Sample design and simulations

We first consider a generalised model composed of a simple optical cavity, with coupling rate 1∕τ e and intrinsic decay rate 1∕τ k0 for the kth mode supported by the system, as illustrated in Fig. 1a. Based on CMT, the coupling efficiency for the kth mode to the cavity can be expressed as the following:

$$\eta =\frac{2{\tau }_{e}/{\tau }_{k0}}{{(1+{\tau }_{e}/{\tau }_{k0})}^{2}}$$ (1)

with details shown in Supplementary Note 1 and ref. 10. The optimised coupling (absorption) occurs when the two rates match each other, similar to impedance matching in the electronic systems. For an optical system with intense disorder, the decay rates (1∕τ k0 ) converge to a constant and consequently the optical energy can be equally distributed to each mode10. Such energy equipartition mechanism equips the disordered system with a broadband response, as shown in Fig. 1b. The upper panel illustrates the situation when the decay rates (colourful arrows) matches the coupling rate (black arrow), leading to high coupling efficiency for all the modes. Consequently, a broadband absorption is achieved, as shown in the lower panel in Fig. 1b. Interestingly, the system can transit to another regime with band-limited response by a simple operation. As illustrated in Fig. 1c, when the decay rate of a specific mode is suddenly reduced, a mismatch between 1∕τ e (black arrow) and 1∕τ k0 (green arrow) is induced with a significant reduction of the coupling efficiency for that mode (upper panel). As a result, the disordered system cannot hold (absorb) that mode but must release it, forming a band-limited reflection or transmision (as shown in the lower panel). In addition, the freedom of selecting modes in different spectral position allows tunability of the system, producing absorption dip at desired position.

Fig. 1: A disordered system with transition from broadband absorption to band-limited reflection/transmission based on coupled mode theory. a Schematic for a generalised optical cavity with coupling rate τ e and intrinsic decay rate τ k0 for the kth mode supported. b Broadband absorption regime. Upper panel: the disorder induces the convergence of intrinsic decay rates, leading to energy equipartition with similar coupling efficiency η for all the modes. Lower panel: broad absorption is achieved, computed with 25 modes with τ k0 converging within [0.8, 1.2]τ e . c Band-limited regime. Upper panel: sudden reduction of intrinsic loss of a specific mode; lower panel: an absorption dip is formed by enlarging the intrinsic decay τ k0 (by 20) of specific mode. d A realistic design matching the CMT model, composed of a disordered plasmonic system on a cavity with plasmonic mirror beneath. The thickness t can tune the intrinsic decay rate of a specific mode, driving the system from a broadband absorption (upper panel, t is too small to hold a mode) to band-limited reflection (upper panel, with a mode confined around the plasmonic structures). Full size image

Inspired by this model, we design a reconfigurable disordered system based on a disordered assembly of plasmonic nanoparticles on top of a planar optical cavity, as demonstrated in Fig. 1d. We choose the material of the cavity as a transparent (lossless) one, so that the light trapped inside would not be dissipated. The plasmonic system is composed of nanometre-sized silver particles with randomised size and position, providing the required disorder. The nanoparticles are deposited on a dielectric spacer above a silver substrate, forming a Fabry-Perot-liked system. A plasmonic mirror is introduced at the backside to improve the confinement of the cavity. More importantly, such a mirror plays an indispensable role to enhance the light-matter interactions. Together with the metallic nanoparticles on top, it reflects the photons and lets them be absorbed by the plasmonic network (more details in Supplementary Note 4). When the spacer thickness is too small to support any Fabry-Perot mode in the visible, the randomised nanoparticles perform as a system with broadband absorption as shown in the upper panel of Fig. 1d, owing to the disorder-induced energy equipartition. The system transforms to another regime if a certain optical mode can be confined, as depicted in the lower panel of Fig. 1d. The Fabry-Perot structure behaves like an external optical environment that can tune the optical response of the system19,20. Here, the configuration of a Fabry-Perot cavity constrains a specific mode inside the dielectric with tiny loss, reducing the absorption of the nanostructures, or equivalently reducing the intrinsic decay rate 1∕τ k0 for the target mode. Owing to the mismatch with the coupling rate, this mode will be reflected instead of being absorbed, generating a band-limited response as desired. Controllability is readily achieved by straightforwardly tuning the spacer thickness to release the light in particular modes. From another point of view, the disordered system reverses the reflection property of the cavity. The photons preserved in the cavity is reflected back when other photons are absorbed by the disordered system, while a pure Fabry-Perot cavity always captures (absorbs) the photons around its resonance.

Next, we implement full-wave simulations based on the 3D FDTD method for the elucidation of the mechanism demonstrated above, which are summarised in Fig. 2. Starting from a periodic array of nanoparticles (Fig. 2a), we embed disorder into the system by introducing fluctuations to both diameters and positions of the nanospheres, represented by a parameter α that describes the degree of disorder (details in the Methods). Here, the thickness-dependent optical response is analysed with three setups with different values of α = 0, 0.2, 0.5, as depicted in Fig. 2a–c. Figure 2d–f demonstrates the corresponding simulated reflection spectra at different thicknesses respectively. When the thickness is small (t = 50 nm), the increasing disorder assists the plasmonic system to trap more photons inside, reducing the reflection as shown in Figs. 2d to f. For the system with intense disorder (α = 0.5), a reflection band is produced with increase of the thickness after a certain transition region (as shown in Fig. 2f); the spectral position of the reflection peak is controlled by the thickness. A comparison among Fig. 2d–f also clarifies the indispensable role of disorder in producing bounded reflection—only the system with broadband absorption capability can release otherwise trapped photons throughout the tunable regime. To provide more details, we illustrate intensity distributions of the electric field on x–z and y–z planes, for both the broadband absorption and tunable reflection regimes in Fig. 2g, h respectively. The size and position distribution of the nanoparticles are fixed to be the same as that of Fig. 2c, but with different spacer thickness. The situation of broadband absorption is shown in Fig. 2g while a spectrally confined reflection band is depicted in Fig. 2h. In Fig. 2g, most photons are trapped inside the plasmonic system and dissipated, with miniscule reflection (above the black dashed line), while a significant portion of the photons are reflected for the case with mismatch between 1∕τ e and 1∕τ k0 (Fig. 2h).

Fig. 2: 3D full-wave simulations for the disordered plasmonic system with deterministic coupling to the optical environment. a–c Schematic diagram of platforms with different disorder parameters α for a α = 0, b α = 0.2, c α = 0.5. d–f The corresponding reflection spectra R as a function of spacer thickness from FDTD simulations. d corresponds to setup in a, e corresponds to setup in b while f corresponds to setup in e. Interestingly, the introduction of disorder creates and narrows the reflection peaks. A prominent transition from broadband absorption to turnable reflection is observed in f. g, h Spatial distributions of the electrical field intensity in x–z and y–z plane for g t = 50 nm and h t = 110 nm. A plane wave is launched at z = 200 nm (blacked dashed lines) along the y direction with wavelength λ = 400 nm. Full size image

Experimental realisation

We realise the proposed system by producing Ag clusters deposited on a transparent dielectric with a gas-phase cluster beam technique21. A LiF spacer layer is placed on the top of a plasmonic mirror, whose thickness can be precisely controlled within nanometre precision through thermal evaporation (more details in the Methods). Figure 3a shows a typical scanning transmission electron microscope (STEM) picture of the disordered system (Fig. 3b) with a 3D reconstruction plot in the inset for more detailed morphology, demonstrating the disorder embedded in both the shape and position of the nanoclusters. Besides the direct visualisation of the randomness from Fig. 3a, we fabricate one sample with a small spacer thickness (~60 nm) in the broadband absorption regime, as shown in Fig. 3b. The experimental results indirectly demonstrate that the system carries strong enough disorder for the convergence of the decay rates τ k0 . At the thickness with a coupling rate τ e matching τ k0 , the system is endowed with strong absorption in a broad spectral region and produces a black colour. Figure 3c summarises the reflection spectra with a gradually increased thickness of the spacer from 60 to 400 nm. With increasing of thickness, a distinct transition is observed in the spectra, as predicted by both the theoretical and numerical analyses. Starting from a small thickness, a broadband absorption is achieved by virtue of energy equipartition. After a transition region, the system enters the regime of tunable reflection, obtaining a band with red-shifted reflection peaks. Another reflection band also appears due to the second-order mode supported by the Ag film/spacer/Ag clusters structure. Figure 3d shows the averaged reflection \(\overline{R}\) in the visible, with corresponding thickness illustrated for different spectra in Fig. 3c. The transition from broadband absorption to tunable reflection is verified by the value of \(\overline{R}\), from a monotonical increase to oscillation. The relationship between reflection peak λ p and thickness of the dielectric layer is plotted in Fig. 3e. According to our theory, the mode confined in the Ag film/spacer/Ag clusters would not be absorbed but released, with its reflection peak located at λ p,CMT = 2n s t∕m, with n s the refractive index of its spacer, t the thickness and m the order of the mode.

Fig. 3: Experimental realisation of the disordered plasmonic system. a A typical STEM picture of Ag nanoclusters. The inset shows the 3D reconstruction plot. Disorder is introduced to the system by the deformed shapes and randomised positions of the clusters. b A sample with a fixed thickness around 60 nm, producing a completely black colour due to the disorder-induced broadband absorption. c The reflection spectra of the structure at different thicknesses, with reflection peaks marked as dashed lines. The transition is experimentally observed from broadband absorption to band-limited reflection with thickness-dependent reflection peaks. d The averaged reflection \(\overline{R}\) from 350 to 750 nm as the thickness varies. e The relationship between the peak shift and thickness from experiment, which shows a good match with our model based on coupled mode theory (red dashed lines). f STEM pictures for the hybrid platform with four different nanoparticle densities. g The corresponding reflection spectra for samples with different densities. The inset shows a photographic image, illustrating the vivid colour generation from the sample with stronger disorder. Full size image

To clarify the indispensable role of disorder for the proposed system experimentally, we fabricated systems with different nanocluster density, as illustrated in Fig. 3f, with coverage rates of 8.5%, 30.8%, 45.2% and 51.8%. When the density is low, the interactions among the clusters are negligable (as the coupling strength between the nanoparticle decays strongly with the increase of the distance), only the disorder in shape plays a dominant role. As the density increases, the interaction (near-field coupling) strength grows nonlinearly, intensifying the disorder embedded in the system. Figure 3g plots the reflection spectra of the samples with three different densities. Note that the amplitude of the reflection spectra is normalised to exclude the effect of absorption variation resulted from the increment of the density of plasmonic nanoclusters. The bandwidth shrinks with increasing density (disorder), agreeing with the result shown in Fig. 2d–f. The inset shows a picture of the samples with colour shining more brilliantly as the density increases. This visually shows the beneficial role that disorder plays.

To demonstrate the ability for continuous tuning, we fabricate a sample with a linearly increased spacer thickness along its diagonal (as depicted in Fig. 4a). Figure 4b provides a photographic image of the sample. After a transition from black at the bottom left corner, we observe a rainbow-like colour variation along the diagonal, demonstrating the high tunability of the reflection peaks. Naturally, such a platform can be used as a novel mechanism for structural colour generation22. Compared with the conventional methodology utilising a deterministic resonant structure to filter (absorb) unwanted light, our spacer thickness here is designed to release photons with a desired colour. From a practical point view, taming the disordered system formed by merely a single step of deposition brings a crucial benefit: it relieves the fabrication from sophisticated and/or time-consuming techniques (such as top-down process based on lithography) required for periodic nanostructures22,23,24. This fact is especially beneficial for colour decoration at large scales. In Fig. 4c, we plot the International Commission on Illumination (CIE) 1931 xy chromaticity diagram, with corresponding chromaticity calculated and marked (black dots) from the reflection spectra in Fig. 3c (see Supplementary Note 8 for more details). The colours have a broad range that can cover most of the standard Red Green Blue (sRGB) gamut (inside the white triangle), providing enticing prospects for colour displays. Quite interestingly, disorder endows the hybrid plasmonic system with a unique feature: randomness inside the system can counterintuitively enhance the resonance controllability (colour coverage). Compared with the CIE diagram from the simulation (Supplementary Note 7), the experimental results outperform the simulated ones, due to stronger disorder generated experimentally from fabrication imperfection, while in the numerical study, the nanoparticles are treated as perfect spheres with different radii for the simplicity of simulation and control of disorder. Further increment of the level of disorder in numerical calculation can enlarge the coverage in the CIE diagram (Supplementary Note 9). However, the intrinsic imperfection in the fabrication endows the realistic structures a better colour availability.

Fig. 4: The versatility of the disordered plasmonic system for colouration. a Schematic of the sample with Ag nanoclusters on a spacer with gradually varying thickness along the diagonal. b Photographic image of the structure depicted in a, with rainbow-like colours starting from black along the diagonal where coupling strength is tuned. c The diverse colour generation in CIE 1931 xy chromaticity diagram with black dots calculated from the spectra from Fig. 3c and the sRGB gamut outlined in white lines. d Digital copy of the painting (Peony Flower). e The fabricated hybrid structure. The scale bar represents 10 mm. Full size image

To highlight the feasibility of colourful pattern generation, we print the Chinese watercolour painting “The Peony Flower” by Baishi Qi, as shown in Fig. 4d–e. The original work is presented in Fig. 4d, containing several ubiquitously used colours such as red, pink, yellow and green. Meanwhile, it also requires black colour (for Chinese calligraphy) that can be easily achieved with our system working in the broadband absorption regime; we note that generating black remains as a formidable task for conventional periodic structures. In our system, all desired colours including black can be formed by simply tuning the spacer thickness to replicate the original work, as shown in Fig. 4e. Coloration at oblique angles is also investigated in Supplementary Note 11.