= Optical energy outside a time gate
{title2=$f_{\rm out}=1-\int_G P(t)\,dt/U$}
\b[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,
$$
U=\int_{\mathbb R}P(t)\,dt,\quad U_G=\int_G P(t)\,dt,\quad
U_{\rm out}=U-U_G,\quad f_{\rm out}=\frac{U_{\rm out}}U.
$$
The time-integral definition is the radiometric one.<International Lighting Vocabulary by CIE>[entry17-21-041, note1, radiant energy as the integral of radiant flux]{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 $f_{\rm out}=0.40$. 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,
$$
\frac{dU_G}{d\delta}=P(b+\delta)-P(a+\delta).
$$
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.
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