Theoretical study of the general order Hermite-Gaussian beam transformation by a phase circular zone hologram, classifies the phase circular zone hologram as a multiple laser beam splitter, if the incidence is off axial. Each of the split components is of the same mode-order and orientation as the incident beam, but are described by different complex parameters. Their waist locations and magnifications are dictated by the position of the manyfold foci of the zone hologram, and for a given diffracting order, satisfy the Self's relations, typical for the beam transformation by ordinary lens. From the theoretical results the Kogelnik's ABCD rules and ray transfer matrices for the circular hologram are defined.
"Phase-circular hologram as a laser beam splitter", Proc. SPIE 1574, Intl Colloquium on Diffractive Optical Elements, (1 October 1991); doi: 10.1117/12.50117; https://doi.org/10.1117/12.50117