# The emergence of low-frequency dual Fano resonances in chiral twisting
metamaterials

Brahim Lemkalli<EMAIL_ADDRESS>Laboratory for the Study of
Advanced Materials and Applications, Department of Physics, Moulay Ismail
University, B.P. 11201, Zitoune, Meknes, Morocco  Muamer Kadic  Institut
FEMTO-ST, UMR 6174, CNRS, Université de Bourgogne Franche-Comté, 25000
Besançon, France  Youssef El Badri  Laboratory of optics, information
processing, Mechanics, Energetics and Electronics, Department of Physics,
Moulay Ismail University, B.P. 11201, Zitoune, Meknes, Morocco  Sébastien
Guenneau  UMI 2004 Abraham de Moivre-CNRS, Imperial College London, SW7 2AZ,
UK  Abdellah Mir  Laboratory for the Study of Advanced Materials and
Applications, Department of Physics, Moulay Ismail University, B.P. 11201,
Zitoune, Meknes, Morocco  Younes Achaoui  Laboratory for the Study of Advanced
Materials and Applications, Department of Physics, Moulay Ismail University,
B.P. 11201, Zitoune, Meknes, Morocco

###### Abstract

In the current work, through a finite element analysis, we demonstrate that a
configuration of chiral cells having syndiotactic symmetry provides dual Fano
resonances at low frequency. From the phononic dispersion and transmission
response, we compare the signature provided by a composite made of chiral
cells to the ones of homogeneous medium, isotactic nonchiral, and isotactic
chiral beams. The study results in an innovative design of a mechanical
metamaterial that induces the Fano resonance at low frequency with a
relatively high quality factor. This might be a significant step forward for
mechanical wave filtering and detection. Performances have been evaluated
using a sensor that will be implemented as a thermometer.

Twisting metamaterials, Fano resonance, Temperature sensor

††preprint: AIP/123-QED

##  I Introduction

In recent years, the emergence of composite-structured materials has heralded
significant advancements in mechanical engineering [1]. As a result, new
generations of man-made materials, known as ”metamaterials,” are created,
allowing mechanical behaviors to be adapted with new characteristics beyond
the intrinsically well known [2]. From an elastodynamic viewpoint, these allow
the manipulation and control of acoustic wave propagation by both mechanisms,
namely local resonance and Bragg scattering [3, 4]. Besides, mechanical
metamaterials are well known by their various exotic parameters in the static
regime, including negative Poisson’s ratio [5], flexibility [6], and twist
conversion [7, 8], which leads to a ”dynamic paradigm,” used today in a wide
range of applications. For instance, auxetic metamaterials were proposed in
order to enhance seismic shielding against surface waves [9]. Besides,
metamaterials with a twist can exhibit a distinct feature called acoustic
activity. This converts the linear polarization of a transverse wave to
circular polarization [10]. Recently, twisting metamaterials demonstrated the
conversion of longitudinal waves into twist waves [11, 12].

In general, the local resonance is caused by the coupling between a discrete
resonance and a continuous state, which causes the appearance of a peak at the
resonance frequency followed or preceded by the dip of the anti-resonance.
This mechanism is a consequence of constructive and destructive interferences,
respectively, previously reported in the field of optics [13]. Since its
discovery more than 60 years ago [13], the prominent Fano resonance has piqued
the interest of scientists due to its asymmetric nature, which is used in some
relevant applications [14] such as filtering [15] and detection [16].

As a mechanical counterpart, this sort of resonance has gained prominence
[17]. Several devices based on mechanical Fano resonance have been developed
in recent years [18], including concentrated pipes [19], Helmholtz resonators
[20], and phononic crystals [21, 22, 23]. However, the dimensions of these
structures, notably phononic crystals, are equivalent to or even larger than
the wavelengths; also, the Fano resonance effect occurs in just one
operational frequency range. Multi-band systems with sub-wavelength dimensions
and a high quality factor at low frequencies remain a major challenge for the
development of multi-band and multi-functional devices [21]. Dual Fano
resonators for low frequencies have recently been developed, employing an
array of units made up of two types of cell units containing multiple
cavities, each with its own specific set of characteristics [24]. These are
based on the emergence of acoustic metamaterials[25] with dimensions smaller
than the wavelength, leading to exceptional elastic wave manipulation
abilities.

Figure 1: Schematics of the beams. (a) The homogenous medium cell. (b) The
nonchiral isotactic cell ($\alpha=0$). (c) The chiral isotactic cell
$\alpha=arctn(\frac{h}{c})$. (d) The chiral syndiotactic cell
$\alpha=arctn(\frac{h}{c})$. (e) The geometrical parameters of the cells, the
two octagonal plates are separated by a distance of
$h=30$$\mathrm{m}\mathrm{m}$ by rods with diameter of
$d=1.2$$\mathrm{m}\mathrm{m}$ inclined by an angle $\alpha$ equal to
$arctn(\frac{h}{c})$ with the side of the octagon equal to
$c=3.9$$\mathrm{m}\mathrm{m}$ and the radii $R_{1}=5.08$$\mathrm{m}\mathrm{m}$
and $R_{2}=3.3$$\mathrm{m}\mathrm{m}$ and $b=1.4$$\mathrm{m}\mathrm{m}$. The
beams have a width of $a=14.4$$\mathrm{m}\mathrm{m}$ and length of $4h$.

In this study, we leverage the design of a metamaterial with a twist to
generate double Fano resonance at low frequency, inspired by the chiral
tacticity in metamaterials [26]. In Section II, we demonstrate numerically
that a chiral syndiotactic cell generates local resonance. By connecting two
cells in such a way that the contact plane between the two cells forms a
mirror, the Fano resonance fingerprint is the direct consequence of the
coupling between a longitudinal continuum and the discrete state of the chiral
unit-cell. In Section III, we propose an application of the dual Fano
resonances to detect temperature changes in water.

##  II Elastodynamic Characteristics

In this section, we analyzed the elastodynamic behavior of four structures in
order to demonstrate the presence of the Fano resonance. These structures have
rectangular beams with a length of $4h$ and a width of $a$ made of two media;
one homogeneous in steel at the beam borders with a length of $h$ and the
other inside, which is a cell in Acrylonitrile Butadiene Styrene (ABS) with a
length of $2h$ alternating the four unit cells. The first cell is purely a
homogeneous medium (Figure 1(a)). The second is a nonchiral isotactic unit
cell throughout two cells composed of non-inclined rods ($\alpha=0$) connected
to octagonal plates (Figure 1(b)). The third is a chiral isotactic unit cell
in which two cells composed of rods inclined by ($\alpha=atan(h/b)$) are
connected to octagonal plates (Figure 1(c)). The fourth cell is a chiral
syndiotactic cell composed of two chiral cells with inclined rods
($\alpha=atan(h/b)$) attached to octagonal plates (Figure 1(d)). These two
cells are connected by a plane of symmetry mirror.

To determine the elastodynamic behaviors of the four beams, we used the
commercial software COMSOL Multiphysics to solve the Navier equations in the
weak form. We considered all the materials used in the simulations as
isotropic linear elastic materials. These are depicted in Table 1.

Table 1:  The materials parameters Materials | Young’s modulus | Poisson’s ratio | Density
---|---|---|---
| ($\mathrm{G}\mathrm{P}\mathrm{a}$) |  | ($\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{m}^{3}$)
Steel | 201 | 0.33 | 7843
ABS | 2.6 | 0.4 | 1020
Figure 2: The phononic dispersion curves along the $x$-direction in the first
Brillouin zone ($\Gamma X$) for the four beams. (a) The homogeneous medium
cell. (b) The nonchiral isotactic cell. (c) The chiral isotactic cell. (d) The
chiral syndiotactic cell. (e) Screenshots of the syndiotactic chiral cell
beam’s eigenmodes at points $A$, $B$, $C$, and $D$.

The first step was to calculate the phononic dispersion curves towards the
$x$-direction and analyze the eigenmodes of the four beams. To elucidate the
mechanisms that govern the interaction of localised modes (flat modes) with
longitudinal modes, which gives rise to the local resonance. We calculated
mode polarization using equation 1, which is represented by the color bar in
the dispersion curves, as depicted in Figure 2.

$p_{yz}=\frac{\iiint\sqrt{|u_{y}|^{2}+|u_{z}|^{2}}dV_{1}}{\iiint\sqrt{|u_{x}|^{2}+|u_{y}|^{2}+|u_{z}|^{2})}dV_{tot}},$
(1)

where $V_{1}$ is the volume of the inner cell and $V_{tot}$ is the total
volume of the beam.

The beam with the cell of homogeneous medium (ABS) (Figure 2(a)) exhibits four
fundamental modes: bending and transverse, which are degenerated in the
present case because of the symmetry in the $yz$-plane, plus the other two
modes: twisting and longitudinal. As illustrated, the first three modes have
their polarization active in the $yz$-plane, with the exception of the
longitudinal mode, which remains fully polarized all along the $x$-direction.
However, when we substitute the homogeneous medium (ABS) with the nonchiral
isotactic cell (Figure 2(b)) in the beam with a length of $2h$, the first two
modes remain degenerate but have shifted down towards low frequencies. The
polarization indicates that the flat modes do not interfere with the
longitudinal mode.

Analogously, the longitudinal mode remains polarized along the $x$-direction
in the isotactic chiral cell (Figure 2(c)), regardless of the fact that the
first two modes have undergone degeneracy lifting (the transverse modes travel
with different velocities in the first Brillouin zone ($\Gamma X$)) as a
consequence of the symmetry in the $yz$-plane. In other words, the effect of
the isotactic chiral cell has no influence on the polarization of the
longitudinal mode (the coupling between the flat modes does not take place
with the longitudinal mode).

However, due to the presence of a distinct symmetry plane inside the
syndiotactic cell, the first two modes do not undergo degeneracy lifting
(Figure 2(d)). On the other hand, there is interference between a localized
mode, polarized in the $yz$-plane, and the longitudinal mode, which causes the
local resonance composed of two resonances, symmetric and anti-symmetric,
which is produced as a result of the coupling of the flat mode with the
longitudinal mode, resulting in the presence of the Fano resonance near $1$
$\mathrm{k}\mathrm{H}\mathrm{z}$, as indicated by the red color of the mode
polarization around this frequency.

To illustrate that the interference between the localized twist mode and the
longitudinal is entirely responsible for the appearance of the local
resonances, screenshots of the modes at the local resonance are displayed in
Figure 2(e). $A$ has the coordinates of ($k$, $\omega$)=($1$, $862.8)$ and
$B$($0$, $1138$). According to the dispersion curve, these two points
represent the initial local resonance. Both images show a localized
displacement in the center of the syndiotactic chiral cell. These images
suggest that at these frequencies, the effect of longitudinal-twist conversion
between the two cells is active, which results in local resonances. Around $3$
$\mathrm{k}\mathrm{H}\mathrm{z}$ of Figure 2(d), we can discern two polarized
modes in the $yz$-plane, indicating the resonance pattern that forms the Fano
resonance. In those, we consider two points $C$ and $D$, as seen in the
screenshots in Figure 2(e). This last Fano resonance is produced by a high-
order twist; the total displacement is localized in three regions of the
syndiotactic chiral cell: in the center and in the middle of the rods, as seen
in the Figure 2(e).

After demonstrating the existence of the Fano resonance in the syndiotactic
chiral cell using eigenvalue analysis, we have investigated the transmission
analysis of a longitudinal wave through these four beams. We added free media
in steel and Perfectly Matched Layers at each extremity. We used equation $2$
to calculate the longitudinal wave transmission along the $x$-direction for
the four beams.

$T=20\times
log_{10}\frac{\iiint|u_{x}|^{2}dV_{output}}{\iiint|u_{x}|^{2}dV_{input}},$ (2)

Figure 3: (a) Transmission spectrum of the homogeneous medium cell in black
color. The nonchiral isotactic cell in blue color. The chiral isotactic cell
in blue color. The chiral syndiotactic cell in red color. (b) Screenshots of
the syndiotactic chiral beam in the anti-resonance and resonance peaks, at
point A of frequency of $970$ $\mathrm{H}\mathrm{z}$, at point B of frequency
of $1192$ $\mathrm{H}\mathrm{z}$, at point C of frequency of $3082$
$\mathrm{H}\mathrm{z}$, and at point D of frequency of $3127$
$\mathrm{H}\mathrm{z}$.

Figure 3(a) depicts the transmission curves of the beams with a homogeneous
medium cell, both isotactic nonchiral and chiral cells, and a syndiotactic
chiral cell, as shown by the black, blue, and red curves, respectively. The
first three beams have no Fano resonances, whereas the fourth one contains two
Fano resonances. This indicates that the syndiotactic chiral structure
exhibits a dual Fano resonances at low frequencies. We evaluated the quality
factor, which corresponds to the ratio of the resonance frequency to the full
width frequency at half-maximum of each peak. The first peak at the resonance
frequency of $1.191$ $\mathrm{k}\mathrm{H}\mathrm{z}$ has a Q-factor of $350$,
while the second occurs at the resonance frequency of $3.127$
$\mathrm{k}\mathrm{H}\mathrm{z}$ has a Q-factor of $11,010$. Figure 3(b) shows
screenshots at resonance and anti-resonance frequencies, which are the peak
and the dip for the two Fano resonances that exist exclusively in the
syndiotactic chiral cell. The first resonance represents the first-order
twist, as indicated by the dispersion curves, while the second resonance is
the high-order twist.

##  III Liquid sensing application

Phononic crystals and metamaterials have gained significant attention in the
realm of sensing, particularly as an innovative resonant platform for
analyzing liquid properties. The general idea is based on the incorporation of
liquid as a constituent in the phononic crystal or within a cavity localized
in the perfect structure [27, 28, 29]. Their sensing functionality, in
particular, is realized through the solid-liquid interaction [30, 31]. Based
on this approach, we propose a sensor based on the chiral syndiotactic cell,
as shown in Figure 4(a), which is a beam composed of two homogeneous media in
steel and a syndiotactic chiral cell in ABS submerged in water with a
dimension two order of magnitude smaller than the geometrical parameters
outlined in Section II. This makes the sensor operates at frequencies around
$100$ $\mathrm{k}\mathrm{H}\mathrm{z}$, which is low frequency in comparison
to the sensors described in the literature.

Figure 4: Chiral syndiotactic beam as liquid sensor. (a) The sensor’s design.
(b) Longitudinal transmission for both peaks as a function of $-1\%$ density
and speed of sound variation.

In order to evaluate the potentiel of the presented sensor to detect changes
in liquid properties, we defined the sensitivity using equation 3, which
quantifies the frequency shift of each resonance peak in response to a slight
change in the liquid properties. Additionally, we use equation 4 to define the
figure of merit (FoM), which evaluates if two nearly identical media can be
distinguished.

$S_{i}=\frac{\Delta f_{i}}{\Delta T},$ (3) $FoM_{i}=\frac{S_{i}\times
Q_{i}}{f_{i}},$ (4)

where $i$ symbolizes peak $1$ or peak $2$, $T$ is the temperature, $Q_{i}$ is
the quality factor of each resonance peak, and $f_{i}$ represents the
resonance frequency for each peak.

In the first step, we assessed the syndiotactic beam sensor’s capabilities
employing investigations given in recent works using phononic crystals, in
which they varied the water density and sound velocity by $1\%$ and determined
the characteristic parameters of each variation [32, 33]. The longitudinal
transmission responses are depicted in Figure 4(b). As shown in Figure 4(b),
the syndiotactic sensor is not highly sensitive to the variations of the sound
speed, but is sensitive to variations in density. We summarize the calculated
parameters of density variation for the two peaks in Table 2.

Table 2:  Frequency, $Q$-factor, sensitivity, and figure of merit of the two peaks for $-1\%$ of density water variation. | Peak 1 | Peak 2
---|---|---
Frequency ($\mathrm{k}\mathrm{H}\mathrm{z}$) | 97.9 | 251.9
Q | 840 | $17\times 10^{3}$
$S_{-1\%\rho}$ ($\mathrm{H}\mathrm{z}\mathrm{/}{kgm^{-3}}$) | 16 | 100
$FoM_{-1\%\rho}$ ($\mathrm{1}\mathrm{/}{kgm^{-3}}$) | 0.14 | 6.74

The first peak at $97.9$ $\mathrm{k}\mathrm{H}\mathrm{z}$ has a $Q$-factor of
$840$ and a sensitivity to density variation of $1\%$ equal to $16$
$\mathrm{H}\mathrm{z}\mathrm{/}{kgm^{-3}}$ and a figure of merit of $0.14$
$\mathrm{1}\mathrm{/}{kgm^{-3}}$. Regarding the second peak around $251$
$\mathrm{k}\mathrm{H}\mathrm{z}$, a $Q$-factor of $17,000$, a sensitivity of
$100$ $\mathrm{H}\mathrm{z}\mathrm{/}{kgm^{-3}}$, and a figure of merit of
$6.74$ $\mathrm{1}\mathrm{/}{kgm^{-3}}$. The parameters obtained for the $1\%$
density variation are comparable to those published in the literature [32,
33]. These parameters show that at low frequencies, the chiral syndiotactic
cell has an interesting feature in the detection of density variation unlike
the sound speed.

In the second step, we used the syndiotactic chiral beam to detect changes in
water temperature, in which the density and the speed of sound depend on the
temperature of water, as depicted in Table 3. We computed the longitudinal
transmission in function of water temperature variation, as illustrated in
Figure 5. The frequencies, quality factors, sensitivities, and FoMs of the two
peaks are all described in Table 4. The sensitivity of each peak was
determined by estimating two successive values of the frequency shift produced
by the temperature change.

Table 3:  The density and speed of sound as a function of water temperature variation. Temperature | Density | Speed of sound
---|---|---
($\mathrm{\SIUnitSymbolDegree}\mathrm{C}$) | ($\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{m}^{3}$) | ($\mathrm{m}\mathrm{/}\mathrm{s}$)
0 | 999 | 1403
10 | 999 | 1447
20 | 998 | 1481
30 | 995 | 1507
40 | 992 | 1526
50 | 988 | 1541
60 | 983 | 1541
Figure 5: The evolution of longitudinal transmission as a function of water temperature variation. (a) The first peak. (b) The second Peak. Table 4:  Frequency, $Q$-factor, sensitivity, and figure of merit of the two peaks for temperature variation in water. | Peak 1 | Peak 2
---|---|---
Frequency ($\mathrm{k}\mathrm{H}\mathrm{z}$) | 98.8 | 253.5
$Q$-factor | 840 | $17\times 10^{3}$
Sensitivity ($\mathrm{H}\mathrm{z}\mathrm{/}{\mathrm{\SIUnitSymbolDegree}C}$) | 3 | 20
FoM ($\mathrm{1}\mathrm{/}{\mathrm{\SIUnitSymbolDegree}C}$) | 0.02 | 1.34

As the water temperature increases, the frequency of the two Fano resonance
peak shifts. Thus, the characteristic frequencies of the two Fano resonance
peaks are sensitive to both the speed of sound and density of water as a
function of temperature. In other words, the chiral syndiotactic beam can
exhibit two peaks that can change the resonance frequency based on the
liquid’s material properties, as illustrated in Figure 5. The first peak has
lower sensitivity than the second peak; however, both peaks have acceptable
characteristics in terms of increasing dimension and using a low frequency
when compared to the phononic sensor, which uses a frequency of hundreds of
$\mathrm{G}\mathrm{H}\mathrm{z}$.

The findings suggest that the syndiotactic chiral beam could be used as a
temperature sensor. Due to the geometric size, two peaks occur at low
frequencies, which indicates that their characteristics are indeed very small.
It should be noted that in order to achieve the highest sensitivity value, it
must use a structure with geometrical parameters scaled by 0.1 compared to the
presented configuration. This means that both frequency response and
sensitivity will be multiplied by a factor of $10$.

##  IV Conclusion

To conclude, we used the finite element method to calculate phononic
dispersion and transmission. This was done in order to demonstrate the
presence of dual Fano resonances in chiral metamaterials with twist.
Furthermore, we proved that the beam with the chiral syndiotactic cell based
on the two octagonal plates exhibited dual Fano resonances at low frequencies,
one at $1$ $\mathrm{k}\mathrm{H}\mathrm{z}$ and the other at $3$
$\mathrm{k}\mathrm{H}\mathrm{z}$. This characteristic has been compared to
other beams with a homogeneous medium, an isotactic nonchiral cell, and an
isotactic chiral cell. The interference of localized twisting and longitudinal
modes, in particular, causes low-frequency local resonances, also known as
Fano resonances. Following that, the presence of dual Fano resonances in
syndiotactic beam metamaterials is used to detect liquid properties such as
water density and sound speed. Finally, this study demonstrated that
syndiotactic beam metamaterials with twist can be used as temperature sensors,
exhibiting considerable sensitivity and quality factors for the proposed size
and for low frequencies.

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# Weak localisation enhanced ultrathin scattering media

R. C. R. Pompe contributed equally to this work Department of Physics,
Bielefeld University, 33615 Bielefeld, Germany  D. T. Meiers∗ Physics
Department and Research Center OPTIMAS, Technische Universität Kaiserslautern,
67663 Kaiserslautern, Germany  W. Pfeiffer  Department of Physics, Bielefeld
University, 33615 Bielefeld, Germany  G. von Freymann  Physics Department and
Research Center OPTIMAS, Technische Universität Kaiserslautern, 67663
Kaiserslautern, Germany  Fraunhofer Institute for Industrial Mathematics ITWM,
67663 Kaiserslautern, Germany

The brilliant white appearance of ultrathin scattering media with low
refractive index contrast and the underlying radiative transport phenomena
fascinate scientists for more than a decade. Examples of such systems are the
scales of beetles of the genus Cyphochilus[1, 2], photonic network structures
[3] or disordered Bragg stacks (DBS) [4, 5]. While previous studies relate the
highly efficient scattering in the scales to the anisotropy of the intra-scale
network and diffusive light transport [6, 7, 11, 12, 10, 8, 9], the coherent
radiation propagation dynamics remained unaccounted for. Here, we identify
different coherent light transport regimes using time and spatially resolved
coherent light scattering spectroscopy. At least 20% of the collected
scattered light originates from weakly localised random photonic modes, in
contrast to solely diffusive light transport assumed to date [6, 7, 8, 9]. The
identification of this significant role of weak localisation in ultrathin
brilliant scattering media establishes a new design paradigm for efficient
scattering optical materials.

Figure 1: Microscopic and ultrafast time-resolved spectroscopy of light
scattered from Cyphochilus scales and microfabricated DBS structures. a,
Scheme of the spectral interference setup (see explanations in the text and in
Methods). b, Photograph of Cyphochilus (left) and disordered Bragg stacks
(DBS, centre closeup) with light microscope images of a single beetle scale
(right top) and DBS (bottom) as insets. c,d, Spatially resolved time domain
amplitude of light scattered from a Cyphochilus scale (c) and DBS (d). The
transition threshold between diffusive regime and resonance radiation as
identified in h are indicated (vertical translucent bar). e,f, Scheme
illustrating how incoming light is scattered in the initial diffusion-like
regime (e) and later via weakly localised photonic modes indicated by closed
pathways (f). The grey structure is a cross section of a Cyphochilus scale
(taken from Wilts et al. [11]). The black overlay on the left side shows the
disordered Bragg stacks. g, Scattered electric field at a single scan position
(white dashed line in c) with indication of the short time Fourier transform
windows used in i (red) and j (blue). h, Wigner distribution function of the
scattered field shown in g. At $105\pm 10$ fs (black line) the dominating
light transport regime changes from diffusion-like to weak localisation
assisted. i,j,k, Fourier spectra of the early time window (i) (-50 to 50 fs,
red in g and h), the later time window (j) (250 to 350 fs window, blue in g
and h) and the total measured time window (k).

In strongly scattering media the description of light propagation as ballistic
transport breaks down and is commonly replaced by diffusive radiation
transport that explains well the observed optical characteristics in numerous
applications [13, 14]. Diffusive radiation transport neglects the coherent
propagation of scattered fields and hence does not account for interference
phenomena in disordered media, which are known to occur for example when weak
localisation gives rise to coherent back scattering [15] or random lasing in
disordered active media [16]. For increased scattering strength coherent back
scattering occurs, when two counter-propagating scattering light paths in the
medium, i.e. the illuminating light and collinear back scattered light,
interfere constructively and giving rise to a peak in the back scattered
intensity, as it was, e.g., reported for Cyphochilus scales [10]. However,
modelling of the brilliant white appearance of Cyphochilus scales still
completely relies on diffusive propagation [6, 7, 8, 9] and thus coherent
effects are neglected. This could hamper tailoring disordered photonic media
since an unambiguously identified scattering mechanism is the basis for
nanostructure design for optimised performance. Using ultrafast time-resolved
light scattering spectromicroscopy [17, 18] we here identify the coherent
light scattering mechanisms for Cyphochilus scales and disordered Bragg stacks
and show that weak localisation in leaky photonic modes significantly
contributes to the brilliant whiteness of these scatterers.

The identification of coherent scattering is significantly facilitated if the
number of interfering pathways is kept small. For example, laser speckles are
most pronounced when only a small area of the scatterer is illuminated.
However, if the detector integrates over sufficiently many different
interfering pathways, the speckles disappear. In this case with exception of
the coherent back scattering peak, the scattering behaviour is often well
explained by diffusive radiation transport theory, although the underlying
transport is coherent. To reduce the number of interfering pathways the
present investigation relies, both, on focused illumination and collection of
scattered light from a small sample volume. Furthermore, coherent propagation
adds a well-defined phase to the scattered fields and thus reconstruction of
the temporal evolution of the scattered electric field provides additional
information on the scattering mechanism.

To systematically study the impact of coherent transport on the whiteness of
the Cyphochilus’ scales, we use the setup shown in Fig. 1a to perform
ultrafast time-resolved light scattering spectromicroscopy on a single
scale[17, 18]. The observations are confirmed for DBS fabricated via direct
laser writing (see Supplementary Information) shown in Fig. 1b. The DBS mimic
the beetle scales, reproduce their known optical properties [4] and allow for
realistic scattering light simulations based on finite-difference time-domain
(FDTD) Maxwell solvers and Monte Carlo (MC) diffusive light transport
simulations.

To achieve the spatial resolution necessary to observe only few interfering
scattering pathways, a parabolic mirror (Fig. 1a, M) focuses a pulsed
Ti:sapphire laser beam down to a $\lesssim 3$ µm spot on the surface of the
sample (Sa) and collects the scattered light under an angle of $\sim 24$°
relative to the specular direction. To filter for intra-scale scattering, i.e.
multiple scattered light components, a cross-polarisation configuration is
used. The illuminated position is scanned by moving the sample using a piezo
stage. Spectral interference [19] between the scattered light pulse (SP) and a
reference pulse (RP) allows for the time reconstruction of the field of the
scattered light (see Methods). The amplitude of the measured electric field
(cf. Fig. 1c and d) shows for both samples essentially the same dynamics, i.e.
spatially varying exponential decay modulated by distinct beating, indicating
interference taking place. As discussed below two different propagation
regimes can be identified in the scattered light signals. Initially diffusion-
like transport (Fig. 1e) dominates, whereas for longer times radiation leaking
from weakly localised photonic modes formed by randomly closed scattering
pathways (Fig. 1f) prevails, which gives rise to the observed beating
behaviour.

To identify the different propagation regimes we analyse the coherent
scattering signal (cf. Fig. 1g) in time and frequency domain by means of the
Wigner distribution function (WDF, see Methods) [20], exemplarily shown in
Fig. 1h for the Cyphochilus scale. For early times broadband features are
present, which reproduce the excitation spectrum when evaluating the short
time Fourier transform (cf. Fig. 1i). At about $105\pm 10$ fs there is a
qualitative change in the spectral content of the WDF, i.e. broad spectral
features are replaced by fine modulations. This time matches closely to the
pulse round trip time (see Methods), i.e. the time a pulse needs to travel
back and forth through the layer assuming a homogeneous, effective medium with
an effective refractive index, as it is commonly done in diffusion
approximation. The spectral modulations stem from multiple sharp resonances,
which become better visible in the short time Fourier transform for later
times (cf. Fig. 1j). The power spectrum illustrates that the signal now
contains spectral peaks independent of the original excitation spectrum,
whereas the scattered light in the initial diffusion-like phase exhibits no
significant modulation. The spectrum for the full measured signal, shown in
Fig. 1k, exhibits spectral peaks on top of a broadband background and thus
reflects the spectral characteristics of both transport regimes.

While the short time Fourier transforms (Fig. 1i,j) allow identifying the
contribution of the different light transport mechanisms over time, this
spectral analysis of resonances lacks of resolution due to the short time
windows. To unambiguously identify the weak localisation assisted scattering
the probability distribution of the resonance lifetimes is investigated
applying full time Fourier transformations. Fig. 2a reveals that the scattered
light spectra possess multiple peaks with varying centre frequency and width
as function of the spatial coordinate. In the incoherent mean of the spectra
over the whole scan (Fig. 2b, grey shaded area) these narrow spectral peaks
average out and reproduce the excitation spectrum (Fig. 2b, dashed line),
macroscopically resulting in the white appearance. Based on peak fitting (Fig.
2b, red curve) we derive the spectral widths of the peaks, which yield a lower
limit for the underlying resonance lifetimes. The distribution of these
lifetimes is displayed in Fig. 2c and follows a log-normal distribution (red
curve), deviating from a normal distribution for longer lifetimes as expected
when localisation effects occur [21]. The tail towards long lifetimes is
associated with the rare occurrence of increasingly localised modes, i.e.
cases where scattering pathways close inside the structure instead of coupling
to loss channels [22].

This identification of weak localisation assisted light scattering is further
supported by FDTD simulations based on the known microstructure of the
Cyphochilus scale [11] (model data provided by courtesy of B. Wilts) and the
DBS. As exemplified in Fig. LABEL:fig:Details_FDTDb and c the local spectra
recorded inside the structures also exhibit sharp resonances. Statistical
analysis of these resonances yields the lifetime distributions shown in Fig.
2d and e, which are in excellent accordance with the experimental results.
Hence, we conclude that the spectral resonances experimentally observed in the
scattered light indeed originate from weakly localised photonic modes
occurring in the same way inside the beetle structure and DBS. The
corresponding spectral features give rise to the observed beating behaviour in
scattered light spectromicroscopy (Fig. 1 c,d).

Figure 2:  Lifetime distribution for weakly localised photonic modes. a,
Spatially resolved light scattering spectra of Cyphochilus scale. b, Spectral
intensity (in blue) for the position indicated by the white line in a. The
excitation spectrum and the incoherent mean spectral intensity over the entire
scan is shown as dashed line and grey shaded area, respectively. Distinct
peaks are identified (exemplified by red curve) and used to estimate the
corresponding photonic mode lifetimes. c, Photonic mode lifetime distribution
derived from the scan displayed in a. d, e, Lifetime distributions obtained
from FDTD simulations of the intra-scale structure [11] (d) and the DBS model
(e). f, Transient average power in the monitor plane perpendicular to the
surface sectioning the DBS model (cf. Fig. LABEL:fig:Details_FDTDa) derived
from FDTD simulation (black curve) and average photon counts in the same plane
calculated by Monte Carlo simulation (grey curve). Both ordinates span the
same orders of magnitudes, making the slopes directly comparable. The non-
exponential decay of the FDTD results is indicated by coloured exponential
slopes with different lifetimes $\tau$. The vertical dashed line indicates the
point in time where both curves start to differ. Inset: The time averaged
local power enhancement in a snippet of the FDTD monitor plane averaged over
the time span indicated by the blue line (170-650 fs).

To further investigate the light propagation inside the structure the spatio-
temporal evolution of the local power (in FDTD simulations) and the photon
counts (in MC simulations) are recorded on a monitor plane sectioning the DBS
perpendicular to the surface (cf. Fig. LABEL:fig:Details_FDTDa). To avoid
artefacts from the lateral periodic boundary conditions (see Methods) a
sufficiently large lateral simulation domain of $20\times 20$ µm² is used.
This ensures that any potential spectral contribution from this periodicity
lies far outside the considered spectral range. In contrast to the rather
complex beetle intra-scale structure the DBS consist of simple building blocks
and thus is used for further simulations to keep the computation time
manageable.

The FDTD simulations (Fig. 2f, black curve) reveal a non-exponential decay
with lifetimes $\tau$ ranging from about 80 fs up to roughly 100 fs. This
directly reflects the lifetime distribution (Fig. 2e) possessing a mean value
around 80 fs, implying that for longer times the longer living photonic modes
dominate the decay. In contrast the MC simulations (Fig. 2f, grey curve) show
a mono-exponential decay with a decay constant of 65 fs (cf. Fig.
LABEL:fig:comparison_lsa), failing to match both the simulated and measured
lifetime distributions. Nevertheless, it is possible to find a set of
parameters such that the MC simulations reproduce for the same layer thickness
the properties of the DBS obtained by FDTD simulations, i.e. reflectance,
transport mean free path and initial shape of the curve. Hence, we conclude
that the initial coherent transport inside the structure can be approximated
as diffusive transport emphasising that there is a diffusion-like scattering
regime despite interference effects may occur. However, beyond about 170 fs
modelling as diffusive transport breaks down and the curve obtained by MC
simulation starts to deviate from the FDTD results. Assuming propagation in an
effective medium approach (as done for the experiment) yields a pulse round
trip time of 160 fs for the 100 fs long pulses applied in the simulations (see
Methods). This coincides well with the time at which FDTD and MC simulations
deviate indicating that the pulse round trip time is indeed a suitable
estimation for the upper limit of the time domain in which diffusion-like
photon transport dominates. For longer times the trapping in weakly localised
photonic modes takes over, which is only captured in the fully coherent FDTD
simulations.

The FDTD simulations provide means to directly visualise the weakly localised
photonic modes inside the DBS structure (inset in Fig. 2f). The time averaged
local power enhancement normalised to the average power (see Supplementary
Information) exhibits distinct, spatially localised hotspots with an up to
three times enhanced local power. These hotspots are associated with antinodes
of weakly localised random photonic modes (as depicted schematically in Fig.
1f) which give rise to the experimentally observed distinct peaks in the
spectra (cf. Fig. 1j). As expected incoherent diffusive photon propagation in
MC simulations do not exhibit any hotspots but an almost constant photon count
enhancement across the monitor plane (cf. Fig. LABEL:fig:comparison_lsc).

Summarising the observations and model simulations we conclude that the
scattering yield is dominated by photon leakage from weakly localised photonic
modes after an initial scattering time window, which can be roughly estimated
as the pulse round trip time in the ultrathin scattering layer treated in an
effective medium approach. Such modes have previously been identified for
systems that exhibit random lasing with coherent feedback [23, 16], but were
not yet identified to significantly contribute to the brilliant whiteness of
ultrathin scattering media. As shown in Fig. 3 scattering via weakly localised
photonic modes is responsible for at least about $20\%$ of the total
scattering and thus is relevant when the scattering efficiency of ultrathin
disordered photonic media are concerned. As indicated in the background
shadings of Fig. 3 the scales and the DBS would appear rather greyish and not
brilliant white, if scattering via leakage from weakly localised photonic
modes would be missing.

In conclusion, we have experimentally shown that the light transport in
scattering, brilliant white structures is dominated initially by a diffusion-
like transport which is surpassed by scattering via leakage from weakly
localised photonic modes after roughly the pulse round trip time in the
ultrathin scattering layer. Leakage from weakly localised modes accounts for
at least 20% of the scattered light, underlining their significance for the
brilliant whiteness of the ultrathin scattering media. This identification of
the coherent weak localisation assisted scattering mechanisms based on time-
resolved scattered light spectromicroscopy could serve, both conceptionally
and methodologically, to gain a better understanding of the transport regimes
in disordered materials and their time dynamics. This is e.g. relevant in
imaging through turbid media for bioimaging applications or random lasing
action in disordered gain media [24, 25, 26]. Furthermore, the here
demonstrated weak localisation feature of the biomimetic DBS relying on a
distorted Bragg reflector design provides a blueprint for tailoring
nanostructures to particularly support random photonic resonances which can
enhance light-matter interaction and therefore may find applications as
materials for efficient solar energy harvesting [17, 27, 28] or sensor
applications, where resonance enhanced absorption is employed to improve
sensitivity [29].

Figure 3: Spatially averaged time-dependent accumulated scattering yields. The
square modulus of the time-resolved scattering fields are averaged over the
recorded positions. This incoherent intensity signal is integrated over time
to yield the time-resolved accumulated scattering yield. The background
shading at $t_{\text{thr}}$ indicates the loss of whiteness if weak
localisation assisted scattering would be absent. a, Accumulated scattering
yield experimentally measured for the Cyphochilus scale. The white vertical
line corresponds to a threshold time of $t_{\text{thr}}$=105 fs, as indicated
in Fig. 1b,h, from which one weak localisation scattering dominates. The
scattering yield from weak localisation is $35\%$ (white horizontal line). b,
Accumulated scattering yield for the simulated DBS, with a threshold time of
$t_{\text{thr}}$=160 fs, as indicated in Fig. 2f. The scattering yield from
weak localisation is $21\%$. c, Accumulated scattering yield experimentally
measured for the fabricated DBS, with a threshold time of $t_{\text{thr}}$=190
fs (see Supplementary Information), as indicated in Fig. 1c. The scattering
yield from weak localisation is $20\%$.

## Methods

Experimental setup. The light source is a mode-locked Ti:sapphire laser
(Femtosource Scientific, Femtolasers Produktions GmbH, Austria) with a centre
wavelength of $\lambda_{0}=780$ nm and spectral full width half maximum (FWHM)
$\Delta\lambda=47$ nm, filtered in s-polarisation relative to the sample. To
achieve microscopic resolution the beam is focused onto the sample by a
parabolic mirror (custom fabricate, Jenoptik, Germany). The sample is moved
via a piezo stage (M-664.164, Physik Instrumente (PI) GmbH & Co. KG, Germany)
in the focal plane to scan the excitation and light collection position. The
parabolic mirror horizontally separates the incoming beam, the specular
reflection and the scattered light under different angles, allowing to select
the measured scattering angle via a blocker aperture. To ensure that only
light that was scattered multiple times is measured, the scattered light is
measured in cross polarisation with a spectrometer (USB 2000, Ocean Optics
Inc., USA).

Phase reconstruction. The time resolution is achieved by phase reconstruction
via spectral interference of the scattered light with a reference pulse.
Therefore the incoming pulse is separated into sample and reference path. The
reference path is delayed relative to the sample pulse and rotated into the
measured p-polarisation. The resulting interference spectrum
$|E_{\text{s}}(\omega)+E_{\text{r}}(\omega)|^{2}=|E_{\text{s}}(\omega)|^{2}+|E_{\text{r}}(\omega)|^{2}+E_{\text{s}}(\omega)E_{\text{r}}^{*}(\omega)\cos(\Delta\varphi(\omega))$
contains the phase difference $\Delta\varphi$ between the two beams. Via
Fourier filtering of the interference spectrum and after correcting for the
phase imbalance of the interferometer the phase effect of the sample alone can
be reconstructed (see Supplementary Information). Since the phase difference
is measured no phase optimisation of the probing pulse is necessary.

Wigner distribution function and Short Time Fourier Transform. The Wigner
distribution function is defined as
$W(t,\omega)=\int^{\infty}_{-\infty}E(t-t^{\prime}/2)E^{*}(t+t^{\prime}/2)\exp{(-i\omega
t^{\prime})}\text{d}t^{\prime}$, where $E$ and $E^{*}$ are the complex
electric field and its complex conjugate respectively. The WDF yields the
highest time-frequency resolution possible. On the other hand it is not a
linear transform, resulting in cross-terms modulating the the WDF. To help
with the interpretation the spectral power of the short time Fourier transform
(STFT), given by
$|S(\tau,\omega)|^{2}=|\int^{-\infty}_{\infty}w(t^{\prime},\tau,\Delta
t,t_{r})E(t^{\prime})\exp{(-i\omega t^{\prime})}\text{d}t^{\prime}|^{2}$,
where $w(t,\tau,\Delta t,t_{r})$ is a Tukey window function [33] centred at
time $\tau$, is used, which as linear transform produces no cross-terms. For
the STFT the spectral resolution is limited by the window width $\Delta t=120$
fs. The window rising time is $t_{r}=30$ fs.

Calculation of the pulse round trip time. For a single photon travelling back
and forth through an effective medium with thickness $l_{\text{s}}$ the
effective round trip time is given by
$t_{\text{eff}}={2l_{\text{s}}}/{v_{\text{eff}}}$. The speed of light inside
the medium is calculated via $v_{\text{eff}}=c_{0}/n_{\text{eff}}$ where the
effective refractive index $n_{\text{eff}}$ is computed using the Maxwell-
Garnett mixing rule [30]. To obtain the limit when all photons within the
pulse length have propagated back and forth through the effective medium, i.e.
the pulse round trip time $t_{\text{prt}}$, the pulse length has to be added
to the effective round trip time of a single photon. This ensures that also
the ‘last’ photon within the pulse length has reached the top of the medium
again. The scale and the simulated DBS structure possess a filling fraction of
$f_{\text{scale}}=31\%$ [12, 8] and $f_{\text{DBS}}=27\%$, respectively and
the refractive index of chitin $n_{\text{chitin}}=1.55$ [31] is used in both
cases. Applying these values in the Maxwell-Garnett mixing rule yields
$n_{\text{eff, scale}}=1.15$ for the scale as well as $n_{\text{eff,
DBS}}=1.13$ for the DBS. Evaluating the effective round trip time with a
sample thickness of $l_{\text{s, scale}}=10$ µm [12] and $l_{\text{s,
DBS}}=7.9$ µm results in $t_{\text{eff, scale}}=77$ fs for the scale and
$t_{\text{eff, DBS}}=60$ fs for the DBS, respectively. The pulse length is
defined as the time span between the pulse front and the point in the pulse
tail where the intensity dropped to $I_{\text{p}}/e^{2}$ with the peak
intensity of the pulse $I_{\text{p}}$. In the experiment the pulse front is
set at the point where the intensity first reaches $I_{\text{p}}/e^{2}$
yielding a pulse length of $t_{\text{pulse, exp}}=29$ fs. In the simulation
the definite pulse front as emitted by the source is used, resulting in a
pulse length of $t_{\text{pulse, sim}}=100$ fs. Thus, pulse round trip times
of $t_{\text{prt, scale}}=106$ fs and $t_{\text{prt, DBS}}=160$ fs are
obtained for the scale and DBS, respectively.

Extraction of lifetimes from spectral peaks. We estimate the intensity
lifetimes of the resonances by $\tau_{l}=1/\Delta\omega$, where $\Delta\omega$
is the spectral intensity FWHM of the peak [22]. To measure the spectral
widths of a peak, it is fitted with a Gaussian (cf. Fig.
LABEL:fig:meth_peakfit). Fitting the individual peaks ignores slope change by
overlapping resonances, thus the resulting lifetimes are accordingly lower
estimates. To identify individual peaks in the frequency-position plane of the
line scans a 2D peak finding routine is used.

Finite-difference time-domain simulations. The FDTD simulations were performed
using the software Lumerical FDTD Solutions (Ansys Inc., USA). In all
simulations a plane wave pulse impinges in the z-direction on the respective
structure (cf. Fig. LABEL:fig:Details_FDTDa). In the z-direction we apply
perfectly matched layers as boundary conditions. In the x- and y-direction we
use periodic boundary conditions to eliminate unwanted absorption in lateral
boundaries due to the finite size of the simulation. For the calculation of
the lifetime distribution we collect the spectra from roughly 3900 distinct
point-shaped frequency monitors placed in the structure model provided by
Wilts et al. [11] and the DBS structure (for model parameters see Ref. [4])
respectively, both occupying a footprint of $7\times 7$ µm² and a height of
$7-8$ µm. For excitation we use a light pulse with a centre wavelength of 780
nm and collect wavelengths between 745 nm and 815 nm approximating the
experimental conditions. The calculation of the time-dependent power
distribution is done for a DBS model based on the same parameters but with a
lateral footprint of about $20\times 20$ µm². A time-domain monitor cross
sectioning the structure in the x-z-plane is applied to record every 1.14 fs
the poynting vector at every monitor grid point over a total simulation time
of 1000 fs. A pulse length of 100 fs is used to obtain a spectral narrow band
excitation with a centre wavelength of 780 nm and a FWHM of 14 nm. The zero
time is set to the time when the pulse front enters the structure.

Monte Carlo Simulation. Monte Carlo simulations are performed using a self-
written Matlab code (The MathWorks Inc., USA) based on the well known
algorithm presented in literature [13, 32]. To match the FDTD simulation
conditions no absorption inside the slab is applied and in lateral direction
periodic boundary conditions are used. As light source (with about 6.8 billion
photons) a plane wave is chosen possessing a temporal profile matching the
temporal power profile of the impinging pulse in FDTD simulations. An
appropriate monitor cross sectioning the slab is placed according to the FDTD
setup. The lateral width of the slab is 12 µm, the height and effective
refractive index are equal to the values given above for the simulated DBS
model. The applied transport mean free path of $l_{\text{t}}=3$ µm is equal to
the one obtained by FDTD simulations (see Supplementary Information). A
scattering mean free path of $l_{\text{s}}=1$ µm is selected reproducing the
FDTD results for short times closely (cf. Fig LABEL:fig:comparison_ls). The
anisotropy factor $g$ is defined via $l_{\text{t}}=l_{\text{s}}/(1-g)$ [6] and
hence determined by the choice of $l_{\text{t}}$ and $l_{\text{s}}$.

## Acknowledgements

We gratefully acknowledge financial support from the German Research
Foundation DFG within the priority program ”Tailored Disorder - A science- and
engineering-based approach to materials design for advanced photonic
applications” (SPP 1839). We thank B. D. Wilts for supplying us with a 3D
computer tomography model of the beetle scales’ inner structure. We thank the
team of the Nano Structuring Centre (NSC) at the Technische Universität
Kaiserslautern for their support with focused ion beam milling and scanning
electron microscopy.

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