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Spatial Kramers–Kronig relations and the reflection of waves

Nature Photonics volume 9, pages 436–439 (2015)Cite this article

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Abstract

When a planar dielectric medium has a permittivity profile that is an analytic function in the upper or lower half of the complex position plane x = x′ + ix″ then the real and imaginary parts of its permittivity are related by the spatial Kramers–Kronig relations. We find that such a medium will not reflect radiation incident from one side, whatever the angle of incidence. Using the spatial Kramers–Kronig relations, one can derive a real part of a permittivity profile from some given imaginary part (or vice versa) such that the reflection is guaranteed to be zero. This result is valid for both scalar and vector wave theories and may have relevance for designing materials that efficiently absorb radiation or for the creation of a new type of anti-reflection surface.

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Figure 1: Wave propagation through inhomogeneous media.
Figure 2: Test of reflection for all angles of incidence by simulating a line source (pointing out of the page) placed within the inhomogeneous permittivity profile shown in Fig. 1b.
Figure 3: In cases where the permittivity has neither poles nor zeros in the upper half position plane, the reflection also vanishes for TM polarization.

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Acknowledgements

S.A.R.H. acknowledges financial support from the EPSRC under programme grant EP/I034548/1 and thanks Scuola Normale Superiore (Pisa) for its hospitality. The authors thank J.B. Pendry, T.G. Philbin, A. Di Falco, J.R. Sambles, E. Hendry, I.R. Hooper, A.P. Hibbins, J.-H. Wu, V. Agranovich and V. Lucarini for discussions. In particular, J.B. Pendry and J.R. Sambles are both thanked for separately pointing out the limit of grazing incidence. The authors thank the anonymous referees for their comments, which have much improved the manuscript.

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Authors and Affiliations

  1. Department of Physics and Astronomy, University of Exeter, Stocker Road, Exeter, UK

    S. A. R. Horsley

  2. Department of Engineering and Information Technology INO-CNR Sensor Lab, Brescia University, Brescia, Italy

    M. Artoni

  3. European Laboratory for Nonlinear Spectroscopy, Sesto Fiorentino, Italy

    M. Artoni

  4. Scuola Normale Superiore and CNISM, Pisa, Italy

    G. C. La Rocca

Authors
  1. S. A. R. Horsley
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  2. M. Artoni
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  3. G. C. La Rocca
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Contributions

S.A.R.H. devised the theory, performed the simulations and wrote the manuscript. M.A. and G.C.L.R. worked on the theory and co-wrote the manuscript.

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Correspondence to S. A. R. Horsley.

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The authors declare no competing financial interests.

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Horsley, S., Artoni, M. & La Rocca, G. Spatial Kramers–Kronig relations and the reflection of waves. Nature Photon 9, 436–439 (2015). https://doi.org/10.1038/nphoton.2015.106

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