# Strong localization of microwaves beyond 2D in aperiodic Vogel spirals

Luis A. Razo-López  Université Côte d’Azur, CNRS, Institut de Physique de Nice
(INPHYNI), France  Geoffroy J. Aubry  Université Côte d’Azur, CNRS, Institut
de Physique de Nice (INPHYNI), France<EMAIL_ADDRESS>Felipe
A. Pinheiro  Instituto de Física, Universidade Federal do Rio de Janeiro, Rio
de Janeiro-RJ, 21941-972, Brazil  Fabrice Mortessagne  Université Côte d’Azur,
CNRS, Institut de Physique de Nice (INPHYNI), France
<EMAIL_ADDRESS>
###### Abstract

We carry out dynamical microwave transport experiments in aperiodic Vogel
spiral arrays of cylinders with high dielectric permittivity. We
experimentally disclose the electromagnetic modal structure of these
structures in real space showing that they simultaneously support long-lived
modes with Gaussian, exponential, and power law spatial decay. This unique
modal structure, which cannot be found in traditional periodic or disordered
photonic materials, is shown to be at the origin of strong localization in
Vogel spirals that survives even in three dimensions. Altogether our results
unveil the manifestations of the rich, unprecedented, spatial structure of
electromagnetic modes supported by aperiodic photonic systems in wave
transport and localization.

A full understanding and control over electromagnetic transport in photonic
media is crucial for the efficient design of optical structures, paving the
way for many applications [1]. At the microscopic level, controlling light
transport in photonic structures involves the ability of not only understand,
but also to engineer the electromagnetic modes that such structures can
support. In the case of traditional periodic and disordered optical
structures, light transport and the underlying electromagnetic modal structure
have been extensively investigated over the years. In periodic photonic
structures the scattering of propagating, extended electromagnetic waves from
Bragg planes is related to the opening of photonic bandgaps at certain
frequencies [2]. In disordered optical systems the interference of multiply
scattered waves may lead to the formation of exponentially localized states
and eventually to the breakdown of light diffusion [3, 4, 5]. This effect, the
optical counterpart of Anderson localization for electrons in solids [6],
strongly depends on dimensionality [7], and in three-dimensional (3D) media
there is no unquestionable observation of light localization transition so far
for disordered systems [8, 9, 10]. When structural correlations are
introduced, as it is the case of two-dimensional (2D) hyperuniform disordered
materials, a richer transport diagram exists that include transparency, light
diffusion, Anderson localization, or full band gaps, depending on the
frequency [11, 12, 13, 14, 15, 16, 13, 17, 18, 19, 20, 21].

As an alternative to periodic and disordered photonic structures, aperiodic
metamaterials designed by means of deterministic mathematical rules have
emerged as a novel material platform for photonic devices [22, 23, 24].
Indeed, these structures exhibit unique optical properties that do not exist
in either periodic or disordered photonic media, such as fractal transmission
spectra [25, 26], subdiffusive transport [27], and light localization
transition [28]. From a technological point of view, these unusual optical
properties have fostered the development of functionalities that also cannot
be found in conventional periodic or disordered structures, including
applications in lasing [29, 30], optical sensing [31, 32, 33], photo-detection
[34], and optical imaging [35]. Among various classes of deterministic
aperiodic photonic media, Vogel spiral arrays single out for its versatility
and the possibility to tailor its structural order [36, 37, 38]. From the
microscopic level the unusual optical properties of aperiodic systems are
enabled by their unique electromagnetic modal structure. For instance,
recently it was theoretically demonstrated that aperiodic Vogel spiral arrays
display a rich spectrum of long-lived and spatially localized quasimodes with
distinctive spatial decay forms, namely Gaussian, exponential, and power law
[39].

Figure 1:  (a) Typical configuration of the experimental microwave setup.
Cylinders are placed between two parallel aluminum plates separated by a
distance $h=13$ mm. A fixed antenna (2) is placed at the origin of the
reference system $(x,y)=(0,0)$ while an other antenna (1) is positioned in the
center of the movable top plate $(x,y)$. (b) Image of the 2D array of
dielectric cylinders. The top plate has been removed to reveal the details of
the sample. (c) Reflected $|S_{11}(\nu)|^{2}$ (blue line) and transmitted
coefficients $|S_{21}(\nu)|^{2}$ (orange line) for the GA spiral at a given
distance $d=\sqrt{x^{2}+y^{2}}$ ($x=75$ mm, $y=75$ mm) from the origin of the
reference system.

In the present letter, we not only experimentally demonstrate that these
characteristic types of electromagnetic modes coexist in Vogel spirals, but
also that this unique electromagnetic modal structure leads to unusual wave
transport phenomena. Indeed, by conducting microwave transport experiments in
Vogel spiral arrays of cylinders with high dielectric permittivity, we unveil
the consequences of this peculiar modal structure on wave transport and
localization. In particular, we show that the presence of long-lived
quasimodes with exponential, power law and Gaussian spatial decays is at the
origin of a very slow decay of the electromagnetic energy that propagates
throughout the arrays—of the same order as in the case of conventional 2D
disordered systems in the regime of Anderson localization. Furthermore, we
experimentally demonstrate that Vogel spirals support localized modes
satisfying the Thouless criterion for Anderson localization in random systems,
despite the fact these structures are not disordered. Importantly, we show
that the Thouless criterion can be fulfilled for modes that are not
necessarily exponentially localized in space, in contrast to what occurs in
the case of Anderson localization in disordered systems. Finally, we
demonstrate the robustness of these long-lived modes against the change in the
dimensionality of the cavity, beyond the 2D limit, preserving their spatial
profiles and quality factors even when the homogeneity of the electric field
in the $z$-direction is broken. Altogether, our findings experimentally
demonstrate the unprecedented electromagnetic modal structure of Vogel spirals
arrays in real space, and reveal its impact in wave transport phenomena and
localization.

Our main sample consists of $N=390$ cylindrical scatterers (dielectric
permittivity $\varepsilon\simeq 45$, radius 3 mm and height 5 mm) disposed in
a cavity made of two parallel aluminium plates which are separated by a
distance $h=13$ mm [see Fig. 1(a)]. Cylinders are placed following a Vogel
spiral array of 140 mm radius with a planar density $\rho\approx 0.65$ cm-2
[see Fig. 1(b)]. This lattice, often called _Golden-Angle spiral_ (GA spiral),
is defined in polar coordinates $(r,\theta)$, as $r_{n}=a_{0}\sqrt{n}$
($a_{0}=6.93$ mm) and $\theta_{n}$ linked to the golden number (see
Supplementary material I). In order to ensure an homogeneous electrical
contact between the scatterers and the bottom plate and thus a good
reproducibility, we covered the bottom plate with a self-adhesive thin plastic
film.

Figure 2: Spatial modal structure and radial profile of representative
eigenmodes with characteristic (a)-(b) power law, (c)-(d) exponential, and
(e)-(f) Gaussian decay. Amplitude maps are normalized such that
$\max(\absolutevalue{E_{\nu}})=1$. Radial decays (pink dots) are obtained by
performing an azimuthal average operation where $r=\sqrt{(x+5)^{2}+y^{2}}$
i.e., $r$ is measured from the geometrical center of the spiral and expressed
in mm. Different radial decays are discriminated by minimizing the sum of
squared residuals.

As sketched in Fig. 1(a), the electric field in the cavity is mapped by the
straight antenna (1) placed at the center of the movable top plate. We measure
all points on a $5\times 5$ mm2 grid, covering the disk region occupied by the
spiral plus one corner of the embedding square [see, e.g., Fig. 2(a)],
resulting in $\sim 3600$ measured points (see Supplementary material II). The
presence of the second straight antenna (2) at the center of the bottom plate
defines the origin of the system coordinates $(x,y)=(0,0)$ and allows to
measure both the complex reflection and transmission signals $S_{11}(\nu)$ and
$S_{21}(\nu)$, respectively, using a Vector Network Analyzer (VNA). Note that
both antennas are linear and perpendicular to the plane of the cavity, thus
imposing a transverse magnetic (TM) polarization (electric field perpendicular
to the plane of the cavity). For a height $h=13$ mm, the empty cavity can be
completely considered as 2D below the cutoff frequency
$\nu_{\mathrm{cut}}=c_{0}/\left(2h\right)\approx 11.5$ GHz, where $c_{0}$ is
the speed of light in air. Beneath this threshold, just the fundamental
transverse magnetic mode TM0 can propagate in air and the field is invariant
along the $z$-axis. Measured positions are scanned in a frequency range
between 5.5 and 15 GHz meaning that both 2D and 3D regimes can be
investigated. Examples of the measured spectra $|S_{21}|^{2}$ as well as
$|S_{11}|^{2}$ are shown in Fig. 1(c) for a given distance $d$ between both
antennas, where the geometrical center of the spiral is placed at
$(x,y)=(-5\,\mathrm{mm},0)$. Vanishing transmission values (reflection values
close to 1) at certain frequencies indicate the presence of bandgaps. Outside
of these gaps, the transmitted signal is a superposition of peaks which are
related with the resonances of the system. The parameters of each resonance
(frequency $\nu$, width $\delta\nu$ and complex amplitude) are extracted by
means of the harmonic inversion technique as described in Refs. [40, 41].
Later, by clustering the amplitudes of the same resonance measured at all
positions, the map of the electric field amplitude $E_{\nu}(x,y)$ is obtained
revealing the spatial structures of each eigenmode of the system [18].

Figure 2 shows the spatial modal structure of three characteristic eigenstates
found in the same experimental GA spiral and their corresponding radial decay.
Specifically, power-law, exponential, and Gaussian radial decays have been
identified by minimizing the sum of squared residuals. Our experimental
results demonstrate that Vogel spirals support a rich variety of long-lived
modes that exhibit different spatial extent and radial decay profiles, hence
confirming recent theoretical predictions [39]. This can be contrasted to
disordered samples, where (Anderson) localized states are always characterized
by an exponential radial decay (see Supplementary material III). The analyzed
long-lived modes found experimentally in this GA spiral sample are distributed
in three frequency windows around $\nu\sim 6.6$ GHz (exponential, power-law
and Gaussian modes), $\nu\sim 8.3$ GHz and $\nu\sim 11.2$ GHz (Gaussian modes)
and are characterized by high quality factors $Q=\nu/\delta\nu$, i.e., low
energy-loss ratios.

Dynamical electromagnetic transport properties can be probed by measuring the
temporal evolution of the energy carried by a certain superposition of modes
by means of the transmission spectra as $E=\sum_{\textrm{all
positions}}\left|\mathcal{F}\left\\{S_{21}\times
F_{f_{0},\Delta\omega}(\nu)\right\\}\right|^{2}$, where
$\mathcal{F}\left\\{\cdot\right\\}$ represents the Fourier-transform and
$F_{f_{0},\Delta\omega}(\nu)$ a Gaussian band pass filter of bandwidth
$\Delta\omega$ centered around $f_{0}$. Figure 3(a) displays the energy as a
function of time for three different filters centered around the frequency of
the modes extracted in Fig. 2 and with $\Delta\omega=0.01$ GHz.

Figure 3:  (a) Evolution of the energy as function of time for three different
frequency centers $f_{0}$ and $\Delta\omega=0.01$ GHz. (b) Characteristic
decay time $t_{0}$ as a function of the frequency. The total frequency range
have been mapped by 472 frequency filters spaced by $\Delta f=0.02$ GHz with
$\Delta\omega=0.01$ GHz. (c) Experimental density of states (DOS). (d)
Experimental Thouless conductance $\expectationvalue{g}$ as a function of the
frequency. The dotted line indicates $\expectationvalue{g}=1$. Frequency
windows with the analyzed long-lived states (exponential, power-law or
Gaussian) or band gaps have been highlighted (violet or grey stripes,
respectively). Solid [dashed] lines in (b), (c) and (d) correspond to the case
$h=13$ mm [$h=17$ mm].

Within these frequency intervals, the presence of modes with high quality
factors leads to very slow energy dynamics. Assuming an exponential decay of
the energy with respect to time, $E\sim\exp\left(-t/t_{0}\right)$, one can fit
a characteristic decay time $t_{0}$ that is closely related to the average
width of the modes contributing to the transport
$\expectationvalue{\delta\nu}\sim 1/t_{0}$. Next, by repeating the previous
analysis in a systematic way, we compute the characteristic decay time $t_{0}$
as a function of the frequency [see Fig. 3(b)]. We focus at frequency ranges
where the spatial structure of the eigenmodes can be properly characterized by
the harmonic inversion/clustering methods. Here, $t_{0}$-regions associated
with frequency windows containing exponential, power-law or Gaussian states
(pink stripes) are characterized by high peaks whose maximum values are of the
order of those found in disordered systems (see Supplementary material III),
whilst $t_{0}$-valleys correspond to short-lived, not spatially localized
states or band gaps.

The existence of band gaps can be investigated by experimentally extracting
the density of states (DOS) that is directly accessible from the intensity of
the reflected signal as $\mathrm{DOS}\approx
1-\expectationvalue{\left|S_{11}\right|^{2}}_{\mathrm{all\,positions}}$ [42,
43]. In Fig. 3(c), we plot the DOS, where two band gaps are observed (grey
stripes). Additionally, no modes are found by the harmonic-
inversion/clustering methods in these frequency intervals whose positions
coincide with two of the valleys previously displayed by the characteristic
decay time. Peaks at the center of the first bandgap are a signature of defect
modes of the system. Additionally, an increment in the number of states
supported by the system is observed above $\nu_{\mathrm{cut}}$.

The Thouless conductance, defined as $g=\delta\nu/\Delta\nu$ (where
$\Delta\nu$ is the spacing between consecutive resonances), is a key quantity
in localization theory and it is used as a fundamental criterion for Anderson
localization in disordered systems. Indeed, the Thouless criterion establishes
that Anderson localization occurs for $g<1$ [44, 45, 46]. Using the two
previously introduced quantities (DOS and $t_{0}$), we extract experimentally
the average Thouless conductance [see Fig. 3(d)] as
$\expectationvalue{g}\sim\mathrm{\expectationvalue{DOS}}_{\Delta
f}/(t_{0}\Delta f)$ where $\expectationvalue{\mathrm{DOS}}_{\Delta f}$ is the
average density of states over the frequency interval $\Delta f$. Figure 3(d)
experimentally demonstrates that non-random systems can fulfil the Thouless
criterion for Anderson localization, originally conceived to characterize
localization in disordered structures, again confirming theoretical
predictions [28, 39]. Our findings confirm that not only eigenmodes
characterized by an exponential spatial decay can satisfy the Thouless
condition, as it occurs for disordered systems, but also other modes with
different spatial decay forms, such as algebraic and Gaussian decays. To the
best of our knowledge localized, long-lived modes that fulfil the Thouless
criterion in non-random arrays with non-exponential spatial decays have never
been experimentally observed so far. This result demonstrates experimentally
the unique modal structure that aperiodic Vogel spirals support, leading to
unusual properties of wave transport and localization. The existence of long-
lived modes with different decay types in the same system means that these
distinct classes of modes will have different sensitivity to the sample
boundaries. As a result, these classes of modes exhibit different evolution of
the transport quantities with respect to the system size, as shown in
Supplementary material IV.

It should be noted that, up to this point, all long-lived modes shown have
been found below the 2D cutoff frequency in air ($\nu_{\mathrm{cut}}\approx
11.5$ GHz), so the electromagnetic field is confined in the plane of the
array. In order to study the robustness of GA spiral modes with respect to the
dimensionality, the distance between both aluminium plates is increased from
$h=13$ mm to $h=17$ mm implying a new cutoff frequency
$\nu_{\mathrm{cut}}\approx 8.8$ GHz. As a result the electromagnetic field is
actually three-dimensional beyond this frequency. Then previous experimental
procedure and data analysis are repeated, thus the characteristic decay time
$t_{0}$, the DOS, and the average Thouless conductance $\expectationvalue{g}$
are extracted and correspond to the dashed lines depicted in Fig. 3(b), (c)
and (d), respectively. $t_{0}$ displays remarkable similarities in both cases,
being just differentiated by a decrease of the characteristic peaks related to
the first localization region (at $\sim 6.6$ GHz) and the formation of a new
peak in the first band gap. The appearance of this peak can be also observed
in the DOS, where the size of the first band gap has been considerably reduced
in its upper part. Additionally, the presence of new modes at lower
frequencies leads to the population of the second band gap that has completely
disappeared, confirming their 3D character. Nevertheless,
$\expectationvalue{g}$ still drops by around one order of magnitude near $\sim
11.2$ GHz, keeping its shape in all frequency windows containing long-lived
modes.

In the last part of this letter we focus in the highest-frequency window in
which long-lived exist ($\sim 11.2$ GHz), and which occurs beyond the new
cutoff frequency. Here the presence of long-lived modes previously predicted
by the analysis of $t_{0}$ and $\expectationvalue{g}$ is verified by the
Harmonic inversion/clustering methods. Four different Gaussian modes
corresponding to this frequency band are shown in Fig. 4 for both cavity sizes
[(a)-(c) $h=13$ mm and (b)-(d) $h=17$ mm]. High quality factors confirm the
existence of long-lived modes even in the case where the 2D confinement of the
electromagnetic field is not due to the geometry of the cavity.

Figure 4:  Spatial modal structure of two Gaussian long-lived modes at $\sim
11.2$ GHz with frequencies [quality factors] (a) $\nu=11.346$ GHz [$Q=5294$],
(b) $\nu=11.396$ GHz [$Q=3899$], (c) $\nu=11.35$ GHz [$Q=5170$] and (d)
$\nu=11.401$ GHz [$Q=4159$] and a distance between plates (a)-(c) $h=13$ mm
and (b)-(d) $h=17$ mm. Amplitude maps are normalized such that
$\max(\absolutevalue{E_{\nu}})=1$.

Indeed, Fig. 4 experimentally proves the robustness of Gaussian long-lived
modes in Vogel spirals against the situation in which the electric field is
inhomogeneous in the $z$ direction. This result also experimentally confirms
previous numerical findings [28] that demonstrate the existence of localized,
long-lived 3D electromagnetic modes supported by 2D Vogel spirals arrays
precisely for $\rho c_{0}^{2}/\nu^{2}>3.5$, corresponding to $\nu<12.9$ GHz in
our experimental system.

In conclusion, we have experimentally revealed the spatial modal structure
supported by quasi-two-dimensional arrays of dielectric cylinders placed
according to aperiodic Vogel spirals and its impact in wave transport. We
showed that these structures support a unique modal structure where long-lived
modes with different radial decay types (exponential, power-law and Gaussian)
coexist, confirming recent theoretical predictions [39]. We also investigate
the impact of these peculiar modal structure on microwave transport properties
by means of the temporal evolution of the energy as well as the characteristic
decay time $t_{0}$ (related with the average resonance width
$\expectationvalue{\delta\nu}$), the density of states, and the Thouless
conductance. This analysis reveals that Vogel spirals exhibit very slow energy
dynamics as a result of this particular modal structure. Indeed, we show that
in frequency windows containing long-lived modes the values of $t_{0}$ are
similar to those found in the Anderson localized regime in traditional 2D
disordered systems. In these same frequency windows the Thouless criterion for
Anderson localization is shown to be fulfilled despite the lack of disorder
and the presence of non-exponentially localized modes. Long-lived modes
supported by Vogel spiral are also proven to be robust against the change in
the dimensionality of the cavity, from 2D to 3D, so that they remain
essentially unperturbed and preserve their spatial profiles and quality
factors even when the homogeneity of the electric field in the $z$-direction
is broken. Our findings not only experimentally demonstrate the unprecedented
electromagnetic modal structure of Vogel spirals arrays in real space and
their robustness against the opening of the cavity, but also their effects in
wave localization and transport phenomena, such as slow energy transport
dynamics.

## Acknowledgements

We thank L. Dal Negro, F. Sgrignuoli, M. Prado, and Y. Chen for enlightening
discussions. F.A.P. acknowledge CNPq, CAPES, and FAPERJ for financial support.
L.A.R.-L. gratefully acknowledges the financial support from CONACyT (Mexico),
through the Grant No.775585, and from the French government, through the
UCAJEDI Investments in the Future project managed by the National Research
Agency (ANR) with the reference number ANR-15-IDEX-0001.

## References

  * Vynck  _et al._ [2021] K. Vynck, R. Pierrat, R. Carminati, L. S. Froufe-Pérez, F. Scheffold, R. Sapienza, S. Vignolini, and J. J. Sáenz, Light in correlated disordered media (2021), arXiv:2106.13892 [physics.optics] .
  * Joannopoulos  _et al._ [2008] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade,  _Photonic Crystals: Molding the Flow of Light_, 2nd ed. (Princeton University Press, Princeton (New Jersey), 2008).
  * Lagendijk  _et al._ [2009] A. Lagendijk, B. v. Tiggelen, and D. S. Wiersma, Fifty years of Anderson localization, Physics Today 62, 24 (2009).
  * Schertel  _et al._ [2019] L. Schertel, O. Irtenkauf, C. M. Aegerter, G. Maret, and G. J. Aubry, Magnetic-field effects on one-dimensional Anderson localization of light, Phys. Rev. A 100, 043818 (2019), 1812.06447 .
  * Laurent  _et al._ [2007] D. Laurent, O. Legrand, P. Sebbah, C. Vanneste, and F. Mortessagne, Localized modes in a finite-size open disordered microwave cavity, Phys. Rev. Lett. 99, 253902 (2007).
  * Anderson [1958] P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109, 1492 (1958).
  * Abrahams  _et al._ [1979] E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions, Phys. Rev. Lett. 42, 673 (1979).
  * Skipetrov and Sokolov [2014] S. E. Skipetrov and I. M. Sokolov, Absence of Anderson localization of light in a random ensemble of point scatterers, Phys. Rev. Lett. 112, 023905 (2014).
  * Sperling  _et al._ [2016] T. Sperling, L. Schertel, M. Ackermann, G. J. Aubry, C. Aegerter, and G. Maret, Can 3D light localization be reached in ‘white paint’?, New J. Phys. 18, 013039 (2016), 1510.08092 .
  * Skipetrov and Page [2016] S. E. Skipetrov and J. H. Page, Red light for Anderson localization, New J. Phys. 18, 021001 (2016), arXiv:1601.07848 [cond-mat.dis-nn] .
  * Torquato and Stillinger [2003] S. Torquato and F. H. Stillinger, Local density fluctuations, hyperuniformity, and order metrics, Phys. Rev. E 68, 041113 (2003).
  * Leseur  _et al._ [2016] O. Leseur, R. Pierrat, and R. Carminati, High-density hyperuniform materials can be transparent, Optica 3, 763 (2016).
  * Ricouvier  _et al._ [2017] J. Ricouvier, R. Pierrat, R. Carminati, P. Tabeling, and P. Yazhgur, Optimizing hyperuniformity in self-assembled bidisperse emulsions, Phys. Rev. Lett. 119, 208001 (2017), arXiv:1711.00719 [cond-mat.soft] .
  * Froufe-Pérez  _et al._ [2016] L. S. Froufe-Pérez, M. Engel, P. F. Damasceno, N. Muller, J. Haberko, S. C. Glotzer, and F. Scheffold, Role of short-range order and hyperuniformity in the formation of band gaps in disordered photonic materials, Phys. Rev. Lett. 117, 053902 (2016).
  * Froufe-Pérez  _et al._ [2017] L. S. Froufe-Pérez, M. Engel, J. J. Sáenz, and F. Scheffold, Band gap formation and Anderson localization in disordered photonic materials with structural correlations, Proceedings of the National Academy of Sciences 114, 9570 (2017).
  * Sgrignuoli  _et al._ [2022] F. Sgrignuoli, S. Torquato, and L. Dal Negro, Subdiffusive wave transport and weak localization transition in three-dimensional stealthy hyperuniform disordered systems, Phys. Rev. B 105, 064204 (2022).
  * Haberko  _et al._ [2020] J. Haberko, L. S. Froufe-Pérez, and F. Scheffold, Transition from light diffusion to localization in three-dimensional amorphous dielectric networks near the band edge, Nat. Commun. 11, 4867 (2020).
  * Aubry  _et al._ [2020] G. J. Aubry, L. S. Froufe-Pérez, U. Kuhl, O. Legrand, F. Scheffold, and F. Mortessagne, Experimental tuning of transport regimes in hyperuniform disordered photonic materials, Phys. Rev. Lett. 125, 127402 (2020), 2003.00913 .
  * Klatt  _et al._ [2022] M. A. Klatt, P. J. Steinhardt, and S. Torquato, Wave propagation and band tails of two-dimensional disordered systems in the thermodynamic limit, Proceedings of the National Academy of Sciences 119, e2213633119 (2022).
  * Froufe-Pérez  _et al._ [2023] L. S. Froufe-Pérez, G. J. Aubry, F. Scheffold, and S. Magkiriadou, Bandgap fluctuations and robustness in two-dimensional hyperuniform dielectric materials, Opt. Express 31, 18509 (2023).
  * Granchi  _et al._ [2023] N. Granchi, M. Lodde, K. Stokkereit, R. Spalding, P. J. van Veldhoven, R. Sapienza, A. Fiore, M. Gurioli, M. Florescu, and F. Intonti, Near-field imaging of optical nanocavities in hyperuniform disordered materials, Phys. Rev. B 107, 064204 (2023).
  * Dal Negro [2022] L. Dal Negro,  _Waves in complex media_ (Cambridge University Press, 2022).
  * Dal Negro and Boriskina [2012] L. Dal Negro and S. Boriskina, Deterministic aperiodic nanostructures for photonics and plasmonics applications, Laser & Photonics Reviews 6, 178 (2012).
  * Dal Negro  _et al._ [2016] L. Dal Negro, R. Wang, and F. A. Pinheiro, Structural and spectral properties of deterministic aperiodic optical structures, Crystals 6, 161 (2016).
  * Sgrignuoli  _et al._ [2020] F. Sgrignuoli, S. Gorsky, W. A. Britton, R. Zhang, F. Riboli, and L. Dal Negro, Multifractality of light in photonic arrays based on algebraic number theory, Communications Physics 3, 106 (2020).
  * Reisner  _et al._ [2022] M. Reisner, Y. Tahmi, F. Piéchon, U. Kuhl, and F. Mortessagne, Experimental observation of multifractality in Fibonacci chains (2022), arXiv:2207.13755 [cond-mat.dis-nn] .
  * Sgrignuoli and Dal Negro [2020] F. Sgrignuoli and L. Dal Negro, Subdiffusive light transport in three-dimensional subrandom arrays, Phys. Rev. B 101, 214204 (2020).
  * Sgrignuoli  _et al._ [2019] F. Sgrignuoli, R. Wang, F. A. Pinheiro, and L. Dal Negro, Localization of scattering resonances in aperiodic Vogel spirals, Phys. Rev. B 99, 104202 (2019), arXiv:1810.01909 [cond-mat.dis-nn] .
  * Noh  _et al._ [2011] H. Noh, J.-K. Yang, S. V. Boriskina, M. J. Rooks, G. S. Solomon, L. Dal Negro, and H. Cao, Lasing in Thue–Morse structures with optimized aperiodicity, Applied Physics Letters 98, 201109 (2011).
  * Vardeny  _et al._ [2013] Z. V. Vardeny, A. Nahata, and A. Agrawal, Optics of photonic quasicrystals, Nature Photonics 7, 177 (2013).
  * Razi  _et al._ [2019] M. Razi, R. Wang, Y. He, R. M. Kirby, and L. Dal Negro, Optimization of large-scale Vogel spiral arrays of plasmonic nanoparticles, Plasmonics 14, 253 (2019).
  * Lee  _et al._ [2010] S. Y. Lee, J. J. Amsden, S. V. Boriskina, A. Gopinath, A. Mitropolous, D. L. Kaplan, F. G. Omenetto, and L. Dal Negro, Spatial and spectral detection of protein monolayers with deterministic aperiodic arrays of metal nanoparticles, Proceedings of the National Academy of Sciences 107, 12086 (2010).
  * Gopinath  _et al._ [2009] A. Gopinath, S. V. Boriskina, B. M. Reinhard, and L. Dal Negro, Deterministic aperiodic arrays of metal nanoparticles for surface-enhanced raman scattering (sers), Opt. Express 17, 3741 (2009).
  * Trevino  _et al._ [2011] J. Trevino, H. Cao, and L. Dal Negro, Circularly symmetric light scattering from nanoplasmonic spirals, Nano Letters 11, 2008 (2011).
  * Huang  _et al._ [2007] F. M. Huang, Y. Chen, F. J. G. de Abajo, and N. I. Zheludev, Optical super-resolution through super-oscillations, Journal of Optics A: Pure and Applied Optics 9, S285 (2007).
  * Dal Negro  _et al._ [2012] L. Dal Negro, N. Lawrence, and J. Trevino, Analytical light scattering and orbital angular momentum spectra of arbitrary vogel spirals, Opt. Express 20, 18209 (2012).
  * Christofi  _et al._ [2016] A. Christofi, F. A. Pinheiro, and L. Dal Negro, Probing scattering resonances of Vogel’s spirals with the Green’s matrix spectral method, Opt. Lett. 41, 1933 (2016).
  * Pollard and Parker [2009] M. E. Pollard and G. J. Parker, Low-contrast bandgaps of a planar parabolic spiral lattice, Opt. Lett. 34, 2805 (2009).
  * Prado  _et al._ [2021] M. Prado, F. Sgrignuoli, Y. Chen, L. Dal Negro, and F. A. Pinheiro, Structural entropy and spatial decay of quasimodes in vogel spirals, Phys. Rev. B 104, 184204 (2021).
  * Main  _et al._ [2000] J. Main, P. A. Dando, D. Belkic, and H. S. Taylor, Decimation and harmonic inversion of periodic orbit signals, Journal of Physics A: Mathematical and General 33, 1247 (2000).
  * Wiersig and Main [2008] J. Wiersig and J. Main, Fractal Weyl law for chaotic microcavities: Fresnel’s laws imply multifractal scattering, Phys. Rev. E 77, 036205 (2008).
  * Kuhl  _et al._ [2010] U. Kuhl, S. Barkhofen, T. Tudorovskiy, H.-J. Stöckmann, T. Hossain, L. de Forges de Parny, and F. Mortessagne, Dirac point and edge states in a microwave realization of tight-binding graphene-like structures, Phys. Rev. B 82, 094308 (2010).
  * Barkhofen  _et al._ [2013] S. Barkhofen, M. Bellec, U. Kuhl, and F. Mortessagne, Disordered graphene and boron nitride in a microwave tight-binding analog, Phys. Rev. B 87, 035101 (2013).
  * Thouless [1977] D. J. Thouless, Maximum metallic resistance in thin wires, Phys. Rev. Lett. 39, 1167 (1977).
  * Wang and Genack [2011] J. Wang and A. Z. Genack, Transport through modes in random media, Nature 471, 345 (2011).
  * Mondal  _et al._ [2019] S. Mondal, R. Kumar, M. Kamp, and S. Mujumdar, Optical Thouless conductance and level-spacing statistics in two-dimensional Anderson localizing systems, Phys. Rev. B 100, 060201(R) (2019).

Supplemental Material

##  I Definition of Vogel spirals

Vogel spirals are defined by their polar coordinates $(r,\theta)$ as

$\begin{split}r_{n}&=a_{0}\sqrt{n},\\\ \theta_{n}&=n\alpha,\end{split}$ (S1)

where $n=1,2,\dots$ is an integer, $a_{0}$ is a positive constant, and
$\alpha$ is an irrational number. The scaling factor $a_{0}$ sets the particle
separation while $\alpha$ is the divergence angle and determines the constant
aperture between successive point. The angle $\alpha$ is specified as a
function of the irrational number $\xi$ as
$\alpha=2\pi\left[1-\mathrm{frac}(\xi)\right]$ where $\mathrm{frac}(\xi)$ is
the fractional part of $\xi$. When $\alpha$ is irrational, point patterns are
characterized by a lack of both translational and rotational symmetries.

In this work, we focus in the Golden-Angle (GA) Vogel spiral, also known as
“sunflower spiral”, which is obtained by considering $\xi$ as the golden
number $\xi=(1+\sqrt{5})/2$ leading to $\alpha\approx 2.4$ $(137.508^{\circ})$
also called the “golden angle”. Experimentally, 12 samples consisting of
$N=390-34\times i$ ($i=0,1,\cdots,11$) cylinders are placed and measured for
different heights of the cavity ($h=13$ and 17 mm are presented in this work).
All samples are characterized by the same scaling factor $a_{0}=6.93$ mm and
their centers are always located at $(x,y)=(-5~{}\mathrm{mm},0)$ where the
central antenna sets the origin $(x,y)=(0,0)$. Figure S1 shows the 12
different GS experimentally studied in this work.

Figure S1:  Golden-Angle spiral arrays consisting of $N$ cylinders created
with $a_{0}=6.93$ mm and $\xi=(1+\sqrt{5})/2$. The black dot sets the position
of the fixed antenna with respect to the cylinder pattern.

##  II Experimental map

Experimental measurements are carried out over a surface determined by a
circle of 160 mm radius centred in the origin $(x,y)=(0,0)$. In order to
observe the electric field out of the lattice, we superimpose a 165 mm size
square with corners at $(x,y)=(0,0)$, $(x,y)=(165~{}\mathrm{mm},0)$,
$(x,y)=(0,165~{}\mathrm{mm})$ and
$(x,y)=(165~{}\mathrm{mm},165~{}\mathrm{mm})$. The resulting area is mapped in
a regular $5\times 5$ mm2 grid unit cell. Figure S2 shows the experimental map
used to scan the cavity where each grey point represents a measured point and
the orange circle the space occupied for the array of cylinders ($N=390$). The
total of measured positions is $n=3675$.

Figure S2:  Experimental map used to scan the cavity (grey dots). Orange
circle indicates the position of the array of cylinders (GA spiral with
$N=390$).

##  III Disordered system

For the sake of comparison, besides the main analysis carried out in the GA
spiral, we also study the modal structure and microwave transport in a
traditional disordered systems (DS). The disordered point pattern is generated
using the software developed in [14] and considers a set of $N=388$ packing
hard disks of radius $R=3.25$ mm enclosed into a circle of radius 140 mm, thus
the planar density is constant respect to the GA spiral case ($\rho\approx
0.65$ cm-2, see Fig. S3).

Figure S3:  Image of the 2D disordered array of dielectric cylinders. The top
place has been removed to reveal the details of the sample.

Here, $R=3.25$ mm is the size of our experimental cylinders plus the size of
the tube used to place the cylinders. The cavity is mapped using the same
experimental map (see Fig. S2) in the same frequency range (from 5.5 GHz to 15
GHz) than the GA spiral case. Both reflected and transmitted signals are
measured. Data analysis to obtain the eigenmodes of the systems (via the
harmonic inversion/clustering method), the characteristic decay time $t_{0}$
(via the Fourier transform), the Density of States (via the reflected signal)
and the Thouless conductance $\expectationvalue{g}$ (via $t_{0}$ and the
$\expectationvalue{\mathrm{DOS}}$) is performed as explained in the main text.

Figures S4[(a), (d), (g) and (j)] show four eigenmodes found by means of the
harmonic inversion/clustering algorithms in the DS.

Figure S4:  Spatial modal structure, field amplitude spatial autocorrelation
and radial profile of representative eigenmodes in a DS. Amplitude maps are
normalized such that $\max(\absolutevalue{E_{\nu}})=1$, thus
$\max[\mathrm{Cor}(E_{\nu})]=1$. Radial decays (pink dots) are obtained by
performing an angular average in the autocorrelation space ($r$ is measured
from the well defined autocorrelation center).

Since disordered systems lack of a center (in contrast to Vogel spirals),
their eigenmodes are not centered in the system. This makes the field
amplitude spatial autocorrelation [see Fig. S4(b), (e), (h) and (k)], defined
as

$\mathrm{Cor}(E_{\nu})=\left|\mathcal{F}^{-1}\left\\{\left|\mathcal{F}\\{E_{\nu}\\}\right|^{2}\right\\}\right|,$
(S2)

the most suitable quantity to study the radial decay [see Fig. S4(c), (f), (i)
and (l)] [5]. As it can be observed, all eigenmodes present large quality
factors $Q$ and a clear exponential radial decay of their autocorrelation
functions no matter their spatial extension, frequency, the position of their
center, or the amplitude distribution with respect to the center of the mode,
as expected. The localization lenght $\xi_{\mathrm{loc}}$ is extracted by
assuming $\mathrm{Cor}(E_{\nu})\propto\exp\left(-r/\xi_{\mathrm{loc}}\right)$.

The characteristic time $t_{0}$, the DOS and the average Thouless conductance
$\expectationvalue{g}$ are computed and plotted in Fig. S5[(a), (b) and (c),
respectively]. Similarly to the GA spiral case, $t_{0}$ varies from flat
valleys to high peaks. Nevertheless, in contrast to the GA spiral case, no
band gap can be clearly observed in the DOS. The maxima $t_{0}$-values found
in the GA spiral case are $t_{0,M}=40.84$ ns, $t_{0,M}=33.02$ ns and
$t_{0,M}=30.11$ ns for the first and second confined regions, respectively,
while in the disordered case $t_{0,M}=40.58$ ns. Hence we conclude that the
characteristic energy decay time in disordered and aperiodic Vogel spiral
structures is of the same order. Finally, the average Thouless conductance
shows a fast decay of around one order of magnitude in the frequency windows
where exponential eigenmodes are found by the harmonic inversion/clustering
method.

Figure S5:  (a) Characteristic decay time $t_{0}$ as a function of the
frequency. The total frequency range have been mapped by 472 frequency filters
spaced by $\Delta f=0.02$ GHz with $\Delta\omega=0.01$ GHz. (b) Experimental
density of states (DOS). (c) Experimental Thouless conductance
$\expectationvalue{g}$ as a function of the frequency. The dotted line
indicates $\expectationvalue{g}=1$.

##  IV Scaling analysis

To investigate how the transport quantities as well as the modes are affected
by the boundaries of the spiral, the experiment is repeated for 12 different
configurations, and for each one of them, the number $N$ of cylinders in the
array is reduced according to $N=390-34\times i$ ($i=0,1,\cdots,11$). In Fig.
S6, we show [(a) and (b)] the characteristic decay time $t_{0}$, [(c) and (d)]
the normalized density of states, and [(e) and (f)] the average Thouless
conductance $\expectationvalue{g}$ as a function of the number of cylinders
and frequency in two frequency windows containing the analyzed long-lived
modes (from 6 to 7 GHz and from 10.5 to 11.5 GHz). The spatial structure of
three different eigenmodes with characteristic (g) power-law, (h) exponential,
and (i) Gaussian decays found by the harmonic inversion/clustering methods are
also shown in Fig. S6. Note that below a certain threshold $N$ [(g) $N\simeq
84$, (h) $N\simeq 186$, (i) $N\simeq 288$], the number of resonances recovered
by the harmonic inversion is found to be insufficient to form a cluster and
subsequently to reveal the spatial structure of the modes.

Figure S6:  (a)-(b) Characteristic decay time $t_{0}$ as a function of the
frequency and of the number of cylinders. The total frequency range has been
mapped by 47 frequency filters spaced by $\Delta f=0.02$ GHz with
$\Delta\omega=0.01$ GHz. (c)-(d) Experimental normalized density of states
(DOS). (e)-(f) Experimental Thouless conductance $\expectationvalue{g}$.
Spatial modal structure as a function of the number of cylinders of modes with
(g) power-law ($\nu=6.586$ GHz), (h) exponential ($\nu=6.646$ GHz), and (i)
Gaussian ($\nu=11.357$ GHz) radial decay. The grey circles show the boundary
of the samples.

Around $\sim 6.6$ GHz, Fig. S6(a) and (e) show that the localization
signatures earlier analyzed (high $t_{0}$ values and $\expectationvalue{g}<1$)
remain unperturbed even for lattices with a reduced number of cylinders
$N\simeq 84$, while Fig. S6(c) shows that the density of states start to lose
its structure and becomes flat below $N\simeq 152$. Here the leaking of the
wave out of the spiral is driven by the absence of certain cylinders needed to
support the long-lived modes. This fact is illustrated by the power-law and
exponential modes [see Fig. S6(g) and (h), respectively] which cannot be
recovered by our analysis whenever the typical system size becomes smaller
than the noticeable modal size when $N\simeq 84$ and $N\simeq 186$,
respectively. At higher frequency, Gaussian modes are found to have larger
sizes, and are therefore more sensitive to changes in the system boundaries so
that they can only exist for larger systems $N\geq 288$. This value of $N$
also determines a critical value above which Gaussian modes with high $t_{0}$
($\expectationvalue{g}<1$) values disappear [Fig. S6(b)]. Nevertheless, the
DOS remains unperturbed even for larger systems ($N\simeq 186$).