 Emission-spectrum mismatch in nail fluorescence subtraction

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.