  Space-time (ST) wave packets are propagation-invariant pulsed optical beams
whose group velocity can be tuned in free space by tailoring their
spatio-temporal spectral structure. To date, efforts on synthesizing ST wave
packets have striven to maintain their propagation invariance. Here, we
demonstrate that one degree of freedom of a ST wave packet -- its on-axis
spectrum -- can be isolated and purposefully controlled independently of the
others. Judicious spatio-temporal spectral amplitude and phase modulation
yields ST wave packets with programmable spectral changes along the propagation
axis; including red-shifting or blue-shifting spectra, or more sophisticated
axial spectral encoding including bidirectional spectral shifts and
accelerating spectra. In all cases, the spectral shift can be larger than the
initial on-axis bandwidth, while preserving the propagation-invariance of the
other degrees of freedom, including the wave packet spatio-temporal profile.
These results may be useful in range-finding in microscopy or remote sensing
via spectral stamping.
