Controlling electromagnetic fields at boundaries of arbitrary geometries

Jonathon Yi Han Teo, Liang Jie Wong, Carlo Molardi, and Patrice Genevet

Phys. Rev. A 94, 023820 – Published 9 August 2016

Abstract

Rapid developments in the emerging field of stretchable and conformable photonics necessitate analytical expressions for boundary conditions at metasurfaces of arbitrary geometries. Here, we introduce the concept of conformal boundary optics: a design theory that determines the optical response for designer input and output fields at such interfaces. Given any object, we can realize coatings to achieve exotic effects like optical illusions and anomalous diffraction behavior. This approach is relevant to a broad range of applications from conventional refractive optics to the design of the next-generation of wearable optical components. This concept can be generalized to other fields of research where designer interfaces with nontrivial geometries are encountered.

Received 3 March 2016

DOI:https://doi.org/10.1103/PhysRevA.94.023820

Physics Subject Headings (PhySH)

Imaging & optical processing Photonics

Metamaterials

Atomic, Molecular & Optical

Authors & Affiliations

Jonathon Yi Han Teo¹, Liang Jie Wong¹, Carlo Molardi¹, and Patrice Genevet^2,*

¹Singapore Institute of Manufacturing Technology, Agency for Science, Technology and Research (A*STAR), 71 Nanyang Drive, Singapore, 638075, Singapore
²Centre de Recherche sur l'Hétéro-Epitaxie et ses Application, Université Côte d'Azur, CNRS, Rue Bernard Gregory, Sophia-Antipolis, 06560 Valbonne, France

^*patrice.genevet@crhea.cnrs.fr

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Vol. 94, Iss. 2 — August 2016

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Figure 1
The surface susceptibility of an interface that conforms to an object can be designed to produce an arbitrary, user-specified response. In (a), a cat is visually transmogrified into a rat as an incident light $E^{inc}$ is converted at the object interface to produce a user-specified virtual holographic image $E^{ref}$ in the far field. This way, the actual geometry of the object no longer restricts the properties of the reflected light. Our proposed concept exploits the relationship between (b) a given ambient coordinate system and (c) the coordinate system that conforms to the geometry of the boundary. $χ_{e, m}$ and $χ_{e, m}^{^{'}}$ denote the optical response of the interface quantities called surface susceptibilities tensors in these coordinate systems.
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Figure 2
(a) A 2D planar metasurface of subwavelength thickness $δ ≪ λ$ can transmit any incident optical field at a specific angle by imposing a gradient of phase discontinuity. For planar interfaces, GSTC boundary conditions readily apply, and the surface susceptibility tensors can be calculated. (b) The local coordinate system of the surface follows its local curvature, and therefore it changes with the position along the interface. Boundary conditions of the fields are obtained in the coordinate system of the interface and are therefore position dependent. To produce an effect equivalent to that in (b), the surface susceptibilities of the optical interface have to be engineered to account for the effect of the physical distortion. The dashed blue lines denote the equiphase fronts of the electromagnetic fields.
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Figure 3
Schematic describing the mathematical method used to rewrite the electromagnetic field discontinuities in the language of differential geometry. (a) We define three disjointed three-manifolds (three-dimensional manifolds), $M^{+}, M^{-}$ , and $S$ . We identify the boundaries of $M^{\pm}$ with the boundaries of $S$ to form a single manifold $M = M^{+} \cup S \cup M^{-}$ . $S$ has subwavelength thickness $δ ≪ λ$ . $N$ is the unit vector field normal to $S$ . To derive the boundary conditions from Maxwell's integral equations, we consider (b) an Amperian loop and (d) a Gaussian pillbox. (c) $γ^{\pm}$ and ${γ |}_{S}$ are defined such that $γ = γ^{\pm} {\cup γ |}_{S}$ , and $γ^{S^{\pm}}$ is defined such that $γ^{S^{\pm}} {\cup γ |}_{S} = γ^{S}$ forms a closed loop crossing $S$ . (e) $\partial K^{\pm} = \partial K \cap M^{\pm}$ , each having outward-pointing unit normals $N_{K}^{\pm}$ . Define $\partial k^{S^{\pm}}$ such that together with $\partial K \cap S$ they form a Gaussian pillbox inside $S$ ; let $N_{K}^{S}$ be the outward-pointing unit normal of this Gaussian pillbox.
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Figure 4
(a) Mirror surface cloak described by $f (u, v) = cos u + cos v$ reflects an incident plane wave at an angle of $θ$ with respect to the $z$ axis to form a plane wave propagating at the same angle $θ$ for all orientation of the surface. (b) The relation between the angle of the incoming and outgoing waves can be imposed at the interface by considering the condition that the difference between the propagation phase shift of two light rays of the incident wave front $Δ ϕ_{i}$ , impinging on the surface at points separated by a distance ( $Δ u, Δ v$ ), and the propagation phase shift after the interface $Δ ϕ_{t}$ is exactly compensated by the phase shift introduced at the interface at those points. However, local phase retardations are not the only physical quantities to take into consideration. Because the orientation of the surface, specifically the orientation of the unit vector field $N$ normal to $S$ , changes as one moves across it, the induced dipole moments at the surface are excited and radiate differently depending on the location. To account for the aforementioned changes in phase, amplitude, and polarization, susceptibility tensors are calculated from Eqs. (37) and (38, 39, 40, 41). (c) and (d) Conventional and anomalous gratings, respectively. (e) and (f) Light incident on a metasurface grating in (d) can blaze the diffraction towards a single order (black curves in e and f), but it can also refract light at any other user-specified angle (red curves). Solid (dashed) curves represent the imaginary (real) part of the susceptibility tensors. (g) The calculation of the real part of the electric fields for free-standing metasurfaces in air, that is, when the refractive indices on either side of the boundary are set to be equal.
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Figure 5
Combining the use of conformal boundary optics and transformation optics can be useful for achieving complete, geometry-independent control over the behavior of light at interfaces and in bulk media.
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