 Paul D. Hale

Paul D. Hale is a metrology researcher named with Tracy S. Clement and Dylan F. Williams on a 2001 NIST paper about optical receiver frequency response. It compares frequency-domain optical heterodyne detection with time-domain measurements using an oscilloscope, including corrections for the electrical connection and finite optical stimulus.^[ref]

An oscilloscope trace need not be the receiver's isolated impulse response. In an original idealized model of linear time-invariant systems, let $r(t)$ be the receiver impulse response, $p(t)$ the optical pulse envelope and $o(t)$ the readout impulse response. Ignoring mismatch for this illustration, the measured waveform is the convolutionA broader measured pulse can therefore reflect the receiver, the stimulus or the readout. If all three envelopes are normalized Gaussian functions, with independent convolution widths, their variances add:For illustrative widths ${\sigma }_y=5\mathrm{ps}$, ${\sigma }_p=3\mathrm{ps}$ and ${\sigma }_o=\sqrt{7}\mathrm{ps}$, the inferred receiver width is ${\sigma }_r=3\mathrm{ps}$, rather than $5\mathrm{ps}$. These are chosen model values, not results reported by Hale.

The example motivates characterizing the stimulus and readout as well as the specimen. Dividing by $P(f)O(f)$ where that product is nearly zero also amplifies measurement noise, so a formal Fourier transform inversion does not guarantee a reliable corrected response. This is why measurement uncertainty belongs alongside the quoted frequency response.