= Spectral phase in pulse compression
{title2=$\phi(\Omega)=\phi_0+\phi_1\Omega+\phi_2\Omega^2/2+\cdots$}
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 $e^{-i\omega_0t}$: the physical <electric field> is proportional to the real part of $A(t)e^{-i\omega_0t}$ for a fixed transverse mode. With <angular frequency> offset $\Omega=\omega-\omega_0$, fix the <Fourier transform> convention
$$
\widetilde A(\Omega)=\int_{-\infty}^{\infty}A(t)e^{i\Omega t}\,dt,\qquad
A(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\widetilde A(\Omega)e^{-i\Omega t}\,d\Omega.
$$
Write $\widetilde A=B e^{i\phi}$ with real $B\geq0$. A dispersive element adds <spectral phase>
$$
\phi(\Omega)=\phi_0+\phi_1\Omega+\frac{\phi_2}{2}\Omega^2+\frac{\phi_3}{6}\Omega^3+\cdots.
$$
Constant <phase> rotates the envelope; the linear term delays it by $\phi_1$. The <group delay> is $d\phi/d\Omega$. The quadratic coefficient $\phi_2$ 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:
$$
\phi_{2,\rm total}=\phi_{2,\rm stretch}+\phi_{2,\rm amplifier}+\phi_{2,\rm compress}.
$$
Setting this sum to zero does not cancel <third-order dispersion>, $\phi_3$, 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>,
$$
U=\int |A|^2\,dt=\frac{1}{2\pi}\int B^2\,d\Omega.
$$
For a smooth decaying spectrum, the <Fourier transform> of $tA$ is $-i\,d\widetilde A/d\Omega$. Consequently the power-weighted mean arrival time and <variance> obey
$$
\bar t=\frac{\int B^2\phi'\,d\Omega}{\int B^2\,d\Omega},\qquad
\sigma_t^2=\frac{\int (B')^2\,d\Omega}{\int B^2\,d\Omega}
+\frac{\int B^2(\phi'-\bar t)^2\,d\Omega}{\int B^2\,d\Omega}.
$$
At 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.
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