Mathieu Space-Time Wave Packets

Nondiffractive space-time wave packets are optical pulses whose engineered spatiotemporal spectra allow them to propagate without diffraction or dispersion. In this work, we introduce Mathieu nondiffractive space-time wave packets, an exact family of smooth, confined pulses free of the singularities and discontinuities that often appear in exact analytic nondiffractive space-time wave-packet solutions. The construction is based on Lorentzian elliptic coordinates in space-time, for which the Lorentzian semifocal scale 𝑓 controls a continuous transition from a light-cone-like hyperbolic geometry to an increasingly Cartesian-like regime. We characterize the wave packets through both their space-time profiles and spectral-line amplitudes. These results provide a coordinate-based framework for constructing and describing broader families of nondiffractive space-time fields.