# Spectroscopic Ellipsometry and Optical Modelling of Structurally Colored Opaline Thin-Films
## 2 Materials and Methods
### Modelling and Simulation  
We note that there are marked differences between the approach used here for the structurally colored periodic arrays and those standard methodologies used for dielectric layers with resonant oscillator absorption (e.g., Drude model) [32].  
To further elucidate the roles of both order/disorder and index-contrast in the polarimetric properties of POs, we have developed an optical multilayer quasi-model of the opaline structures, as developed from our earlier studies of the development of structural color [33]. Previous characterizations of POs have repeatedly confirmed the intuitive packing of the (111) plane at the top surface next to the PET laminate, and the subsequent stacking/layering normal to the plane. Hence representing the in-plane periodicity of the opals, rather than their exact three-dimensional structure, the model thus consists of non-absorbing \(\lambda/4n\) layer-pair stacks of the high and low \(n\) components giving resonant reflectivity at central \(\lambda_{0}\) corresponding to the experimental data, as illustrated in Figure 1d. Whilst the quasi-model focuses on generic matching to optical properties; as a rough consistency check when \(\lambda_{0}=\) 560 nm, we can compare the implied PS layer of thickness 89 nmto the core particle radius of 85 nm and the implied PEA layer thickness of 95 nm to the shell-to-shell thickness of \(\sim\)80 nm.  
Coherent superposition at the reflectance peak is disrupted by the cumulative phase shifts from thickness fluctuations (disorder) since the light penetrates to depths of 10-100 layers in such low \(\Delta n\) systems (i.e., Bragg length, \(L_{\text{B}}\)). Indeed, the model is restricted to 15 alternating pairs of layers, which corresponds to a total depth of around 3 microns, which is safely beyond the values for \(L_{\text{B}}\), as calculated in our earlier simulation reports [28]. The layers are also modified to have an inbuilt disorder, whereby there is a Gaussian-distributed thickness variation of width (standard deviation) \(\delta\), which accounts for deviations from perfect crystalline order within the photonic crystal. As a quantitative example, a value of \(\delta=15\) nm for \(\lambda/4\) layer thicknesses of 150 nm would represent a fractional disorder of 10%, and so on. The reflectivity amplitudes (\(r_{p}\) and \(r_{s}\)) of these model structures are calculated at specified angles of incidence using a transfer-matrix method averaged over many cycles [34], and the ellipsometric parameters, \(\Delta\) and \(\Psi\), are then determined across the spectral using the standard definitions as above. In order to give meaningful outputs that are representative of the time- and spatially-averaged bulk properties, we apply a \"Monte-Carlo\" type approach to average out the inherently chaotic variations over a number of cycles/iterations, until an acceptably stable and reproducible output is reached.  
Whilst both \(\delta\) and refractive-index contrast, \(\Delta n\), may be varied within this model, the values of the effective refractive index (\(\overline{n}\approx 1.52\)) and \(\Delta n\) (\(\approx\)0.02) may be set based on the semi-empirical micrographic analysis from [PERSON] et al., whereby the periodicity and amplitude in the \(\Delta n\) contrast have been ascertained as a function of sample depth [17; 29]. This value of \(\Delta n\) is lower than the notional \(n_{\text{PS}}\)-\(n_{\text{PEA}}\) value, on account of how the layer-by-layer stacking of the intercalated spheres translates into the one-dimensional interlayer model. The parameter \(\delta\) may then be varied in order to match the simulated ellipsometric parameters with the experimental data, and this is seen to give realistic spectral profiles with Lorentzian FWHM values of 50-60 nm. The fitting of these simulated outputs to appropriate analytical functions is detailed below in the Section 3.