Strong Coupling beyond the Light-Line

Abstract

Strong coupling of molecules placed in an optical microcavity may lead to the formation of hybrid states called polaritons; states that inherit characteristics of both the optical cavity modes and the molecular resonance. Developing a better understanding of the matter characteristics of these hybrid states has been the focus of much recent attention. Here, as we will show, a better understanding of the role of the optical modes supported by typical cavity structures is also required. Typical microcavities used in molecular strong coupling experiments support more than one mode at the frequency of the material resonance. While the effect of strong coupling to multiple photonic modes has been considered before, here we extend this topic by looking at strong coupling between one vibrational mode and multiple photonic modes. Many experiments involving strong coupling make use of metal-clad microcavities, ones with metallic mirrors. Metal-clad microcavities are well-known to support coupled plasmon modes in addition to the standard microcavity mode. However, the coupled plasmon modes associated with a metal-clad optical microcavity lie beyond the light-line and are thus not probed in typical experiments on strong coupling. Here we investigate, through experiment and numerical modeling, the interaction between molecules within a cavity and the modes both inside and outside the light-line. Making use of grating coupling and a metal-clad microcavity, we provide an experimental demonstration that such modes undergo strong coupling. We further show that a common variant of the metal-clad microcavity, one in which the metal mirrors are replaced by distributed Bragg reflector also show strong coupling to modes that exist in these structures beyond the light-line. Our results highlight the need to consider the effect of beyond the light-line modes on the strong coupling of molecular resonances in microcavities and may be of relevance in designing strong coupling resonators for chemistry and materials science investigations.

Strong coupling between confined light fields and ensembles of molecules is attracting increasing attention owing to the potential such coupling offers as a new tool to control molecular properties, especially chemical and material properties.¹⁻⁸ In strong coupling, the interaction between the molecules and the confined light field, for example, those associated with an optical microcavity, leads to the formation of new hybrid states called polaritons, states that inherit both molecular and photonic properties.⁹ The key features offered by this approach are the very significant control over polariton energy levels and the coherent, delocalized nature of the polariton states. The effects of strong coupling on a number of processes have been investigated, for example, chemical reaction rates¹⁰ and exciton transport.¹¹ In addition, polaritons have been the focus of attention for room-temperature condensation phenomena¹²⁻¹⁴ and for lasing.¹⁵⁻¹⁷ A full understanding of the hybrid light–matter polariton states is important in all of these investigations, and much attention has been devoted recently to understand better the matter aspects of this type of light–matter interaction.¹⁸⁻²² Here we highlight the need for a better understanding of the role the different optical modes supported by typical cavity structures play in strong coupling. As we show below, typical microcavities support more than one mode at the frequency of the material resonance, whereas previous work has considered only one of these modes. The typical microcavities to which we refer are planar Fabry–Perot-type cavities where two mirrors trap light inside a planar region in which the molecules of interest are placed. Two types of mirror are in common use, metallic thin films²³ and multilayer dielectric (distributed Bragg reflectors (DBR))^9,24 or a combination of the two.²⁵

Many experiments on strong coupling of organic molecules, both in the visible and infrared spectral regions, make use of metal-clad microcavities.²⁶ Metal-clad microcavities are attractive because they are generally easy to fabricate and because they offer good field confinement,²³ thus, increasing the extent of the modification of the energy levels through the so-called Rabi-splitting. Strong coupling is usually investigated between the lowest-order cavity mode and an excitonic^9,14,16,23 or vibronic^1,27 molecular resonance. The dispersion of microcavity modes is well-known^28,29 and easily determined through angle-resolved measurements of, for example, reflectance and photoluminescence.^30,31 However, metal-clad microcavities also support a coupled surface plasmon mode.^32,33 Although previous work has investigated strong coupling between this mode and the excitonic resonances,³⁴⁻³⁷ it appears to have been largely overlooked in the context of vibrational resonances. Similarly, a number of authors have considered the effect of strong coupling excitonic resonances to multiple photonic modes,^38,39 with interesting conclusions that we will return to in the discussion. Here we extend this topic by looking at the simultaneous strong coupling of one vibrational resonance and to photonic modes that lie inside and outside the light-line; this work thus compliments that of George et al.⁴⁰ who looked at the strong coupling of a vibrational resonance to multiple photonics modes, all of which were within the light-line.

The coupled plasmon mode may be understood as a hybrid mode associated with what would otherwise be two degenerate surface plasmon modes if the metallic mirror surfaces of the cavity were far apart. When the mirrors are separated by a wavelength or less the fields of these modes overlap, and two new coupled modes may form.³² One of the coupled modes is the standard lowest-order TM mode of a microcavity, which we label here the TM₀ mode, the other is the coupled plasmon mode,⁴¹ which we label the TM_–1 mode. This coupled plasmon mode is sometimes known as a gap mode plasmon,^42,43 a particle-on-film plasmon,⁴⁴⁻⁴⁶ or a MIM plasmon (metal–insulator–metal).^32,47,48 The TM₀ mode also has a TE-polarized equivalent, the TE₀ mode, which will also be discussed below.

The important characteristic of the coupled plasmon mode for the present work is that it has no lower frequency cutoff, enabling an outstanding degree of field confinement to be achieved. Indeed, this combination of field confinement and no cutoff has been exploited recently to achieve single molecule strong coupling, by placing a molecule in a gap only a few nm thick, between a planar metal film and an overlying metal particle.⁴⁴ (For this particle-on-film configuration, the particle geometry breaks the translational symmetry so that the coupled plasmon mode in this case lies inside the light-line.) However, the focus of the present report is not the attainment of single molecule strong coupling, but rather ensemble strong coupling involving large numbers of molecules, that is, situations where chemical and material properties might be changed.^4,7,49,50 For ensemble strong coupling, “standard” microcavities are generally employed that support the usual cavity (TM₀) mode. It is in the context of these “standard” microcavities that we consider the role of the coupled plasmon (TM_–1) mode on strong coupling with molecular vibrational resonances.

We show below that the coupled plasmon mode, though so far not seen in vibrational strong coupling experiments, is nonetheless present, and that it too is strongly coupled to the molecular resonance. Since such modes are beyond the light-line we need to provide some kind of momentum matching, here we do this by introducing a grating structure into the cavity, allowing the coupled plasmon mode to be momentum matched to incident light and, thus, probed by grating scattering.⁵¹ Furthermore, we show that beyond-the-light-line modes also exist for DBR-based (distributed Bragg reflector) cavities.

Results and Discussion

In this work we make use of strong coupling between the molecular vibrational resonance of the C=O bond in the polymer poly(methyl methacrylate) (PMMA) at 1732 cm^–1 and the resonant infrared modes of suitable metallic microstructures. Strong coupling involving vibrational resonances^1,27,52−61 has been less intensively studied than that of excitonic resonances; however, vibrational resonances have allowed strong coupling to be observed in liquids⁶² and in transition metal complexes,⁶¹ as well as in liquid crystals.⁶³ In addition, strong coupling of vibrational resonances has been reported both to catalyze and inhibit chemical reactions,¹⁰ and to allow the control of the nonlinear infrared properties.⁶⁴ For the present work, vibrational resonances offer the benefit of narrow line width transitions and the advantage of involving longer wavelengths, thus, simplifying some of the fabrication tolerances required.

In Figure 1 we summarize the structures and modes that form the basis of the investigation reported here. The top row shows schematics of the structures considered, the lower row shows numerically calculated dispersion plots to indicate the modes supported by the different structures. We look at two types of confined light field, those associated with the surface plasmon mode of a single metallic interface, Figure 1a, and those associated with a planar metal microcavity, Figure 1b,c. For the microcavity, we consider two different cavity thicknesses, one that places the usual cavity cutoff at the same frequency as the molecular resonance, Figure 1b, and the other (thinner) only supports the usual lowest order cavity mode for frequencies much higher than the molecular resonance, Figure 1c.

Figure 1.

Structures and modes investigated. Top row, structures considered; middle row, dispersion plots for zero oscillator strength; lower row, dispersion plots for finite oscillator strength. Top row, schematics: (a) A single planar gold metal film (30 nm) overlaid with a 2 μm polymer PMMA film, supported on a CaF₂ substrate; this structure is used to investigate the surface plasmon mode; (b) Standard microcavity: a 2 μm thick film of PMMA is sandwiched between two 30 nm thick planar gold mirrors, the substrate is CaF₂; (c) A microcavity below the usual cutoff: this structure is the same as for (b), except that here the PMMA thickness is 100 nm, the substrate is CaF₂. Middle row, dispersion for zero oscillator strength: based on numerical calculations for the structures shown in the top row, showing the dispersion of the TM-polarized modes supported by these structures. Calculations were based on Fresnel coefficients, and the absolute value of the complex p-polarized amplitude transmission coefficient is shown as a function of frequency (wavenumber) and in-plane wavevector. For the data shown in this row, the oscillator strength was set to zero. Bottom row, dispersion for finite oscillator strength. For the dispersion plots (middle and lower rows), the modes indicated are SP, surface plasmon; TM₀, standard cavity mode; TM_–1, coupled plasmon mode. The horizontal white dashed line in each dispersion plot indicates the position of the molecular resonance. Also shown in each dispersion plot are two light lines: these are the air light-line, blue-dashed; and the polymer light-line (assuming the C=O resonance is absent), green-dashed. Note that to calculate the dispersion data shown here, we set the refractive index of the superstrate and the substrate to be n = 10; we did this to avoid the calculated data showing surface plasmon modes associated with the metal/air and metal/substrate interfaces. Further calculations (see Supporting Information) indicate that making this choice does not significantly alter the dispersion of the modes in which we are interested.

The dispersion of the modes supported by these structures is also shown in Figure 1. Here, a Fresnel coefficient formalism⁶⁵ was used to calculate the absolute value of the p-polarized amplitude transmission coefficient, |t_p|, which we have plotted as a function of frequency and in-plane wavevector. (The in-plane wavevector, k_x, is the wavevector component in the plane of the structure.) For these calculations, the PMMA molecular resonance at 1732 cm^–1 was represented by a single Lorentzian oscillator; details of this model, together with other parameters used, are given in the Methods section. The middle row shows the dispersion for the uncoupled modes, achieved by setting the oscillator strength for the PMMA to be zero. In the lower row, the oscillator strength is restored so as to indicate the coupling.

The surface plasmon (SP) mode is considered in the left-hand column. The structure consists of a CaF₂ substrate coated with a 30 nm thick planar layer of gold, completed by adding a 2 μm thick film of the polymer PMMA, Figure 1a. The data in Figure 1d show the dispersion of the surface plasmon mode, indicated by the bright region. In this plot, the oscillator strength has been set to zero, as it has in every plot in this row. This mode lies, as expected, just beyond the air light-line (indicated as a blue dashed line). Importantly, when we turn the molecular resonance back on, that is, reset the oscillator strength, then a clear anticrossing of this mode is seen where the mode’s frequency matches the frequency of the C=O molecular resonance, shown in Figure 1g. This anticrossing is the result of strong coupling between the surface plasmon mode and the C=O resonance. Although the surface plasmon mode is beyond the light-line, it is not the main focus of the investigation reported here. However, we have begun by examining this mode so as to provide a familiar starting point, strong coupling to surface plasmon modes has been seen before, access to the beyond the light-line SP mode being possible by prism⁶⁶ and grating coupling.⁶⁷ We also note that grating coupling has been used to observe strong coupling between waveguide modes that lie beyond the light-line and excitons in quantum wells,⁶⁸ while prism coupling has been used to observe similar coupling in a polymer waveguide.⁶⁹ Strong coupling between vibrational molecular resonances and surface plasmon modes has also been observed before.^58,70,71 Next we shift our attention to perhaps the most studied cavity for strong coupling involving molecular resonances, the metal clad microcavity; the remainder of Figure 1 is devoted to this system.

The structure of the metal-clad microcavity is shown in Figure 1b; a cavity has been formed by adding an upper layer (30 nm) of gold to the structure used for the surface plasmon investigation (Figure 1a). For this central column of Figure 1, the cavity thickness was chosen to be 2 μm so as to place the usual fundamental cavity mode resonance close to the molecular resonance of the C=O bond at 1732 cm^–1. The dispersion of the TM-polarized modes supported by this structure are shown in Figure 1e, where, as for Figure 1d, the absolute value of the p-polarized transmission amplitude coefficient has been plotted as a function of frequency and in-plane wavevector. Two modes are seen in these data, and both show an anticrossing when the molecular resonance is turned on, Figure 1h. Let us first concentrate on the fundamental cavity mode (TM₀), seen for lower values of the in-plane wavevector, that is, inside the light-line. The anticrossing seen here is again a result of strong coupling, this time between the fundamental cavity mode and the molecular resonance; a combination that has been the workhorse of many strong coupling experiments,^{1,27,57,59,72} yielding a very clear splitting that produces upper and lower hybrid polariton modes. These hybrid polariton modes lie inside the air light-line (the blue dashed line in Figure 1h). The second mode lies beyond the air light-line and, as discussed above, does not have a cutoff, that is, it has no lower frequency limit. The dispersion of this second mode looks similar to the surface plasmon mode seen in Figure 1g; this is perhaps not so surprising since this mode is the coupled plasmon (TM_–1) mode.

The coupled plasmon mode also shows clear evidence in Figure 1h of strong coupling, a fuller analysis, including an evaluation of the appropriate Hopfield coefficients, will be discussed below. In previous work on vibrational strong coupling employing metal-clad microcavities this mode was not seen because it lies beyond the light-line.

In the remainder of this paper we will make use of grating coupling to gain access to this mode; in this way we are able to show that it is strongly coupled, and we are thus led to consider what role it might play in the way molecules within the cavity behave in the strong coupling regime. Before we move to the experimental investigation of microcavities, we want to consider what happens when we reduce the cavity thickness so that the usual fundamental mode is shifted well above the molecular resonance in frequency, so that the (TM_–1) mode becomes the only mode supported by the structure in the frequency range of interest. The dispersion of the mode present in such a cavity is shown in Figure 1f. The structure we consider, shown in Figure 1c, is the same as for Figure 1b, except that the thickness of the PMMA has been reduced, from 2 μm down to 100 nm. Despite the very subwavelength thickness of this cavity, it is still the vibrational (IR) mode that is of interest here. Although the fundamental cavity mode (TM₀) is no longer seen, the coupled plasmon mode is still present, lying at higher in-plane wavevector values than for the thicker cavity, Figure 1e, a direct result of the greater degree of field confinement in this thinner cavity. Importantly, a clear anticrossing signature is seen, Figure 1i, again the result of strong coupling. This result will be discussed in more detail below.

Now that we have established the structures and modes for study, we next consider their experimental investigation. To access modes beyond the air light-line, we make use of grating coupling. We do this by making (one of) the gold film(s) in the form of a 1D stripe array, see Figure 2b,e. Despite their discontinuous nature, metal stripe arrays of this kind generally still support propagating modes (except near band-edges) and have been used effectively in this way before;^73,74 further details are included in the Supporting Information. Details of the sample fabrication, including the gratings, are given in the Methods section. Figure 2 provides the key information associated with using gratings to couple to the modes we wish to investigate. The central column, Figure 2b,e, shows the structures we consider: Figure 2b is a 1D gold stripe array that allows us to explore the surface plasmon mode; Figure 2e is a metal-clad microcavity where the lower mirror is in the form of a 1D stripe array, again to allow us to explore the cavity coupled plasmon mode. In the left-hand column, we show indicative dispersion plots for the two structures. Here we have simply made use of the Fresnel-based data shown similar to those shown in Figure 1g,i, except that now we have plotted two copies of the data, one shifted by an in-plane wavevector that matches the grating wavevector, −k_g/2π = 2127 cm^–1, the other reversed in wavevector and also shifted, this time by +k_g; these two sets thus represent the effect we expect grating scattering to have on the transmission of incident light. For details about the way a grating modifies dispersion diagrams, the reader is referred to ref (75).

Figure 2.

Modes supported by structures incorporating gratings. Top row, surface plasmon (SP) mode. Bottom row, “below cutoff” microcavity (TM_–1) mode (the coupled plasmon mode). Central column, grating structures; shown here are schematics of upper (b), 1D metal grating supporting a surface plasmon mode, on top of a CaF₂ substrate, overlaid by a 1 μm layer of PMMA; lower (e), a microcavity incorporating a 1D metal grating as the lower mirror, the cavity is filled by a 1 μm layer of PMMA. The gold films for both structures were 30 nm thick. Left-hand column (a, d), calculated grating-scattered dispersion plots. These data were produced by taking data similar to Figure 1g,I and applying both +k_g and −k_g grating scattering, so as to produce Figure 1a and d, respectively; details are given in the text. The grating period was taken as 4.7 μm, for which k_x/2π = 1/λ_g = 2127 cm^–1. The ±k_g scattered air and PMMA light-lines are shown as yellow and light-blue dashed lines, respectively. Right-hand column (c, f): these dispersion plots are divided into two halves. In the left half, data calculated using COMSOL are shown, in the right half, experimentally measured data are shown. The maximum polar angle for these data is 18°. Details of the grating profile are provided in the Supporting Information. The ±k_g scattered air and PMMA light-lines are again shown as light-blue and green dashed lines, respectively. Note that in (c) and (f) the calculated data have been multiplied by a factor of 0.5, see main text.

Our choice of grating wavevector (grating period = 4.7 μm) was made so as to place the anticrossing region of the modes we are interested in close to an in-plane wavevector of zero, that is, close to normal incidence. In the right-hand column of Figure 2 we show a combination of numerically simulated and experimental data. For both plots in the right-hand column, Figure 2c,f, the left-hand half of the plot shows data calculated numerically using COMSOL, while the right-hand half shows experimental data. Note that in Figure 2c,f the calculated data have been multiplied by a factor of 0.5 to allow easy comparison with the experimental data. (The experimental transmission is roughly a factor of 2 weaker than calculated, most likely due to differences between the metal thickness used in the calculations and that obtained from the samples.) For the experimental data we measured the infrared transmission of our samples using Fourier transform infrared spectroscopy (see Methods). By measuring transmittance spectra as a function of the angle of incidence (in the x–z plane, see Figure 2) a dispersion plot can be constructed;^1,27,76 such plots are shown in Figure 2c,f.

For both the surface plasmon mode, Figure 2c, and the coupled plasmon mode associated with the cavity below cutoff, Figure 2f, we see anticrossings in the experimental data that agree well with what we expect from our indicative Fresnel-based modeling (Figure 2a,d) and from our more thorough numerical calculations (Figure 2c,f). The mode splitting we observe can also be seen in line-spectra extracted from the experimental data shown in Figure 2c,f; these line-spectra, shown in Figure S10 (Supporting Information) show that in both cases the splitting significantly exceeds the mode widths.

We note at this point that vibrational (IR) strong coupling via the coupled plasmon mode can also be achieved for very thin cavities, as is the case for excitonic (visible) resonances.^44,46 In the Supporting Information, we show, in Figure S2, the result of calculating, using the Fresnel-based approach, the Rabi-splitting for a system similar to that shown in Figure 2e as a function of cavity thickness, we see in Figure S3 that the splitting is largely independent of cavity thickness at least down to ∼30 nm, that is, cavities as thin as roughly 1/100λ. The independence of the extent of the Rabi-splitting on cavity thickness may be understood as follows. The strength of the Rabi-splitting depends on the product of the net dipole moment associated with the molecular resonance and the electric field strength associated with the mode being occupied by one vacuum fluctuation. This strength may be written as²

1

where Ω_R is the angular frequency extent of the Rabi splitting, N is the number of molecules in the mode volume V, ε is the permittivity of the medium, ω₀ is the angular frequency of the resonance, and μ is the dipole moment associated with the molecular resonance. The fields associated with the coupled plasmon mode span the full extent of the cavity (unlike a single interface plasmon mode) thus, as the cavity is made thicker the mode volume V also increases. At the same time the number of molecules in the cavity also rises due the increase in volume of the cavity-filling material. These two effects thus cancel out in eq 1, leading to the observed independence of Rabi-splitting on cavity thickness, a finding similar to that recently observed for thicker “standard” multimode cavities.⁷⁷

Having established that vibrational strong coupling may occur between molecular resonances and modes beyond the light-line we can begin to assess the implications this may have. One way to explore this is to make use of a coupled oscillators model to look at the contribution the different cavity modes and the molecular resonance make to the polaritons. This is a well-known approach^25,72 that we now adopt to look at a cavity close to the usual cutoff, that is, the situation shown in Figure 1h. We chose this particular situation since a cavity close to cutoff is the commonly used system in many strong coupling experiments. As a first step we represent the dispersion of the TM₀ and TM_–1 modes analytically, see Methods. As has been investigated recently,^78,79 there are two types of coupled oscillator models that can be applied to situations where there is one molecular resonance and several photonic resonances, the situation we have here. The first is what one might have expected from the usual (N + 1) × (N + 1) matrix (Hamiltonian) used to describe one photonic mode and several molecular resonances, where N is the number of photonic modes involved and the 1 refers to the single molecular resonance. The second is a 2N × 2N matrix (Hamiltonian), previously discussed in the context of the orthogonality of the photonic modes.⁷⁸ In this 2N model, each photonic mode is separately coupled to the molecular resonance. The (N + 1) model can, in its simplest form, be represented by the following matrix equation:

2

In eq 2, ω_TM0(k_x) is the frequency of the cavity (TM₀) mode; ω_TM–1(k_x is the frequency of the coupled plasmon (TM_–1) mode; and ω_C=O is the frequency of the vibrational resonance; E_L,M,U are the eigenvalues of the characteristic matrix (left) and are the frequencies of the three hybrid polaritons, indicated by the subscripts, L for the lower polariton, M for the middle polariton, and U for the upper polariton; a, b, and c are the (Hopfield) coefficients of the eigenvectors, and they are subject to the condition |a|² + |b|² + |c|² = 1, the individual coefficients giving the contribution of the TM₀ mode, the TM_–1 mode, and the vibrational resonance to the three polaritons, L, M, and U. Finally, Ω₁ is the coupling strength (Rabi-splitting) for the interaction between the TM₀ and the molecular resonance, Ω₂ is the coupling strength between the TM_–1 mode and the molecular resonance. Meanwhile, the 2N model can, in its simplest form, be represented as

3

In eq 3, E_L1,U1,L2,U2 are the eigenvalues of the characteristic matrix (left) and are the frequencies of the four hybrid polaritons, indicated by the subscripts, L1 (U1) for the lower (upper) polariton associated with the cavity mode; and L2 (U2) for the lower (upper) polariton associated with the coupled plasmon mode; d, e f and g are now the (Hopfield) coefficients of the eigen-vectors, subject again to the condition |d|² + |e|² + |f|² + |g|² = 1, the individual coefficients giving the contribution of the TM₀ mode, and the vibrational resonance to the first two polaritons, L1 and U1, and the contribution of the TM_–1 mode and the vibrational resonance to the second two polaritons, L2 and U2. Finally, Ω_α is the coupling strength (Rabi-splitting) for the interaction between the vibrational resonance and the TM₀ mode, while Ω_β is the coupling strength between the vibrational resonance and the TM_–1 mode. The results obtained from solving for the eigenvalues and coefficients of eqs 2 and 3 and comparing them with the reflectivity (Fresnel) data show that eq 3 provides a much better match than eq 2. The comparison is shown in Figure S12 of the Supporting Information. In Figure 3 we show the results of a match between solutions of eq 3 and the reflectivity (Fresnel) data, see panels (d) and (e).

Figure 3.

Coupled oscillators model. Here we consider the structure shown in Figure 1b. The left column is for TE polarization. (a) The result of calculating the reflectance (Fresnel) to produce a dispersion diagram, upon which we have superposed the data from a simple two coupled oscillators model, one that includes only the cavity (TE₀) mode and the vibrational resonance; for clarity, we show the coupled oscillator-only data in (b). (c) The associated Hopfield coefficients are shown. For TM polarization (right column), we now have two modes, the TM₀ cavity mode and the TM_–1 coupled plasmon mode to consider. As discussed in the main text, the results are best matched by using a 2N coupled oscillator model. The results of this model are superposed on the reflectance data in (d), and for clarity are shown on their own in (e). (f) Again, the associated Hopfield coefficients are shown.

Before discussing the results of the coupled oscillator analysis and the coupled plasmon mode in more depth, we first want to look again at the difference between simply considering the standard cavity mode and combination of the standard cavity mode and the beyond-the-light-line coupled plasmon mode. One way to investigate the standard cavity mode on its own is to consider the case of TE polarized light, plasmon modes are TM-polarized and so do not couple to TE-polarized light. The left-hand column of Figure 3 is for TE polarized light, the right-hand column for TM-polarized light. In panel (a) we show the calculated (Fresnel) TE reflectance for the cavity shown in Figure 1b. In these data the expected anticrossing between the TE₀ mode and the vibrational resonance is clearly seen. In panel (b) we show the result of matching a simple coupled oscillator model (one vibrational resonance and one cavity mode) to the reflectance data in panel (a), there is broad agreement between the calculated polariton dispersion (reflectance) and the coupled oscillator model. In panel (c) we show the associated Hopfield coefficients. What we see here is typical of the kind of picture frequently displayed in the context of molecular strong coupling to a planar microcavity mode.

Let us return now to the combination of the standard cavity mode and the coupled plasmon mode. We can do this for the same system (the one shown in Figure 1b) but now switch from TE to TM polarization. This change in polarization allows us to couple to the cavity mode and the coupled plasmon mode. The results from eq 3 are shown in panel (e) and are superposed on the reflectance data in panel (d). From the coupled oscillator models we find Ω_α ≈ Ω_β ≈ 150 cm^–1. A similar interaction strength was used to match the TE data (left column, Figure 3). These splittings compare with the measured Gaussian fwhm of a bare resonance cavity mode of 45 cm^–1 (±2 cm^–1), the C=O molecular resonance width of 60 cm^–1 (±5 cm^–1), and the coupled plasmon mode line width of 60 cm^–1 (±5 cm^–1), confirming that we are within the strong coupling regime.^80,81

Let us now return to consider the consequences that might arise from the molecular resonance having more than one photonic mode with which it may couple. Several possibilities can be readily identified.

First, in many phenomena associated with strong coupling the role of the reservoir of so-called “uncoupled resonators” or “dark states” is often discussed. Many of these dark states occur at high in-plane wavevectors which are only rather weakly coupled to the usual cavity mode. The range of wavevectors over which the lower polariton branch of the coupled oscillator model makes physical sense is not unlimited, as pointed out by Agranovich and Lidzey.²⁹ The upper wavevector limit on the polariton wavevector calculated by Agranovich and Lidzey far exceeds the range over which the cavity (TM₀/TE₀) mode is the only significant mode, it easily encompasses wavevector values appropriate to the TM_–1 mode. What effect such hybridization with the TM_–1 mode might have on the reservoir and, thus, on the phenomena for which the reservoir properties are important is not clear; further work is required.

Second, we need to consider the spatial location of the molecules involved in the different polaritons, and the orientation of their dipole moments. This is because the cavity (TM₀) mode and the coupled plasmon (TM_–1) mode have rather different field profiles. In Figure 4d,e, we show the calculated field distributions associated with these two modes. For completeness we also show the field distribution for the TE polarized lowest order cavity mode, TE₀ (Figure 4b). From the data shown in Figure 4b, d, and e, we see that molecules distributed across the cavity may couple to each of the TE₀, TM₀, and TM_–1 modes, albeit that the effectiveness of the coupling as a function of molecular position and dipole orientation will be different for the different modes. We also note that, for the parameters chosen here, the strength of the field for the three different modes (TM₀, TE₀, and TM_–1) is very similar, we thus expect a similar degree of Rabi splitting. The data in Figure 4 indicate that the net electric field near the cavity mirrors is significantly greater for the TM_–1 mode, so that molecules near the mirrors will preferentially couple to this mode; specifically, while the z-component of the electric field (see Figure 4e) is reasonably constant across the cavity, the x-component is significantly stronger near the metal surfaces than in the center. Note also that there are subtle differences between the dispersion of the TE₀ and TM₀ modes,⁸² and in addition, when the dipole moment associated with a molecular resonance is oriented in the plane of the cavity, the polarization of the modes that the molecules may couple to also becomes important.⁸³ The consequences of these spatial and orientational variations on, for example, polaritonic chemistry, need to be explored.

Figure 4.

Field profiles. Top row: the absolute value of the complex TE-polarized and TM-polarized amplitude transmission coefficients in the absence of the molecular resonance (vibrational mode) is shown in (a) and (c), respectively, as a function of frequency (wavenumber) and in-plane wavevector. The remaining plots show the time-averaged electric fields. (b) The TE₀ mode, here with E_y and E_z in blue and red, respectively. (d, e) The TM₀ and TM_–1 modes, with E_x and E_z in blue and red, respectively. The field profiles were calculated at a frequency of 1732 cm^–1, and for the following in-plane wavevector values, TM₀ and TE₀ modes at ∼138 cm^–1 and TM_–1 mode at ∼2450 cm^–1. The cavity-filling material was taken to be PMMA, see the Methods section.

Third, in the emerging field of polaritonic chemistry vibrational resonances are particularly important, and our results show that strong coupling of vibrational resonances to multiple photonic modes is not yet fully understood, notably because of the seemingly empirical choice that needs to be made between an N + 1 and a 2N coupled oscillator situation. We infer from this that, as for an excitonic resonance coupled to multiple photonic modes,⁷⁹ a better model is required for the coupling between a vibrational resonance and multiple photonic modes, with possible implications for the coupling dynamics. The implications regarding the coupling of molecules to multiple photonic modes is thus more complex than simple pictures indicate and is again an area in need of further work.

Lastly we note that strong coupling of a vibrational resonance with the TM_–1 mode shows that one can study and harness vibrational-polariton phenomena in microcavities that are very thin, thin enough that the usual cutoff frequency is in the range of excitonic resonances, offering the prospect of simultaneous control over excitonic and vibrational resonances.

Finally, while the focus of the results presented so far has been to look at strong coupling between ensembles of molecules and the optical modes supported by metal clad cavities, it is interesting to see whether something similar happens for another common class of microcavity where distributed Bragg reflectors (DBR) are used rather than metals to form the mirrors of the cavity.⁸⁴ In what follows we will see that while these DBR-based cavities do not support coupled plasmon modes; they do support modes well beyond the light-line, and these modes may also lead to strong coupling. Muallem et al.⁸⁵ presented results on strong coupling between molecular vibrational resonances in PMMA and the cavity mode of a DBR-based cavity. They showed that strong coupling to the standard cavity mode occurs using these cavities, much as it does for the usual metal-based cavities. However, their investigation did not go beyond the light-line. Here we extend this investigation to regions beyond the air light-line. Figure 5a shows a schematic of the structure we consider, employing ZnS and Ge DBR mirrors (see Methods for further details). In Figure 5b we show a dispersion plot for this structure, calculated using the Fresnel-based approach we used for Figures 1 and 4. As for the metal-clad cavities discussed above, in addition to strong coupling of the standard cavity mode we see that there is also strong coupling between the molecular resonance and the mirror modes of the DBR that lie well beyond the light line. These beyond the light-line modes of the DBR-based cavity are associated with fields that, in addition to being confined to the central region of the cavity, have some of their field distribution in the different layers that make up the DBR; for recent examples, see ref (86).

Figure 5.

DBR-based cavity. (a) Schematic showing DBR cavity structure. (b) Dispersion plot based on Fresnel-type calculations. The absolute value of the TM-polarized Fresnel coefficient is shown as a function of frequency (wavenumber) and in-plane wavevector. The light-blue and green dashed lines represent the air and PMMA light-lines, respectively. The molecular resonance is shown as a horizontal dashed white line. Layer thicknesses and material parameters for the DBR mirrors are given in the Methods section.

In summary, our results show that in addition to the usual cavity mode of metal-clad cavities, strong coupling also arises due to the coupled plasmon modes present in such structures. Similar considerations apply to beyond the light-line modes in DBR cavities. While much of the focus in exploring the fundamentals of polaritonic chemistry is quite rightly targeted at the material aspect of the polaritons, our results indicate that these beyond the light-line modes should also be taken into account when looking at how strong coupling may be used to alter/create molecular properties via strong coupling.

Methods

Sample Fabrication

The desired 1D grating structures were produced by electron beam lithography (EBL). structures. We used the e-beam resist PMMA (950 K A9), which we spun at 4000 rpm onto a square 20 mm × 20 mm CaF₂ substrate so as to obtain a thickness of ∼400 nm. The substrate was then heated to 180 °C for a duration of 10 min so as to remove any remaining solvent. For the EBL, an ∼20 nA beam current was employed to write the desired pattern. Following exposure, the resist was developed using a MEK + MIBK + IPA mixture for 40 s. A thin gold film of 30 nm was then deposited by thermal evaporation. We then used a lift-off process to leave the desired 1D gold stripe grating, see Figure 2b,e. Where appropriate, the next step was to add a layer of PMMA. Then, for the cavity structures, these steps were then followed by 30 nm gold film. The gratings had a period of 4.7 μm, the slot being 1.0 μm wide. For an AFM image of the gold grating, see Supporting Information, Figure S1.

FTIR Measurements

The IR transmission of the samples was determined using an FTIR setup (Fourier Transform Infrared spectroscopy, Bruker V80). To acquire dispersion curves spectra were acquired for a range of incident angles, typically in the range −18° to +18°. All measurements were performed with a spectral resolution of 8 cm^–1 and an angular resolution of 2°. To improve the signal-to-noise, averaging over 128 scans was carried out. An example of the measured transmittance spectra is shown in Figure 2c,f.

Numerical Modeling

To model the response from our structures, we employed finite-element-modeling through the use of COMSOL Multiphysics. An example in Figure 2c, for which the COMSOL calculations involved a modeling volume that comprised a 3 μm layer of CaF₂ overlain with a 30 nm gold grating, covered by 1 μm layer of PMMA and finally followed by 3 μm layer of air. Periodic boundary conditions were added in the grating (x) direction. For the meshing a minimum mesh element size of 0.22 nm was used, while the maximum mesh element size was 185 nm, a curvature factor of 0.2 was used to smooth the vertices so as to better represent the fabricated samples.

Material Parameters

For the frequency-dependent permittivity of both gold and PMMA, we made use of Drude-Lorentz,

4

and Lorentz oscillator,

5

models, respectively. For gold, we used parameters taken from Olmon et al.;⁸⁷ specifically, ω_p = 1.29 × 10¹⁶ rad s^–1 and γ = 7.30 × 10¹³ rad s^–1, with ε_b = 1.0. For PMMA, we took single oscillator parameters taken from Shalabney et al.;¹ specifically, ω₀ ≡ 3.28 × 10¹⁴ rad s^–1 and γ ≡ 2.45 × 10¹² rad s^–1, with f₀ = 0.0165 and ε_b = 1.99. The parameters for silicon in the infrared are based on data compiled by Edwards⁸⁸ and are taken to be ε = 11.76 + 0.001i, while for air we took ε = 1.0.

Coupled Oscillator Models

For the coupled oscillator models, we took the dispersion of the cavity mode, E_TM0(k_x) to be , and for the coupled plasmon mode .

DBR Mirrors

To model the DBR mirrors, the refractive index and thickness of the mirror layers were taken tobe Ge thickness, 360 nm, n = 4.01; ZnS thickness 645 nm, n = 2.24.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsphotonics.0c00552.

• Grating morphology, thickness dependence of Rabi-splitting with the TM_–1 mode, field profiles of the TE₀, TM₀, and TM_–1 modes, effect of superstrate index on dispersion of coupled surface plasmon mode, calculated dispersion of continuous and discontinuous (stripe array) one-dimensional gold grating structures, dispersion of DBR cavity modes for zero oscillator strength, line spectra, field profiles of the TE₀, TM₀ modes at higher k_x, and oscillator models for one molecular resonance and several photonic resonances (PDF)

The authors declare no competing financial interest.