# Second Harmonic Generation from Grating-Coupled Hybrid Plasmon-Phonon
Polaritons

Marcel Kohlmann  Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg
4-6,14195 Berlin, Germany  Institut für Physik, Universität Greifswald, Felix-
Hausdorff-Str. 6, 17489 Greifswald, Germany  Christian Denker  Institut für
Physik, Universität Greifswald, Felix-Hausdorff-Str. 6, 17489 Greifswald,
Germany  Nikolai C. Passler  Fritz-Haber-Institut der Max-Planck-Gesellschaft,
Faradayweg 4-6,14195 Berlin, Germany  Jana Kredl  Institut für Physik,
Universität Greifswald, Felix-Hausdorff-Str. 6, 17489 Greifswald, Germany
Martin Wolf  Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg
4-6,14195 Berlin, Germany  Markus Münzenberg  Institut für Physik, Universität
Greifswald, Felix-Hausdorff-Str. 6, 17489 Greifswald, Germany  Alexander
Paarmann  Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg
4-6,14195 Berlin, Germany<EMAIL_ADDRESS>
###### Abstract

Polaritons can provide strong optical field enhancement allowing to boost
light-matter interaction. Here, we experimentally observe enhancement of mid-
infrared second-harmonic generation (SHG) using grating-coupled surface phonon
polaritons of the 6H-SiC surface. In our experiment, we measure the SHG along
the polariton dispersion by changing the incidence angle of the excitation
beam. We observe hybridization between the propagating surface phonon
polaritons and localized plasmon resonances in the gold grating, evidenced by
the modification of the polariton dispersion as we change the area ratio of
grating and substrate. Design options for engineering the plasmon-phonon
polariton hybridization are discussed. Overall, we find a rather low yield of
polariton-enhanced SHG in this geometry compared to prism-coupling and
nanostructures, and discuss possible origins.

††preprint: AIP/123-QED

The sub-diffractional light localization and enhancement of the local optical
fields provided by surface phonon polaritons (SPhPs) Caldwell  _et al._ (2015)
make them an ideal platform for applications relying on increased light-matter
interaction in the mid- to far-infrared (IR). Typical examples include
improved molecular sensingFolland  _et al._ (2020); Zheng  _et al._ (2017a);
Autore  _et al._ (2018); Berte  _et al._ (2018), coherent thermal
emissionGreffet  _et al._ (2002); Wang  _et al._ (2017); Lu  _et al._ (2021),
or nonlinear-optical signal generationRazdolski  _et al._ (2016); Passler  _et
al._ (2017, 2019); Kiessling  _et al._ (2019). Due to their evanescent nature,
exciting and probing SPhPs requires specific experimental schemesFolland  _et
al._ (2019), such as nanotip-based near-field mappingDai  _et al._ (2018); Ma
_et al._ (2021), sub-diffractional nanostructuresCaldwell  _et al._ (2013,
2014), prism couplingNeuner  _et al._ (2009); Passler  _et al._ (2017, 2022),
or grating couplingGreffet  _et al._ (2002); Zheng  _et al._ (2017a). The
latter offers several advantages such as extrinsic resonance tuning via the
incidence angleGreffet  _et al._ (2002), far-field access allowing for easy
device integrationZheng  _et al._ (2017a); Folland  _et al._ (2020) with
additional design options using the grating shapeGreffet  _et al._ (2002);
Folland  _et al._ (2020) and material combination of polaritonic substrate and
line gratingZheng  _et al._ (2017a).

In this work, we demonstrate mid-infrared nonlinear-optical second harmonic
generation (SHG) from SPhPs excited via gold-on-6H-SiC line gratings. We
observe enhancement of the SHG response at the polariton resonance which can
be spectrally tuned through the Reststrahlen band of 6H-SiC by changing the
incidence angle. We discuss possible origins for the rather low SHG yield
observed. By investigating two different samples with varied Au-to-SiC area
ratio, we further observe a modified polariton dispersion which we ascribe to
hybridization of the propagating SPhP modes with localized plasmon resonances
at the edges of the gold bars. Finite element simulations show excellent
agreement with our measurements, and we provide a simple effective medium
model qualitatively explaining the plasmon-phonon polariton hybridization.

Figure 1: Experimental concept for SHG from grating coupled surface polaritons
a) Experimental geometry: wavelength-tunable IR laser pulses polarized in the
$x-z$ plane impinge on the 6H-SiC sample with gold grating lines along the
$y$-direction, allowing excitation of surface polaritons propagating along the
$x$-direction. The specularely reflected IR and SHG intensity are detected
simultaneously. A dip in reflectance and a peak in SHG indicates resonant
surface polariton excitation. b) Optical microscopy image of the $d$=$4.8$
$\mu$m sample with a period $p=7.2$ $\mu$m. c) Grating coupling to SPhPs:
(red-to-blue) angle-of-incidence dispersion of the in-plane momenta for the
$m$=$-$1,0,$+$1 grating orders that allow coupling to the SPhPs at
intersections with the SPhP dispersion (green). d) Zoom into the counter-
propagating ($k_{\parallel}<0$) SPhP branch, highlighting the steep dispersion
expected from the $m$=$-$1 grating order.

The experimental scheme is outlined in Fig. 1a. The IR laser pulses generated
by the IR free-electron laser (FEL) installed at the Fritz Haber Institute
impinge on the sample, and the reflected IR intensity as well as the SHG are
detected. The FEL provides high-power, narrow-band (bandwidth of $\sim 0.3$%),
and wavelength-tunable IR light, and is described in detail
elsewhere.Schöllkopf  _et al._ (2015)

The IR laser beam was mildly focused onto the sample (spot size $\approx$ 600
$\mu$m) with angle of incidence $\alpha$, polarized in the _x-z_ plane
(P-polarized) with the lines of the grating along the _y_ -direction, see Fig.
1a. The reflected fundamental beam as well as the SHG beam emitted in specular
reflection are each detected using home-built pyroelectric and mercury cadmium
telluride detectors (IR associates), respectively, after spectral separation
using IR short-pass filters. SHG and linear reflectance spectra are acquired
by spectral scanning of the FEL wavenumber from 770 to $1020~{}\text{cm}^{-1}$
using the motorized undulator gap.

The gold line grating samples were produced on a c-cut semi-insulating 6H-SiC
substrate using 3D two-photon laser lithography printingHohmann  _et al._
(2015). In this technique, a wavelength-dependent photoresist is uniformly
applied to the substrate, and the grating is patterned by scanning the focus
of an illuminating laser across. After a developing process, a gold layer of
$h$=$200$ nm was deposited with an electron beam evaporator and a final lift-
off was done.. Two separate samples where made, one with grating line width
$d$=$2.4~{}\mu$m and one with $d$=$4.8~{}\mu$m, however, both with the same
period $p$=$7.2~{}\mu$m. An optical microscope image of the $d$=$4.8~{}\mu$m
sample is shown in Fig. 1b.

Propagating polaritons can be excited in these structures within the
Reststrahlen band of SiC between the transversal optical (TO) and longitudinal
optical (LO) phonon frequencies where the dielectric permittivity is
negative.Caldwell  _et al._ (2013); Chen  _et al._ (2014); Razdolski  _et al._
(2016) The dispersion relation $k(\omega)$ of the extraordinary surface phonon
polaritons at the $z$-cut uni-axial crystal surface is given as:Otto (1968)

$k_{SPhP}=\frac{\omega}{c_{0}}\sqrt{\frac{\epsilon_{\perp}\epsilon_{\parallel}-\epsilon_{\parallel}}{\epsilon_{\perp}\epsilon_{\parallel}-1}},$
(1)

where $k$ and $\omega$ are the in-plane momentum and the wavenumber,
respectively, and $\epsilon_{\perp}(\epsilon_{\parallel})$ is the in-plane
(out-of-plane) dielectric function of 6H-SiC.Passler  _et al._ (2017) The
dispersion of forward and backward propagating surface phonon polaritons at
the 6H-SiC surface are plotted as green solid lines in Fig. 1c and d.

Momentum transfer from a line grating, in the simplest description, can be
modelled by adding the grating momentum $G_{m}=\frac{m}{p}$ with the grating
period $p$ and the diffraction order $m$ to the in-plane momentum component of
the incoming light $k_{x}(\alpha)$. This leads to a total in-plane momentum
$k(\alpha,m)=k_{x}(\alpha)+G_{m}$, as illustrated in Fig. 1c for
$m$=$-$1,0,$+$1 for a range of incidence angles $\alpha$ of the incoming
light. Surface polaritons can be excited if:

$k_{SPhP}=k_{\parallel}(\alpha,m)=k_{x}(\alpha)+\frac{m}{p}.$ (2)

Clearly, no coupling is possible for $m$=0 where $k_{\parallel}$ remains
within the light cone. For the period $p=7.2~{}\mu$m chosen in our work, the
$m$=$+$1 order provides access to large momentum modes with small dispersion,
while a steep dispersion is expected for the $m$=$-$1, counter-propagating
diffraction order. The latter is illustrated in Fig. 1d, where a steep
dependence of the polariton resonance wavenumber on incidence angle is
predicted.

Figure 2: Experimental SHG and IR reflectance spectra. SHG intensity (a,c) and
reflectance (b,d) at various incidence angles for $d$=$2.4$ $\mu$m (a,b) and
$d$=$4.8$ $\mu$m (c,d). Dips in the reflectance with associated peaks in the
SHG strongly dispersing with incidence angle (red-to-blue) are assigned to
resonant excitation of surface polaritons via the $m$=$-$1 grating order
(light green shaded area). Non-dispersive features include 6H-SiC substrate
resonances at $\omega_{TO}^{\perp}$ and $\omega_{LO}^{\parallel}$, and the
zone-folded modes $\omega_{zf}$ previously observed in SHG spectroscopy of
SiC.Paarmann  _et al._ (2016); Razdolski  _et al._ (2016) Additionally, a
weakly dispersive feature at $\omega\approx$$940~{}\text{cm}^{-1}$ is
tentatively assigned to surface polariton excitation via the $m$=$+$1 grating
order (light orange shaded area). Please note the logarithmic scale in a) and
c).

Experimental SHG and reflectance spectra at various incidence angles for both
samples are shown in Fig. 2. The SHG spectra feature several resonances: the
peaks at the LO phonon wavenumber $\omega_{LO}^{\parallel}$ and the TO phonon
wavenumber $\omega_{TO}^{\perp}$ are typical for SHG spectroscopy of SiC and
arise due to resonances in the Fresnel transmission and the nonlinear
susceptibility $\chi^{(2)}$, respectively.Paarmann  _et al._ (2015, 2016);
Passler  _et al._ (2017) These features also mark the edges of the
Reststrahlen band of SiC observed in the reflectance spectra, Fig. 2b.
Notably, these features are almost unaffected by the incidence angle. In the
reflectance spectra for $d$=$4.8$ $\mu$m, Fig. 2d, we find the upper
Reststrahlen band edge significantly modified due to the large total area of
gold on the sample surface.

The strongly dispersive peaks in the SHG spectra and dips in the reflectance
spectra between $810~{}\text{cm}^{-1}$ and $920~{}\text{cm}^{-1}$ indicate
resonant excitation of counter-propagating surface polaritons via the $m$=$-$1
grating order, in qualitative agreement with the prediction in Fig. 1d.
Hereby, the amplitude of the SHG enhancement correlates with the magnitude of
the reflectance dip, reaching enhancement values of about 10. An additional
tenfold SHG enhancement is observed for the polariton resonance in spectral
proximity of the zone-folded weak modes of 6H-SiC at
$\omega_{zf}=$$881,886~{}\text{cm}^{-1}$,Bluet  _et al._ (1999); Paarmann  _et
al._ (2016) similar to what was observed for localized polaritons in
subdiffractional nanostructures.Razdolski  _et al._ (2016)

Finally, a weakly dispersive SHG resonance is observed at
$940~{}\text{cm}^{-1}$. For $d$=$2.4$ $\mu$m in Fig. 2a and b, the weak SHG
peak correlates with a small dip in the reflectance, suggesting (relatively
inefficient) excitation of forward-propagating polaritons via the $m$=$+$1
grating order. For $d$=$4.8$ $\mu$m, the corresponding SHG peak is much more
pronounced. In the reflectance spectra for this sample, Fig. 2d, a strong,
broad bleach is observed which, however, cannot be clearly assigned to a
polariton resonance.

To corroborate these experimental findings, we performed finite-element
simulations of our structures using Comsol Multiphysics.Comsol (2020)
Frequency-domain calculations with periodic Floquet boundary conditions
including first order diffraction yielded the frequency-dependent electric
field distributions as well as the reflectance spectra for various incidence
angles, see Fig. 3a,b and c,d, respectively, for the two grating geometries
studied experimentally. Hereby, we used the isotropic dielectric function of
SiCPaarmann  _et al._ (2016) and the Drude model for gold.

For $d$=$2.4$ $\mu$m, the deep dips in the reflectance spectra, Fig. 3c,
dispersing with incident angle suggest near-optimal sample design parameters
for excitation of the counter-propagating polaritons via the $m$=$-$1 grating
order, in excellent qualitative agreement with the experiment. Similarly, the
shallow dip with small dispersion at $940~{}\text{cm}^{-1}$ suggests
excitation of forward-propagating polaritons via the $m$=$+$1 grating order,
though with less efficiency just like in the experiment. We also note the
step-like drop of the reflectance arising at wavenumbers slightly above the
$m$=$-$1 polariton resonance which corresponds to onset of first order
diffraction of the sub-wavelength period grating, _i.e._ , the Wood’s
anomaly.Wood (1902) We did not observe clear signatures of the Wood’s anomaly
in the experiment, and will thus focus on the polariton resonances in the
following.

Figure 3: Simulations of grating coupling to surface polaritons. Electric
field distributions for a) $d$=$2.4$ $\mu$m and b) $d$=$4.8$ $\mu$m at the
respective $m$=$-$1 polariton resonance for $\alpha=30^{\circ}$, exhibiting
both homogeneous as well as strongly localized contributions of the
polaritonic field enhancement. c,d) Calculated reflectance spectra for various
incidence angles (see legend). e,f) Spectra of the fourth power of the field
enhancement driving SHG integrated over the SiC volume,
$\int_{V(SiC)}|E|^{4}$. Light green and light orange shaded areas mark the
spectral ranges for the backward-propagating $m$=$-$1 and forward-propagating
$m$=$+$1 grating orders, respectively.

The electric field distributions for resonant counter-propagating $m$=$-$1
polariton excitation under incidence angle $\alpha=30^{\circ}$ at
$\omega$=$884~{}\text{cm}^{-1}$ and $\omega$=$894~{}\text{cm}^{-1}$ is plotted
in Fig. 3a and b, respectively, for $d$=$2.4$ $\mu$m and $d$=$4.8$ $\mu$m.
Very similar field distributions are found at the counter-propagating
polariton resonance at different incidence angles and at the $m$=$+$1
resonances (both not shown). Fig. 3a and b suggest two major components of the
polaritonic field enhancement; one being relatively homogeneous along both
sides of the SiC/air interface as expected for a propagating polariton, and
another being largely localized in a much smaller volume at the edges of the
gold grating. These latter hot spots already suggest that the gold grating
does not merely serve as a momentum transducer for SPhP excitation, but
instead actively contributes to the formation of the polariton mode.

For $d$=$4.8$ $\mu$m, the reflectance spectra in Fig. 3d behave quite similar
to $d$=$2.4$ $\mu$m in terms of the $m$=$-$1 polariton resonance. However, a
clearly different behavior is observed for $\omega>940~{}\text{cm}^{-1}$,
similar to what was found in the experiment. First, the reflectance drops
sharply at $\omega=$$940~{}\text{cm}^{-1}$ around where the $m$=$+$1 polariton
resonance is predicted. Second, significantly higher reflectance values are
found above the SiC Reststrahlen band, indicative of an optical response being
strongly modified by the presence of the gold grating in this range.
Specifically interesting is the rapid flip from peak to dip from
$\alpha<30^{\circ}$ to $\alpha>30^{\circ}$ at the upper edge of the SiC
Reststrahlen band, $\omega=$$965~{}\text{cm}^{-1}$, yet again in excellent
qualitative agreement with the experimental reflectance.

Figure 4: Hybridization of grating-coupled surface polaritons a) Experimental
(circles) and simulated (dashed lines) dispersion of $m$=$-$1 grating coupled
polaritons extracted from the dip positions in the reflectance spectra for
$d=$2.4 $\mu$m (blue) and $d=$4.8 $\mu$m (red). The data for $d=$2.4 $\mu$m
agrees well with the theoretical dispersion, Eq. 2, while a systematic blue
shift is observed for $d=$4.8 $\mu$m. b) Comparison of the simulated $d=$4.8
$\mu$m dispersion with effective medium modelling using Eq. 3 with
$\omega_{p}$=$350~{}\text{cm}^{-1}$ (lower grey edge) and
$\omega_{p}$=$850~{}\text{cm}^{-1}$ (upper grey edge), matching the simulated
$m$=$-$1 dispersion at $\alpha$=15∘ and for $\alpha>$25∘, respectively. This
can be interpreted as a change in the effective plasma frequency $\omega_{p}$
required to reproduce the simulation along the dispersion. This behavior
correlates well with the effective mode lengthMaier (2006) $L_{eff}^{Au}$
within the gold shown in c), such that the number of charge carriers
participating in the mode reduces for larger momenta (smaller incidence
angles).

With the overall good agreement of the simulated reflectance spectra with the
experiment, we can confidently use the simulation results to further analyze
the SHG spectra. A microscopic description of SHGRazdolski  _et al._ (2016) in
this structure is beyond the scope of this work. Instead, we here estimate the
contribution of the polaritonic field enhancement to the SHG yield by
integrating the fourth power of the electric field magnitude across the SiC
volume, $\int_{V(\mathrm{SiC})}|E|^{4}$. By doing so, we neglect any phase
relation between the local sources of nonlinear signal and the resulting
impact on the far-field emission, as well as the tensorial nature of the
nonlinear susceptibility.

The spectral dependence of $\int_{V(\mathrm{SiC})}|E|^{4}$ is plotted in Fig.
3e,f. These simulated data resemble the experimental SHG spectra, see Fig. 2a
and c, with remarkable precision, not only in terms of the spectral position
of the resonances, but also in terms of the relative amplitudes between peaks
and background. Even the pronounced enhancement of the SHG yield at the
$m$=$+$1 polariton resonance between $d$=$2.4$ and $4.8$ $\mu$m is almost
perfectly reproduced in the simulations. Notably, the SHG resonance at the TO
phonon frequency $\omega_{TO,\perp}=$$797~{}\text{cm}^{-1}$ arises due to the
resonance in the nonlinear susceptibility of SiCPaarmann  _et al._ (2015,
2016) and would, thus, not be expected to appear in the field enhancement
where instead a deep dip reports on the Fresnel anti-resonancePaarmann  _et
al._ (2016) at the TO phonon.

Notably, we observe a blue-shift of the dispersive $m$=$-$1 polariton
resonance between the $d$=2.4 and 4.8 $\mu$m geometries, both in the
experiment and in the simulations. This is shown in Fig. 4a where we plot the
polariton resonance positions we extracted from the dips in the reflectance
spectra. For each grating line width, experiment and simulations agree very
well. For reference, we also plot the theoretical dispersion, Eq. 2.

For $d$=$2.4$ $\mu$m, the experimental and simulated dispersion agree almost
perfectly with the theoretical prediction, suggesting that here the polariton
mode is dominated by the SiC properties and the influence of the gold is
rather weak. In contrast, for $d$=$4.8$ $\mu$m the dispersion is stretched to
higher wavenumbers,_i.e._ , the blue-shift increases for larger
$|k_{\parallel}|$, reaching values of $>$$15~{}\text{cm}^{-1}$ at
$\alpha=15^{\circ}$. Clearly, the simple model of momentum transfer by the
grating coupler, Eq. 2, fails to accurately describe the dispersion, which we
ascribe to mode hybridization between the SPhP and the local response of the
free charge carriers at the gold grating etches,_i.e._ , localized plasmons.

A similar behavior of plasmon-phonon polariton hybridization has been observed
recently in studies of surface polaritons in heavily doped
semiconductorsShalygin  _et al._ (2019); Janonis  _et al._ (2020), van der
Waals heterostructures,Dai  _et al._ (2015); Woessner  _et al._ (2014); Huber
_et al._ (2016) and metallic nanostructures placed on polar crystals slabsHuck
_et al._ (2016); Pons-Valencia  _et al._ (2019), with varying degrees of
coupling strength. For the case of doped semiconductors, the plasmon-phonon
coupling can be described analytically.Shalygin  _et al._ (2019) Inclusion of
a Drude term in the SiC dielectric function describing the free charge carrier
response results in an additional pole in the dielectric function, while the
pole at the LO phonon frequency is blue-shifted. In consequence, the
Reststrahlen-band broadens and the polariton dispersion is stretched.Shalygin
_et al._ (2019); Janonis  _et al._ (2020)

We can use this simple model for an effective medium description of the Au
grating-induced plasmonic contribution to the polariton dispersion, by adding
a Drude term to the dielectric function of SiC:

$\epsilon_{\perp(\parallel)}^{\mathrm{eff}}(\omega)=\epsilon_{\perp(\parallel)}^{SiC}(\omega)-\epsilon_{\infty,\parallel(\perp)}^{SiC}\left(\frac{\omega_{p}^{2}}{\omega^{2}+i\gamma\omega}\right),$
(3)

where $\epsilon_{\perp(\parallel)}^{SiC}(\omega)$ and
$\epsilon_{\infty,\parallel(\perp)}^{SiC}$ are the in-plane (out-of-plane)
intrinsic dielectric function and high-frequency permittivity of SiC,
$\omega_{p}$ the plasma frequency and $\gamma$ the plasma damping. The
resulting polariton dispersion is plotted in Fig. 4b for
$\omega_{p}$=$350~{}\text{cm}^{-1}$ and $\omega_{p}$=$850~{}\text{cm}^{-1}$
with $\gamma$=$0.1\omega_{p}$. While the former matches the simulated
dispersion for large momenta (small $\alpha$), the latter better describes the
initial slope of the dispersion (larger $\alpha$). Thus, the effective medium
description using a single plasma frequency cannot sufficiently describe the
plasmon-phonon polariton hybridization.

One possibility to explain this discrepancy could be the different magnitude
of the plasmon contribution to the hybrid mode along the dispersion. To test
this hypothesis, we evaluated the effective mode volumeMaier (2006) within the
Au from the finite-element simulations Fig. 3. In this picture, the different
spatial extent of mode penetration into the gold results in a different
effective number of charge carriers actively participating in the mode,
leading to a different effective plasma frequency in Equ. 3. Indeed, the
extracted effective mode lengths $L_{\mathrm{eff}}^{\mathrm{Au}}$ shown in
Fig. 4c agree very well with our observations: For $d$=$2.4$ $\mu$m we find
only a mild change of $L_{\mathrm{eff}}^{\mathrm{Au}}$ along the dispersion.
For $d$=$4.8$ $\mu$m, however, $L_{\mathrm{eff}}^{\mathrm{Au}}$ increases
significantly with $\alpha$, i.e., the mode penetrates deeper and deeper into
the gold as the polariton momentum is reduced towards the light line.

We note that frequency shifts of the polariton resonances can additionally
occur due to radiative coupling, which typically results in a broadening and
simultaneous red-shift of the polariton resonances.Passler  _et al._ (2017)
Fig. 3d indeed suggests increasing radiative coupling with decreasing $\alpha$
(increasing $|k_{\parallel}|$) for the $m$=-1 resonance, where the reflectance
dip amplitudes decrease while the dip widths increase. Such a behavior is in
fact identical to the critical coupling effect in the Otto-type prism coupling
configuration.Passler  _et al._ (2017) These red-shifts cannot, however,
explain the blue-shift of the hybrid polariton resonance observed in Fig. 4a.
Careful design of the grating structure might allow to simultaneously
optimize, e.g., plasmon hybridization and radiative coupling across the
dispersion. For instance, a thickness gradient in the gold film across each
grating stripe would provide an additional design parameter to independently
control both quantities.

Overall, the polaritonic SHG enhancement we find here using the grating
coupling configuration is much less than what we previously observed in
subdiffractional nanostructuresRazdolski  _et al._ (2016) and using Otto-type
prism couplingPassler  _et al._ (2017), where in both cases the polariton
resonances dominated the SHG spectra, and where we generally observed larger
SHG signal amplitudes. In fact, we expect that formation of hot spots would
boost the SHG in contrast to homogeneous polariton field distributions typical
for prism couplingPassler  _et al._ (2017) because of the fourth order
electric field scaling of the nonlinear signal, in contrast to our
observations. In principle, interference between the hot spot emissions as
well as shadowing by the gold could reduce the far-field coupling of the SHG.
However, the excellent agreement of the simulations in Fig. 3e,f with the
experimental SHG spectra suggests that the reduced yield occurs already in the
generation process. Further experiments and simulations including SHG far-
field coupling are needed to clarify the lack of efficiency of the SHG in the
grating geometry.

In summary, we demonstrated infrared SHG from grating-coupled polaritons using
gold gratings on 6H-SiC. We show that the polariton resonance can be widely
tuned using the incidence angle in the counter-propagating $m$=$-$1 grating
order. Unfortunately, we find the yield of the polaritonic enhancement of the
SHG to be rather small, compared to previous work using prism-coupled or
localized phonon polaritons. Further, we find that the polariton dispersion
can be significantly modified using different width-to-gap ratios of the
grating, which we ascribe to hybridization of the propagating phonon polariton
modes with localized plasmon resonances in the gold grating. We qualitatively
describe the modification of the hybrid mode dispersion with an effective
medium description which also accounts for the variation of the effective mode
volume within the gold along the dispersion. We propose that this platform is
ideal to engineer the hybrid mode dispersion. Conveniently, gold gratings can
be employed for any polar crystal surface, providing a versatile tool for
probing hybrid polaritons in a wide range of polariton materials.

###### Acknowledgements.

The authors wish to thank S. Gewinner and W. Schöllkopf for operating the FEL.

## Data Availability Statement

The data that support the findings of this study are openly available in
Zenodo at http://doi.org/10.5281/zenodo.6880979.

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