# Spectroscopic Ellipsometry and Optical Modelling of Structurally Colored Opaline Thin-Films
## 3 Results
### Simulation  
As an illustration of the simulation outputs, Figure 3 shows the ellipsometric parameters \(\Delta(\lambda)\) and \(\Psi(\lambda)\) at incidence angle \(\theta_{l}=50^{\circ}\) and with low disorder (\(\delta=0\)). In order to extract parameter metrics from these simulations which can be meaningfully compared to experiments, appropriate fitting functions were developed for the \(\Delta\) and \(\Psi\) plots vs. wavelength (or frequency/energy). The parameter \(\Psi\) may be conveniently fitted to a Lorentzian resonance lineshape  
\[\Psi(\omega)=y_{0}+\frac{A}{\left(\omega-\omega_{0}\right)^{2}+B}, \tag{2}\]  
where the ratio of constants \(A/B\) is indicative of the amplitude of resonance at \(\omega_{0}\), and the FWHM may be evaluated as \(2\sqrt{B}\). As per the expected Kramer-Kronig consistent relationship between the Lorentzian \(\Psi\) spectrum and the dispersive \(\Delta\) spectrum [35], these metrics may also be extracted from a fitting, thus  
\[\Delta(\omega)=\frac{{y_{0}}^{\prime}+\frac{A^{\prime}}{\left(\omega-\omega_{ 0}\right)^{2}+B^{\prime}}}{1/\left(\omega-\omega_{0}\right)}. \tag{3}\]  
Particularly within these conditions of low disorder, the simulations also show the periodic modulation phenomena, with some similarity to the effects noted experimentally. However, the fringes here tend to occur over the entire wavelength range, rather than being prominent only in the infra-red end of the spectrum. At wavelengths outside of the resonance condition, this overall structure presents a dielectric layer, which is around  
Figure 2: Experimental data showing ellipsometric parameters delta (\(\Delta\), left) and psi (\(\Psi\), right) as a function of wavelength. Results from polymer opal films with 0, 5, 20, and 40 BIOS shear processing steps are shown in (**a–d**), respectively. 3 microns thick, and where interference phenomena may arise from the reflections at the substrate and superstrate interfaces. The disorder uncertainly introduced at higher values of \(\delta\) causes the fringing to become less pronounced (as is also seen in the experimental data trends). A rough calculation of thickness \(L\), based on the spacing of fringes (\(\Delta\lambda\)) being 50 nm in the near-IR at 900 nm, and taking into account the refractive index \(\overline{n}\) of 1.52 and incidence angle of 50\({}^{\circ}\), gives \(L=\lambda^{2}\sin\theta_{1}/2n\Delta\lambda=3.4\) microns, which is acceptably close to the notional value. The fringing in the experimental data (Figure 2) shows a tighter arrangement with smaller \(\Delta\lambda\), reflecting the greater thicknesses of the \"real\" sample structures. To attain a perspective of how the global simulation inputs can be varied to achieve a matching with the experimental data, in Figure 4, simulation outputs of line fitting of the ellipsometric parameters at spectral resonance are shown as the refractive index contrast (\(\Delta n\)) and Gaussian disorder factor (\(\delta\)) are varied. As expected, increasing \(\Delta n\) leads to a sharper optical resonance, with increasing amplitude and commensurate narrowing, which are both a strong function of \(\Delta n\) in the pertinent 0.15-0.25 range (Figure 4a). Increasing \(\delta\) rather has the effect of reducing the resonance amplitude and increasing the linewidth (Figure 4b). As \(\delta\) increases past around 25 nm (corresponding to a fractional layer-thickness variation of 27%), the spectral feature can no longer be clearly discerned. Figure 4: Simulation outputs of Lorentzian line fitting of the \(\overline{\
u}\) ellipsometric parameter at spectral resonance. In (**a**), refractive index contrast (\(\Delta n\)) is varied for a model structure where the Gaussian disorder factor is held constant at \(\delta=10\) nm. In (**b**), similar simulation outputs are displayed as \(\delta\) is varied for a model structure where \(\Delta n\) is held constant at 0.02. Transfer matrix models are averaged over 150 cycles, and indicative smoothing-spline fits are added to each of the plots.