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Different kinds of spiral planar circularly polarized (CP) antennas are presented. These antennas are based on an interaction between a cylindrical surface-wave excited by an omnidirectional probe and a inhomogeneous surface impedance with a spiral pattern. The surface impedance interaction transforms a bounded surface wave into a circularly polarized leaky wave with almost broadside radiation. The problem is studied by adiabatically matching the local 2D solution of a modulated surface-impedance problem to the actual surface. Analytical expressions are derived for the far-field radiation pattern; on this basis, universal design curves for antenna gain are given and a design procedure is outlined. Two types of practical solutions are presented, which are relevant to different implementations of the impedance modulation: i) a grounded dielectric slab with a spiral-sinusoidal thickness and ii) a texture of dense printed patches with sizes variable with a spiral-sinusoidal function. Full wave results are compared successfully with the analytical approximations. Both the layouts represent good solutions for millimeter wave CP antennas.