= Aperture-dependent diffuse transmission in nail measurements
{title2=$T(a)=T_d+T_s a^2/(a^2+r^2)$}
\b[<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 fraction
$$
\eta=\frac{\int_0^{\theta_m}\cos\theta\sin\theta\,d\theta}{\int_0^{\pi/2}\cos\theta\sin\theta\,d\theta}
=\sin^2\theta_m=\frac{a^2}{a^2+r^2}.
$$
If 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.
\Image[/nail-aperture-transmission.png]
{title=An unchanged hypothetical nail specimen gives different measured transmission as collection radius changes}
{description=Original constant-radiance geometric model at distance 10 mm. The direct fraction is 4 percent and the diffuse fraction is 8 percent. These are stipulated values, not measured human-nail data.}
An <aperture-scan separation of direct and diffuse nail transmission> needs distinct acceptances and a stable angular model. In the ideal two-aperture case,
$$
T_s=\frac{T(a_2)-T(a_1)}{\eta_2-\eta_1},\qquad T_d=T(a_1)-\eta_1T_s.
$$
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.<The transmission of optical radiation through human nails by S. G. Parker and B. L. Diffey>[Abstract, collection apparatus]{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>.
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