Emergent scattering regimes in disordered metasurfaces near critical packing

Abstract

Disordered metasurfaces provide a versatile platform for harnessing near- and far-field scattered light. Most research has focused on either particulate topologies composed of individual, well-identified metaatoms or, to a lesser extent, semi-continuous aggregate topologies without well identified inclusions. Here, we uncover an intermediate ‘critical packing’ regime characterized by metasurface morphologies in which a significant fraction of metaatoms begin to connect. We experimentally demonstrate that, at this threshold, the properties of the scattered light abruptly change and, via a statistical quasinormal mode analysis, interpret this change as a marked transition in the statistics of the photon density of states. Unlike percolation in semicontinuous metal films, this transition affects not only the specular but also the diffuse components of the scattered light in a profound way. Our results introduce critical packing topologies as a novel design strategy for manipulating the spectral and angular characteristics of light using ultrathin optical coatings. Emergent functionalities include colour shifts in diffuse light driven by multiple scattering and surface whitening, with potential applications in display technologies—for example, to reduce glare in electronic screens.

Introduction

In solid-state systems, rapid changes in the electronic density of states (DoS) are often associated to spectacular phenomena, such as phase transitions^1,2,3. In random photonic materials, the analogue of the electronic DoS is played by the photon density of states (PDoS) and the states are quasinormal modes (QNMs) with complex energies, owing to non-Hermiticity induced by leakage or absorption⁴. The PDoS contains crucial information on the optical transport properties of the material^{5,6,7,8,9,10,11}, with important implications on light localization^12,13,14, bandgap formation^15,16 or lasing in the presence of gain with long-lived quasi-extended modes¹⁷. In this work, we report a phase-transition-like phenomenon in the context of dielectric disordered metasurfaces, revealing a close photonic analogue to some behaviour observed in electronic systems.

Optical metasurfaces are artificial surfaces in which the constituent inclusions (or metaatoms) are meticulously engineered from dielectric or metallic subwavelength elements to achieve tailored optical properties and functionalities^4,18,19. Among these, disordered metasurfaces hold a prominent place, both from fundamental^20,21 and applied^22,23 perspectives. They have led to new designs for optical encryption^24,25, light extraction^26,27, transparent displays²⁸, low-emissivity coatings²⁹, non-iridescent colouration^30,31,32,33, visual appearance design³⁴, and optical diffuser³⁵, among others.

This work introduces disordered metasurfaces operating at the transition between particulate and aggregate topologies, where metaatoms begin to connect (Fig. 1). In this unexplored regime, referred to as critical packing hereafter, we theoretically predict a sudden transition in the statistical distribution of QNMs when morphological parameters like the metaatom density or spatial correlation are varied. We experimentally verify that this transition results in significant changes in the behaviours of both specular and diffuse light, offering new opportunities for creating ultrathin fade-resistant coatings designed to implement advanced functionalities, unattainable through multilayer thin films^{22,23,30,31,32,33,34,36,37}.

Fig. 1: Metasurfaces transitioning around critical packing.

The figure illustrates the transition from particulate (blue) to semicontinuous aggregate (red) topologies through critical packing topologies (purple), where approximately half of the nanoparticles touch each other. All metasurfaces are created using electron beam lithography by writing disordered arrays of squares by adjusting three key fabrication knobs: the size, density, and spatial correlation of the squares (see “Methods” section).

The phase transition in our disordered metasurfaces shares similarities with the well-known system of semicontinuous metal films^38,39, yet it also reveals significant differences. Semicontinuous metal films consist of deeply subwavelength (typically $< 20$ nm) nanoislands, which exhibit much larger absorption cross-sections than scattering cross-sections and are arranged with limited short-range order. In contrast, our metasurfaces exhibit a different hierarchy: scattering dominates over absorption, and we achieve highly precise short-range spatial correlations. These differences help explain why semicontinuous metal films are predominantly studied for their absorption properties and specular responses²³, while in our metasurfaces, the spectral and angular behaviour of diffuse light plays a central role. Furthermore, the optical properties of semicontinuous metal films are often analysed through a geometric phase transition, where small, disconnected clusters merge into larger, connected clusters—ultimately reaching a percolation threshold that spans the entire system. In our metasurfaces, we also observe a geometric phase transition; however, here, the transition occurs when the metaatoms connect locally, forming small clusters well before a percolation threshold with spanning clusters is reached.

Hereafter, we focus on a set of disordered metasurfaces consisting of silicon resonant metaatoms on an optically inert glass substrate. This simple system is intentionally selected to isolate the effects that arise near the transition between particulate and aggregate topologies only. As illustrated in Fig. 1, we exploit three key control knobs—metaatom size, density, and packing fraction—of particulate metasurfaces to gradually explore the transition between particulate and aggregate structures. The central idea of this work is to examine the sudden changes in the optical response of metasurfaces around critical packing by tuning these variables.

Results

Manifold visual appearances around critical packing

To experimentally investigate the light scattering transition between particulate and aggregate topologies, we designed a series of disordered metasurfaces written in a negative resist using electron beam lithography and then transferred into a 145 nm—thick silicon layer on a glass substrate (Fig. 2a and “Methods” section). The metasurfaces are circular with a 300 ${\rm{\mu }} {{\rm{m}}}$ diameter. They are fabricated on the same sample to reduce fabrication variability. All the electron beam patterns consist of random arrays of square features, whose centres are positioned using a Poisson disk sampling algorithm⁴⁰. The $\left(x,y\right)$ coordinates of the centres are formatted into the graphic data system (GDS) files.

Fig. 2: A great variability of visual diffuse and specular responses is obtained with a restricted set of manufacturing parameters.

a SEM image of a metasurface. The scale bar represents 500 nm. b Ping-pong ball setup used to visualise the diffuse component of the BRDF upon illumination with a focused supercontinuum laser beam. c Photographs of the ping-pong ball show the diffuse responses for 27 metasurfaces under 45° incident illumination, with nominal square features of side length ${\ell}=95,130$ and $170\,{{\rm{nm}}}$, densities $\rho={\mathrm{2,5}}$ and $10\,{{{\rm{\mu }}}{{\rm{m}}}}^{-2}$, and packing fractions $p={\mathrm{0.1,0.3}}$ and $0.5$. The two white dots indicate the incident and specular directions. Specular colours recorded at 10° incidence for the metasurfaces are displayed as uniformly coloured rectangles. SEM images depict the arrangement, size, and density of the metaatoms. Note that the metasurface with ${\ell}=130\,{\mathrm{nm}}$, $p=0.5$ and $\rho=2\,{{{\rm{\mu }}}{{\rm{m}}}}^{-2}$ has been damaged during sample handling.

We vary three control knobs of the square features, their density, size and disorder correlation. Structural correlation is controlled by setting a minimum interparticle distance, $a$, between particle centres, resulting in an effective surface coverage or ‘packing fraction’ defined as $p=\rho \pi {a}^{2}/4$.

This process yields a comprehensive set of 27 metasurfaces. The designed patterns feature squares with lateral sizes ${\ell}=95,130$ and $170\,{{\rm{nm}}}$, densities $\rho={\mathrm{2,5}}$ and $10\,{{{\rm{\mu }}}{{\rm{m}}}}^{-2}$, and packing fractions $p={\mathrm{0.1,0.3}},$ and $0.5$. These values represent nominal target parameters, but different topological states emerge during fabrication due to spatial overlaps between the square features at certain densities, sizes, or disorder correlations. Scanning electron microscope (SEM) images of all metasurfaces are shown in Fig. 2c, with a few previously displayed in Fig. 1. The metasurfaces can be categorised into three topological states: particulate, aggregate, and critical-packing topologies, where particles begin to touch.

Both the specular and diffuse reflection responses of the metasurfaces are measured using a gonio-spectrometer setup and a slightly focused supercontinuum laser. After normalising the measurements with the diffuse response of a reference diffuser, which offers high diffuse and Lambertian reflectance, we derive the bidirectional reflectance distribution function (BRDF)⁴¹. This multidimensional function characterises how the metasurfaces scatter light across all viewing directions, depending on wavelength, incident angle and polarisation. In addition to these quantitative measurements, the light scattered off the metasurfaces is visualised on a centimetre-scale hemispheric diffusive screen.

A halved ping-pong ball serves as the screen, with an elongated band of apertures along the plane of incidence to prevent blocking the incident beams and their specular reflections (Fig. 2b). This simple setup provides a direct visualisation of the colour and scattered intensity produced by the metasurfaces, offering a qualitative representation of their visual appearances in reflection. Photographs of the ping-pong ball, capturing the diffuse light from all 27 metasurfaces, are taken using a Canon EOS 1000D DSLR. Specular reflection is also photographed by aligning the camera with the specular direction. Different camera settings are used for specular and diffuse light images to prevent saturation, but the settings remain consistent across all recordings, allowing for direct intercomparison of the images.

The 27 × 2 photographs capturing both the diffuse components of scattered light at 45° incident angle and the specular components at a 10° incident angle under unpolarized light illumination are displayed in Fig. 2c. They are presented alongside corresponding SEM images of the metasurfaces, with the specular light photographs appearing as rectangular colour boxes positioned just to the left of the SEM images.

Particulate metasurfaces are primarily located in the leftmost column and the upper portions of the central and rightmost columns of Fig. 2c. With the exception of the singular metasurface in the upper right corner, which will be discussed separately, they exhibit two dominant, vivid hues, primarily determined by the resonance frequency of individual metaatoms. For uncorrelated disorder, $p=0.1$, the colour and brightness of the diffuse light remain independent of the observation direction. In contrast, for correlated disorder, $p=0.5$, brightness is reduced around the specular direction.

This trend is a direct manifestation of the engineered correlations, quantitatively captured by the static structure factor, ${S}_{r}$ (see Supplementary Note 1, Supplementary Fig. 1d and ref. ²³). For weakly correlated patterns ($p=0.1$), ${S}_{r}$ is nearly constant, resulting in the observed uniform, isotropic scattering. As correlation increases (e.g. $p=0.5$), a pronounced dip develops in ${S}_{r}$ at small in-plane wavevectors (${q}_{{{\rm{||}}}}\approx 0$), which directly causes the reduction of diffuse light around the specular direction. Concurrently, a strong peak emerges for ${q}_{{{\rm{||}}}}\approx 2\pi \sqrt{\rho }$, creating the bright, ring-shaped halo of enhanced scattering. The dispersive nature of this peak, where its angular position depends on wavelength, is what gives the halo its distinct colour. Furthermore, as the particle density $\rho$ increases from $2$ to $5$ ${\rm{\mu }} {{{\rm{m}}}}^{-2}$ (moving from the left to the middle column in Fig. 2c), the primary peak of the structure factor shifts to shorter wavelengths (Supplementary Fig. 1b). This shift directly explains why the coloured halo appears to expand and change hue, providing another knob for tuning the visual appearance³⁴. These observations are further analysed in Supplementary Note 1, which presents additional variations in vivid hues obtained by adjusting the metaatom size parameter.

Metasurfaces operating near critical packing appear in the middle and lower sections of the second and third columns. They display pastel hues with low saturation, incorporating a significant amount of white light. Some of these metasurfaces, particularly at packing fractions of 0.5 and 0.3, exhibit an angle-dependent brightness, with a noticeable reduction of diffuse light around the specular direction. This confirms that correlation in the initial electron-beam pattern continues to affect scattering properties, even as the metaatom overlap is significant.

Aggregate metasurfaces, which form at high density and low packing fraction, are primarily found in the lower part of the third column. They are characterised by high lightness, close to white, with soft hues that slightly vary between the specular and backward directions. Unlike particulate metasurfaces, this angular-dependent colouring does not stem from correlated disorder, as aggregate metasurfaces emerge from a weakly correlated configuration and do not exhibit a reduction in diffuse light intensity around the specular direction. A closer inspection reveals pronounced shifts in hue as the system approaches and crosses critical packing, including both a progressive blue-shift and a general whitening of the diffuse light. These spectral trends are not immediately obvious in Fig. 2 alone, but will be quantitatively analysed in detail in the following sections.

Specular colours tend to be relatively dull for aggregate metasurfaces, as well as for particulate metasurfaces with low densities, $\rho=2$ and $5\,{{\mu} {{\rm{m}}}}^{-2}$, where blue-green or brown hues dominate. However, bright colours emerge near critical packing at a density of $10\,{{\mu}}{{{\rm{m}}}}^{-2}$, and these colours differ significantly from those observed in diffuse light. This distinction arises from the fundamentally different physical processes governing the specular and diffuse components of the scattered light. Specular reflection is formally associated with the average field in the material, which propagates in an effective homogeneous medium and is therefore subject to classical thin-film interference effects. This can lead to significant variations of the specular colour with small changes of metasurface parameters and illumination conditions. Diffuse reflection, on the other hand, is associated with the fluctuations of the scattered field around the average. Diffuse colours are mostly driven by the individual scattering elements dressed by the interaction with their neighbours, namely, for particulate metasurfaces, by dressed Mie resonances and particle pair correlations. Colour variations with the incident or scattered angles are often milder for the diffuse component than for the specular one.

Overall, the photographs in Fig. 2c and Supplementary Fig. 1 illustrate a striking diversity of visual appearances, including angular-dependent intensity, variations in lightness, whitening effects, distinct diffuse and specular colours, and changes in chroma as density and correlation vary. What makes this diversity particularly remarkable is the transition from vivid hues to softer tints near critical packing. Equally noteworthy is that this broad range of optical effects is achieved using simple, manufacturable samples on a neutral silica substrate. The fact that all samples share identical raw materials, uniform square-shaped e-beam patterns, and a fixed etching depth underscores the profound role of the nanoscale topology itself in generating such diverse appearances.

Statistical quasinormal mode analysis around critical packing

Let us now examine the cause of the sudden changes in the specular and diffuse reflectance arising as metasurfaces transition around critical packing. By analogy with solid-state systems, we consider the PDoS. Because of leakage in the air clads and absorption in silicon, the QNM frequencies are complex valued, and we thus define the PDoS as the normalised number of photon states available at a particular energy and given quality factor.

Modal analyses of waves in disordered systems often rely on point scatterers with electric dipole resonances to linearise the eigenvalue problem⁴². This restriction is completely relaxed hereafter by using full-field electromagnetic simulations and a supercell approach with Bloch boundary conditions in the lateral $x$ and $y$-directions and perfectly matched layers (PMLs) in the $z$-direction (see “Methods” section). To ensure the accuracy of our predictions, we systematically test them by progressively increasing the supercell size.

Our smaller supercells consist of approximately 30 silicon metaatoms and have a lateral size of $1.2\times 1.2\,{{\mu}}{{{\rm{m}}}}^{2}$. The metaatoms' side length is ${\ell}=100\,{{\rm{nm}}}$, and their height is $145\,{{\rm{nm}}}$. All metasurface patterns are generated using a Poisson disc sampling algorithm⁴⁰ for a fixed filling fraction of 0.2. The QNMs are computed and normalised using the freeware MAN (Modal Analysis of Nanoresonators)⁴³. For computational efficiency, we assume a fixed silicon permittivity of $20-0.5i$, an average value derived from literature data⁴⁴ for the visible spectrum ($\lambda=400$ ${{\rm{to}}}$ $750$ ${{\rm{nm}}}$); the negative imaginary part signifies optical loss under the ${e}^{i\omega t}$ time-harmonic convention used by our solver. Additionally, we approximate the electromagnetically inert silica substrate as air, introducing a planar symmetry at $z=0$, which reduces the computational time. Based on this assumption, we classify modes as even or odd according to the symmetry or asymmetry of the x-component of the QNM electric field. The PDoS maps are computed from a total of ≈7600 QNMs over 40 independent realizations, providing a statistically stable dataset. Additional QNM maps can be found in Supplementary Note 2.2.

To unify the varied structural configurations that arise from tuning the metaatom size, density, and correlation, we introduce an effective parameter, the metaatom overlap rate $\widetilde{O}$ defined as the fraction of metaatoms which are touching at least one neighbour. This fraction is calculated directly from the random pattern generated for the electron beam writer with the Poisson disc sampling algorithm. The overlap rate provides a direct measure of the structural connectivity and serves as a fundamental variable to track the system’s progression from the particulate ($\widetilde{O}=0$) to the aggregate regime.

Figure 3a summarises our key computational results for the PDoS maps for various topologies. To provide a comprehensive picture, each map uses two distinct colour scales: the main heatmap visualises the PDoS itself (right colour bar, logarithmic scale), while the horizontal colour bands above the maps represent the averaged QNM excitation probability (left colour bar, linear scale). The four panels show the progression through the different regimes: (i) particulate $\widetilde{O}=0\%$, (ii) and (iii) critical packing $\widetilde{O}=10\%$ and $27\%$ and (iv) aggregate $\widetilde{O}=49\%$. The corresponding packing fractions are (i) $p=0.5$, (ii) $p=0.2$, (iii) $p=0.16$ and (iv) $p=0.1$.

Fig. 3: Evolution of the PDoS when transitioning through critical-packing topologies.

All metasurface patterns are generated using a Poisson disc sampling algorithm⁴⁰ for Si metaatoms with lateral size ${\ell}=100\,{{\rm{nm}}}$ and height of $145\,{{\rm{nm}}}$. The filling fraction is 0.2. a PDoS maps of even electromagnetic QNMs for increasing overlap rate $\widetilde{O}$: (i) particulate ($\widetilde{O}=0\%$), (ii, iii) critical packing ($\widetilde{O}=10\%$ and $27\%$), and (iv) aggregate ($\widetilde{O}=49\%$). The corresponding packing fractions are $p=0.5$, $0.2$, $0.16$ and $0.1$, respectively. Colour bands above each map represent the averaged QNM excitation coefficients (see “Methods” section), while cyan squares mark the dominant single-metaatom modes—MDz, EDx and EQ. b Representative field maps showing the $x$-component of the QNM electric field (see Supplementary Fig. 2.8 for details). The black inset shows the MDz, EDx and EQ QNM fields in the ${xy}$-plane. c $\rho -p{{\mathscr{-}}}{\ell}$ phase diagram distinguishing particulate (blue) and aggregate (red) topologies. d Quantitative analysis of the PDoS evolution shown in (a) from fragmented clouds at small $\widetilde{O}$ values to a unique spanning cloud at $\Phi \approx 1$. Insets show the largest connected component for each PDoS map.

The spectral range covered by the maps corresponds to the visible region, within which the silicon subwavelength metaatoms exhibit several distinct low-loss resonances. These include a quadrupolar-electric (EQ) mode, a degenerate dipolar-electric (EDx and EDy) mode, and a dipolar-magnetic (MDz) mode. These modes are highlighted with bright cyan squares on the maps; the $x$-component of their electric fields in the ${xy}$-plane are shown in the black inset in Fig. 3b.

The maps in Fig. 3a reveal a clear redistribution of the QNMs in the complex frequency plane as the metasurfaces transition through critical coupling, from isolated clouds in (i) to a single, regular cloud in (iv).

PDoS diversity

For particulate metasurfaces (i), the PDoS map appears as a patchwork of distinct QNM clouds, each centred predominantly around the individual QNM frequencies. One prominent example is the high-$Q$ ($\sim 80$) cloud at ${\ell}/\lambda \sim 0.16$, which primarily results from the hybridisation of MDz modes that mostly radiate in-plane⁴⁵. A representative QNM from this group is Mode 1 in Fig. 3b(i). This mode exhibits a highly ordered field distribution localised in air, which is reminiscent of leaky guided resonances at ${k}_{\parallel }=0$ in grating waveguides⁴⁶. Additional QNM clouds emerge between the real frequencies of the individual modes due to hybridisation between different mode types—for instance, the cloud spanning $0.17{{ < }}{\ell}/\lambda < 0.2$ is driven by the interaction between EDx and MDz modes.

A distinguishing feature of the particulate PDoS map is the emergence of pronounced pseudogaps, with a notable example at ${\ell}/\lambda \approx 0.21$. These gaps arise from destructive interference among waves scattered by the strongly correlated metaatoms, analogous to the formation of photonic band gaps in crystals. The analogy is reinforced by the mode profiles: QNMs just below the gap localise their fields primarily within silicon (as in dielectric bands), whereas those just above the gap concentrate their fields in surrounding air (as in air band)⁴⁵.

At the other end, for aggregate topologies (iv), metaatoms organise into extended chains—comparable in size to the incident wavelength—that merge into irregular clusters. Due to substantial polydispersity in cluster size and morphology, the resulting QNMs are more sparsely and almost uniformly distributed across the visible spectrum. This leads to a single, broad QNM cloud with smaller $Q$-factors, in agreement with the fact that larger metaatoms are better radiators. This broad cloud contrasts sharply with the fragmented PDoS of particulate topologies and arises from the complete closure of the pseudogaps—a hallmark of the profound reorganisation of the electromagnetic mode structure at critical packing. Typical QNM field profiles are shown in Fig. 3b(iv) as Modes 4 and 5. Their electric-field distributions differ markedly from those of isolated Si nanopost resonances: instead of being confined within the posts, the fields form ‘hot spots’ predominantly along the boundaries of the irregular Si clusters.

At critical packing, (ii) and (iii), the PDoS exhibits characteristics intermediate between the particulate and aggregate regimes. Even within the particulate topology, hybridised modes can shift by several linewidths from the resonance frequencies of isolated Si metaatoms, signalling the onset of strong inter-metaatom coupling. As the metaatoms begin to overlap, this coupling becomes the dominant factor shaping the PDoS, as seen in panels (ii) and (iii). In (ii), additional QNMs appear at lower frequencies, reflecting the formation of larger effective scatterers. In (iii), new hybridised modes emerge to fill the pseudogaps, marking the breakdown of the destructive interference that defines the particulate state.

Collectively, these features identify critical packing as the transitional regime where the PDoS undergoes a fundamental reorganisation. The nature of this transitional regime is exemplified by Mode 3 in Fig. 3b(ii), which exhibits a hybrid character: on isolated metaatoms (left side of the unit cell), the internal fields show a high degree of phase coherence, with each particle exhibiting a similar field profile. In contrast, when the metaatoms begin to connect, this local order breaks down, and the field profile becomes non-uniform, signalling the onset of strong, disorder-induced hybridisation.

QNM excitation probability

While the appearance of the PDoS map is informative, it is equally important to identify which modes are preferentially excited. The QNM excitation probability densities are represented by the colour bands above the PDoS maps in Fig. 3a and as colour dots in Supplementary Figs. 2.2–2.4, ranging from blue (weak excitation) to red (strong excitation). These probability densities are computed by averaging the QNM excitation coefficients over narrow spectral intervals (see “Methods” section for the detailed procedure), thereby highlighting the spectral regions of the PDoS that are most effectively excited.

A striking feature of the particulate topology (i) is the strongly bimodal nature of the excitation distribution: only QNMs within two narrow frequency bands, near the EDx and EQ resonances, have significant excitation. At critical packing (ii) and (iii), this bimodality persists, though the peaks broaden and the maximum excitation strength diminishes. In the aggregate regime (iv), the excitation distribution becomes more heterogeneous, forming a multicoloured pattern that reflects generally weak-to-intermediate excitations across a broader frequency range. This trend aligns with the equipartition theorem^47,48, which suggests that, in highly disordered or chaotic systems, energy becomes uniformly distributed among all accessible degrees of freedom.

The near-field distributions in Fig. 3b provide further insight into these excitation trends. High cooperativity, where many metaatoms exhibit induced polarisations with similar phases, leads to the strong, coherent scattering required for efficient excitation by the incident plane wave. This is clearly seen in the highly excited Mode 2. Conversely, the loss of long-range coherence in the aggregate regime, exemplified by the disordered field patterns of Modes 4 and 5, results in weaker, less efficient excitation. The hybrid nature of the critical packing modes, such as Mode 3, naturally corresponds to the intermediate excitation strengths observed during the transition.

Supplementary Note 2.3 examines the odd-symmetric modes, where ${\widetilde{E}}_{x}\left(z\right)=-{\widetilde{E}}_{x}\left(-z\right)$, and similarly identifies the critical packing transition as a pivotal point in the evolution of mode structure and excitation behaviour.

Phase-transition diagram

The appearance of the metasurfaces in Fig. 2 and the PDoS in Fig. 3a both suggest that the transition occurs when the metaatoms start to connect, irrespective of the specific trajectory chosen to transition, whether by varying $\rho,p,$ or $\ell$. Consequently, the metaatom overlap rate $\widetilde{O}$ is expected to be a universal, trajectory-independent parameter for tracking the phase change.

Figure 3c confirms this hypothesis. It provides a phase diagram that delineates the boundaries between particulate (blue) and aggregate (red) topologies across the full $\rho -p{{\mathscr{-}}}{\ell}$ morphological parameter space. The critical-packing regime is highlighted by two constant-$\widetilde{O}$ contours, corresponding to $10{{\rm{\%}}}$ (dashed) and $30{{\rm{\%}}}$ (solid) overlap rates. This phase diagram underscores that the critical-packing regime is trajectory independent and emerges as a general feature governed by the combined effects of all the morphological parameters.

To further reveal the impact of the transition in the PDoS maps, we further introduce the order parameter, $\Phi$, defined as the largest connected–component fraction. This metric, widely used in the statistical analysis of complex networks⁴⁹, measures the degree of fragmentation: $\Phi$ ranges from 0 to 1, with small values indicating highly fragmented networks.

Applied to probability density functions such as the PDoS, $\Phi$ is obtained by binarising the PDoS maps and computing the ratio of the largest cloud area to the total area occupied by all clouds. In other words, $\Phi$ represents the fraction of “active” PDoS pixels that belong to the largest connected cloud⁴⁹. Figure 3d plots $\Phi$ as a function of the overlap rate $\widetilde{O}$, showing a sigmoidal evolution toward unity, with a sharp transition near $\widetilde{O}=20\%$. This evolution marks a shift from fragmented QNM clouds separated by band gaps to a single, spanning cloud that extends across the entire visible spectrum. The coloured insets along the curve highlight this evolution, illustrating the largest connected region in each map.

Intuitively, as adjacent metaatoms begin to connect, the likelihood increases that additional metaatoms will also link up, leading to the rapid formation of larger clusters. This, in turn, results in a significant shift in PDoS connectivity. Thus, although Fig. 3a, d follow one specific trajectory in parameter space (with only $p$ varied), the abrupt transition in PDoS connectivity—and the associated pseudogap closure or optical response—should be considered a universal feature. This transition is expected to occur for any path that passes through the critical-packing regime. This universality is likely to hold when the optical wavelength is comparable to the metaatom size ($\lambda \sim n{\ell}$, where $n$ is the refractive index of the metaatoms), although the precise visual appearance (e.g. hue or saturation) may still depend on the specific trajectory taken.

Blue shift of diffuse light in dense particulate metasurfaces

Strong modal coupling between localised resonances is a key factor driving the widespread use of subwavelength metaatoms⁴⁶, particularly when their optical near fields strongly overlap^50,51. This principle is well established and widely exploited in ordered systems—such as small clusters of nanoresonators or periodic arrays—to finely tune the structural colours arising from collective resonances^52,53.

In contrast, the behaviour of disordered ensembles of nanoresonators remains less well understood. Due to the random spatial arrangement, the polarisations induced by multiple scattering fluctuate and tend to statistically average out^23,54, weakening coherent interactions. Consequently, strong coupling effects in disordered metasurfaces often manifest more subtly^54,55.

Figure 4 highlights a striking and unexpectedly blue shift that stands out as particularly remarkable in comparison. The shift is displayed for an incidence angle of 30°, with three ping-pong ball images obtained for $p=0.5$ and increasing densities. At a low density of $\rho=2\,{{\mu}}{{{\rm{m}}}}^{-2}$, multiple scattering is minimal. The diffuse light appears cadmium green, with an angle-dependent hue marked by a suppressed diffusion near the specular reflection direction and a bright halo marked with a dashed arc line. These features—specular suppression and halo enhancement—are not central to the present discussion and are addressed in Supplementary Note 1.

Fig. 4: Blue shift induced by multiple scattering just below the critical packing threshold.

Three particulate metasurfaces are presented, each designed with the same metaatom size (${\ell}=95$ ${{\rm{nm}}}$) and packing fraction ($p=0.5$), but with increasing densities: a $\rho=2$ ${\rm{\mu}} {{{\rm{m}}}}^{-2}$; b $\rho=5\,{{\mu}}{{{\rm{m}}}}^{-2}$; c $\rho=10$ ${\mu} {{{\rm{m}}}}^{-2}$. Measured and simulated in-plane BRDF maps are presented in the middle and right columns. The inclined streaks visible in the experimental data resemble speckle patterns, which arise from the small dimensions of the metasurfaces. Details of the algorithm employed to compute the correction factor can be found in Supplementary Fig. 3. Note that $C\left({{{\bf{k}}}}_{{{\rm{s}}}},{{{\bf{e}}}}_{{{\rm{s}}}},{{{\bf{k}}}}_{{{\rm{i}}}},{{{\bf{e}}}}_{{{\rm{i}}}}\right)\approx 1$ for $\rho=2$ and $5\,{{\mu}}{{{\rm{m}}}}^{-2}$. The correction factor matters only for the larger density, resulting in a blue shift of the BRDF peak. The ping-pong balls and the BRDF maps are collected at a 30° incidence angle. Similar results are observed at 15° and 45° incidence angles.

As the density increases to $5\,{{\mu}}{{{\rm{m}}}}^{-2}$, the halo shifts to larger angles. The shift is accompanied by a modest increase in brightness and the onset of a perceptible bluing effect. At a high density of $10\,{{\mu}}{{{\rm{m}}}}^{-2}$ just below the critical packing threshold, a vivid sapphire blue emerges. Further increases in density would lead to a whitening of the diffuse light due to metaatom overlapping, as discussed in the next section.

Given that the shape and size of the metaatoms are precisely controlled during fabrication, the blue shift is attributed to direct electromagnetic interaction between metaatoms, particularly through evanescent and radiative couplings that are enhanced at higher densities. In most studies of cooperative scattering in particulate systems, particles are treated as point scatterers with resonant electric polarizabilities, where multiple scattering is modelled through dipole-dipole interactions, as seen in research on cold atomic clouds^56,57 and micro- and nanoscale discrete disordered media^58,59.

Instead, our high-index, 145 nm-tall metaatoms support two electric dipolar modes, two magnetic dipolar modes, and one electric quadrupolar mode within the visible spectral range, see Fig. 3b. While it is possible to incorporate these multipoles using a closed set of equations⁶⁰, deriving a closed-form expression for diffuse light intensity that accounts for both multiple scattering and multipolar interactions is notoriously challenging, as it requires solving for the field-field correlation function²¹.

To model the blue-shift effect, we employ a recent model that incorporates multiple scattering effects in a mean-field manner^34,36, without requiring explicit consideration of the multipolar response. In this model, the diffuse contribution to the BRDF is expressed as:

$${f}_{{{\rm{diff}}}}=\rho \frac{d{\sigma }_{{{\rm{s}}}}}{d\Omega }\left({{{\bf{k}}}}_{{{\rm{s}}}},{{{\bf{e}}}}_{{{\rm{s}}}},{{{\bf{k}}}}_{{{\rm{i}}}},{{{\bf{e}}}}_{{{\rm{i}}}}\right){S}_{r}\left({{{\bf{k}}}}_{{{\rm{s}}}}-{{{\bf{k}}}}_{{{\rm{i}}}}\right)C\left({{{\bf{k}}}}_{{{\rm{s}}}},{{{\bf{e}}}}_{{{\rm{s}}}},{{{\bf{k}}}}_{{{\rm{i}}}},{{{\bf{e}}}}_{{{\rm{i}}}}\right),$$

(1)

where the first three terms—the density, the form factor $\frac{d{\sigma }_{{{\rm{s}}}}}{d\Omega }$ which describes how a single isolated metaatom scatters light and the structure factor ${S}_{r}$—neglect the electromagnetic interaction between metaatoms. Only the final term, the correction factor $C$, accounts for this interaction and is therefore responsible for the observed blue shift. In Eq. (1), ${{{\bf{k}}}}_{{{\rm{i}}}}$ and ${{{\bf{k}}}}_{{{\rm{s}}}}$ denote the incident and scattered wavevectors, and ${{{\bf{e}}}}_{{{\rm{i}}}}$ and ${{{\bf{e}}}}_{{{\rm{s}}}}$ represent the corresponding polarisations.

Without delving into the details provided in Supplementary Note 3, the correction factor is theoretically inferred in the model from a mean-field estimation of the field driving every scatterer in the presence of multiple scattering. Through energy conservation arguments, it is possible to show that the sum of the specular transmittance (${T}_{{{\rm{s}}}}$) and reflectance $({R}_{{{\rm{s}}}})$ plays a key role in inferring the mean field. In ref. ³⁴, these specular coefficients were inferred theoretically using advanced homogenization theories⁶¹.

Here, we instead use measured values of ${T}_{{{\rm{s}}}}$ and ${R}_{{{\rm{s}}}}$, improving resilience to topological and material variations and minimising potential inaccuracies in the homogenization theories. The spectral and angular maps of $(1-{T}_{{{\rm{s}}}}-{R}_{{{\rm{s}}}})$, used to infer the correction factor, are shown in Supplementary Fig. 3a. They are recorded with a 2-degree step variation in the incident angle and averaged over TE and TM incident light. Note that the maximum of ($1-{T}_{{{\rm{s}}}}-{R}_{{{\rm{s}}}}$) is blue-shifted for the denser metasurface for $\rho=10\,{{\mu}}{{{\rm{m}}}}^{-2}$, compared to the lower densities. Consequently, the correction factor peaks near this wavelength (Supplementary Note 3), suggesting that the averaged local field exciting the metaatoms is maximised due to electromagnetic interaction.

To quantitatively assess the model accuracy, we measure the BRDF of the metasurfaces and compare the experimental data with theoretical predictions from Eq. (1). The intermediate steps for calculating the correction factor and subsequently deducing the BRDF are detailed in Supplementary Fig. 3, which additionally includes colour maps of $C\left({{{\bf{k}}}}_{{{\rm{s}}}},{{{\bf{e}}}}_{{{\rm{s}}}},{{{\bf{k}}}}_{{{\rm{i}}}},{{{\bf{e}}}}_{{{\rm{i}}}}\right)$ as functions of wavelength and scattering angle.

The agreement between theory and experiment is notably good. The model effectively captures the BRDF increase and the resonant blue shift at high density. Remaining discrepancies likely stem from inaccuracies in modelling the metaatom shape and size, as well as moderate variations in these parameters, which significantly influence metaatom resonances.

The pronounced blue shift observed here is notably different compared to previously reported coupling effects induced by multiple scattering in high-density disordered metasurfaces^54,55. This shift points to a novel mechanism for harnessing structural colouration in disordered metasurfaces, paving the way for innovative design strategies that move beyond conventional periodic architectures.

Whitening effect across the critical packing transition

The previous effect arises for metasurfaces with tightly packed non-overlapping metaatoms. As the fabrication parameters are further tuned, the critical packing threshold is reached, and the diffuse light shows important hue variations.

These variations are illustrated in Fig. 5a for a fixed metaatom size ${\ell}=95\,{{\rm{nm}}}$. Two transitions from particulate to aggregate topologies are highlighted, either by decreasing the packing fraction from $p=0.5$, to $0.3$ and then $0.1$ (transition (i)) or by increasing the density from $\rho={\mathrm{2,5}}$ and then $10\,{{\mu}}{{{\rm{m}}}}^{-2}$ (transition (ii)), causing the non-iridescent blue and green diffuse colours of the ping-pong ball images to converge into a white hue. The whitening is confirmed by in-plane BRDF measurements showing the spectral and angular dependence of the diffuse light for an incident angle of 30° (Fig. 5a). As we cross critical packing, the initial spectral distributions of the particulate topologies initially centred in the blue-green spectral region suddenly broaden, covering almost uniformly the entire spectral range from 420 ${{\rm{nm}}}$ to 700 ${{\rm{nm}}}$.

Fig. 5: Whitening of diffuse light above critical packing.

a The BRDF and ping-pong ball images are obtained for a fixed metaatom size ${\ell}=95\,{{\rm{nm}}}$, by increasing the density or decreasing the spatial correlation for an incident angle of 30°. b CIE 1931 chromaticity diagram providing the colour coordinates (dark crosses) of the diffuse light in the direction perpendicular to the metasurfaces. The two blue dots indicate the white points for two standard illuminants, D65 and D50. In Supplementary Note 6, we present the same diffuse colours in the CIELAB 3D colour space, where brightness information (L*) is explicitly included. c Spectrum of $1-{T}_{{{\rm{s}}}}-{R}_{{{\rm{s}}}}$ for the whitest metasurfaces of the set of 27 metasurfaces and for ${\theta }_{i}=10^\circ$. The brightest one shown with a red colour corresponds to $p=0.1,$ ${\ell}=170\,{{\rm{nm}}}$ and $\rho=10\,{{\mu}}{{{\rm{m}}}}^{-2}$ in Fig. 2c.

To quantitatively assess the purity of the white hue of the spectrally broad BRDF, we map the ping-pong ball images onto the CIE 1931 chromaticity diagram. This diagram is widely used to correlate reflection or transmission spectra with the colours perceived by human vision. The two transitions are indicated by the trajectories (i) and (ii) in Fig. 5b, where we additionally mark the white point with coordinates (0.3127, 0.3290) with a blue circle. This point represents a standard reference for many colour applications with the D65 illuminant. Since our supercontinuum illuminant is different from the average daylighting, we also mark the white point with the D50 illuminant used in graphic arts, making it easier to evaluate the purity of the observed white, which is very close to the two marks. This trend is further quantified in Supplementary Note 6, where the 3D CIELAB representation confirms the trajectories’ convergence toward the white point in chromaticity, while also explicitly showing the marked increase in lightness (L*) that accompanies the visual effect.

The whitening phenomenon can be intuitively understood through the QNM analysis in Fig. 2. It can be attributed to an inhomogeneous broadening of aggregate topologies, where particles overlap and even cluster. Then, numerous modes, which typically cover all visible frequencies, are significantly excited by collimated beams, scattering light at all visible frequencies in all directions.

A key performance metric for white paint or high haze surfaces is the brightness, which quantifies the fraction of the light that is effectively diffused. High brightness values are typically achieved with materials that exhibit strong scattering, often involving complex multilayer structures or thick particulate coatings^21,62. Figure 5c presents the spectral dependence of the diffuse light—actually, $1-{R}_{s}-{T}_{s}$ includes $\approx 15\%$ Si absorption in the blue − for the two metasurfaces that generate the most pronounced whitening. Both metasurfaces have minimal correlation ($p=0.1$) and maximum density ($\rho=10\,{{\mu}}{{{\rm{m}}}}^{-2}$), with the only difference being the metaatom sizes, ${\ell}=95$ and $170\,{{\rm{nm}}}$. The metasurface with the larger metaatom size forms larger aggregates, resulting in a brightness approaching $80\%$ across the entire visible spectrum.

While this value is lower than the brightness achieved by certain animals⁶³ or ultra-white paints⁶⁴, the 145-nm-thick metasurfaces strike a relevant balance between brightness and thickness. This makes them a promising choice for compact, photonic CMOS-compatible devices in applications like materials science and lighting, particularly for reducing glare in electronic displays.

Broadband mirror effect at critical packing

The two preceding effects are observed for diffuse light. Despite their stark contrast in nature, both diffuse and specular lights are influenced by the PDOS and the QNM excitation probabilities. Thus, much like diffuse light, specular light is expected to exhibit rapid variations at critical packing.

This expectation is confirmed in Fig. 6, where we analyse the specular reflection spectra of five metasurfaces transitioning from particulate to aggregate topologies by increasing the silicon post size and disorder correlation. As expected, we first observe a colour transition, from forest green to a dark shade, passing through a distinct scarlet red at critical packing.

Fig. 6: Specular reflectance when transitioning through critical coupling.

a Transitioning through critical packing by increasing the metaatom size of particulate metasurfaces and further reducing the disorder correlation of aggregate metasurfaces. The reflected colour transitions from forest green to a dark shade with a hint of grey; at critical packing, it exhibits a scarlet red (it is teal green in transmission). b (i) Specular reflectance spectra ${R}_{s}$ near normal incidence (${\theta }_{i}=10^\circ$) for the five metasurfaces shown in (a). (ii) ${R}_{{{\rm{s}}}}/{T}_{{{\rm{s}}}}$ ratio between the specular reflectance and transmission for the three metasurfaces operating around critical packing (the same colours as in (i) are used). (iii) Specular reflectance at critical packing for different incidences, ${\theta }_{i}=10^\circ,30^\circ$ and ${50}^{\circ }$. In (ii) and (iii), the dashed curves represent computational results for an equivalent 145 nm-thick Si film on glass. The nominal metaatom density for all metasurfaces is $10\,{{\mu}}{{{\rm{m}}}}^{-2}$.

Further insight is gained from the reflectivity spectra in Fig. 6b(i). For the particulate metasurface with the smallest metaatoms, the specular reflectivity ${R}_{s}$ remains weak across the visible spectrum, peaking below $10\%$. This metasurface primarily transmits light coherently, with a specular transmission ${T}_{s} > 80\%$ for $\lambda > 500\,{{\rm{nm}}}$. Similarly, the aggregate metasurface with minimal correlation exhibits weak specular reflectivity. Here, the polydispersity of cluster sizes and shapes, combined with their weakly correlated positions, suppresses coherence, leading to dominant diffuse scattering across all wavelengths (Fig. 5).

This dual behaviour explains why aggregates can appear bright white in diffuse reflection while remaining dark in the specular direction. The broad continuum of resonant states ensures efficient scattering of all visible wavelengths into diffuse channels, giving rise to whiteness. At the same time, the randomised phases of these scattered waves suppress specular reflection. The effect is robust across particle sizes large enough to scatter efficiently in the visible, as illustrated in the last three rows of the third column of Fig. 2c.

Between these extremes, the intermediate metasurfaces exhibit significant changes in specular reflectivity as they approach critical packing. The peak reflectivity first increases, reaching $50\%$ in the red, before declining. Notably, the ratio ${R}_{{{\rm{s}}}}/{T}_{{{\rm{s}}}}$ at critical packing is particularly high, reaching $\approx 15$ (Fig. 6b(ii)).

A large coherent reflection of $50\%$ is a singular observation. It is neither explained by existing models nor intuitively anticipated by considering the morphology of the critical-packing metasurface in Fig. 6a. To contextualise this observation, we compare the metasurface reflectance with that of a benchmark highly coherent system: a Si film on glass with the same 145 nm thickness. The Si-film reflectance computed using tabulated refractive index data of silicon⁴⁴ are shown with dashed curves in Fig. 6b(ii–iii). It exhibits different characteristics, such as an absence of a marked peak for ${R}_{s}/{T}_{s}$ and a reflectance nearly independent of the incident angle. Interestingly, the maximum reflectance of $60\%$ is only slightly higher than that of the metasurface, highlighting the singular capability of the critical-packing metasurface to coherently reflect light despite its strong heterogeneities.

No robust theoretical framework currently exists for modelling metasurfaces near critical packing^{21,38,46,59,65,66}. However, to achieve a qualitative insight into the singular properties of the critical packing metasurface, we attempted to infer effective parameters using effective medium theory⁴⁶. For the two particulate metasurfaces, predictions based on an Extended–Maxwell–Garnett model⁶⁷ align well with specular reflectance measurements (Supplementary Fig. 4), by using the electric and magnetic dipole Mie coefficients of the individual metaatoms. As expected, the model is less quantitative at critical packing, where it significantly underestimates the reflection peak at 670 nm (Supplementary Fig. 4). While limited confidence can be placed in the model within this regime—where it exceeds its range of validity—we note an increasing role of the magnetic dipole resonance as the metaatom sizes increase to reach critical packing, where we find that the reconstructed permittivities and permeabilities are both remarkably close to zero at the peak wavelength.

The QNM analysis further supports the significance of magnetic effects. At critical packing, the excitation probability distribution for even-symmetry QNMs in Fig. 3a(ii) peaks around the electric dipolar modes of individual metaatoms, while the odd-symmetry QNM PDOS in Supplementary Fig. 2.5 peaks from magnetic dipolar modes. Since both figures indicate a single dominant QNM with comparable excitation efficiency—such as mode 3 for the even-symmetry QNM in Fig. 3a(ii)—we expect that the metasurface at critical packing exhibits both significant effective permeability and permittivity.

Discussion

By tuning key technological parameters—such as particle size ${\ell}$, density $\rho$, and packing fraction $p$—in silicon metasurfaces, we have investigated the rapid evolution of specular and diffuse optical responses across disordered structures transitioning from particulate to aggregate topologies. A host of interesting visual effects can transpire at the transition, such as a pronounced blue-shift in diffuse colour, a whitening of the diffuse light and a singular mirror effect, which is not intuitively anticipated from a morphology perspective. These results mark a significant advancement in a largely unexplored design space²³.

Our theoretical model of this transition reveals that, at critical packing, the resonant modes undergo a rapid evolution, accompanied by a geometrical phase transition in the PDoS. The model further highlights that the overlap rate $\widetilde{O}$ between metaatoms can serve as an effective unified parameter to track the transition in the full $\rho -p{{\mathscr{-}}}{\ell}$ parameter space. It also predicts that the transition occurs within the range $10\% < \widetilde{O} < 30\%$.

It is important to distinguish between critical packing and percolation in semicontinuous metal films, both of which involve geometrical phase transitions. In disordered metal metasurfaces, percolation refers to the formation of spanning clusters⁶⁸ that connect the entire sample, resulting in a sharp decrease in sheet resistance. These nearly continuous systems behave similarly to homogeneous films, dominated by extended networks of hot spots, which play a key role in absorption and nonlinear optics^38,39.

In contrast, the critical-packing phase transition examined here affects local connectivity, where only a small number of metaatoms are interconnected, forming small clusters. This occurs much earlier than percolation, long before a continuous film is established. The morphology remains particulate or aggregate, and the abrupt shift in optical behaviour arises from scattering, mode hybridisation, and diffusion rather than percolative transport. Thus, while this transition is reminiscent of percolation due to its geometrical phase transition, critical packing represents a distinct regime that broadens the design space for disordered metasurfaces.

The use of high-index-contrast Si subwavelength metaatoms with a few distinct low-loss resonances is central in the present work. By comparison, low-index-contrast metaatoms, such as those fabricated from polymers, are expected to exhibit much weaker optical effects. Metallic metaatoms, on the other hand, behave quite differently: at critical packing, they support numerous additional resonances with intense fields localised in the nanogaps between nearly touching particles. This produces a PDoS landscape distinct from that shown in Fig. 3a. Because these extra modes are lossy and span a broad spectral range, absorption strongly influences the critical-packing transition, diminishing the visibility of certain effects—such as the blue shift—that we observe in dielectric metasurfaces. However, these reductions may be offset by the striking emergence of metallic clusters.

Similarly, variations in the morphology of the metaatoms—such as changes in size or shape—do influence their resonance and thus the resulting colour. However, these variations do not hinder the observation of the transition phenomena. In particular, the pronounced dispersity of appearances in Fig. 2 is expected to remain robust against morphological fluctuations. A noteworthy special case arises with elongated particles, especially when their orientation is not random. In such situations, anisotropy is preserved within the metasurfaces, and new polarisation-dependent effects may emerge in both diffuse and specular light.

A deeper statistical characterisation of the QNMs, including spectral and spatial fluctuations of the PDoS, would represent an exciting avenue for future investigation. Such an analysis could reveal additional signatures of the transition and provide complementary insights beyond the mean statistics considered here⁶⁹.

We expect that metasurfaces engineered at critical packing will serve as platforms for a broad spectrum of applications, spanning both fundamental research and technological implementation, by enabling precise control over light–matter interactions. Moreover, their compatibility with scalable fabrication methods—such as soft lithography⁷⁰ and bottom-up colloidal lithography^71,72—and their robustness to imperfections make them promising for large-area production.

Methods

Sample fabrication

The metasurfaces were fabricated by etching structures in a 150 nm-thick polycrystalline silicon (Si-poly) layer. First, Si-poly was deposited on both sides of a 4-inch fused silica (FS) wafer by LPCVD at 605 °C. Then, Si-poly was removed from one face of the FS wafer using fluorine-based based reactive ion etching (F-RIE). The sample was then covered by a 160 nm-thick layer of a negative resist (maN2405—Micro Resist Technology) followed by a 40 nm-thick conductive layer (ELECTRA92—AllResist) to avoid charging effects during subsequent electron beam lithography.

Samples were exposed by EBL with a beam energy of 20 keV at a current of 135 pA with a stepsize of 4 nm and a nominal dose of 150 ${{\rm{\mu }}}{{\rm{C}}}/{{{\rm{cm}}}}^{2}$. Resist was developed using MF-CD−26 (Micropposit) at 20 °C for 50 s, rinsed with deionised water for 1 min and dried with nitrogen. Negative maN resist patterns were transferred in the Si-poly layer using F-RIE, and the remaining resist mask was removed by oxygen plasma.

Depending on density and applied dose, one can vary the size of the squares in the resist after development using proximity effects. Increasing the dose can then lead to fusion of squares as it can be seen in Figs. 1, 2, 5 and 6.

Visual observation with the ping-pong ball

A $200\,{{\mu}}{{\rm{m}}}$ supercontinuum laser beam (Leukos, Rock 400) is slightly focused onto the metasurface samples. A semi-translucent half-ping-pong ball is positioned at the centre of the metasurface, serving as a hemispherical screen for the diffusely reflected light. The images of the ping-pong ball are recorded with a Canon EOS 1000D camera mounted on an optical table with fixed exposure times for each series. Full details on the characterisation of diffuse and specular light, including the diffusive transmission properties of the ping-pong ball, are provided in Supplementary Note 5.

BRDF characterization

BRDF measurements are conducted using a custom-built goniospectrophotometric setup equipped with a supercontinuum laser and a 1 mm-diameter optical fibre connected to a spectrometer (Ocean Insight, HDX). The incidence (${\theta }_{i}$) and scattering $\left({\theta }_{s},{\phi }_{s}\right)$ angles are precisely controlled by three stepper motor rotation stages (Newport URS75, URS150, and SR50CC).

To ensure measurements in the visible range, the unpolarized supercontinuum laser beam is filtered using a short-pass filter (Schott KG-1). The incident laser radiant flux is measured within the same setup, with the fibre detector aligned to capture the focused laser beam.

Electromagnetic computational results

The QNMs are computed for a null in-plane wavevector with the QNMEig solver of the freeware MAN⁴³. This solver automatically identifies a large number of modes near a user-defined complex frequency. The computed eigenvectors include both QNMs and numerical PML modes. To eliminate PML modes, we implement a systematic procedure detailed in Supplementary Note 2.1. After filtering, we normalise the QNMs^43,73 and rank them based on their excitation coefficient⁴³

$$|{\alpha }_{m}| \sim |2\left({\varepsilon }_{{Si}}-{\varepsilon }_{b}\right){Q}_{m}{{\int}{\int}{\int}}{{{\bf{E}}}}_{b}\cdot {\widetilde{{{\bf{E}}}}}_{m}{d}^{3}r|,$$

(2)

for a normally incident plane wave at a frequency matching the real part of the QNM frequency. Here, ${\varepsilon }_{b}=1$ and ${\varepsilon }_{{Si}}=20-0.5i$ denote the background and silicon relative permittivities, respectively. The negative sign of the imaginary part reflects COMSOL’s use of the ${e}^{i\omega t}$ time-harmonic convention, where a negative imaginary component corresponds to material loss. The spatial overlap integral between the incident electric field ${{{\bf{E}}}}_{b}$ and the normalised QNM electric field ${\widetilde{{{\bf{E}}}}}_{m}$ is evaluated within the silicon posts of the supercell.

Figure 3a using even-symmetric QNMs collected from 40 independent metasurface realisations. Twenty of these 40 realisations are displayed in Supplementary Figs. 2.2−2.4, where each dot represents a QNM complex frequency, with the colour encoding the excitation coefficients (blue for low values, red for high). In each realisation, only a small subset of QNMs is strongly excited. On average, approximately 190 QNMs are computed per realisation, resulting in a total of around 7600 QNMs used to estimate the PDoS maps. Additional realisations for odd-symmetric modes at critical packing density are presented in Supplementary Fig. 2.5, with the corresponding PDoS map shown in Supplementary Fig. 2.6.

Importantly, note that consistent results are observed for supercells four times larger (Supplementary Figs. 2.7 and 2.8), reinforcing our confidence that the statistical data from the 1.2 × 1.2 ${\mu } {{{\rm{m}}}}^{2}$ supercells are sufficiently converged to support our conclusions on the transition of the PDoS across particulate, critical-packing, and aggregate topologies.