# Hyperspectral Molecular Orientation Mapping in Metamaterials
## 2 Experimental: Setups and Samples  
### Four-Polarisation Method  
The four polarisation method was demonstrated for IR absorbance mapping using linearly polarised light and measuring transmittance \(A(\theta)=-\lg(T(\theta))\) at angles \(\theta=0\), \(\frac{\pi}{4}\), \(\frac{\pi}{2}\), \(\frac{3\pi}{4}\)[19]. Three (out of four) Stokes parameters \(S_{0,1,2}\) can be obtained from the transmitted intensities \(I_{\theta}\): \(S_{0}=I_{\pi/2}+I_{0}\) (full intensity), \(S_{1}=I_{0}-I_{\pi/2}\) and \(S_{2}=I_{\pi/4}-I_{3\pi/4}\) (the normalised set is \(s_{0}=1\), \(s_{1}=S_{1}/S_{0}\), \(s_{2}=S_{2}/S_{0}\)). Different reflectivity and absorbance for the \(E_{x}\) and \(E_{y}\) components at the interface cause polarisation to become elliptical with the azimuth angle (orientation) \(\psi\) of an ellipse determined by \(\psi=\frac{1}{2}\arctan(S_{1}/S_{2})\). The phase angle \(\delta\) between the two perpendicular \(E_{x}\) and \(E_{y}\) components determines the ellipticity, which can be extracted when the \(S_{3}\) parameter is determined for which the intensity of the right-circular (RC) and left-circular (LC) polarisations has to be measured \(S_{3}=I_{RC}-I_{LC}\). Then, the \(\tan\delta=S_{3}/S_{2}\) and the ellipticity \(\chi=\frac{1}{2}\arctan(S_{3}/(S_{1}+S_{2}))\). For the four polarisation method, only linearly polarised light is used.  
Further development of the method allows separation of birefringence \(\Delta n\) and dichroism \(\Delta x\) caused by changes in transmitted intensity where the complex refractive index is \(\bar{n}=n+i\kappa\)[20, 21]. This can be achieved by virtue of angular dependence of transmittance \(T(\theta)\), which has a \(2\theta\) angular dependence for birefringence and \(\theta\) for absorbance. The absorbance follows a \(\cos\theta\) dependence ([PERSON] law, 1729), while the birefringence has \(\cos(2\theta)\) dependence ([PERSON] law, 1809). The two laws are accounted for by multiplication [20]. The \(\cos(2\theta)\) dependence of \(\Delta n\) reflects the fact that \(T\) has a minimum when the light's E-field is aligned with, or is perpendicular to, the two parallel polarizers. The absorbance meanwhile is always defined by the E-field projection onto the axis of the dipole transition, hence the \(\cos\theta\) dependence for \(\Delta x\).  
Anisotropy of absorbance due to orientation of the absorbers (dipoles) is defined by the [PERSON] law \(\cos^{2}\theta\) and has dependence of \(\cos(2\theta)\) (i.e., \(\cos^{2}\theta=[\cos(2\theta)+1]/2\)). By fitting the angular dependence of transmittance \(A(\theta)\propto Amp\times\cos(2\theta-2\theta_{\rm O})\), the amplitude of absorbance \(Amp\) and the initial orientation angle \(\theta_{\rm O}\) of the dipoles are obtained from the best fit of \(T\) at four angles \(\theta=0\), \(\frac{\pi}{4}\), \(\frac{\pi}{2}\), \(\frac{3\pi}{4}\)[19] (more angles can be used for the fit [21]). Only the data from fits with a regression coefficient \(R^{2}>96\%\) were selected for plotting as in previous the study [21]. The four polarisation method was used for characterisation of biopolymers and tissue [18, 22].