Aperture-dependent diffuse transmission in nail measurements ($T(a)=T_d+T_s{a}^2/(a^2+r^2)$)

Aperture-dependent diffuse transmission in nail measurements is the change in measured transmittance caused by accepting different outgoing directions from a nail plate. The material can remain unchanged while the collected fraction changes. The relevant quantity is the optical acceptance, not aperture area alone.

For an original geometric model, take a small emitting region with constant diffuse radiance, viewed by a coaxial circular opening of radius $a$ at distance $r>0$. Its limiting angle obeys $\tan {\theta }_m=a/r$. Projected-area weighting gives the accepted diffuse fractionIf the direct beam is fully collected, a direct fraction $T_d$ and a total diffuse fraction $T_s$ produce $T(a)=T_d+\eta T_s$. This assumes geometric propagation and negligible finite-source and diffraction effects. In particular, the direct-beam assumption is not valid for an arbitrarily small physical opening.

At stipulated $r=10$ mm, $T_d=0.04$ and $T_s=0.08$, openings of radius 5 and 10 mm collect $\eta =0.20$ and 0.50. The respective measured fractions are 0.056 and 0.080, although total transmittance is 0.12 in both cases. Doubling the aperture radius quadruples its area but does not quadruple the collected diffuse fraction.

An aperture-scan separation of direct and diffuse nail transmission needs distinct acceptances and a stable angular model. In the ideal two-aperture case,For the stated values, the two measurements recover 0.08 and 0.04. Nearly equal acceptances amplify measurement differences by a large factor. Changing distance or source size during the scan would change the model as well as the acceptance.

The primary nail experiment used an integrating sphere with a monochromator and photodiode to measure transmitted radiation.^[ref] The sphere provides a different angular collection arrangement from this opening model; it does not establish the assumed diffuse radiance law. A useful comparison characterizes each detector geometry before interpreting a signal difference as a change in the nail plate.

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.