= Known-input calibration of a pulse detector
{title2=$y=g(P_{\rm ref}*h)$}
\b[Known-input calibration of a pulse detector] uses a characterized optical waveform to separate the incident signal from the receiver response. In a linear, baseline-corrected model,
$$
y(t)=g(P_{\rm ref}*h)(t),\qquad \int h(t)\,dt=1,
$$
where $P_{\rm ref}$ is known input power, $h$ a normalized <impulse response> and $g$ the overall <responsivity>. Recording the complete response gives
$$
\int y(t)\,dt=gU_{\rm ref}.
$$
Thus a known reference energy calibrates the area gain without requiring the input to be an ideal impulse. Shape calibration additionally uses the reference waveform.
For nonnegative normalized profiles with finite second moments, define $p=P_{\rm ref}/U_{\rm ref}$ and $q=y/\int y$. Since $q=p*h$, their temporal variances satisfy $\sigma_q^2=\sigma_p^2+\sigma_h^2$. In an original chosen example, an input standard deviation $0.60\,\mathrm{ns}$ and recorded deviation $1.00\,\mathrm{ns}$ imply $\sigma_h=0.80\,\mathrm{ns}$. Treating that finite input as a delta impulse would incorrectly assign the full $1.00\,\mathrm{ns}$ to the detector. This <finite-input correction in pulse-detector calibration> concerns standard deviations; a universal subtraction rule for arbitrary FWHM values does not follow. Negative ringing also invalidates the nonnegative-profile interpretation used for the variance identity.
NIST's primary procedure illuminated the detector with a pulse much shorter than the detector response and characterized detector, cable and oscilloscope together at the relevant gain and signal levels.<Calibration Service for Low-level Pulsed-Laser Radiometers at 1.06 µm: Pulse Energy and Peak Power by Rodney W. Leonhardt>[§§6.1–6.2; Appendix C.1–C.2, printed pp. 36–37, Figure C1 and measurement procedure]{ref} Such a known input breaks the <optical tail and detector-response ambiguity> described by the parent article.
In the <frequency domain>, the same model is $Y=gP_{\rm ref}H$. Division can estimate $H$ where the input spectrum is known and sufficiently nonzero; small input components amplify measurement noise. A short reference pulse is useful because it excites a broad frequency range, but detector linearity, baseline, area coverage and reference uncertainty still need checking.
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