OpticalBIC_SI.tex

# Emergent resonances in a thin film tailored by optically-induced small
permittivity asymmetries

Rodrigo Berté  Contributed equally to this work  Chair in Hybrid Nanosystems,
Nanoinstitut München, Fakultät für Physik, Ludwig-Maximilians-Universität
München, 80799 München, Germany  Thomas Possmayer  Contributed equally to this
work  Chair in Hybrid Nanosystems, Nanoinstitut München, Fakultät für Physik,
Ludwig-Maximilians-Universität München, 80799 München, Germany  Andreas Tittl
Chair in Hybrid Nanosystems, Nanoinstitut München, Fakultät für Physik,
Ludwig-Maximilians-Universität München, 80799 München, Germany  Leonardo de S.
Menezes  Chair in Hybrid Nanosystems, Nanoinstitut München, Fakultät für
Physik, Ludwig-Maximilians-Universität München, 80799 München, Germany
Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco
50670-901, Brazil  Stefan A. Maier  School of Physics and Astronomy, Monash
University, Clayton, Victoria 3800, Australia  The Blackett Laboratory,
Department of Physics, Imperial College London, London SW7 2AZ, United Kingdom

##  I Abstract

Resonances are usually associated with finite systems - the vibrations of
clamped strings in a guitar or the optical modes in a cavity defined by
mirrors. In optics, resonances may be induced in infinite continuous media via
periodic modulations of their optical properties. Here we demonstrate that
periodic modulations of the permittivity of a featureless thin film can also
act as a symmetry breaking mechanism, allowing the excitation of photonic
quasi-bound states in the continuum (qBICs). By interfering two ultrashort
laser pulses in the unbounded film, transient resonances can be tailored
through different parameters of the pump beams. We show that the system offers
resonances tunable in wavelength and quality-factor, and spectrally selective
enhancement of third harmonic generation. Due to a fast decay of the
permittivity asymmetry, we observe ultrafast dynamics, enabling time-selective
near-field enhancement with picosecond precision. Optically-induced
permittivity asymmetries may be exploited in on-demand weak to ultrastrong
light-matter interaction regimes and light manipulation at dynamically chosen
wavelengths in lithography-free metasurfaces.

##  II Introduction

While individual atoms display spectral lines CondonBook1951 , electronic and
phononic interactions in extended matter generate broad optical responses
Fox2002 . Bulk excitonic resonances are a prominent exception, but their
wavelength is predefined near the band edges of the given material Kazi2014 ,
severely limiting the optical bandwidth available for applications. A new
paradigm in optical resonances appeared with the ability to pattern materials
below or in the same size scale as the wavelength of light NovotnyBook , into
flat (metasurfaces Kildishev2013 ) and into three-dimensional (metamaterials
Kadic2019 ) structures, in which optical functionalities are not limited by
the weakly-dispersive macroscopic permittivity of its constituents. However,
their optical response is intimately linked to the dimensions of the
comprising resonators established during fabrication.

Optical modes in infinite featureless media have been shown to arise from
periodic modulations of the permittivity of thin films in the seminal
theoretical works on guided-mode resonances (GMRs) Wang1990 ; Magnusson1992 ;
Wang1993 , in which narrower linewidths were observed for vanishing
permittivity modulations ($\Delta\varepsilon\rightarrow 0$). Periodic
modulations, recently explored in time varying metamaterials Galiffi2022 ,
have also been extensively investigated using laser beam interference in
dynamic gratings induced in featureless films EichlerBook . Such induced
gratings are commonly measured via diffraction effects that are almost
invariant for small changes in the wavelength of the probing beam. Here we
demonstrate that periodic permittivity changes act as a symmetry breaking
mechanism, enabling all-optical tailored resonances. These correspond to
symmetry-protected permittivity-asymmetric quasi-bound states in the continuum
($\varepsilon$-qBICs), for which higher quality-factors ($Q$) correlate with
smaller permittivity asymmetries Liu2020 ; Yu2022 ; Berte2023 ; Li2023 , akin
to GMRs Wang1990 ; Magnusson1992 ; Wang1993 . But while GMRs are excited in
hybrid grating/waveguide structures with predefined periodicities Quaranta2018
, we demonstrate a higher control over $\varepsilon$-qBICs (in wavelength, $Q$
and amplitude) in an all-optical manner without any in-plane geometrical
constraints.

Arising in the context of electronic wavefunction localization
NeumannWigner1929 , BICs are a set of phenomena demonstrated in elastic
Lim1969 , acoustic Parker1966 , hydrodynamic Ursell1951  and optical
Plotnik2011  systems, where particular oscillatory excitations remain
spatially localized in spite of co-existing in the continuum of possible
propagating modes. True BICs can only exist in systems devoid of intrinsic
losses and that are spatially infinite in at least one dimension Hsu2016 .
Among the different types of BICs Hsu2016  are those whose symmetry is
incompatible with that of propagating modes Koshelev2018 , thus termed
symmetry-protected BICs. As true BICs cannot be measured, their observable
counterpart in finite photonic systems, called quasi-BICs (qBICs) Rybin2017 ,
are usually probed via geometrical perturbations in the unit cell of
metasurfaces Fedotov2007 ; Koshelev2018 ; Kuhner2022 ; Noda2001 ; Hirose2014 ;
Liu2019 . Recently, qBICs have been proposed and theoretically investigated
Liu2020 ; Yu2022 ; Berte2023 ; Li2023  and experimentally demonstrated
Berte2023  in metasurfaces that are geometrically symmetric under an in-plane
inversion, but permittivity asymmetric under the same spatial transformation
($C_{2}$ or $(x,y)\rightarrow(-x,-y)$).

In a similar fashion to dynamic gratings, we are able to induce a permittivity
symmetry breaking in an in-plane unbounded homogeneous medium by interfering
two $\approx$ 180 fs ultrashort laser pulses on a thin film (Fig. 1, and Fig.
S4-3 of the Supplementary Information - SI - for the optical setup used in the
pump-probe experiments).

Figure 1: Permittivity symmetry breaking of continuous media and tailored
$\varepsilon$-qBICs. (top left) An unbounded thin film of continuous
translational symmetry features a symmetry-protected BIC state, which does not
couple to a far-field impinging radiation (flat optical response). (top right)
Scheme of the symmetry breaking through the interference of two $\approx$ 180
fs pulses on the a-Si thin film (of height $h$) deposited on a borosilicate
cover glass. The pump beams interfere at an angle $\theta$ generating a 1D
grating pattern of bright (BF) and dark fringes (DF) that excites electron-
hole pairs ($e^{-}/h^{+}$) mainly at the BFs via absorption (middle panel).
These excited carriers impose a periodicity $p_{y}$ in the film permittivity
(bottom panel), whose 1D unit cell is asymmetric under an in-plane inversion
($C_{2}$ or $(x,y)\rightarrow(-x,-y)$) transformation. The optically-induced
metasurface is able to temporarily sustain an $\varepsilon$-qBIC (Fano-shaped
optical response). (bottom) Tailored resonances that can be induced in the
unpatterned film by changing different parameters of the pump beams.

Conversely to dynamic gratings, we probe the induced permittivity modulation
at wavelengths longer than the induced periodicity, for which no diffraction
effect is expected (other than those at zero order) Magnusson1992 ;
EichlerBook . Laser beam interference has been used for imaginary refractive
index-symmetry breaking and ultrafast laser output control in perovskite
metasurfaces, albeit with a resonance wavelength predefined by the fabricated
array of holes Huang2020 . Here, we periodically modulate the real part of the
permittivity of an unstructured amorphous silicon (a-Si) film in space via a
preferential carrier excitation in the bright fringes of the generated
interference pattern (see Fig. S6-3 of the SI) Sjodin1998 . This modulation
breaks the original continuous translational symmetry of the system and
imposes a periodic 1D unit cell that is asymmetric under in-plane inversion
(Fig. 1 top right) due to the anisotropic carrier distribution. The set of
these periodic cells form an optically-induced metasurface, allowing the film
to sustain $\varepsilon$-qBICs. Employing this continuous, in-plane unbounded
medium, we demonstrate significant optical modulation at normal incidence and
ultrafast enhancement of nonlinear effects at selected wavelengths.

##  III Results

To achieve symmetry breaking, modulations in the permittivity of a thin film
need to be induced locally by the pump beams. These perturbations can arise
from a range of processes, including the Kerr effect, thermal expansion,
molecular reorientation and photorefraction. Some of the most significant
changes can however be introduced by changing the electronic energy states,
i.e. generating optically-pumped free electrons and holes in a high-index
semiconductor. a-Si is a suitable material for optical excitation and can be
deposited in thin films with standard equipment. To prove the feasibility of
the metasurface, a 45 nm thick a-Si film was deposited on a borosilicate glass
substrate (Fig. 1, see Methods for fabrication details). The optically-induced
permittivity modulation on it is then produced by interfering two ultrafast
($\approx$ 180 fs) laser pulses at an angle $\theta$, which produces bright
and dark interference fringes with a pitch along the y-axis related to the
pump wavelength $\lambda_{pump}$ via EichlerBook :

$p_{y}=\frac{\lambda_{pump}}{2\sin\big{(}\theta/2\big{)}}$ (1)

To reach regimes with significant changes in the film’s permittivity, these
pulses with energies up to 130 nJ are focused to a $1/e^{2}$ waist radius of
25 $\mu$m.

###  III.1 Measuring all-optical $\varepsilon$-qBICs

Since the resonances appear isolated in time after the pump pulses arrive on
the substrate, the induced transmittance changes are measured with
synchronized white light pulses at variable time delays. By subtracting the
intrinsically flat response of the a-Si film, it is possible to spectrally
resolve the time-dynamics of only the induced resonance. As seen in Fig. 2a,
when the white light pulse is spatially and temporally superimposed with the
grating, a significant ($>$30%) reduction of differential transmittance
($\Delta T/T$) is observed in an isolated wavelength range, which correlates
to the simulated response of an induced $\varepsilon$-qBIC under strong
permittivity asymmetry (see below). This strong wavelength dependence is not
expected from the theory of thin dynamic gratings EichlerBook .

Changing the time delay of the probe pulses reveals a rich dynamic in the
temporal evolution of the $\varepsilon$-qBIC, related to the carrier dynamics
in the film. The induced resonance, at first, builds up during the first 200
fs (blue-shifting to $\approx$ 780 nm) as the pump pulses are absorbed mainly
at the bright fringes, progressively breaking the permittivity symmetry of the
film. This resonance shows a transmittance modulation of $>30\%$ for times 0.1
ps $<t<$ 0.3 ps (see Fig. S7-3 of the SI), until it slightly red-shifts and
mostly disappears within approximately 1 ps. Sub-ps dynamics has been
previously reported on high carrier densities in bulk crystalline Si generated
via pulsed interference patterns at small angles (implying an optical grating
with very large pitch, $\approx$ 13 $\mu$m) Sjodin1998 . However, relaxation
times of the optical response, due to hot carriers relaxation to the band
edges, had an opposite trend with pulse energy than we have observed (see
below), and without any signs of strong wavelength dependence or attribution
to resonant phenomena.

The decay shown in Fig. 2a is attributed to the migration of excited carriers
from the bright to the dark fringes, as this $\approx$ 1 ps dynamic is too
fast for a full carrier recombination in the thin film (see, for example,
measurements of the carrier dynamics without the metasurface in Fig. S8-3 of
the SI) Sokolowski2000 . This carrier migration, in turn, reduces the
permittivity asymmetry of the unit cell, quantified by the asymmetry parameter
$\alpha_{\varepsilon}\stackrel{{\scriptstyle\text{def}}}{{=}}\Delta\varepsilon/\varepsilon$.
As the red-shift continues towards the symmetry-protected BIC state
(horizontal dashed line in Fig. 2a top - defined by $p_{y}$, the substrate,
the film height and unperturbed permittivity), also due to an increased
refractive index attributed to carrier recombination, a second spectral
feature appears. At $t\approx$ 4 ps, a long-lived resonance emerges that lasts
for $>$ 1000 ps, attributed to trapped carriers at the bright fringes that
sustain a small permittivity asymmetry (implying $\alpha_{\varepsilon}\neq 0$)
in the metasurface. The spectra (cross-sections of top panel) and the
corresponding $Q$-factors at different times are shown in the bottom left and
bottom right panels in Fig. 2a, respectively.

A few aspects play a role in these dynamics. As mentioned, the excited
carriers diffuse from the bright to the dark fringes, reducing the
permittivity asymmetry of the system. Smaller values of $\alpha_{\varepsilon}$
shift the resonance closer to the BIC state and are correlated with higher
$Q$-factors Berte2023 . Besides, the excited free electrons and holes
recombine and the film becomes less lossy, being able to sustain higher-$Q$
$\varepsilon$-qBICs. This explains the appearance and red-shift of the second
spectral feature observed in the differential transmittance measurements, and
the respective increase in $Q$ for longer times, as shown in Fig. 2a (bottom
right).

To prove the spectral tunability of the metasurface, measurements were
performed at different pump wavelengths, leading to different periodicities
$p_{y}$ of the permittivity modulation (see Eq. 1). The spectral response at
the highest modulation for each $\lambda_{pump}$ is shown in Fig. 2b, along
with their respective numerically-calculated spectra. By red-shifting the pump
wavelength from 410 nm to 450 nm with a fixed interference angle
$\theta=58^{\circ}$, we observe a redshift of the resonance from 805 nm to 875
nm (see Fig. S9-3 of the SI for the full differential transmittance maps). By
only varying the periodicity in simulations we are able to reproduce
numerically the experimental spectra in Fig. 2b. The optical responses can
then be calculated, with good agreement, using solely the unit cell size of
the metasurface and a modified complex permittivity of a-Si as a function of
the employed pump pulse energy (see below and the Numerical calculations in
Methods for details). We are here neglecting the different carrier excitation
efficiencies between the pump wavelengths in the formation of the metasurface.

In addition to this spectral tunability, we expect that different asymmetries
lead to different $Q$-factors of the resonance. In this system, the asymmetry
can be continuously adjusted by changing the incident pump pulse energy, which
however also introduces different losses and changes the overall refractive
index of the medium. To evaluate the response of the film and, thus, the
behavior of resonances, we have therefore measured the transmission spectra at
different pump pulse energies ($E_{p}$) between 4 nJ to 130 nJ. As seen in
Fig. 2c (top), higher $E_{p}$ values blue-shift the resonance from the
assigned BIC state (vertical dashed line) and also increase their
transmittance modulation (film response without the metasurface has not been
subtracted). Additionally, slower decay times are observed for lower pump
energies, due to the interplay between a slower carrier diffusion and lower
losses due to an overall smaller excited carrier density (see the complete
differential transmittance maps of the $\varepsilon$-qBICs as a function of
the pump pulse energy in Fig. S11-3 of the SI). A similar trend is observed
for a pump wavelength of 410 nm (Figs. S12-3 and S13-3 of the SI). Therefore,
by merely changing the pump pulse energy not only the wavelength and amplitude
of resonances can be modified, but also their $Q$-factor and decay times.

Figure 2: Experimental all-optical $\varepsilon$-qBICs in a featureless thin
film. a Temporal evolution of the $\varepsilon$-qBIC from a 45 nm thick a-Si
film measured at 130 nJ of pulse energy per arm ($\lambda_{pump}=$ 420 nm, the
response of the film without a metasurface has been subtracted from the signal
for clarity). Schemes on top of the differential transmittance map depict
$-\alpha_{\varepsilon}$ and the free-carrier concentration ($[e^{-}],[h^{+}]$)
of bright (BF) and dark fringes (DF) at different times. The resonance red-
shifts towards the symmetry-protected BIC state (horizontal - on the top
panel, and vertical - on the bottom left panel, dashed lines, arbitrarily
positioned for guiding purposes) for longer times. Bottom right panel shows
$Q$ vs $t$. $Q$-factors increase as the permittivity symmetry is restored
($\alpha_{\varepsilon}\rightarrow 0$) and losses are reduced due to carrier
recombination. b Tunability of $\varepsilon$-qBICs as function of
$\lambda_{pump}$. (left) Spectra measured at $\theta=58^{\circ}$ and 40 nJ
laser pulse energy per arm. (right) Corresponding spectra obtained from
simulations by varying $p_{y}$ of the metasurface unit cell (corresponding
$\lambda_{pump}$ shown on the right). Parameters
$\varepsilon_{r,E_{p}\rightarrow\infty}=$ 0.6, $\varepsilon^{M}_{i}=$ 0.25,
$\varepsilon_{i,E_{p}\rightarrow\infty}=$ 1.15, $k_{b}=$ 0.04/nJ, $k_{d}=$
0.001/nJ and $E_{p}=$ 40 nJ as defined in Eq. 2 were used in all simulations.
See Fig. S18-3 of the SI for the permittivity model as a function of $E_{p}$.
c Symmetry breaking versus pump pulse energy per arm $E_{p}$ (in nJ, shown on
the right, $\lambda_{pump}=$ 420 nm, see Fig. S11-3 for full differential
transmittance maps). (bottom) Experimental (circles) and FEM-calculated (red
squares) $Q$s vs $E_{p}$. An inverse-square dependence ($Q\propto
1/E_{p}^{2}$) is shown as a dashed grey line. FEM calculations employed
identical parameters used in simulations of the spectra shown in b. d Electric
near-field polarization (arrows) and electric near-field enhancement
($E/E_{0}$) of the quasi-symmetry-protected resonance. A 400 nm-wide section
of the metasurface along the $x$ axis is shown.

A hallmark of symmetry-protected qBICs is the inverse-square dependence of the
$Q$-factor with the asymmetry parameter $\alpha$, as derived from a
perturbative treatment of the BIC state Koshelev2018 ; Berte2023 . This
behavior occurs for a system in which the losses are dominated by radiative
processes (i.e. photon emission through coupling to the continuum of states
induced by geometrical/permittivity asymmetries). This implies that either
non-radiative losses (absorption) are completely absent, which is an ideal
case, or can be neglected when compared to radiative losses. If the $Q$-factor
is described by a reciprocal sum of radiative ($r$) and non-radiative ($nr$)
quality-factors ($1/Q=1/Q_{r}+1/Q_{nr}$), $Q_{nr}\rightarrow\infty$, and thus
$Q=Q_{r}$. As the pump pulse energy is our symmetry-breaking tuning parameter,
it is worthy investigating how $Q$ behaves as a function of $E_{p}$, and if
$E_{p}$ can play an analogous role of geometrical Koshelev2018  or in-plane
momentum perturbations Lee2012  in symmetry-protected qBICs (i.e. $Q\propto
1/E_{p}^{2}$).

In the bottom panel of Fig. 2c we show the calculated (red squares) and the
experimental (circles) values of $Q$ as a function of $E_{p}$. The
calculations for the ideal lossless metasurface versus $-\alpha_{\varepsilon}$
is shown in Fig. S14-3 of the SI. As expected, when dealing with a system
devoid of intrinsic losses $Q$ of the metasurface follows an inverse-square
dependence with $\alpha_{\varepsilon}$ ($Q\propto\alpha_{\varepsilon}^{2}$),
in agreement with theoretical predictions Yu2022 ; Berte2023 ; Li2023 . This
corresponds to a situation where the permittivity difference between the
bright and dark fringes in the 1D unit cell is gradually increased, and the
losses in the grating only occur due to radiative processes as the metasurface
resonance progressively couples to propagating states in the continuum. When
the experimental $Q$ (see Fig. S15-3 for fitted data) is plotted against
$E_{p}$, however, an obvious deviation from a $1/E_{p}^{2}$ behavior (dashed
grey line) is observed, which implies that losses are present and the
asymmetry parameter $\alpha_{\varepsilon}$ is not a mere linear function of
$E_{p}$ in the measured range. In spite of these deviations, we may reproduce
the observed behavior by resorting to a simple model of the permittivity of
bright and dark fringes as a function of $E_{p}$.

We may model the dispersive permittivity of the thin Si film
($\varepsilon^{Si}_{b(d)}(\omega,E_{p})$) in the bright (dark) fringes as a
function of the pump pulse energy $E_{p}$ as:

$\begin{split}\varepsilon^{Si}_{b(d)}(\omega,E_{p})=\varepsilon^{Si}_{r}(\omega)\times&\bigg{[}\varepsilon_{r,E_{p}\rightarrow\infty}+(1-\varepsilon_{r,E_{p}\rightarrow\infty})e^{-k_{b(d)}E_{p}}\bigg{]}\quad+\\\
&\quad
i\bigg{[}\varepsilon^{Si}_{i}(\omega)+\varepsilon^{M}_{i}+\varepsilon_{i,E_{p}\rightarrow\infty}(1-e^{-k_{b(d)}E_{p}})\bigg{]}\end{split}$
(2)

where $\varepsilon^{Si}_{r}(\omega)$ and $\varepsilon^{Si}_{i}(\omega)$ are
the real and the imaginary permittivities, respectively, of the unperturbed
a-Si film as measured via ellipsometry (Fig. S17-3 of the SI),
$\varepsilon^{M}_{i}$ a parameter that accounts for losses due to the finite
size of the metasurface, $\varepsilon_{r,E_{p}\rightarrow\infty}$ the real
(fractional) and $\varepsilon_{i,E_{p}\rightarrow\infty}$ the (additive)
imaginary saturation values of the permittivity for $E_{p}\rightarrow\infty$,
and $k_{b(d)}$ a constant that defines the exponential changes in the bright
(dark) fringes. Three important aspects are represented in this model: (i) At
very small pulse energies, $E_{p}\rightarrow 0$, losses are limited by the
imaginary permittivity of the film and the metasurface size
($\mathfrak{Im}({\varepsilon^{Si}_{b(d)}(\omega,E_{p}\rightarrow
0)})\approx\varepsilon^{Si}_{i}(\omega)+\varepsilon^{M}_{i}$); (ii) at low
pulse energies, a linear change in the permittivity is obtained
($e^{-k_{b(d)}E_{p}}\approx 1-k_{b(d)}E_{p}$); (iii) at high pulse energies,
$E_{p}\rightarrow\infty$, a saturation regime is achieved
($\varepsilon^{Si}_{b(d)}(\omega,E_{p}\rightarrow\infty)=\varepsilon^{Si}_{r}(\omega)\times\varepsilon_{r,E_{p}\rightarrow\infty}+i(\varepsilon^{Si}_{i}(\omega)+\varepsilon^{M}_{i}+\varepsilon_{i,E_{p}\rightarrow\infty})$,
Fig. S18-3 of the SI).

This simple model is able to reproduce the trend in the measured $Q$s with a
fairly good agreement. Counterintuitively, the model predicts a reduction in
the asymmetry parameter ($\alpha_{\varepsilon}$) for higher pump pulse
energies, as the bright fringe optical response to the pump beams saturates
while the dark fringe permittivity is further reduced with increasing pump
absorption (Fig. S18-3 of the SI). In addition, the model predicts a
saturation of $Q$ at low pulse energies ($Q\approx 70$, where losses are
dominated by the finiteness of the metasurface, $\varepsilon^{M}_{i}$).
Although changing $E_{p}$ allows us to navigate through the parameter space of
asymmetries, assessing the system’s response to minute values of
$\alpha_{\varepsilon}$ and probing an eventual $Q\propto 1/E_{p}^{2}$
dependence requires the generation of larger metasurfaces than we are
currently able with our optical setup. Besides, as the pump pulses have a
Gaussian-shaped intensity distribution from its center, different carrier
densities are excited at different positions in the optical 1D grating,
leading to different permittivity asymmetries as we move from the center of
the grating towards its edge. Naturally, by representing the metasurface using
the unit cell only we are neglecting this spatial variation of
$\alpha_{\varepsilon}$, nonetheless being able to reproduce the observed
experimental behavior (see Fig. S19-3 of the SI for the spectral response as a
function of $E_{p}$). Further characterization of the all-optical
$\varepsilon$-qBICs (pump and probe polarization dependence, etc) can be found
in Figs. S20-3-S23-3 of the SI.

###  III.2 Spectrally-selective third harmonic generation enhancement via
$\varepsilon$-qBICs

$\varepsilon$-qBICs excited via optically-induced metasurfaces are expected to
carry the same functionalities of resonator-based metasurfaces. One of the
most utilized properties of metasurfaces featuring resonances is their optical
near-field enhancement, which can be used to boost phenomena that require
strong light-matter interaction, like nonlinear effects Tonkaev2023 ;
Zograf2022 . Symmetry (centrosymmetry) breaking via DC electric fields has
been long used to induce second-harmonic generation in centrosymmetric
materials Terhune1962 , although it is not a spectrally-selective method. As a
proof-of-principle, we demonstrate how the induced in-plane symmetry breaking
can be used to enhance the third harmonic generation (THG) at specific
wavelengths in a bare silicon film within a narrow timeframe.

For that, we generate a grating at 515 nm pump, which we now probe with pulses
of tunable narrow bandwidth (fundamental beam) instead of the supercontinuum
white light. The longer pump wavelength increases the $p_{y}$ of the
metasurface, red-shifting the $\varepsilon$-qBIC further into the infrared
range, and therefore its THG closer to the visible. In Fig. 3a the THG
response of the bare film around 340 nm ($\lambda_{fundamental}=$ 1020 nm) is
compared when both pump beams are either turned on (solid red) or turned off
(dashed black). A 234% enhancement of the THG signal is obtained when the pump
beams are illuminating the sample and breaking the in-plane inversion
permittivity symmetry of the thin film. The corresponding THG signals as a
function of time are shown in Fig. 3b and Fig. 3c. The spot at $\approx$ 343
nm in Fig. 3b is due to a four-wave mixing process, disappearing in the
absence of the pump beams (Fig. 3c, see SI for further characterization). The
decay of the THG enhancement clearly follows an exponential trend (fitted grey
dashed line) instead of a Gaussian temporal profile expected from an
enhancement due to a mixing of ultrafast beams. This $\approx$ 1 ps dynamic is
temporally similar to the excitation of $\varepsilon$-qBICs as shown in Fig.
2a.

To evaluate the spectral response of the nonlinear effect, sweeps at different
probe wavelengths were taken and referenced with sweeps where the pump pulses
were cross-polarized, which switched off the grating contrast. Given the
different collection efficiencies at different wavelengths of our optical
setup, we have normalized the THG output (see Figs. S28-3-S28 in the SI for
the raw data). By comparing the THG signals with ($0^{\circ}$) and without
($90^{\circ}$) the metasurface, we observed a preferential enhancement for
probe wavelengths that are spectrally overlapped with the $\varepsilon$-qBIC
(grey shadowed area in Fig. 3d, whose spectra was shifted to $1/3$ of its
value to overlap with the THG output). Here, a $>$ 100% enhancement was
obtained for $\lambda_{fundamental}=$ 1040 nm ($\lambda_{THG}\approx$ 347 nm)
in the presence of the metasurface, when compared to the $90^{\circ}$-rotated
polarization of one of the pump beams, despite all incident radiation of pump
and probe having the same pulse energy in both situations. The lower
enhancement is attributed to the smaller pump energy used for this set of
measurements ($E_{p}$ = 50 nJ vs $E_{p}$ = 75 nJ in Fig. 3a).

Figure 3: Spectrally-selective third-harmonic generation enhancement using
$\varepsilon$-qBICs. a THG signal ($E_{p}=$ 75 nJ) in the presence (pump on)
and absence (pump off) of the optical grating ($\lambda_{pump}=$ 515 nm)
providing the permittivity symmetry breaking. A 234% enhancement of THG (at
340 nm, $\lambda_{fundamental}=$ 1020 nm) is obtained when a metasurface is
generated in the thin film (solid red line) compared to when both pump beams
are blocked (dashed black). b THG signal vs delay line position, whose cross-
sections in time are shown in panel a. c Corresponding THG signals as function
of time at 339 nm. The enhancement provided by the $\varepsilon$-qBIC decays
exponentially ($1/e\approx$ 0.72 ps, fit shown as a grey dashed line). See
Fig. S24-3 for full data. d Normalized THG signal for $0^{\circ}$ (blue
curves) and $90^{\circ}$ (dashed light red) relative pump polarizations as a
function of the fundamental beam wavelength (from $\lambda_{fundamental}=$ 970
nm to $\lambda_{fundamental}=$ 1090 nm). A preferential THG enhancement is
observed when the fundamental is overlapped with the $\varepsilon$-qBIC
(transmittance shown as a gray shadowed area, right-hand side axis).
Parameters $\varepsilon_{r,E_{p}\rightarrow\infty}=$ 0.75,
$\varepsilon^{M}_{i}=$ 0.25, $\varepsilon_{i,E_{p}\rightarrow\infty}=$ 1.05,
$k_{b}=$ 0.03/nJ, $k_{d}=$ 5$\times 10^{-4}$/nJ, $E_{p}=$ 50 nJ and $\theta=$
51∘ (implying $p_{y}=$ 598 nm) were used in the transmittance calculation. A
$>$ 100% enhancement is obtained for $\lambda_{fundamental}=$ 1040 nm.
$\varepsilon$-qBIC spectrum was shifted to a third of its value ($\lambda/3$)
to overlap with THG signals.

##  IV Discussion

We have shown that a simple unpatterned dielectric amorphous film can yield
tailored resonances when its permittivity is periodically modulated in space,
breaking its original translational and in-plane inversion symmetries. Optical
resonances do not require then to be pre-defined and constrained by etching
and deposition fabrication procedures that provide the boundary conditions for
typical spatially-finite resonators. $\varepsilon$-qBICs demonstrated here in
the near infrared can easily be extended further into the visible and towards
the mid- and far-infrared, where coupling to excitonic resonances Weber2023
and molecular vibrational/rotational levels could be performed Xie2023 ;
Tittl2018 .

The emergence of the symmetry-protected quasi-BICs in the unbounded medium
cannot be easily, nor intuitively, derived from the optical properties of the
thin film, i.e., regardless how much one knows about its atomic and electronic
band structure that define its optical properties, the symmetry-protected qBIC
does not follow trivially from this knowledge Anderson1972 .

Periodic permittivity variations in space are also a core concept in
distributed Bragg reflectors (DBRs) and photonic crystals (PhCs) Rayleigh1888
; Yablonovitch1987 ; John1987 . But, conversely to DBRs and PhCs, where
confinement of light deteriorates with a smaller contrast of refractive
indices, here higher localization and, thus, higher $Q$-factors are achieved
with smaller spatial permittivity differences. This facilitates the dynamic
modulation of devices, as it reduces - or, in principle, completely
circumvents - the need for large alterations in optical properties of
materials with a given input to significantly change their optical response.
The minute values of electro-optical coefficients of most materials here play
to our advantage, and a number of substances which were previously dismissed
as “optically inert” can now be incorporated into the set of electro-optical
materials, including silicon. Besides free-carriers, any permittivity-change
mechanism can be employed for symmetry breaking and the excitation of
$\varepsilon$-qBICs in a thin film, be it via linear (Pockels) Abel2019  or
nonlinear (Kerr) electric-field effects Boydbook , ion implantation Basu2010 ,
temperature differences Plotnik2011 , etc.

The broad optical responses in condensed matter imply that any given input
aimed to modulate the permittivity of materials would also affect it in a
broadband fashion Fox2002 ; Soref1987 . For this reason, optical signals that
are encoded in different frequencies within an optical fiber (wavelength-
division multiplexing) need to be physically separated in demultiplexing
operations. By leveraging the strong and spectrally-localized resonances that
arise from small perturbations in permittivity symmetries in an otherwise
optically-flat platform, we may modulate/filter the propagation of a small set
of wavelengths independently, leaving the propagation of others frequencies
nearly undisturbed. Selective optical modulation using $\varepsilon$-qBICs
across all optical telecommunication bands (from the high-energy edge of the O
band at 1260 nm to the low-energy end of the L band at 1625 nm) can, in
principle, be easily performed in the same thin silicon film.

As we have demonstrated, several degrees of freedom can be used to tailor the
desired resonances, such as pump pulse energy and wavelength, relative delay,
incidence angle and polarization of the grating-forming beams. Another
advantage of a homogeneous medium relative to pre-fabricated metasurfaces of
resonators is the absence of losses due to corners, edges and fabrication
imperfections. This makes defect-free crystalline films a sort of ultimate
platform for on-demand high-$Q$ modes. We have shown the excitation of
$\varepsilon$-qBICs through a rather lossy symmetry-breaking mechanism,
although it allowed us to exploit large values of the asymmetry parameter
$\alpha_{\varepsilon}$ and a way to assess ultrafast carrier dynamics and
spatial distribution in solid state systems EichlerBook . Besides, selectively
enhancing nonlinearities in a platform whose frequency-conversion efficiency
is not constrained by phase-matching requirements, but instead governed by the
induced resonances, provides great flexibility when exploiting nonlinear
phenomena. Crucial questions concern what practical limits this platform can
achieve, and what other asymmetries may be exploited Liu2020 ; Yu2022 ;
Huang2020 . The features of active metasurfaces may be expanded to include
permittivity symmetry breaking in continuous media.

##  V Conclusion

In conclusion, we have demonstrated that an unbounded, featureless thin film
with an intrinsically flat optical response can sustain resonances at desired
wavelengths, amplitudes and $Q$-factors when a particular permittivity
asymmetry is induced in it via optical means, i.e., by interfering two
ultrafast laser beams at an angle. We have argued that the nontrivial
emergence of strong, frequency-dependent, optical responses follows directly
from the symmetry breaking of the original continuous translational and in-
plane inversion permittivity symmetries of the film.

Experiments were performed in one of the simplest CMOS-compatible platforms,
namely, an amorphous Si (a-Si) film over borosilicate glass. Transmittance
modulations as high as 35% and $Q\approx 80$ for long-lasting resonances
(observed due to carrier trapping) were achieved via carrier excitation in the
semiconductor, an inherently lossy symmetry-breaking mechanism. Significant
improvements are expected in modulation amplitudes and in $Q$-factors of
$\varepsilon$-qBICs if a crystalline Si film, inherently less lossy, is used
instead, by pumping at longer wavelengths and also by exploiting larger
metasurfaces than we have used here. We have also demonstrated a sub-ps $>$
200% enhancement of THG in the bare Si film at wavelengths corresponding to
the $\varepsilon$-qBIC excitation. Again, the inherently lossy mechanism of
symmetry breaking due to carrier excitation inhibits larger enhancements of
the third harmonic than we have observed, a process whose efficiency should
scale enormously for larger $Qs$ (proportional to the dwelling time of the
photons in the film) and due to increasing near-field enhancement.

We believe that $\varepsilon$-qBICs in featureless continuous media have a
strong connection with GMRs in unpatterned thin films Wang1990 ; Magnusson1992
; Wang1993 . Indeed, a $Q\propto 1/\alpha^{2}(\theta)$ dependence and the
symmetry-protected nature of GMRs have been observed in silicon cuboid
metasurfaces at THz frequencies due to an incidence angle($\theta$)-induced
symmetry breaking Han2019 . However, while GMRs are essentially invariant in
one dimension, all-optical $\varepsilon$-qBICs of higher dimensionality could
be excited Liu2020 ; Yu2022 ; Berte2023 ; Li2023 ; Huang2020 . Besides, GMRs
have required so far featureless thin films to be coupled to corrugated
structures in resonant waveguide gratings Quaranta2018  or to bear inclusions
in high-contrast gratings Sturmberg2015 , both already exploited in several
applications. The addition of corrugations or inclusions, however, inevitably
defines the periodicity of the structure and thus its resonant wavelength,
limiting the spectral flexibility of the design. All-optical
$\varepsilon$-qBICs here demonstrated do not have this structural limitation.

We hope that the all-optical excitation of $\varepsilon$-qBICs will provide
new ways of dynamically manipulating electromagnetic radiation at required
wavelengths in a single lithography-free platform and bring new possibilities
for light technologies. These may encompass applications as varied as optical
filters Magnusson1992 , $Q$-switching McClung1967 , holography, focusing and
steering of optical beams Malek2020  and also thermal emission control
OvervigPRX2021  through the interplay of local and nonlocal responses, chiral
Overvig2021  and orbital angular momentum Wang2020  manipulation, generation
of second- and higher-order harmonics Tonkaev2023 ; Zograf2022 , optical
modulators Reed2010 , tunable lasing Kodigala2017 ; Noda2001 ; Hirose2014 ;
Huang2020 , sensing Tittl2018 , coupling to vibrational Xie2023  and excitonic
resonances Weber2023  in controllable weak to ultrastrong light-matter
interaction regimes Kockum2019  and polaritonic condensation Berghuis2023 .

##  VI Methods

###  VI.1 Numerical calculations

Thin film spectra were calculated through finite-difference time domain
solutions of Maxwell’s equations using the commercially available software
Lumerical (Ansys). A normal incidence, linearly-polarized (along the $x$-axis)
plane wave propagating in the $-z$ direction illuminated the thin film.
Periodic boundary conditions were used to represent the in-plane ($xy$)
unbounded media. Power monitors were used for transmittance calculations, and
perfectly matched layers (PMLs) domains at the top and bottom for the
absorption of propagating waves. A dispersionless refractive index of $n=1.45$
was employed for the borosilicate substrate, while the dispersive complex
permittivity of the amorphous silicon (a-Si), as measured via ellipsometry was
used for the thin film (see Supplementary Information for data, Fig. S17-3).
The dispersive complex permittivity of a-Si was then modified with the model
proposed in Eq. 2 for bright and dark fringes (Fig. S18-3 of the SI). The
modified permittivity was considered homogeneous in each fringe.
Eigenfrequency solutions of Maxwell’s equations were used for $Q$-factor
calculations and were performed using the commercially available RF module of
the finite element solver COMSOL Multiphysics with identical boundary
conditions as described above. In both types of simulations we have
represented the optically-induced metasurface using only the unit cell (Fig.
1, top right). This approximation reduces the computational resources required
for simulating and predicting the desired optical responses of the thin film
when compared to a simulation of the full grating domain (tens of $\mu
m\times$tens of $\mu m$). By simulating the unit cell alone we have then
neglected spatial permittivity variations from the center towards the outer
edges of the metasurface, which would be inevitably induced by the Gaussian
intensity profiles of the pump beams used to generate the grating.

###  VI.2 Sample fabrication

a-Si was deposited onto borosilicate glass substrates via PECVD. Samples were
annelead at 700∘ C for 90 s to improve film quality.

###  VI.3 Optical measurements

A collinear optical parametric amplifier (OPA, ORPHEUS-HP) pumped by a pulsed
Yb:KGW Pharos laser system (Light Conversion Ltd) of maximum repetition rate
of 200 kHz generated outputs of pulse duration $\approx$ 180 fs. For time-
dependent transmittance measurements, the 515 nm invariant beam output of the
OPA was focused onto a 5-mm-thick sapphire plate and used to generate the
supercontinuum probe light beam (see spectrum in the SI, Fig. S5-3), while the
mechanically-chopped wavelength tunable OPA output was sent through a 50/50
beam splitter to generate the metasurface. A motorized optical delay line was
used to introduce controlled time differences between the pump and
supercontinuum pulses. One of the pump beams passed through a controlled half-
wave plate and a neutral density wheel for adjustable attenuation. The
grating-generating pump beams and the supercontinuum light were slightly
focused onto the sample using lenses of focal length equal to 100mm and 75mm,
respectively. Transmittance measurements were carried out with a lock-in
detection system (Stanford Research Instruments) by modulating the wavelength
tunable OPA output at $<2$ kHz frequency using an optical chopper. A
spectrograph (PI Acton SP2300, Princeton Instruments) coupled to an avalanche
photodiode (Thorlabs APD440A) was used for spectral characterization of the
supercontinuum probe light transmitted by the sample. THG measurements were
performed by using the invariant 515 nm output of the OPA to generate the
metasurface, while the variable output (previously used as the pump beams) was
used as the fundamental beam to generate the third harmonic response in the
thin film.

##  VII Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

##  VIII Acknowledgments

We acknowledge financial support from the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under Grant Nos. EXC 2089/1 - 390776260
(Germany’s Excellence Strategy and TI 1063/1 - Emmy Noether Program), the
Bavarian State Ministry of Science, Research, and Arts through the program
“Solar Technologies Go Hybrid (SolTech)” and the Enabling Quantum
Communication and Imaging Applications (EQAP) project. Funded by the European
Union (ERC, METANEXT, 101078018). Views and opinions expressed are however
those of the author(s) only and do not necessarily reflect those of the
European Union or the European Research Council Executive Agency. Neither the
European Union nor the granting authority can be held responsible for them.
S.A. Maier additionally acknowledges the Lee-Lucas Chair in Physics and the
Australian Research Council. We thank local research clusters and centers such
as the Center of Nanoscience (CeNS) for providing communicative networking
structures.

##  IX Authors contributions

R.B. conceived the project and performed the theoretical and numerical
analysis. R.B., T.P. and L. de S.M. conceived the experimental demonstrations.
T.P. and L. de S.M. assembled the optical setup and performed the
measurements. R.B. and T.P. performed the experimental data analysis. R.B., L.
de S.M., A.T. and S.A.M. supervised the project. R.B. wrote the manuscript.
All other authors contributed to the manuscript improvement.

##  X Competing interests

The authors declare no competing interests.

##  XI Additional information

Supplementary Information is available for this paper. Correspondence and
requests for materials should be addressed to R.B<EMAIL_ADDRESS>muenchen.de) and A.T. (A.Tittl@physik.uni-muenchen.de).

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Supplementary Information for
Emergent resonances in a thin film tailored by optically-induced small
permittivity asymmetries

##  XII Optical setup

Figure S4-3: Scheme of the optical setup employed in the generation of the
optically-induced metasurface and detection of $\varepsilon$-qBICs. The
spectrum of the supercontinuum probe beam is shown in Fig. S2.

##  XIII Spectrum of the supercontinuum probe beam

Figure S5-3: Spectrum of the supercontinuum probe beam generated from the 515
nm OPA output ($P_{515nm}=$ 210 mW) in a sapphire nonlinear crystal. The
spectrum was filtered with a long-pass 650 nm optical filter prior to
measurements. The generated supercontinuum beam is heavily chirped, which has
been manually corrected in the post-processing of data for analysis when
necessary.

##  XIV Optical response versus path (time) delay between pump pulses

Figure S6-3: Optical micrographs ($\lambda=430nm$) without (left), and with
temporal delays between pulses induced by a 40 $\mu$m (or 40 $\mu$m/3$\times
10^{8}$ m/s = 133 fs, center) and by an 80 $\mu$m (or 266 fs, right) increase
in the optical path of one of the pump beams. The 1D optical grating
progressively disappears as the pulses arrive in the thin film at increasingly
different times. (bottom) Temporal evolution of $\varepsilon$-qBICs as a
function of the optical path delay (shown in the insets). As the 1D optical
grating disappears, so does the induced resonance, and only the flat optical
response of the thin film is observed.

##  XV Temporal cross sections of $\varepsilon$-qBICs

Figure S7-3: Temporal cross-sections of data shown in Fig. 2a of the main
manuscript, for 0.1 ps $<t<$ 0.5 ps. Although the main resonance lasts for
$\approx$ 1 ps, the transmittance modulation is larger than 30% (shadowed
area) only between 0.1 ps and 0.3 ps. The cross-sections at different probe
delays are shifted by 30% for better visualization.

##  XVI Optical response of the film versus pump pulse energy in the absence
of the optically-induced metasurface

Figure S8-3: (left) Differential transmissivity at 800nm as a function of time
for a 90∘-rotated relative polarization between the pump beams (a condition
where no 1D optical grating, thus no metasurface, is formed). Increasing pump
pulse energies are shown in darker red colors. $\lambda_{pump}=$ 420 nm.
(right) Respective maximum differential transmissivity vs pump pulse energy
per arm (at $t=$ 0 ps). Dashed line shows the exponential fit ($\Delta
T/T=a_{0}+a_{1}e^{-kE_{p}}$) of the data. The saturation of the optical
response and a deviation from a linear behavior are clearly observed for pulse
energies per arm larger than 65 nJ.

##  XVII Temporal evolution of $\varepsilon$-qBICs as a function of the pump
wavelength

Figure S9-3: Temporal evolution of $\varepsilon$-qBICs as a function of the
pump wavelength (insets). Resonances red-shift for longer pump wavelenghts,
corresponding to larger periodicities $p_{y}$ of the metasurface. All
resonances have a similar temporal profile, lasting for $\approx$ 1 ps and
shifting towards the symmetry-protected BIC state for longer times, as the
permittivity symmetry is restored due to carrier migration/recombination. Film
response without the metasurface was subtracted to highlight resonances.

##  XVIII Optical grating periodicity versus pump wavelength

Figure S10-3: Micrographs of optical 1DGs: larger periodicities ($p_{y}$) are
notably formed for longer excitation wavelengths $\lambda_{pump}$ for the same
incidence angle $\theta=58^{\circ}$.

##  XIX Optical response versus the pump pulse energy

Figure S11-3: Temporal evolution of $\varepsilon$-qBICs as a function of the
pump pulse energy per arm (indicated in the insets), $\lambda_{pump}=$ 420 nm.
Resonances red-shift towards the symmetry-protected BIC state as the pump
power is reduced.

Figure S12-3: Temporal evolution of $\varepsilon$-qBICs as a function of the
pump pulse energy per arm (insets), $\lambda_{pump}=$ 410 nm. Similar to the
$\lambda_{pump}=$ 420 nm case, resonances red-shift towards the symmetry-
protected BIC state and decrease in amplitude modulation as the pump power is
reduced.

Figure S13-3: Spectra (left) and respective $Q$-factors (right) of
$\varepsilon$-qBICs as a function of the pump pulse energy per arm ($E_{p}$ in
nJ, indicated on the right hand side of the corresponding spectrum),
$\lambda_{pump}=$ 410 nm. Resonances blue-shift away from the symmetry-
protected BIC state and increase in transmittance modulation for larger pump
powers, being 80 nJ per pump arm the upper power limit of our setup for this
wavelength.

##  XX $Q$ versus the asymmetry parameter for a lossless metasurface

Figure S14-3: Finite element method (FEM)-calculated $Q$ of a lossless
metasurface (triangles) vs the negative (due to a decreasing permittivity with
increasing free-carrier concentration) of the asymmetry parameter
$\alpha_{\varepsilon}$, in agreement with the theoretical prediction (dashed
grey line, $Q\propto 1/\alpha_{\varepsilon}^{2}$).

##  XXI Fitting of differential transmittance data

Figure S15-3: Fitting of differential transmittance for different pump pulse
energies (insets), $\lambda_{pump}=$ 420 nm. A Breigt-Wigner-Fano (BWF)
function $\frac{\Delta
T}{T}=T_{0}+\frac{H\big{(}1+\frac{\lambda-\lambda_{c}}{qw}\big{)}^{2}}{1+\big{(}\frac{\lambda-\lambda_{c}}{w}\big{)}^{2}}$
was used to fit (dashed lines) the data (open squares), where $H$ is the
height, $q$ is the Fano asymmetry parameter, $\lambda_{c}$ is the central
wavelength of the resonance and $w$ its width. The resulting $Q$-factor is
calculated as $Q=\lambda_{c}/2w$.

Figure S16-3: Fitting of differential transmittance for different pump pulse
energies (insets), $\lambda_{pump}=$ 410 nm. A Breigt-Wigner-Fano (BWF)
function was used to fit (dashed lines) the data (open squares).

##  XXII Permittivity data of silicon films

Figure S17-3: Real ($\varepsilon_{r}^{Si}$, solid line) and imaginary
($\varepsilon_{i}^{Si}$, dashed line) parts of the permittivity of the
fabricated a-Si films as a function of the wavelength, measured via
ellipsometry.

##  XXIII Permittivity and permittivity asymmetry as a function of the pump
pulse energy

Figure S18-3: Real (multiplicative, left) and imaginary (additive, center)
permittivity terms of bright and dark fringes of the metasurface as a function
of the pump pulse energy per arm ($E_{p}$). (right) $-\alpha_{\varepsilon}$
versus $E_{p}$. Values were obtained from the model shown in Eqn. 2 of the
main manuscript. The parameters values used in model are
$\varepsilon_{r,P\rightarrow\infty}=0.6$, $\varepsilon^{vM}_{i}=0.25$,
$\varepsilon_{i,P\rightarrow\infty}=1.15$, $k_{b}=0.04$/nJ and
$k_{d}=0.001$/nJ. $-\alpha_{\varepsilon}$ decreases for large pulse energies
due to an absorption saturation of the bright fringes.

##  XXIV Spectrum versus pump pulse energy

Figure S19-3: Finite-difference time-domain (FDTD) method-calculated
transmittance spectra as a function of the pump pulse energy ($E_{p}$) per arm
(shown on the right). Permittivity parameters used in calculations are the
same as stated in the caption of Fig. 2 of the main manuscript.

##  XXV Optical response versus the polarization of one of the pump beams

Figure S20-3: Effect on the $\varepsilon$-qBICs of the relative polarization
(shown in the insets) between the pump beams, controlled via a half-wave plate
positioned in one of the pump arms (as shown in Fig. 1c of the main
manuscript). As the relative polarization is increased towards an orthogonal
configuration, the optically-induced metasurface generation is hindered, and
the mode excitation is less effective. Measurements performed at
$\lambda_{pump}=$ 420 nm and 130 nJ pulse energy per arm.

Figure S21-3: Temporal cross-sections at $t=$ 0 ps of modulated transmittance
as a function of the relative polarization between the pump beams (selected
angles shown along curves as inset).

##  XXVI Temporal evolution of $\varepsilon$-qBICs as a function of the probe
polarization

Figure S22-3: Temporal evolution of $\varepsilon$-qBICs as a function of the
probe polarization (insets) relative to the $x$-axis of the metasurface (along
the ridges). Coupling to the $\varepsilon$-qBIC reduces as the relative
polarization angle increases. When only one of the pump arms illuminates the
thin film (a condition in which no metasurface is formed) no resonance is
observed, even at a 0∘ relative polarization. Measurements performed at pump
wavelength $\lambda_{pump}=$ 420 nm and 16 nJ pulse energy per arm.

Figure S23-3: Temporal cross-sections at $t=$ 0 ps of differential
transmittance versus probe polarization (inset) relative to the $x$-axis of
the metasurface (along the ridges, see Fig. 1 of the main manuscript).
Coupling to the $\varepsilon$-qBIC reduces as the relative polarization angle
increases.

##  XXVII Third-harmonic generation for different pump beams configurations

Figure S24-3: (top) Third harmonic generation (THG) signal versus delay line
position (in ps) for different pump configurations (insets).
$\lambda_{fundamental}=$ 1020nm. THG enhancement is observed following the
formation of the metasurface (0∘, left) and the excitation of an
$\varepsilon$-qBIC, while a nearly constant signal is obtained for cross-
polarized (90∘, center) and in the absence of the pump beams (Off, right).
Spot at $\approx$ 343nm in the left and central panels corresponds to the
four-wave mixing signal (FWM,
$\nu_{fwm}=\nu_{515nm}+\nu_{515nm}-\nu_{1020nm}$), a process due to the
overlap of the ultrafast pump and probe beams. (bottom) Time trace of the THG
signal at 339 nm for the corresponding configurations. THG enhancement decays
exponentially (dashed grey line, $1/e\approx$ 0.72 ps), along with the quasi-
symmetry-protected resonance excited at low pump powers, not being a process
limited by the Gaussian overlap of ultrafast beams. For cross-polarized pump
beams (90∘, center), a slight decrease in the THG signal is observed,
attributed to increased losses in the film induced by the absorption of the
pump beams. Pump pulse energy $E_{p,515nm}=$ 75 nJ per arm, probe pulse energy
$E_{p,1020nm}=$ 5 nJ.

Figure S25-3: Third harmonic generation (THG) signal at higher probe pulse
energy versus delay line position (in ps) for different pump configurations
(insets). $\lambda_{fundamental}=$ 1020nm. (bottom) Corresponding THG signal
as a function of time at $\lambda_{THG}=$ 339 nm. A similar trend observed at
low powers is obtained at high powers, with a faster exponential decay of the
THG enhancement process (grey dashed line, $1/e\approx$ 0.24 ps).
$E_{p,515nm}=$ 50 nJ per arm, $E_{p,1020nm}=$ 9 nJ.

##  XXVIII Third-harmonic generation versus probe beam polarization

Figure S26-3: THG and FWM signals vs delay line position (in ps) as a function
of the relative polarization between the probe and the pump beams (inset).
$\lambda_{fundamental}=$ 1020nm. The efficiency of THG (centered at $\approx$
339 nm) decreases as the relative polarization angle between the probe and the
pump beams increases. Red spot along the THG vertical strip corresponds to the
excitation of an $\varepsilon$-qBIC and the enhancement of the THG signal. The
efficiency of FWM ($\nu_{fwm}=\nu_{515nm}+\nu_{515nm}-\nu_{1020nm}$, at
$\approx$ 343 nm), on the other hand, increases with increasing relative
polarization. When only one pump arm is used, the efficiency of THG increases
(relative to the 90∘ case, due to smaller losses), albeit with no enhancement,
and no FWM signal is observed. $E_{p,515nm}=$ 50 nJ per arm, $E_{p,1020nm}=$ 9
nJ.

Figure S27-3: TGH and FWM signals as a function of the relative polarization
betweem pump and probe beams. A reduction of the THG signal (peak at $\approx$
339 nm) occurs for larger relative polarizations while the red-shifted
competing FWM process output increases. $E_{p,515nm}=$ 50 nJ per arm,
$E_{p,1020nm}=$ 9 nJ.

##  XXIX Third-harmonic generation versus probe beam wavelength

Figure S28-3: THG and FWM signals vs probe wavelength (inset) for a 0∘ (blue)
and a 90∘ (dashed red) relative polarization between pump beams
($\lambda_{pump}=$ 515 nm). THG and FWM red- and blue-shift, respectively
(crossing at $\lambda_{probe}=$ 1030 nm), for longer probe wavelengths. Note
the logarithmic scale in the $y$-axis (Counts).

Figure S29-3: THG and FWM signals vs delay line position (in ps) as a function
of the probe wavelength (inset). Here a 0∘ relative polarization between the
pump beams is used (thus forming the metasurface). Note the logarithmic scale
in the colormap (counts). $E_{p,515nm}=$ 50 nJ per arm, $E_{p,probe}=$ 9 nJ.
The simultaneous arrival of beams (red spot along the THG vertical strip)
changes in time for $\lambda_{probe}=$ 1040 nm and $\lambda_{probe}=$ 1070 nm
as the OPA changes its output beam between signal and idler. The fainter
vertical stripes corresponds to both signal and idler beams being present,
leading to two closely-spaced THG signals.

Figure S30-3: THG and FWM signals vs delay line position (in ps) as a function
of the probe wavelength (inset). Here a 90∘ relative polarization between the
pump beams is used (absence of the metasurface), reducing the harmonic
generation efficiency. Note the logarithmic scale in the colormap (Counts).
$E_{p,515nm}=$ 50 nJ per arm, $E_{p,probe}=$ 9 nJ.

Figure S31-3: THG and FWM signals vs delay line position (in ps) as a function
of the probe wavelength (inset) in the absence of the pump beams. As the 515
nm photons are not present, the FWM signal vanishes. Note the logarithmic
scale in the colormap (Counts). $E_{p,probe}=$ 9 nJ.