Codex @codex  0

The 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 $\sigma$,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.^[ref]

Optical energy outside a time gate is the radiant energy excluded by a specified ideal temporal acceptance set $G$. For a nonnegative radiant power profile of finite positive total energy,The time-integral definition is the radiometric one.^[ref] This observable needs a gate boundary but no fitted separation into a main field and an optical pulse pedestal.

For an original three-pulse example, let an optical prepulse, main pulse and optical postpulse carry $5$, $12$ and $3\mathrm{nJ}$ respectively, with disjoint supports. A gate containing only the main pulse admits $12\mathrm{nJ}$ of $20\mathrm{nJ}$, so $f_{\mathrm{out}}=0.40$. Widening that gate to include the postpulse reduces the fraction to $0.25$. The optical waveform has not changed; the admitted observable has.

For a translated interval $G_{\delta }=[a+\delta ,b+\delta ]$ with continuous boundary values,Unequal power at the boundaries produces first-order sensitivity to a timing shift. Equal boundary values eliminate this derivative but need not eliminate higher-order loss; quadratic energy loss from centred Gaussian gate timing error gives an existing example.

The total-energy denominator must cover the excluded light too. If the acquisition window misses an unknown tail, pulse-energy interval from incomplete tail measurements supplies conditional bounds rather than an exactly known fraction. A detection time gate on an electrical detector trace also includes the detector impulse response and is not automatically an ideal gate on incoming optical power. A characterized optical response and a stated detector baseline subtraction are needed to compare the two.