Dynamic effective anisotropy: Asymptotics, simulations, and microwave experiments with dielectric fibers

Abstract

We investigate dynamic effective anisotropy in photonic crystals (PCs) through a combination of an effective medium theory, which is a high-frequency homogenisation (HFH) method explicitly developed to operate for short-waves, numerical simulations and through microwave experiments. The HFH yields accurate prediction of the effective anisotropic properties of periodic structures when the typical wave wavelength is on the order of the array pitch, specifically we investigate a square array of pitch 2 cm consisting of dielectric rods of radius 0.5 cm and refractive index n = √ 6 within an air matrix. This behaves as an effective medium, with strong artificial anisotropy, at a frequency corresponding to a flat band emerging from a Dirac-like point in transverse magnetic (TM) polarization (i.e. longitudinal electric field parallel to the rod axis); highly-directive emission for an electric source placed inside this PC is predicted at critical frequencies. The artificial anisotropy can be shown to coincide with a change of character of the underlying effective equation from isotropic to unidirective, with markedly different values for the effective equation tensor. In transverse electric polarization, we note an even more radical change of character of the underlying effective equation from elliptic to hyperbolic near a frequency at which an inflection point occurs in dispersion curves. Delicate microwave experiments are performed in both polarizations for such a PC consisting of 80 rods and demonstrate that a directive emission in the form of a + (resp. an X) is indeed seen experimentally at the predicted frequency 9.5 GHz in TM polarization (resp. 6 GHz in TE polarization). These are clearly dynamic effects since in the quasi-static regime the PC just behaves as an isotropic medium.

Asymptotics of dynamic lattice Green’s functions

“…This should, as in the two-dimensional case [8], motivate experimental studies and motivate studies of the analogous examples in three-dimensional continuum systems.…”

“…We are also motivated by wave phenomena in structured continuum mechanics that show strong directional anisotropy when forced close to specific frequencies and this behaviour is observed in optics for photonic crystals [2,9], in microwave experiments, simulations and asymptotics involving arrays of dielectric fibres [8], and in elastic lattices and frames [12]. For two-dimensional mass-spring lattices, [22], showed strongly directional cross-like responses close to critical frequencies and more recent authors [3,31,32] have demonstrated and investigated this behaviour further with [14] connecting the strong anisotropy in these model discrete systems to the continuum observations and interpreting this in terms of a change in character of the e↵ective equations from elliptic to hyperbolic; the highly directional behaviour then being along the characteristics of the hyperbolic e↵ective equations.…”

Elastic Wave Control Beyond Band-Gaps: Shaping the Flow of Waves in Plates and Half-Spaces with Subwavelength Resonant Rods

“…In fact, it has been recently realized that many novel features of hyperbolic metamaterials such as superlensing and enhanced spontaneous emission (Poddubny et al, 2013) could be achieved thanks to dynamic anisotropy in photonic (Ceresoli et al, 2016) and phononic crystals (Colquitt et al, 2011;Antonakakis et al, 2014b). For instance, the high-frequency homogenization theory (Craster et al, 2010) establishes a correspondence between anomalous features of dispersion curves on band diagrams with effective tensors in governing wave equations: flat band and inflection (or saddle) points lead to extremely anisotropic and indefinite effective tensors, respectively, that change the nature of the wave equations (elliptic partial differential equations can turn parabolic or hyperbolic depending upon effective tensors).…”

Front Matter

“…The elastic waves carry the heat (phonons). It is worth remarking that, subsequent to pioneering work by Bensoussan, Lions, and Papanicolaou in Chapter 4 of their book [14], there has been a resurgence of interest in Foreword xvii high-frequency homogenization at stationary points in the dispersion diagram, which may be local minima or maxima, or even saddle points [15][16][17][18][19][20][21]. The wave is a modulated Bloch wave and modulation satisfies appropriate effective equations.…”