# Highly confined incident-angle-robust surface phonon polariton bound states
in the continuum metasurfaces

Lin Nan<EMAIL_ADDRESS>[  Andrea Mancini<EMAIL_ADDRESS>muenchen.de [  Thomas Weber  [  Geok Leng Seah  [  Emiliano Cortés  [  Andreas
Tittl<EMAIL_ADDRESS>[  Stefan A. Maier  [

###### Abstract

Squeezing light into subwavelength dimensions is vital for on-chip integration
of photonic technologies. One approach to overcome the diffraction limit is
coupling light to material excitations, leading to polariton states. Here, we
showcase how low-loss mid-infrared surface phonon polaritons enable
metasurfaces supporting quasi-bound states in the continuum (qBICs) with
deeply subwavelength unit cells. Utilizing $100\text{\,}\mathrm{nm}$ thick
free-standing silicon carbide membranes, we achieve highly confined qBIC
states with a unit cell volume $\sim 10^{4}$ times smaller than the
diffraction limit. This results in remarkable robustness of the platform
against the incident angle that is unique among qBIC systems. We also
demonstrate vibrational strong coupling with a thin layer of spin-coated
molecules, leveraging the small mode volume. This work introduces phononic
qBICs as a novel ultra-confined nanophotonic platform, paving a way for the
miniaturization of mid-infrared devices for molecular sensing and thermal
radiation engineering.

Equal] These authors contributed equally Chair in Hybrid Nanosystems,
Nanoinstitute Munich, Faculty of Physics, Ludwig-Maximilians-Universität
München, 80539 München, Germany Equal] These authors contributed equally Chair
in Hybrid Nanosystems, Nanoinstitute Munich, Faculty of Physics, Ludwig-
Maximilians-Universität München, 80539 München, Germany Equal] These authors
contributed equally Chair in Hybrid Nanosystems, Nanoinstitute Munich, Faculty
of Physics, Ludwig-Maximilians-Universität München, 80539 München, Germany
NTU] School of Materials Science Engineering, Nanyang Technological
University, 639798, Singapore, Singapore LMU] Chair in Hybrid Nanosystems,
Nanoinstitute Munich, Faculty of Physics, Ludwig-Maximilians-Universität
München, 80539 München, Germany LMU] Chair in Hybrid Nanosystems,
Nanoinstitute Munich, Faculty of Physics, Ludwig-Maximilians-Universität
München, 80539 München, Germany Australia] School of Physics and Astronomy,
Monash University, Clayton, Victoria 3800, Australia Department of Physics,
Imperial College London, London SW7 2AZ, United Kingdom

##  1 Introduction

Borrowed from a concept developed in quantum mechanics 1, photonic bound
states in the continuum (BICs) are localized states with formally infinite
lifetimes at energies within the radiation continuum 2, 3. While true BICs are
dark modes, coupling to far-field radiation can be achieved by geometrical
detuning of the structure from the BIC state, opening a radiative channel
leading to the emergence of so-called quasi-bound states in the continuum
(qBICs). Of particular interest are “symmetry-protected” qBICs within
metasurfaces, where coupling to the continuum results from the deliberate
symmetry breaking of the unit cell structure. These qBICs boast remarkable
attributes, including high and tunable quality factors ($Q$ factor) controlled
by the asymmetry parameter 4, 5, readily adjustable resonance frequency
through geometric size scaling 6, and robustness against fabrication errors 7,
8. These compelling attributes have driven diverse applications of symmetry-
protected qBICs, including but not limited to mid-infrared vibrational
spectroscopy 6, 9, 10, refractive index sensing 11, 12, enhanced light-matter
interactions 13, 14, photocatalysis 15, lasing 16, 17, 18, and thermal
radiation engineering 19, 20

The $Q$ factor of qBICs is governed by both the radiation leakage (depending
on the degree of asymmetry) and the inherent non-radiative decay due to
material absorption. To enhance the $Q$ factor, the simplest strategy is to
utilize low-loss dielectrics with negligible intrinsic dissipation 6, 21, 14.
Recent endeavors have introduced plasmonic qBICs to amplify surface field
enhancement, albeit with reduced $Q$ factors owing to increased losses from
electron-electron scattering in noble metals 22, 10. Polar dielectrics
supporting surface phonon polaritons (SPhPs) can effectively combine the
benefits of both material platforms: high field enhancement due to the
excitation of surface polaritons in the Reststrahlen (RS) band, and high $Q$
factors due to the slow phonon-phonon scattering mechanism determining their
decay 23, 24, 25, 26, 27, 28. Furthermore, qBICs in both dielectric and
plasmonic metasurfaces typically feature lateral unit cell sizes on the order
of the resonance wavelength, hindering their miniaturization 29, 30, 31. While
deeply sub-diffractional unit cells volumes
($V_{\mathrm{diff}}=(\lambda/2)^{3}$ as imposed by the diffraction limit) are
in principle feasible in plasmonic qBICs, practical realization is impeded by
high losses in the high confinement region where
$\operatorname{Re}(\varepsilon)\lesssim 0$. Owing to the intrinsic longer
SPhPs lifetimes, “phononic” qBICs can conversely efficiently operate in the
sub-wavelength regime at frequencies where
$\operatorname{Re}(\varepsilon)\lesssim 0$.

In this work, we demonstrate for the first time deeply sub-wavelength SPhPs-
driven symmetry-protected qBICs metasurfaces with a minimum unit cell volume
that is $10^{4}$ times smaller than the free-space diffraction limited volume
$V_{\mathrm{diff}}\sim\lambda^{3}/10$ at the resonant frequency. We fabricate
the structures on commercially available large-area (up to few millimeters)
free-standing CMOS-compatible silicon carbide (SiC) membranes 32, 33, where
the $Q$ factor is maximized thanks to the symmetric environment provided by
the absence of a solid substrate 34. We confirm the qBIC character by
amplitude and phase resolved maps obtained through transmission scattering-
scanning near-field optical microscopy (sSNOM). We further investigate the
unique robustness of phononic qBICs against tilt in the illumination angle,
which heavily shifts the resonance frequency in dielectric and plasmonic
systems. Additionally, we demonstrate enhanced mid-infrared (IR) light-matter
interactions via surface-enhanced IR absorption and vibrational strong
coupling between the qBIC resonance and a spin-coated polymer. Polyethylene
glycol (PEG) is chosen as the target molecule as it features a fingerprint
vibrational mode lying within the SiC RS. Our work introduces phononic qBICs
as a novel nanophotonic platform and paves the way for its implementation in
mid-IR applications such a as molecular sensing 35, 36, 37 and thermal
radiation engineering 38, 39, 40.

##  2 Results

To highlight the advantages brought by our BIC-driven SiC phononic
metasurface, we compare it with analogous dielectric and metal structures,
namely silicon (Si) and gold (Au) supporting a qBIC resonance at the same
wavelength ($\sim$ $11\text{\,}\mathrm{\SIUnitSymbolMicro m}$). We consider
metasurfaces made of elliptical perforations milled in a film (either Si, Au
or SiC) where the radiative channel to enable far-field coupling of the qBIC
is activated by the tilting of the ellipses with respect to each other 6. The
most striking difference between the three materials is in the unit cell
volume as shown in Fig. 1a, where the SiC structures are remarkably smaller.
The real part of the permittivity for the dielectric, plasmonic, and phononic
cases is shown in Fig. 1b. For the dielectric metasurface (with
$\operatorname{Re}(\varepsilon_{\mathrm{Si}})\sim 10$), the lateral size of
the unit cell, although sub-wavelength, is typically comparable to the
resonance wavelength, and its minimum thickness is on the order of half a
micron. The plasmonic, i.e., gold structure, allows for a thickness reduction
by one order of magnitude. However, the lateral size remains the same as the
dielectric metasurface because at mid-infrared frequencies noble metals behave
almost as perfect electric conductors 41
($\operatorname{Re}(\varepsilon_{\mathrm{Au}})\ll 0$). On the other hand, our
$100\text{\,}\mathrm{n}\mathrm{m}$ thick SiC metasurface supports qBIC
resonances even for deeply sub-wavelength unit cells due to the negative,
near-zero value of the permittivity
($\operatorname{Re}(\varepsilon_{\mathrm{SiC}})\lesssim 0$), allowing extreme
miniaturization of the device. The quantitative comparison of the unit cell
volume obtained from full-three dimensional electromagnetic simulations in
Fig. 1c (see Methods for more details) shows that the Au unit cell is an order
of magnitude smaller than that of Si, while the SiC unit cell is a further
order of magnitude smaller. Moreover, the maximum field enhancement at
resonance is approximately twice as large in both the plasmonic and phononic
metasurfaces compared to that reached in the dielectric structure (Fig. 1d).

Figure 1: Highly confined surface phonon polariton driven qBICs. a Sketch of
the unit cell for inverse metasurfaces made of Si (dielectric), Au (metal) and
SiC (polar dielectric) showing a qBIC resonance at
$11\text{\,}\mathrm{\SIUnitSymbolMicro m}$. b Real part of the permittivity
for Si, Au, and SiC. Comparison of the unit cell volume c and maximum field
enhancement d for the three metasurfaces at the qBIC resonance (the inset in c
shows zoomed-in view of only Au and SiC). e Simulated spectra for a SiC
“negative” metasurface with different ellipses tilting angles $\theta$ under
excitation with polarization along the $y-$ (blue curves) and $x-$ (red
curves) directions. The meaning of “negative” unit cell is illustrated on the
right. f Coating the SiC metasurface with a thin film of molecules having a
vibrational mode matching the qBIC frequency induces a splitting of the
transmission spectra as a result of enhanced light-matter coupling.

Simulated transmission spectra for a SiC metasurface at various ellipses
tilting angles $\theta$ are shown in Fig. 1e. Each spectrum is computed for
two polarizations, aligned with the two orthogonal sides of the unit cell.
While a thick SiC layer (several microns) is highly reflective in the RS band
24, 27, our $100\text{\,}\mathrm{nm}$ membranes are below the skin depth and
thus highly transparent 32, except for a dip around
$800\text{\,}{\mathrm{cm}}^{-1}$ due to the TO phonon absorption. In the band
from $875\text{\,}{\mathrm{cm}}^{-1}$ to $925\text{\,}{\mathrm{cm}}^{-1}$, we
observe a series of dips for both polarizations and for all $\theta$, which we
associate with modes existing in the single elliptical perforations. At
$830\text{\,}{\mathrm{cm}}^{-1}$, we observe a resonance that is associated
with a qBIC as it disappears for
$\theta=$0\text{\,}\mathrm{\SIUnitSymbolDegree}$$ and exists only for one of
the polarizations 6, 7, 21. In conventional “positive” metasurfaces (i.e. made
of antennas placed on a substrate), the net dipole moment, $p_{\mathrm{net}}$,
induced by the antennae tilting is along the $x-$direction of the unit cell
due to the difference in the gap distances. Consequently, to excite the qBIC,
the polarization of the excitation beam has to be along the $x-$direction.
According to Babinet’s principle, in a “negative” metasurface (i.e. holes in
an extended film), the polarization has to be flipped by
$90\text{\,}\mathrm{\SIUnitSymbolDegree}$. Therefore, in our structures, the
qBIC is activated when the electric field is along the $y-$direction 30.

As a proof-of-concept application of phononic qBICs, we show how our
structures can be used to enhance light-matter interactions at mid-IR
frequencies. In Fig. 1f, we sketch the effect of coating the metasurfaces with
a thin layer of molecule with a vibrational mode matching the qBIC resonance.
The transmission spectrum of the target molecule, PEG in our case, is
represented by a solid green curve denoted as $\omega_{\mathrm{vib}}$, while
the spectrum of the bare metasurface is shown in purple. The coupling between
the molecule and the metasurface results in a dip in the spectrum, indicated
by the dashed curve, which corresponds to the absorption of the vibrational
mode. This dip is commonly observed in surface-enhanced infrared absorption
(SEIRA) 42, 43, 44. In the case of very efficient light-matter coupling, the
regime of strong-coupling can be achieved, resulting in mixed light-matter
states where a clear distinction between the two modes cannot be made anymore
45, 36, 46, 47, 48, 13. In the conventional definition, the strong coupling
regime is reached when the energy splitting between the two resonances is
larger than their individual linewidths.

Figure 2: Far- and near-field characterization of SPhPs-driven qBICs. a
Illustration of the metasurface unit cell with relevant geometrical parameters
(top) and tilted SEM image of a portion of a fabricated array (bottom). Scale
bar: $500\text{\,}\mathrm{n}\mathrm{m}$. b Sketch of the polarization
direction in the transmission FTIR measurements. c Simulated (top) and
experimental (bottom) far-field transmission spectra for a metasurface with
$S=1$ and varying tilting angle $\theta$. d Radiative $Q$ factor extracted
from the TCMT fit of the experimental data in c. e Sketch of the sSNOM setup
for the near-field mapping. f Simulated (top) amplitude (left) and phase
(right) of the out-of plane component of the surface electric field at the
qBIC resonance frequency. The near-field amplitude and phase measured with the
sSNOM are shown in the bottom. g Same as in f, but at a frequency of a higher
order mode that does not show a qBIC character. Scale bar:
$500\text{\,}\mathrm{n}\mathrm{m}$.

We confirm the realization of qBIC states in SiC metasurfaces through a
combination of far-field and near-field experiments. The fabrication process
begins with a standard electron-beam lithography step for the creation of a
chromium hard mask, followed by a reactive ion etching treatment to mill holes
into the membranes (see Methods for more details). A sketch of a single unit
cell with relevant geometrical parameters is shown in Fig. 2a together with a
SEM image of a portion of one of the fabricated arrays. The long and short
axes of the ellipses are set to be $a=$1190\text{\,}\mathrm{n}\mathrm{m}$$ and
$b=$560\text{\,}\mathrm{n}\mathrm{m}$$, while the edges of the unit cell are
$P_{\mathrm{x}}=$2100\text{\,}\mathrm{n}\mathrm{m}$$ and
$P_{\mathrm{y}}=$1610\text{\,}\mathrm{n}\mathrm{m}$$. These dimensions are
defined as the scaling factor $S=1$. While adjusting the thickness of the
material offers a means to tune the position of the qBIC resonance frequency
(see Supporting Information LABEL:SiC_200nm), altering the scaling factor is a
more convenient and efficient method for tuning the resonance of qBIC
metasurfaces.6, 21. The theoretical spectra shown in Fig. 1e is validated
through transmittance measurement on our structures with Fourier-transform
infrared (FTIR) microscopy (see Methods section for more details). To
demonstrate the existence of the qBIC resonance, we maintain all geometrical
parameters of the unit cell constant but solely vary the tilting angle
$\theta$ between the elliptical holes. Following Babinet’s principle, we
polarize the electric field of the incident beam along the short axis of the
unit cell 30 as sketched in Fig. 2b. When changing the ellipse tilting angle
$\theta$, we observe a series of dips in both simulated (top) and experimental
(bottom) transmission spectra in Fig. 2c. The lowest-energy dip is associated
with the TO-phonon absorption from the bare SiC membrane. At higher energy,
the qBIC resonance can be identified as it becomes optically active only for
$\theta>$0\text{\,}\mathrm{\SIUnitSymbolDegree}$$. As expected, we observe an
increase of the modulation depth and linewidth of the resonance with
increasing $\theta$ due to the strengthening of the radiative dissipation
channel introduced by the structure asymmetry. Higher-energy dips are
associated with higher order resonances of the elliptical meta-atoms as they
are present even at $\theta=$0\text{\,}\mathrm{\SIUnitSymbolDegree}$$. The
close match between experimental and simulated spectra confirms the accuracy
of our theoretical model and interpretation. Interestingly, we observe a
narrowing of the TO phonon band-width with increasing $\theta$ and qBIC
modulation depth (see Supporting Information LABEL:Unit_cell_modulation),
indicating the presence of coupling between the two.

To further confirm the qBIC characteristic of the resonance, we fit the
experimental spectra with a temporal coupled mode theory (TCMT) model (for
more details see Supporting Information LABEL:TCMT). From this fit, we can
extract the radiative part of the resonance $Q$ factor $Q_{\mathrm{rad}}$ as
shown in Fig. 2d. As a general feature of symmetry-protected BIC-driven
metasurfaces, $Q_{\mathrm{rad}}$ is proportional to the inverse square of the
asymmetry parameter, which for a metasurface made of elliptical antennas can
be defined as $\sin^{2}(\theta)$ 4. The $Q_{\mathrm{rad}}$ values extracted
from the TCMT model fits well with the $1/\sin^{2}(\theta)$ function,
confirming the qBIC character of the resonance.

To study the field-enhancement distribution and ultimately confirm the qBIC
nature of the observed resonance, we carry out nanoscale mapping with a
scattering scanning near-field optical microscope (sSNOM) as sketched in Fig.
2e. We perform the measurements in transmission mode, which is ideal for the
investigation of resonators due to reduced tip-sample coupling and normal-
incidence illumination 49, 50. Briefly, a mid-infrared beam from a tunable
mid-IR laser source is focused from below by a parabolic mirror on the tip of
an atomic force microscope (AFM). The tip scatters the incident light, which
is collected by an off-axis upper parabolic mirror. The signal is demodulated
at higher harmonics $n\geq 2$ of the tip oscillation frequency for far-field
background suppression and is analyzed with a pseudo-heterodyne interferometer
for extraction of the amplitude and phase response (more details in the
Methods).

For the near-field mapping, we use a metasurface with $S=1$ and
$\theta=$30\text{\,}\mathrm{\SIUnitSymbolDegree}$$, of which the far-field
response is included in Fig. 2c. Results at frequencies where the structure
supports a qBIC ($\omega=$834\text{\,}{\mathrm{cm}}^{-1}$$) and a single
elliptical hole resonance ($\omega=$877\text{\,}{\mathrm{cm}}^{-1}$$) are
reported in Fig. 2f and g respectively. For each frequency, we show in the top
row the simulated amplitude (left) and phase (right) of the out-of plane
component of the field $E_{z}$ at the SiC surface. The corresponding near-
field data (demodulation order $n=2$) are shown in the bottom row,
demonstrating good agreement between simulated and experimental maps. A small
discrepancy can be seen in the experimental amplitude maps, where high field
hot-spots appear at the antenna gap that have no counterpart in the
simulation. We attribute this effect to the combination of non-perfect normal
illumination and residual tip-sample coupling which can impact the measured
amplitude, but have negligible effect on the phase profile. We use the
topography to mask the optical data in the region of the perforations, where
the mechanical tip-sample interaction is unreliable due to the absence of a
substrate. Interestingly, we can clearly distinguish the qBIC resonance in
Fig. 2f as the phase jumps are in the $x-$direction, which is perpendicular to
the polarization of the exciting beam. For the non-qBIC mode shown in Fig. 2g,
the phase jumps are, as expected for a dipolar-type mode, along the
polarization direction (in the $y-$direction). The combination of far-field
and near-field measurements shown in Fig. 2 allows us to characterize and
unequivocally confirm the qBIC nature of the deeply sub-wavelength resonances
we engineer in SiC phononic metasurfaces.

Figure 3: Angle-robustness of deeply subwavelength qBICs. a Experimental setup
for transmission measurement with various sample tilting angles $\psi$. b
Simulated and experimental spectra for a metasurface with $S=1$ and
$\theta=$40\text{\,}\mathrm{\SIUnitSymbolDegree}$$ under effectively tilted
illumination at multiple angles from normal incidence to
$\psi=$50\text{\,}\mathrm{\SIUnitSymbolDegree}$$.

In conventional qBIC systems (both plasmonic and dielectric), the resonance
position is heavily dependent on the excitation beam incident angle 51, 30.
For this reason, refractive objectives with low numerical apertures are
generally required to obtain good modulation depths in qBIC systems. A
standout feature of our phononic qBIC is its dispersionless nature, which can
already be indirectly inferred from the pronounced resonance dips we observe
in our far-field spectra (see e.g. Fig. 2c) obtained with off-axis
illumination from a Cassegrain objective, the use of which generally degrades
the measured resonance modulation 30. To confirm this observation, we mount
our samples on a tilting stage allowing us to measure transmission spectra for
different incident angles with respect to the sample normal direction as shown
in Fig. 3a. To ensure that we probe the metasurfaces response with a spread of
angles averaged over a single value, we block half of the upper objective.
Strikingly, simulations and measurements for a metasurface with $S=1$ and
$\theta=$40\text{\,}\mathrm{\SIUnitSymbolDegree}$$ demonstrate the clear
independence of all the modes to the illumination angle $\psi$ (Fig. 3 b).
Especially, the consistent qBIC resonance positions and their unvarying
modulation depths underscore the superior angle-robustness of our system. The
only appreciable variation of the spectra can be seen around
$970\text{\,}{\mathrm{cm}}^{-1}$ due to the excitation of the so-called
Brewster mode at the epsilon-near-zero point 52. This interpretation is
confirmed by the appearance of this transmission dip only for TM-polarized
excitation (for more details see Supporting Information LABEL:TM_TE). We
attribute the angle-robustness of our design to the uniquely deeply sub-
wavelength character of the phononic qBIC, which implies neighboring unit
cells are illuminated with an almost identical phase even for highly tilted
illumination. We numerically confirm that a positive phononic qBIC metasurface
is also angle independent, while an analogous metallic metasurface is strongly
dispersive both in the positive and negative configurations (see Supporting
Information LABEL:Au_SiC_sim_angle). This observation excludes the attribution
of the angle-independence to the particular metasurface geometry and points to
the generality of the phenomena for sub-wavelength unit cells.

Figure 4: Vibrational strong coupling of SiC qBICs with PEG. a Experimental
and simulated transmission spectra of bare SiC metasurfaces with scaling
factors ranging from 0.4 to 1.5. b Schematic of vibrational strong coupling.
The BIC resonance is spectrally overlapped with the absorption band of PEG,
resulting in a splitting of the BIC mode into upper and lower polaritonic
states. c Experimentally and simulation resolved anti-crossing of with a
thickness of $30\text{\,}\mathrm{nm}$ PEG covered SiC metasurfaces. The dotted
lines denote the polariton dispersion calculated with extracted parameters
from TCMT fitting and a Rabi splitting of $11.5\text{\,}{\mathrm{cm}}^{-1}$.

The unique combination of deeply sub-diffractional unit cell size and high
field enhancement of our phononic qBICs can be exploited to boost light-matter
interactions in the mid-IR. As the near-field coupling strength between
photonic modes and material excitations is inversely proportional to the
square root of the mode volume 53, qBIC metasurfaces with extremely small
effective mode volumes ($V_{\mathrm{eff}}$) are an ideal platform for highly
efficient light-matter interactions. We select polyethylene glycol (PEG) as a
prototype molecule that features a characteristic C-H bending mode at
$\omega=$842\text{\,}{\mathrm{cm}}^{-1}$$ matching SiC’s RS band. PEG is a
technologically relevant water-soluble polyether compound widely employed in
pharmaceutical, cosmetics and manufacturing industries 54. In order to match
the qBIC and PEG resonance frequencies, we study the tunability of our
metasurfaces via scaling factor tuning, where all the unit cell geometrical
parameters are scaled by a single multiplicative factor $S$. As the phononic
qBIC resonance can only exist inside the material RS band, we observe in Fig.
4a a highly nonlinear frequency shift in the experimental (left) and simulated
(right) transmission spectra of a metasurface with
$\theta=$40\text{\,}\mathrm{\SIUnitSymbolDegree}$$. The qBIC resonance
positions can only asymptotically approach the TO-phonon even for large $S$
values and analogously are bounded from above by the LO-phonon energy. This
second effect is not addressed in our experimental study as saturation of the
qBIC resonance position for decreasing $S$ is appreciable only for very small
unit cell sizes below the resolution limit of standard nanofabrication methods
(see Supporting Information LABEL:small_UC_limit). The smallest metasurface
fabricated for this study corresponds to $S=0.25$. Importantly, the qBIC
tunability extends both above and below the PEG vibrational mode, allowing a
comprehensive investigation of the light-matter coupling strength.

To enable effective near-field interaction, we cover our SiC metasurfaces with
a layer of PEG with an estimated thickness of $30\text{\,}\mathrm{nm}$ by spin
coating (see Supporting Information LABEL:PEG_thickness). The effect of the
PEG layer on the qBIC resonance is shown in Fig. 4b. When the frequencies of
both modes overlap, a splitting of the qBIC dip can be observed in the
transmission spectra, corresponding to the creation of two polariton branches
of different energies. For numerical modeling and fitting of the coupled
states, the dielectric function of PEG was calculated from Kramers-Kronig
consistent fits to transmission measurements of films of different thicknesses
on a CaF2 substrate. The extracted linewidth of the PEG absorption band was
$\gamma_{\mathrm{PEG}}=$3.59\text{\,}{\mathrm{cm}}^{-1}$$, which we fix for
the TCMT fits used to extract the coupling strength. In Fig. 4c, we report
experimental (left) and simulated (right) transmission spectra for the coupled
system when varying the metasurface scaling factor. Two changes in the optical
response of our SiC metasurface can be observed: (i) a spectral red-shift of
the qBIC resonances due to the change in the local refractive index of the
environment, resulting in a reduction of the scaling factor needed to achieve
spectral overlap between qBIC and PEG, and (ii) a band-splitting at the
absorption peak of PEG around $842\text{\,}{\mathrm{cm}}^{-1}$ into upper and
lower polaritonic branches as illustrated in Fig. 4b. Significantly, the
absorption fingerprint of the molecule on the metasurfaces is strongly
enhanced in comparison to the bare CaF2 substrates, which can be attributed to
the pronounced near-field enhancement of the qBIC mode. Detuning the qBIC mode
from the PEG absorption by varying the scaling factor $S$ reveals the typical
anti-crossing pattern characteristic of strongly coupled resonances. In the
numerical simulations we fill the elliptical holes of the membrane completely
and add a $30\text{\,}\mathrm{nm}$ thick layer on the top side of the
membrane, resulting in excellent agreement with our experimental results. To
quantify the strength of the vibrational coupling, we fit transmittance
spectra with our TCMT model (for more information on the fitting procedure see
Supporting Information LABEL:TCMT_vibrational_coupling), which allows for the
extraction of the dispersion and linewidth of the qBIC from coupled spectra,
which would be unattainable from mere polariton dispersion fits.

As can be seen in Fig. LABEL:Fig_SI_TCMT_cmp_res_pos_fits, the extracted qBIC
dispersion from simulations follows a similar quadratic and continuous
dependence on the scaling factor that we observe from experimental
measurements of the bare metasurfaces. Because the thickness of the PEG varies
slightly depending on the scaling factor in experiment, we instead approximate
the qBIC dispersion with a linear relationship. Furthermore, we simplify the
system by assuming a constant qBIC linewidth over the sampled scaling factor
range, which is consistent to commonly utilized fits to extract the Rabi
splitting of anti-crossing spectra. The damping rate of the qBIC is extracted
to be $\gamma_{\mathrm{BIC}}=$7.57\text{\,}{\mathrm{cm}}^{-1}$$ and the near-
field coupling strength at the point of spectral overlap between qBIC and PEG
to be $g=$6.08\text{\,}{\mathrm{cm}}^{-1}$$ via interpolation at a scaling
factor of $S=0.435$. In order for the system to be in the strong-coupling
regime, the exchange of energy between BIC and PEG needs to be faster than the
constituting resonances can dissipate energy into their individual loss
channels 53, 13. In other words, the frequency separation of the polariton
peaks (the vacuum Rabi splitting $2g$) must be larger than the sum of the
linewidths of BIC and PEG. This is commonly referred as the strong coupling
criterion $c_{1}=2g/(\gamma_{\mathrm{BIC}}+\gamma_{\mathrm{PEG}})>1$, which we
evaluate for our system to be $c_{1}=1.09$. Because of the mismatch of the BIC
and PEG linewidths, the peak splitting is actually given by the Rabi splitting
$\Omega_{\mathrm{R}}=2\sqrt{g^{2}-(\gamma_{\mathrm{BIC}}-\gamma_{\mathrm{PEG}})^{2}/4}=$11.5\text{\,}{\mathrm{cm}}^{-1}$$,
which is slightly lower than $2g$ and a stricter criterion is given by
$c_{2}=\Omega_{\mathrm{R}}/(\gamma_{\mathrm{BIC}}+\gamma_{\mathrm{PEG}})>1$
with $c_{2}=1.03$. Both criteria are fulfilled and show that we reach the
strong coupling regime. Finally, we want to remark that thanks to the low mode
volume of phononic qBICs we are able to reach the strong-coupling regime even
with the relatively weak vibrational peak of PEG when compared with molecules
routinely used for mid-IR vibrational strong-coupling such as SiO2 and PMMA
46, 47, 48, 55 (see Supporting Information LABEL:molecules_stronger_coupling).
These observations highlight the potential of phononic qBICs for the
investigation and exploitation of mid-IR strong coupling at the nanoscale.

##  3 Conclusions

In this work, we reported the pioneering realization of a phonon polariton-
mediated qBIC metasurface. The qBIC nature was unequivocally assessed through
comprehensive far- and near-field characterizations, supported by rigorous
numerical simulations. By utilizing a highly dispersive low-loss polaritonic
material, we achieved extreme miniaturization by shrinking the unit cell size
down to $\sim 1.5\cdot 10^{-4}$ of the volume $\lambda_{0}^{3}/10$ occupied by
the free-space wavelength. This remarkable miniaturization led to the
observation of a striking incident-angle invariant behavior on our
metasurfaces, a highly unconventional characteristic for a symmetry-protected
BIC metasurface 51, 30. This result paves a way for the utilization of
phononic qBIC metasurfaces, and sub-wavelength polaritonic systems in general,
in applications requiring consistent performances over a wide range of
incident angles such as in stealth technologies 56 and omnidirectional
radiative cooling 57, 58. Moreover, the strong confinement of the mode volume
facilitated the emergence of vibrational strong coupling between a weak
resonator, exemplified by PEG, and our qBIC system. While phonon polariton
vibrational strong coupling has been shown with hBN antennas obtained from
exfoliated flakes 36, we report here its first realization in a large-area
CMOS-compatible platform. The use of SiC is therefore suited for direct
applications in SEIRA 43, 44, nonlinear mid-IR optics 59, 60 and for the
control of chemical-reaction pathways through strong-coupling engineering 61,
62 at the nanoscale.

## Methods

### Sample preparation

Metasurfaces were fabricated on free-standing
$100\text{\,}\mathrm{n}\mathrm{m}$ SiC membranes supported by Si frame that we
purchased from Silson Ltd. Fabrication follows the conventional top-down
nanofabrication method. First, the electron-beam resist, SX AR-N 8200.06, was
exposed with
$10\text{\,}\mathrm{m}\mathrm{i}\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{n}$
aperture under $30\text{\,}\mathrm{K}\mathrm{V}$ acceleration voltage,
followed by the deposition of chromium as a hard-mask for reactive ion etching
(RIE) via electron-beam deposition. SF6 based RIE was performed to perforate
the SiC membrane followed by wet-etching process for the removal of the hard-
mask.

For the vibrational coupling measurements, 2000K PEG was purchased from Sigma
aldrich and dissolved into methoxybenzene to achieve
$12.5\text{\,}\mathrm{m}\mathrm{g}\mathrm{/}\mathrm{m}\mathrm{L}$. The PEG
solution was spin-coated onto the suspended SiC metasurfaces. The PEG
thickness on SiC metasurfaces was estimated by interpolating the thickness-
absorption correlated curve acquired from ellipsometry and Fourier transform
infrared (FTIR) spectroscopy on bulk substrates (for more detials, see
Supporting Information LABEL:PEG_thickness.)

### Fourier Transform Infrared spectroscopy

Transmission spectra of the metasurfaces were acquired using a VERTEX 80v FTIR
spectrometer (Bruker) in conjunction with a HYPERION 3000 microscope (Bruker).
Measurements were performed utilizing a reflective microscope objective
(Newport), also known as Cassegrain objective, featuring a numerical aperture
(NA) of 0.4 and 15× magnification covering a range of polar angles from
$12\text{\,}\mathrm{\SIUnitSymbolDegree}$ to
$23.6\text{\,}\mathrm{\SIUnitSymbolDegree}$. A liquid nitrogen-cooled mercury
cadmium telluride detector recorded the spectra in transmission mode. The
sample tilting was achieved by mounting the sample on a custom-made sample
holder that can be rotated along central axis that is perpendicular to the
effective beam path. Custom-made aperture was added onto the objective over
the half of the transmitted beam for the tilted measurements.

### Transmission sSNOM

Near-field spectra were obtained with a commercial sSNOM set-up (Neaspec)
equipped with a pseudo-heterodyne interferometer to obtain amplitude and phase
resolved images. The light source used in the experiments was an optical
parametric oscillator (OPO) laser (Stuttgart Instruments) powered by a pump
laser at $\lambda=$1035\text{\,}\mathrm{nm}$$ with $40\text{\,}\mathrm{MHz}$
repetition rate and $\approx$500\text{\,}\mathrm{fs}$$ pulses. The linearly
polarized MIR output was obtained by difference frequency generation in a
nonlinear crystal between the signal and idler outputs of the OPO. The beam
frequency bandwidth was reduced using a monochromator. The beam was loosely
focused at normal incidence by a parabolic mirror positioned below the sample.
The near-field signal was scattered by a metal-coated (Pt/Ir) atomic force
microscope tip (Arrow-NCPt, Nanoworld) oscillating at a frequency
$\Omega\approx$280\text{\,}\mathrm{kHz}$$, and collected by a second off-axis
parabolic mirror positioned above the sample. The tapping amplitude was set to
$\approx$80\text{\,}\mathrm{nm}$$ and the signal demodulated at the second
harmonic $2\Omega$ for background suppression. Before focusing, half of the
light was redirected towards a pseudo-heterodyne interferometer used to
retrieve both amplitude and phase of the signal. The light scattered by the
tip was recombined with the interferometer reference arm by a second beam-
splitter and directed towards a nitrogen-cooled mercury cadmium telluride
infrared detector.

### Simulations

Electromagnetic simulations were performed with a commercial solver (CST
Studio) in frequency domain. The dielectric function of SiC was modeled as
follows:

$\varepsilon(\omega)=\varepsilon_{\infty}\left(1+\frac{\omega_{LO}^{2}-\omega_{TO}^{2}}{\omega_{TO}^{2}-\omega^{2}-i\gamma\omega}\right)$
(1)

with $\omega_{TO}=$797\text{\,}{\mathrm{cm}}^{-1}$$,
$\omega_{LO}=$973\text{\,}{\mathrm{cm}}^{-1}$$, $\varepsilon_{\infty}=6.6$ and
$\gamma=$6.6\text{\,}{\mathrm{cm}}^{-1}$$. The PEG dielectric function was
modeled with a single Lorentzian as:

$\varepsilon_{\mathrm{PEG}}(\omega)=\varepsilon_{\mathrm{PEG},\infty}+\frac{A_{\mathrm{PEG}}}{\omega_{\mathrm{PEG},0}^{2}-\omega^{2}-2i\gamma_{\mathrm{PEG}}\omega}$
(2)

where $\varepsilon_{\mathrm{PEG},\infty}$ is set to 2.25,
$\omega_{\mathrm{PEG},0}=$842.4\text{\,}{\mathrm{cm}}^{-1}$$,
$\gamma_{\mathrm{PEG}}=$3.59\text{\,}{\mathrm{cm}}^{-1}$$ and the oscillator
strength $A_{\mathrm{PEG}}=$8026\text{\,}{\mathrm{cm}}^{-2}$$. The factor of
two in the denominator is due to the convention chosen to be comparable to our
TCMT model, where $\gamma$ refers to the half width at half maximum of a
Lorentzian.

The metasurface response was simulated by using periodic Floquet boundary
conditions and plane-wave excitation. The electric field distributions were
extracted at the top layer of the SiC film.

### Acknowledgments

This project was funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under grant numbers EXC 2089/1–390776260 (Germany’s
Excellence Strategy) and TI 1063/1 (Emmy Noether Program), the Bavarian
program Solar Energies Go Hybrid (SolTech) and the Center for NanoScience
(CeNS). Funded by the European Union (METANEXT, 101078018 and NEHO,
101046329). L.N. and E.C acknowledge funding from the European Commission
through the ERC Starting Grant CATALIGHT (802989) and DAAD (57573042). 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.M. additionally acknowledges
the Lee-Lucas Chair in Physics and the EPSRC (EP/W017075/1).

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