To understand the mathematical problem we are dealing with, consider the general expression connecting the electromagnetic D and B fields to E and H, via the material parameters ε, µ, ξ and ζ,

D = ε . E + ¹⁄_C ξ . H

B = µ . H + ¹⁄_C ζ . E

Or equivalently the connection between the stress σ and strain u tensors, for an acoustic wave in an elastic material:

σ = -C : u

(note that we are restricting the discussion to linear materials).

For an electromagnetic material we have to specify the 4 different matrices ε, µ, ξ and ζ, which amounts to 4⨉3²=36 parameters, and for an elastic material we must specify the elements of the rank four tensor C_ijkl which amount to 3⁴=81 parameters. There are various symmetries that can be applied to reduce the number of independent parameters, but considering that the material can also vary arbitrarily in space and time, we have an impossibly vast number of possible materials to explore.

If we want to answer a question such as “which materials don’t reflect any waves?”, or “how can I make an atom emit light more quickly?” then a computer can only give us a partial answer - we need to develop mathematical techniques to explore this expansive parameter space.

In recent years we have been involved in the development of several such techniques. Our initial interest was in the use of differential geometry to design electromagnetic materials, a technique known as “transformation optics” [1,2], which famously led to the design and (approximate) realization of an electromagnetic “invisibility cloak” [3]. Amongst other things, we contributed a text book on this subject [4] and applied this theory to find a new way to realise previously impossible lenses in planar waveguides [5].

More recently we have been developing techniques that, despite being often inspired by transformation optics, go beyond it. A common thread in our work is to adapt approaches from areas outside of classical wave physics and apply them to design materials. For instance, we used the formalism of Bohmian mechanics (a sadly maligned interpretation of quantum mechanics) to design a class of electromagnetic materials where geometrical optics is not an approximation, but contains all the information about the wave [6]. In a similar spirit we used a known relationship between the Schrodinger equation and the Kortweg-de Vries equation of fluid dynamics to design absorptive anti-reflection coatings, as well as invisible media exhibiting both absorption and gain [7].

We also recently used a trick often found in general relativity, where - for the purposes of finding an exact solution - the spatial coordinates are allowed to be complex numbers. Using this idea we have recently found a very general way to design reflectionless materials [8,9,10], resulting in several experiments [11,12], and a new way to eliminate reflection from disordered media [13]. This method of understanding wave propagation using complex spatial coordinates also turns out to be useful for understanding the conditions necessary for a wave to be forced to propagate in only one direction [14]. Our interest is not only confined to electromagnetic and acoustic waves. With Professor Volodymyr Kruglyak we are also applying our ideas to control spin waves in magnetic materials [15].

1. J. B. Pendry, D. Schurig and D. R. Smith, “Controlling Electromagnetic Fields”, Science 312, 1780 (2006).
2. U. Leonhardt and T. G. Philbin, “General Relativity in electrical engineering”, New Journal of Physics 8, 247 (2006).
3. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies”, Science 314, 977 (2006).
4. U. Leonhardt and T. G. Philbin, “Geometry and Light: The Science of Invisibility”, Dover (New York) (2010).
5. S. A. R. Horsley, I. R. Hooper, R. C. Mitchell-Thomas and O. Quevedo-Teruel, “Removing singular refractive indices with sculpted surfaces” Sci. Rep. 4, 4876 (2014).
6. T. G. Philbin, “Making geometrical optics exact” J. Mod. Opt. 61, 552-557 (2014).
7. S. A. R. Horsley, “The KdV Hierarchy in Optics” J. Opt. 18 085104 (2016).
8. S. A. R. Horsley, M. Artoni and G. C. La Rocca, “Spatial Kramers-Kronig relations and the reflection of waves” Nat. Phot. 9, 436 (2015).
9. S. A. R. Horsley, C. G. King and T. G. Philbin, “Wave propagation in complex coordinates” J. Opt. 18 044016 (2016).
10. S. A. R. Horsley and S. Longhi, “Spatiotemporal deformations of reflectionless potentials” Phys. Rev. A 96, 023841 (2017).
11. W. Jiang, Y. Ma, J. Yuan, G. Yin, W. Wu, S. He, “Deformable metamaterial absorber”, Laser & Photonics Reviews, 1600253 (2017).
12. D. Ye, C. Cao, T. Zhou, J. Huangfu, G. Zheng and L. Ran, “Observation of reflectionless absorption due to spatial Kramers-Kronig profile”, Nat. Com. 8 51 (2017).
13. C. G. King, S. A. R. Horsley and T. G. Philbin, “Perfect Transmission through Disordered Media”, Phys. Rev. Lett. 118, 163201 (2017).
14. S. A. R. Horsley, “Unidirectional wave propagation in media with complex principal axes”, Phys. Rev. A 97, 023834 (2018).
15. N. J . Whitehead, S. A. R. Horsley, T. G. Philbin, A. N. Kuchko and V. V. Kruglyak, “Theory of linear spin wave emission from a Bloch domain wall”, Phys. Rev. B 96, 064415 (2017).