Conformal transformation of Dyakonov surface waves into bound states of cylindrical metamaterials

Veerachart Kajorndejnukul, David Artigas, and Lluis Torner

Phys. Rev. B 100, 195404 – Published 5 November 2019

Abstract

We put forward the use of transformation optics to map surface waves that exist as one-dimensional modes supported by anisotropic structures into bound states in two-dimensional geometries. Specifically, we show the conformal mapping of Dyakonov waves existing in infinite planar surfaces separating birefringent media into bound modes supported by a cylindrical structure made of suitable metamaterials. In contrast with the original Dyakonov waves, the resulting fiber-like modes are highly dispersive, may exist as fundamental as well as higher-order states, feature helical wave fronts, and exhibit a lower and upper frequency cutoff. The program we put forward can be applied to all wave phenomena currently known to occur only in planar geometries in different types of anisotropic media.

Received 1 August 2019
Revised 21 October 2019

DOI:https://doi.org/10.1103/PhysRevB.100.195404

Physics Subject Headings (PhySH)

Metamaterials Optical vortices Plasmonics Transformation optics

Atomic, Molecular & Optical

Authors & Affiliations

Veerachart Kajorndejnukul¹, David Artigas ^1,2,*, and Lluis Torner ^1,2

¹ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
²Department of Signal Theory and Communications, Universitat Politecnica de Catalunya, 08034 Barcelona, Spain

^*david.artigas@icfo.eu

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Vol. 100, Iss. 19 — 15 November 2019

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Images

Figure 1
Conformal transformation of one-dimensional surface states. (a) and (b) conformal map used to transform DSWs existing at planar interfaces of an anisotropic substrate and suitable isotropic cladding (c) into bounds modes existing at the interface of a cylindrical metamaterial (d). Amplitude (e) and phase (g) of the original DSW mapped into an amplitude (f) and helical phase (h) with charge $m = + 4$ in the transformed space.
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Figure 2
(a) Radial distribution of the relative permittivity and permeability tensor elements of the cylindrical metamaterial obtained by using Eqs. (6) and (7), with $n_{o} = 1.25, n_{e} = 2$ , and $n_{c} = 1.3$ in the original space. (b) The inset shows a potential practical realization constructed by distributing rods with radius $r_{a}$ in a hexagonal lattice with unit cell dimension $a$ made with different electric and magnetic properties in the core and cladding regions with a radially varying filling factor embedded in a host dielectric core and cladding with the same refractive indices as in the original space. The changing radius of the rods, normalized with respect the cell unit size, are shown as a blue line, corresponding to rods composed of magnetic dielectrics with $μ_{r} = 6$ and $ɛ_{r} = 20$ . The changing rods radius in the cladding is shown as red lines, and corresponds to rods composed of magnetic metals with $μ_{r} = - 10$ and $ɛ_{r} = - 20$ .
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Figure 3
Electric-field amplitudes of the generalized two- dimensional Dyakonov-like surface states. Electric-field amplitudes of each component, (a) $E_{ρ}$ , (b) $E_{ϕ}$ and (c) $E_{Z}$ , obtained by direct field mapping described by Eq. (3). (d)–(f) numerical FE results. The black circular line represents the interface between the core and the cladding regions located at $ρ = R = 1 μ m$ . The comparison between the two sets of results was performed over an equally sized window.
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Figure 4
Mode dispersion of the two-dimensional generalized Dyakonov-like modes. (a) Angular dispersion for the effective index $N$ of the original DSW. The lower $(θ_{c, \min})$ and upper $(θ_{c, \max})$ cut-off angles indicate angular positions for radiation into the cladding $(γ_{c} = 0)$ and substrate $(γ_{e} = 0)$ , respectively. (b) Wavelength dispersion of the transformed DSW showing the effective index, $N_{Z}$ , for different modes with order $m$ (blue solid line). Blue markers indicate FE results. The grey dashed lines provide the lower and upper cut-off wavelengths, $λ_{c, \min, m}$ and $λ_{c, \max, m}$ , which are related with $θ_{c, \min}$ and $θ_{c, \max}$ , respectively, through Eq. (2). The orange dashed line denotes the threshold for surface guiding. At wavelength above the orange line, DSWs in the original space are mapped into highly localized field at the core center. Power flow calculated with the FE of two EM modes (c) below $(λ = 0.633 μ m)$ and (d) above $(λ = 0.75 μ m)$ the threshold for surface guiding [indicated by the gold markers in (b)].
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Figure 5
Normalized localization length $L / L_{o}$ of the two-dimensional generalized Dyakonov-like states as a function of a normalized core radius $R / λ_{0}$ of the cylindrical structure.
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