## Appendix A

Absorption measurements are generally straightforward when light takes a direct path through the medium under investigation (

Figure A1). In the presence of scattering events while traversing the sample, however, the travel path may become significantly longer than the thickness itself, adding a non-trivial degree of complication to the extraction of an absolute absorption coefficient due to the difficulty of knowing this exact path length. Simply using the sample thickness as the travel length indeed leads to a severe overestimation of the absorption coefficient. Furthermore, given that the average path length depends on the number of scattering events which extend the travel, the overestimation of the absorption will be more severe for samples with lower RIT.

**Figure A1.**
Schematic representation of the absorption measurement setup. The light source was a tunable filtered supercontinuum laser and the sample was inserted into the integrating sphere at an angle to keep the specular reflection inside the sphere.

**Figure A1.**
Schematic representation of the absorption measurement setup. The light source was a tunable filtered supercontinuum laser and the sample was inserted into the integrating sphere at an angle to keep the specular reflection inside the sphere.

Without an exact knowledge of the light travel distance through each sample, in order to extract a first estimate, a series of calculations was performed based on standard reflection and refraction laws. Each sample was mounted in the integrating sphere such that the angle of incidence of the light from the fiber was approximately 40.5° from the surface normal. This angle was introduced intentionally so as to capture the specular reflection of the incident beam within the integrating sphere, rather than letting this reflection re-exit the input port. Snell’s law may be used to calculate the angle at which the primary beam refracts into the material:

where the first refractive index is that of air

${n}_{i}={n}_{air}=1$, the angle of incidence

${\theta}_{i}=40.5\xb0$, the refracted material index

${n}_{r}$ is taken as the average for alumina,

${n}_{r}=\frac{1}{2}\left({n}_{e}+{n}_{o}\right)$, and the angle of refraction in the alumina is

${\theta}_{r}$. Calculating this angle yields

${\theta}_{r}=21.5\xb0$, again as measured from the sample normal. Whereas in the usual surface-normal RIT measurements the relevant light path is simply the sample thickness, here the nonzero angles of incidence and refraction imply that the distance

$L$ that the primary beam travels is related to the thickness

$d$ by

$d=L\mathrm{cos}{\theta}_{r}$.

The distance that light travels before a scattering event may be estimated from the inverse of the RIT loss coefficient

${\gamma}_{tot}$. As this total measured loss coefficient also implicitly includes the absorption term, determining the distance between scattering events in this manner already represents an approximation. Nonetheless, one can consider that the amount of light entering the sample is reduced from the incident power by the first surface reflection

${R}_{1}$. The reflectance itself, however, is in general a function of the angle of incidence and the light polarization. The transverse-electric and transverse-magnetic reflection polarization components, denoted as s-polarized and p-polarized, are written as:

From these formulae, it is easily seen that at normal incidence, where ${\theta}_{1}={\theta}_{2}=90\xb0$, the cosine terms equal unity and the reflection coefficients simplify to the familiar difference over the sum of indexes, $R={R}_{s}={R}_{p}={\left(\frac{{n}_{1}-{n}_{2}}{{n}_{1}+{n}_{2}}\right)}^{2}$. Using the refractive indexes of air and alumina at the angle of incidence of 40.5°, the above expressions yield ${R}_{s}=0.135$ and ${R}_{p}=0.033$. The value used for the surface reflection in this treatment is considered as the average of these two components, or ${R}_{1}=0.084$. The fraction of the total incident power entering the sample is thus $1-{R}_{1}$, some further fraction of which—as given by the RIT fraction—continues in the same direction as given by ${\theta}_{r}$ for a distance L. The rest, by definition, undergoes a scattering event which changes its direction, a distance given by the inverse scattering coefficient ${\gamma}^{-1}$.

After light is scattered from its primary

${\theta}_{r}=21.5\xb0$ direction through the sample, here it is considered equally likely that the photons will depart in any angular direction afterwards. In a further approximation, only forward scattering will be included in this treatment. This results in an important simplification regarding the range of scattered angles to be calculated. As a result that each angle represents a different path length traveled between the scattering event and the second (exit) surface of the sample, these path lengths must be integrated over the range of angles considered. For forward scattering, the upper angular limit is defined by the critical angle

${\theta}_{c}$ for total internal reflectance:

where in this case

${n}_{2}={n}_{air}=1$ since the light would be passing from alumina back into air. Using the average of the extraordinary and ordinary indexes for alumina,

${n}_{1}=\frac{1}{2}\left({n}_{e}+{n}_{o}\right)$, the total internal reflection angle becomes 34.6°. All of the light incident upon the second surface is then subject to an angle-dependent degree of reflection. These different contributions are summarized schematically in

Figure A2, with the incident amount from the source denoted by

${I}_{0}$.

**Figure A2.**
Schematic illustration of the different scattering contributions to the total path length.

**Figure A2.**
Schematic illustration of the different scattering contributions to the total path length.

The light entering the sample, $\left(1-{R}_{s1}\right){I}_{0}$, in principle, travels on average a distance given by the absorption-corrected inverse scattering coefficient, ${\gamma}^{-1}$, before changing direction. This value is related to the one measured from the RIT experiments by $\gamma ={\gamma}_{tot}-\alpha $. The distance from the scattering event to the second surface normal, determined solely from geometric considerations, is related to the “RIT” travel length $L$ and scattering length $\gamma $ by $\left(L-{\gamma}^{-1}\right)\mathrm{cos}{\theta}_{r}$. More generally, for an arbitrary scattering angle θ, the distance traveled by a photon between the scattering event, and the second surface is given by $l=\left(L-{\gamma}^{-1}\right)\frac{\mathrm{cos}{\theta}_{r}}{\mathrm{cos}\theta}$.

The amount of light exiting the alumina-air interface may be considered as two separate terms: one for the primary beam that is refracted back to the original direction

${\theta}_{i}$ as measured in an RIT experiment, denoted here as

${I}_{RIT}$; and a second consisting of the integrated power of all the light scattered,

${I}_{scatt}$. As a result that the interface reflection is angle-dependent, the transmitted

${I}_{RIT}$ term may be specified as:

where

$\gamma $ represents the grain-scattering and pore-scattering contributions. The second term,

${I}_{2}$, must be integrated over the angular range up to the total internal reflectance angle for forward scattering:

The intensity of light exiting the second sample surface can thus be considered as the sum of these terms weighted by the RIT fraction, i.e., $I={f}_{RIT}{I}_{RIT}+\left(1-{f}_{RIT}\right){I}_{2}$. It must be noted that this expression neglects all the light that is either back-scattered or forward-scattered outside of the range given by the total internal reflectance angle $\pm {\theta}_{TIR}$, as well as light reflected at the second interface back into the alumina sample. These factors represent corrections to the ${I}_{2}$ term beyond the first-order path length calculation described above, and can be accounted for qualitatively by adding an ad-hoc term inversely proportional to the dimensionless $\alpha {l}^{\prime}$ in the above sum.

Including the reflection from the first air-alumina interface, the total intensity of light detected in the integrating sphere becomes:

Practically, because this expression is transcendental for the absorption implicitly through the exponential of the RIT term as well as explicitly in the ${I}_{2}$ and multiple-reflection-scattering terms, it must be solved numerically. Compared to the absorption calculated from the uncorrected path length, i.e., calculating $\alpha $ based only on the as-is measured intensity, the value of $\alpha d$ is reduced by approximately 0.1. For the least transparent sample considered, the reduction of the optical depth from about $\alpha d=0.8$ to $\alpha d=0.7$ represents a change of about 12%, while for the most transparent sample having an unadjusted depth of about $\alpha d=0.1$, this represents a correction of nearly 100% of the uncorrected value.