (40)-(41) leads to 1 ℓs = ρ 16π2 ∫ |P(u)T0(kru, kru ′)e′|2 S(kr(u− u′))du (108) where e′ is the polarization vector perpendicular to u′. This expression is valid for a spherical particle of arbi- trary size. Also note that T0 is the transition operator of the particle in the host medium, evaluated for the in- cident and scattered wavevectors in the effective medium (i.e., at the wavenumber kr). Equation (108) can even- tually be reformulated using the form factor F (q) = k2r dσ dΩ (q), (109) where q = kr|u−u′| and the differential scattering cross- section is now defined as dσ dΩ (q) = 1 16π2 |P(u)T0(kr|u− u′|)e′|2 . (110) This leads to 1 ℓs = 2πρ k4r ∫ 2kr 0 F (q)S(q)qdq. (111) We observe that the approaches based on the average field and the average intensity lead to the same final expressions [Eqs. (103) and (111), respectively] for non- absorbing media. A similar derivation using Eqs. (40), (43) and (44) fi- nally leads to a closed-form expression of the transport mean free path, 1 ℓt = πρ k6r ∫ 2kr 0 F (q)S(q)q3dq. (112) Equations (111) and (112) seem to be the most widely- used expressions in the literature on light scattering and transport in correlated disordered media (Fraden and Maret, 1990; Reufer et al., 2007; Rojas-Ochoa et al., 2004; Saulnier et al., 1990). These two expressions, originally obtained from phenomenological arguments, have been derived here within a rigorous theoretical framework. D. Summary and further remarks The literature on multiple light scattering theory is vast and many approaches have been developed through- out the years. We adopted here a unique theoretical framework that can handle both families of systems de- scribed by a continuous permittivity that randomly fluc- tuates in space or by a set of identical particles that are randomly arranged in space. This has the great benefit of highlighting the main physical principles behind the role of spatial correlations on light scattering and transport, as well as the underlying approximations. Analytical ex- pressions for the characteristic lengths were obtained in the weak extinction limit (krℓe ≫ 1, with kr the effective wave number and ℓe the extinction mean free path), for statistically homogeneous, isotropic and non-absorbing media. These final expressions are provided in Table I in their most general form. The model for continuous permittivity media pre- sented in Sec. II.B does not make any specific assumption on the size or shape of a particular scattering element, and may thus be applied to different types of microstruc- tures. Several expressions for the scattering mean free path ℓs were derived from the average field or from the average intensity, assuming or not weak fluctuations, tak- ing or not the long-wavelength limit [Eqs. (55), (73), (79) and (82)]. All expressions eventually take the same form, reported in Table I, thereby unveiling the fundamental relation between structural correlations and light scat- tering and transport. In the case of strong fluctuations [Eqs. (73) and (79)], the expression to use depends on the choice of the “reference” homogeneous medium around which the permittivity fluctuates: the former is asso- ciated to the Bruggeman mixing rule and the latter to the Maxwell-Garnett mixing rule. The common ground that links these two approaches, as discussed for instance by Mackay and Lakhtakia (2020), is often overlooked. The model for particulate media presented in Sec. II.C applies specifically to disordered assemblies of identical particles, wherein the contributions of the individual par- ticles (possibly exhibiting a resonant behavior) and of the particle spatial arrangement on light scattering can be formally separated. A first expression for the scat- tering mean free path ℓs was obtained from the average field in the long-wavelength limit [Eq. (103)]. A compari- son with Eq. (79) unveils a fundamental concept, namely that a fluid of tiny identical particles with a fluctuating density (e.g., a fluid of molecules) can be assimilated to a continuous medium with a fluctuating permittivity (or polarizability). Indeed, inserting Eqs. (104) and (105) in Eq. (103), taking the limit qa → 0 (very small parti- cles) and comparing the resulting expression for 1/ℓe with Eq. (79) leads to the equivalence δ2αhα(q) ≡ α2 pρS(q) in reciprocal space. In real space, using Eq. (C10) in Ap- 18 Scattering (= Extinction) Transport Fluctuating permittivity media 1 ℓs = k40 8πk2r ϵ2b∆ 2 x ∫ 2kr 0 P (q/2kr)hx(q)qdq 1 ℓt = k40 16πk4r ϵ2b∆ 2 x ∫ 2kr 0 P (q/2kr)hx(q)q 3dq Monodisperse particulate media 1 ℓs = 2πρ k4r ∫ 2kr 0 F (q)S(q)qdq = ρ ∫ 4π dσ dΩ (θ)S(θ)dΩ 1 ℓt = πρ k6r ∫ 2kr 0 F (q)S(q)q3dq = ρ ∫ 4π dσ dΩ (θ)S(θ)(1− cos θ)dΩ TABLE I Analytical expressions for the scattering and transport mean free paths, ℓs and ℓt, in non-absorbing media with correlated disorder. These expressions were obtained from the average field and/or the average intensity, using different models (fluctuating permittivity or identical particles). ∆x and hx describe the amplitude of the fluctuation and the two-point correlation function of material descriptor x, respectively, ϵb is the auxiliary background permittivity, and kr is wavenumber associated to the real part of the effective index. Hence, for weakly fluctuating media [Eqs. (55), (82) and (83)], one should set ∆x = δϵ, hx = hϵ, ϵb = ϵav and kr = k0 √ ϵav. For strongly fluctuating media, one should set ∆x = δα, hx = hα, ϵb = ϵBG and kr = k0 √ ϵBG, when using the Bruggeman mixing rule [Eq. (73)], or ∆x = fLδα with fL = (∂ϵMG/∂ρ)/(ϵhα0), hx = hα, ϵb = ϵh and kr = k0 √ ϵMG, when using the Maxwell-Garnett mixing rule [Eq. (79)]. For monodisperse particulate media [Eqs. (103), (111) and (112)], the form factor F (q) can be expressed directly in terms of the differential scattering cross- section dσ dΩ (q) [Eq. (109)], leading to the simple expressions given on the last line, see also Eqs. (118) and (119) in Sec. IV.A.1. Remarkably, the fact to recover the same expressions with different approaches highlights a fundamental relation between structural correlations and light scattering and transport, that is independent of the choice of a specific effective medium approximation. pendix C.2 leads to ⟨∆ρ(r)∆ρ(r′)⟩α2 p ≡ ⟨∆α(r)∆α(r′)⟩ /v2, (113) with ∆ρ(r) = ρ(r)− ⟨ρ(r)⟩, the particle density fluctua- tion around the average density (⟨ρ(r)⟩ ≡ ρ using our no- tation). This is the reason why Rayleigh’s and Einstein’s quantitative results on the mean free paths (explaining the blue color of the sky) were essentially identical (Ein- stein, 1910; Rayleigh, 1899). A second yet identical expression for ℓs was obtained from the average intensity [Eq. (111)] under the assump- tion that a solution for the real part of the effective in- dex exists. The correspondance between the two expres- sions highlights again how structural correlations impact light scattering, independently of the effective medium description. Importantly, this and the expression for the transport mean free path ℓt [Eq. (112)] are not restricted to the long-wavelength limit, thereby explaining their popularity in studies on resonant scattering and trans- port in photonic liquids and glasses [Sec. IV.A]. To reach the expressions reported in Table I, we as- sumed the disordered media to be non-absorbing every- where in space, ϵ(r) ∈ R. This makes that extinction is driven by scattering, leading to 1/ℓe = 1/ℓs. To end this section, let us motivate this initial choice and explain how absorption might change the picture. In practice, absorption reduces the number of scat- tering events a wave can experience in a disordered medium and hinders wave interference phenomena be- tween multiply-scattered waves. Although the absorp- tance of a medium has been proposed as a means to identify Anderson localization of light (John, 1984), the mesoscopic optics community has mostly been interested in the study of strongly scattering materials with negli- gible absorption, which has naturally led to considering high-index dielectrics like Si and Ge in the near-infrared range, and TiO2 in the visible range. Many experiments have also been made with lower-index materials with neg- ligible absorption in the visible range, including polymers (polystyrene, PMMA) – well-suited to direct laser writ- ing (see Sec. III.E) – and SiO2. Thus, our assumption of non-absorbing media is valid in most cases of interest. The theoretical problem of multiple light scattering in presence of absorption has received relatively little atten- tion compared to its non-absorbing counterpart. Gener- ally speaking, the extinction length in a scattering and absorbing medium is simply given by 1/ℓe = 1/ℓs+1/ℓa, with ℓa the absorption mean free path, that yet remains to be determined. For particulate media, the absorption may come from the particles themselves, in which case it is incorporated in the transition operator T0 of the individual particle, and from the host medium, in which case the wavenum- ber kh should have a non-zero imaginary part due to ab- sorption. Initial efforts to take correlations into account 19 were made in the 1990s (Kumar and Tien, 1990; Ma et al., 1990), but were limited to very small (Rayleigh) parti- cles. A multiple-scattering model for the average field in assemblies of spherical particles of arbitrary size has been developed by Durant et al. (2007), leading to analytical expressions of the effective wavenumber (directly related to the extinction mean free path) as a function of the pair correlation function g2(r). The approach has later been generalized by Mishchenko (2008) to arbitrary particu- late media, including non-spherical particles and poly- dispersity. The theory unfortunately does not provide analytical expressions for the absorption mean free path as a function of correlations. This limitation can be lifted via a model for the average intensity, as shown, though for dipolar particles only, by Wang and Zhao (2018b) un- der the QCA. The authors of the latter study find that short-range correlations have a weaker effect on the ab- sorption mean free path than on the scattering mean free path. The case of fluctuating permittivity media with ab- sorption has been tackled only recently, to our knowledge, by Sheremet et al. (2020), who relied a diagrammatic de- scription of multiple scattering for the average field and average intensity to reach an analytical expression for the absorbed power as a function of structural correla- tions. Although the theory is made specifically for scalar waves in fluctuating media with short-range correlations, it leads to the same conclusion as above on the impact of correlations on the scattering and absorption lengths. III. STRUCTURAL PROPERTIES OF CORRELATED DISORDERED MEDIA Correlated disordered media can exhibit a rich variety of complex morphologies, which will impact light scatter- ing and transport in many different ways. This section is concerned with the statistical description of the struc- tural properties of these materials. For a more complete and thorough description, we recommend the textbook by Torquato (2013). After comparing the quantities describing spatial cor- relations and derived in the previous section on realistic systems [Sec. III.A] and introducing a fundamental re- lation between fluctuations of particle number and spa- tial correlations in point patterns [Sec. III.B], we define the main classes of correlated disordered media accord- ing to their pair correlation function g2(r) and structure factor S(q) [Sec. III.C]. We then review the main tech- niques to numerically generate correlated disordered ma- terials and simulate their optical properties [Sec. III.D]. Experimental fabrication techniques for correlated disor- dered media are then presented [Sec. III.E]. The section is concluded with a brief summary of the experimental techniques to characterize structural correlations in real materials [Sec. III.F]. A. Continuous permittivity versus particulate models in practice As shown in the previous section, light scattering in correlated disordered media can be described either by a continuous permittivity model that relies on a spa- tial permittivity correlation function or by a particulate model that relies on a two-point correlation function in the specific case of localized permittivity variations. To start this section, it is interesting to compare these two pictures in practical cases. We consider a classical and very relevant example for photonics, that is a 2D assembly of impenetrable disks (diameter a) at two different packing fractions p = 0.10 and 0.50, see Fig. 3. The disk packings were gener- ated with a compression algorithm (Skoge et al., 2006) that will be briefly described in Sec. III.D. The top-left panel of Fig. 3 shows the permittivity correlation function gϵ(|r−r′|) = hϵ(|r−r′|)+1 for the disk packings. This cor- relation function was introduced to describe media with a fluctuating continuous permittivity, but can in fact be applied to any type of microstructure given its general- ity. The function first displays a rapid decrease of cor- relation followed by regular, vanishing oscillations. The very short-range correlation here is mostly associated to the finite size of the disks and the oscillations, which are stronger for higher packing fractions, are mostly a signa- ture of correlations between neighboring particles. The difficulty to formally distinguish distinct types of corre- lations is a downside of the generality of hϵ. FIG. 3 Description of structural correlations for hard disk (diameter a) packings at packing fractions p = 0.10 (black curves) and p = 0.50 (blue curves). (Left) Correlation func- tion gϵ(r) = hϵ(r) + 1 describing correlations due to the finite-size of the disks and to their positional correlation, for two different topologies: (top) a packing of disks and (bot- tom) a continuous network. (Right) Pair correlation function g2(r) = h2(r) + 1, which describe only the positional correla- tion between the disk centers. The generated disk positions can also serve as a basis to generate more complex structures. An example often encountered in photonics is based on Delaunay tessela- 20 tions (described in Sec. III.D). In the bottom-left panel of Fig. 3, we show the permittivity correlation function for these “inverted” structures, as generated from the disk packing above. Strikingly, the behavior of the correlation functions are very similar to those of the disk packing. The major difference is observed at small distances due to a very different morphology, but the curves become indistinguishable for |r− r′| ≳ a. Finally, we can consider the pair-correlation function g2(|r− r′|), applicable specifically to ensembles of identi- cal scatterers. The results are shown in the right panel of Fig. 3. The pair-correlation function for the disk packing is zero for |ra − rb| between 0 to 2R due to the impen- etrability of the disks, and exhibits strong oscillations indicating structural correlations in the relative position between particles. Note that the amplitude of the oscil- lations is much larger than for gϵ(|r− r′|). This simple comparison shows that the use of the two- point permittivity correlation is hardly sufficient to dis- tinguish different types of correlated disordered media. In fact, the permittivity correlation of the inverted disk structure at p = 0.50 would be strictly identical to that of the direct structure by definition, while light scattering would evidently be markedly different. This is a strong indication that scattering is both dramatically affected by the local morphology of the system, which yields op- tical resonances, and by structural correlations in the rel- ative position between scattering elements. Since a “scat- tering element” is not well defined for inverted structures such as connected networks, for clarity and simplicity, we will focus on the particulate description of scattering me- dia in the remainder of this section. B. Fluctuation-correlation relation The description of point patterns underlying the struc- ture of correlated disordered media is central, and many descriptors may be used in general. An important attribute of point patterns is the vari- ance of the number N of points contained within a win- dow Ω with volume V . This quantity has a long his- tory, several derivations are found for both continuous and discrete disorder models (de Boer, 1949; Landau and Lifshitz, 1980; Martin and Yalcin, 1980; Ornstein and Zernike, 1914; Torquato and Stillinger, 2003; Van Kra- nendonk and Sipe, 1977). Considering a spherical win- dow of radius R for simplicity, probabilistic calculations eventually lead to a closed-form expression for the vari- ance of N (Torquato and Stillinger, 2003) ⟨N2(R)⟩ − ⟨N(R)⟩2 ⟨N(R)⟩ = 1 + ρ ∫ h2(r)Λ(r;R)dr, (114) where Λ(r, R) = vint2 (r;R)/V is the intersection volume vint2 (r;R) of two windows separated by r normalized to the window volume V = 4 3πR 3. In the limit of large windows, one finds that lim R→∞ ⟨N2(R)⟩ − ⟨N(R)⟩2 ⟨N(R)⟩ = lim |q|→0 S(q), (115) = 1 + ρ ∫ h2(r)dr,(116) which corresponds to the simplified definition given in Appendix C.2. Equations (114)-(116) are remarkable in that they de- scribe the spatial fluctuations in the number of points in the pattern from its pair correlation between points – or equivalently, its structure factor near 0, which is a measurable quantity (e.g., by small-angle scatter- ing, see Sec. III.F). A Poisson point pattern pN = ⟨N⟩N exp [−⟨N⟩] /N ! yields ⟨N2⟩ = ⟨N⟩ + ⟨N⟩2, which, as expected, corresponds to a fully uncorrelated system with h2(r) = 0 or lim|q|→0 S(q) = 1. Implementing structural correlations at constant density ρ therefore re- sults into weaker or stronger point density fluctuations. Negative correlations are obtained when ρ ∫ h2(r)dr < 0, leading to a sub-Poissonian fluctuations, while positive correlations are obtained when ρ ∫ h2(r)dr > 0, lead- ing to a super-Poissonian fluctuations. In the literature, such structures are sometimes denoted as negatively- and positively-correlated, respectively (Davis and Mi- neev, 2008). As illustrated in Fig. 4, they correspond to situations in which the points either repel or attract themselves. As we will see in Sec. IV, the impact of neg- ative and positive correlations on optical transport are markedly different. FIG. 4 Illustration of the fluctuation-correlation relation with three point patterns: (left) A negatively-correlated dis- ordered medium, where points tend to repel themselves; (mid- dle) A Poisson point pattern, that is uncorrelated; (right) A positively-correlated disordered medium, wherein clustering is present. The variance of the number of points in a spherical window Ω of radius R is related to the total pair correlation function h2 via Eq. (114). C. Classes of correlated disordered media Figure 5 summarizes the most important classes of cor- related disordered media and their properties, that we 21 will now describe specifically. Note that this panel is non- exhaustive – other families of correlated disordered me- dia exist, such as paracrystals (Hosemann, 1963), char- acterized by regular point patterns deformed on scales typically larger than the distance between neighboring points. Our focus here is on the classes that led to a substantial body of work in optics and photonics. 1. Short-range correlated disordered structures Consider a volume containing a disordered ensemble of mobile, impenetrable particles (i.e., a fluid of hard par- ticles) at a low density. With increasing particle density, the particles tend to organize themselves to fill space. In this regime of low to moderate densities, the sys- tem exhibits no structural correlation in the long range yet the impenetrability of the particles impose a short- range correlation that increases with the packing frac- tion (Hansen and McDonald, 1990). As shown in Fig. 5 (1st column), short-range structural correlations give rise to decaying oscillations in the pair correlation function g2. The most likely distance to find a neighboring parti- cle is given by the position of the first peak and the decay of the higher-order peaks, which is generally rapid, al- lows defining a correlation length. In reciprocal space, such oscillations are also observed. When increasing short-range correlations, the structure factor goes from a flat response around 1 to sharper peaks whose ampli- tudes decrease with increasing q. Short-range structural correlations can be described, for instance, by analyti- cal or semi-analytical solutions of the Ornstein-Zernike equation using the so-called Percus-Yevick approxima- tion (Percus and Yevick, 1958; Wertheim, 1963) in three dimensions and the Baus-Colot approximation in two di- mensions (Baus and Colot, 1987), respectively. At higher densities, these models become less accurate, although, for slightly polydisperse systems, errors appear to can- cel, and predictions by Percus-Yevick approximation can describe experimental data up to random close packing or jamming (Frenkel et al., 1986; Scheffold and Mason, 2009). Short-range correlated disordered systems constitute the primary class found in colloidal suspensions with isotropic interactions, since short-range correlations stem from the impenetrability of particles in suspension. The behavior of other repulsive particles, such as charge- stabilized particles, can often be mapped onto the isotropic hard-sphere case (Gast and Russel, 1998; Pusey and Van Megen, 1986). The recent advent of col- loids interacting via sticky patches could open a path- way towards more complex structures through self- assembly (He et al., 2020). In general, short-range structural correlations are not limited to sphere assemblies but can also be encoded in connected networks (Florescu et al., 2009; Liew et al., 2011; Muller et al., 2013), in which case the individual scattering centers are more difficult to identify. This form of correlated structure is widespread in natural photonic structures such as bird feathers (Saranathan et al., 2012) and very popular in artificial photonic structures fabri- cated by top-down techniques (Liew et al., 2011; Muller et al., 2013, 2017). Dry foams are promising candidates for correlated network structures that can be made by self-assembly (Klatt et al., 2019; Maimouni et al., 2020; Ricouvier et al., 2019). 2. Polycrystalline structures For a disordered ensemble of identical hard particles, one reaches a liquid-crystal coexistence at about 49% and a purely crystalline phase for concentrations above 54.5% (Gast and Russel, 1998; Pusey and Van Megen, 1986; Pusey, 1991; Zhu et al., 1997). The equilibrium structure appears to be face-centered-cubic but hexag- onal close packed structures are also observed and are found to be at least meta-stable (Pusey et al., 1989). This liquid to crystal phase transition, also known as the Kirkwood-Alder transition (Gast and Russel, 1998), is purely driven by the higher entropy of the crystalline phase compared to the liquid phase. The densest pack- ing of monodisperse spheres in three dimensions is ap- proximately 74%, also referred to as the close-packing of equal spheres. Monodisperse particles usually assemble in finite-sized crystal clusters. These clusters are ran- domly arranged and form a polycrystalline material (As- tratov et al., 2002; Yang et al., 2010b), see Fig. 5 (2nd column). In the bulk, crystallites are formed by homo- geneous nucleation throughout the sample (Pusey et al., 1989). The size of the crystal clusters is then typically several tens of µm, much larger than the particle diam- eter ∼ λ but smaller than the usual sample size, which is typically in the millimeter to centimeter range. The radially-averaged pair correlation function exhibits peaks indicating the position of the nth-order neighboring par- ticles, as well as minima approaching zero. Positional correlations vanish for distances exceeding the size of the crystal clusters. Similarly, the structure factor shows well-defined Debye-Scherrer rings due to Bragg scattering from randomly oriented crystal planes, that can be iden- tified in light scattering (Pusey et al., 1989), similarly to powder diffraction in X-ray crystallography. The formation of clusters of regular arrays in fluids of hard particles is strongly influenced by the polydisper- sity of the particles, since particles of very different sizes do not naturally arrange in a crystal. Indeed, for hard sphere fluids with a polydispersity larger than 6-12%, crystallization is avoided in three dimensions (Pusey, 1987). The spheres remain disordered and particles enter a solid glass phase at about 58%. The glass can be further compressed until the spheres ’jam’ forming what is known 22 FIG. 5 Classes of correlated disordered media. (From top to bottom) Illustration of a correlated disordered medium; Pair correlation function; Structure factor; SEM image of a fabricated correlated disordered structure. (From left to right) Disordered short-range correlated structures, SEM image adapted with permission from (Garćıa et al., 2008); Polycrystalline structure, SEM image adapted with permission from (Salvarezza et al., 1996); Imperfect ordered structures, SEM image adapted with permission from (Garcia et al., 2012); Disordered hyperuniform structures, SEM image adapted with permission from (Haberko and Scheffold, 2013); Disordered hierarchical structures, SEM image adapted with permission from (Burresi et al., 2012). as a “randomly closed packed” or “maximally jammed structure” in the literature (Torquato et al., 2000). The presence of some hidden structural order, crystalline pre- cursors or locally favoured structures in the glass and jammed phase is still being discussed (Zhang et al., 2016). Due to the unavoidable finite polydispersity, exper- imental realizations of crystalline photonic structures based on colloidal suspensions are the exception rather than the rule even at high packing fractions. By care- ful synthesis of colloidal particles made from polystyrene or silica (SiO2), it is however possible to induce crys- tallization rather easily (Salvarezza et al., 1996). These materials are usually polycrystalline and display some defects and stacking faults to a varying degree. Poly- crystalline structures are also observed in natural pho- tonic structures, such as opals. Interestingly, relatively little is known about the comparison of scattering and light transport between random-close-packed assemblies of spheres and polycrystalline materials (Yang et al., 2010b), in particular when the size of crystallites is grad- ually reduced to smaller length scales. 3. Imperfect ordered structures The two previous classes of correlated disorder were obtained by “adding order” into a fully-disordered (un- correlated) system. Materials with correlated disorder can also be obtained starting from the other limit, that is a periodic system with random perturbations, see Fig. 5 (3rd column). In systems of infinite size, both the pair-correlation function g2 and the structure factor S are characterized by a series of Dirac peaks located at r−r′ = u1a1+u2a2+u3a3 with ui ∈ Z and ai the lattice vectors, and G = v1b1 + v2b2 + v1b2 with vi ∈ Z and bi the reciprocal lattice vectors, respectively (Joannopou- los et al., 2011; Kittel, 1976). If the position of a point of the lattice is randomly shifted (e.g., with normal dis- tribution) around its nominal position, this results in a broadening of the Dirac peaks with a width that depends on the disorder amplitude. By contrast with the pre- 23 vious classes of disordered systems, disorder in such a periodic-on-average structure does not impact the corre- lation length, which remains infinite. A striking conse- quence of this is that the structure factor is characterized by Dirac peaks of vanishing width (for systems of infinite size) and decreasing amplitude with increasing wavenum- ber q on top of a diffuse background. The latter, which is due to the random nature of the point pattern, equals 1 for large values of q and quadratically goes to zero when q goes to zero, as discussed, for instance, by Klatt et al. (2020). Real systems are however never exactly periodic, due to fabrication imperfections and in practice this also leads to a finite correlation length (Koenderink et al., 2005; López, 2003; Meseguer et al., 2002; Nelson et al., 2011). A plethora of studies of ordered photonic crystal struc- tures with imperfections, both numerical and experimen- tal, can be found in the literature (Soukoulis, 2012). De- fects were also added intentionally, either at random or selected positions, to study the interplay between defect states, density of states, wave tunneling and percolation, random diffuse scattering, and directed Bragg scatter- ing of light (Aeby et al., 2021; Florescu et al., 2010; Garćıa et al., 2009). Moreover, the interaction between the band structures and defect scattering is fascinating, since it might lead to other critical coherent transport phenomena such as Anderson localization of light (John, 1987). Defect states can also be introduced in a photonic crystal to deliberately implement a particular function, such as optical sensing applications, lasing, or optical cir- cuitry (Joannopoulos et al., 2011; Nelson et al., 2011; Soukoulis, 2012). 4. Disordered hyperuniform structures One of the important characteristics of point patterns is how the number of points contained in a given volume fluctuate with various disorder realizations (Torquato, 2013). This quantity is related to the notion of spatial uniformity. For a Poisson point process, one shows from Eq. (114) that the variance in the number of points N contained in a d-dimensional sphere of radius R grows as the sphere volume (i.e., ⟨N2⟩ − ⟨N⟩2 = ⟨N⟩ ∼ Rd). This result holds for many disordered point patterns. By contrast, the same analysis performed on a periodic pat- tern shows that the variance grows with the surface of the sphere, ⟨N2⟩ − ⟨N⟩2 ∼ Rd−1. In a founding work, Torquato and Stillinger (2003) proposed to define a gen- eral class of point patterns, dubbed “hyperuniform”, the property of which is to exhibit point number fluctuations scaling as the surface of the window, that is slower than expected for usual disordered media. Hyperuniformity encompasses periodic, quasi-periodic but also – very in- terestingly in the framework of this review – a subclass of disordered systems, see Fig. 5 (4th column) for an il- lustration and (Torquato, 2018) for a recent review. It was observed numerically that maximally jammed pack- ings of spheres and platonic solids tend to a hyperuniform structure (Donev et al., 2005; Jiao and Torquato, 2011; Zachary et al., 2011). While such long-range fluctuations can hardly be observed on the pair-correlation function, hyperuniform point patterns can be recognized from the behavior of the structure factor at low values lim q→0 S(q) = 0. (117) Of particular interest in photonics are so-called “stealthy” hyperuniform structures, for which S(q) = 0 for 0 < q ⩽ qmax, where qmax may be set to an arbi- trary value. The region of zero structure factor is often followed by oscillations similar to those found in short- range disordered correlated media (Froufe-Pérez et al., 2016). The concept of hyperuniformity in photonics has first been introduced in a numerical study by Florescu et al. (2009). Important efforts have been put since then on the fabrication of hyperuniform disordered systems, which could be achieved so far by lithography in 2D (Man et al., 2013) and 3D (Muller et al., 2013), block copolymer as- sembly (Zito et al., 2015), emulsion routes (Piechulla et al., 2018, 2022; Ricouvier et al., 2017; Weijs et al., 2015), and spinodal solid-state dewetting (Salvalaglio et al., 2020). 5. Disordered fractal structures In all classes of disordered point patterns discussed above, the average number of points N contained in a d-dimensional sphere of radius R is expected to grow as ⟨N⟩ ∝ Rd - by doubling the observation radius for a 3D point pattern, the number of points increases by a fac- tor 23 = 8. This scaling is however not a general rule. Introduced by Mandelbrot (1967), the concept of frac- tals encompasses systems for which the power-law scal- ing of the mass with the system size does not have the Euclidean dimension as an exponent. More specifically, for fractal point patterns, we have ⟨N⟩ ∝ Rdf , where df is a non-integer fractal dimension. Fractality has a dra- matic impact on the structure, as illustrated in Fig. 5 (5th column). First, it is statistically self-similar, mean- ing that the structure is statistically identical whatever the scale on which it is looked at (though lower and up- per bounds are always met in practice). Second, it ex- hibits enormous local density fluctuations and high la- cunarity (Allain and Cloitre, 1991), meaning that both very dense and very empty regions are found. As a re- sult, the pair correlation function can be shown to de- cay as a power-law as g2(r) ∼ r−α and similarly for the structure factor, S(q)− 1 ∼ |q|−(d−α) assuming that 0 < α < d. Depending on the process of structure for- 24 mation, one can directly relate the exponent α with the fractal dimension df. For instance, clustering described by the Soneira-Peebles model gives α = d−df, leading to S(q) − 1 ∼ |q|−df (Soneira and Peebles, 1977). It is im- portant to note that real systems generally exhibit lower and upper bounds in their fractal nature. This implies that the power-law decays are observed on finite range (g2(r) eventually goes to 1 at large r). Fractal disordered optical materials are encountered in a wide variety of colloidal aggregates that form nat- urally for certain charged particles (Meakin, 1987) as well as in certain emulsions (Bibette et al., 1993). They can also be designed in a laboratory by inserting spac- ing particles with a size distribution that covers several orders of magnitude in a statistically-homogeneous disor- dered medium (Barthelemy et al., 2008; Bertolotti et al., 2010a). D. Numerical simulation of correlated disordered media Numerical simulations of the complex heterogeneous morphologies play a key role in colloidal chemistry and soft matter physics. For light scattering studies, mod- elled structured materials are taken as input data for solving Maxwell’s equations. Here, we present some stan- dard numerical approaches that have been used in the lit- erature to generate correlated disordered structures and simulate their optical properties. 1. Structure generation Random packings of hard spheres in different dimen- sions is of great interest due to their structural and ther- modynamic properties. The numerical generation of such ensembles plays a key role in research, especially in the case of random close packings (RCP) (Parisi and Zam- poni, 2010; Song et al., 2008; Torquato and Stillinger, 2010). When the packing fraction of the system is kept below a few tens of percent, a random sequential absorp- tion (RSA) model (Widom, 1966) is suitable to generate large (non-equilibrium) ensembles in any dimensionality. In the RSA model, new points are randomly added to the system following a uniform distribution. The new point is rejected if closer than a given distance to any of the previous points in the pattern. A careful management of the coordinates storage in appropriate structures lead to very efficient algorithms but the computational time nevertheless diverges due to high rejection rate when ap- proaching the maximum possible RSA filling fraction, be- ing about 54% and 38% for disks in 2D (Wang, 2000) and spheres in 3D (Meakin and Jullien, 1992), respectively. This limitation has been lifted by Zhang and Torquato (2013), who proposed a precise algorithm to generate a saturated RSA configuration within a finite time. In order to achieve larger packing fractions up to the jamming packing (at a filling fraction ϕ ≃ 64% in 3D) several approaches leading to efficient algorithms have been developed. The Lubachevsky-Stillinger (or com- pression) algorithm (Lubachevsky and Stillinger, 1990) generates random packings of any physically-realistic ϕ by placing a set of N particles of vanishingly small size in a closed or periodic domain at random, and then letting the particles grow at a given rate, move and collide (elas- tically), until the desired ϕ is reached. This algorithm has been successfully used in the study of sphere packings in any dimensionality (Skoge et al., 2006) and is largely used in photonics (Conley et al., 2014; Froufe-Pérez et al., 2016). An approach named “ideal amorphous solids” was also proposed by Lee et al. (2010) to realize maximally random jammed packings of polydisperse particles. The method relies on building aggregates of touching spheres by placing spheres one by one around a center of mass. Enlarging the kind of correlations encountered in sphere packings requires the use of interaction potentials beyond the hard sphere model. Simple two-body interac- tion potentials such as Lennard-Jones together with stan- dard Monte-Carlo techniques have been used to gener- ate assemblies of scatterers in different phases (De Sousa et al., 2016). Two and three-body interaction models like the Stillinger-Weber model (Stillinger and Weber, 1985) can be used to generate fully connected dielectric networks showing the same statistical structural proper- ties (coordination and angle statistics) as amorphous sil- icon or diamond. Stealthy hyperuniform point patterns have been generated numerically using a suitable pair- wise, long-range potential in the real space (Froufe-Pérez et al., 2016). Quite often, the structures are generated using molecular dynamics. Being a very vast and mature field, various softwares are nowadays available, including NAMD (Phillips et al., 2005) and CHARMMS (Brooks et al., 2009), which are both extensively used in the field of chemistry and biochemistry. Other software packages include HOOMD-blue (Anderson et al., 2008), which is implemented for GPU, and LAMMPS (LAMMPS, 2019), which exploits massive parallelization (Plimpton, 1995). Instead of using constraints in real space as in the case of hard spheres, targeted interaction potentials can be ob- tained by imposing constraints on the structure factor in reciprocal space (Uche et al., 2004), which allowed realiz- ing, for instance, stealthy hyperuniform structures (Bat- ten et al., 2008; Florescu et al., 2009). A more general approach to the problem is obtained using constrained Fourier transforms as collective coordinates (Kim et al., 2018). Materials forming a continuous correlated disordered network are very relevant on different levels (Wright and Thorpe, 2013). On the one hand, a network presents the necessary structural stability required by different fabri- cation methods (Gaio et al., 2019). On the other hand, the topology of the network apparently plays an impor- 25 tant role in the emergence of different optical proper- ties such as photonic gaps in disordered networks (Flo- rescu et al., 2009; Weaire, 1971). Besides the Stillinger- Weber model considered above, there are different pro- tocols described in the literature to generate continuous random networks. The Wooten-Winer-Weaire (WWW) algorithm (Wooten et al., 1985) considers a collection of points and bonds connecting pairs of points. The ini- tial network, that can be ordered, is randomized after a number of bond reassignments followed by a relaxation of the structure (for instance following the Stillinger-Weber interaction potential). In this way, accurate predictions of the electronic structure, bond geometry statistics and atomic structure of amorphous semiconductors are ob- tained (Barkema and Mousseau, 2000). Replacing the chemical bonds by dielectric rods leads to amorphous di- electric materials (Edagawa, 2014). A protocol to generate strongly correlated continuous random networks was first proposed by Florescu et al. (2009) in two dimensions and used by Liew et al. (2011) in three dimensions. The idea is to create a uniform topology network starting from an arbitrary point pat- tern. The Delaunay tessellation (Watson, 1981) is con- structed from the seed point pattern. By definition each Delaunay cell is surrounded by 3 (in 2D) or 4 (in 3D) neighbors. The protocol indicates that the centroids of neighboring triangles (2D) or tetrahedrons (3D) are linked. This connected network shows a uniform con- nectivity since each node of the network is linked to the same number of neighbors. When the seed pattern is strongly correlated, for instance using random closed or stealthy hyperuniform packings, the resulting dielectric network presents interesting photonic properties, for in- stance complete gaps in its density of states (Florescu et al., 2009; Froufe-Pérez et al., 2016; Liew et al., 2011) in 2D and 3D. The optical properties of these structures will further be discussed in Sec. V. 2. Electromagnetic simulations No numerical method has been specifically developed to model the optical properties of correlated disordered structures. Generic electromagnetic methods are being used instead, the choice of a specific method depending on the type of structure to simulate, the quantity of in- terest and the computational load. We refer the reader to Gallinet et al. (2015); Wriedt (2009b) for an overview of the computational techniques used in photonics and light scattering, and to the internet portal Scattport by Wriedt (2009a), which conveniently provides a large collection of freely available software packages dedicated to light scat- tering problems. The most widely used numerical methods for the quantitative analysis of 2D and 3D correlated disor- dered media are (i) the T-matrix method and its vari- ants (Mishchenko et al., 1996), (ii) the finite-difference time-domain (FDTD) method (Sullivan, 2013), and (iii) the planewave expansion (PWE) method (Ho et al., 1990; Johnson and Joannopoulos, 2001). The T-matrix method, initially proposed byWaterman (1965) and further developed over the years (Mishchenko et al., 1996), is probably the most adapted to solve light scattering problems by particulate media, including in layered environments (Kristensson, 1980; Mackowski, 2008; Videen, 1991). The method essentially relies on the possibility to decompose the incident and scattered fields around a particle as a superposition of vector spher- ical wave functions (VSWFs). Formally, the T-matrix re- lates the amplitude coefficients of the incident wave func- tions to those of the scattered wave functions, and the multiple-scattering problem is solved with a high degree of analyticity by making use of the translation addition theorem for VSWFs (Cruzan, 1962; Stein, 1961). Several publicly available codes exist, among which the long- established MSTM (Mackowski and Mishchenko, 2011; Mackowski, 2022) and the more recent GPU-parallelized CELES (Egel et al., 2017) and SMUTHI (Egel et al., 2021), which have been used to model large disordered clusters of particles, as in (Aubry et al., 2017; Yazhgur et al., 2021). In assemblies of very small scatterers, the T-matrix of the individual elements reduces to their electric polariz- ability and the electromagnetic interaction between scat- terers can be described directly with the dyadic Green tensor, Eq. (15). More straightforward to implement than the T-matrix method, the so-called coupled dipoles method (CDM) (Foldy, 1945; Lax, 1952) is a standard to test new concepts or theoretical models, for instance, on homogenization (Schilder et al., 2017), light emis- sion statistics (Pierrat and Carminati, 2010; Sapienza et al., 2011) or mesoscopic transport regimes (Leseur et al., 2014). For resonant scatterers with high qual- ity factors (e.g., cold atoms), the coupled dipoles equa- tions in absence of an incident field become a linear non- Hermitian eigenvalue problem (Rusek et al., 1996), whose solutions are the so-called quasinormal modes (QNMs) of the system with complex-valued frequencies (Ching et al., 1998). The statistical properties of QNMs pro- vide information on collective phenomena, like polari- tonic modes (Schilder et al., 2016) and the Anderson transition (Monsarrat et al., 2022; Sgrignuoli et al., 2022; Skipetrov and Sokolov, 2014). The FDTD method is instead the most popular choice for non-particulate structures (e.g., connected networks). In essence, the method provides numerical solutions of the time-dependent Maxwell’s equations with discretized space and time partial derivatives (Yee, 1966) over a (nec- essarily) finite volume and for a certain duration. Quan- tities related to light emission, scattering, transport and localization can be computed using appropriate bound- ary and initial conditions on the fields and sources, see 26 (Scheffold et al., 2022; Yamilov et al., 2022) for very re- cent examples. The FDTD method is extremely versa- tile but the requirement to discretize the entire space for large disordered structures and perform simulations over long times implies a high computational load, which is yet mitigated by efficient parallelization. Many commer- cial and non-commercial software packages are nowadays available. Among those, MEEP is a powerful, maintained and open-source solution that is extensively used by the community (Oskooi et al., 2010). Last but not least, the PWE method is the most common choice for identifying photonic gaps in non- absorbing (non-dispersive) dielectric structures. In short, the method solves the source-free wave propagation equa- tion with periodic boundary conditions, expanding the fields and space-dependent permittivity in a Fourier se- ries in reciprocal space, to solve the photonic band struc- ture of the geometry (Ho et al., 1990). Primarily applied to photonic crystals, the use of the supercell approach (see Sec. V.A) on large disordered structures generated with periodic boundary conditions, can provide quanti- tative information on the photonic density of states (Flo- rescu et al., 2009). De facto, the standard tool used by the community is the MIT Photonic Bands (MPB) open-source software package (Johnson and Joannopou- los, 2001). E. Fabrication of correlated disordered media Here, we present an overview of the different strategies and important design parameters for the experimental fabrication of strongly scattering correlated disordered media. We mainly focus on the fabrication of 3D materi- als but note that many of the concepts discussed here also hold for 2D materials. We will illustrate the fabrication concepts with a few examples but will not attempt to provide a comprehensive overview of this field of materi- als research, which is beyond the scope of our work. We note that the fabrication of disordered correlated pho- tonic materials faces the same challenges than other op- tical metamaterials such as photonic crystal circuits or other 3D arrangements of structural units (Soukoulis and Wegener, 2011). The trade-offs one needs to consider are simplicity, freedom of design, speed or throughput, accu- racy and resolution. Those parameters vary enormously and therefore no one-method-fits-all fabrication route can be singled out. The range of interest to observe strong scattering and coherent phenomena due to structural correlations is when typical length scales of the structure are on the order of the wavelength (typically half a wavelength) in the medium. This mandates, first of all, the sub-micron structuring of dielectric materials on length scales com- parable to the wavelength of light with a refractive index contrast (n/nh)− 1 ≫ 0.1. In practice, finding the opti- cal material properties is also influenced by the fact that the effective refractive index neff is often higher than the nominal background material index nh which tends to reduce the scattering and transport coefficients (Naraghi et al., 2015; Reufer et al., 2007; Schertel et al., 2019b). As a general rule, the higher the space filling fraction of the scattering material the higher neff ≥ nh and the stronger this effect. The optimum space filling fraction is often found around ϕ ∼ 0.3 which is much lower than the space filling fraction obtained naturally by randomly packing spheres ϕ ∼ 0.64. In addition to these funda- mental scattering parameters, structural correlations are key for the design and fabrication of optimally white ma- terials. In general we can distinguish global structural proper- ties, that is, properties that can be expressed by sta- tistical averages and a corresponding structure factor S(q), and local properties related the local topology, fill- ing fraction and the scatterer morphology. In the fol- lowing, we describe different approaches that have been used to fabricate disordered and strongly scattering me- dia. Structural correlations then appear naturally or by design. Figure 6 summarizes the most important fabrication methods of correlated disordered media, that we will de- scribe consecutively in detail below. Thermal or assisted self-assembly are bottom-up processes driven by a com- bination of entropy and external forces, such as grav- ity. Equilibrium and non-equilibrium self-assembly de- sign routes following predefined pathways are frequently found in nature but are also increasingly considered as al- ternatives in the laboratory. Top-down approaches based on lithography come in many flavors and take advantage of powerful technology at hand. Lithography is very pow- erful for structuring two dimensional materials but only more recently significant progress has been made to fabri- cate 3D structured materials on submicron length scales. We will discuss some of the strengths and limitations of the different methods. 1. Jammed colloidal packing Dispersing submicron sized colloidal particles in a sol- vent phase with a lower index of refraction is the most common and most simple way to fabricate a strongly scattering, disordered medium. Ubiquitous examples are white paints, often based on a dispersion of submicron TiO2 or polymer latex particles, or milk. For uniform suspensions of spherical particles, structural correlations appear naturally owing to the interactions between the particles which can be longer range DLVO-type double- layer repulsion or short-range excluded volume inter- actions The preparation is fairly simple and only re- quires some command over the stability of the suspen- sion to avoid the formation of very large aggregates or 27 FIG. 6 Overview of fabrication methods. (From left to right) Jammed colloidal packing, where the colloids deposit via gravity assisted methods, or where they assemble in confined spaces due to local interactions. Thermodynamically driven assembly, where the system goes through a phase-separation and rearrangement, driven by entropy or by pre-programmed interactions as for example using DNA strands. Optical and electron lithography, where samples are fabricated by direct sculpturing of a material, using optical or electron beams to modify the local physical and chemical properties (subtractive) or where material is locally added to the structure via selective polymerisation or deposition (additive). Courtesy from Mélanie M. Bay (University of Cambridge, UK). flocks (Galisteo-López et al., 2011). The degree of struc- tural correlations can be controlled by the composition in particle volume fraction, electrolyte and type of solvent. Colloidal particles can also be processed as powders which provides a higher refractive index contrast to air (nair h = 1), as compared to solvent based dispersions (e.g., nwater h = 1.33), but offers less control over the microstruc- ture. The statistically well defined structure in a liquid can be transferred to a solid film by film drying often preceded by sedimentation or centrifugation, see Fig. 6 (left column, top) (Reufer et al., 2007). Such a colloidal film has the structural properties of a frozen colloidal liquid at random close packing conditions. For identical spheres, this results in pronounced short-range correla- tions. It is however often difficult to avoid crystallization. To avoid the formation of crystallites one can employ size polydispersity, the pre-formation of aggregates or the use of non-spherical particles, but this usually leads to a reduction of structural correlations. Photonic crystals with controlled disorder can also be fabricated by com- bining spherical colloids of two different polymers, and to selectively etch one after the crystal deposition (Peng and Dinsmore, 2007). In this way, controlled defects in an otherwise periodic lattice are formed (Garćıa et al., 2009). Optimizing this fabrication process has been sub- ject of active research (Garćıa et al., 2007). Finally, densely-packed colloidal aggregates, typically of micron-sized spherical shapes, can be realized by se- lective solvent evaporation or spray-drying (Manoharan et al., 2003; Moon et al., 2004; Vogel et al., 2015; Yazhgur et al., 2021; Yi et al., 2003), see Fig. 6 (left column, bottom). These so-called “photonic balls”, which may be composed of dielectric or metallic particles and be suspended in air or in a solvent, have been used to re- alize angle-independent structural colors (Park et al., 2014), artificial (meta) materials (Dintinger et al., 2012) or micron-sized random lasers (Ta et al., 2021). The finite size of the photonic balls breaks the translational invari- ance and, for small photonic balls, this leads to additional metaball-scattering contribution. Structural correlations within the balls are likely to depend on the size of the aggregate and and the quenching rate. Crystallisation of the surface layer is often observed. 2. Thermodynamically-driven self-assembly Colloidal self-assembly proceeds via a random process that arranges prefabricated scatterer of a given size a in space. The fabrication process is stochastic and driven by 28 thermal motion or external fields such as gravity and the resulting structures are relatively simple. In contrast, bi- ology and recent DNA based nano-fabrication processes rely on well defined fabrication pathways or cascades that can be programmed which leads to beautiful and complex optical materials in nature (Prum et al., 1998; Vignolini et al., 2012). In biology, it is known that many species are able to produce optical materials that show color or whiteness with optimal morphology and structural cor- relations, short or long ranged (Burresi et al., 2014; Luke et al., 2010; Prum et al., 2009). The physical mecha- nisms that underlies the assembly of photonic structures in living organism is still not understood (Dufresne et al., 2009; Onelli et al., 2017; Prum et al., 2009; Wilts et al., 2019). Even without a complete understanding of the biological processes, it is possible to use such architec- tures as materials for biotemplating. To this end, the biological material is used as a template or cast for a syn- thetic material with a high refractive index such as TiO2 (Galusha et al., 2010). Analogous three-dimensional ar- chitectures have been produced on the tens of nanometer- scale via block-copolymers self-assembly (Stefik et al., 2015), and the interplay between order and disorder on a slightly larger-scale (few hundreds of nanometers) have been shown to be controllable via block-copolymers brush systems (Song et al., 2018), see Fig. 6 (middle column, top). Another very promising route is based on DNA- nanotechnology (He et al., 2020). DNA-origami tech- niques, invented a decade ago (Rothemund, 2006), are considered one of the breakthroughs in nanotechnology. Recently, methods for making micrometre-scale DNA- Origami objects have been developed (Zhang and Yan, 2017), see Fig. 6 (middle column, bottom). The use of DNA-origami or bioinspired assembly techniques is still in its infancy. It is however the only fabrication route that may possibly allow the design of complex, corre- lated disordered three-dimensional optical materials in the visible range owing to the nanoscale control over the fabrication process. 3. Optical and e-beam lithography Despite the rapid advances in nano-assembly, such as DNA origami, it is still difficult and often impossible to fabricate tailored disordered optical materials at will. In particular optimized structures designed in silico cannot be readily transferred into real materials yet using such approaches. Lithography is an established and powerful alternative to self-assembly. Its leading performance is unchallenged in the fabrication of two dimensional ma- terials, such as silicon, owing to the decades of optimiza- tion in the semiconductor industry. The resolution of deep UV based optical lithography is now at 10− 20 nm (Sanders, 2010). The use of a predefined photographic mask means that this is a highly parallelized method and the resolution can be reached over a large area, such as entire 30 cm silicon wavers. High resolution optical lithography however has a very high start-up cost for in- strumentation and for the fabrication of individual photo masks. E-beam lithography is a serial fabrication tool with similar resolution capacity. It is versatile and can fabricate any 2D structure but it is much slower and thus not suited for high-output volumes (Altissimo, 2010). Early attempts in the late 1990s focused on the fabrica- tion of structured photonic materials in 2D for visible and near-infrared wavelengths using lithographic patterning followed by reactive ion etching to produce long air holes in high index materials (Krauss et al., 1996; Zoorob et al., 2000). It is very challenging to generalize the use of these powerful 2D methods for the fabrication of 3D materials. Small sized three-dimensional infrared photonic crystal on a silicon wafer were reported based on stacking sev- eral layers of 2D structures, fabricated with fairly stan- dard microelectronics fabrication technology (Lin et al., 1998). In principle, this approach can also be applied to correlated disordered materials but owing to its ex- treme cost and complexity as well as limitations in size it has not been widely used. More recently, the etching of air rods has been applied to fabricate 3D hole-arrays using 3D masks (Grishina et al., 2015). This method is in an early stage of development and the evaluation of the optical performance of the materials obtained is still in progress, nonetheless, it offers potential also for the template-free, direct fabrication of correlated disordered 3D photonic materials with a very high refractive index contrast. The inherent limitations of conventional colloidal self- assembly strategies have led to the development of a class of 3D high resolution lithography tools in the late 1990s and the early 2000s known as direct laser writ- ing (DLW) (Deubel et al., 2004; Sun et al., 1999). The most popular implementation of direct laser writing is based upon the development of the two-photon micro- scope by Denk et al. (1990). Using a focused femtosecond pulsed laser two photons are absorbed simultaneously in the focal spot, but not elsewhere, owing to the highly nonlinear absorption cross section. In microscopy, the re-emission of a photon is used for imaging, in direct laser writing the absorbed energy is used to initiate a chemical reaction in the photoresist. By scanning a near infrared fs-pulsed laser beam in 3D, a polymeric structure can be written with a resolution of approximately 200nm laterally and 500nm axially. The resolution is limited by the point spread function of the microscope objective and the two photon cross section as well as the photore- sist. Recently, it was shown that the resolution can be further enhanced using a stimulated-emission-depletion (STED) microscopy inspired approach (Fischer and We- gener, 2011; Klar et al., 2014) or by controlled heat- induced shrinkage of polymeric network structures (Aeby 29 et al., 2022).. DLW has been used to fabricate poly- mer templates for a variety of optical metamaterials such as woodpile photonic crystals, quasicrystals and polariz- ers (Deubel et al., 2004; Gansel et al., 2009; Ledermann et al., 2006; Soukoulis and Wegener, 2011). It has also been instrumental for the experimental realization of 3D correlated disordered network materials, based on hy- peruniform point patterns or other types of disordered correlated photonic materials (Renner and Von Frey- mann, 2015). Despite its power and versatily, the DLW method also suffers from imperfections due to shrinkage of the polymer structure during development and defor- mations (Deubel et al., 2004; Haberko et al., 2013; Ren- ner and Von Freymann, 2015). Moreover, due to the relatively low refractive index of the polymer photore- sist (n ≃ 1.5), it is usually necessary to transfer the cast or template into another, higher index, material such as TiO2 or silicon. This can be done using single or double inversion protocols which can be parallelized (Marichy et al., 2016; Muller et al., 2013, 2017; Staude et al., 2010; Tétreault et al., 2006). Therefore, such single or dou- ble inversion of the template is, in principle, not a time limiting step in the fabrication protocol. However, the complex chemical and etching procedures needed often lead to an incomplete infiltration (and thus a lower re- fractive index) (Marichy et al., 2016; Staude et al., 2010), a general deterioration of the quality of the structure and additional surface roughness (Muller et al., 2017). F. Measuring structural correlations Structural correlations in disordered photonic media can be measured using microscopy, tomography and scat- tering. Scattering can only be employed if the mate- rials structure is translationally invariant and isotropic or aligned in a well-defined direction. This is the case for colloidal photonic liquids and glasses or randomly close packed particles or rods which are correlated on short length scale but are statistically uncorrelated (at least asymptotically) on large length scales (Garćıa et al., 2007; Reufer et al., 2007; Rojas-Ochoa et al., 2004). A challenge is the fact that the material has to be fairly transparent to the used radiation and therefore light is not a suitable probe, unless some form of refractive index matching or clearing is possible. In the latter case, confocal microscopy has also been applied success- fully (Haberko et al., 2013). Optical materials are usu- ally fairly transparent to neutrons or X-rays. Ultra Small Angle Neutron and X-Ray scattering instruments are in principle suitable for this task and available at large scale facilities (Bahadur et al., 2015) but these experiments are difficult and time-consuming and they are thus not rou- tinely carried out to measure structural correlations in complex photonic media. Small Angle Neutron Scatter- ing (SANS) has been used successfully to measure the structure factor of photonic liquids composed or rela- tively small colloids in suspension (Rojas-Ochoa et al., 2004). The direct visualization of the materials local and global structure is often more useful or, for many novel systems, even required. To this end, electron mi- croscopy is routinely applied often in tandem with fo- cused ion beam milling and cutting. More recently, X- ray imaging and X-ray tomography have been developed as non-invasive tools for the real space characterization of correlated photonic materials (Grishina et al., 2019; Wilts et al., 2018). Another promising route to study the internal structure of 3D photonic materials is de- structive tomography using ion beam milling or etching techniques in conjuction with electron or atomic force microscopy (Burresi et al., 2014; Magerle, 2000). IV. MODIFIED TRANSPORT PARAMETERS The primary effect of structural correlations is to mod- ify the light scattering and transport parameters. This section offers a survey of the theoretical predictions and experimental observations of modified transport prop- erties due to structural correlations. We first focus on colloidal systems and photonic materials, typically char- acterized by short-range correlations (i.e., negatively- correlated) [Sec. IV.A]. We discuss optical transparency and enhanced single backscattering phenomena on the basis of the theory developed in Sec. II, and survey progress on resonant and Bloch-mediated scattering. In a second part, we describe the markedly different transport properties of materials with large-scale heterogeneities (i.e., positively-correlated) [Sec. IV.B]. Transport in such systems requires a generalization of the radiative trans- fer equation and can become anomalous in presence of a fractal heterogeneity. A. Light scattering and transport in colloids and photonic materials 1. Impact of short-range correlations: first insights Let us start this section by examining the expressions derived for the scattering and transport mean free paths for assemblies of spherical particles, Eqs. (111) and (112). In deriving these expressions, we have assumed that an effective permittivity ϵeff for the system can be defined, leading to an effective wavenumber kr = k0Re[neff] and a scattering wavevector q = kr|u − u′|, where u and u′ are the scattered and incident directions. The form fac- tor is given by F (q) = k2r dσ dΩ (q) [Eq. (109)], where dσ dΩ is the differential scattering cross-section of the individual particle in the host medium evaluated at the wavenum- ber kr. The structure factor S(q) is the key quantity describing structural correlations for particulate media. Figure 7(a) shows the structure factor predicted within 30 the Percus-Yevick approximation for hard spherical par- ticles (Wertheim, 1963) at different filling or packing fractions p = (π/6)a3ρ, where a is the particle diame- ter and ρ is the particle density. Increasing the density and/or the particle diameter leads to short-range corre- lations characterized by a reduction of S at small values of q (gray-shaded area), the emergence of a peak slightly above qa = 2π (Liu et al., 2000), and oscillations with a decaying amplitude at larger values of qa. FIG. 7 Impact of structural correlations on light scattering and transport in colloids. (a) Structure factor S of a hard- sphere liquid for three different packing fractions p = π/6a3ρ, with a the particle diameter and ρ the particle number density. The gray-shaded area indicates the low scattering wavenumber range where the structure factor is strongly di- minished. (b) Angular and spectral response of the structure factor taking q = 4π/λr sin θ/2 and θ the scattering angle for p = 0.4. Exact backscattering (θ → π) is particularly pro- nounced when λr ≈ 2a. (c) Ratio of the scattering mean free paths neglecting structural correlations (ℓ0s ) and considering structural correlations (ℓs), and (d) scattering anisotropy pa- rameter g without and with structural correlations, obtained in the realistic case of particles of diameter a = 100 nm and refractive index np = 1.6 (e.g., polystyrene) in a host medium with index nh = 1.33 (e.g., water). The observed features are directly linked to the structure factor. The red-shaded area in panel (d) highlights the range where single scattering is dominantly backward, implying ℓt < ℓs. The effect of these short-range correlations on light scattering can be apprehended by rewriting the scatter- ing wavevector as q = 2kr sin θ/2 with θ the scattering angle. Insightful expressions for the scattering and trans- port mean free paths can in fact be derived from this change of variables, leading to 1 ℓs = ρ ∫ 4π dσ dΩ (θ)S(θ)dΩ, (118) and 1 ℓt = ρ ∫ 4π dσ dΩ (θ)S(θ)(1− cos θ)dΩ. (119) respectively, where Ω is the solid angle. Similarly, the scattering anisotropy parameter is given by g = ∫ 4π dσ dΩ (θ)S(θ) cos θdΩ∫ 4π dσ dΩ (θ)S(θ)dΩ . (120) The structure factor can thus be seen as a quantity that modifies the scattering diagram dσ dΩ (θ) of the individual particle due to far-field interference. In absence of corre- lations (S = 1), the scattering mean free path is simply given by ℓ0s = (ρσs) −1 with σs = ∫ 4π dσ dΩ (θ)dΩ the scat- tering cross-section of an individual particle. Figure 7(b) shows the structure factor for p = 0.4 ex- pressed as a function of a/λr with λr = λ/Re[neff] and θ. The most remarkable features here are the systematic re- duction of scattering around the forward direction θ ≈ 0 and strong increases in the backward direction θ ≈ π at specific frequencies, especially near qa = 2π, similarly to the Bragg condition in crystals. We apply Eqs. (118)-(120) to a practical situation, namely spherical polystyrene particles (np = 1.6) with diameter 100 nm dispersed in water (nh = 1.33). We use the CPA to get the effective refractive index (Soukoulis et al., 1994), resulting in neff ≈ 1.44 on the entire wave- length range considered here, although the actual choice of the effective medium theory is of little importance for such low-index contrast systems. Figures 7(c)-(d) show the variation of scattering efficiency ℓ0s/ℓs, and the scat- tering asymmetry parameter g in the limit of an uncor- related medium (asymmetric scattering is then entirely due to the particle alone) and for a strongly correlated system, p = 0.4. Two main conclusions can be drawn here. First, structural correlations lead to a reduction of the scattering efficiency, that is particularly pronounced in the low frequency range, where the wavelength is much larger than the characteristic length of the system. Thus, an incident wave propagates ballistically on longer dis- tances (on average). Second, the angular dependence up to the first peak in the structure leads to a negative scattering anisotropy parameter g, meaning that light is predominantly scattered backward, leading to ℓt < ℓs. Both these effects have been observed experimentally, as 31 reported below. We considered here particles with a fairly low-index contrast to emphasize the role of short-range structural correlations on light scattering and transport. The range of optical properties is significantly enriched when con- sidering the possibility of having spectrally sharp Mie resonances in high refractive index contrast materials or longer-range structural correlations, as will be discussed hereafter (Sec. IV.A.4 and IV.A.5). 2. Enhanced optical transparency The impact of structural correlations on light scatter- ing in colloids emerged in the 1950s when it was noticed that the light intensity scattered either by protein solu- tions (Doty and Steiner, 1952) or by collagen fibrils in the cornea stroma (Maurice, 1957) was not following the behavior expected for small scattering elements uncor- related in position. In the celebrated article by Mau- rice (1957), it was supposed that a periodic organiza- tion of the fibrils was at the origin of a surprising optical transparency. Later works shown theoretically that this transparency could be explained by short-range corre- lated disorder (Benedek, 1971; Hart and Farrell, 1969; Twersky, 1975). More recently, dense nanoemulsions, a kind of synthetic mayonnaise made from smaller than usual oil droplets with a diameter around 50 nm, have been shown to be much more transparent than more di- lute suspensions ϕ ∼ 0.1 of the same droplets (Graves and Mason, 2008). A transparency window has been observed in scattering fibrillar collagen matrices as a function of collagen concentration (Salameh et al., 2020). All these observations are explained by the strongly reduced scat- tering efficiency observed in the long-wavelength regime and shown in Fig. 7(c). The notion of transparency relies on the proportion of ballistic light after a sample and therefore depends on the ratio between the extinction mean free path ℓe, or scat- tering mean free path ℓs in absence of absorption, and the sample thickness L. Thus, materials exhibiting short- range correlated disorder unavoidably become opaque for very large thicknesses. The question of whether this con- clusion holds for stealthy hyperuniform media, for which the structure factor strictly equals 0 on a range of scat- tering wavevectors q, naturally follows. The perturbative expansion of the intensity vertex (or equivalently of the phase function) up to the second order [Eq. (106)] pre- dicts that scattering is completely suppressed for such media [Eq. (111)]. Scattering may however occur due to the higher-order terms. Taking these into account leads to the definition of a criterion for optical transparency that reads (Leseur et al., 2016) L ℓ0s ≪ krℓ 0 s , (121) derived here for point scatterers (ℓs = ℓt). This shows that stealth hyperuniformity does not completely sup- press scattering. Optical transparency can be achieved in situations in which an uncorrelated disordered medium would be opaque (L/ℓ0s ≫ 1) but only provided that the ratio ℓ0s/λr is sufficiently large. 3. Tunable light transport in photonic liquids Spherical colloids are often considered as big atoms in soft matter physics (Poon, 2004). From this viewpoint, each colloidal particle takes the place of an atom that is interacting with its peers via specific colloidal inter- actions. For colloids in suspensions these interactions are often tunable both in strength and sign, such as the well-known double layer repulsion between charged mi- crospheres suspended in salty water. Thus, depending on the volume fraction occupied by the particles and the in- teraction strength, different colloidal phases can be found such as correlated liquids, entropic glasses, jammed pack- ings, or a crystal (Pusey and Van Megen, 1986). Early experiments of light scattering by charged par- ticles, typically made of polystyrene or PMMA, were ini- tiated in the mid 1970s, notably by Brown et al. (1975), who could measure the structure factor of colloidal sus- pensions of subwavelength particles beyond the first peak by conventional light scattering. The impact of structural correlations on light transport in the multiple-scattering regime was later studied by Fraden and Maret (1990) and Saulnier et al. (1990), who reported transmission and coherent backscattering measurements of the trans- port mean free path in optically thick materials composed of resonant (wavelength-scale) particles at various pack- ing fractions, see also (Kaplan et al., 1994; Rojas-Ochoa et al., 2002; Sbalbi et al., 2022; Yazhgur et al., 2021). Both works observed an increase of the transport mean free path due to structural correlations. A further step forward was made by Rojas-Ochoa et al. (2004), where it was shown that a fine control over structural correlations via Coulomb repulsion could induce a strong wavelength dependence of the optical properties of colloidal liquids and even negative values of the scattering anisotropy pa- rameter, g < 0 (i.e., ℓt < ℓs). In such “photonic liquids”, the strong spectral variations of transport parameters make that samples of intermediate optical thicknesses and/or partly absorbing become structurally colored in reflection. Note the overall excellent agreement between experiments and theoretical predictions based on direct measurements of S(q) with small angle neutron scatter- ing (SANS) (Rojas-Ochoa et al., 2004). Initial works (Fraden and Maret, 1990; Rojas-Ochoa et al., 2004; Saulnier et al., 1990) have not considered an effective index to correct the scattering wave number q. Fortunately, the outcome of doing so does not signifi- cantly impact the results due to the low index contrast. 32 4. Resonant effects in photonic glasses Photonic glasses are solid materials composed of closed-packed dielectric spheres, with size comparable to the wavelength of light, arranged in a disordered way (Garćıa et al., 2007). This is usually achieved by intentional colloidal flocculation and subsequent deposi- tion. The mono-dispersity of the building blocks that compose them induces Mie resonances, which remain observable in the closely-packed systems (Aubry et al., 2017). The resonances are all the stronger as a higher in- dex contrast is achieved by evaporation of the host liquid. Besides, when the scattering material is solid, material stability is ensured by physical contacts between neigh- boring particles, thereby resulting in stronger short-range correlations compared to photonic liquids, and strong near-field interaction between particles. The latter im- pacts both the magnitude and frequency of the Mie res- onance of the individual sphere (Sapienza et al., 2007) and can transmit more light than expected from classi- cal scattering theory (as developed in Sec. II) (Naraghi et al., 2015). The strong short-range correlation and near-field inter- actions makes the modelling of realistic photonic glasses very challenging. Recent works (Aubry et al., 2017; Schertel et al., 2019b) have argued that the effect of the near-field coupling on transport in photonic glasses could be captured by defining an effective wave number kr with an index obtained from the energy-density coherent potential approximation (ECPA) (Busch and Soukoulis, 1995). This appears in contradiction with our rigorous derivation of Eqs. (111) and (112), which required ne- glecting near-field interaction between particles. Besides, it is surprising that an approach based on the evaluation of the energy density would correctly predict the aver- age field phase velocity. The most advanced formalism to date to describe scattering and transport by dense, particulate media possibly with high-index materials is the quasicrystalline approximation (QCA), exploited re- cently by Wang and Zhao (2018a) to study the interplay between Mie resonances and structural correlations. Numerical simulations can be useful to validate theo- retical models and provide physical insight. For example, the strong-contrast formulas derived by Torquato and Kim (2021) for two-phase composites have been tested with FDTD simulations in the case of 2D and 3D dense packings of spheres with hard-sphere (equilibrium) and stealthy hyperuniform correlated disorder, showing in passing the existence of a transparency window up to a finite wave number in the latter. The complexity of the relation between structural correlations and light trans- port was evidenced in a recent work by Pattelli et al. (2018), where a graphics processing unit (GPU) imple- mentation of the T-matrix method (Egel et al., 2017) was used to investigate scattering by large assemblies of par- ticles on a wide range of parameters. Simulations reveal that, given the wavelength and the particles size and re- fractive index, the shortest transport mean free path is obtained at intermediate degrees of correlations and par- ticle densities. Although much remains to be understood, photonic glasses and all resonant dense scattering media have demonstrated their great versatility and efficiency to har- ness light scattering and transport, with interesting ap- plications in, for instance, structural colors and random lasing, as will be discussed in Sec. VI. 5. Modified diffusion in imperfect photonic crystals An extreme case of correlated disordered media is that of a disordered photonic crystals, in which long range order is established by the almost-periodic struc- ture and scattering can be induced by imperfections, defects or (intentional) contamination with additional scattering elements. In a crystal, light propagation is dictated by the photonic band diagram which maps the frequency-wavevector relation of propagating Bloch modes (Joannopoulos et al., 2011). Perfectly periodic structures are typically characterized by strong variations of the group velocity and the formation of partial (or even complete) photonic gaps corresponding to a lack of prop- agating states. The scattering cross-section of a defect typically increases with the reduction of the group veloc- ity and light will scatter only where propagating states exist, therefore very anisotropically. Multiple scattering and transport of light are expected to be strongly affected, while a more quantitative predic- tion requires a precise modelling of the kind of scattering and the crystal topology. Pioneering experiments on co- herent backscattering (Huang et al., 2001; Koenderink et al., 2000) and diffuse light transport (Astratov et al., 1995; Vlasov et al., 1999) in photonic crystals searched for signatures of Bloch-mode mediated scattering but have merely shown standard light diffusion (Aeby et al., 2021; Koenderink et al., 2005; Rengarajan et al., 2005). Single light scattering in a disordered photonic crystal have been measured, with clear modification of the scattering mean free path around the bandgap (Garćıa et al., 2009), re- flection studies have shown anisotropic scattering (Haines et al., 2012), while dynamical studies have shown ex- ceptionally reduced diffusion constants (Toninelli et al., 2008). Instead of relying on natural imperfections in otherwise ordered photonic crystal, correlated disordered media can be made by creating lattice vacancies in photonic crys- tals (Garćıa et al., 2011). In these structures transport and scattering mean free path and the diffusion constant have been measured to present strong dispersion (Garćıa et al., 2011). The transition from order to disorder in the structure and its impact on the transport parameters is still an active field of research (Schöps et al., 2018), which 33 has also motivated the development of hyperuniform ma- terials, where such a transition can be driven by a single parameter, as will be discussed in Sec. V. B. Anomalous transport in media with large-scale heterogeneity We have been concerned so far with systems for which the distribution pN for the number of scatterers in a win- dow of volume V ≫ 1/ρ has a small variance, thereby making the system appear quite homogeneous on the scale of tens or hundreds of scatterers. Here, we will be concerned with disordered systems exhibiting large- scale heterogeneities leading to a large variance, also known as positively-correlated systems, as described in Sec. III. Such systems are ubiquitous in nature, a well- known example being cloudy atmospheres (Marshak and Davis, 2005). The density of droplets in suspension in clouds can indeed fluctuate over orders of magni- tude. As illustrated in the right panel of Fig. 4, one may find very sparse as well as denser regions, imply- ing a strongly fluctuating scattering efficiency. Research on the topic has experienced numerous developments, most notably in the framework of transport theory in so- called non-Markovian stochastic mixtures, also known as non-classical transport theory (Pomraning, 1991). For a recent and thorough review of the literature on non- classical transport, we refer the reader to d’Eon (2022). As we shall now see, despite the absence of coherent inter- ference effects between neighboring scatterers, such long- range correlations have a dramatic impact on transport. 1. Radiative transfer with non-exponential extinction The first element to describe radiative transfer is ex- tinction. Equations (11) and (12) in Sec. II impose that the coherent intensity | ⟨E⟩ |2 should decay exponentially on an average distance given by the extinction mean free path ℓe. In strongly heterogeneous media, however, one may anticipate that the decay will be slower than exponential. An intuitive explanation is that spatially- extended non-scattering or weakly scattering regions pro- mote trajectories much longer than the average decay length (i.e., the extinction mean free path). Discus- sions on non-exponential extinction in strongly hetero- geneous media date back to the mid-twentieth century with studies on neutron propagation in pebble bed reac- tors (Behrens, 1949; Randall, 1962) and light absorption in suspensions of photosynthesizing cells (Duyens, 1956; Rabinowitch, 1951). The topic gained further attention with later studies on radiative transfer in cloudy atmo- spheres (Davis and Marshak, 2004; Natta and Panagia, 1984; Városi and Dwek, 1999) and, more recently, in the framework of computer graphics (Bitterli et al., 2018; FIG. 8 Impact of large-scale heterogeneity on transport in multiple-scattering media. (a) Sketch of a transport process in a statistically homogeneous medium (gray shaded area). Within radiative transfer, transport can be described as a random walk process with exponentially-decaying step length distribution. For thick media, transport is well described by the diffusion equation. (b) Sketch of transport in a scattering medium containing large non-scattering regions (white disks). Transport is driven by long steps, making the step length distribution no longer exponential. For certain systems with fractal heterogeneity, such as Lévy glasses (Bertolotti et al., 2010a), transport can experience a transient superdiffusive behavior. Jarabo et al., 2018). A common approach to describe the non-exponential decay of the coherent intensity, proposed by several au- thors about two decades ago (Borovoi, 2002; Kostinski, 2001, 2002; Marshak et al., 1998), consists in describing the heterogeneous medium as local “patches” or clusters of particles exhibiting a varying average extinction rate (or equivalently average particle densities). To describe this heuristic model, let us define a position-dependent particle density ρ(r) = ⟨N(r)⟩/V , with ⟨N⟩ the average number of scatterers in volume V . We consider a system that is dilute at all points of space (ρ(r)λ3 ≫ 1) such that radiative transfer applies. The key point of the approach is to assume that the distribution of number N of parti- cles in the volume V , and consequently the distribution of number of extinction events, follows a Poisson distribu- tion, pN |⟨N(r)⟩ = ⟨N⟩N exp [−⟨N⟩] /N !. The patchiness leads to variations of ⟨N(r)⟩ via a distribution p⟨N⟩. The distribution of extinction counts in a volume V should then be pN = ∫ ∞ 0 pN |⟨N(r)⟩)p⟨N⟩d⟨N⟩ = ∫ ∞ 0 ⟨N⟩N exp [−⟨N⟩] N ! p⟨N⟩d⟨N⟩, (122) The relation with the classical extinction (Beer-Lambert) law is established by noting that the probability to cross the volume with no extinction event over a depth z is 34 given by p0, and invoking the law of large numbers with ⟨N⟩ = z/ℓe. Taking p⟨N⟩ = δ(⟨N⟩ − β) and the ballistic transmission Tb = | ⟨E⟩ /E0|2 of a planewave along the z-direction through a medium leads to Tb(z) ≡ p0(z) = exp[−⟨ρ⟩σez], (123) where we have set β = ⟨ρ⟩σez and σe is the extinction cross-section. By contrast, the use of Γ or fractional Pois- son distributions lead to asymptotic power-law decays with varying exponents m (Casasanta and Garra, 2018; Kostinski, 2001) Tb(z) ∼ (1 + β)−m. (124) One key aspect of course is the determination of an ac- tual function in realistic systems. Important efforts have notably been dedicated to the determination of particle density distribution in clouds (Kostinski and Jameson, 2000). Slower-than-exponential decays of the coherent intensity have also been observed in photosynthetic cul- tures (Knyazikhin et al., 1998). Very importantly, the radiative transfer equation [Eq. (38)] has been generalized to account for arbitrary non-exponential extinction. Defining a probability den- sity function fs of the random step length s as fs(s) ≡ Tb(s)/ ∫∞ 0 Tb(s)ds, one reaches a generalized (scalar) ra- diative transfer equation (Larsen and Vasques, 2011)[ ∂ ∂s + u ·∇r +Σe(s) ] I(r,u, s) = δ(s)γ ∫ p(u,u′)Σe(s ′)I(r,u′, s′)ds′du′, (125) where γ = ℓe/ℓs is the single-scattering albedo (the prob- ability to be scattered upon an extinction event) and I(r,u, s) now depends on the step length s via Σe(s) = fs(s) 1− ∫ s 0 fs(s′)ds′ . (126) Equation (38) is recovered by taking fs(s) = exp[−s/ℓe]/ℓe. A few remarks are in order. First, the distribution fs(s) should have a finite mean as to al- low the definition of a mean free path ℓe. Second, one of the key features of transport with non-exponential step length distributions is the fact that it is a non- Markovian process (i.e., implying memory in the con- struction of individual steps), contrary to the classical Beer-Lambert law which is a Markovian (memoryless) process (exp[x+y] = exp[x] exp[y]). Third, Eq. (125) de- scribes transport in the volume of a medium. Care should be taken on its applicability to bounded domains, since an incorrect treatment of the initial steps (light enter- ing the medium) can result in a breaking of reciprocity. An extension of the formalism to bounded domains has been proposed by d’Eon (2018). Finally, – and this as- pect has not been significantly emphasized previously – the formalism assumes an “annealed” disorder, mean- ing that the medium is randomized after each scattering event. As we will see below, correlations between suc- cessive scattering events due to a “quenched” disordered potential can have a significant impact on transport. Along similar lines, radiation transport can be effi- ciently modelled numerically in arbitrary geometries via random-walk (Monte Carlo) simulations either in hetero- geneous media with spatially-varying scattering param- eters (Audic and Frisch, 1993; Boissé, 1990; Glazov and Titov, 1977) or in statistically homogeneous media using arbitrary step length distributions, including those with diverging second moment (Nolan, 2003). The latter are known as Lévy walks (Zaburdaev et al., 2015) and have been proposed as a tool to describe radiation transport in clouds (Davis and Marshak, 1997), leading to enhanced ballistic transmission and transmitted intensity fluctua- tions. 2. From normal to super-diffusion In classical radiative transfer, the average (incoherent) intensity is expected to follow the laws of diffusion af- ter many scattering events, Eq. (45). Physically, diffu- sion is related to the Brownian motion of many indepen- dent moving elements (i.e., random walkers). As long as the second moment of the step length distribution fs(s) is finite, the central limit theorem shows that the aver- age step length converges towards a normal distribution, eventually leading to a diffusive process, characterized by a mean square displacement ⟨r2(t)⟩ ∼ 2dDt. Under these circumstances, the presence of large heterogeneities does not prevent the diffusion limit but leads to a modified diffusion constant. From a simple isotropic random walk consideration (Ben-Avraham and Havlin, 2000), it is pos- sible to show that the diffusion constant for a step length distribution fs(s) is given by (Svensson et al., 2013) D = v 2d E[s2] E[s] , (127) with E[X] is the expectation value of the random vari- ableX. This expression is apparently not well known and yet very interesting. It shows that the fluctuations in the step length are as important as the mean step length. In practice, any correlated system exhibiting slower-than- exponential decay will experience an increased diffusion constant. The known expression D = vℓt d is only recov- ered for an exponentially-decaying function of fs(s) and using the similarity relation ℓt = ℓs/(1− g). A fundamentally different behavior is observed when heterogeneities are so strong that they make the second moment of fs(s) diverge. This is the case for power-law decays fs(s) ∼ s−(α+1) with α < 2, defining the so-called 35 Lévy walks. By virtue of the generalized central limit theorem (Gnedenko and Kolmogorov, 1954), one shows that the average step length should follow an α-stable Lévy distribution, which is identically heavy-tailed. Lévy walks lead to superdiffusive transport, characterized by a mean-square displacement growing faster than linear with time (Zaburdaev et al., 2015) ⟨r2(t)⟩ ∼ tγ , with 1 < γ ≤ 2. (128) Lévy statistics and anomalous diffusion are widespread in science, from the random displacement of molecules in flows (Solomon et al., 1993) to the foraging strategy of animals (Bartumeus et al., 2005). While early studies had already evidenced modified path length distributions of light in fractal aggregates of particles (Dogariu et al., 1992, 1996; Ishii et al., 1998), the first experiments aim- ing to control the anomalous diffusion of light in disor- dered systems have been initiated by Barthelemy et al. (2008). So-called Lévy glasses are fabricated by incorpo- rating in a disordered medium containing small scatter- ing particles, a set of transparent, non-scattering spheres with sizes ranging over orders of magnitude acting as spacers (see the last panel in Fig. 5). By controlling the distribution of sphere diameters and assuming sin- gle scattering in the interstices between the spheres and annealed disorder, one can control the step length dis- tribution p(s) in the medium (Bertolotti et al., 2010a). Latest time-resolved experiments on Lévy glasses showed indeed a (transient) superdiffusive light transport (Savo et al., 2014). Lévy statistics in light transport has also been observed in hot atomic clouds (Araújo et al., 2021; Baudouin et al., 2014; Mercadier et al., 2009), as a result from Doppler broadening (Baudouin et al., 2014; Pereira et al., 2004), not from structural correlations. An important aspect of transport in Lévy glasses is the fact that the disorder is frozen or quenched. Classi- cal transport models assume annealed disorder, in the sense that there is no correlation between successive scattering events – a photon “sees” a new structure af- ter each scattering event. In real samples, successive steps are however not independent. There exists cor- relations due to the large empty regions. As shown by theory and experiments on scattering powders con- taining large monodisperse voids (Svensson et al., 2014), quenched disorder leads to an effective reduction of the diffusion constant compared to annealed disorder. The impact of quenched disorder in Lévy-like systems has been subject to several numerical and theoretical investi- gations (Barthelemy et al., 2010; Beenakker et al., 2009; Buonsante et al., 2011; Burioni et al., 2010, 2012, 2014; Groth et al., 2012), eventually showing that the actual observation of superdiffusive transport in finite-size sys- tem (hence with truncated step-length distribution) re- quires a proper finite-size scaling analysis and packing strategy (Burioni et al., 2014). V. MESOSCOPIC AND NEAR-FIELD EFFECTS The interplay of order and disorder in photonic struc- tures not only impacts light transport but also promotes strong coherent effects, resulting in the emergence of sometimes unexpected phenomena for disordered sys- tems. This section is devoted to the main mesoscopic and near-field phenomena that have attracted attention in the past decades, namely the opening of photonic gaps in dis- ordered systems [Sec. V.A], transitions between various mesoscopic transport regimes [Sec. V.B], non-universal speckle correlations [Sec. V.C] and large local density of states fluctuations [Sec. V.D]. We attempt to provide a clear picture of the current understanding in the field. A. Photonic gaps in disordered media Photonic gaps are one of the most striking manifesta- tions of structural parameters on optical transport. Simi- larly to electronic gaps in semiconductors, a photonic gap corresponds to a spectral range in which no propagating modes exist. The concept of photonic gap is known in op- tics since the early works on (one-dimensional) thin-film optical stacks (Yeh et al., 1988), emerging as a conse- quence of the periodic modulation of the refractive index on the wavelength scale. The idea was generalized in the late 1980s to two and three-dimensional periodic struc- tures (John, 1987; Yablonovitch, 1987) and has been at the heart of research in optics and photonics for about two decades. The interest in photonic gaps largely comes from the possibility to engineer defects states with high- quality factors and wavelength-scale confinement, open- ing unprecedented opportunities to control spontaneous light emission and light propagation for applications in all-optical integrated circuits (Joannopoulos et al., 2011). Probably because of the convenience of Bloch’s theo- rem and the development of numerical methods exploit- ing periodicity to solve Maxwell’s equations, it is widely believed in the optics and photonics community that the opening of photonic gaps requires the refractive index variation to be periodic in space (i.e., the structure to exhibit long-range periodic correlations). Early works investigating the impact of structural imperfections on optical properties however realized, by drawing a par- allel with semiconductor physics were similar questions have been tackled (Phillips, 1971; Thorpe, 1973; Weaire, 1971), that certain gaps could persist even in absence of periodicity (Chan et al., 1998; Jin et al., 2001), thanks to local (Mie or short-range correlated) resonances (Li- dorikis et al., 2000). Later reports on photonic gaps in 3D disordered structures exhibiting short-range correla- tions (Edagawa et al., 2008; Imagawa et al., 2010; Liew et al., 2011) and the proposition that hyperuniformity was a requirement for photonic gaps (Florescu et al., 2009) greatly stimulated the community to unveil the re- 36 lation between local morphology and structure, and the opening of spectrally wide gaps (Froufe-Pérez et al., 2016; Klatt et al., 2019; Ricouvier et al., 2019; Sellers et al., 2017). 1. Definition and identification of photonic gaps in disordered media The notion of photonic gap is intimately linked with that of density of states (DOS). Formally, the DOS de- scribes the spectral density of eigenmodes in the medium (i.e., the solutions of the source-free Maxwell’s equations) around frequency ω. For instance, the DOS of a closed and non-absorbing system with volume V is simply ρ(ω) = 1 V ∑ m δ(ω − ωm), (129) where ωm is the frequency, that is the eigenvalue, as- sociated to resonant mode m. For non-dissipative sys- tems, this frequency is real. In this framework, a pho- tonic gap thus corresponds to a spectral region wherein ρ(ω) = 0, and can therefore easily be found from an eigenmode analysis. In the case of disordered media, a classical strategy, illustrated in Fig. 9a in the case of parallel dielectric cylinders in TM polarization, is to em- ploy the PWE method (Ho et al., 1990; Johnson and Joannopoulos, 2001), briefly presented in Sec. III.D.2, on a large periodic supercell. The system being conser- vative, a photonic gap is easily recognized as a spectral region containing no propagating modes [Fig. 9b]. A pho- tonic gap can also be identified by time-domain simula- tions in real space (FDTD) using the order-N spectral method (Chan et al., 1995). Special attention should be given to whether the gap persists in the thermody- namic limit (i.e., when the system size increases) – an as- pect that has been overlooked until recently (Klatt et al., 2022). This approach can only be numerical as the DOS is not directly accessible experimentally and real systems are always of finite size and open. The latter makes that the DOS can in fact never be strictly equal to zero. A second approach, which may now be performed ex- perimentally (Aubry et al., 2020; Leistikow et al., 2011; Lodahl et al., 2004; Sapienza et al., 2011), consists in performing a finite-size scaling analysis of the emitted power (or spontaneous emission rate) of a quantum emit- ter embedded in a finite-size systems, see Fig. 9c for an illustration. The power Pem emitted by a dipole source with moment p = pu, rigorously derived from Maxwell’s equations (Carminati et al., 2015; Novotny and Hecht, 2012), reads Pem = πω2 4ϵ0 |p|2ρe(r,u, ω), (130) where ρe is the projected local density of states (LDOS) FIG. 9 Signatures of photonic gaps in short-range correlated ensembles of dielectric rods in TM polarization. The rods have a permittivity ϵ = 11.6, a radius r = 0.189a, are placed in air and are packed by RSA at a surface filling fraction f = 11.2% (number density n = a−2). (a) A disordered en- semble of rods is generated in a square region with periodic boundary conditions. (b) Photonic band structure with a gap that correspond to an absence of eigenmodes in a finite spec- tral range a/λ. Numerically, the eigenmodes can be computed using the planewave expansion method with the supercell ap- proach. The band structure was calculated here for a supercell containing N = 100 rods. (c) A photonic gap can be identi- fied by monitoring the spontaneous emission rate of a dipole source in the center of the system for varying system sizes. The emitter here is always placed in air. (d) Finite-size scal- ing of the spontaneous emission rate (or LDOS) for systems containing 25, 50 and 100 rods. ⟨ρe⟩ is the projected LDOS averaged over disorder configurations and ρ0 is the projected LDOS in air. A gap leads to a strong damping of spontaneous emission with system size. (in units of s.m−3) defined as ρe(r,u, ω) = 2ω πc2 Im [u ·G(r, r, ω)u] . (131) G(r, r′, ω) is the total Green tensor in the structured en- vironment. Decomposing it as a sum of the Green tensor in the homogeneous background and the Green tensor due to the fluctuating permittivity, one readily under- stands that the suppression (resp., enhancement) of the LDOS results from destructive (resp., constructive) inter- ference at the dipole position between the field radiated in the homogeneous background and the field scattered by the heterogeneities. Equation (131) is general, yet it does not explicitly de- pend on the actual states of the system. To gain some 37 physical insight, it is possible to express G in terms of the eigenmodes of the system, see Appendix D for more details. The eigenmodes of open (non-Hermitian) sys- tems, also known as quasinormal modes (QNMs) (Ching et al., 1998; Lalanne et al., 2018), are described by com- plex frequencies ω̃m = ωm − iγm/2 and normalized fields Ẽm(r), where the non-zero imaginary part stems from leakage. Physically, the existence of a photonic gap trans- lates into the absence of resonant modes in the volume of the medium: the resonant modes may only be confined to the boundaries of the medium (i.e., on a length scale of the order of the extinction (scattering) mean free path). Thus, their excitation by a source deep inside the sys- tem, the LDOS and the resulting emitted power are all expected to tend towards zero with increasing size in the photonic gap, while it should remain quite unchanged in presence of propagating modes [Fig. 9d]. Despite the conceptual simplicity of this second strat- egy, care should be taken with the interpretation of emit- ted power (or spontaneous emission decay rate) spectra measurements. Indeed, as we will see in Section V.D, LDOS fluctuations can be enormous in complex media, depending considerably on the local environment around the emitter position as well as on the emitter orientation. The interplay between near-field interaction and the far- field radiation has been studied by FDTD simulations in finite-size photonic crystals by Mavidis et al. (2020). Ad- ditionally, in disordered systems, only average quantities acquired over a large set of disorder realizations are sta- tistically relevant. This raises a second difficulty related to the fact that quantum emitters like quantum dots have a finite size, thereby inducing a local spatial correlation, and they are usually not distributed uniformly in all ma- terials composing the complex medium (e.g., a semicon- ductor and air). Thus, the configurational average of the LDOS for a real emitter will often not be strictly equal to the average LDOS, as may be computed numerically for instance. Although it seems reasonable to assume that the average LDOS should converge towards the DOS in the limit of infinite system size, it appears that the link between photon emission statistics and the existence of photonic gaps has only been established phenomenologi- cally to date. 2. Competing viewpoints on the origin of photonic gaps Discussions on the origin on the photonic gaps in dis- ordered media started to emerge in the late 1990s, in- spired by earlier works on electronic gaps in (periodic and amorphous) semiconductors. Two main mechanisms have been identified. The first, generally accepted, mechanism is that pho- tonic gaps build up from interference between counter- propagating waves on a periodic lattice, thereby placing long-range structural correlations at the core of the pic- ture. It is the photonic analog of the nearly free electron model in solid-state physics (Kittel, 1976). Formally, the Bloch modes – the eigenmodes of periodic systems – re- sult from a coupling between forward and backward prop- agating planewaves on the periodic lattice (Yeh et al., 1988). In spectral gaps, they form stationary patterns that do not carry energy (in the lossless case) due to a backscattering phenomenon with precise phase-matching condition. Their propagation constant is complex, lead- ing to a damping of an incident wave in the specular direction, without scattering. Photonic (band) gaps in 1D media, or in one particular direction in a 2D or 3D photonic crystal (Spry and Kosan, 1986), can exist even for vanishingly small refractive index contrasts. Om- nidirectional gaps in higher dimensions requires higher contrasts and a finely-optimized structure and morphol- ogy (Joannopoulos et al., 2011). Because such spectral gaps are created by long-range periodicity, they are ex- pected to be very sensitive to lattice deformations. The second proposed mechanism is that photonic gaps are formed by coupled resonances between short-range correlated neighboring scatterers. It is the photonic analog of the tight-binding model in solid-state physics, developed in particular to explain the origin of the electronic density of states of amorphous semiconduc- tors (Weaire, 1971). Intuitively, similarly to the level repulsion observed in a pair of coupled resonances, inter- action between nearest neighbors in ensembles of iden- tical resonators may “push” the states of the coupled system away from the resonant frequency. This is typi- cally obtained with high refractive index Mie scatterers at moderate densities in low refractive index media. In this picture, the interaction between distant resonators, and thus long-range structural correlations, are irrele- vant. As a consequence, one expects photonic gaps to exist in both periodic and disordered systems, provided that the resonances of the individual scatterers and the coupling coefficient between scatterers remain nearly con- stant throughout the entire structure. Care should how- ever be taken regarding the analogy with the electronic tight-binding model due to the polarized nature of light waves (Monsarrat et al., 2022). A different, complementary viewpoint on this second mechanism is provided by considering the effective ma- terial parameters of assemblies of resonant objects (La- gendijk and van Tiggelen, 1996). In particular, the effec- tive permittivity ϵeff is predicted to exhibit a polaritonic response that possibly becomes negative in its real part for sufficiently strong resonances and high density. Hav- ing Re [ϵeff] < 0 implies that Im [neff] > Re [neff], corre- sponding to a coherent field propagating in the effective medium that is overdamped, as in a metal. Stealthy hyperuniform structures (be they ordered or disordered) suppress scattering in the long-wavelength regime (up to the second order in the expansion of the intensity ver- tex). This implies that Im [ϵeff] ≃ 0. Thus, one arrives 38 to a situation where propagation is damped by coupled resonances and scattering is suppressed by structural cor- relations. This describes a system behaving as a homo- geneous medium with no propagating states, that is a system exhibiting a photonic gap. 3. Reports of photonic gaps in the litterature The formation of photonic gaps very much depends on the dimensionality of the system and on the light po- larization, for both periodic and disordered media. For instance, early works using numerical simulations with scalar waves have suggested that 3D face-centered cubic lattices of dielectric spheres could exhibit an omnidirec- tional gap, but vector wave calculations disproved this prediction (Ho et al., 1990). Figure 10 shows various ex- amples of disordered photonic structures exhibiting pho- tonic gaps. It was suggested and demonstrated numerically al- ready about two decades ago that photonic gaps in 2D ensembles of dielectric (e.g., silicon) rods in TM polariza- tion (electric field normal to the propagation plane) are created by the strong electric dipole resonance of indi- vidual rods (Jin et al., 2001). Those gaps were actually observed previously in an experiment on light localiza- tion (Dalichaouch et al., 1991) and interpreted as “ves- tiges” of the photonic band diagram of the periodic sys- tem. The structures do not need to be hyperuniform to exhibit gaps, but require a reasonable amount of short- range correlations. It is interesting to note that both the first and second gaps (in periodic arrays) are actu- ally due to the same electric dipole resonance, while the intermediate conduction band is associated to the mag- netic dipole resonance (Vynck et al., 2009). In TE po- larization (electric field in the propagation plane), a sim- ilar resonant behavior leading to a gap was pointed out by O’Brien and Pendry (2002) for very high index mate- rials, but the gap closes for typical dielectric materials in the optical regime. 2D inverted structures made of circular air holes in dielectric exhibit photonic gaps that, by comparison, are much more sensitive to lattice deformations (Yang et al., 2010b), suggesting that periodicity, at least on a few periods, is required. It was shown however that connected networks made of thin dielectric walls on a stealthy hyperuniform pattern are favorable to ex- hibit a photonic gap in TE polarization (Florescu et al., 2009). First reports in non-periodic arrays were made on quasiperiodic structures (Chan et al., 1998). This re- sult is more unexpected than for the direct structures since one cannot define a unique scattering element in this case. Nevertheless, short-range correlations tend to homogenize locally the size distribution and shape of air pores, which, surrounded by dielectric walls in TE polarization, could be seen as nearly identical resonant scatterers. Though short-range correlations appeared being sufficient to open a photonic gap (Froufe-Pérez et al., 2016), a recent numerical investigation by Klatt et al. (2022) showed that the apparent gap of many non- stealthy-hyperuniform structures actually closes at suffi- ciently large system sizes. This supports the conjecture that three attributes – hyperuniformity, high degree of stealthiness (χ-parameter) and bounded holes – are nec- essary for a photonic gap to exist in the thermodynamic limit. A greater challenge is to form photonic gaps in 3D disordered media. Studies nowadays tend to agree that the best solution for the purpose are 3D connected net- works, basically consisting of air pores of nearly identical size surrounded by an array of dielectric rods (Edagawa et al., 2008; Imagawa et al., 2010; Liew et al., 2011; Yin et al., 2012). Recently, Sellers et al. (2017) put forward the idea of “local self-uniformity” to explain the forma- tion of wide gaps. Such structures strongly ressemble foams (Klatt et al., 2019; Ricouvier et al., 2019), which suggests the possibility to fabricate them with bottom- up techniques (Bergman et al., 2022; Maimouni et al., 2020). Important efforts are underway to demonstrate experimentally 3D photonic gaps in the optical regime. First results on samples realized by direct laser writing and double inversion to increase the refractive index con- trast indicate a depletion of transmission (Muller et al., 2017). This feature could recently be pushed down close to the telecom wavelengths at 1.5 µm by heat-induced shrinkage of the network polymer template prior to sili- con coating (Aeby et al., 2022). FIG. 10 . Photonic structures lacking long-range order that were shown to exhibit large photonic gaps. (a) 2D stealthy hy- peruniform structure exhibiting a gap for both TM and TE polarizations. The TM gap is due to the resonances of the dielectric rods and the TE gap to the air pores surrounded by dielectric holes. Adapted with permission from (Florescu et al., 2013). (b) 3D amorphous diamond structure exhibit- ing an omnidirectional photonic gap. The structure consists in a network of dielectric rods forming air pores of compa- rable sizes. Adapted with permission from (Edagawa et al., 2008). (c) First experimental realizations of 3D disordered structures potentially exhibiting a photonic gap at optical fre- quencies. The silicon photonic medium was realized by direct laser writing followed by a double inversion process. Adapted with permission from (Muller et al., 2013). 39 B. Mesoscopic transport and light localization Mesoscopic transport in disordered systems refers to a regime wherein interferences between multiply-scattered waves lead to significant transport parameter deviations compared to classical approaches such as radiative trans- fer. Coherent effects, at the mesoscopic length scales, of- ten lead to statistical distributions that are much broader and more complex than those expected from thermody- namic considerations. Signatures of mesoscopic effects, such as large sample-to-sample transmittance fluctua- tions, non-self-averaging transport parameters or long- range speckle intensity correlations, may still be visible on macroscopic scales provided that the signal has a suf- ficiently long coherence length compared to the char- acteristic lengths of the system. If many concepts in mesoscopic physics have been developed in the context of electronic transport (Akkermans and Montambaux, 2007; Altshuler et al., 2012; Mello et al., 2004; Sheng, 2006), research on classical waves brought a great deal of new ideas and challenges to the topic (Rotter and Gigan, 2017), stimulated by the unique possibility to engineer the scattering materials at the subwavelength scale. One of the most fascinating phenomena in meso- scopic physics of classical waves is the Anderson local- ization (Anderson, 1958), see (Lagendijk et al., 2009) for an historical overview of the topic and (Abrahams, 2010) for more technical details. The phenomenon takes its roots in the so-called weak localization effect, which de- scribes a small reduction of the diffusion constant (com- pared to that predicted from radiative transfer) due to interference between counter-propagating waves. This ef- fect requires reciprocity to hold (van Tiggelen and May- nard, 1998), which is generally the case in non-magnetic optical materials. Strong (Anderson) localization is ob- tained by a progressive renormalization of the diffusion constant that eventually leads to a complete halt of trans- port, as described by the self-consistent diagrammatic theory due to Vollhardt and Wölfle (1982, 1980). In open finite-size systems, the localized regime is char- acterized by exponentially-decaying transmittance (van Tiggelen et al., 2000), anomalous time-dependent re- sponse (Skipetrov and van Tiggelen, 2006), large trans- mitted speckle intensity fluctuations (Chabanov et al., 2000) and multifractality of the field (Mirlin et al., 2006). A transition between extended and localized regimes is expected in three-dimensional (3D) systems when the scattering mean free path becomes comparable with the effective wavelength in the medium, krℓs ≈ 1, also known as the Ioffe-Regel criterion (Ioffe and Regel, 1960). Experiments on high-index semiconductor powders and photonic glasses, which offer amongst the smallest ℓs in optics, have failed to provide evidence of light local- ization (Scheffold et al., 1999; Scheffold and Wiersma, 2013; Skipetrov and Page, 2016; Sperling et al., 2013; Wiersma et al., 1997), contrary to studies on ultrasounds in elastic networks (Hu et al., 2008) and matter waves in optical potentials (Jendrzejewski et al., 2012; Kondov et al., 2011). It turned out that the key role of polar- ization for electromagnetic waves and near-field effects had been largely underestimated (Bellando et al., 2014; Cobus et al., 2022; Naraghi et al., 2015; Skipetrov and Sokolov, 2014), thereby placing a finer engineering of the local morphology – and of structural correlations – at the heart of the problem. In the literature, the challenge of reaching a local- ized regime in 3D in optics appears closely related to that of creating a photonic gap. In a founding work, John (1987) proposed that a slight disorder in a peri- odic medium exhibiting a photonic gap would promote Anderson localization near the gap edge, where some (but not all) propagation directions are inhibited. An- derson localization occurs in the band and differs in that sense from classical light confinement, where defect (cav- ity) modes – or bound states – are formed in the gap. This distinction has remained somewhat fuzzy in the lit- erature in optics. Localized modes have been observed via numerical simulations in randomly-perturbed peri- odic inverse opals (Conti and Fratalocchi, 2008), where it was found that the strongest light localization was obtained at an optimal degree of disorder [Fig. 1(d)], as well as in amorphous diamond structures (Imagawa et al., 2010) [Fig. 1(e)], but their precise nature is un- clear. The transition between extended and Anderson- localized regimes (outside the photonic gap) has been evidenced only recently in a numerical study on dis- ordered hyperuniform structures thanks to a statistical analysis based on the self-consistent theory of localiza- tion (Haberko et al., 2020; Scheffold et al., 2022). Al- though the effect of the kind of structural correlation on mesoscopic transport remains to be clarified, disorder engineering has clearly given a new hope for the experi- mental observation of 3D Anderson localization of light. Light localization in two-dimensional (2D) disordered systems has experienced much less difficulties in com- parison. Theoretical arguments developed for electronic transport (Abrahams et al., 1979) let us expect that all waves be localized on some length scale ξ in two di- mensions independently of the scattering strength of the medium. Despite the absence of a “true” transition, 2D systems have been very appealing because they can be fabricated, characterized (structurally and optically) and modelled much more easily than their 3D counterpart. The first report of localization of classical waves date back to Dalichaouch et al. (1991) with a study of mi- crowave propagation in high-index dielectric cylinders in TM polarization, where a link with photonic gaps was already made. The first experimental demonstration of Anderson localization in the optical regime was obtained by Schwartz et al. (2007) in photonic lattices consisting of evanescently coupled parallel waveguides wherein local- ization occurs in the transverse direction (De Raedt et al., 40 1989). In this configuration, the electromagnetic problem is mapped onto the time-dependent Schrödinger equation where the propagation direction plays the role of time, enabling the exploration of many interesting problems in condensed matter physics (Rechtsman et al., 2013; Segev et al., 2013; Weimann et al., 2017). 2D Anderson localization has later been reported for in-plane prop- agation of near-infrared light in suspended high-index dielectric membranes perforated by disordered patterns of holes (Riboli et al., 2011). Quantum dots incorpo- rated in the membrane are excited locally by a near-field probe and their photoluminescence is collected by the same probe at the same position. A post-treatment al- lows recovering spatial and spectral information on the resonant modes of the system (Riboli et al., 2014). Structural correlations have not been considered specifically in these early works, probably because they were not necessary to observe localized modes in 2D. Nevertheless, they can impact mesoscopic transport in mainly two ways: First, by creating a photonic gap in the vicinity of which localized modes appear, as shown ex- perimentally in photonic lattices with short-range corre- lations (Rechtsman et al., 2011) and randomly-perturbed periodic hole arrays in dielectric membranes (Garcia et al., 2012). The latter study shows that the stronger confinement in periodic systems is obtained with an opti- mal level of disorder, similarly to (Conti and Fratalocchi, 2008). Second, by modifying the scattering and transport parameters of disordered systems, which in turn modify the localization length ξ, as suggested by Conley et al. (2014). In 2D, small changes of structural correlations may induce variations of ξ over orders of magnitude, since this quantity is expected to grow exponentially with the mean free path (Abrahams et al., 1979; Sheng, 2006). This allows moving very easily from a quasi-extended regime to a localized regime in finite-size systems. Nu- merical simulations performed for TE-polarized waves in 2D disordered patterns of holes in dielectric confirm this possibility, although only a qualitative agreement with theoretical predictions is obtained. Note also that the study covers a short-range correlation up to the onset of polycrystallinity, which might affect localization. The variety of mesoscopic transport regimes in 2D dis- ordered media was investigated recently by Froufe-Pérez et al. (2017), who proposed a transport phase diagram, shown in Fig. 11, for 2D stealthy hyperuniform structures made of high-index cylinders in TM polarization. Uncor- related media (small values of χ) experience the standard behavior with quasi-extended and localized regimes de- pending on the scattering strength of the cylinders and the system size. At the opposite, strongly correlated me- dia (high values of χ) are very transparent at low frequen- cies due to the suppressed single scattering over a finite range of scattering wavenumbers and exhibit a photonic gap (i.e., zero DOS in infinite media) at intermediate frequencies near the resonant frequency of the cylinder. The gap is surrounded by a low-DOS region containing weakly-coupled resonant states (defect modes) and the Anderson-localized regime. The phase diagram has been validated numerically (Froufe-Pérez et al., 2017) and experimentally in the microwave regime (Aubry et al., 2020). This diagram is specific to the considered system (including system size) and polarization. Nevertheless, it is quite representative of the different transport regimes that may be observed in correlated disordered media. FIG. 11 Correlation-frequency (χ − ν) transport phase dia- gram for 2D disordered hyperuniform media. The system is stealthy hyperuniform array of high-index dielectric rods and the wave is TM-polarized. χ is the degree of stealthiness (Bat- ten et al., 2008), νa/c ≡ a/λ is a reduced frequency with a the mean distance between scatterers (related to the cylinder density). Five transport regimes may be identified, as dis- cussed in the main text. Note that the transition between photon diffusion (quasi-extended regime) and Anderson lo- calization depends on system size. Adapted with permission from (Froufe-Pérez et al., 2017). Localization of light in 2D ensembles of resonant scat- terers in TE polarization encounters similar difficul- ties as in the 3D case due to the vectorial nature of light (Máximo et al., 2015). A recent theoretical study by Monsarrat et al. (2022) on hyperuniform patterns of high-quality-factor resonant dipole scatterers shows that localization of TE-polarized waves can occur at moder- ate scatterer densities concomitantly with the opening of a pseudogap, provided that a sufficiently high degree of short-range correlation is implemented. In essence, imposing a typical distance between resonant scatterers enables efficient destructive interference of vector waves, which leads to a depletion of the density of states and promotes the formation of localized states. Analytical expressions for the density of states and the localization length are rigorously established and found to agree well with numerical simulations. Clearly, the generalization of this theoretical framework to 3D resonant systems could contribute to unveiling the microscopic mechanisms be- hind the 3D Anderson localization of light. 41 C. Near-field speckles on correlated materials Upon scattering by one specific realization of a disor- dered medium, a speckle pattern is formed (Goodman, 2007). Universal intensity statistics are found in far-field speckle patterns, that are independent on the microscopic features of disorder. When speckle patterns are observed in the near field (i.e., at a distance from the output sur- face smaller than the wavelength of the incident light), the statistical properties of the speckle become depen- dent on the statistical features of the medium itself. In particular, as we will see, near-field speckles may exhibit direct signatures of the presence of spatial correlations in the scattering medium (Carminati, 2010; Naraghi et al., 2016; Parigi et al., 2016). 1. Intensity and field correlations in bulk speckle patterns A standard observable in the study of speckles is the correlation function of the intensity fluctuations δI at two different points r and r′, defined as ⟨δI(r) δI(r′)⟩ = ⟨I(r) I(r′)⟩ − ⟨I(r)⟩⟨I(r′)⟩, (132) with the intensity I(r) = |E(r)|2. As a measure of the degree of correlation of the intensity, one uses the nor- malized correlation function CI(r, r′) = ⟨δI(r) δI(r′)⟩ ⟨I(r)⟩⟨I(r′)⟩ (133) which, in terms of the field amplitude, is a fourth- order correlation function. In the weak-scattering regime krℓs ≫ 1, the field is a Gaussian random variable. In- deed, the field at any point in the speckle results from the summation of a large number of independent scatter- ing sequences, leading to Gaussian statistics by virtue of the central-limit theorem (Goodman, 2015). Moreover, in a statistically homogeneous and isotropic medium, and far from sources, the speckle pattern can be considered to be unpolarized. In these conditions, the intensity cor- relation function factorizes in the form (Carminati and Schotland, 2021) CI(r, r′) = ∑ i ∣∣CE ii (r, r ′) ∣∣2 , (134) where CE ij is the (i, j) component of the normalized cor- relation function between two vector components of the field, or normalized coherence tensor, defined as CE(r, r′) = ⟨E(r)⊗E∗(r′)⟩√ ⟨I(r)⟩ √ ⟨I(r′)⟩ . (135) Recall that the spatial correlation of the field in the numerator is described by the Bethe-Salpeter equation [Eq. (23)]. Let us first consider the simplest model of an in- finite medium illuminated by a point source at posi- tion r0. For large observation distances (|r − r0| ≫ ℓs and |r′ − r0| ≫ ℓs), one can derive the following gen- eral result (Carminati and Schotland, 2021; Dogariu and Carminati, 2015; Vynck et al., 2014) CE(r, r′) = 2π kr Im⟨G(r, r′)⟩, (136) where ⟨G⟩ is the averaged Green function in the medium. This form of the field correlation function is always found under general conditions of statistical homogeneity and isotropy of the field (Setälä et al., 2003). In order to characterize the field spatial correlation av- eraged over the polarization degrees of freedom, one often introduces the degree of spatial coherence γE(r, r′) = Tr [ CE(r, r′) ] . (137) In an infinite medium, and for short-range correlation with krℓc ≪ 1, with ℓc the correlation length of disorder, it is known that (Carminati et al., 2015; Carminati and Schotland, 2021) γE(r, r′) = sinc (krR) exp[−R/(2ℓs)] , (138) where R = |r − r′|. This expression takes the same form as that initially derived for scalar waves in (Shapiro, 1986). In an infinite medium, for a Gaussian and unpo- larized speckle pattern, the field and intensity correla- tion functions have a range limited by the wavelength λr = 2π/kr and by the scattering mean free path ℓs. The impact of structural correlations in the medium on speckle correlations (of the field or intensity) can oc- cur on different levels. First, the value of ℓs is directly dependent on the degree of correlation of disorder. Sec- ond, the general shape of the field correlation function can also be substantially modified when near fields can- not be ignored in either the illumination process (e.g., under excitation by a localized source inside the medium or close to its surface), or the detection process (e.g., de- tection at subwavelength distance from the surface). An example is discussed in the next subsection. 2. Near-field speckles on dielectrics A speckle pattern observed at subwavelength distance from the surface of a disordered medium (near-field speckle) exhibits statistical properties that may strongly differ from the universal properties of far-field speckles. In the case of near-field speckles produced by rough sur- face scattering, it is known that in the single-scattering regime the spatial correlation function of the near-field intensity is linearly related to the spatial autocorrelation function of the surface profile (Greffet and Carminati, 42 1995). In the case of speckles produced by volume mul- tiple scattering, the degree of spatial coherence can be evaluated in a plane at a distance z from the sample sur- face, in regimes ranging from the far field to the extreme near field (Carminati, 2010). For z ≫ λ, we obtain γE(r, r′) = sinc(k0ρ) , (139) where ρ is the distance separating the two observation points r and r′ in a plane at a constant z (parallel to the sample surface). The width δ of the correlation function, that measures the average size of a speckle spot, is limited by diffraction and scales as δ ∼ λ/2. At subwavelength distance from the medium surface, near fields are domi- nated by quasi-static interactions. The scale of variation of the field is driven by geometrical length scales, and no more by the wavelength (Greffet and Carminati, 1997; Novotny and Hecht, 2012). Characterizing the structure by the correlation length ℓc, and assuming ℓc ≪ λ, we can distinguish two regimes. For ℓc ≪ z ≪ λ, we have γE(r, r′) = 1− ρ2/(8z2) [1 + ρ2/(4z2)]5/2 , (140) showing that δ ∼ z due to quasi-static (evanescent) near fields. Finally, in the regime ℓc ≃ z ≪ λ (extreme near field), we obtain γE(r, r′) = M ( 3 2 , 1, −ρ2 ℓ2c ) (141) where M(a, b, x) is the confluent hypergeometric func- tion, which here takes the form of a function decaying from 1 to 0 over a width δ ≃ ℓc. In summary, according to the theory in (Carminati, 2010), we expect the speckle spot size to decrease in the near field as the distance z to the surface, and to saturate at a size on the order of the correlation length of the medium. The dependence of the speckle spot size at short dis- tance can be probed experimentally using scanning near- field microscopy (SNOM). Studies have been reported in (Apostol and Dogariu, 2003, 2004; Emiliani et al., 2003). The behavior described above has been confirmed recently by Parigi et al. (2016), and the main result is summarized in Fig. 12. The measurement provides the intensity correlation function CI(r, r′), the width of which, according to Eq. (134), can be qualitatively com- pared to that of the degree of spatial coherence γE . By recording near-field speckle images at different distances z from the surface of sample with correlated disorder, the dependence of the speckle spot size δ on the distance to the surface can be extracted. The result is displayed in Fig. 12c. The decrease δ in the near field regime is clearly visible, as well as the non-universal dependence at very short distance (the two curves correspond to two samples with different structural correlations). Retrieving information on structural correlations of FIG. 12 Signatures of structural correlations on near-field speckles. (a) Scanning electron microscope image of the sur- face of a typical sample, consisting of several layers of silica spheres in a partially ordered arrangement. (b) Example of speckle image recorded with a scanning near-field optical mi- croscope at a wavelength λ = 633 nm. (c) Measured correla- tion length δ in the speckle pattern versus the distance z to the sample surface, in the distance range for which the far- field to near-field transition is observed. The vertical dashed line corresponds to z = λ. The black and red curves corre- spond to two samples with an average diameter of the silica spheres d = 276 nm and d = 430 nm. The very short-distance behavior is expected to depend on the level of short-range or- der in the sample (size and local organization of the spheres in space). Adapted with permission from (Parigi et al., 2016). disordered media from optical measurements is also pos- sible via a stochastic polarimetry analysis of the scat- tered light (Haefner et al., 2008). As shown by Haefner et al. (2010), the local anisotropic polarizabilities of a complex material generally depend on the volume of ex- citation (which may be controlled, for instance, via a near-field probe). It turns out that one can define a length scale corresponding to a maximum degree of local anisotropy, that is characteristic of the material morphol- ogy. This length scale has been evidenced in numerical simulations (Haefner et al., 2010) but not yet experimen- tally to our knowledge. D. Local density of states fluctuations The modification of the spontaneous emission rate from quantum emitters due to electromagnetic interac- tion with a structured environment is one of the ma- jor achievements in optics and photonics in the past decades (Pelton, 2015). As briefly discussed in Sec. V.A, this effect is formally described by the local density of states (LDOS) that is expressed as a function of the Green tensor G(r, r) at the origin [Eq. (131)]. Expect- edly, the LDOS should be highly sensitive to the local en- vironment with which it interacts, especially in the near field. The near-field interaction regime has been initially de- scribed using numerical simulations of LDOS distribu- 43 tions inside disordered media and a single-scattering the- ory (Froufe-Pérez et al., 2007). The model system is a spherical domain with radius R, filled with subwave- length dipole scatterers. The LDOS is calculated at the center of the domain and surrounded by a spherical ex- clusion volume of radius R0. The length scale R0 is a microscopic length scale that characterizes the local envi- ronment (R0 can be understood as the minimum distance to the nearest scatterer). It was shown that the statisti- cal distribution of the LDOS is strongly influenced by the proximity of scatterers in the near field, and by the lo- cal correlations in the disorder (Cazé et al., 2010; Leseur et al., 2017). As in the case of near-field speckle, this is a consequence of quasi-static near-field interactions that make the LDOS sensitive to the local geometry. Statistical distributions of LDOS in strongly scatter- ing dielectric samples have been measured experimen- tally at optical wavelength. The approach consists in dispersing fluorescent nanosources inside a scattering ma- terial (Birowosuto et al., 2010; Sapienza et al., 2011). Ex- periments mimicking the model systems studied theoret- ically use powders made of polydisperse spheres of high- index material (such a ZnO at wavelength λ ∼ 600− 700 nm). An example of measured LDOS distributions is shown in Fig. 13. The LDOS distribution (top panel), inferred from the distribution of the decay rate Γ of nanoscale fluorescent beads, exhibits a high asymmet- ric shape with a long tail that is a feature of near- field interactions. This experiment confirms the sensi- tivity of LDOS fluctuations to the local environment in a volume scattering material in the multiple scattering regime. Comparison with numerical simulations (bottom panel) demonstrates the substantial role of the micro- scopic length scale R0 on the shape of the distributions. Disordered metallic films made by deposition of no- ble metals (silver or gold) on an insulating substrate (glass) are also known to produce large near-field inten- sity fluctuations close to the percolation threshold. On the surface of such materials, the near field intensity lo- calizes in subwavelength domains (hot spots) (Laverdant et al., 2008; Seal et al., 2005; Shalaev, 2007). The near- field LDOS exhibits enhanced spatial fluctuations in this regime, that reveal the existence of spatially localized modes (Carminati et al., 2015; Cazé et al., 2013; Krach- malnicoff et al., 2010). Disordered metallic films close to percolation are an example of nanoscale disordered ma- terials in which correlations in the disorder substantially influence the optical properties. Being the LDOS very sensitive to small changes in the local environment of the emitter, the study of the statis- tics of LDOS, accessible through the decay rate Γ, can be related to the structural properties of a dynamical sys- tem of interacting, and hence correlated, scatterers. As an example, it has been numerically demonstrated that the statistical distributions of single emitter lifetimes in a scattering medium can evolve from a unimodal distribu- FIG. 13 Impact of structural correlations on LDOS fluctua- tions. Top: Measured statistical distributions of the spon- taneous decay rate Γ ∝ ρ (LDOS) of fluorescent beads (nanosources with 20 nm diameter) in a ZnO powder with transport mean free path ℓt = 0.9 µm. A scanning electron microscope image of the sample is shown on the left, together with a schematic view of the illumination/detection geome- try. The asymmetric shape of the statistical distribution of LDOS and the long tail is a signature of near-field interactions occuring inside the sample. Bottom: Numerical simulations of the statistical distribution of the normalized decay rate Γ/Γ0 = ρ/ρ0 of a dipole emitter placed at the center of a disordered cluster mimicking the ZnO powder. The emitter is surrounded by an exclusion volume with radius R0. This length scale describes local correlations in the positions of the scatterers in the sample. Blue curves correspond to an exclusion radius R0 = 0.14 µm while red curves correspond to an exclusion radius R0 = 0.07 µm (the two curves in the same color correspond to two different densities of scatterers). The simulation demonstrates the substantial influence of R0 (near-field interactions and local correlations in the disorder) on the shape of the distribution. Adapted with permission from (Sapienza et al., 2011). tion to a different one when the system undergoes a phase transition. The regions of phase coexistence in small sys- tems often turn out to be dynamical phase switching regions, where the entire systems switches between the two phases (Berry et al., 1984; Briant and Burton, 1975; Honeycutt and Andersen, 1987; Labastie and Whetten, 1990; Wales and Berry, 1994). The signature of the phase switching regime in the Γ statistics can be dramatic, since the distribution can be bimodal in the phase switching regime regions while unimodal in the pure phases. This striking behavior can be related to the statistics of neigh- boring scatterers surrounding the emitter and is not sig- naled by other light transport properties such as scatter- ing cross section statistics for instance (de Sousa et al., 44 2016). Bimodal distributions of LDOS have been also described for emitters embedded in single layers of disor- dered but correlated lattices (de Sousa et al., 2014). Figure 14 shows numerical predictions for a system of ∼ 1000 resonant point dipoles interacting through a Lennard-Jones potential and tightly confined within a spherical volume (de Sousa et al., 2016). The system is kept at a temperature corresponding to the liquid- gas transition. Due to strong finite size effects, the sys- tem is not in phase coexistence but rather switches ran- domly between the two phases. We see that the emit- ter decay rates are strongly correlated to the energetic state of the system, leading to two clearly-distinguishable modes. Thus, slight differences in structural correlations can clearly be identified by a statistical analysis of decay rate measurements. FIG. 14 Signature of structural phase transition in LDOS statistics. Monte Carlo sampling of energy per particle nor- malized to the energy minimum ε (top) and decay rate nor- malized to the vacuum one Γ0 (bottom) for a single emitter placed at the center of a tightly confined system of resonant point scatterers interacting via Lennard-Jones potential. The temperature is such that the systems entirely switches be- tween two phases randomly. On the bottom, the correspond- ing decay rates distributions are represented for the low en- ergy branch (red), high energy branch (blue). The gray area shows the sum of both distributions. Adapted with permis- sion from (de Sousa et al., 2016). Recent experiments have shown strongly inhibited spontaneous emission in systems undergoing an order- disorder phase transition (Schöps et al., 2018). The formation of clusters exhibiting short-range correlations leads to a strong suppression of emission that is appar- ently comparable to that of an ordered structure. VI. PHOTONICS APPLICATIONS The considerable advances in nanofabrication in the past decades have opened new opportunities in the en- gineering of disordered materials at the subwavelength scale. In this section, we describe the main applications of correlated disordered media in optics and photonics, namely in light management [Sec. VI.A], random lasing [Sec. VI.B] and visual appearance design [Sec. VI.C]. The interested reader will find more examples of photonic ap- plications of correlated disorder in the recent review ar- ticle by Cao and Eliezer (2022). A. Light trapping for enhanced absorption Enhancing the interaction of light with matter is of paramount importance for various applications, includ- ing photovoltaics, white light emission and gas spec- troscopy. The enhanced light-matter interaction gener- ally translates into a stronger light absorption, be it ex- ploited for photocurrent generation, converted into emis- sion by fluorescence, or simply monitored. The most popular light trapping strategy for thick (L ≫ λ) bulk materials rely on randomly textured sur- faces, acting as Lambertian diffusers to efficiently spread light along all directions within the medium for an arbi- trary incoming wave (Green, 2002; Yablonovitch, 1982). Structural correlations on random rough surfaces pro- vide angular and spectral control over scattering (Mar- tins et al., 2013), described via the so-called Bidirectional Scattering Distribution Function (BSDF) (Stover, 1995). Volume scattering constitutes an interesting alternative to surface scattering, as multiple scattering tends to in- crease the interaction between light and matter (Ben- zaouia et al., 2019; Mupparapu et al., 2015; Muskens et al., 2008; Rothenberger et al., 1999). Quite counter- intuitively, one should note that the average path length for a Lambertian illumination in non-absorbing media is independent of the scattering strength of the mate- rial (Pierrat et al., 2014) and equivalent to the surface scattering light trapping, as predicted from the equipar- tition theorem (Yablonovitch, 1982) and as verified ex- perimentally recently (Savo et al., 2017). The absorption efficiency therefore depends strongly on the ratio between scattering and absorption. The benefit of structural cor- relations on light absorption in disordered media has only been considered recently by Bigourdan et al. (2019) and Sheremet et al. (2020), who showed by theory and nu- merical simulations that stealthy hyperuniform patterns of absorbing dipolar particles enhance the overall absorp- tion of the medium (compared to the uncorrelated sys- tem) close to an upper bound. Stimulated by technological development in next- generation photovoltaic panels, considerable efforts have been dedicated to light trapping in thin films (L ≈ λ) in the past two decades, exploiting coherent phenom- ena as a new means for enhancing light-matter interac- tion (Fahr et al., 2008; Gomard et al., 2013; Mokkapati and Catchpole, 2012). Fundamentally, coupling between an incident planewave and a resonant mode in the lay- 45 ered medium is enabled by fulfilling a matching condition between the projected wavevectors parallel to the inter- face k|| in the two media (Yu et al., 2010). For periodic photonic crystals, this condition is found for leaky Bloch modes having wavevectors in the light cone kB,|| < k0nstr, where nstr is the refractive index of the superstrate or substrate. In periodic structures, the absorption peaks are spectrally narrow and strongly depend on kB,||. A further improvement can be obtained by creating imper- fections that broaden the spectral and angular response, leading to an overall improved optical efficiency (Oskooi et al., 2012; Peretti et al., 2013). For disordered media, the quantity of interest is the so-called spectral function, defined as (Sheng, 2006), ρs(k, ω) = 2ω πc2 Im [Tr⟨G(k, ω)⟩] , (142) which is the average density of states resolved in spatial frequencies and is obtained from the Fourier transform of the average Green tensor ⟨G(r, r′, ω)⟩. At a given fre- quency, the spectral function typically exhibits a peak centered on the effective wavevector in the disordered medium, kr, and a width that is inversely proportional to the extinction mean free path, see Fig. 15. As shown by Vynck et al. (2012), short-range correlations allow a fine tuning of the spectral function, including in the radiative zone, eventually leading to a spectrally and angularly op- timal light absorption (Bozzola et al., 2014; Pratesi et al., 2013). It has been suggested that stealthy hyperuniform structures can lead to even higher overall absorption effi- ciency (integrated over a spectrum of interest) compared to short-range correlated and periodic media (Liu et al., 2018). Experiments on correlated disordered hole pat- terns (Trompoukis et al., 2016), nanowire arrays exhibit- ing fractality on some scale (Fazio et al., 2016), complex nanostructured patterns (Lee et al., 2017) and hyperuni- form structures (Piechulla et al., 2021b; Tavakoli et al., 2022) have clearly demonstrated the benefit of disorder engineering for light trapping. Finally, let us point out that the coupling process be- tween free space and thin-film layers is very relevant also in the optimization of light extraction from light-emitting devices like organic LEDs (Gomard et al., 2016), where correlated disordered photonic structures could be real- ized on large scales, for instance, by inkjet-printing of polymer blends (Donie et al., 2021). B. Random lasing Random lasers, where light is trapped in the gain medium by multiple scattering, offer new possibilities for efficient lasing architectures. The disordered matrix folds the optical paths inside the medium by multiple scatter- ing, effectively increasing the probability of stimulated emission, which in turns provides optical gain and the FIG. 15 Process of light coupling and decoupling between a thin dielectric membrane and free-space modes. (a) Sketch of a photonic structure with correlated disorder containing sev- eral leaky resonant modes (QNMs). The QNMs are described by different complex frequencies and are coupled to free-space modes. (b) Spectral functions of a short-range correlated dis- ordered photonic structure at different frequencies. At low frequencies, the spectral function is a narrow peak. The value at k|| = 0 provides information on the coupling efficiency at normal incidence. At higher frequencies, the peaks broaden as a result from stronger scattering and reach higher values for small wavevectors, indicating more efficient coupling. amplification that triggers lasing (Cao, 2005; Wiersma, 2008), see Fig. 16. FIG. 16 Conventional vs Random lasing. While a conven- tional laser (left) is usually composed of a two-mirrors cavity which defines the optical modes, a random laser (right) ex- ploits the confinement by multiple scattering to enhance the probability of stimulated emission. It lases on the “speckle” modes of the disordered medium, either delocalized (bottom left) or localized (bottom right). In both lasers lasing occurs when the gain is larger than the losses, above a certain pump- ing threshold energy, when stimulated emission becomes the dominant emission process. Lasing peaks can appear in both diffusive or localized regimes, but are easily washed out by temporal or spatial averaging in diffusive media. Adapted with permission from (Sapienza, 2019). Its functioning principle is the same as in conventional lasing but without the need for carefully aligned optical elements. The emission of a random laser is also surpris- 46 ingly coherent, with photon statistics close to that of nor- mal laser emission (Florescu and John, 2004), with strong mode coupling (Türeci et al., 2008) non-trivial modes organization (symmetry replica breaking) (Ghofraniha et al., 2015) and unbounded (Lévy distributed) intensity fluctuations (Uppu and Mujumdar, 2015). Due to the volumetric and (on average) isotropic nature of its lasing patterns a random laser is expected to feature β-factors close to one (i.e., a threshold-less behaviour (van Soest and Lagendijk, 2002)). The result (Fig. 16) is an opaque medium in which laser light is generated along random paths in all directions, and over a broad spectral range, with complex temporal profiles (Leonetti et al., 2011). In the multiple scattering regime, the random lasing threshold can be related to a critical volume or size, such that lasing action can only be achieved for sample sizes larger than this critical dimension. Similar to the criti- cal volume in a neutron bomb, the critical size ensures that the photons sustain net amplification and therefore that the light emerging from the sample is mostly due to spontaneous emission. For a three-dimensional scatter- ing medium, with isotropic scattering and no correlation, embedded in a slab geometry, the critical thickness Lcr has been calculated by using the radiative transfer equa- tion (Pierrat and Carminati, 2007) and is solution of 1 ℓs − 1 ℓg = π (Lcr + 2z0) tan (πℓs/(Lcr + 2z0)) (143) where z0 = 0.7104ℓs is the extrapolation length and ℓg the net-gain length. Eq. (143) reduces to Lcr = π √ ℓsℓg/3 in the diffusive limit with z0 = 0. For anisotropic scat- tering, we get Lcr = π √ ℓtℓg/3. In typical samples, the critical length is of the order of 1 to 100 µm (e.g., Lcr ∼ ℓt with ℓt ∼ 4 µm in (Caixeiro et al., 2016) and Lcr ∼ 300 ℓs in (Froufe-Pérez et al., 2009)). The initial scattering architectures for lasing have been 3D disordered semiconductors powders or randomly fluc- tuating colloids in solution, which can be well thought of and described as a random cloud of dipoles (i.e., with- out any correlation). Pure randomness is the assump- tion which simplifies the complexity of the problem to make it treatable with theoretical models. Despite the many successes of this type of uncorrelated disorder, a new generation of disordered lasing architectures, with more robust and collective light-trapping schemes (Got- tardo et al., 2008) and new topologies (Gaio et al., 2019) has emerged. In particular, spatial correlations between scatterers is a very effective approach for tuning the spectral prop- erties, the number of lasing modes and their thresh- old by designing photonic band-edge states at the po- sition of the gain. For example, localized modes near the edge of a (2D) photonic gap have been exploited for random lasing (Liu et al., 2014) and single-mode oper- ation has been achieved in compositionally disordered photonic crystals (Lee et al., 2019). The role of the gap edge has been highlighted in semiconductor membranes with pseudo-random patterning (Yang et al., 2010a), ran- domly mixed photonic crystals (Kim et al., 2010) and amorphous network structures (Wan et al., 2011), while in photonic amorphous structures, the short-range order improves optical confinement and enhances the quality factor of lasing modes (Yang et al., 2011). Modelling lasing action in correlated disordered media is often a challenge. In particular lasing occurs for the modes with highest net gain, often escaping the trans- port models which instead deal with the average inten- sity. While a full-wave solution of the Maxwell’s equa- tions, coupled to the dynamics of the gain, as for exam- ple Maxwell-Bloch models (Conti and Fratalocchi, 2008) would contain all the relevant phenomena, it is very hard to implement in realistic samples due to its com- putational requirements. Advanced ab-initio theoretical models have been developed. Let us mention the self- consistent laser theory (Ge et al., 2010; Türeci et al., 2008), which relies on a decomposition of the lasing field on a basis of resonant modes, and the Euclidean matrix theory by Goetschy and Skipetrov (2011), which relies on analytical predictions for the random Green’s matrix of a system. Alternatively, more simplified models that neglect the coherence of the modes and describes trans- port with the radiative transfer equation (or within the diffusion approximation) can be used. The diffusion ap- proximation stems from the initial proposal by Letokhov (1968) and simplifies the calculations significantly (Gaio et al., 2015; Wiersma and Lagendijk, 1996). These mod- els can be extended to include scattering correlations, to modify the scattering and gain parameters, following the theory described in Sec. II. C. Visual appearance 1. Photonic structures in nature Living organisms produce a vast variety of photonic mechanisms to modulate their visual appearance by ex- ploiting a wide range of biopolymers and architectures. Colors produced by these organisms are referred to as structural colors as they are mainly influenced by the nanostructural features of the materials, rather than pig- ments. The lack of consistent methods and tools of analy- sis, as well as the incredibly large number of species show- ing different architectures however makes it difficult to categorize natural photonic structures. Distinct species use different materials, structures and strategies for as many biological functions (to attract mates, hide from predators or act as a defence mechanism) (Seago et al., 2008; Vignolini et al., 2012; Whitney et al., 2009; Wilts et al., 2014). Another degree of difficulty arises from the fact that such natural architectures are often hierarchical 47 and their visual appearance depends on several factors, including geometrical features, the addition of absorbing pigments and finally, the visual system for which such structures are build (different animals and insects have different perceptions). Only a limited selection of explanative examples will be discussed and analyzed hereafter. The reader should keep in mind that this represents a minimal fraction of the ef- forts that have been done in this field to systematically characterize and categorize natural photonic structures. For consistency with the scope of this review, we only mention in passing the case of one-dimensional disordered multilayered structures, which are found in certain bee- tles (Del Rio et al., 2016; Hunt et al., 2007; Onelli et al., 2017), butterflies (Bossard et al., 2016), leaves (Vignolini et al., 2016) and algae (Chandler et al., 2017). Imper- fect one-dimensional grating structures, which play an important role for flowers to enhance signalling to polli- nators (Moyroud et al., 2017), for instance, will also not be discussed further. FIG. 17 (a) Kingfisher, the angular independent blue coloration of the bird feather is the result of a corre- lated 3D structure, which is shown in the SEM image in (b). Image courtesy of PIXABAy and B. Wilts, respec- tively. See (Stavenga et al., 2011) for more information. (c) Pachyrhynchus sarcitis Weevils, the blue colored spot in the weevil exoskeleton are the results of a polydomain dia- mond photonic structures, shown in the SEM image on (d). Adapted with permission from (Chang et al., 2020). The most widespread family of (2D or 3D) correlated disorder in nature is the short-range correlation, which generally aims at producing angular-independent col- orations. Short-range correlated structures are found in butterflies (Prum et al., 2006), bacteria (Schertel et al., 2020), and many animals. Probably the most famous examples are the structures found in the feathers of the eastern bluebird Cotinga maynana (Prum et al., 1998) and of the Kingfisher (Stavenga et al., 2011), which present a short-range correlation of keratin fibrillary net- work but also several other one such as the I. puella (Noh et al., 2010). Correlated ensembles of collagen spheres producing an angle-independent color are found in avian skin (Prum and Torres, 2003) and mammalian skin (pri- mates) (Prum and Torres, 2004). Interestingly, these types of structures are always producing blue and green colorations in nature (Hwang et al., 2021; Jacucci et al., 2020; Jeon et al., 2022; Magkiriadou et al., 2014). Many examples of polycrystalline structures are ex- ploited for coloration in insect wings using three- dimensional photonic crystal structures such as diamond and gyroids. In these cases, polydomains provide a more angular independent response which might again be func- tional for signalling and camouflaging (Michielsen and Stavenga, 2007). Examples are the diamond-like struc- tures observed inside the scales of the weevils Lamprocy- phus augustus (Galusha et al., 2008), Entimus imperialis (Wilts et al., 2012) and Pachyrhynchus weevils (Chang et al., 2020). Similarly, three-dimensional gyroid struc- tures are found in many lycaenid and papilionid butter- flies by (Michielsen and Stavenga, 2007) and in various species such as C. rubi (Michielsen et al., 2009; Schröder- Turk et al., 2011), C. remus, P. sesostris (Wilts et al., 2011), and T. opisena (Wilts et al., 2017). Distinct from the above cases, anisotropic network-like structures can be optimized to enhance whiteness. Sev- eral natural examples exist, see (Jacucci et al., 2021) for a recent review. A notable example of optimized whiteness is found on the scales of the beetle genus Cyphochilus, which shows a brilliant white coloration while being only 5-7 µm thick (Burg et al., 2019; Burresi et al., 2014; Luke et al., 2010; Vukusic et al., 2007; Wilts et al., 2018). Op- tical and anatomical studies confirm that anisotropy of the random polymeric network structure in such beetles scales are crucial for scattering optimisation at low re- fractive indices (Cortese et al., 2015; Jacucci et al., 2019; Lee et al., 2020; Utel et al., 2019). 2. Synthetic structural colors The ability to control visual appearance with corre- lated photonic structures, both in terms of color and scat- tering response, is critical in photonic pigments. With the improvements in the fabrication techniques, it is now possible to assemble such photonic materials cheaply and on a large scale and therefore, their use to replace traditional pigments is becoming a reality (Goerlitzer et al., 2018; Lan et al., 2018; Saito et al., 2018). Of particular interest here are short-range correlated struc- tures (Shi et al., 2013). The simplest way to achieve such materials in films consists in a rapid drying of col- loidal suspensions to form photonic glasses (Forster et al., 48 2010; Garćıa et al., 2007; Schertel et al., 2019a). Com- bined with additive manufacturing techniques, one can fabricate complex-shaped objects exhibiting diffuse col- ors (Demirörs et al., 2022). Several types of colloidal par- ticles, functionalisation and matrices have been proposed to enlarge the color palette with these structures (Forster et al., 2010; Ge et al., 2015; Häntsch et al., 2019; Kim et al., 2021; Schertel et al., 2019a) and several tricks have been proposed to improve color contrast and appearance (Häntsch et al., 2021; Hwang et al., 2020) also exploiting absorbing species, such as carbon black (Takeoka et al., 2013). However, all these approaches have so far only been capable of providing only faint blue and green col- ors. In order to expand the visible palette toward red hues, core-shell photonic glasses (Kim et al., 2017; Shang et al., 2018) and inverse photonic structures have been exploited (Zhao et al., 2020) – however their color pu- rity and the reflected intensity remain limited (Jacucci et al., 2020). Short-range crystalline systems with care- fully tuned domain orientations or the geometry of the system might allow to overcome this issue (Song et al., 2019). The design of structural colors with artificial materials also raises questions about their predictability with the- oretical models or numerical methods, and the inverse design of artificial materials. The starting point to predict a color is the computa- tion of the reflectance or transmittance spectra of the disordered material. These spectra are then weighted by the spectral power distribution of the illuminant and by color matching functions for the chromatic response of the observer, to be finally projected onto a specific color space (Ohta and Robertson, 2006), such as CIE 1931 XYZ. The computation of the reflectance and trans- mittance spectra is evidently the most tedious step. Most studies have relied on finite-difference time-domain (FDTD) simulations (Taflove and Hagness, 2005), for instance for 3D particulate media (Dong et al., 2010) and porous dielectric networks (Galinski et al., 2017), yet at the cost of heavy computational loads (although this is mitigated by efficient parallelization). Analyti- cal expressions based on diffusion theory have also been used (Schertel et al., 2019a), but care should be taken on the validity of diffusion approximation (L/ℓt ≫ 1). A good alternative is to rely on Monte Carlo light transport simulations (Alerstam et al., 2008; Wang and Jacques, 1992) – the numerical counterpart of radiative transfer – wherein positional correlations can be taken into account analytically via Eqs. (111) and (112) and assuming that an effective index can be defined. This numerical ap- proach was used to unveil the importance of the pack- ing strategy of photonic glasses on their color saturation and angle-dependence (Xiao et al., 2021), explore effi- ciently the parameter space (Hwang et al., 2021), and investigate the potential of random dispersions of pho- tonic balls (Yazhgur et al., 2022b) for coloring applica- tions (Stephenson et al., 2023). The inverse design of structural colors is, by compar- ison, still in its infancy. The aforementioned Monte- Carlo approach by Hwang et al. (2021) has been com- bined with Bayesian optimization to determine the ex- perimental parameters required to reach a target color. Powerful topology optimization techniques (Jensen and Sigmund, 2011), also known as adjoint methods, have been used recently to design complex dielectric network materials creating targeted colors in reflection (Andkjær et al., 2014; Auzinger et al., 2018). Although the role of structural correlations on coloration is implicit in this case, a subsequent structural analysis of the optimal de- signs could lead to the definition of recipes for materials creating vivid colors. VII. SUMMARY AND PERSPECTIVES Research on disorder engineering in optics and photon- ics has considerably grown in the past decade stimulated by the advent of new concepts and applications. In this last section, we attempt to identify some of the most promising developments for future research along with the theoretical and experimental challenges that will need to be tackled. A. Near-field-mediated mesoscopic transport in 3D high-index correlated media Multiple light scattering in disordered media has been treated for many years as a process wherein the vector nature of light could be simplified either by keeping its transverse component only, as we have done in Sec. II, or by treating light just as a scalar wave (Akkermans and Montambaux, 2007). Whereas these approximations may be well justified in dilute media (for both) and far from any polarized source in an opaque medium (for the latter), it recently turned out that the importance of the longitudinal component, which appears in the near- field regime, on mesoscopic transport in dense systems has been largely underestimated so far (Cobus et al., 2022; Escalante and Skipetrov, 2017; Monsarrat et al., 2022; Naraghi et al., 2015; Naraghi and Dogariu, 2016; Skipetrov and Sokolov, 2014). We believe that this as- pect deserves full attention from the community. A first attempt to incorporate the longitudinal component in the theory has been proposed by van Tiggelen and Skipetrov (2021) for random ensembles of resonant point scatter- ers, giving physical ground to the existence of near-field channels in light transport. Near-field interaction pro- cesses are dramatically impacted by subwavelength-scale structural correlations, as we have seen in Sec. V.C, and developing a theoretical framework to describe radiative transfer in arbitrary correlated media including the near- 49 field contribution would certainly be an important step forward. Related to this are the determination of effective ma- terial parameters for dense, resonant disordered media and their use to describe light scattering and transport, which are still matter of investigation (Aubry et al., 2017; Yazhgur et al., 2021, 2022a), as quantitative agreement with experiments and numerics has remained difficult to reach with classical models. On this aspect, let us point out the recent works by Gower et al. (2019a,b), demonstrating that multiple coherent waves with differ- ent wavenumbers (at fixed frequency) should actually contribute to the average field. This may have important consequences for scattering by finite-size systems (Gower and Kristensson, 2021) and raises the question of whether these multiple waves are affected in a similar way by structural correlations. The prominent role played by the precise morphol- ogy of 3D disordered media on the emergence of pho- tonic gap and Anderson-localized regimes also deserves clarification. 3D high-index connected (foam-like) struc- tures appear as the best candidates for this purpose ac- cording to numerical simulations (Haberko et al., 2020; Imagawa et al., 2010; Klatt et al., 2019; Liew et al., 2011; Sellers et al., 2017) but the underlying physical mechanisms have remained difficult to grasp, thereby calling for further theoretical advances (Scheffold et al., 2022). In addition to near-field effects, future works may need to consider high-order n-point correlation functions (with n > 2) in the description of structural character- istics (Torquato, 2013; Torquato and Kim, 2021) as well as high-order diagrams in the multiple-scattering expan- sion (Vollhardt and Wölfle, 1980), which can contribute significantly in strongly correlated media as shown re- cently (Leseur et al., 2016). Numerical investigations will continue in parallel, and we believe that progress would greatly benefit from the development of numerical meth- ods to solve electromagnetic problems on large systems more efficiently (Bertrand et al., 2020; Egel et al., 2021, 2017; Lin et al., 2022; Valantinas and Vettenburg, 2022). Experimental demonstrations of photonic gaps and 3D Anderson localization of light in the optical regime have remained out of reach until now and would be great scientific milestones. The main challenge to overcome at this stage is the fabrication of 3D connected struc- tures with finely-tuned correlated morphologies with suf- ficiently high refractive indices (ideally offering an in- dex contrast above 3) and sufficiently large thicknesses (L ≫ ℓt). The steady progress on bottom-up approaches such as bio-templating (Galusha et al., 2010), DNA- origami (Zhang and Yan, 2017) and microfluidic-based foam processing (Maimouni et al., 2020) gives hope for first successful realizations in the next few years. As a longer-term objective, the design and fabrication of 3D stealthy hyperuniform media would be an outstanding result. Ultimately, the availability of such high-index nanostructured materials will unlock the possibility to explore experimentally the physics of mesoscopic phase transitions (Evers and Mirlin, 2008) for (polarized) elec- tromagnetic waves. B. Mesoscopic optics in fractal and long-range correlated media Light propagation in positively-correlated media is characterized by a non-exponential decay of the coherent intensity. As discussed in Sec. IV.B, materials exhibiting fractal heterogeneities in the form of non-scattering re- gions of varying sizes can lead in certain conditions to a superdiffusive behavior (Burioni et al., 2014; Savo et al., 2014). Anomalous transport processes (Klages et al., 2008) and dynamics on fractal networks (Nakayama et al., 1994) have a long history, but optical studies on fractal media have so far been mostly concerned with structure factor measurements in the single scattering regime (Lin et al., 1989) (note that the optical properties of semicontinuous metal films near percolation, on which there exists a vast litterature (Shalaev, 2007), strongly rely on near-field plasmonic effects and not on light trans- port). Coherent optical phenomena in “Lévy-like” media have been sparsely addressed until now (Burresi et al., 2012). Multiple scattering formalisms have been ex- tended to media described by fractal dimensions (Akker- mans et al., 1988; Wang and Lu, 1994) or exhibiting su- perdiffusion (Bertolotti et al., 2010b), disregarding how- ever several difficulties related to the definition of the self-energy and the effective index (Tarasov, 2015), and perhaps more importantly to the quenched nature of dis- order (Barthelemy et al., 2010; Burioni et al., 2014). All in all, the development of a rigorous ab-initio theory for multiple light scattering in strongly-heterogeneous mate- rials would be a formidable achievement. Numerical and experimental studies on 1D and quasi- 1D Lévy-like systems have revealed anomalous conduc- tance fluctuations and scaling (Ardakani and Nezhad- haghighi, 2015; Fernández-Maŕın et al., 2014; Lima et al., 2019). Higher-dimensional systems are likely to exhibit a similarly rich physics, as illustrated recently (Chen et al., 2022), which yet remains to be explored. One example is the critical dimension of 2 above which the Anderson transition exists (Abrahams et al., 1979) that may be lowered, depending on the fractality or lacunarity of the system. Optical experiments and numerical simulations could be performed in this regard on high-index planar photonic structures similarly to Riboli et al. (2014), giv- ing access to LDOS statistics, or to Yamilov et al. (2014) for transmittance and internal light intensity measure- ments. An alternative route for the experimental study of mesoscopic phenomena in long-range correlated sys- tems could rely on disordered photonic network (Gaio 50 et al., 2019) – an optical implementation of random graphs (Janson et al., 2011) –, wherein light propa- gates through 1D waveguides and is scattered at the waveguide vertices. The waveguide lengths and ver- tices connectivity thus take the role of structural cor- relations. The network is a low-dimensional medium embedded in three-dimensional space, and allow to de- sign light transport and the optical modes. Complex networks with finely-controlled parameters can be fab- ricated by self-assembly (Gaio et al., 2019), by standard lithography techniques or implemented on macroscopic systems (Lepri et al., 2017). C. Towards novel applications The sensitivity of the LDOS to the local environment discussed in Sec. V.D makes quantum emitters interest- ing optical probes of nanostructured materials (Pelton, 2015). Many studies have reported the dramatic im- pact of the local morphology of a complex medium on the spontaneous emission statistics from neighboring flu- orescent molecules or quantum dots (Birowosuto et al., 2010; Granchi et al., 2022; Krachmalnicoff et al., 2010; Riboli et al., 2017; Sapienza et al., 2011; de Sousa et al., 2016). A key question to address will be whether optical measurements mediated by near-field probes could reveal statistical information on an unknown morphology, which could be extremely interesting for the remote monitoring of structural phases deep inside a 3D volume (de Sousa et al., 2016). This would require a deep understanding on the relation between subwavelength-scale correlations and near-field phenomena. In addition to LDOS mea- surements, measuring the cross spectral density of states (CDOS) (Cazé et al., 2013), which describes mode con- nectivity in structured media and could be obtained from coherence measurements on the light emitted from two classical or quantum dipole sources (Canaguier-Durand et al., 2019), may bring precious additional information. Spatially-resolved intensity correlations in the optical regime have recently been measured in the bulk of a dis- ordered medium using pairs of emitters separated by dis- tances controlled via DNA strings (Leonetti et al., 2021), unveiling already a very rich physics. First CDOS mea- surements have recently been realized in the microwave regime (Rustomji et al., 2021). The coherent control of light waves in disordered media is an important branch of research in multiple light scat- tering that has been powered in recent years by wavefront shaping techniques (Rotter and Gigan, 2017). We have seen in Sec. V that structural correlations can result into strong spectral variations of scattering, transport and lo- calization, suggesting that correlated disorder could yield higher degrees of spectral and spatial control, with appli- cations in optical imaging. Disordered media are also be- ing exploited as an unconventional platform for quantum walks and quantum state engineering (Defienne et al., 2016; Leedumrongwatthanakun et al., 2020). These are very delicate experiments that require lossless materials, so far attempted in multimode fibers, but which could benefit in the future from correlated disordered media for multiplexing and spectral resolution. The design of visual appearance is another aspect of re- search on correlated disorder that has considerably grown in importance in recent years. As beautifully illustrated by many diverse examples in the living world, the inter- play or order and disorder has a direct impact on ap- pearance at macroscopic scales, yielding increased trans- parency or whiteness, iridescent or non-iridescent colors [Sec. VI.C]. The numerical modelling of realistic corre- lated materials, considering for instance local imperfec- tions (Chung et al., 2012) and large-scale random varia- tions (Chan et al., 2019) in certain ordered systems, will be an essential development of the field in the near fu- ture. Our perception of objects indeed relies on many at- tributes of visual appearance (Hunter and Harold, 1987) – not only color, but also gloss, haze, translucency, tex- ture, etc. –, which are affected by multiple scattering and are rarely considered in full. Understanding how optical properties created by structural correlations at the microscopic scale translate into visual effects at the macroscopic scale is a great and exciting challenge for the coming years, which could be enabled by merging concepts and techniques from coherent light scattering and computer graphics (Guo et al., 2021; Musbach et al., 2013; Vynck et al., 2022). Beyond appearance, correlated disordered media will play an important role in thermal management, for example for radiative cooling (Wang and Zhao, 2020), where broadband light control is needed from inexpensive self-assembled media. Correlated dis- ordered materials could be used to realize multifunc- tional materials, where optical (visual) properties could be combined with desired thermal, electrical, mechan- ical or tribological functionalities. Last but not least, efforts should be amplified to develop and promote low- carbon footprint, ecologically-responsible material syn- thesis, which can be best achieved in disordered assem- blies as those discussed in this review. ACKNOWLEDGMENTS We are grateful to Eugene d’Eon (NVIDIA, New Zealand), Aristide Dogariu (CREOL, University of Cen- tral Florida, USA), Akhlesh Lakhtakia (Penn State Uni- versity, USA) and Lorenzo Pattelli (INRiM, Italy) for fruitful discussions and precious feedbacks on our orig- inal manuscript. KV acknowledges funding from the french National Agency for Research (ANR) via the projects “NanoMiX” (ANR-16-CE30-0008) and “Nano- Appearance” (ANR-19-CE09-0014-01). This work has received support under the program “Investissements 51 d’Avenir” launched by the French Government. This research was supported by the Swiss National Science Foundation through project numbers 169074, 188494 (FS) and 197146 (LSFP). RS and SV acknowledge fund- ing by the Engineering and Physical Sciences Research Council (EPSRC). SV acknowledges the European Re- search Council (ERC-2014-STG H2020 639088). Appendix A: Green functions in Fourier space 1. Dyadic Green tensor We consider a statistically homogeneous and translationally-invariant medium. The dyadic Green tensor in a uniform background medium with auxiliary permittivity ϵb is given by Eq. (15) and related to its Fourier transform as Gb(r− r′) = 1 (2π)3 ∫ Gb(k)e ik·(r−r′)dk. (A1) The Green tensor in Fourier space is given by Gb(k) = [ k21− k⊗ k− k2b1 ]−1 (A2) = [ −k2b k⊗ k k2 + ( k2 − k2b )( 1− k⊗ k k2 )]−1 (A3) = 1 k2b [ −k⊗ k k2 + k2b k2 − (kb + i0)2 ( 1− k⊗ k k2 )] . (A4) The small imaginary part i0 introduced here is relevant in integrals involving Gb(k). Using Eq. (33), the Green tensor can finally be rewritten as Gb(k) = 1 k2b [ −k⊗ k k2 + PV { k2b k2 − k2b }( 1− k⊗ k k2 )] +iπδ(k2 − k2b) ( 1− k⊗ k k2 ) . (A5) 2. Dressed Green tensor Following Eq. (57), the Green tensor can be decom- posed into local and non-local terms, the latter being also known as the Lorentz propagator. 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theories Scattering and transport mean free paths for resonant scatterers Summary and further remarks Structural properties of correlated disordered media Continuous permittivity versus particulate models in practice Fluctuation-correlation relation Classes of correlated disordered media Short-range correlated disordered structures Polycrystalline structures Imperfect ordered structures Disordered hyperuniform structures Disordered fractal structures Numerical simulation of correlated disordered media Structure generation Electromagnetic simulations Fabrication of correlated disordered media Jammed colloidal packing Thermodynamically-driven self-assembly Optical and e-beam lithography Measuring structural correlations Modified transport parameters Light scattering and transport in colloids and photonic materials Impact of short-range correlations: first insights Enhanced optical transparency Tunable light transport in photonic liquids Resonant effects in photonic glasses Modified diffusion in imperfect photonic crystals Anomalous transport in media with large-scale heterogeneity Radiative transfer with non-exponential extinction From normal to super-diffusion Mesoscopic and near-field effects Photonic gaps in disordered media Definition and identification of photonic gaps in disordered media Competing viewpoints on the origin of photonic gaps Reports of photonic gaps in the litterature Mesoscopic transport and light localization Near-field speckles on correlated materials Intensity and field correlations in bulk speckle patterns Near-field speckles on dielectrics Local density of states fluctuations Photonics applications Light trapping for enhanced absorption Random lasing Visual appearance Photonic structures in nature Synthetic structural colors Summary and perspectives Near-field-mediated mesoscopic transport in 3D high-index correlated media Mesoscopic optics in fractal and long-range correlated media Towards novel applications Acknowledgments Green functions in Fourier space Dyadic Green tensor Dressed Green tensor Derivation of Eq. (28) Configurational average for statistically homogeneous systems Particle correlation functions Fluctuations of the number of particles in a volume Local density of states and quasinormal modes References