= Optical tail and detector-response ambiguity
\b[Optical tail and detector-response ambiguity] arises because a measured temporal trace combines the optical waveform with the receiver's dynamics. In a calibrated <linear time-invariant system>,
$$
s(t)=(P*h)(t),
$$
where $P$ is incident <radiant power> and $h$ the unit-area detector <impulse response>. One trace does not generally identify both unknown functions. The parent <bounding unobserved optical energy with an exponential tail model> gives an impulsive-input example; two finite-width profiles provide another exact ambiguity.
Define $h_\tau(t)=\tau^{-1}e^{-t/\tau}$ for $t\geq0$, and zero otherwise. Consider model A with optical power $P=Uh_{2\,\mathrm{ns}}$ and detector response $h_{10\,\mathrm{ns}}$, and model B with these two time constants exchanged. By commutativity of <convolution>, both give exactly
$$
s(t)=\frac{U}{8\,\mathrm{ns}}
\left(e^{-t/(10\,\mathrm{ns})}-e^{-t/(2\,\mathrm{ns})}\right)
\quad(t\geq0).
$$
Their optical energies after $20\,\mathrm{ns}$ are nevertheless $Ue^{-10}$ and $Ue^{-2}$. For original chosen $U=100\,\mathrm{nJ}$, those are about $0.00454$ and $13.53\,\mathrm{nJ}$. Identical recorded tails therefore coexist with very different true late optical energy.
\Image[/optical-tail-detector-factorization.png]
{title=Different optical tails and detector responses can produce the same measured pulse}
{description=Original unit-area exponential response models. Model A has a2ns optical time constant and10ns detector time constant; model B exchanges them. Both incident pulses have100nJ total energy. The measured convolution is identical despite different optical energy after20ns.}
{source=optical-tail-detector-factorization.py}
<Known-input calibration of a pulse detector> supplies information missing from this inverse problem. NIST measured the detector/cable/oscilloscope response with optical excitation much shorter than the response before calculating waveform distortions.<Calibration Service for Low-level Pulsed-Laser Radiometers at 1.06 µm: Pulse Energy and Peak Power by Rodney W. Leonhardt>[Appendix C.1–C.2, printed pp. 36–37, Figure C1 and bandwidth-correction procedure]{ref} The example above is an independent mathematical construction. A fitted electrical decay constant alone cannot establish a global optical-tail envelope.
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