= Optical fiber cutback measurement
{title2=$\widehat\alpha_{\rm dB}=10\log_{10}(P_s/P_\ell)/(L_\ell-L_s)$}
An <optical fiber cutback measurement> estimates distributed <optical fiber attenuation> by comparing a long guide with the short input portion left after cutting the output end back. Keep the launch untouched. For the following original functional model, fix <wavelength>, source output, input alignment and collected output cone; subtract a dark reading and keep the <photodetector> in its linear range. Renewed end-face quality and changed <normal mode> populations can still spoil the comparison. The diagram describes a measurement principle, not Kao's historical short-glass apparatus.
Let $L_\ell>L_s$ be lengths in kilometres, $P_\ell,P_s$ the corresponding positive dark-subtracted readings, and $C$ the same launch and detection factor. A uniform guide obeys
$$
P_\ell=C10^{-\alpha_{\rm dB}L_\ell/10},\qquad P_s=C10^{-\alpha_{\rm dB}L_s/10},\qquad
\widehat\alpha_{\rm dB}=\frac{10}{L_\ell-L_s}\log_{10}\!\frac{P_s}{P_\ell}.
$$
Here $\alpha_{\rm dB}$ and its estimate are numerical values in $\mathrm{dB/km}$. Dividing eliminates $C$; taking a <logarithm> proves the formula. A changed $C$ instead adds an error $10\log_{10}(C_s/C_\ell)/(L_\ell-L_s)$. Reconnecting the launch between readings defeats this cancellation.
For original chosen values $L_s=0.1\,\mathrm{km}$, $L_\ell=2.1\,\mathrm{km}$, $P_s=100\,\mathrm{\mu W}$ and $\alpha_{\rm dB}=0.5\,\mathrm{dB/km}$, the predicted long reading is $P_\ell=100\,10^{-0.1}\simeq79.4328\,\mathrm{\mu W}$. The ratio gives $1\,\mathrm{dB}$ for the removed $2\,\mathrm{km}$ and hence $0.5\,\mathrm{dB/km}$.
Suppose an unremoved additive detector background contributes $b=5\,\mathrm{\mu W}$ to both readings. Then the apparent coefficient is
$$
\widetilde\alpha_{\rm dB}=\frac{10}{2}\log_{10}\!\frac{100+5}{79.4328+5}\simeq0.47339\,\mathrm{dB/km}.
$$
This is a <bias of an estimator>: a common additive background pushes the ratio toward one, whereas a common multiplicative factor cancels. For independent small zero-mean errors with <standard deviations> $\sigma_s,\sigma_\ell$ on the dark-subtracted readings, a first-order <Taylor expansion> gives
$$
\sigma_{\widehat\alpha}\simeq\frac{10}{(L_\ell-L_s)\ln10}\sqrt{\left(\frac{\sigma_s}{P_s}\right)^2+\left(\frac{\sigma_\ell}{P_\ell}\right)^2}.
$$
Choosing both <standard deviations> as $1\,\mathrm{\mu W}$ gives $\sigma_{\widehat\alpha}\simeq0.03491\,\mathrm{dB/km}$. A shared estimated dark offset introduces <covariance>, so it requires a different uncertainty calculation. Near zero signal, the <logarithm> and this linear approximation become unreliable.
\Image[/optical-fiber-cutback-background-bias.png]
{title=<Optical fiber cutback measurement>: fixed launch and additive-background bias}
{description=Original functional schematic and calculated example. The launch stays fixed while the remote end is cut back. The graph uses the stipulated powers and lengths above; it compares dark-subtracted readings with a common positive unremoved background. No plotted point is a historical measurement.}
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