Experimental observation of electromagnetic wave localization in Vogel spirals

. We present experiments of microwave transport in planar Vogel spirals arrays of high permittivity dielectric constant. Despite the lack of disorder, wave transport in certain frequency regions is dominated by localized modes. We characterize these modes spatially, and find that in contrary to disorder induced Anderson localization, their radial decay does not only decay exponentially, but some modes are found to decay according to a power law or to a Gaussian profile. Nevertheless, by extracting experimentally the Thouless conductance, we find that the region where these Gaussian and power law modes exist are regions of low Thouless conductance, similarly to what is expected for Anderson localization. This study unveil the rich modal structure associated with these aperiodic point patterns, and pave the way toward a better understanding of wave localization in general.


Introduction
Wave propagation in ordered or disordered media can lead to various transport phenomena depending on the short or long range order present in the structures.Among those phenomena, we can cite the photonic bandgap [1], structural colors that are observed in colloidal packings [2], Anderson localization which can be observed in the presence of disorder [3], and recent studies that unveiled other properties like transparency in stealthy hyperuniform materials [4].Understanding and being able to predict the transport in such photonic media is therefore crucial for many different applications.Here, we focus on wave localization properties in photonic media that are neither fully ordered, nor disordered.Vogel spirals are deterministic aperiodic point patterns in which point n is defined in polar coordinates (r, θ) by r n = a 0 √ n and θ n = nα with a 0 a positive constant and α is an irrational number.An example of such a point pattern is shown in Fig. 1.It has been recently numerically shown that electric dipoles placed on a planar Vogel spiral point pattern support light localization even when the full 3D vector radiation pattern of the dipoles is taken into account [5].This result has to be put in parallel with the fact that Anderson localization in a fully disordered 3D point pattern of electric dipoles has been shown * e-mail: geoffroy.aubry@cnrs.frto be impossible [6], and Anderson localization of light in a 3D medium has never been experimentally observed [7].Due the the absence of disorder in Vogel spirals, Wave localization in these structures should therefore not be confused with genuine Anderson localization where disorder is a key ingredient in the theory [8].Nevertheless, the authors of Ref. [5] characterize this localization using tools that are usually linked to Anderson localization like the Thouless conductance or the level spacing statistics.

Experimental setup
Recently, we developed an experimental microwave platform for studying the effect of structural correlations between dielectric scatterers on wave transport in planar geometries in the TM polarization, i.e., when the electric field is perpendicular to the plane of the cavity [9].This platform is ideally suited for studying how dielectric scatterers placed on a Vogel spiral could foster new localization regimes.For this, we placed N = 390 cylindrical scatterers (dielectric permittivity ε = 45, radius 3 mm, height 5 mm) on an aluminium plate following a golden spiral point pattern (defined by α = 1 + √ 5 /2 the golden mean) with a 0 = 6.93 mm.A vertical antenna (1) is placed at the center of the spiral, and a mobile aluminium plate with a second antenna (2) is placed 13 mm above the base plate.For all the positions of the mobile antenna on a 5×5 mm 2 grid, we measure both the complex transmission spectra between the two antenna S 12 (ν), and the complex reflection spectra for the mobile antenna S 22 (ν) between ν = 5.5 and 15 GHz using a Vector Network Analyzer (VNA).The gray regions correspond to the bandgaps, the pink regions to the frequency ranges where power law, exponential or Gaussian modes were found.

Results
As described in Ref. [9], we extract all resonances and their parameters (frequency f 0 , complex amplitude and width δν) in each transmission spectrum using the harmonic inversion technique, and then cluster all the resonances in order to retrieve a spatial map of every mode E f 0 (x, y) of the system.Three modes are shown in Fig. 2, together with their spatial decay ⟨E(r)⟩.These measurements together with the best fits show that these structures support localized modes for the electromagnetic waves similar to the ones numerically observed in the electric dipoles Vogel spirals.The localized modes are characterized by their radial decay, which-in contrast to genuine Anderson localization-are not only exponential, but also Gaussian or power-law.These latter are characteristic of this new type of localization found in Vogel spirals.The Thouless conductance, g = δν/∆ν, with ∆ν the mode spacing, is used to further characterize the transport in these spirals.Experimentally, we extract a moving average ⟨g⟩ ∆ f ( f ) = ⟨δν⟩ ∆ f ⟨∆ν⟩ ∆ f at each frequency f ± ∆ f /2 of this quantity.We use the fact that ⟨∆ν⟩ ∆ f ∼ ∆ f / ⟨DOS⟩ ∆ f with ⟨DOS⟩ ∆ f the average density of states over ∆ f , and ⟨δν⟩ ∆ f ∼ 1/ ⟨t 0 ⟩ ∆ f with ⟨t 0 ⟩ ∆ f the average energy decay time in the system.To obtain t 0 , we first Fourier transform the transmission spectra around a frequency f to retrieve the temporal evolution of a Gaussian pulse centered at f , transmitted from the central antenna to the moving antenna.The energy lifetime in the whole system as a function of the frequency is obtained by integrating spatially the signals over the whole cavity.This allows to extract the characteristic decay time t 0 ( f ).Finally, This quantity is plotted in Fig. 2. The frequency intervals highlited in gray are the bandgaps, while the intervals in pink are those where the exponential, power law or Gaussian modes are found.We note that in these regions, the Thouless conductance drops well below 1, similarly to what is expected in the case of genuine Anderson localization, despite the lack of disorder in our system.

Conclusion
By studying the localization properties in Vogel spirals, we not only describe yet another specific system, but we show that tools used up to now to characterize disorder induced wave localization can be used in non-disordered systems.We foresee that this will turn out to have some implications in the understanding of genuine Anderson localization.

Figure 2 .
Figure 2. Spatial modal structure and their radial decay of (a)-(b) a power law mode, (c)-(d) an exponential mode, and (e)-(f) a Gaussian mode.The Thouless conductance ⟨g⟩ is plotted below.The gray regions correspond to the bandgaps, the pink regions to the frequency ranges where power law, exponential or Gaussian modes were found.