= Emission-spectrum mismatch in nail fluorescence subtraction
{title2=$\Delta P_t=(b-b_0)P_f/a$}
\b[<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. Define
$$
b=\int k(\lambda)f(\lambda)\,d\lambda,\qquad
b_0=\int k(\lambda)f_0(\lambda)\,d\lambda.
$$
For a current $i=aP_t+bP_f$, subtraction of $b_0P_f$ gives
$$
\widehat P_t-P_t=\frac{P_f}{a}\int k(\lambda)\bigl[f(\lambda)-f_0(\lambda)\bigr]\,d\lambda.
$$
Although $\int(f-f_0)\,d\lambda=0$, 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 $k_2=0.050\,\mathrm{A/W}$ in two nonoverlapping bands. If fraction $q$ of fluorescent power lies in the first band, $b(q)=k_1q+k_2(1-q)$. The assumed fraction $q_0=0.80$ gives $b_0=0.25\,\mathrm{A/W}$, whereas the true $q=0.20$ gives $b=0.10\,\mathrm{A/W}$. At $P_f=20\,\mu\mathrm W$ and transmitted-band responsivity $a=0.30\,\mathrm{A/W}$, 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>.<The transmission of optical radiation through human nails by S. G. Parker and B. L. Diffey>[Abstract, fluorescence correction and apparatus]{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.
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