Abstract
Lenses are of interest for the design of directive antennas and multi-optics instruments in the microwave, terahertz and optical domains. Here, we introduce an optical problem defined as the complement of the well-known generalized Luneburg lens problem. The spherically symmetric inhomogeneous lenses obtained as solutions of this problem transform a given sphere in the homogeneous region outside of the lens into a virtual conjugate sphere, forming a virtual image from a real source. An analytical solution is proposed for the equivalent geodesic lens using the analogy between classical mechanics and geometrical optics. The refractive index profile of the corresponding inhomogeneous lens is then obtained using transformation optics. The focusing properties of this family of lenses are validated using ray-tracing models, further corroborated with full-wave simulations. The numerical results agree well with the predictions over the analyzed frequency bandwidth (10–30 GHz). This virtual focusing property may further benefit from recent developments in the fields of metamaterials and transformation optics.
Highlights
Lenses are of interest for the design of directive antennas and multi-optics instruments in the microwave, terahertz and optical domains
In the case of the so-called Luneburg lens[8], one sphere is on the surface of the lens and the conjugate sphere is at infinity, corresponding to the perfect collimation of a spherical wave emerging from a point source
Including the extension formulated by Eaton, who suggested to remove “the restriction imposed by Luneburg that the emerging rays be parallel to the axis of symmetry of the system”[9] for the particular case of a conjugate sphere at infinity, one can schematically represent the problem to be solved with Fig. 1a
Summary
Lenses are of interest for the design of directive antennas and multi-optics instruments in the microwave, terahertz and optical domains. The bounded version of this medium that is truncated at the normalized radius found more interest and is generally called Maxwell’s fish-eye lens This latter design is a particular case of the more general problem formulated about a century later by Luneburg[8] where each point of a given sphere has a perfect image on another concentric sphere, the object and its image being in an homogeneous region outside or on the surface of the spherically symmetric inhomogeneous lens. The Rinehart–Luneburg lens has a simple closed-form expression when its rotationally symmetric profile is defined using the arc length measured from the symmetry axis as a function of the lens radius This equivalence may be seen as an early implementation of conformal mapping, or transformation optics (TO)[16,17], with Kunz[18] extending the particular transformation described by Rinehart to a more general equivalence between two rotationally symmetric surfaces. This problem is introduced here as the complement of the generalized
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
