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As transformation optics matures and evolves into technology, one may reasonably ask whether any significant new insights are to be had at the conceptual level. In this talk I will focus upon two such issues:
1) Reflectionless media: Using transformation optics we can design, quite trivially, a medium that has zero reflection for all incidence angles and all polarizations. While it is clear how to interpret this result in terms of ‘impedance matching’ (wherein only the components parallel to the interface contribute to the impedance), it is less clear how a continuously varying medium produced via transformation optics (e.g. the standard cloaking transformation) can be interpreted as being ‘impedance matched’, since there is then no interface. This leads us on to some general considerations for the design of zero-reflection media.
2) Curvature: The relationship between transformation optics and curvature has been re-examined, and has yielded a surprise. It is now well established that transformation optics does not induce curvature, since any deformation of the electromagnetic field is accompanied by a deformation of the metric, ensuring that parallel rays remain parallel. But there is a striking, and as far as I am aware, unexplored similarity between the constitutive map of macroscopic electromagnetics, and Riemannian curvature. Even in flat vacuum, where the ‘constitutive map’ is simply the Hodge dual, it is possible to define a Riemann tensor purely in terms of the vacuum permittivity, ε_0 and the vacuum permeability, μ_0. This ‘hidden’ curvature is not curvature with respect to physical space, but is an abstract curvature that is constant throughout physical space, even for flat vacuum. Transformation Optics can then distort this latent curvature yielding a structure that is both locally non-trivial, and inhomogeneous. Curvature has been present all the time!
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