# Enhancing vibrational light-matter coupling strength beyond the molecular
concentration limit using plasmonic arrays

Manuel Hertzog  Department of Chemistry and Molecular Biology, University of
Gothenburg, Kemigården 4, 412 96, Gothenburg, Sweden  Battulga Munkhbat  Denis
G. Baranov  Timur O. Shegai  Department of Physics, Chalmers University of
Technology, 412 96, Gothenburg, Sweden<EMAIL_ADDRESS>Karl Börjesson
Department of Chemistry and Molecular Biology, University of Gothenburg,
Kemigården 4, 412 96, Gothenburg, Sweden<EMAIL_ADDRESS>
###### Abstract

Vibrational strong coupling is emerging as a promising tool to modify
molecular properties, by making use of hybrid light-matter states known as
polaritons. Fabry-Perot cavities filled with organic molecules are typically
used, and the molecular concentration limits the maximum reachable coupling
strength. Developing methods to increase the coupling strength beyond the
molecular concentration limit are highly desirable. In this letter, we
investigate the effect of adding a gold nanorod array into a cavity containing
pure organic molecules, using FT-IR microscopy and numerical modeling.
Incorporation of the plasmonic nanorod array, that acts as artificial
molecules, leads to an order of magnitude increase in the total coupling
strength for the cavity filled with organic molecules. Additionally, we
observe a significant narrowing of the plasmon linewidth inside the cavity. We
anticipate that these results will be a step forward in exploring
vibropolaritonic chemistry and may be used in plasmon based bio-sensors.

###### keywords:

Vibropolariton, Strong coupling, Plasmonic, Polaritonic chemistry

##  1 Introduction

Strong light-matter coupling has attracted considerable attention in the past
couple of years due to the potential applications it offers in physical and
chemical sciences.1, 2, 3, 4 For example, strong coupling of organic molecules
has been shown to modify the rate of a photoisomerization reaction,5, 6
increase electronic transport,7 and expand the length scale of Förster energy
transfer.8, 9, 10 Not to mention other effects of strong coupling such as
selective manipulation of excited states,11 suppression of photo-oxidation 12
or reducing photodegradation in polymers.13 Recently, vibrational strong
coupling has come into focus as a promising physical tool to control molecular
properties. Since the first experimental evidences of vibrational strong
coupling (VSC) in both solid14, 15, 16 and liquid states,17 the field has
expanded considerably,18, 19, 20, 21, 22, 23, 24, 25 and it has been shown to
alter reaction kinetics,26, 27, 28, 29 control reaction selectivity,30 allow
for intermolecular energy transfer,31 and modification of enzyme activity.32
Recent progresses in chemical reactions making use of polaritons were
summarized by Hirai et al.33 In order to significantly impact chemical
reactivity, theoretical investigations has demonstrated that a high coupling
strength is required34, 35 and a recent experimental study has shown a non-
linear relationship between the coupling strength and thermodynamics of a
chemical reaction.29

Strong light-matter coupling is achieved by interfacing molecules with
confined electromagnetic field of resonant cavities tuned to a molecular
transition. When a molecular vibrational transition is in the strong coupling
regime, two new hybrid states, known as polaritons, are formed,36 separated in
energy by the so-called Rabi splitting $\hbar\Omega_{\text{R}}$.
Traditionally, polaritons have been engineered with the use of planar
cavities, such as Fabry-Perot resonators confining the electromagnetic fields
between two mirrors.14 Reaching large coupling strengths usually requires
saturating the cavity volume with the molecular material, and additionally
aligning the transition dipole moments with the cavity vacuum field. 37, 38,
39

Plasmonics offers an alternative route to strong coupling by confining light
down to subwavelength scales with the use of metallic nanoparticles and
nanocavities.40, 41, 42 Only a tight region of space around the metallic
cavity needs to be filled with molecules in order to form polaritons.43
Plasmon resonances of nanoparticles are tunable from the UV to the IR range,
and can be used for molecular sensing in the IR range due to nanoscale mode
volumes.44, 45, 46, 47 However, to improve the sensitivity, new approaches for
narrowing the plasmon linewidth are desirable.

The magnitude of the Rabi splitting for a given molecular transition is
proportional to the square root of the molecular concentration and the filling
factor. Furthermore, the maximal achievable magnitude of the Rabi splitting in
the case of saturated mode volume is ultimately bounded by the bulk Rabi
splitting $\hbar\Omega_{\text{R}}\sim\sqrt{f}$ (with $f$ being the transition
oscillator strength 48), which is independent on the cavity type.49, 50, 51
Therefore, new approaches are required to increase the Rabi splitting in order
to maximize the effect of the confined electromagnetic field onto molecules,
as shown by theoretical studies.52, 34, 53

In this Letter, we utilize a hierarchical coupling between a Fabry-Perot
cavity, a vibrational absorption band of an organic molecule, and a localized
surface plasmon resonance in the mid-IR regime to go beyond the Rabi splitting
imposed by the maximal concentration limit 54. First we show that the coupling
of the plasmon and the FP cavity results in an order of magnitude decrease in
plasmon’s linewidth, an observation rationalized by reduced radiative losses
from the plasmon in the cavity 55. Then, by using numerical and analytical
modelling in conjunction to our experimental data, we show a five to nine-fold
increase in the total coupling strength, indicating that the plasmon act as an
artificial molecule that increases the molecular coupling strength.

##  2 Results & discussions

Figure 1: (a) Conceptual diagram of the system in this study. (b) Schematic
view of the Fabry-Perot cavity. Fourier transform infrared spectroscopy
microscopy was used to either probe the small area where gold nanorods were
deposited (beam 1) or where no rods where present (beam 2). (c) Molecular
structure of 4-butylbenzonitrile (top) and hexanal (bottom). Highlighted in
orange are the functional groups responsible for the vibrational band coupled
to the FP cavity. (d) SEM picture of the gold nanorods deposited on a glass
substrate.

In this study, we report a method to increase the coupling strength above the
limit of $\sqrt{C}$ using a hybrid system composed of a Fabry-Perot (FP)
cavity, an organic molecule, and a localized surface plasmon, in a fashion
similar to the one introduced by Bisht et al.54 using two-dimensional
transition metal dichalcogenides in the visible regime. All three entities are
tuned to the same resonance frequency, thereby coupled amidst themselves,
creating hybrid polaritons (Figure 1a). The Fabry-Perot cavities used in the
following experiments are composed of IR-transparent substrates (CaF2 and
ZnSe) coated with $10\text{\,}\mathrm{nm}$ of gold (Figure 1b; see SI for
experimental methods). The physical distance between the gold mirrors, ranging
from 11 to $16\text{\,}\mathrm{\SIUnitSymbolMicro m}$, was controlled using a
polymer spacer. The cavities were designed with two inlets allowing to inject
liquids. The quality factor of an empty cavity was 28, and the free spectral
range 496 cm-1 (Figure 2a). We choose hexanal and 4-butylbenzonitrile as the
organic molecules for this study (Figure 1c). Both are liquids at room
temperature and were processed in neat form. The molecular vibrations of
interest are the C=O stretching mode of hexanal and the C#N stretching mode of
4-butylbenzonitrile. These are strong absorption bands centered around
$1724\text{\,}{\mathrm{cm}}^{-1}$ and $2225\text{\,}{\mathrm{cm}}^{-1}$, with
full width at half maxima (FWHM) of 27 and $10\text{\,}{\mathrm{cm}}^{-1}$,
respectively (Figure 2b and 2c). The surface plasmon was provided by gold
nanorod arrays (size ca.
$100\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}$), which where
deposited on top a spacer composed of $450\text{\,}\mathrm{nm}$ of SiO2 to
minimize interference from the gold mirror (Figure 1b). Five different arrays
were made, all having equal thickness and spacing but lengths ranging from
1100 to $1500\text{\,}\mathrm{nm}$, which gave plasmon resonances covering the
energy of both two molecular vibrations and the Fabry-Perot mode (Figure 2d).
The plasmon resonances were broad (FWHM = 621–1120 ${\mathrm{cm}}^{-1}$) due
to radiative losses. This is an intrinsic feature of plasmon arrays in the
mid-IR, hampering its use.

Figure 2: (a) Transmission spectrum of an empty Fabry-Perot cavity. (b) ATR
spectrum of hexanal (purple) showing the C=O absorption band around
$1724\text{\,}{\mathrm{cm}}^{-1}$, and transmission spectrum of the FP cavity
filled with hexanal (orange). The value of $\hbar\Omega_{\text{R}}$ is
$101\text{\,}{\mathrm{cm}}^{-1}$. (c) ATR spectrum of 4-butylbenzonitrile
(purple) showing the C3N absorption band around
$2225\text{\,}{\mathrm{cm}}^{-1}$, and transmission spectrum of the FP cavity
filled with 4-butylbenzonitrile (orange). The value of
$\hbar\Omega_{\text{R}}$ is $46\text{\,}{\mathrm{cm}}^{-1}$. (d) Reflection
map of the five different gold nanorod arrays.

We will first describe the coupling between the cavity and the two organic
compounds, then the cavity and plasmon, and lastly, the complete coupled
system with the FP cavity, the plasmon and the organic molecules. Figure 2b
and c show a clear normal mode splitting of the vibrational absorption band of
the two molecules when placed inside the FP cavity at resonant conditions.
This indicates strong coupling between the cavity mode and the carbonyl group
of hexanal (Figure 2b), and the cavity mode and the nitrile group of
4-butylbenzonitrile (Figure 2c). The resulting formation of vibro-polaritons
gives a measured Rabi splitting of $101\text{\,}{\mathrm{cm}}^{-1}$ and
$46\text{\,}{\mathrm{cm}}^{-1}$ for the C=O mode and the C#N mode with the
cavity, respectively. The Rabi splittings are larger than both the FWHM of the
bare molecular vibrations and the cavity mode, which constitutes further
evidence that our system is indeed in the strong-coupling regime. Furthermore
the ratio between the coupling strength and the bare transition energy of the
vibrations are 2.93% for the C=O band and 1.03% for the C#N band.

Let us now consider the plasmonic arrays inside the FP cavity. With increasing
rod length, the plasmon resonance shifts to lower energies (Figure 2d), and
the nanorod length therefore provides a way of controlling the cavity-plasmon
detuning. Moreover, due to intrinsic anisotropy of the rods, the microcavity-
plasmon polaritons are observed only along the long axis of the rods, while
along the short axis, the bare cavity is recovered. This polariton anisotropy
can be probed using polarization-resolved transmission spectroscopy. As shown
in Figure 3a, the two original FP cavity modes that can be observed in the
given spectral range couple to the plasmonic array mode, forming exactly
$2+1=3$ microcavity-plasmon polaritonic states, denoted with dot-dashed lines.
These new modes exhibit almost linear dispersive behavior because of the
underlying dispersion of the plasmonic mode with the nanorod length.

Furthermore, we observe an order of magnitude decrease in the linewidth of the
resulting polariton compared to the bare plasmon. Similar narrowing was
observed in our previous experiments with plasmonic arrays ultra-strongly
coupled with FP cavities in the visible to mid-IR range.55 The linewidth of
closely packed metallic nanorods in the IR region is dominated by radiative
losses; but when placed inside a closed cavity, the radiation from the
nanorods does not instantaneously leave the cavity, instead bouncing between
the mirrors and thus reducing the total resonance linewidth. Consequently, the
FWHM of the plasmon polaritons drops to ca. $70\text{\,}{\mathrm{cm}}^{-1}$
for all the rods’ lengths, as a result of significantly suppressed radiative
damping. Observing that the FWHM of the FP modes are $84\pm 7.3$
${\mathrm{cm}}^{-1}$, the linewidth of the cavity is the main limitation of
the linewidth of the plasmon polariton (Table S2).

By adding molecules into the cavity, a third resonant component is introduced
to the hybrid system. The hexanal C=O band is on resonance with the longest
nanorods (Figure 3b), whereas the 4-butylbenzonitrile C#N band is on resonance
with the shortest nanorods (Figure 3c). This time, more FP modes are present
in the same spectral range due to the larger background refractive index of
the molecular solution compared to air. Dot-dashed lines indicate the
dispersions of the resulting cavity-plasmon-molecule polaritonic modes,
whereas the dashed vertical lines indicate the respective molecular
resonances. In the next section, we will uncover the polaritonic origin of
these linearly-dispersing modes with the use of the analytical Hamiltonian
formalism. Furthermore, the polaritonic linewidths are again reduced by the
cavity mirrors, showing that the FP cavity mode limits the polariton linewidth
irrespectively of the medium inside the cavity.

In order to gain further understanding of the experimental results, we
performed numerical modelling of the coupled systems with the use of the
finite-difference time domain (FDTD) method in a commercial software
(Lumerical Figure 3d-f and Figures S1-3). Simulated normal-incidence
transmission spectra demonstrate a good agreement with the experimental
spectra for both plasmonic structures, and composite plasmon-molecule ones.
Specifically, the observed polaritonic modes linearly disperse with the rods
length, however, the dispersion is less pronounced than for uncoupled rods
(gray dots). This is a consequence of intermixing between highly dispersive
plasmonic modes and non-dispersive FP modes (all arrays were placed in the
same FP cavity).

Generally, a system of $N$ distinct oscillators produces $N$ polaritonic modes
upon coupling (as long as the system is not exactly at an exceptional point).
In our case, we start with $M$ bare cavity modes, the plasmonic array mode,
and the molecular resonance, so that one could expect emergence of $M+2$ new
eigenstates. However, the simulated spectra strongly suggest that the
molecules and the plasmons join in a single effective oscillator, which adds
only one additional polaritonic mode to the system, as one can clearly see
from the transmission dispersions in Fig. 3 (aside from the parasitic
uncoupled FP modes). This confirms that even a system with slightly detuned
plasmons and molecules does behave as a system with a single additional
compound plasmonic-molecular oscillator.

Figure 3: (a)-(c) Transmission maps of the gold rods inside the Fabry-Perot
cavity, containing air, hexanal, or 4-butylbenzonitrile, respectively. All
three were measured with a polarizer along the long axis of the rods. The blue
dash-dot lines highlight the newly formed polaritonic states, the residual
vertical modes are artifact due to non-ideal polarization alignment. (d)-(f)
Simulated spectra of the same systems (with the cavity thicknesses of
$10.8\text{\,}\mathrm{\SIUnitSymbolMicro m}$,
$8.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and
$10.4\text{\,}\mathrm{\SIUnitSymbolMicro m}$, respectively). The gray dashed
line indicates the absorption band of interest of the molecules, and the gray
dots indicate the plasmon absorption maximum.

Next, to ensure that the observed dispersions are a result of the interaction
between the three components in the hybrid system, transmission spectra of the
system with a polarizer perpendicular to the nanorods’ long axis were
measured. The results are shown in Figure S5 (numerical modeling in Figure
S1). As expected, when the contribution of the plasmonic array is removed by
the polarizer, the transmission spectra are the same as when probing the
system beside the array (Figure 2b and c). Likewise, the values of
$\hbar\Omega_{\text{R}}$ with a perpendicular polarizer are
$46\text{\,}{\mathrm{cm}}^{-1}$ and $101\text{\,}{\mathrm{cm}}^{-1}$ for the
C#N and C=O vibrations, respectively, which are the same values as we observed
when when probing next to the plasmonic array.

##  3 Theoretical analysis

In order to extract coupling strengths and confirm that the plasmonic array
acts as an artificial molecule to enhance the total coupling strength of the
system, we turn to theoretical analysis of the experimental results. This
analysis is essentially based on the coupled harmonic oscillator algebra in
the simplest possible implementation. Furthermore, this analysis requires
several rather crude assumptions, which may be false in general, but are
satisfactory for the goal of extracting the collective coupling constants.

In view of the above remark, we describe the cavity by a set of $M$ orthogonal
Fabry-Perot eigenmodes, each coupling to the molecular resonance with a
certain coupling constant. As we are far from the ultrastrong coupling regime
in this case, the interacting system can be described by multimode coupled-
harmonic oscillator Hamiltonian, which under the rotating wave approximation
takes the form (see Methods):

$\hat{H}_{mol}=\sum_{m=1}^{M}{\hbar\omega_{m}\hat{a}_{m}^{\dagger}\hat{a}_{m}}+\hbar\omega_{0}\hat{b}^{\dagger}\hat{b}+\sum_{m=1}^{M}\hbar
g_{m}(\hat{a}^{\dagger}_{m}\hat{b}+\hat{a}_{m}\hat{b}^{\dagger})$ (1)

where $\hat{a}$ and $\hat{b}$ are the annihilation operators of the $m$-th
cavity mode and that of the molecular resonance, respectively, and $g_{m}$ is
the coupling constant.

The coupled Fabry-Perot system exhibits transmission peaks at its polaritonic
resonances, corresponding to the eigenvalues of the Hamiltonian (Eq. 1).
Therefore, we estimate the cavity-molecule coupling strength by fitting the
energies of the transmission peaks by the eigenvalues of Hamiltonian Eq. 1
accounting for $M=20$ lowest cavity modes (see Methods for the details of the
fitting procedure). Since in the multi-mode system the coupling strength is
dependent on the FP mode order and the frequency, to make a reasonable
comparison we estimate the coupling strength at zero detuning: that is, with
the particular FP mode, (near-)resonant with the molecular transition.

By fitting the Hamiltonian eigenvalues to the positions of measured
transmission peaks (see Figure S6), we obtain the hexanal-cavity system zero-
detuning coupling strength (with the 7-th FP mode of a
$14.25\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick cavity) of about
$41\text{\,}{\mathrm{cm}}^{-1}$. Remarkably, this value is close to the bulk
polariton coupling strength of hexanal
$g_{bulk}=\omega_{p}\sqrt{f/\varepsilon_{\infty}}/2\approx 49$
${\mathrm{cm}}^{-1}$, which describes the photon-molecule interaction strength
in an unbounded homogeneous molecular medium.56 Similarly, for the
4-butylbenzonitrile-cavity system, we obtain the zero-detuning coupling
strength (with the 11-th mode of a $16.15\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ thick cavity) of about $18\text{\,}{\mathrm{cm}}^{-1}$, which is also
comparable to the respective bulk coupling strength of 4-butylbenzonitrile
($21\text{\,}{\mathrm{cm}}^{-1}$).

Figure 4: Hamiltonian analysis of the experimental data. (a) Fitting of the
measured transmission peaks of the coupled nanorod-cavity system (circles)
with eigenvalues of the multimode JC Hamiltonian (lines). (b) The same as (a)
for the hybrid hexanal-nanorod system. (c) The same as (a) for the hybrid
4-butylbenzonitrile-nanorod system.

In a similar way, we analyze the cavity-nanorod system by replacing the single
collective molecular resonance with a single mode plasmonic mode (see
Methods). By fitting the measured dispersion of transmission peaks with the
Hamiltonian eigenvalues, we obtain the energy spectrum of the coupled system,
Figure 4(a). For as long as the plasmonic array is located in a specific
horizontal plane $z=z_{0}$ of the cavity, the coupling strength strongly
disperses with the mode order $m$ as the result of alternating electric field
nodes and anti-nodes of the cavity’s standing wave (see Methods). Considering
the 4-th FP mode of a $9.7\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick cavity
resonant at about $2000\text{\,}{\mathrm{cm}}^{-1}$ (which is close to both
molecular resonances), we obtain the nanorod array coupling strength to that
cavity mode of about $172\text{\,}{\mathrm{cm}}^{-1}$. To help appreciate the
appearance of anti-crossings in this multi-mode system, we present the
resulting dispersion of the Hamiltonian eigenvalues in the full spectral range
in Fig. S7, which clearly demonstrate a set of anti-crossings between the
polaritonic states.

Finally, we switch to the complete cavity-plasmon-molecule systems. The key
idea of our experiment is to show that plasmonic nanorods can act as
artificial molecules with the oscillator strength significantly exceeding
molecular ones, joining the real molecules in a single collective oscillator,
and thus boosting the coupling strength and Rabi splitting. Therefore, we
model the full three-component structures with the same Jaynes-Cummings
Hamiltonian, in which the plasmon-molecule hybrid is described as a single
collective oscillator (see Methods). By fitting the measured dispersion of
transmission peaks of the hybrid plasmon-hexanal-cavity system with the
Hamiltonian eigenvalues, Figure 4b, we obtain a resonant coupling strength
(with the 5-th FP mode of a $8.0\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick
cavity and $z=500$ nm) of about $220\text{\,}{\mathrm{cm}}^{-1}$. Despite the
number of assumptions made in the analysis, the Hamiltonian fit shows a good
agreement with measured transmission peaks. Similarly, for the hybrid
plasmon-4-butylbenzonitrile-cavity system we obtain a resonant coupling
strength of $202\text{\,}{\mathrm{cm}}^{-1}$ (with the 8-th FP mode of a
$10.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick cavity), Figure 4c. Full
spectral range dispersions obtained for these samples, shown in Fig. S7, again
clearly demonstrate a set of anti-crossings between polaritonic states, which
within the experimentally accessible domain appear as straight lines.

The simple analysis performed above suggests that the molecules and the
nanorods act indeed as a collective oscillator with the effective coupling
strength very approximately given by the sum of the two individual coupling
strengths, $g_{eff}\sim g_{pl}+g_{mol}$. Thus, the presence of the plasmon
boosts the effective coupling strength of 4-butylbenzonitrile and hexanal with
the cavity mode by 10 times and 5 times, respectively. The results of our
experiment and its subsequent analysis suggest that using the plasmon
resonance with large oscillator strength indeed allows to boost the effective
coupling strength beyond the bulk limit bounded by the molecular
concentration. Of course, this addition of the plasmonic meta-atoms does not
modify the molecular oscillator strength per se, but rather modifies the
effective polaritonic spectrum of the hybrid system in the vibrational strong
coupling regime. This in turn may potentially affect chemical reactions whose
rate was claimed to depend on the vacuum Rabi splitting in the recent
literature. 30

##  4 Conclusion

In summary, our hybrid Fabry-Perot cavities show that the addition of a
plasmonic array to the standard molecular vibro-polaritonic system increases
the total coupling strength by almost an order of magnitude for a nitrile
absorption band and five times for a carbonyl absorption band. Increasing the
coupling strength beyond the molecular concentration limit, dismantle the
crucial obstacle for reaching the ultra-strong coupling regime using organic
molecules. Furthermore, precisely controlling the coupling strength, not only
with the molecular concentration, but also with the density of the plasmonic
array, allows molecules at small concentrations to reach the strong coupling
limit in a complementary fashion than the cooperating vibrational strong
coupling reported by Lather et al.26 One can in a sense view the plasmonic
array as a form of "catalyst", that enables any on-resonance molecular
transition, regardless of molecular concentration and transition dipole moment
strength, to reach the strong coupling regime. Furthermore, the cavity reduces
the radiative damping from the plasmon, sharpening the polariton linewidth
with more than an order of magnitude. Together with the spectral tuning
ability, such sharp linewidths may allow for mode-selective chemical sensing
in the mid-IR. The approach described here is not limited to infrared
transitions, but can also be transposed to electronic transitions. For these
reasons, we suggest that our hybrid system will be an ideal platform to
explore the promising potential of polaritonic chemistry, the ultra-strong
coupling regime, as well as provide an approach to mode-selective mid-IR
sensing.

B.M., D.G.B. and T.O.S. acknowledge financial support from the Swedish
Research Council (under VR Miljö project, grant No: 2016-06059) and the Knut
and Alice Wallenberg Foundation. K.B. and M.H. acknowledge financial support
from the European Research council (ERC-2017-StG-757733) and the Knut and
Alice Wallenberg Foundation (KAW 2017.0192).

The supporting information contains the following sections.

  * •

S1: Methods

  * •

S2: Additional figures

## References

  * Ebbesen 2016 Ebbesen, T. W. Hybrid Light–Matter States in a Molecular and Material Science Perspective. _Acc. Chem. Res._ 2016, _49_ , 2403–2412
  * Ribeiro et al. 2018 Ribeiro, R. F.; Martínez-Martínez, L. A.; Du, M.; Campos-Gonzalez-Angulo, J.; Yuen-Zhou, J. Polariton chemistry: controlling molecular dynamics with optical cavities. _Chem. Sci._ 2018, _9_ , 6325–6339
  * Flick et al. 2018 Flick, J.; Rivera, N.; Narang, P. Strong light-matter coupling in quantum chemistry and quantum photonics. _Nanophotonics_ 2018, _7_ , 1479–1501
  * Hertzog et al. 2019 Hertzog, M.; Wang, M.; Mony, J.; Börjesson, K. Strong light–matter interactions: a new direction within chemistry. _Chem. Soc. Rev._ 2019, _48_ , 937–961
  * Schwartz et al. 2011 Schwartz, T.; Hutchison, J. A.; Genet, C.; Ebbesen, T. W. Reversible Switching of Ultrastrong Light-Molecule Coupling. _Phys. Rev. Lett._ 2011, _106_ , 196405
  * Hutchison et al. 2012 Hutchison, J. A.; Schwartz, T.; Genet, C.; Devaux, E.; Ebbesen, T. W. Modifying Chemical Landscapes by Coupling to Vacuum Fields. _Angew. Chem. Int. Ed._ 2012, _51_ , 1592–1596
  * Orgiu et al. 2015 Orgiu, E.; George, J.; Hutchison, J. A.; Devaux, E.; Dayen, J. F.; Doudin, B.; Stellacci, F.; Genet, C.; Schachenmayer, J.; Genes, C.; Pupillo, G.; Samorì, P.; Ebbesen, T. W. Conductivity in organic semiconductors hybridized with the vacuum field. _Nat. Mater._ 2015, _14_ , 1123–1129
  * Coles et al. 2014 Coles, D. M.; Somaschi, N.; Michetti, P.; Clark, C.; Lagoudakis, P. G.; Savvidis, P. G.; Lidzey, D. G. Polariton-mediated energy transfer between organic dyes in a strongly coupled optical microcavity. _Nat. Mater._ 2014, _13_ , 712–719
  * Zhong et al. 2016 Zhong, X.; Chervy, T.; Wang, S.; George, J.; Thomas, A.; Hutchison, J. A.; Devaux, E.; Genet, C.; Ebbesen, T. W. Non-Radiative Energy Transfer Mediated by Hybrid Light-Matter States. _Angew. Chem. Int. Ed._ 2016, _128_ , 6310–6314
  * Zhong et al. 2017 Zhong, X.; Chervy, T.; Zhang, L.; Thomas, A.; George, J.; Genet, C.; Hutchison, J. A.; Ebbesen, T. W. Energy Transfer between Spatially Separated Entangled Molecules. _Angew. Chem. Int. Ed._ 2017, _56_ , 9034–9038
  * Stranius et al. 2018 Stranius, K.; Hertzog, M.; Börjesson, K. Selective manipulation of electronically excited states through strong light–matter interactions. _Nat. Commun._ 2018, _9_ , 2273
  * Munkhbat et al. 2018 Munkhbat, B.; Wersäll, M.; Baranov, D. G.; Antosiewicz, T. J.; Shegai, T. Suppression of photo-oxidation of organic chromophores by strong coupling to plasmonic nanoantennas. _Sci. Adv._ 2018, _4_ , eaas9552
  * Peters et al. 2019 Peters, V. N.; Faruk, M. O.; Asane, J.; Alexander, R.; Peters, D. A.; Prayakarao, S.; Rout, S.; Noginov, M. A. Effect of strong coupling on photodegradation of the semiconducting polymer P3HT. _Optica_ 2019, _6_ , 318–325
  * Shalabney et al. 2015 Shalabney, A.; George, J.; Hutchison, J.; Pupillo, G.; Genet, C.; Ebbesen, T. W. Coherent coupling of molecular resonators with a microcavity mode. _Nat. Commun._ 2015, _6_ , 5981
  * Long and Simpkins 2015 Long, J. P.; Simpkins, B. S. Coherent Coupling between a Molecular Vibration and Fabry–Perot Optical Cavity to Give Hybridized States in the Strong Coupling Limit. _ACS Photonics_ 2015, _2_ , 130–136
  * Simpkins et al. 2015 Simpkins, B. S.; Fears, K. P.; Dressick, W. J.; Spann, B. T.; Dunkelberger, A. D.; Owrutsky, J. C. Spanning Strong to Weak Normal Mode Coupling between Vibrational and Fabry–Pérot Cavity Modes through Tuning of Vibrational Absorption Strength. _ACS Photonics_ 2015, _2_ , 1460–1467
  * George et al. 2015 George, J.; Shalabney, A.; Hutchison, J. A.; Genet, C.; Ebbesen, T. W. Liquid-Phase Vibrational Strong Coupling. _J. Phys. Chem. Lett._ 2015, _6_ , 1027–1031
  * Hertzog and Börjesson 2020 Hertzog, M.; Börjesson, K. The Effect of Coupling Mode in the Vibrational Strong Coupling Regime. _ChemPhotoChem_ 2020, _4_ , 612–617
  * Menghrajani et al. 2019 Menghrajani, K. S.; Fernandez, H. A.; Nash, G. R.; Barnes, W. L. Hybridization of Multiple Vibrational Modes via Strong Coupling Using Confined Light Fields. _Adv. Opt. Mater._ 2019, _7_ , 1900403
  * Dunkelberger et al. 2019 Dunkelberger, A. D.; Grafton, A. B.; Vurgaftman, I.; Öney O. Soykal,; Reinecke, T. L.; Davidson, R. B.; Simpkins, B. S.; Owrutsky, J. C. Saturable Absorption in Solution-Phase and Cavity-Coupled Tungsten Hexacarbonyl. _ACS Photonics_ 2019, _6_ , 2719–2725
  * Pietron et al. 2019 Pietron, J. J.; Fears, K. P.; Owrutsky, J. C.; Simpkins, B. S. Electrochemical Modulation of Strong Vibration–Cavity Coupling. _ACS Photonics_ 2019, _7_ , 165–173
  * Xiang et al. 2019 Xiang, B.; Ribeiro, R. F.; Li, Y.; Dunkelberger, A. D.; Simpkins, B. B.; Yuen-Zhou, J.; Xiong, W. Manipulating optical nonlinearities of molecular polaritons by delocalization. _Sci. Adv._ 2019, _5_ , eaax5196
  * Menghrajani et al. 2019 Menghrajani, K. S.; Nash, G. R.; Barnes, W. L. Vibrational Strong Coupling with Surface Plasmons and the Presence of Surface Plasmon Stop Bands. _ACS Photonics_ 2019, _6_ , 2110–2116
  * Menghrajani and Barnes 2020 Menghrajani, K. S.; Barnes, W. L. Strong coupling beyond the light-line. _ACS Photonics_ 2020, _7_ , 2448–2459
  * Takele et al. 2020 Takele, W. M.; Wackenhut, F.; Piatkowski, L.; Meixner, A. J.; Waluk, J. Multimode Vibrational Strong Coupling of Methyl Salicylate to a Fabry–Pérot Microcavity. _J. Phys. Chem. B_ 2020, _124_ , 5709–5716
  * Lather et al. 2019 Lather, J.; Bhatt, P.; Thomas, A.; Ebbesen, T. W.; George, J. Cavity Catalysis by Cooperative Vibrational Strong Coupling of Reactant and Solvent Molecules. _Angew. Chem. Int. Ed._ 2019, _58_ , 10635–10638
  * Hirai et al. 2020 Hirai, K.; Takeda, R.; Hutchison, J. A.; Uji-i, H. Modulation of Prins Cyclization by Vibrational Strong Coupling. _Angew. Chem. Int. Ed._ 2020, _132_ , 5370–5373
  * Pang et al. 2020 Pang, Y.; Thomas, A.; Nagarajan, K.; Vergauwe, R. M. A.; Joseph, K.; Patrahau, B.; Wang, K.; Genet, C.; Ebbesen, T. W. On the Role of Symmetry in Vibrational Strong Coupling: The Case of Charge-Transfer Complexation. _Angew. Chem. Int. Ed._ 2020, _132_ , 10522–10526
  * Thomas et al. 2020 Thomas, A.; Jayachandran, A.; Lethuillier-Karl, L.; Vergauwe, R. M.; Nagarajan, K.; Devaux, E.; Genet, C.; Moran, J.; Ebbesen, T. W. Ground state chemistry under vibrational strong coupling: dependence of thermodynamic parameters on the Rabi splitting energy. _Nanophotonics_ 2020, _9_ , 249–255
  * Thomas et al. 2019 Thomas, A.; Lethuillier-Karl, L.; Nagarajan, K.; Vergauwe, R. M. A.; George, J.; Chervy, T.; Shalabney, A.; Devaux, E.; Genet, C.; Moran, J.; Ebbesen, T. W. Tilting a ground-state reactivity landscape by vibrational strong coupling. _Science_ 2019, _363_ , 615–619
  * Xiang et al. 2020 Xiang, B.; Ribeiro, R. F.; Du, M.; Chen, L.; Yang, Z.; Wang, J.; Yuen-Zhou, J.; Xiong, W. Intermolecular vibrational energy transfer enabled by microcavity strong light–matter coupling. _Science_ 2020, _368_ , 665–667
  * Vergauwe et al. 2019 Vergauwe, R. M. A.; Thomas, A.; Nagarajan, K.; Shalabney, A.; George, J.; Chervy, T.; Seidel, M.; Devaux, E.; Torbeev, V.; Ebbesen, T. W. Modification of Enzyme Activity by Vibrational Strong Coupling of Water. _Angew. Chem. Int. Ed._ 2019, _58_ , 15324–15328
  * Hirai et al. 2020 Hirai, K.; Hutchison, J. A.; Uji-i, H. Recent Progress of Vibropolaritonic Chemistry. _ChemPlusChem_ 2020, _85_ , 1981
  * Galego et al. 2019 Galego, J.; Climent, C.; Garcia-Vidal, F. J.; Feist, J. Cavity Casimir-Polder Forces and Their Effects in Ground-State Chemical Reactivity. _Phys. Rev. X_ 2019, _9_
  * Hiura and Shalabney 2019 Hiura, H.; Shalabney, A. A Reaction Kinetic Model for Vacuum-Field Catalysis Based on Vibrational Light-Matter Coupling. _ChemRxiv_ 2019, chemrxiv.9275777.v1
  * Khitrova et al. 2006 Khitrova, G.; Gibbs, H. M.; Kira, M.; Koch, S. W.; Scherer, A. Vacuum Rabi splitting in semiconductors. _Nat. Phys._ 2006, _2_ , 81–90
  * Hertzog et al. 2017 Hertzog, M.; Rudquist, P.; Hutchison, J. A.; George, J.; Ebbesen, T. W.; Börjesson, K. Voltage-controlled switching of strong light–matter interactions using liquid crystals. _Chem. Eur. J._ 2017, _23_ , 18166–18170
  * Berghuis et al. 2020 Berghuis, A. M.; Serpenti, V.; Ramezani, M.; Wang, S.; Rivas, J. G. Light–Matter Coupling Strength Controlled by the Orientation of Organic Crystals in Plasmonic Cavities. _J. Phys. Chem. C_ 2020, _124_ , 12030–12038
  * Roux et al. 2020 Roux, F. L.; Taylor, R. A.; Bradley, D. D. C. Enhanced and Polarization-Dependent Coupling for Photoaligned Liquid Crystalline Conjugated Polymer Microcavities. _ACS Photonics_ 2020, _7_ , 746–758
  * Savasta et al. 2010 Savasta, S.; Saija, R.; Ridolfo, A.; Stefano, O. D.; Denti, P.; Borghese, F. Nanopolaritons: Vacuum Rabi Splitting with a Single Quantum Dot in the Center of a Dimer Nanoantenna. _ACS Nano_ 2010, _4_ , 6369–6376
  * Manjavacas et al. 2011 Manjavacas, A.; de Abajo, F. J. G.; Nordlander, P. Quantum Plexcitonics: Strongly Interacting Plasmons and Excitons. _Nano Lett._ 2011, _11_ , 2318–2323
  * Schlather et al. 2013 Schlather, A. E.; Large, N.; Urban, A. S.; Nordlander, P.; Halas, N. J. Near-Field Mediated Plexcitonic Coupling and Giant Rabi Splitting in Individual Metallic Dimers. _Nano Lett._ 2013, _13_ , 3281–3286
  * Chikkaraddy et al. 2016 Chikkaraddy, R.; de Nijs, B.; Benz, F.; Barrow, S. J.; Scherman, O. A.; Rosta, E.; Demetriadou, A.; Fox, P.; Hess, O.; Baumberg, J. J. Single-molecule strong coupling at room temperature in plasmonic nanocavities. _Nature_ 2016, _535_ , 127–130
  * Rodrigo et al. 2015 Rodrigo, D.; Limaj, O.; Janner, D.; Etezadi, D.; de Abajo, F. J. G.; Pruneri, V.; Altug, H. Mid-infrared plasmonic biosensing with graphene. _Science_ 2015, _349_ , 165–168
  * Limaj et al. 2016 Limaj, O.; Etezadi, D.; Wittenberg, N. J.; Rodrigo, D.; Yoo, D.; Oh, S.-H.; Altug, H. Infrared Plasmonic Biosensor for Real-Time and Label-Free Monitoring of Lipid Membranes. _Nano Lett._ 2016, _16_ , 1502–1508
  * Singh 2016 Singh, P. SPR Biosensors: Historical Perspectives and Current Challenges. _Sens. Actuators, B_ 2016, _229_ , 110–130
  * Leitis et al. 2019 Leitis, A.; Tittl, A.; Liu, M.; Lee, B. H.; Gu, M. B.; Kivshar, Y. S.; Altug, H. Angle-multiplexed all-dielectric metasurfaces for broadband molecular fingerprint retrieval. _Sci. Adv._ 2019, _5_ , eaaw2871
  * Novotny and Hecht 2006 Novotny, L.; Hecht, B. _Principles of Nano-Optics_ ; Cambridge University Press, 2006
  * Törmä and Barnes 2014 Törmä, P.; Barnes, W. L. Strong coupling between surface plasmon polaritons and emitters: a review. _Rep. Prog. Phys._ 2014, _78_ , 013901
  * del Pino et al. 2015 del Pino, J.; Feist, J.; Garcia-Vidal, F. J. Quantum theory of collective strong coupling of molecular vibrations with a microcavity mode. _New J. Phys._ 2015, _17_ , 053040
  * Baranov et al. 2017 Baranov, D. G.; Wersäll, M.; Cuadra, J.; Antosiewicz, T. J.; Shegai, T. Novel Nanostructures and Materials for Strong Light–Matter Interactions. _ACS Photonics_ 2017, _5_ , 24–42
  * Climent et al. 2019 Climent, C.; Galego, J.; Garcia-Vidal, F. J.; Feist, J. Plasmonic Nanocavities Enable Self-Induced Electrostatic Catalysis. _Angew. Chem. Int. Ed._ 2019, _58_ , 8698–8702
  * Triana and Herrera 2020 Triana, J.; Herrera, F. Self-Dissociation of Polar Molecules in a Confined Infrared Vacuum. _10.26434/chemrxiv.12702419.v1_ 2020,
  * Bisht et al. 2018 Bisht, A.; Cuadra, J.; Wersäll, M.; Canales, A.; Antosiewicz, T. J.; Shegai, T. Collective Strong Light-Matter Coupling in Hierarchical Microcavity-Plasmon-Exciton Systems. _Nano Lett._ 2018, _19_ , 189–196
  * Baranov et al. 2020 Baranov, D. G.; Munkhbat, B.; Zhukova, E.; Bisht, A.; Canales, A.; Rousseaux, B.; Johansson, G.; Antosiewicz, T. J.; Shegai, T. Ultrastrong coupling between nanoparticle plasmons and cavity photons at ambient conditions. _Nat. Commun._ 2020, _11_ , 5561
  * Hopfield 1958 Hopfield, J. J. Theory of the Contribution of Excitons to the Complex Dielectric Constant of Crystals. _Phys. Rev._ 1958, _112_ , 1555–1567


# Rigidity for piece-wise smooth circle maps and certain GIETs

Przemysław Berk, Frank Trujillo