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An advanced focusing scheme, called a “flying focus,” uses a chromatic focusing system combined with a broadband laser pulse with its colors arranged in time to propagate a high intensity focus over a distance that can be much greater than its Rayleigh length while decoupling the speed at which the peak intensity propagates from its group velocity. The flying focus generates a short effective pulse duration with a small diameter focal spot that co- or counter-propagates along the optical axis at any velocity. Experiments validating the concept measured subluminal (−0.09c) to superluminal (39c) focal spot velocities with a nearly constant peak intensity over 4.5 mm. Experiments that increased the peak intensity above the ionization threshold for gas demonstrated ionization waves propagating at the velocity of the flying focus. These ionization waves of any velocity overcome several laser-plasma propagation issues, including ionization-induced refraction. The flying focus presents opportunities to overcome current fundamental limitations in laser-plasma amplifiers, laser wakefield accelerators, photon accelerators, and high-order frequency conversion.An advanced focusing scheme, called a “flying focus,” uses a chromatic focusing system combined with a broadband laser pulse with its colors arranged in time to propagate a high intensity focus over a distance that can be much greater than its Rayleigh length while decoupling the speed at which the peak intensity propagates from its group velocity. The flying focus generates a short effective pulse duration with a small diameter focal spot that co- or counter-propagates along the optical axis at any velocity. Experiments validating the concept measured subluminal (−0.09c) to superluminal (39c) focal spot velocities with a nearly constant peak intensity over 4.5 mm. Experiments that increased the peak intensity above the ionization threshold for gas demonstrated ionization waves propagating at the velocity of the flying focus. These ionization waves of any velocity overcome several laser-plasma propagation issues, including ionization-induced refraction. The flying focus presents opportunities to overcome ...