 Optical tail and detector-response ambiguity

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,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 $t\ge 0$, and zero otherwise. Consider model A with optical power $P=U{h}_{2\mathrm{ns}}$ and detector response $h_{10\mathrm{ns}}$, and model B with these two time constants exchanged. By commutativity of convolution, both give exactlyTheir optical energies after $20\mathrm{ns}$ are nevertheless $U{e}^{-10}$ and $U{e}^{-2}$. 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.

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.^[ref] The example above is an independent mathematical construction. A fitted electrical decay constant alone cannot establish a global optical-tail envelope.