Excitation-band selectivity in nail fluorescence corrections ($i=a{P}_t+b{P}_f$)

Excitation-band selectivity in nail fluorescence corrections describes how well the detection system distinguishes transmitted illumination from light emitted by the nail plate. Selecting the illumination with a monochromator does not force every detected photon to remain in that spectral band: fluorescence can contribute at other wavelengths.

For an original linear detection model, write the transmitted and emitted spectral powers as $P_{t}h(\lambda )$ and $P_{f}f(\lambda )$, where both spectral shapes integrate to one. If an optical filter has transmission $\tau (\lambda )$ and a detector has spectral responsivity $R(\lambda )$, its current after subtraction of dark current isThe coefficients $a,b$ are spectrum-weighted responsivities, with units of amperes per watt. A selective filter can reduce $b$ while also reducing $a$, so an uncorrected smaller current does not establish lower specimen transmittance.

For stipulated $a=0.30A/W$, $b=0.020A/W$, $P_t=80\mu \mathrm{W}$ and $P_f=20\mu \mathrm{W}$, current is $24.4\mu \mathrm{A}$. Assigning all of it to transmitted illumination gives $i/a=81.33\mu \mathrm{W}$, a 1.67% relative overestimate. These chosen coefficients illustrate leakage; they are not measured filter characteristics for nails.

Emission-spectrum mismatch in nail fluorescence subtraction matters when the assumed $f(\lambda )$ differs from the specimen's emission. If analysis subtracts $b_0{P}_f$ while the true coefficient is $b$, thenAt the same powers and $a$, using $b_0=0.010A/W$ instead of 0.020 leaves a $0.667\mu \mathrm{W}$ positive bias. A precisely known total emitted power would not eliminate this spectral mismatch.

The historical nail-transmission experiment explicitly corrected for fluorescence.^[ref] The integral above is an explanatory extension, not a reconstruction of its unpublished correction coefficients. An experimental test would characterize detector/filter response and compare detection bands while keeping illumination and collection geometry in nail-transmission measurements fixed. Spectral rejection and angular collection constrain different parts of the measurement.

Emission-spectrum mismatch in nail fluorescence subtraction is the error caused by correcting a detector reading with an assumed fluorescence spectrum that differs from the actual emitted spectral shape. Knowing total fluorescent power does not generally determine fluorescent detector current, because the detector/filter response varies with wavelength.

Let $f(\lambda )$ be the true normalized emission spectrum, $f_0(\lambda )$ the assumed normalized spectrum and $k(\lambda )=R(\lambda )\tau (\lambda )$ the product of spectral responsivity and optical filter transmission. DefineFor a current $i=a{P}_t+b{P}_f$, subtraction of $b_0{P}_f$ givesAlthough $\int (f-f_0)d\lambda =0$, the response-weighted integral need not vanish. Equal total emitted powers can therefore require different corrections.

An original two-band example makes the effect visible. Suppose the effective responsivity is $k_1=0.30$ and $k_2=0.050A/W$ in two nonoverlapping bands. If fraction $q$ of fluorescent power lies in the first band, $b(q)=k_{1}q+k_2(1-q)$. The assumed fraction $q_0=0.80$ gives $b_0=0.25A/W$, whereas the true $q=0.20$ gives $b=0.10A/W$. At $P_f=20\mu \mathrm{W}$ and transmitted-band responsivity $a=0.30A/W$, the transmitted-power estimate is biased by $-10\mu \mathrm{W}$. A true $80\mu \mathrm{W}$ would be reported as $70\mu \mathrm{W}$ after that excessive subtraction.

These are stipulated spectral shapes and coefficients, not measured nail spectra. If $k$ were constant over the entire emission band, normalization would make this particular shape error vanish. A selective system generally sacrifices that invariance to distinguish transmitted excitation light from emission. Responsivity characterization and spectral measurement must therefore accompany a proposed subtraction.

The primary nail experiment explicitly included a correction for fluorescence while measuring transmission with a monochromator, integrating sphere and photodiode.^[ref] That observation motivates the distinction between emitted power and detected current; it does not specify the two-band correction model above. Fixing an assumed spectrum in software does not establish that the specimen actually has that spectrum.