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Choose an initially transform-limited pulse with Gaussian power and complex envelopeHere ${\sigma }_0$ 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 ${\tau }_0\Delta \nu =2\ln 2/\pi \simeq 0.4413$, where both widths are power FWHMs.

Multiplication by $e^{i{\varphi }_2{\Omega }^2/2}$ preserves spectral power and energy. Substituting ${\varphi }^{\prime }={\varphi }_2\Omega$ into the moment identity givesThis is an exact result for a Gaussian spectrum with quadratic spectral phase. It also follows by completing the square in its inverse Fourier transform.

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 $-3000\mathrm{f}{\mathrm{s}}^2$ recovers $30\mathrm{fs}$; leaving $300\mathrm{f}{\mathrm{s}}^2$ gives $40.85\mathrm{fs}$. All phase-only cases retain $1\mathrm{nJ}$.

If GDD is cancelled but ${\varphi }_3=20000\mathrm{f}{\mathrm{s}}^3$ remains, the pulse is asymmetric. Gaussian spectral moments giveThus 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 profileThis specifies finite gain bandwidth, with no saturation or extra amplifier phase. Multiplying two Gaussian spectra givesFor chosen ${\sigma }_g={\sigma }_{\Omega }$, the amplified spectrum is narrower by $\sqrt{2}$, the shortest Gaussian power FWHM becomes $30\sqrt{2}=42.43\mathrm{fs}$, and the total energy is $G_{0}U/\sqrt{2}$. 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.^[ref]

Use a complex envelope $A(t)$ normalized so that $|A(t){|}^2=P(t)$; its units are ${\mathrm{W}}^{1/2}$. The optical carrier convention is $e^{-i{\omega }_{0}t}$: the physical electric field is proportional to the real part of $A(t)e^{-i{\omega }_{0}t}$ for a fixed transverse mode. With angular frequency offset $\Omega =\omega -{\omega }_0$, fix the Fourier transform conventionWrite $\tilde{A}=B{e}^{i\varphi }$ with real $B\ge 0$. A dispersive element adds spectral phaseConstant phase rotates the envelope; the linear term delays it by ${\varphi }_1$. The group delay is $d\varphi /d\Omega$. The quadratic coefficient ${\varphi }_2$ is group delay dispersion, abbreviated GDD, with units of time squared. Positive GDD makes higher frequencies arrive later with this convention. It is accumulated dispersion of a component; material group velocity dispersion is its corresponding coefficient per unit length.

To undo a quadratic stretcher spectral phase, the compressor must supply opposite GDD. Include the amplifier and all intervening optics in the sum:Setting this sum to zero does not cancel third-order dispersion, ${\varphi }_3$, or still higher orders. Residual spectral phase can produce asymmetry and satellite pulses even when the quadratic term vanishes. A diffraction grating system can make path length depend on frequency; matching the complete spectral phase over the occupied spectrum is more demanding than merely choosing opposite quadratic coefficients.

There is a useful proof of what phase correction can accomplish. By Parseval's identity,For a smooth decaying spectrum, the Fourier transform of $tA$ is $-id\tilde{A}/d\Omega$. Consequently the power-weighted mean arrival time and variance obeyAt fixed spectral amplitude, constant group delay minimizes this root mean square duration. Such a pulse is a transform-limited pulse, up to a time shift and constant phase. The nonnegative second term measures temporal broadening from unequal arrival times; the first term remains even after perfect phase correction.