Manipulating disordered plasmonic systems by external cavity with transition from broadband absorption to reconfigurable reflection

Disordered biostructures are ubiquitous in nature, usually generating white or black colours due to their broadband optical response and robustness to perturbations. Through judicious design, disordered nanostructures have been realised in artificial systems, with unique properties for light localisation, photon transportation and energy harvesting. On the other hand, the tunability of disordered systems with a broadband response has been scarcely explored. Here, we achieve the controlled manipulation of disordered plasmonic systems, realising the transition from broadband absorption to tunable reflection through deterministic control of the coupling to an external cavity. Starting from a generalised model, we realise disordered systems composed of plasmonic nanoclusters that either operate as a broadband absorber or with a reconfigurable reflection band throughout the visible. Not limited to its significance for the further understanding of the physics of disorder, our disordered plasmonic system provides a novel platform for various practical application such as structural colour patterning.

Supplementary Note 1: more details of system modelling based on coupled mode theory The modes inside the optical cavity can be expressed as: with ω k the frequency, 1/τ k the decay rate (inverse lifetime) considering both the coupling and intrinsic loss 1/τ k = 1/τ k0 + 1/τ e . E k = |a k | 2 is the energy store in the k th mode. It obeys the relationship as the following (1): with S(t) the input pump. While the reflection R k and transmission T k for k th mode obeys (1): For a CW pump S(t) = e iωt , the a k can be solved as: For the total energy H stored in the cavity can be derived from equation (5): Considering the integral yields significant contributions only for ω ≈ ω k , the equation can be simplified to: The power P k = ∂|a k | 2 ∂t transferred into the k th mode can be also expressed as (from energy conservation): Combined with equations (3,4), the absorbance of the mode (coupling efficiency) η = P k /|S| 2 can be expressed as: η = 2τ e /τ k0 (1 + τ e /τ k0 ) 2 (9) .
When the system attaches to an external cavity which supports specific modes ω j , the corresponding decay rate 1/τ j0 is perturbed. The confinement of mode ω j in the external cavity mitigates the light-matter interaction in the original system, reducing the decay rate 1/τ j0 for the setup shown in the lower panel of Fig.1d in the main text. Despite the difficulties in writing an analytical solution of the variation of the rate (due to disorder), the decay rate qualitatively decreases, causing the mismatch to τ e . Consequently, the mode ω j supported by the cavity is released outside (absorbance η reduces ).

Supplementary Note 2: analysis for different types of disorder
In the main text, we analysis the case for the disorder in both size and position of the Ag clusters, to match the experimental configurations. Here we also implement simulations with either disorder in the size or the position to clarify the contributions from different types of disorder.
The results are summerized in Fig. S1. Similarly, The disorder in either shape or position may lead to the reflection band formations (Fig. S1C-D) at different thickness. However, the reflection bandwidth is increased for single type of disorder, which also verified by the reduced absorption A = 1-R at t=50nm according the mechanism we discussed in the main text.
This phenomenon verifies that the disorder level determines the sharpness of the reflection band, as demonstrated in the main text. In the fabrication process, more complicated disorder in shape and position is intrinsically induced, driving the system to a strong chaotic regime which is desired here.

Supplementary Note 3: simulations with different random seeds
To show the reflection tunability is a general effect resulted from the interaction between external cavity and the disordered plasmonic system other than a specific arrangement, we implement additional simulations with different random seeds (U 1 , U 2 and U 3 ). Figure S2 summarises the reflection spectra with three different sets of random seeds, with similar results compared with Supplementary Note 4: indispensable role of both disorder system and external cavity to achieve good tunability To demonstrate the tunability is achieved with both disorder system and the external cavity ( Fig. S3A-B), we implement additional simulations as the control groups, which is summarised in Fig. S3.  The fixed shutter has a square opening that only allows a section of the sample to be exposed to the evaporated materials at any given time. As the sample is constantly moving during the evaporation with a speed of v, the exposed section of the sample is decreasing, resulting in a gradient morphology of the film, as schematically shown in Fig. S6.

Gas-Phase Ag Cluster beam deposition
Gas-phase cluster beam deposition process was used to deposit Ag clusters on the surface of Ag film/spacer layer structure (5). In this fabrication, Ag clusters were generated in a magnetron plasma gas aggregation cluster source and deposited on substrates directly, as shown in Supplementary Note 6: sample characterizations STEM characterizations of the Ag clusters Figure S7B shows the HAADF-STEM image as-deposited Ag clusters. As seen, Ag clusters are randomly distributed on the substrate and form numerous closely spaced cluster-assembling areas. These randomly distributed Ag clusters on the substrate resulted in the formation a disordered plasmonic system. Figure S7C shows

SEM characterizations of the hybrid system
The structure of the Ag film/LiF/Ag cluster sample was characterized by scanning electron microscopy (SEM, Hitachi S4800). Figure S8A show the optical photographs of blank silica, Ag film/LiF sample and Ag film/LiF (60nm)/Ag cluster sample. Figure S8B show the optical pho-

Supplementary Note 8: details of color projection to CIE chromaticity diagram
According to The International Commission on Illumination (CIE), there is a standard method of specifying the color of illuminants or materials by tristimulus values X, Y , Z (8): with P (λ) the is spectral power, K a scaling factor. x(λ), y(λ), z(λ) color-matching functions (CMFs) correspond to three kinds of cone cells of human eyes in short ( "S", 420nm -440nm), middle ( "M", 530nm -540nm), and long ( "L", 560nm -580nm) wavelengths. For the reflective colors, the spectral power can be decomposed to into two terms P (λ) = S(λ) · R(λ), where R(λ) is the reflectivity and S(λ) is the relative power of the illuminant shining on the system.
The tristimulus values X, Y , Z are normalized to CIE xy color space, with x,y containing the only chromaticity information (brightness is missing due to normalization): Here, we choose color-matching functions as the CIE 1931 2 o Standard Observer Observer function, as demonstrated in Fig. S10A. For the illuminant in the above section, we select the CIE Standard Illuminant D65 as S(λ) (9), which corresponds to average daylight and has a correlated colour temperature of approximately 6500 K (as shown in Fig. S10B). We also calculate the CIE xy chromaticity diagram with a different S(λ) (9), that is Illuminant A representing typical, domestic, tungsten-filament lighting, as plotted in in Fig. S10B. The results are demonstrated in Fig. S10C, with a color shift to red compared to Fig.1h

Supplementary Note 9: an improved match for experimental results
In the main text, we demonstrate the trend that the reflection band narrows when the disorder increases. To demonstrate the mechanism, we simply model the nanoparticles as a perfect sphere and assume that the disorder level in shape and position is the same. Figure  Here we introduce more disorder to the simulation, i.e., treating the nanoparticle as an ellipsoid instead of a sphere. Now there are three degrees of freedom -lengths of the three principal axes instead of the diameter of the sphere. The length of in x axis is defined as And the rest of two lengths ( in y and z) is d y,z = d x [1 + α s y,z U y,z ]. Again, U x,y,z is independent uniform distribution in [-1, 1] while α s x,y,z is a control parameter for the disorder in shape. α p i is another control parameter for the disorder in position as defined in eq.
(3-4) in the main text. Figure S12 summarised the numerical results for a disordered plasmonic network with α s x,y,z = 0.25 and α p i = 0.3. The schematic of the platform is shown in the inset of Figure S12A, demonstrating the deformation from spheres to ellipsoids that introduces disorder in the other two degrees. The enhancement of the level of disorder is verified by the absorption spectrum around the critical coupling (t=60nm), as shown in Fig. S12A. The reflection spectra and the corresponding CIE diagram is shown in Fig. S12B and C, separately. An improved match to experimental data ( Fig. 3 and Fig. 4) is achieved owing to the disorder enhancement.

Supplementary Note 11: Reflection at oblique angles
We first investigate the situation with incident light at oblique angles, as demonstrated in Fig.S14A.
The spectra with different incident angle θ inc are summarised in Fig. S14B. Due to the sensitivity of the spacer effective thickness, the reflection deviates from normal incident as θ inc increases. The spectrum maintains its feature when θ inc < 20 o , which is further clarified by images of the reflected light beam demonstrated in Fig. S14C. The diffusive reflection under the normal incidence of the sample is also measured, with the setup illustrated in Fig.S15A. The reflection spectra shown in Fig.S15B are normalised to exclude the attenuation. The spectra of the diffusive light at different angles remain almost the same. The spectrum beyond θ c > 6 o is not available due to the minute intensity.