Flying focus

A flying focus is an optical pulse whose position of maximum focusing varies with time because different frequencies focus at different axial locations and arrive at different times. For a broad incident Gaussian beam, $f(\omega )=f_0+\beta (\omega -{\omega }_0)$ and spectral phase $\psi =\eta (\omega -{\omega }_0{)}^2$ give $z=f(\omega )$ and $t={\psi }^{\prime }(\omega )+z/c$, hence $v_f/c=[1+2\eta c/\beta {]}^{-1}$. This is the velocity of a focusing pattern, which can exceed $c$ or be negative without individual portions of the field propagating faster than light.