Strong coupling between various kinds of material excitations and optical modes has recently shown potential to modify chemical reaction rates in both excited and ground states. The ground-state modification in chemical reaction rates has usually been reported by coupling a vibrational mode of an organic molecule to the vacuum field of an external optical cavity, such as a planar Fabry–Pérot microcavity made of two metallic mirrors. However, using an external cavity to form polaritonic states might (i) limit the scope of possible applications of such systems and (ii) might be unnecessary. Here, we highlight the possibility of using optical modes sustained by materials themselves to self-couple to their own electronic or vibrational resonances. By tracing the roots of the corresponding dispersion relations in the complex frequency plane, we show that electronic and vibrational polaritons are natural eigenstates of bulk and nanostructured resonant materials that require no external cavity. Several concrete examples such as a slab of the excitonic material and a spherical water droplet in vacuum are shown to reach the regime of such cavity-free self-strong coupling. The abundance of cavity-free polaritons in simple and natural structures points at their relevance and potential practical importance for the emerging field of polaritonic chemistry, exciton transport, and modified material properties.

Strong coupling is a distinct regime of light–matter interactions realized when resonant optical modes and material excitations (electronic, vibrational, etc.) exchange energy faster than they lose it to the environment. This fast energy exchange gives rise to new quasiparticles: polaritons.1,2

Following the achievement of strong coupling, fundamental questions have arisen, such as whether polaritonic states could influence material and chemical properties.3,4 The possibility of affecting chemical reactions in the excited state (i.e., photochemistry) seems to be quite well understood today. Several experimental studies show that strong coupling can affect photo-chemical processes,5–7 and these results are essentially in agreement with theoretical descriptions.8–14 Thermally activated ground-state chemical reactions in the vibrational strong coupling (VSC) regime seem to be more controversial. In this case, just a few experimental reports exist,15–18 and there is far less conclusive agreement with theory.19–26 

An example of this controversy is given by common intuition, by stating that ground-state chemical reactions are local, i.e., the chemistry at position A does not depend on any far-away position B. Nevertheless, recent experimental studies15–18 suggest that such a local approximation to the chemistry might not always hold. Contemporary empirical observations3,4,27 indicate that the chemistry can be affected by resonant, collective, and non-local variables. For example, the reaction probability at a specific position depends on the far away presence (or absence) of metallic mirrors forming a Fabry–Pérot (FP) cavity. Specifically, it depends on the set of resonant optical modes that the mirrors form. This set of resonant optical modes and its relevance to chemistry are especially important, since they distinguish polaritonic chemistry from other common non-local but non-resonant contributions, such as screening, electrostatic, and solvent polarity effects.28–30 Moreover, while standard chemical theory states that the reaction rate depends on the concentration of molecules, concentration-dependent collective Rabi splitting influences the rate further by modifying the energy levels. Although this train of thought is heavily debated in the community, mainly due to a lack of theoretical understanding of the recent experimental observations, we believe that the potential of this new research frontier is quite high. Of course, further experiments and theoretical developments are needed to clarify this and other subtle issues.

Without trying to resolve the discrepancy of the ground-state chemistry in the VSC regime at this point, here, we are willing to focus on another potentially relevant issue. The so far experimentally reported observations of chemistry modifications under VSC exclusively use FP cavities comprised of two metallic mirrors.15– 18 Then, a natural question is if the role of the mirrors in these experiments is only to confine electromagnetic modes in order to form polaritons or if it goes beyond that. It is thus highly important to discuss the concept of “cavity” in the strong coupling regime and understand its relevance to polaritonic chemistry.

Traditionally, one requires an external cavity to reach strong coupling, such as a FP cavity, a plasmonic nanocavity, or a photonic crystal cavity, schematically visualized in Fig. 1(a). Assuming that the role of the cavity is solely to confine the optical field, we argue that using external cavities may limit the scope of potential applications, as such cavities are impractical.31 There also exist a number of the so-called “open” cavities, which might be more accessible and useful,1,2,6 but also require an external object to provide the optical mode confinement. Here, we show that to achieve strong coupling, the cavity does not need to be external. Instead, electronic or vibrational excitations in bulk or nanostructured materials can self-couple to an optical mode sustained by its own geometry, as illustrated in Fig. 1(b). We consider a generic medium with a Lorentz dielectric function, motivated by the fact that optical properties of most materials can be modeled as such.

FIG. 1.

(a) Examples of architectures for traditional polaritons based on a separate cavity mode provided by a Fabry–Pérot resonator, a plasmonic resonator, or a photonic crystal cavity. (b) Examples of self-hybridized polaritons based on slabs, long nanorods, and spheres made of resonant (excitonic/Lorentzian) materials.

FIG. 1.

(a) Examples of architectures for traditional polaritons based on a separate cavity mode provided by a Fabry–Pérot resonator, a plasmonic resonator, or a photonic crystal cavity. (b) Examples of self-hybridized polaritons based on slabs, long nanorods, and spheres made of resonant (excitonic/Lorentzian) materials.

Close modal

It is well known that objects with a refractive index different from their surrounding can sustain localized optical eigenmodes.32 The characteristics of the modes depend on the geometry and refractive indices of the involved objects and its surroundings. If such an optical resonance is found in proximity to the material resonance, these optical and material resonances can hybridize and, provided that the oscillator strength of the transition is high, give rise to polaritons. The Rabi splitting of such self-hybridized polaritons can approach (but cannot exceed) the so-called bulk polariton splitting, which is a function of oscillator strength only and does not depend on parameters of the cavity. Such self-coupled or cavity-free polaritons have been realized in many relevant systems, such as slabs of hexagonal boron nitride (hBN), transition metal dichalcogenides, perovskites, and plasmonic nanoparticles.33–40 However, their relevance for the polaritonic chemistry and modified material properties has not risen in the community.

Originally, the problem of material resonance from the quantum optical light–matter coupling perspective has been solved by Hopfield in his seminal work in 1958.41 Hopfield considered a three-dimensional (3D) continuous semiconductor and modeled it using a quantum coupled harmonic oscillator model but did not consider materials shaped into a specific object(s), which can sustain a set of discrete well-defined electromagnetic modes.

As we will show here, cavity-free polaritons are found in a great variety of structures, ranging from bulk materials to spherical microdroplets [Fig. 1(b)]. The sole existence and apparent abundance of such cavity-free polaritons indicate their applicability in polaritonic chemistry and, more generally, in any polariton-induced modifications of material properties.30 

We start by revisiting the standard light–matter interaction Hamiltonian in Hopfield’s formulation.41 This formulation is essentially a coupled harmonic oscillator problem approached from the quantum optical perspective. We note that this is not exactly the same model as the famous Jaynes–Cummings or quantum Rabi models, which deal with two-level quantum emitters (QEs) coupled to a cavity mode. However, within the weak excitation regime, this formulation is sufficiently appropriate even for QEs and even more so within the scope of this study, since we utilize a macroscopic Lorentz-resonant description of matter.

The full Hopfield Hamiltonian for a cavity mode oscillating at a frequency ωc interacting at a rate g with a material transition at a frequency ω0, written in the Coulomb gauge and including the diamagnetic term, reads

Ĥ=ω0b^b^+ωcââ+g(â+â)(b^+b^)+g2ω0(â+â)2,
(1)

where b^ and â are the annihilation operators of the material transition and the cavity mode, respectively, and g is the coupling constant.42 Two eigenvalues (resonant transitions) of this Hamiltonian are given by

ω±=ωc2+4g2ωc/ω0+ω02±(ωc2+4g2ωc/ω0+ω02)24ωc2ω022.
(2)

With the material oscillators homogeneously occupying the cavity mode volume V, the coupling strength g=μρV/3Evacω0/ωc (the factor 1/3 accounts for their isotropic orientation). Thus, the coupling strength depends on the transition dipole moment of the material transition, μ, and the volume density of these dipoles (oscillators), ρ, as well as the vacuum electric field of the cavity, Evac=ωc/2εε0V. When the coupling strength is large enough (with respect to the losses, which are neglected in the above description for simplicity), two polaritons with different energies [Eq. (2)] are formed, which has been explained in great detail before.2,43

The two main ingredients for achieving strong coupling are (i) an optical cavity with a large quality factor (Q) and a small mode volume (V) and (ii) an electronic (or vibrational) transition with a high transition dipole moment.2 Following these suggestions, there has been extensive research to obtain the best optical cavities, ranging from high-finesse FP cavities composed of distributed Bragg reflector (DBR) mirrors to plasmonic nanocavities. Many of these cavities are covered by Baranov et al.2 and some are depicted in Fig. 1(a).

In the examples above, we see that a stand-alone cavity is important for realizing the standard cavity-quantum electrodynamics (QED) scenario. That is the case of a single-QE strongly coupled to an external optical cavity. Such a scenario is challenging to realize with the cavity-free approach because a single QE problem cannot be treated in terms of a macroscopic Lorentz dielectric function even with the Hopfield Hamiltonian, as we do below, but instead requires a true cavity-QED approach, which we do not focus on here (these approaches are extensively covered in the literature1,44).

An optical mode is a concept that is not limited to the optical resonators and cavities discussed above. An optical mode is a solution of the source-free Maxwell equations in a given geometry. Thus, a cavity can be made of any material, including the material that makes up the relevant electronic or vibrational transitions themselves. These are precisely the scenarios we focus on in Secs. III A–III D.

Let us start by revisiting bulk polaritons, that is, hybrid light–matter states of an unbounded dielectric medium homogeneously filled with resonant transitions. This situation has been described by Hopfield in the celebrated 1958 paper41 (see also the work of Mills and Burstein45). Instead of a localized eigenmode of a cavity, the photonic modes in this scenario are represented by plane waves propagating in the unbounded dielectric medium with permittivity ε. Those waves have a continuous spectrum ω = kc/n, with k being the vacuum wave vector, c being the speed of light, and n=ε being the refractive index of the medium.

In the classical electromagnetic formalism, an isotropic resonant medium can be described by a Lorentzian permittivity,

ε(ω)=ε+fωP2ω02ω2iγω,
(3)

with ωP=ρe2/3ε0m being the plasma frequency, where ρ is the volume density of oscillators and 1/3 accounts for their isotropic orientation, e and m are the electron charge and mass, respectively, ω0 and γ are the resonance frequency and linewidth, and f is the oscillator strength. The latter is expressed via the microscopic parameters of the medium as f=2mω0e2μ2, where μ is the transition dipole moment of the material oscillator (see Sec. VI). For simplicity, in Eq. (3) and throughout this study (except for the case of hBN), we deal with only one resonance with an isotropic random orientation of transitions. Extensions of this simplified model can describe more complicated cases, e.g., by including several Lorentz resonances and anisotropy.

Bulk polaritons correspond to roots of the dispersion equation,

kcωε(ω)=0.
(4)

One can look for solutions of this equation either with complex ω and real k or complex k and real ω. These two types are both appropriate solutions of Maxwell’s equations but suited for different purposes.46 We are interested in complex-ω solutions of the dispersion equation, since they better reflect the transient decay of hybrid light–matter states of the coupled system.

Quantitative characteristics of the problem, such as the vacuum Rabi splitting and the magnitude of the polaritonic gap, can be obtained from the Hamiltonian formulation, which neglects exciton dissipation.41 Following the standard procedure of quantizing the electromagnetic field in free space, let us consider a quantization box of volume L3 with periodic boundary conditions.47 The vacuum electric field of a photon with frequency ωc reads Evac=ωc/2εε0L3. Recalling the expression for the oscillator strength f and plasma frequency ωP of a Lorentz medium, we express the resulting coupling strength as gC=(ωP/2)fω0/εωc. The vacuum Rabi splitting (with zero losses) is then obtained as the difference between the two polariton energies [Eq. (2)] for the zero-detuned photon (ωc = ω0),

ΩR=2g2+ω02+2gg2+ω022g2+ω022gg2+ω02=2g0,
(5)

where g0=(ωP/2)f/ε is the zero-detuning coupling strength.

The upper edge of the polariton gap is obtained by calculating the upper polariton energy in the limit k = 0 and is ω̃0=ω02+4g02, where ω̃0 is the re-normalized resonance frequency. The lower edge of the gap is found by calculating the lower polariton energy in the limit k, and is exactly the uncoupled exciton energy ω0. This results in the polaritonic gap

Δpol=ω02+4g02ω0,
(6)

which simplifies to Δpol2g02/ω0 under the assumption of 4g02/ω021, which holds in the standard strong coupling regime. In the formalism of dielectric permittivity, this bandgap can be interpreted as the Reststrahlen band of the material: the domain of energies wherein the real part of the permittivity becomes negative, thus forbidding propagation of plane waves.41 

The presence of losses in the dielectric formalism makes the eigenfrequencies complex and changes their dispersion; the original Hopfield model of bulk polaritons does not include any dissipation channels. Subsequent works extended Hopfield’s model to account for exciton decay by considering coupling between them and a continuum bath of harmonic oscillators.48 Such a model predicts that bulk polaritons appear if and only if the condition

fωP2>εγ2/4
(7)

holds. It is easy to see that this condition is equivalent to (ωP/2)f/ε=g0>γ/4, which is the strong coupling criterion in the standard Jaynes–Cummings model with a lossless cavity mode.48,49 Thus, by knowing the parameters of the material resonance in Eq. (3), it is possible to estimate the bulk coupling strength at zero detuning g0 and compare it to γ/4 in order to conclude whether or not this specific material supports bulk polaritons.

Figures 2(a)–2(c) show the dispersion (k vs real part of ω) of bulk polaritons [obtained as roots of Eq. (4)] of a medium described by the permittivity in Eq. (3), with typical values for TDBC J-aggregates (ω0 = 2.11 eV, γ = 0.1 eV, and ε = 2.15)50 and varying oscillator strengths. First, for fωP2=0.0045eV2, corresponding to a strongly diluted J-aggregate, we observe the mode attraction typical for the weak coupling regime [Fig. 2(a)]. In this regime, the energies of the eigenmodes of the medium slightly deviate from the bare values, but their dispersions cross near the zero detuning region.

FIG. 2.

Bulk polaritons. The upper panels show the dispersion of bulk J-aggregate polaritons considering the complex-k and real-ω solutions in a red solid line and the complex-ω and real-k eigenmodes in circles for (a) weak coupling with fωP2=0.0045eV2, (b) strong coupling with fωP2=0.445eV2, and (c) ultrastrong coupling with fωP2=3.116eV2, which displays a polaritonic gap shaded in blue. The dashed lines denote the dispersion of the uncoupled exciton (yellow at 2.11 eV) and the light dispersion in vacuum (gray). The lower panels show the trajectories of the complex frequency roots for the same values in the three regimes: (d) weak, (e) strong, and (f) ultrastrong coupling. The location of the uncoupled exciton (ω0/2) is marked as a yellow star. The Rabi splitting is marked by orange arrows. The color bar represents the value of k in μm−1.

FIG. 2.

Bulk polaritons. The upper panels show the dispersion of bulk J-aggregate polaritons considering the complex-k and real-ω solutions in a red solid line and the complex-ω and real-k eigenmodes in circles for (a) weak coupling with fωP2=0.0045eV2, (b) strong coupling with fωP2=0.445eV2, and (c) ultrastrong coupling with fωP2=3.116eV2, which displays a polaritonic gap shaded in blue. The dashed lines denote the dispersion of the uncoupled exciton (yellow at 2.11 eV) and the light dispersion in vacuum (gray). The lower panels show the trajectories of the complex frequency roots for the same values in the three regimes: (d) weak, (e) strong, and (f) ultrastrong coupling. The location of the uncoupled exciton (ω0/2) is marked as a yellow star. The Rabi splitting is marked by orange arrows. The color bar represents the value of k in μm−1.

Close modal

For a larger oscillator strength fωP2=0.445eV2 (see the work of Balasubrahmaniyam et al.),50 one can clearly see an anticrossing between the two polaritonic modes, with the vacuum Rabi splitting around 455 meV. We calculate the vacuum Rabi splitting as the energy separation along the real frequency axis between polaritonic modes obtained at a zero detuning between the electronic resonance of the material and the bare cavity mode (in turn, obtained by setting f = 0). This choice corresponds to nearly 50%/50% exciton/photon polariton wave-function amplitudes, and we use this definition throughout the text. On the same plot, we also show dispersion of complex-k/real-ω solutions (in a red solid line) found simply as k=ε(ω)ω/c. The two solutions coincide away from the resonance but differ in the vicinity of ω0. While the two branches of complex-ω solutions are disconnected, the two branches of the complex-k solutions are connected by the region of anomalous dispersion.

For even larger oscillator strength, fωP2=3.116eV2 (we note here that this value is unrealistically large for J-aggregates and we give it here only for the sake of theoretical argument; there may be other material systems where such oscillator strengths are feasible), the dispersion shows a larger vacuum Rabi splitting of about 1.2 eV and also displays a polaritonic gap—a region of frequencies with no propagating modes within it, shown in Fig. 2(c).41,44,51 The corresponding complex-k solutions, at the same time, span over the whole range of frequencies but become highly damped within the polariton gap by acquiring a large imaginary part of the wave vector k (see Fig. S1).

In addition to the anticrossing in the kω plane, it is instructive to inspect trajectories of the roots of the dispersion in Eq. (4) in the complex-ω plane. Figures 2(d)–2(f) show trajectories of the complex frequency roots for the three cases of weak, strong, and ultrastrong coupling. In the weak coupling, fωP2<ε(γ/2)2, the “photonic” root of the equation crosses the exciton position and acquires a small imaginary component in the zero detuning region, while the “excitonic” root (the one originating at the complex exciton frequency, marked with a yellow star in Fig. 2) moves along a finite curve around ω0/2. However, in the strong and ultrastrong coupling regimes, the low-energy solution terminates exactly at the exciton energy, while the upper solution originates on the other side of the polaritonic gap [Figs. 2(e) and 2(f)].

In closing this part, we want to emphasize that bulk polaritonic states and the whole anticrossing picture exist in an optically large piece of a resonant material without any external cavity.

The examples given above clearly illustrate that a large (compared to the resonance wavelength) unstructured excitonic or phononic material already possesses polaritonic modes, provided that the oscillator strength of the material exceeds the threshold value. Reducing the dimension of the system (by breaking the translational invariance in one, two, or all three dimensions) does not break this picture: the reduced system still possesses optical modes (either complex-frequency resonant states or real-frequency guided modes) that couple with the material resonance.

Let us first reduce the bulk material in one dimension (along the z-axis to be specific) and consider a slab with a Lorentzian permittivity ε(ω) in vacuum. The eigenmodes of this system can be quantified by the (real) in-plane momentum kx, ky.52 In the absence of the material resonance, the modes above the light line are FP resonances with complex frequencies (not to be confused with real-frequency leaky modes);53 and the modes below the light line are proper waveguide modes with real frequencies.54 Upon inclusion of the material resonance, it couples to modes both below and above the light line.

Figure 3(a) presents the eigenfrequency spectrum of TE-polarized eigenmodes of a 200 nm thick J-aggregate slab with fωP2=0.445eV2 obtained as roots of the characteristic Eq. (13) (see Sec. VI). Such a J-aggregate was also analyzed in Figs. 2(b) and 2(e) corresponding to the bulk strong coupling case. It shows the real part of eigenfrequencies vs the in-plane momentum kx on top of the reflection spectrum calculated above the light line. Eigenfrequencies of the slab are found numerically as poles of the reflection coefficient in the lower half plane of the complex frequency (see Sec. VI). Remarkably, spectral positions of FP modes above the light line do not always coincide with reflection dips, as shown in Fig. 3(a) for polaritonic modes and in Fig. S3 for bare modes (f = 0) of a dielectric slab. This behavior comes from non-resonant contributions of other poles of the dielectric slab, particularly the fundamental waveguide mode present at any kx.52 This mismatch should be always kept in mind in the analysis of experimental or simulated spectra of coupled exciton–polariton systems.

FIG. 3.

Polaritons in dielectric slabs. The upper panels show dispersion of the Fabry–Pérot (FP) and guided modes over the reflection spectra of TE polarized light of (a) a 200 nm thick J-aggregate slab with fωP2=0.445eV2 and (b) with fωP2=3.116eV2 and (c) a 1.75 μm thick slab of hBN with fωP2=0.064eV2. The lower panels show the trajectories, in the complex-ω plane, of the polaritons resulting from the first guided mode of the slab (TE1) hybridizing with the resonance in the same materials as above: (d) J-aggregates with low f—squares, (e) J-aggregates with high f—diamonds, and (f) hBN—down-pointing triangles for hybridization with TE1 and circles for that with TE2. The color bar represents kx. The Rabi splitting with TE1 is marked with a purple arrow. The uncoupled modes are shown in dashed and dotted lines.

FIG. 3.

Polaritons in dielectric slabs. The upper panels show dispersion of the Fabry–Pérot (FP) and guided modes over the reflection spectra of TE polarized light of (a) a 200 nm thick J-aggregate slab with fωP2=0.445eV2 and (b) with fωP2=3.116eV2 and (c) a 1.75 μm thick slab of hBN with fωP2=0.064eV2. The lower panels show the trajectories, in the complex-ω plane, of the polaritons resulting from the first guided mode of the slab (TE1) hybridizing with the resonance in the same materials as above: (d) J-aggregates with low f—squares, (e) J-aggregates with high f—diamonds, and (f) hBN—down-pointing triangles for hybridization with TE1 and circles for that with TE2. The color bar represents kx. The Rabi splitting with TE1 is marked with a purple arrow. The uncoupled modes are shown in dashed and dotted lines.

Close modal

The 200 nm thick slab with fωP2=0.445eV2 does not show any noticeable splitting above the light line [circles in Fig. 3(a)]: the coupling strength is not sufficient to overcome the large radiative decay of the FP mode. Below the light line, however, there is a clear anti-crossing between the exciton and the TE1 waveguide mode with a magnitude of Rabi splitting around 420 meV, which is smaller than the 455 meV of bulk Rabi splitting previously mentioned.

TE-polarized eigenmodes of a 200 nm thick J-aggregate slab with a higher oscillator strength fωP2=3.16eV2 demonstrate anticrossing both below and above the light line [Fig. 3(b)]. This J-aggregate was analyzed in Figs. 2(c) and 2(f) and corresponds to the bulk ultrastrong coupling case. In this case, the magnitude of the Rabi splitting (purple arrows in the rest of the panels) cannot be obtained because the upper polariton has a cut-off given by the light line and the corresponding guided eigenmode does not exist.

Next, we demonstrate self-hybridized polaritons in another practically relevant system: a slab of hexagonal boron nitride (hBN). hBN exhibits two prominent vibrational modes at 169 meV and 95 meV, respectively.55 We focus on TE-polarized modes, which couple only with the in-plane vibrations of hBN at 169 meV. The in-plane permittivity component of hBN can be described by a single Lorentz term with ε = 4.45, fωP2=0.064eV2, and γ = 1 meV, which approximates the experimental data in the relevant spectral range [Fig. S1(g)].55 

Figure 3(c) shows eigenfrequencies of TE-polarized eigenmodes of a 1.75 μm thick hBN slab. Clearly, there are anticrossings both below (circles and down-pointing triangles) and above (squares and triangles) the light line, thanks to the large oscillator strength of hBN’s vibrational transition. For the same reason, we also observe the lower polariton branch of the second FP mode above the light line, shown in blue triangles, and lower polariton of the TE2 waveguide mode below the light line (shown in orange circles). The Rabi splitting of the TE1 waveguide mode, which occurs at about kx = 1.5 μm−1, reaches a value of ≈116 meV, which is also below the bulk Rabi splitting, 2g0 ≈ 120 meV. Remarkably, below the light line, polaritons formed by interaction of the fundamental modes TE1 and TE2 with the phonon show pronounced anticrossing, even though the TE2 mode dispersion in a dielectric slab without the exciton does not approach the exciton energy [see Fig. 3(c)].

This behavior becomes more evident when we look at trajectories of the eigenmodes’ poles of the slab in the complex-ω plane. It turns out, that even for a vanishingly small oscillator strength fωP2, there are countably many poles in an arbitrarily small vicinity of the complex exciton frequency ω0/2 [see Fig. S4(b)], where the Lorentz permittivity ε(ω) has a simple pole [shown in Fig. S4(c)].56,57 Each of these poles represents an eigenmode of the Lorentzian slab: either a radiative mode or a localized waveguide mode, depending on the position of the pole frequency with respect to the complex plane cut.

The origin of this cluster of poles can be appreciated by considering the Lorentzian permittivity ε at complex frequencies [Fig. S4(c)]. One can clearly see that just below the complex exciton frequency, ω0/2, the permittivity takes arbitrarily large values, thus, in theory, allowing for the existence of higher-order FP and waveguide modes that are otherwise inaccessible at real frequencies, where the permittivity is bounded by |ε(ω)|ε+fωP2/γω0.

Because these poles are located so close, they usually merge into a single spectral feature in real-frequency spectral responses, often attributed to “uncoupled molecules.”58,59 In fact, as the complex frequency plane reveals, this feature stems from a whole new multitude of eigenmodes located in the small vicinity of the complex exciton energy (Fig. S4). Of course, only a finite number of these eigenmodes are physical, since the permittivity becomes highly non-local and loses its physical meaning as soon as the wavelength inside the material λ/Re(ε(ω)) at complex ω approaches the microscopic length scale of the medium near the pole.

Complex-plane trajectories also clearly reveal the Rabi splitting. Above the light-line, trajectories of the eigenmodes are limited by the low frequency cutoff on the one side and by the light line on the other side and, for this reason, are difficult to interpret (Fig. S5). Below the light line, however, only the upper polariton branches are bounded by the light line on the low energy side [Fig. 3(d)] and the resulting trajectories are very similar to those for bulk polaritons [see Fig. 2(e)]. Qualitatively similar pictures of anticrossing in the ωkx space and in the complex-ω plane are found for TM-polarized eigenmodes (Fig. S6).

Since the waveguide modes of a dielectric slab have virtually infinite lifetime in contrast to radiative FP modes, anticrossing below the light line appears at a much smaller oscillator strength. For an electrically thin film, we can establish an approximate analytical criterion of strong coupling for waveguide modes. Assuming an electrically thin dielectric film without the Lorentzian transition (k0Lε1, where L is the thickness of the film), the dispersion of the TE1 waveguide mode is approximately

kxk0(1+δk),  δk=(k0L)2(ε1)/8,
(8)

with δk being a small parameter for a thin film (see Sec. VI). The vacuum field of the bare TE waveguide mode at the center of the dielectric slab can be estimated as Evac=ωc/2ε0a2W, where W=1/I(kz,1)1k02δk is the waveguide mode’s effective width. Multiplying the vacuum field by the square root of the number of oscillators ρa2L in the volume a2L and the dipole moment μ, we find the coupling strength (at zero detuning, ωc = ω0 and k0 = ω0/c),

g=ωP2(k0L)fε12.
(9)

Comparing the above g to γ/4, we find the threshold oscillator strength required for anticrossing below the light line in a dielectric film,

fωP2>γ22(k0L)2ε1.
(10)

This condition may help estimate the minimal thickness L of an excitonic film required for the emergence of polaritonic modes below the light line for a given oscillator strength.

By reducing the system in one more dimension (say, along the y-axis), we arrive at one possessing translational invariance only in a single direction (along the x-axis). Thanks to this symmetry, eigenmodes of the one-dimensional system are characterized by a real-valued momentum kx. Similar to the 2D case, eigenmodes with kx > ω/c are guided modes with real frequencies; eigenmodes with kx < ω/c are radiating resonances with complex frequencies.

For simplicity, let us focus on 1D systems with a circular cross section. The states with kx = 0 (normal-incidence resonances of the cylinder) are classified by their polarization state (TE or TM) and the azimuthal number m.60 Eigenmodes with kx ≠ 0 split into TE and TM-polarized solutions only for m = 0 (the monopole harmonic).61 For m > 0, TE and TM polarizations couple, giving rise to hybrid modes named EHnm and HEnm. The HE11 mode (the dipole mode) is the only mode of a circular cylindrical dielectric waveguide not having a cutoff61 [see down-pointing traingles in Figs. S7(a) and S7(c)].

As a practical example, we demonstrate self-hybridized polaritons in infinite circular cylinders of a perovskite CsPbCl3. This material is described by a Lorentz permittivity with ε = 3.7, ω0 = 3 eV, γ = 87 meV, and fωP2=0.54eV2 that approximates the experimental data37 [see Fig. S1(a)]. Polaritons and polaritonic emission in long whiskers made of perovskites of various families have been demonstrated in a number of works.36,62

Figure 4 presents the eigenfrequency spectrum of TM-polarized monopole (m = 0) eigenmodes [obtained as roots of the characteristic Eq. (16); see Sec. VI] for the CsPbCl3 infinitely long circular cylinder with a radius of 250 nm. The resulting spectrum clearly shows anticrossing above the light line (involving the second order radiating mode) and below the light line (involving the TM0,1 waveguide mode). The anticrossing below the light line features a Rabi splitting of about 312 meV, which compares to bulk Rabi splitting of CsPbCl3 of 382 meV. Complex-ω trajectories of eigenfrequencies exhibit a familiar behavior with upward frequency pulling to the real axis above the light line and downward pulling below the light line [Figs. 4(b) and 4(c)]. Qualitatively similar eigenfrequencies are obtained for TE-polarized monopole modes (Fig. S8) and for the dipole TEM modes (Fig. S7).

FIG. 4.

Polaritons in infinite circular cylinders. (a) Dispersion of the polaritons given with the TM radiating modes of the CsPbCl3 cylinder, with a radius of 250 nm, above (first mode in circles and second mode in diamonds) and waveguide modes below the light line (first mode in squares and second in down-pointing triangles). Trajectories of the hybrid modes with (b) radiating and (c) waveguide modes in the complex-ω plane. The color bar shows the value of kx and the colored arrows show the Rabi splitting above (orange) and below (purple) the light line. The uncoupled optical modes are shown by dashed lines.

FIG. 4.

Polaritons in infinite circular cylinders. (a) Dispersion of the polaritons given with the TM radiating modes of the CsPbCl3 cylinder, with a radius of 250 nm, above (first mode in circles and second mode in diamonds) and waveguide modes below the light line (first mode in squares and second in down-pointing triangles). Trajectories of the hybrid modes with (b) radiating and (c) waveguide modes in the complex-ω plane. The color bar shows the value of kx and the colored arrows show the Rabi splitting above (orange) and below (purple) the light line. The uncoupled optical modes are shown by dashed lines.

Close modal

Finally, if we reduce the system in one more dimension, we end up with a compact scatterer supporting a discreet spectrum of localized eigenmodes.63,64 Because of the lack of translational invariance in any direction, all eigenmodes of the scatterer are complex-frequency radiating solutions.65 

To keep the analysis simple, we consider the most symmetric compact scatterer: a sphere whose eigenmodes are the well-known Mie resonances.66 These resonances split into TE- and TM-polarized modes, which are further classified by the angular index l and the radial number N. l = 1 modes are usually referred to as the dipole ones, l = 2 is referred to as the quadrupole ones, and so on. A detailed description of whispering gallery exciton–polaritons in GaAs spheres has been presented by Platts et al.33 Because of the lack of translational invariance, the anticrossing in spheres cannot be seen by scanning the eigenmode’s momentum k. Instead, one usually varies the radius of the sphere, which causes the eigenmodes to move in the ω-space.

As a practically interesting example of self-hybridized polaritons in spheres, we consider liquid water droplets. Liquid water has a strong vibrational resonance around 420 meV with the oscillator strength of fωP2=0.035eV2.67 Therefore, one may naturally expect formation of vibrational polaritons in Mie resonant micrometer-sized water droplets. We describe water by a Lorentz permittivity with ε = 1.75, ω0 = 0.42 eV, γ = 0.048 eV, and fωP2=0.035eV2 that approximates the experimental data [see Fig. S1(d)].67 

Figure 5(a) shows eigenfrequencies of TE1,1 (magnetic dipole) eigenmodes of a water sphere as a function of its radius r. The trajectory of the TE1,1 eigenmode (the first index indicates the angular number of the mode l and the second index stands for the radial number of the mode N) shows a clear anticrossing with a magnitude of about 30 meV for a sphere with r ≈ 1.1 μm [Fig. 5(c)]. This value is significantly lower than the corresponding bulk Rabi splitting for water ωPf/ε100 meV due to high radiative loss of the TE1,1 mode. The small value of the Rabi splitting in this case is also manifested in the shape of the dispersion curves, similar to the exceptional point (EP) regime with the square root dependence of eigenenergies on the perturbation parameter (such as the radius).68 

FIG. 5.

Polaritons in water spheres. The upper left panels show eigenfrequency dependence on the radius of a water sphere for (a) TE1,1 and (c) TM1,2 modes. The lower left panels show the trajectories of the complex −ω roots for the same modes (b) TE1,1 and (d) TM1,2 by varying the radius (shown in the colormap). The poles at zero detuning are noted in the trajectories as the vertices of the orange rectangle joined by the dashed line. The Rabi splitting is the width of such a rectangle in Re(ω). The right panels show the extinction efficiency, Qext, for water droplets of radius (e) r = 1 µm, which is too small to show strong coupling, and (f) r = 2.5 µm, which is large enough to show the splitting in Qext. Both plots are included in (g) where a range of sizes is covered.

FIG. 5.

Polaritons in water spheres. The upper left panels show eigenfrequency dependence on the radius of a water sphere for (a) TE1,1 and (c) TM1,2 modes. The lower left panels show the trajectories of the complex −ω roots for the same modes (b) TE1,1 and (d) TM1,2 by varying the radius (shown in the colormap). The poles at zero detuning are noted in the trajectories as the vertices of the orange rectangle joined by the dashed line. The Rabi splitting is the width of such a rectangle in Re(ω). The right panels show the extinction efficiency, Qext, for water droplets of radius (e) r = 1 µm, which is too small to show strong coupling, and (f) r = 2.5 µm, which is large enough to show the splitting in Qext. Both plots are included in (g) where a range of sizes is covered.

Close modal

We find that 1.1 μm is the smallest radius for which polaritonic modes appear in water spheres. Although the TM1,1 (electric dipole) eigenmode of smaller r ≈ 600 nm spheres with ε = 1.75 matches the vibrational resonance position, it has a much lower Q-factor, which prevents strong coupling in spheres of this size (Fig. S9).

Nevertheless, the TM1,2 mode of r ≈ 1.7 μm spheres does demonstrate a clear anticrossing with the water vibration [Figs. 5(b) and 5(d)], producing a Rabi splitting of about 70 meV, which approaches the bulk splitting of 100 meV. Complex-ω trajectories in both cases reveal the characteristic eigenfrequency pulling to the real axis [Figs. 5(c) and 5(d)] originating from large radiative losses of the bare optical mode compared to the exciton linewidth. With the increase in radius of the sphere, new eigenmodes corresponding to higher orbital and radial numbers (also having a higher Q-factor) will match the vibrational resonance position and show additional anticrossings and polaritonic modes.

We further investigate how the attainable Rabi splitting in water spheres depends on the eigenmode polarization and the orbital number. Figure 6(a) shows the Rabi splitting for TE and TM eigenmodes as a function of the orbital number l. One can see that the Rabi splitting of TE modes systematically exceeds that of TM modes for all orbital numbers, but both gradually approach the bulk water splitting, 2g0, which for water is ∼100 meV (see Table S1).

FIG. 6.

Limit of Rabi splitting. Normalized Rabi splitting for all the materials as a function of the optical mode for all the geometries discussed above: (a) spheres, (b) infinite cylinders, and (c) slabs. The guided modes are shaded in green, and the radiating modes are shaded in blue. The dimensions of the structures and the values of 2g0 are found in Sec. V of the supplementary material.

FIG. 6.

Limit of Rabi splitting. Normalized Rabi splitting for all the materials as a function of the optical mode for all the geometries discussed above: (a) spheres, (b) infinite cylinders, and (c) slabs. The guided modes are shaded in green, and the radiating modes are shaded in blue. The dimensions of the structures and the values of 2g0 are found in Sec. V of the supplementary material.

Close modal

We also illustrate how polaritonic modes of micrometer-sized water spheres manifest themselves in extinction spectra at real frequencies. For droplets of 1 µm radius, no splitting is observed in extinction, as shown in Fig. 5(e). On the other hand, Fig. 5(f) shows observable splitting for a 2.5 µm radius droplet. The false color plot in Fig. 5(g) clearly demonstrates an avoided crossing in the extinction efficiency of a sphere as function of its radius r, Qext = σext/πr2 (see Sec. VI), which is the quantity most easily accessible with optical techniques from a macroscopic ensemble of microspheres. An important note to make is that the anticrossing in extinction spectra, being the sum of all the eigenmodes, significantly overestimates the splitting. Absorption and scattering efficiencies, whose sum constitutes extinction, calculated in a wider range of water droplet parameters, are shown in Fig. S9.

Coincidentally, micrometer-sized droplets of liquid water that support vibrational self-polaritons are quite ubiquitous in nature. For example, a common size of water droplets in fogs, mists, clouds, or steams falls exactly into the range of 1 μm–10 μm that is needed for the formation of vibrational Mie self-polaritons. Due to surface tension, these droplets adopt a spherical geometry, thus forming an ideal natural platform for the observation of vibrational self-polaritons in strong and even ultrastrong coupling regimes. Note that the bulk polariton splitting of liquid water falls into the vibrational ultrastrong coupling regime, since g0H2O0.05 eV and g0H2O/ω0H2O=η0H2O0.12>0.1, which satisfies the ultrastrong coupling criterion.26 

In this section, we turn our attention to the apparent fact that nanostructures supporting cavity-free polaritons have a certain critical size below which no polaritonic behavior is observed. For example, water spheres below ≈1 μm in radius will not support polaritons. Such size-limiting behavior is, of course, not universal to water and should be observed in all Lorentz materials [e.g., rmin ≈ 98 nm for perovskite and ≈180 nm for J-aggregates in Fig. 6(a)]. Neither is it exclusive to the spherical geometry. For dielectric slabs, where the fundamental waveguide modes do not have a cutoff, the physical reason for the critical size is the delocalization of the electric field of the mode and a resulting weak coupling strength. For circular cylinders (also their monopole modes) and spheres, the size bound stems from the fact that the fundamental eigenmodes of electrically small systems are far blue-detuned with respect to the material transition resonance, the case in which efficient hybridization is not possible. Finally, we would like to mention that in the opposite case of exceedingly large 0D–2D objects, the result is trivial and reduces to the bulk case, which has no critical upper limit.

From the material-science perspective, the existence of the critical size suggests that the polaritonic behavior and hence modified material properties are supported only within a certain size range, given by the material resonant properties, its surrounding(s), and its shape.

It is also worth noting that, from the computational point of view, the particles of about the critical size, although they are on the order of several tens to several hundreds of nanometers, contain a large number of oscillators (atoms or molecules). Thus, it is challenging to self-consistently capture these collective cavity-free polaritons using atomistic methods, such as density functional theory,23,69 due to their high computational cost.

The diversity of realizations of strong coupling, both using cavities (Fig. 1) and in the cavity-free format (Figs. 2–5), raises a question about the limit of g and ΩR that one can observe using Lorentz materials. A particularly important question is whether it is the material or the cavity that plays the decisive role in this limit.

To shed light on this issue, we compare the bulk polariton coupling strength to the other realizations for the specific Lorentz material considered above (Fig. 6). From the presented data, it appears that the lossless bulk polariton splitting, 2g0=ωPf/ε (see Table S1 for the values for each material), indeed establishes an ultimate limit on vacuum Rabi splitting achievable with a given Lorentz material with any cavity mode. Indeed, a cavity of mode volume V supporting a localized eigenmode can host only as many as N = transitions. Assuming that they all couple to the mode with an identical coupling strength, which is certainly an overestimation, the resulting Rabi splitting is seen to be bound from top by the bulk polariton coupling. The lossless bulk polariton splitting is, in turn, limited from above by setting f = ε = 1, which gives ΩRmax=2g0max=ωP. Thus, the maximum bulk polariton splitting is principally limited from above by the volume density of oscillators—ρ (also a combination of natural constants).

It is also instructive to consider the maximum ratio between the coupling strength and the resonance frequency ηmax=g0max/ω0=ωP/2ω0. This ratio plays an important role when the coupling strength approaches the transition frequency. Specifically, when η > 0.1, one usually distinguishes the ultrastrong coupling (USC) regime, while when η > 1, one distinguishes the deep strong coupling (DSC) regime. Taking in mind the principal limits of the coupling strength, the parameter η is seen to be intrinsically related to the ratio of plasma and resonant frequencies in Lorentz materials. This ratio can be both below one and significantly above one (in principle, unbound) depending on the material. However, one should keep in mind that in the limit of DSC, the light and matter components of the polaritonic states actually decouple, presenting nearly uncoupled optical mode and material excitation as the eigenstates of the system.70 This decoupling can be seen as the result of the screening by the diamagnetic term. In order to maintain equal cavity/matter fractions, one would have to tune the matter transition in resonance with the renormalized cavity mode. The same decoupling occurs in the dipole gauge, where the matter resonance experiences a de-polarization shift due to dipole–dipole interaction of the electronic subsystem.71 In fact, considering the ratio between the coupling constant and the renormalized resonance frequency η=g0/ω̃0=g0/ω02+4g02, one can see that the latter is principally bound by η ≤ 1/2.72 This hard limit indicates impossibility of reaching DSC with Lorentz media (if normalization is performed with respect to the renormalized frequency).

Since the Rabi splitting is limited from above by the bulk polariton coupling strength, having a separate cavity at best allows confining polaritons in space but does not increase the coupling strength due to the above limitation. As we showed above, the optical mode confinement can be attained without a separate cavity in the first place, with the exciton (or phonon) material itself playing the role of a confining resonator.

The bulk polariton splitting limit may serve as practical guidance for the selection of the material platform in strong coupling experiments. This limit is the function of the material only and does not depend on the nature of the cavity. For any practical realization of strong coupling, the observed Rabi splitting can only approach that of the bulk, 2g0=ωPf/ε, an estimation of which is straightforward (requires no complicated simulation). Thus, a material with a high enough oscillator strength will naturally support polaritonic eigenstates in all studied 0D–3D regimes.

In summary, we have shown that polaritonic states are natural and ubiquitous to bulk materials and nanostructures that can be described by a generic Lorentz resonance(s). Such polaritons are universally encountered in systems that do not involve any stand-alone cavities or metallic mirrors. In these cavity-free systems, boundaries of the Lorentz medium play the role of the mirrors that allow formation of well-defined optical modes, which, in turn, couple to the resonant transition. We demonstrated such cavity-free polaritonic states in 3D-bulk materials, 2D-slabs, 1D-cylinders, and 0D-spheres. Concrete examples of real material systems including spherical water droplets, perovskite nanocylinders, and slabs of J-aggregates and hBN were provided.

These findings might have important implications for the recently observed modification of material and chemical properties seen in electronic and vibrational strongly coupled systems. In particular, from the classical electromagnetism point of view, as well as the Hopfield Hamiltonian treatment, the cavity-free polaritons are no different from the more standard microcavity polaritons containing dense molecular layers. Thus, all observations relevant to the cavity polaritons could just also be applicable to the cavity-free polaritons presented here, provided that it is indeed the polaritonic nature of those modes that is responsible for chemistry and material science modifications (other reasons such as the usage of highly polarizable metallic mirrors as was done in nearly all experimental studies so far might also play a role31). However, in the cavity-free case, the exact polaritonic eigenmode picture depends on the size and shape of the resonant material and its environment. Thus, if we accept the hypothesis that polaritonic states themselves (not the presence of metallic mirrors) are able to modify material and chemical properties, then these material and chemical properties should be dependent on the size and shape of this material, in agreement with the cavity-free polaritonic eigenmode picture. Clearly, such a cavity-free approach to modified material properties is beneficial from the practical point of view, as many of these structures exist naturally and is thus trivial to fabricate. A common concrete example considered here are the water droplets (encountered in fogs, mists, and clouds), which have a sharp cutoff below the radius of about 1 μm. Smaller water droplets will not support vibrational Mie self-polaritons and thus may experience different chemical properties in comparison to bigger droplets that support vibrational Mie self-polaritons and to bulk water that supports bulk vibrational polaritons. Similar consequences may apply to J-aggregates, hBN, etc., as well as to any Lorentz material with a high enough oscillator strength of the relevant (electronic or vibrational) transition. This reasoning applies also to polariton-assisted modification of other material-related properties. We look forward to experimental tests of these predictions.

The permittivity of a resonant medium containing electronic or vibrational transitions can be obtained by calculating the electric dipole polarizability of a single such transition and combining it with the volume density of homogeneously distributed transitions ρ in the medium.

The dipolar polarizability of a two-level system with the transition dipole moment μ in a weak external field can be written as

α^TLS=f^e2/ε0mω02ω2iγω,   f^=2mω0e2μμ,
(11)

where f^ is the transition’s oscillator strength, m is the electron mass, e is the electron charge, and ⊗ is the outer product73 (note ε0 in the expression for polarizability, as required in SI units). Combining this with the volume density of the two-level systems ρ and assuming random orientation of transition dipole moments, one easily obtains the expression for the polarization density and permittivity of the medium,

ε=ε+fωP2ω02ω2iγω,
(12)

where ε is the non-resonant background permittivity, ωP2=ρe2/(3ε0m) is the plasma frequency, and the factor 1/3 accounts for random isotropic orientation of dipoles in the medium.

Eigenfrequencies of a planar slab of thickness L are found as poles of the reflection coefficient in the complex frequency plane. The reflection coefficient of a TE or TM polarized plane wave with the wavenumber k0 = ω/c and in-plane momentum kx reads

RTE,TM=rTE,TM(1e2ikz,2L)1rTE,TM2e2ikz,3L,
(13)

where

rTE=kz,1kz,2kz,1+kz,2,  rTM=kz,1kz,2/ε(ω)kz,1+kz,2/ε(ω),
(14)

with kz,1=k02kx2 and kz,2=ε(ω)k02kx2 being the z-components of the wave vector in vacuum and dielectric, respectively.

kz,1 has a branching point at kx = k0, and a cut needs to be made. The field of the eigenmode outside the slab is described by the phase factor e±ikz,1z, where the plus and minus sign is applied in the regions z > L/2 and z < − L/2 correspondingly. The radiating modes above the light line should have R(kz,1)>0, while proper waveguide modes have I(kz,1)>0. Therefore, we make the branch cut along the negative imaginary axis through ω = −i∞ and choose the Riemann sheet of the square root function such that R(kz,1)>0 above the light line and I(kz,1)>0 below the light line. This choice ensures that we find radiating eigenmodes and proper (localized) waveguide modes. Choosing another Riemann sheet of the square root in the definition of kz,1 results in dispersion of improper (diverging) modes below the light line. The choice of sheet for kz,2 is not important because the slab contains both waves with ±kz,2.

Dispersion of TE waveguide modes in a dielectric film is given by the roots of the characteristic equation 1rTE2e2ikz,2L=0. This is a transcendental equation and, in a general case, cannot be solved analytically. However, in the case of an electrically thin film (k0Lε1), the equation can be linearized and an approximate solution of the fundamental TE1 mode can be found.

For an electrically thin film, dispersion of the TE1 mode is sandwiched between the light line of the environment (k0 = ω/c) and that of the slab (k=εω/c) and follows very close to the environment light line; therefore, we can write the propagation constant of the guided mode as kx = k0 + δk, where δkk0. The z components of the wave vectors in the environment and in the slab can be expanded as follows:

kz,1=k02kx2i2k0δk,
kz,2=εk02kx2k0ε11δk(ε1)k0.

Note that kz,1 scales as O(δk), whereas the leading correction to kz,2 scales as O(δk). The exponential phase factor can be expanded in a Taylor series e2ikz,2L1+2ikz,2L. Plugging these expansions into the characteristic equation, we obtain

kz,1kz,2kz,1+kz,22(1+2ikz,2L)1=0,i(kz,122kz,1kz,2+kz,22)L=2kz,1.

Keeping only the leading power O(δk) of the kx expansion and the constant term O(1) in this equation, we find δk=k03L2(ε1)/8, and the approximate dispersion relation of the TE1, correspondingly, can be written as

kxk0+k03L2ε18.
(15)

TE (TM) modes of an infinitely long cylinder are defined as those having their electric (magnetic) field strictly perpendicular to the cylinder axis. Eigenfrequencies of TE01 and TM01 (monopole transverse electric and transverse magnetic, respectively) waveguide modes of an infinitely long circular cylinder of radius a are found as roots of the following characteristic equations:60,61

εkρ,2J1(kρ,2a)J1(kρ,2a)1kρ,1H1(kρ,1a)H1(kρ,1a)=0,    (TM)
(16)
1kρ,2J1(kρ,2a)J1(kρ,2a)1kρ,1H1(kρ,1a)H1(kρ,1a)=0,    (TE)
(17)

where J1(z) and H1(z)=H1(1)(z) are Bessel and Hankel functions of the first kind, respectively, and kρ,1=k02kx2 and kρ,2=ε(ω)k02kx2 are the radial wavenumber of the eigenmode in vacuum and dielectric, respectively.

The definition of kρ,1 requires a branch cut at kρ,1 = k0. Radial dependence of the eigenmode field outside the waveguide is given by the factor H1(kρ,1ρ). Therefore, making the branch cut going along the negative imaginary axis through ω = −i∞ and choosing the Riemann sheet yielding Re(kρ,1) > 0 above the light-line results in the spectrum of radiating eigenmodes and localized waveguide modes.60 

Eigenfrequencies of the HE11 (dipole) mode are found as roots of the following characteristic equation:61 

1kρ,2J1(kρ,2a)J1(kρ,2a)1kρ,1H1(kρ,1a)H1(kρ,1a)1kρ,2J1(kρ,2a)J1(kρ,2a)  1kρ,1H1(kρ,1a)H1(kρ,1a)=kxk0(ε1)k02a2kρ,12kρ,222.    (HE11).
(18)

TE (TM) modes of a sphere are defined as those having their electric (magnetic) field strictly perpendicular to the radial direction. Eigenfrequencies of TE and TM polarized modes with orbital number l are found as roots of the corresponding characteristic equations,

ψl(nx)ξl(x)nξl(x)ψl(nx)=0,    (TE)
(19)
nψl(nx)ξl(x)ξl(x)ψl(nx)=0.    (TM).
(20)

Scattering coefficients an and bn of the sphere of a sphere of radius a and permittivity ε = n2 in vacuum in the basis of TE and TM polarized spherical harmonics are given by the classical Mie solution,66 

an=nψl(nx)ψl(x)ψl(x)ψl(nx)nψl(nx)ξl(x)ξl(x)ψl(nx),
(21)
bn=ψl(nx)ψl(x)nψl(x)ψl(nx)ψl(nx)ξl(x)nξl(x)ψl(nx),
(22)

where x = k0a, ψl(x) = xjl(x), and ξl(x)=xhl(1)(x) are Ricatti–Bessel functions and jl(x) and hl(1)(x) are spherical Bessel and Hankel functions of the first kind, respectively.

The extinction and scattering efficiencies of the sphere are given by the sum over all harmonics,66 

Qext=2x2n=1(2n+1)R(an+bn),
(23)
Qsca=2x2n=1(2n+1)(|an|2+|bn|2).
(24)

See the supplementary material for the additional mode analysis, permittivities of the materials, and scattering and absorption spectra of water droplets calculated in a wider size range.

A.C., D.G.B., and T.S. acknowledge the financial support from the Swedish Research Council (VR Project Grant No. 2017-04545 and VR Miljö Grant No. 2016-06059). T.J.A. acknowledges support from the Polish National Science Center (Project Grant No. 2019/34/E/ST3/00359). Analytical calculations were supported by the Russian Science Foundation (Grant No. 19-79-00362).

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

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