= Gaussian gate centring error
{c}
{title2=$d=t_c-t_0$}
The \b[Gaussian gate centring error] is $d=t_c-t_0$, the time displacement between a gate centre and a Gaussian pulse centre. The <Gaussian pulse time-gate energy fraction> gives the full displaced-gate formula. Equal positive and negative errors collect equal energy in that symmetric model, so an energy deficit alone does not reveal the direction of the misalignment.
A focused expansion explains sensitivity close to the optimum. Let $p$ be the centred <energy-normalized pulse profile>, let $a=T_g/2$, and define $F(d)=\int_{d-a}^{d+a}p(t)\,dt$. For the Gaussian standard deviation $\sigma$,
$$
F'(0)=0,\qquad F''(0)=-\frac{2a}{\sigma^2}p(a),\qquad
F(d)=F(0)-\frac{ap(a)}{\sigma^2}d^2+O(d^4).
$$
Thus the leading <quadratic energy loss from centred Gaussian gate timing error> is second order, rather than linear in $d$.
For an original example, take a gate as wide as the power FWHM $\tau$. Its centred fraction is $0.760968$. An error $d=0.10\tau$ gives $0.748042$ from the full formula; the quadratic approximation gives $0.747945$. For $\tau=10\,\mathrm{ns}$ this is a $1\,\mathrm{ns}$ displacement, with gate width and total pulse energy unchanged.
A delay scan of a gated readout can locate the maximum, provided the source and gate widths remain stable. <Pulse timing jitter> produces a distribution of offsets rather than a single deterministic $d$. Also distinguish a gate on optical power from one on a detector's electrical waveform: NIST measured detector <impulse responses> and used <convolution> to assess pulse reproduction.<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, short-pulse response measurement and numerical convolution]{ref}
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