= Gain bandwidth limit on a laser pulse
{title2=$\Delta t\sim1/\Delta\nu,\quad\Delta\nu=c(1/\lambda_{\min}-1/\lambda_{\max})$}
A <transform-limited pulse> needs coherent phases across its contributing <frequency> range. A laser's usable <gain bandwidth> therefore sets a duration scale through the <time-bandwidth product>, as displayed. The wavelength interval must first be converted to frequency; the simple conversion $c\Delta\lambda/\lambda^2$ is only a narrow-band approximation. Numerical constants depend on pulse shape and width convention: a Gaussian intensity envelope has $\Delta t_{\rm FWHM}\Delta\nu_{\rm FWHM}=2\ln2/\pi\simeq0.441$. A hard gain interval is not automatically a spectral FWHM. <Mode locking> can arrange the longitudinal modes' phases, while <group velocity dispersion> must be controlled to approach the transform limit. The cavity round-trip time sets pulse spacing, rather than this bandwidth-limited width.
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