## Appendix A  
The index of refraction equations for water, calcite, quartz and cellulose used in this paper are given below. Calcite and crystalline quartz are birefringent materials with two orientation dependent indices of refraction, the extraordinary index for propagation along the direction of the optical axis and the ordinary index for propagation orthogonal to the optical axis. These were measured by [PERSON] [8]. \[n_{co}^{2}-1=0.73358749+\frac{0.96464345\;\lambda^{2}}{\lambda^{2}-0.019432520 3}+\frac{1.8283145\;\lambda^{2}}{\lambda^{2}-120.}\;, \tag{21}\]  
\[n_{ce}^{2}-1=0.35859695+\frac{0.82427830\;\lambda^{2}}{\lambda^{2}-0.0106689543 }+\frac{0.14429128\;\lambda^{2}}{\lambda^{2}-120.}\;, \tag{22}\]  
\[n_{qo}^{2}-1=0.28604141+\frac{1.07044083\;\lambda^{2}}{\lambda^{2}-0.01005859 97}+\frac{1.10202242\;\lambda^{2}}{\lambda^{2}-100.}\;, \tag{23}\]  
\[n_{qe}^{2}-1=0.28851804+\frac{1.09509924\lambda^{2}}{\lambda^{2}-0.0102101864 }+\frac{1.15662475\;\lambda^{2}}{\lambda^{2}-100.}\;, \tag{24}\]  
\[n_{cl}^{2}-1=\frac{1.124\;\lambda^{2}}{\lambda^{2}-0.011087}\;, \tag{25}\]  
\[n_{xq}=1.31405-2.02\times 10^{-6}T_{c}^{2}+\frac{0.01586-4.23\times 10^{-6}T_{ c}}{\lambda}-\frac{0.004382}{\lambda^{2}}+\frac{0.0011455}{\lambda^{3}}\;, \tag{26}\]  
The wavelength in the expressions above is in microns. \(n_{co}\) and \(n_{ce}\) are, respectively, the ordinary and extraordinary index of calcite. \(n_{qo}\) and \(n_{qe}\) are the ordinary and extraordinary index of crystalline quartz. \(n_{cl}\) is the index of pure solid cellulose measured by [PERSON] et al. [9]. \(n_{xq}\) is the index of pure water measured by [PERSON] [10] which is strictly only valid to its full accuracy between 0.4 to 0.7 microns. In that last expression, \(T_{c}\) is the temperature in degrees centigrade. However, we need for our approach an expression for the index of water that is valid in the near IR. [PERSON] et al. (1998) have proposed such an expression valid from 0.2 to 2.5 microns. We have verified that it does match with the available data and the Quan and Fry formula over its range of applicability. \[\frac{\left(n_{2s}^{2}-1\right)}{\left(n_{2s}^{2}+2\right)}\left(\frac{1}{p} \right)=a_{0}+a_{1}\overline{p}+a_{2}\overline{T}+a_{3}\overline{\lambda^{ \top}}+\frac{a_{4}}{\overline{\lambda}^{2}}+\frac{a_{5}}{\left(\overline{ \lambda}^{2}-\overline{\lambda}_{w}\right)}+\frac{a_{6}}{\left(\overline{ \lambda}^{2}-\overline{\lambda}_{w}\right)}+a_{7}\overline{p}^{2} \tag{10}\]  
In this expression we have:  
\[\overline{T}=\frac{T_{k}}{273.15}\]  
\[\overline{\lambda}=\frac{\lambda}{0.589}\]  
\[\overline{p}=\frac{\rho}{1\;\mathrm{g\;cm}^{3}}\]  
Formula (10) is the one we use in this paper because of the extended range we require. We have verified that it matches to one part in a thousand the index formula given by [PERSON] and [PERSON] and that it tracks closely the available experimental data on water index in the near IR and UV. If we assume that the orientation of the optical axis of the calcite and quartz grains is random, we need to compute the resulting average index as follows. The ordinary index \(n_{o}\) is the same no matter the angular orientation of the incoming ray with the optical axis of the crystal. The extraordinary index \(n_{e}\) varies as a function of the angles with respect to the optical axis \(n_{e}(\varphi,\theta)\). The shape of the variation is this spheroid defined by:  
\[n_{ce}^{2}-1=0.35859695+\frac{0.82427830\;\lambda^{2}}{\lambda^{2}-0.0106689543 }+\frac{0.14429128\;\lambda^{2}}{\lambda^{2}-120.}\;, \tag{22}\]0  
The light-wave propagation vector is \(k\) and its angular frequency is \(\omega\) with the speed of light being given by \(c\). The optical axis is along the z direction. Transforming to cylindrical coordinates the spheroid is symmetrical about the angle \(\varphi\) and elliptical in \(\theta\). If we assume that the distribution of the optical axis is random we can derive an expression for the mean extraordinary index.