Energy-normalized pulse profile ($p(t)=P(t)/U,\int p(t)dt=1$)

An energy-normalized pulse profile separates an optical pulse's temporal shape from its total laser pulse energy. For nonnegative radiant power $P(t)$ and finite $U=\int P(t)dt>0$, define $p(t)=P(t)/U$. Then $p$ has units inverse time and integrates to one. The radiant energy in a time interval is $U$ times the integral of $p$ over that interval.^[ref]

For an original two-lobe example, let $p=1/(8\mathrm{ns})$ during $0<t<2\mathrm{ns}$ and $p=3/(8\mathrm{ns})$ during $5<t<7\mathrm{ns}$, with zero values elsewhere. The two integrals are $1/4$ and $3/4$. Choosing $U=4\mathrm{nJ}$ gives powers $0.5$ and $1.5\mathrm{W}$, and energies $1$ and $3\mathrm{nJ}$. Multiplying $U$ by two doubles both powers without changing $p$.

The temporal energy centroid of an optical pulse is $\int tp(t)dt$ when this moment exists. Here it is $(1/4)(1)+(3/4)(6)=4.75\mathrm{ns}$, in the gap between lobes. An energy centroid need not be a location of maximum power.

Mathematically, $p$ is a probability density function, but its weighting here is energy over time. It is not automatically a measured photon-arrival distribution, particularly when photon energies vary. A zero-energy trace cannot be normalized this way, and a noisy baseline-corrected trace with negative excursions is not automatically a nonnegative density. The Gaussian pulse energy and peak power article gives one normalized shape family.

The 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,The normalization uses radiant energy as integrated radiant flux.^[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 $q=p\ast h$, direct integration with $t=s+u$ givesThus 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.^[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 isA 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.