Aperture-scan separation of direct and diffuse nail transmission ($Y_j=T_d+{\eta }_j{T}_s$)

Aperture-scan separation of direct and diffuse nail transmission estimates the directly collected and diffusely transmitted fractions of light by changing detector acceptance while keeping the nail plate and illumination fixed. Separation depends on a stated angular model: a change in aperture is an experimental control, not a direct label attached to every photon.

In the original constant-diffuse-radiance model, let $Y_j=T_d+{\eta }_j{T}_s+{ϵ}_j$, where $T_d$ is the fully accepted direct fraction, $T_s$ the total diffuse fraction and ${\eta }_j$ the known diffuse acceptance. For a coaxial circular opening, the geometric approximation gives ${\eta }_j=a_j^2/(a_j^2+r^2)$. Two distinct acceptances giveThe distinction between these components is identifiable within this model only when the acceptances differ.

Suppose ${\eta }_1=0.20$, ${\eta }_2=0.50$, $Y_1=0.056$ and $Y_2=0.080$. The fitted fractions are ${\hat{T}}_s=0.08$ and ${\hat{T}}_d=0.04$. If independent signal errors each have standard uncertainty $\sigma =0.002$, with acceptance uncertainty temporarily neglected, linear propagation givesandThe resulting uncertainties are about 0.00943 and 0.00359. Their negative covariance matters when adding the components: the total-transmission uncertainty is about 0.00629, rather than the value obtained by incorrectly treating the fitted components as independent.^[ref]

Additional apertures give a straight-line fit of $Y$ against $\eta$, with intercept $T_d$ and slope $T_s$. Weighted least squares can use independently characterized signal uncertainties. Systematic curvature would challenge the assumed angular law or fixed direct-beam acceptance. A plausible line alone does not prove that a finite source, changing geometry or residual fluorescence has been modeled correctly.

The primary nail-transmission study collected transmitted light using an integrating sphere, monochromator and photodiode.^[ref] The aperture scan here is an original proposed comparison, not a procedure or angular scattering law reported in that experiment.