Debye representation of dispersive focused waves

We report a matrix-based diffraction integral that evaluates the focal field of any diffraction-limited axisymmetric complex system in the paraxial regime. This diffraction formula is a generalization of the Debye integral, here accommodated to broadband problems. The Fresnel number is reformulated to guarantee that the focal region is entirely within the region of validity of the Debye approximation when this parameter largely exceeds unity. Several examples are examined in detail, one of them exhibiting in-focal-plane compensation of the spatial dispersion. This simple formalism opens the door for the analysis and design of focused beams with arbitrary angular dispersion.