= Chirped pulse amplification energy budget
{title2=$U=\int P(t)\,dt$}
Let $P(t)$ be the pulse's <power> integrated over its transverse section, and $U$ its <energy>. Its <peak power> is $\max_t P(t)$. For a chosen Gaussian power envelope with <full width at half maximum> $\tau$,
$$
P(t)=P_{\rm pk}e^{-4\ln2\,t^2/\tau^2},\qquad
U=P_{\rm pk}\tau\sqrt{\frac{\pi}{4\ln2}},\qquad
P_{\rm pk}=\sqrt{\frac{4\ln2}{\pi}}\frac{U}{\tau}.
$$
The numerical shape factor is approximately $0.9394$. It matters when comparing quoted <peak powers>; the estimate $U/\tau$ assumes a shape and a width convention.
If an otherwise unchanged pulse is stretched by a factor $S$ without loss, its <energy> stays fixed and its <peak power> falls by $S$. For energy gain $G$ and compressor energy transmission $\eta$, ideal restoration of the original shape gives
$$
U_{\rm out}=\eta G U_{\rm in},\qquad
\frac{P_{{\rm pk},\,\rm out}}{P_{{\rm pk},\,\rm amplified}}=\eta S.
$$
These relations require a specified shape and preserved spectrum; they do not model amplifier saturation. For the diagram's chosen $1\,\mathrm{nJ}$, $30\,\mathrm{fs}$ seed, $S=9.2959$, $G=1000$ and $\eta=0.8$, the successive <peak powers> are $31.31\,\mathrm{kW}$, $3.369\,\mathrm{kW}$, $3.369\,\mathrm{MW}$ and $25.05\,\mathrm{MW}$. The last pulse contains $0.8\,\mathrm{\mu J}$: the compressor increases <peak power> while losing some <energy>.
<Fluence> is time-integrated <light intensity>, $F(x,y)=\int I(x,y,t)\,dt$, measured in $\mathrm{J\,m^{-2}}$. For a uniform beam of area $\mathcal A$, $F=U/\mathcal A$ and $I_{\rm pk}=P_{\rm pk}/\mathcal A$. At fixed transverse profile, lossless temporal stretching lowers peak <light intensity> but leaves <fluence> unchanged. Absorbed <energy>, heat removal, beam size, coatings and pulse duration still constrain optical components. For nonoverlapping identical pulses at repetition <frequency> $f_r$, the average <power> is $U f_r$; it is a different quantity from <peak power>.
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