  Fourier optics enforces a tradeoff between length and narrowness in
electromagnetic wavepackets, so that a narrow spatial focus diffracts at a
large divergence angle, and only infinitely wide beams can remain
non-diffracting. We show that it is possible to bypass this tradeoff between
the length and narrowness of intensity hotspots, and find a family of
electromagnetic wavepackets that abruptly focus to and defocus from
high-intensity regions of any aspect ratio. Such features are potentially
useful in scenarios where one would like to avoid damaging the surrounding
environment, for instance, to target tumors very precisely in cancer treatment,
drill holes of very precise dimensions in laser machining, or trigger nonlinear
processes in a well-defined region. In the process, we also construct the first
closed-form solutions to Maxwell's equations for finite-energy electromagnetic
pulses. These pulses also exhibit intriguing physics, with an on-axis intensity
peak that always travels at the speed of light despite inherent diffraction.
