= Gaussian chirped pulse compression
{title2=$\tau(\phi_2)=\tau_0\sqrt{1+(4\ln2\,\phi_2/\tau_0^2)^2}$}
Choose an initially <transform-limited pulse> with Gaussian <power> and complex envelope
$$
A_0(t)=\sqrt{\frac{U}{\sqrt{2\pi}\sigma_0}}\,e^{-t^2/(4\sigma_0^2)},\qquad
\tau_0=2\sqrt{2\ln2}\,\sigma_0.
$$
Here $\sigma_0$ is the standard deviation of the power-weighted time distribution; $\tau_0$ is its <full width at half maximum>. Neither is a field-amplitude width. The spectral amplitude is proportional to $e^{-\sigma_0^2\Omega^2}$, so the spectral power has <standard deviation> $\sigma_\Omega=1/(2\sigma_0)$. The Gaussian <time-bandwidth product> in ordinary <frequency> is $\tau_0\Delta\nu=2\ln2/\pi\simeq0.4413$, where both widths are power FWHMs.
Multiplication by $e^{i\phi_2\Omega^2/2}$ preserves spectral power and <energy>. Substituting $\phi'=\phi_2\Omega$ into the moment identity gives
$$
\sigma_t^2=\sigma_0^2+\frac{\phi_2^2}{4\sigma_0^2},\qquad
\tau(\phi_2)=\tau_0\sqrt{1+\left(\frac{4\ln2\,\phi_2}{\tau_0^2}\right)^2}.
$$
This is an exact result for a Gaussian spectrum with quadratic <spectral phase>. It also follows by completing the square in its inverse <Fourier transform>.
\Image[/gaussian-chirped-pulse-spectral-phase-compression.png]
{title=Phase cancellation recovers a Gaussian pulse, while residual phase and narrowed bandwidth limit it}
{description=Original complex-envelope <Fast Fourier transform> calculations for a chosen $30\,\mathrm{fs}$ power FWHM, $1\,\mathrm{nJ}$ pulse. The stretcher adds $3000\,\mathrm{fs^2}$ of <group delay dispersion>. Separate output cases show exact cancellation, $300\,\mathrm{fs^2}$ residual GDD, $20000\,\mathrm{fs^3}$ residual <third-order dispersion>, and a Gaussian <optical gain> profile. The finite-gain curve divides out the central-frequency gain $G_0$ and retains the resulting energy change. All curves are calculations, not measured pulse traces.}
For these chosen inputs, $\sigma_0=12.7398\,\mathrm{fs}$. The stretched pulse has $\tau=278.88\,\mathrm{fs}$ and <root mean square> width $118.4282\,\mathrm{fs}$. Adding $-3000\,\mathrm{fs^2}$ recovers $30\,\mathrm{fs}$; leaving $300\,\mathrm{fs^2}$ gives $40.85\,\mathrm{fs}$. All phase-only cases retain $1\,\mathrm{nJ}$.
If GDD is cancelled but $\phi_3=20000\,\mathrm{fs^3}$ remains, the pulse is asymmetric. Gaussian spectral moments give
$$
\bar t=\frac{\phi_3\sigma_\Omega^2}{2},\qquad
\sigma_t^2=\sigma_0^2+\frac{\phi_3^2\sigma_\Omega^4}{2}.
$$
Thus the centroid is $15.4033\,\mathrm{fs}$ and the <root mean square> duration is $25.2354\,\mathrm{fs}$. Converting this last value to a FWHM with the Gaussian factor would be wrong: the power envelope is no longer Gaussian. The calculation's integrated <energy> agrees in time and <frequency> through <Parseval's identity>, while its centroid and <variance> agree with these independent moment formulas.
A real amplifier also changes spectral amplitude. For a simple small-signal example, stipulate an intensity gain profile
$$
G(\Omega)=G_0 e^{-\Omega^2/(2\sigma_g^2)},\qquad
\widetilde A_{\rm amplified}=\sqrt{G(\Omega)}\,\widetilde A_{\rm input}.
$$
This specifies finite <gain bandwidth>, with no saturation or extra amplifier <phase>. Multiplying two Gaussian spectra gives
$$
\frac{1}{\sigma_{\Omega,\rm out}^2}=\frac{1}{\sigma_\Omega^2}+\frac{1}{\sigma_g^2}.
$$
For chosen $\sigma_g=\sigma_\Omega$, the amplified spectrum is narrower by $\sqrt2$, the shortest Gaussian power FWHM becomes $30\sqrt2=42.43\,\mathrm{fs}$, and the total <energy> is $G_0U/\sqrt2$. Perfect <spectral phase> correction cannot recreate the suppressed spectral amplitude. Real <gain narrowing> and saturation require a material-dependent amplifier model; a constant multiplier is only a limited idealization. <Gérard Mourou>'s 2018 lecture discusses <gain narrowing> as a limit on the shortest amplified pulses.<Extreme Light Physics and Application by Gérard Mourou>[Extreme Light Physics and Application, p.105, §II.E: finite gain bandwidth and gain narrowing]{ref}
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