We propose and implement a geodesic half-Maxwell fish-eye lens antenna. The lens was optimized using an inhouse physical optics code adapted for generalized geodesic lenses. The final antenna design was validated with commercial electromagnetic simulation software. The antenna combines a modulated geodesic half-Maxwell fish-eye lens and a transition to a linear flare, which is needed to preserve the linear polarization in the aperture. The antenna prototype, designed to operate in Ka-band, was manufactured with CNC milling and measured in an anechoic chamber. The design provides continuous beam scanning thanks to a mechanically actuated feed. Promising beam scanning properties are demonstrated in an angular range of ±45°with scan loss below 3 dB, as well as good frequency stability from 26 GHz to 32 GHz. Since the antenna is fully metallic, its radiation efficiency is high (approximately 90%).