Far-zone focusing lenses designed by complex coordinate transformations

In this letter, a transformation optics approach involving complex coordinates is presented for designing far-zone focusing lenses fed by a single radiating electromagnetic source. In contrast to conventional coordinate transformation (CT) methods that tailor only the real parts of the material parameters, a spatial amplitude tapering function containing either gain or loss is introduced within the lens, which provides an extra degree of freedom in controlling both the amplitude and phase distributions at the aperture of the lens, resulting in a controllable far-field radiation pattern. The concept is validated by full-wave lens simulations with parametric studies carried out to investigate the impact of both the geometrical and material properties of the lens on the far-field patterns. It is then demonstrated that the lens designs can be further optimized to control the sidelobe levels (peak as well as minor lobes) while maintaining a relatively high gain. Finally, it is shown that the lens exhibits a far-field radiation pattern comparable to that of a linear array with uniform element spacing, but is unidirectional rather than bidirectional. By providing a means to control both the amplitude and phase distribution across the lens aperture, the proposed complex CT design approach provides a pathway to constructing directive emitting devices with radiation properties similar to conventional antenna arrays but with only a single feeding source.