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.

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.

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.

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.

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.

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.