Enhanced Absorption of Waves in Stealth Hyperuniform Disordered Media

We study the propagation of waves in a set of absorbing subwavelength scatterers positioned on a stealth hyperuniform point pattern. We show that spatial correlations in the disorder substantially enhance absorption compared to a fully disordered structure with the same density of scatterers. The non-resonant nature of the mechanism provides broad angular and spectral robustness. These results demonstrate the possibility to design low-density materials with blackbody-like absorption.

Enhancing the absorption of waves is of paramount importance in a number of applications, including solar heating [1], photovoltaics [2], molecular spectroscopy and sensing [3], acoustic insulation [4,5], seismology [6], or ocean waves [7]. In optics, perfect absorption of coherent light has been demonstrated long-ago on designed periodic structures supporting surface plasmons [8]. The conception of metallic nanostructures to enhance absorption has become a strategy to increase the efficiency of photovoltaic conversion or photodetection of visible or near infrared light, taking advantage of the broadband nature of plasmon resonances. A drawback is the strong absorption in the metal itself, which reduces the absorption enhancement in the region of interest (e.g., the active semiconductor). Optimized nanostuctures also require costly nanofabrication techniques, and can be highly sensitive to imperfections. The concept of coherent perfect absorption (CPA) has been generalized, showing that total absorption at a given frequency can be reached in any absorbing material provided that the incident wavefront has been spatially shaped to match an absorption eigenmode [9,10]. Coupled to wavefront shaping techniques, CPA permits total absorption of light in complex disordered media [11,12]. Resonant absorption in designed nanostructures and CPA both require a coherent (shaped or unshaped) incident wavefront. This makes the absorption process sensitive to changes in the incident wave (direction, spectrum or polarization), or to a reduction of its degree of coherence, thus limiting the range of targeted applications.
Alternatively, absorption of natural light in disordered materials has been put forward recently, mostly in the context of photovoltaics [13][14][15]. Nevertheless, a strategy to optimize real disordered materials and maximize the absorbance remains to be defined. Although it is known that structural correlations in the disorder (e.g., correlations in the positions of scatterers dispersed in a homogeneous medium) have a strong impact on the scattering properties [16][17][18][19][20][21][22][23], the influence on absorption has been addressed only recently, in particular cases [24][25][26][27].
In this Letter, we study the potential of disordered media with controlled structural correlations in the design of efficient absorbers of light (and more generally waves) over a broad spectral and angular range. First, we demonstrate theoretically the existence of an upper bound for the absorbed power, and discuss the underlying physical picture. This picture naturally dictates a strategy to engineer disorder materials with optimized absorption. Second, based on exact numerical simulations of wave scattering, we show that hyperuniform materials, a specific class of correlated disordered materials, permit to reach an absorption level close to the predicted upper bound, with substantial spectral and angular robustness. The results demonstrate the impact of structural correlations on absorption, offering new possibilities for the design of disordered materials with engineered properties.
Upper bounds for the absorbed power in photonic materials have been discussed in previous studies. One of them applies to materials made of discrete scatterers with a fixed position in space, the degree of freedom being the polarizabilities of the individual scatterers [28]. Since in the present work we focus on the degrees of freedom offered by the spatial distribution of scatterers, this upper bound is not directly applicable. Another upper bound was derived for a medium with given material properties (dielectric function) and a tunable geometry [29]. Since our objective is to define strategies to optimize the absorption in a disordered medium described statistically, we propose another approach to derive an upper bound that directly applies to the statistically averaged absorbed power. As we will see, the mechanism behind the existence of this upper bound gives insight for the definition of strategies to reach this bound, and optimize the level of absorption by designing the statistical properties (spatial correlations) of the disordered medium.
Let us consider a disordered medium embedded in a volume V with external surface S, illuminated by a monochromatic incident electric field with complex amplitude E 0 . From Poynting's theorem, and writing the total field E = E 0 +E s with E s the scattered field, energy consevation can be written in the form P e = P a + P s [30]. In this expression, P e = Re V j · E * 0 d 3 r/2 is the extinction power, j being the induced current density in volume V , P a = Re V j·E * d 3 r/2 is the absorbed power, and P s = Re S (E s × H * s ) · n S d 2 r/2 is the scattered power, H s being the scattered magnetic field and n S the outward normal on S. Since we are interested in a statisical description of the disordered medium, we introduce the ensemble average denoted by · · · , and write the fields and current density as a sum of an average value and a fluctuating part: X = X + δX with X ∈ {E, E s , H s , j} (note that δX = 0, and that E 0 = E 0 since E 0 is deterministic). Introducing these expansions into the expressions of the extinction, absorbed and scattered powers, they can be cast in the form P = P + P , where P is the component involving average quantities X and P the component involving fluctuations δX (see Supplemental Material, Sec. I). Noting that P e = P e , energy conservation becomes on average P e = P a + P a +P s + P s . An important result of multiple scattering theory states that the averaged field obeys a wave equation in an effective homogeneous medium [31,32]. Energy conservation for the averaged field can be written P e = P a + P s (see Supplemental Material, Sec. I). Combining the two preceding equations leads to P a + P s = 0. As a result, the averaged absorbed power can be written Note that P s ≥ 0 since it can be reduced to the integration of |δE s | 2 over a closed-surface encompassing the medium in the far field. For the sake of simplicity, we now assume that the medium is a slab of finite thickness and illuminated by a plane-wave E 0 which will be the situation of interest in the following. We consider the volume V to be a portion of the slab with cross section Σ. In that simple case, the power P a absorbed in the effective homogeneous medium (as seen by the averaged field) cannot exceed the incident power P 0 = Σ Π 0 · n Σ d 2 r where Π 0 is the incident Poynting vector and n Σ the inward normal on Σ. We finally have The derivation of this upper bound for the averaged absorbed power in a disordered medium embedded in a slab geometry is the first result in this Letter. This upper bound suggests three strategies to increase the absorbed power: (1) Inhibit P s while keeping P a approximately constant, (2) for a fixed P s , maximize P a as close as possible to P 0 , or (3) inhibit P s and maximize P a . Strictly speaking, finding the statistial classes of disorder that maximize the absorbed power is challenging. Indeed, P a and P s depend on transport parameters such as the effective refractive index and the scattering and absorption mean free paths, that have a complex dependence on the structural correlations of disorder [33]. In the following we will constrain the problem by considering media made of absorbing discrete scatterers dispersed in a transparent background, and characterized statistically by the spatial correlation in the positions of the scatterers. Furthermore, we will consider the special class of stealth hyperuniform disorder, that is known to produce P s ≪ P 0 for large wavelengths even in dense materials [22].
Hyperuniform point patterns are such that the variance of the number of points within a sphere of radius R increases slower than the average number of points when R tends to infinity [34]. For a set of N points with positions r j , this property is equivalent to saying that the structure factor vanishes when |q| → 0. First introduced for their interest in close packing processes [34], hyperuniform distributions of scatterers (hereafter denoted by hyperuniform materials) also produce materials of interest in wave physics, in particular due to their ability to produce bandgaps even in absence of periodicity [35][36][37][38][39][40][41]. Stealth hyperuniform materials is a specific class for which S(q) strictly vanishes on a domain Ω of typical size K around q = 0 [42,43]. The size K actually controls the degree of spatial correlations. For large K, the system is very constrained, generating short and long-range order. For K → 0, structural correlations are relaxed and the system tends to a fully disordered material. The degree of order in the pattern is usually measured by the ratio χ = [K/(2π)] d /(2dρ), with ρ the density of points and d the dimension of space. χ has to be understood as the number of constrained degrees of freedom (DOF) normalized by the total number of DOF. χ = 0 corresponds to an uncorrelated pattern, while χ = 1 characterizes a perfect crystal with infinite range correlations [42].
In the single-scattering regime, the scattered intensity is directly proportionnal to the structure factor S(q), and a stealth hyperuniform material does not scatter light for wavelengths satisfying λ > 8π/K, which corresponds to scattering wavevectors q lying in the domain Ω. In the multiple-scattering regime, transparency also holds for small q, provided that the effective scattering mean-free path ℓ H s is larger than the system size L [22] (we use the superscript H for quantities characterizing the hyperuniform material). In this regime, we have P H s ≪ P 0 , leading us to the idea that hyperuniform materials made with absorbing scatterers could be used to optimize the absorbed power, along the line denoted by strategy (1) above.
In order to support this idea, we present numerical simulations of wave scattering in two-dimensional model materials made of subwavelength electric-dipole scatterers distributed on a hyperuniform point pattern. To proceed, we generate 60 configurations of thickness L and transverse size 3L, following the algorithm in Ref. 42 (also described in the Supplemental Material of Ref. 22). A value χ = 0.44 is chosen, permitting to simulate large systems with a limited number of scatterers, although the behaviors discussed below are also observed for smaller values of χ. For simplicity we consider electromagnetic waves with an electric field perpendicular to the plane contain-ing the scatterers. Each scatterer is described by its elec- Here c is the speed of light in vacuum, ω 0 is the resonance frequency, Q = ω 0 /Γ is the quality factor, Γ = Γ R + Γ NR is the linewidth, with a radiative and a non-radiative contribution (absorption), and η = Γ R /Γ is the quantum efficiency (or albedo). From the polarizability, we can deduce the scattering cross-section σ s = k 3 0 |α(ω)| 2 /4, the extinction cross-section σ e = k 0 Im[α(ω)] and the absorption cross-section σ a = σ e − σ s of the scatterers, with k 0 = ω/c = 2π/λ the wavenumber in vacuum. Given a number density ρ of scatterers, we introduce as references the scattering and absorption meanfree paths in the independent scattering approximation (ISA, also known as Boltzmann approximation), defined as ℓ B s = (ρσ s ) −1 and ℓ B a = (ρσ a ) −1 . The disordered medium is illuminated from the left by a plane-wave at normal incidence (as shown schematically in the inset in Fig. 1). Maxwell's equations are solved using the coupled-dipoles method [44], and once the electric field illuminated each scatterer is known, the total absorbed power P H a is readily calculated (see Supplemental Material, Sec. II, for more details on the numerical procedure). Repeating the calculation on the set of generated configurations, an ensemble average is performed to obtain P H a . Running the same simulations on a set of configurations with uncorrelated scatterers, with the same density ρ, leads to a calculation of the average power in a fully disordered material P U a (we use the superscript U for quantities characterizing the uncorrelated material).
The dependence of P H a and P U a on the ISA absorption optical thickness L/ℓ B a is shown in Fig. 1 (a). The parameter L/ℓ B a = ρσ a L is chosen here as a measure of the intrinsic absorption level, independently of the spatial correlations in the medium. The power absorbed by the hyperuniform material is substantially larger than that absorbed in the fully disordered material with the same density. As shown in Fig. 1 (b), we observe a maximum enhancement P H a / P U a ≃ 1.8 for a thickness L ≃ ℓ B a . The propagation regimes in the hyperuniform and disordered structures can be analyzed, in order to get physical insight into the mechanism responsible for the absorption enhancement. The effective scattering mean free path in the non-absorbing hyperuniform structure is estimated to be ℓ H s ≥ 750 ℓ B s , based on a computation of the average field inside the medium (see Supplemental Material, Sec. III). This confirms that hyperuniform correlations, with the set of parameters chosen in Fig. 1, lead to transparency by suppressing the scattered power [22]. Moreover, we have verified numerically that the fluctuations of the absorbed power also vanish (see Supplemental Material, Sec. IV). We can conclude that the hyperuniform material in this regime behaves as an effective homogeneous and absorbing medium. The average absorbed power can be calculated analytically considering a homogeneous slab, with a complex refractive index given by the effective refractive index of the hyperuniform structure, also deduced from the computation of the average field inside the slab (see Supplemental Material, Sec. V). The result of the analytical calculation is represented in Fig. 1 (a), and nicely fits the numerical simulation, confirming the picture of an homogenization process. Regarding the uncorrelated medium, scattering is not suppressed, but the absorption curve can be fairly reproduced using a Monte Carlo simulation of intensity transport, describing the multiple scattering process in a medium with an effective refractive index n U eff = 1 + ρα(ω), and scattering and absorption meanfree paths ℓ U s = ℓ B s and ℓ U a = ℓ B a , as shown in Fig. 1 (a). It is also interesting to note that, although spatial correlations in the hyperuniform structures have a huge impact on the scattering mean free path, they weakly affect the absorption mean free path. This result was already put forward in Refs. 24 and 25 for hard spheres correlations, and is confirmed by numerical simulations for hyperuniform disorder (see Supplemental Material, Sec. VI). Indeed, with the parameters used in Fig. 1, we have ℓ H a ≃ 0.7ℓ B a . This calls for a simple random walk picture, that explains qualitatively the behavior of P H a and P U a . For the hyperuniform medium, we have ℓ H a ∼ ℓ B a < L ≪ ℓ H s , meaning that the photons travel allong the distance L without being scattered, with a high probability to be absorbed before escaping. For the uncorrelated medium, ℓ U s ∼ ℓ B s < ℓ U a ∼ ℓ B a < L, and a non-negligible fraction of photons are backscattered before being absorbed.
From the validity of the intensity transport picture, we also expect the absorption enhancement to be robust against changes in the direction of incidence and the illumination frequency. This is confirmed by the dependence of P a on the incidence angle θ 0 and the frequency ω displayed in Fig. 2. The dependence of P H a on θ 0 [ Fig. 2 (a)] is weak (and weaker than that observed for P U a ), except at large angles where finite-size effects start to play an important role (the transverse size of the medium is limited to 3L in the simulation). Using broadband scatterers (quality factor Q = 3), we obtain a large absorption enhancement on a broad frequency range [ Fig. 2 (b)]. The bandwidth with large absorption corresponds to the frequency range for which the condition ℓ H a ∼ ℓ B a < L ≪ ℓ H s remains satisfied. Finally, it is interesting to use the analytical models to analyze the absorption enhancement at very large optical thicknesses, that are not accessible to the full-wave numerical simulation. For the hyperuniform structure, we can calculate the power absorbed in an effective homogeneous slab with thickness L. We assume a dilute medium in order to neglect the index mismatch at the interfaces, and consider an absorption length ℓ H a ≃ ℓ B a (see Supplemental Material, Sec. VI). For the uncorrelated medium, we estimate the absorbed power using the diffusion approximation valid for L ≫ ℓ B s (see Supplemental Material, Sec. VII). Although not shown for brevity, the enhancement curves have the same shape as that in Fig. 1 (b), with an enhancement reaching P H a / P U a ≃ 7 for L = 600 ℓ B s and L ≃ ℓ B a . We can even estimate the maximum enhancement that can be expected using the analytical models (see Supplemental Material, Sec. VIII for details on the calculation). We obtain (L/ℓ B a ) optimum = 1.256 and In summary, we have demonstrated theoretically the existence of an upper bound for the absorbed power in a disordered medium, and analyzed the potential of hyperuniform materials for the design of strong absorbers of light (and other kinds of waves). Hyperuniform correlations can enhance the absorbed power very close to the upper bound limit, and the non-resonant nature of the mechanism provides broad angular and spectral robustness. Robustness to positional variations of the scatterers is also expected, since random displacements do not seem to destroy hyperuniformity [45]. Together with the development of self-assembly processes able to produce hyperuniform architectures [46,47], the results in this Letter could guide the design of disordered correlated materials with blackbody-like absorption. Let us consider a random variable X for which a statistical average exists and is denoted by X . From that, we define the fluctuation δX for each statistical realization by the deviation to the average: In the case of disordered media, this expansion can be applied for example to the electric field: where the statistics is performed here over all possible spatial configurations of the disordered medium. From this expansion, it is easy to compute the average intensity which reads where we have used by definition δE = 0. I ans I are respectively called the ballistic and the diffuse intensity. This expansion holds for any quadratic quantity such as the extinction, absorbed and scattered powers used in the main text: We thus have P e (ω) = P e (ω) + P e (ω) = P e (ω), P a (ω) = P a (ω) + P a (ω), where we have used E 0 = E 0 and δE 0 = 0 since the incident field is deterministic.

A. Ballistic beam
From Eq. (3), we see that the ballistic intensity is directly given by the average field. Thanks to the multiple scattering theory, we can show that this field propagates in a homogeneous medium with an effective refractive index n eff (ω) [1,2]. The imaginary part of this refractive index gives the attenuation of the average field as it propagates in the medium. This attenuation is called extinction and the typical distance over which the ballistic intensity is attenuated is the extinction mean-free path ℓ e . We can also show that this attenuation is given by absorption (mean-free path ℓ a ) and losses by scattering (mean-free path ℓ s ). We have the relation: The different powers relative to the effective medium read: where j(r, ω) = −iω P(r, ω) We also have P e (ω) = P a (ω) + P s (ω).
Equation (12) together with Eq. (14) are at the root of the computation of the absorbed power in the stealth hyperuniform medium (see Sec. V).

B. Diffuse beam
Regarding the computation of the diffuse intensity of Eq. (3), the problem is more complex. For large optical thicknesses, we can show that it is governed by a diffusion equation which is at the root of the computation of the absorbed power in the uncorrelated structure (see Sec. VII).

II. NUMERICAL COMPUTATION OF THE AVERAGE ABSORBED POWER
The numerical computation is done through the following process schematized in Fig. 1: (1) First, we generate 20 disordered configurations containing 90000 points in a square of size 3L. For the hyperuniform point patterns, we use the algorithm of Ref. 3 and uncorrelated point patterns are simply generated with a uniform random process. (2) To mimic a slab geometry and improve the statistical average, the generated configurations are divided into three bands of thickness L and length 3L. After dressing all points with a polarizability α(ω), Maxwell's equations are solved in these configurations for a plane-wave illumination from the left and using a coupled-dipoles method [4]. In this formalism and for 2D scalar waves, the exciting field on scatterer i is given by where E 0 is the incident field and G 0 is the Green function in vacuum. It connects the field to its point dipole source and is given by where H (1) 0 is the Hankel function of zero order and first kind. The computation of the exciting fields consists in solving the linear system given by Eq. (16). (3) Finally, to avoid finite-size effects again, the absorbed power is computed only for scatterers belonging to the central square of volume V using the relation and a statistical average is performed using the 60 different realizations of the disorder to get P a /P 0 .

III. AVERAGE FIELD EVOLUTION AND EFFECTIVE REFRACTIVE INDEX FIT FOR THE STEALTH HYPERUNIFORM STRUCTURE
To compute analytically the power absorbed by the stealth hyperuniform structures, we have to compute the evolution of the average field which allows us also to estimate the effective refractive index n H eff . The system of interest is depicted in Fig. 2. It is illuminated by a planewave at normal incidence given by First, we compute numerically the average field for several depths inside the slab. For that purpose, we compute for each configuration the field at any position using the expression E(r, ω) = E 0 (r, ω)+ k 2 0 α(ω) and the expressions of the exciting fields given by Eq. (16). Then we perform the statistical average. The numerical result is finally fitted with a standard Fabry-Perot approach which leads to the theoretical expression of the average field given by where r H eff = r exp(in H eff k 0 L), γ = r exp(2in H eff k 0 L), r = (n H eff − 1)/(n H eff + 1) is the Fresnel amplitude coefficient in reflection and t = 2/(n H eff + 1) is its counterpart in transmission. The fit is done using an optimization algorithm, based on a (global) genetic algorithm combined with a (local) Newton algorithm [5]. Note that in the most general case, the effective refractive index is non-local but it can be very well approximated by a simple complex constant for statistically homogeneous dilute systems. Figure 3 gives two examples for a stealth hyperuniform structure. From the effective refractive index, we get the where n H eff ′′ is the imaginary part of the effective refractive index.
In Fig. 3 (a), there is no absorption and the extinction mean-free path can be assimilated to the scattering mean-free path. We clearly see the very slow decay of the intensity of the average field. which means that ℓ H s ≫ L and the structure is transparent. The fit gives for Fig. 3 (a) ℓ H s ∈ [750, 1200]ℓ B s . This large window results from large incertainties because of a small decay. On the contrary, absorption is present in Fig. 3 (b). In that case, we cannot discriminate between the scattering and the absorption processes through the evolution of the average field. However, as pointed out in Sec. IV, the fluctuations of the absorbed power vanish, which probably proves that ℓ H s remains large compared to the size of the system L. Thus the imaginary part of the refractive index gives here a direct access to the absorption length. The fit gives for Fig. 3

IV. STANDARD DEVIATION OF THE ABSORBED POWER DISTRIBUTION
To check the transparent character of the stealth hyperuniform structure for the set of parameters used in this work, we can compare the standard deviation of the absorbed power distribution for both types of correlations (hyperuniform and fully disordered system). Figure 4 shows the computed ensemble standard deviation of the power absorbed σ Pa relative to the ensemble mean power absorbed P a . It indicates clearly the strong reduction of σ Pa for a stealth hyperuniform medium in the transparency regime, compared to an uncorrelated medium with the same intrinsic parameters, typically by one to two orders of magnitude. Furthermore, its amplitude in the hyperuniform case is more than two orders of magnitude smaller than P a , which indicates that P H a ≃ P H a . The power associated to absorption by the average field suffices to describe the absorption of a hyperuniform medium in the transparency regime for most applications. This relatively low standard deviation of the power absorbed is interpreted by the fact that a stealth hyperuniform medium in the transparency regime suppresses most of the ensemble fluctuations of the fields by definition of this regime. One may then expect that all the observables are described by the average fields. Note also that in the case of uncorrelated media, the larger the optical thickness L compared to ℓ U s , the closer P s → P 0 and typically the larger σ P U a relatively to P U a .

V. AVERAGE ABSORBED POWER FOR THE STEALTH HYPERUNIFORM STRUCTURE
Even in the presence of absorption, ℓ H s ≫ L and the stealth hyperuniform structure remains transparent. This means that the fluctuations of the field vanish and P H s ∼ 0 leading to P H a = P H a . Thus the power absorbed by the system is given by where R = |r| 2 is the factor of reflectivity in intensity of a single interface. We finally have which is the expression used in Fig. 1 (a) of the main text (with a fitted refractive index). parameters of interest such as the ISA optical thickness L/ℓ B s , the order degree χ, or disorder strength k 0 ℓ B s . As the extinction length is only given by the absorption length ℓ H a , the weak variability observed in the curves shows that ℓ H a is weakly affected by the presence of structural correlations. This is shown here for stealth hyperuniform media but it was already stressed for hardspheres correlations [6,7].

VI. ABSORPTION MEAN-FREE PATH AND HYPERUNIFORMITY
Nevertheless, a dummy fit by an exponential shows that the absorption mean free path deviates approximately by a factor ℓ H a /ℓ B a ≃ 1/1.4 ≃ 0.7 compared to the ISA absorption mean-free path and the normalized absorbed power is not exactly unity for very large absorption optical thickness. This comes from the fact that small corrections have to be taken into account due to the correlations in the scatterers positions, but also from the rather high density of scatterers (chosen due to computational constraints) and index mismatch at the interface [as shown by Eq. (26)].
For very dilute systems, the index mismatch can be neglected and assuming ℓ H a ∼ ℓ B a , we obtain a simplified fully analytical expression for the average power absorbed by the hyperuniform structure given by which is the expression used to evaluate the average absorption power in the stealth hyperuniform case of Fig. 6. Note that the assumption ℓ H a ∼ ℓ B a is a convenient approximation but is not strictly correct as specified above.

VII. AVERAGE ABSORBED POWER FOR THE UNCORRELATED STRUCTURE
In this section, we derive an analytic expression of the average power absorbed by an uncorrelated cloud using the diffusion approximation. Compared to Monte Carlo simulations, the derivation leads to a simple formula of the estimated gain for very large optical thicknesses. The diffusion approximation is valid (1) whenever the size of the cloud L is much larger than the scattering meanfree path ℓ B s , (2) under the weak absorption condition ℓ B s ≪ ℓ B a and (3) in dilute systems such that k 0 ℓ B s ≫ 1. The system is considered diluted enough to neglect index mismatch at the interfaces and to set ℓ U s = ℓ B s and ℓ U a = ℓ B a . Finally, as we consider uncorrelated point dipoles scatterers, the transport mean-free path is ℓ * = ℓ B s . The diffusion equation for the 2D slab geometry and for a plane-wave illumination at normal incidence is given by [8] where u is the energy density of the diffuse part of the beam and is the ballistic source term. To take into account properly the boundary conditions, we use u(z = L) + z 0 d u dz (z = L) = 0, where z 0 /ℓ B s = π/4 is the 2D extrapolation length [2]. The solution of this set of equations reads where and ℓ = ℓ B a ℓ B s /2. The power absorbed by the slab is given by where u(z) = u(z) + u(z) is the average energy density. We have u(z) = P (z)/c, thus we obtain which is the expression used to evaluate the average absorption power in the uncorrelated case of Fig. 6. To estimate the asymptotic maximum gain that we can get, we use Eq. (27) for the stealth hyperuniform structure. Regarding the uncorrelated system, we need to simplify Eq. (36) under the conditions L/ℓ B e ≫ 1, L/ℓ ≫ 1. By keeping only the second order terms of the expansion in ℓ B s /ℓ B a we obtain The second term may be used to estimate the relative error commited by keeping only the first term. For instance, for ℓ B a = 40ℓ B s , the relative error is approximately 25 % and for ℓ B a = 240ℓ B s , the relative error is about 10 %.
We then keep only the first term and we define the gain through the relation The maximum gain is obtained when From this expression, we predict a gain of the order of 4 for L = 200ℓ B s and 7 for L = 600ℓ B s (see Fig. 6). It is important to note that Eq. (40) gives only an order of magnitude of the maximum gain and refinements are needed to be more accurate by taking into account the real effective refractive index for both the hyperuniform and the uncorrelated structures and the multiple scattering process beyond the diffusion approximation.