Helicon sources are known to produce high-density plasmas and have found many applications. Different types of antenna have been used for helicon excitation but none of them generate a radio-frequency (rf) field that matches the helicon wave field determined by the dispersion equation. We show that this match can be obtained to a very good approximation by using a birdcage type antenna. Our plasma experiments show that a helicon regime with electron densities up to 5x10(12) cm(-3) is obtained for very low rf power injection (typically 200 W), and at an unusual operating pressure up to 25 Pa.