= Temporal energy centroid of an optical pulse
{title2=$t_E=\int t p(t)\,dt$}
The \b[temporal energy centroid of an optical pulse] is its energy-weighted mean time. For nonnegative <radiant power> $P(t)$ with finite positive energy $U$ and finite absolute first moment,
$$
t_E=\frac{\int_{-\infty}^{\infty}tP(t)\,dt}{U}
=\int_{-\infty}^{\infty}t p(t)\,dt,\qquad p(t)=P(t)/U.
$$
The normalization uses radiant energy as integrated radiant flux.<International Lighting Vocabulary by CIE>[entry 17-21-041, note 1]{ref} The centroid is a moment of the <energy-normalized pulse profile>; it need not coincide with the largest power value, a threshold crossing or the middle of the FWHM interval.
For an original two-component example, let ninety percent of the energy belong to a symmetric narrow component centred at zero and ten percent to a symmetric component centred at $20\,\mathrm{ns}$. Linearity of the integral gives $t_E=0.9(0)+0.1(20)=2\,\mathrm{ns}$. Weak delayed energy can move the centroid even when the dominant peak remains near zero. Multiplying the whole power trace by a positive constant leaves $t_E$ unchanged; translating it by $t_0$ shifts its centroid by $t_0$.
A detector response can shift the measured centroid too. Suppose a normalized <impulse response> $h$ and normalized optical profile $p$ have finite absolute first moments. For their <convolution> $q=p*h$, direct integration with $t=s+u$ gives
$$
\int tq(t)\,dt
=\iint(s+u)p(s)h(u)\,ds\,du=t_p+t_h.
$$
Thus a causal exponential detector response of time constant $2\,\mathrm{ns}$ shifts the recorded centroid by $2\,\mathrm{ns}$ in this model. NIST's pulsed-radiometer characterization explicitly measures detector <impulse responses> and uses <convolution> to assess reproduced waveforms.<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, printed pp. 13–15]{ref} The centroid identity is the independent moment calculation above.
<Baseline drift> can cause a separate shift. Suppose a true pulse has $U=1\,\mathrm{nJ}$ and $t_E=0$, but a positive residual baseline of $1\,\mathrm{mW}$ remains across the analysis window from $-10$ to $90\,\mathrm{ns}$. Its added energy is $0.10\,\mathrm{nJ}$, centred at $40\,\mathrm{ns}$. The resulting centroid is
$$
\frac{(1\,\mathrm{nJ})(0)+(0.10\,\mathrm{nJ})(40\,\mathrm{ns})}{1.10\,\mathrm{nJ}}
=3.636\,\mathrm{ns}.
$$
A waveform measurement should therefore record the time window, baseline estimate and detector timing response with its reported centroid. A negative background-subtracted trace also needs care: it is not automatically a nonnegative energy distribution.
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