Strong coupling

The organic microcavities studied here, consisted of a ~λ/2 spin-cast thin film of BODIPY-G1 dye uniformly dispersed in a polystyrene matrix (see Supplementary Note 1) that was positioned between two distributed Bragg reflectors (DBRs) consisting of 8 and 10 pairs of SiO 2 /Nb 2 O 5 placed on the top and bottom of the structure, respectively. For more information about sample fabrication, see Methods. We have found that the spin-casting process used to deposit the organic film results in a gradual increase of film thickness towards the edges of the substrate (bottom mirror). We use this non-uniformity to access a broad range of exciton–photon detuning (δ).

We measure angle-resolved reflectivity of a typical microcavity, as shown in Fig. 1a. Here both upper and lower polariton branches can be observed as local minima in the broad DBR reflectivity stop-band that are split around the BODIPY-G1 peak absorption wavelength at 507 nm. For further experimental details, see Methods. We plot the energy of these modes as a function of angle, creating a dispersion plot as shown in Fig. 1b (red squares). These data are superimposed over a false-color plot of polariton photoluminescence intensity obtained under non-resonant excitation at 400 nm in the linear excitation regime. We fit the upper and lower polariton branches in Fig. 1b using a two coupled-oscillator model13,14, and obtain a vacuum Rabi splitting (\(\hbar {\mathrm{\Omega }}_0\)) of ~116 meV and an exciton–photon detuning of −160 meV (further details are given in Supplementary Note 2).

Fig. 1: Strong light-matter interaction in the dye-filled microcavity. a Angle-dependent reflectivity spectra of the microcavity recorded at different angles exhibit clear anti-crossing at the exciton resonance energy E x (gray dotted line) and indicate the formation of lower (LPB, gray dashed line) and upper (UPB, gray dashed line) exciton–polariton branches. Polariton dispersion relation in b is plotted by combing the data of photoluminescence imaging acquired in a Fourier space (rainbow color density plot in a log scale) with the polariton states extracted from angle-dependent reflectivity measurements (red squares). Error bars are within the marker size and correspond to standard deviations of the best-fit results of angle-dependent reflectivity spectra. Fitting curves of the LPB and UPB, together with the cavity mode E c and energy of exciton resonance E x are shown as a white dashed curves. Full size image

Blueshifts in polariton condensates

Recent studies have shown that a number of the BODIPY family of molecular dyes undergo polariton condensation/lasing following non-resonant optical excitation6,7. To demonstrate condensation using BODIPY-G1, we record the dispersion of polariton photoluminescence emission as a function of excitation density using single excitation pulses in a transmission configuration (see Methods). The excitation laser used provides 2 ps pulses at 400 nm having a horizontal polarization. Figure 2a shows the time-integrated polariton photoluminescence distributed across the lower polariton branch below condensation threshold. Figure 2b shows microcavity emission above-threshold, where it can be seen that polariton photoluminescence collapses to the bottom of the lower polariton branch. In Fig. 2c we plot the photoluminescence intensity and the full-width at half-maximum of the emission linewidth (right axis, in red) integrated over ±1° (±0.2 μm−1) around normal incidence versus excitation density. The corresponding energy shift and the degree of linear polarization of the emission spectrum are also shown in Fig. 2d, e, respectively. At an excitation density of ~6 mJ cm−2 (120 μJ cm−2 of absorbed pump fluence), we observe a rapid increase of the photoluminescence intensity, a concomitant linewidth narrowing from 1.6 to 0.25 nm and a step-like increase of the degree of linear polarization and a step-like blueshift of the emission spectrum. Such blueshifts of the polariton emission wavelength occurring around a lasing threshold are commonly considered a hallmark of polariton condensation. In inorganic semiconductor microcavities, such energy shifts originate from the repulsive interparticle Coulomb exchange interactions between Wannier–Mott excitons15. However, such interactions are in principle precluded in polaritons created using molecular semiconductors as a result of the highly localized nature of Frenkel excitons16,17.

Fig. 2: Non-equilibrium polariton condensation. Normalized E,k polariton-population (photoluminescence) images recorded using Fourier-space imaging, below, at 0.8P th (a) and above condensation threshold, at 1.4P th , (b). Dashed red curves show lower polariton branch dispersion in the linear regime. c Photoluminescence intensity at k || ~ 0, integrated into the range over ±0.2 μm−1 (black squares) and full-width at half-maximum (FWHM, red squares) and d energy of the ground polariton state versus pump power. The superlinear increase and 10-fold linewidth narrowing together with high-energy shift of the polariton ground state observed above the threshold are commonly-recognised features of non-equilibrium polariton condensation. e Degree of linear polarization (blue squares) as a function of pump power, indicating that the condensate inherits the polarization of the pump beam. Solid curves in c–e represent numerical simulations of time-integrated photoluminescence, the ground state energy, and degree of linear polarization as functions of pump power, respectively. The red dashed curve in c is a guide for the eye. Error bars in c, d determined as standard deviations of the best-fit results of integrated photoluminescence intensity at different pump powers. Error bars in d are within the marker size. Error bars in e correspond to standard deviations of DLP obtained from photoluminescence intensities measured for orthogonal polarizations. Full size image

To explore the mechanism behind such blueshifts, we examine the contribution of various nonlinear optical phenomena, omnipresent both in the strong- and weak-coupling regimes, namely the intracavity optical Kerr effect, the gain-induced frequency pulling, and interparticle interactions.

Optical Kerr effect

A step-like increase in the intensity of the electric field inside the cavity at condensation threshold (determined at the lower polariton mode wavelength) could potentially shift the resonance through a change in the nonlinear refractive index of the intracavity material by means of the conventional Kerr effect. To examine the contribution of the E-field induced difference in the refractive index, we measure the optical nonlinearities of the bare intracavity medium using both a closed- and open-aperture Z-scan technique18 (for details, see Supplementary Note 3 and Methods). Figure 3a shows an open-aperture Z-scan transmission recorded at two pulse energies, probing the imaginary part of the nonlinear susceptibility. We find that at the lower incident pulse energy of 9.5 nJ, we do not observe any nonlinear change of the absorption. We note that at the foci, the 9.5 nJ excitation pulse induces an electric-field intensity of 17 GW cm−2; a value that is approximately an order of magnitude higher than the electric-field intensity at the anti-node within the microcavity at 1.4 P th (~2 GW cm−2) (calculation is shown in Supplementary Note 4). At the considerable higher intensity of 779 GW cm−2 at the foci of the beam, we observe an optical nonlinearity in the form of reverse saturable absorption.

Fig. 3: Nonlinear optical susceptibility of the neat film. a Open aperture and b closed-aperture Z-scan data measured at two different incident pump energies 9.5 nJ (black circles) and 438 nJ (red rhombs). The fitting curves (red dashed) in a, b correspond to \({\mathrm{{Im}}}\left[ {\chi ^{\left( 3 \right)}} \right] = 1.71 \times 10^{ - 20}\,\mathrm{m}^{2}\,\mathrm{V}^{- 2}\) and \({\mathrm{{Re}}}\left[ {\chi ^{\left( 3 \right)}} \right] = 2.17 \times 10^{ - 20}\,{\mathrm{{m}}}^{2}\,{\mathrm{V}}^{ - 2}\), respectively. Error bars correspond to the standard deviation of a single Z-scan measurement on a glass substrate. Full size image

The closed-aperture Z-scan measurements presented in Fig. 3b provides a measure of the real part of the nonlinear optical susceptibility. Here, we do not detect any change in the nonlinear refractive index when pumping at the lower incident pulse energy. However, at the higher incident intensity of 779 GW cm−2, we find that the material exhibits a weak self-focusing effect from which we determine a positive nonlinear refractive index (n 2 ) of ~1.89 × 10−14 cm2 W−1 (calculation is shown in Supplementary Note 4). We note that even if such high electromagnetic fields could be generated within a microcavity, the positive value of n 2 that we determine would induce a redshift. We conclude, therefore, that the optically induced change of the intracavity nonlinear refractive index is not responsible for the blueshift observed at the condensation threshold.

Gain-induced frequency pulling

We now consider whether the gain-induced frequency pulling could be responsible for the blueshift at the condensation threshold. This mechanism is expected to be particularly important in negatively detuned microcavities in which the polariton mode has a large photonic fraction. We characterize the spectral distribution of the optical gain by recording amplified spontaneous emission (ASE) from a control (non-cavity) film of BODIPY-G1 dye molecules dispersed in a polystyrene matrix (see Supplementary Note 1 and Methods). A typical ASE spectrum is plotted in Fig. 4a [red line], where it can be seen that the emission (corresponding to the peak of optical gain) is peaked around 2.272 eV (545.8 nm).

Fig. 4: Blueshifts and the optical gain spectrum. a The amplified spontaneous emission spectrum and the blueshift, ΔE, extracted from the binning of scattering plot (b) that shows single-shot blueshift realizations across the whole sample area. c Vacuum Rabi splitting \(\hbar {\mathrm{\Omega }}_0\) and d detuning δ extracted from the multiple angle-dependent reflectivity measurements carried out across the sample area. The Rabi splitting remains constant across the whole-sample area with an average value of (116 ± 1.5) meV (red dashed curve). e The exciton fraction \(\left| {X_{k_\parallel = 0}} \right|^2\) of the polariton wave function at \(k_\parallel = 0\), calculated from \(\hbar {\mathrm{\Omega }}_0\) and δ shown above in c and d, respectively. Red dashed curves in d and e correspond to analytical fit functions for δ and \(\left| {X_{k_\parallel = 0}} \right|^2\), respectively. Error bars in a correspond to standard deviations of the blueshift (y-axis) and lower polariton state (x-axis) calculated for each binning range. Error bars in c–e correspond to standard deviations of the best-fit results of the angle-dependent reflectivity measurements. Full size image

We explore the extent to which gain-induced frequency pulling affects the condensate’s blueshift by tuning the frequency of the lower polariton branch across the optical gain spectrum. Such tunability in the lower polariton branch wavelength is possible through the variation in the thickness of the intracavity film across the sample. This effect allows us to explore polariton condensation over a broad range of exciton–photon detuning conditions. Figure 4b shows the measured energy shift for ~400 single-shot measurements of polariton condensation at a wide range of different polariton ground state energies. For each measurement, the energy shift is defined by comparing the energy of the emission below and above the threshold. Here, we avoid averaging over the intensity fluctuations of the laser by utilizing a single-shot dispersion imaging technique. In Fig. 4a, we superimpose the measured blueshift using the Sturge binning rule with the amplified spontaneous emission spectrum (for details see Supplementary Note 2). It is apparent that at condensation threshold, the recorded energy shifts are always positive, and thus we conclude that the blueshift is not induced by gain frequency pulling. Indeed, if the gain-induced frequency pulling considerably contributes to polariton energy shifts, one would expect a negative sign of the shift observed for the left side of the ASE peak instead. However, we systematically observe polariton blueshifts regardless of the side of the gain peak.

Polariton interactions

We now investigate the possible contribution of polariton–exciton and pair-polariton scattering in determining the observed blueshift at the condensation threshold. In semiconductor microcavities containing Wannier–Mott excitons, the experimentally observed energy shifts (ΔE) are attributed to a combination of pair-polariton (g p−p N p ) and polariton–exciton (g p−x N x ) interaction terms. This is summarized by the following equation12,19:

$${\mathrm{\Delta }}E = g_{\mathrm{{p - p}}} \cdot N_{\mathrm{p}} + g_{\mathrm{{p - x}}} \cdot N_{\mathrm{x}},$$ (1)

where the pair-polariton scattering interaction constant can be related to the exciton–exciton scattering constant (g x−x ) using \(g_{\mathrm{{p - p}}} = g_{\mathrm{{p - x}}} \cdot \left| X \right|^2 = g_{\mathrm{{x - x}}} \cdot \left| X \right|^4\), where X is the amplitude of the exciton fraction that is mixed into a polariton state, and N p and N X are the polariton and exciton reservoir densities, respectively. Since the occupancy of polaritons at the condensation threshold does not depend on the exciton fraction, the measured dependence of the energy shift versus the square of the amplitude of the exciton fraction (|X|2) should reveal whether pair-polariton or polariton–exciton interactions dominate the blueshift.

To determine the dependence of the measured blueshift shown in Fig. 4b versus the exciton fraction, we need first to describe the dependence of the experimentally measured emission frequency of the polariton state on its exciton fraction. The latter depends on the exciton–photon detuning (δ) and vacuum Rabi splitting through \(| {X_{k_\parallel = 0}} |^2 = \frac{1}{2}( {1 + \frac{\delta }{{\sqrt {\delta ^2 + \left( {\hbar {\mathrm{\Omega }}_0} \right)^2} }}} )\). To avoid any excitation density-dependent energy shifts of the lower polariton branch, we perform white light, angle-resolved reflectivity measurements across the available detuning range. We fit the linear polariton dispersions by varying the vacuum Rabi splitting and the exciton–photon detuning, while keeping the exciton energy and the effective refractive index of the intracavity layer constant (see Supplementary Note 2). Figure 4c, d plot the fitted values of vacuum Rabi splitting and exciton–photon detuning vs the energy of the polariton state. This analysis indicates that the vacuum Rabi splitting is virtually invariant across the whole sample area and has an average value of (116 ± 1.5) meV, with the exciton–photon detuning spanning the range 120 meV, δ ϵ [−240, −120] meV. From this, we plot the dependence of \(| {X_{k_\parallel = 0}} |^2\) on the energy of the polariton state, as shown in Fig. 4e.

Using this approach, we can also determine the dependence of the measured energy shift, ΔE, on \(| {X_{k_\parallel = 0}} |^2\), shown in Fig. 5. This indicates that the energy shift of the polaritons on condensation has a sub-linear dependence on \(| {X_{k_\parallel = 0}} |^2\); a result that firmly precludes pair-polariton scattering as the underlying mechanism for the observed blueshift and suggests that polariton–exciton scattering is also unlikely; here the former process would result in a quadratic dependence on \(| {X_{k_\parallel = 0}} |^2\) and the latter on a linear dependence (see Eq. (1)).

Fig. 5: The blueshift ΔE versus exciton fraction. The dependence of the blueshift on the exciton fraction is calculated by binning the scattering plot of single-shot blueshift realizations across the whole accessible detuning range and taking into account the dependence of exciton fraction on the ground polariton state energy. The dashed black line is the best-fit result by power-law \({\mathrm{\Delta }}E\sim ( {| {X_{k_\parallel = 0}} |^2} )^\beta\)with variable parameter β = 0.7. Error bars correspond to standard deviations of the blueshift (y-axis) and exciton fraction (x-axis) calculated for each binning range. Full size image

In the absence of pair-polariton interactions and for a constant exciton fraction/detuning (as expressed by Eq. (1)), we expect that polariton–exciton interactions should lead to a linear energy shift with increasing excitation and thus exciton density. At the condensation threshold, stimulated relaxation from the exciton reservoir to the polariton ground state would lead to clamping of the exciton density and, therefore, of the energy shift. However, to date, all non-crystalline semiconductor microcavities undergo a nearly step-like increase of polariton blueshift at condensation threshold3,4,5,6,7,8,9, as shown in Fig. 2d, subject to the accuracy of the measured excitation density. Thus, the step-like dependence precludes polariton–exciton interactions as the driving mechanism for the observed blueshifts; a conclusion that is also corroborated by the sub-linear dependence of the energy shift on \(| {X_{k_\parallel = 0}} |^2\). Such a conclusion is also consistent with the high degree of localization of Frenkel excitons on a single molecule, as such exciton localization is expected to dramatically weaken Coulomb exchange interactions and suppress interparticle scattering.

Saturation of molecular optical transitions

In the following, we propose and experimentally verify that the observed blueshifts are due to quenching of the Rabi splitting and a nonlinear change of the cavity refractive index n eff . Both mechanisms are a consequence of the same nonlinear process, namely saturation of molecular optical transitions. Owing to the Pauli-blocking principle, excited (i.e., occupied) states cannot be filled twice, thus effectively reducing the oscillator strength of Frenkel excitons. Therefore, occupied states do not contribute to optical absorption at the exciton resonance that in turn reduces the Rabi splitting4,20 through the relation

$$\hbar {\mathrm{\Omega }} = \hbar {\mathrm{\Omega }}_0\sqrt {1 - \frac{{2(n_{\mathrm{x}} + n_{\mathrm{p}})}}{{n_0}}}$$ (2)

(for a microscopic theory of Rabi quenching see Supplementary Note 5). Here, Eq. (2) describes the quenching of the vacuum Rabi splitting, \(\hbar {\mathrm{\Omega }}_0\), as a function of the total number of excitations, namely the sum of excitons and polaritons n x + n p , where n 0 is the total number of molecules contributing to strong coupling. Since the optical pump results in a saturation of the molecular optical transitions that contribute to strong coupling, we expect a partial quenching of the Rabi splitting; an effect that results in a measurable blueshift of the lower polariton mode with increasing excitation density. We note here that only a small fraction of molecules in the intracavity layer are strongly coupled to the cavity mode (\(f_{\mathrm{c}} = \frac{{n_0}}{{n_{\mathrm{{tot}}}}}\) is the fraction of coupled molecules), as was suggested by Agranovich et al.21. Therefore, a renormalization of the light-matter interaction constant originates exclusively from strongly coupled molecules, while the remaining weakly coupled molecules do not contribute to the blueshift through the Rabi quenching mechanism. However, non-resonant pumping leads to a uniform excitation of molecules across the intracavity volume and equally populates both strongly and weakly coupled molecules. Thus, the large amount of weakly coupled molecules dispersed in the cavity can contribute to the blueshift via the renormalization of the cavity mode energy that occurs from the decrease of the intracavity effective refractive index, n eff : a consequence of the quenching of the oscillator strength for the molecules' optical transition. Analogously, the effect of the carrier density-dependent nonlinear refractive index change on the polariton dispersion was recently shown in inorganic ZnO microcavities22. In weakly coupled microcavities, mode energy shifts induced by refractive index changes of the intracavity material have been used as a probe for the measurement of optical nonlinearities23.

The change in refraction under quenching of the oscillator strength is inherent to the causality principle, the Kramers–Kronig relation, that couples the real and imaginary parts of the complex dielectric function24. The relation predicts a decrease of the refractive index above the induced absorption resonance and an increase below the resonance, resulting in an anomalous dispersion that usually appears within the width of an optical transition. We address the problem of refractive index change by general Kramers–Kronig analysis:

$$n\left( \omega \right) = \frac{1}{\pi }{\mathrm{{PV}}}\mathop {\smallint }

olimits_{ \!\!- \infty }^{ + \infty } \frac{{k(\omega {^\prime})}}{{\omega - \omega {^\prime}}}{\mathrm{d}}\omega {^\prime},$$ (3)

where PV stands for integration over the Cauchy principal value, and k(ω) is an extinction coefficient, being an imaginary part of the refractive index.

In order to calculate the real part of the complex refractive index using Eq. (3), one needs to know the k(ω′), which is related to the absorption spectrum. We use the absorption spectrum of the bare film to extract k(ω); for details, see Supplementary Note 6. As the absorption spectrum can be perfectly decomposed by a couple of Gaussian distributions centered at 2.446 and 2.548 eV for the excitonic energy and its vibronic replica, respectively, it is quite convenient to calculate Eq. (3) through the known Hilbert transformation of Gaussians in the form of the weighted sum over the Dawson functions:

$$n\left( \omega \right) = - \frac{2}{{\sqrt \pi }}\mathop {\sum }\limits_i \,A_i \cdot F\left[ {\frac{{(\omega - \omega _{o,i})}}{{\sigma _i\sqrt 2 }}} \right],$$ (4)

where the imaginary part is taken in the form of the sum over the Gaussian distributions \(k\left( \omega \right) = \mathop {\sum }

olimits_i \,A_i{\mathrm{e}}^{\frac{{ - (\omega - \omega _{o,i})^2}}{{2\sigma _i^2}}},\) accordingly, and \(F\left[ {\frac{{(\omega - \omega _{o,i})}}{{\sigma _i\sqrt 2 }}} \right]\) is the Dawson function (integral) with an argument \(\frac{{(\omega - \omega _{o,i})}}{{\sigma _i\sqrt 2 }}\).

Equation (4) describes the anomalous dispersion that naturally appears on the average effective cavity refractive index n eff = 1.81 as the consequence of the molecular optical transitions, see Supplementary Note 2. Therefore, with decreasing the imaginary part, one can observe a reduction in the real part of the refractive index over the lower energy sideband. Figure 6a shows the imaginary and real parts of the complex refractive index as a function of energy. Note that Δn is positive on the high-energy side of the exciton resonance and negative on the low-energy side.

Fig. 6: Kramers–Kronig analysis. a Real (red) and imaginary (blue) parts of the complex refractive index plotted for the case of unperturbed (solid) and saturated (dashed) molecular transitions, respectively, where we saturate the transition by 10%. b The ratio \(\rho = \frac{{{\mathrm{\Delta }}E_{\mathrm{{LPB}}}^{\mathrm{c}}}}{{{\mathrm{\Delta }}E_{\mathrm{{LPB}}}^{\mathrm{\Omega }}}}\) shows the relative contributions of the cavity mode energy renormalization (numerator) and the Rabi-quenching term (denominator) to the overall polariton blueshift as a function of exciton–photon detuning. Full size image

The change in refractive index is evident over a range of energies on either side of the resonance that induces a correspondent energy shift of the cavity mode E c by a value of ΔE c . For small changes of \(\Delta n \ll n_{\mathrm{{eff}}}\), one can approximate the energy shift with

$${\mathrm{\Delta }}E_{\mathrm{c}} \cong - E_{\mathrm{c}}\frac{{{\mathrm{\Delta }}n}}{{n_{\mathrm{{eff}}}}},$$ (5)

where n eff = 1.81 is the effective cavity refractive index.

Equation (5) describes the blueshift of the lower polariton dispersion due to the change that occurs in the cavity refractive index from the saturation of weakly coupled molecular optical transitions. The net effect of both the quenching of the vacuum Rabi splitting and cavity mode renormalization on the blueshift ΔE LPB is given by

$$\Delta E_{\mathrm{{LPB}}} = 1/2\cdot \left\{ {E_{\mathrm{x}} + E_{\mathrm{c}}\left( {1 - \frac{{\Delta n}}{{n_{\mathrm{{eff}}}}}} \right) - \sqrt {\left( {E_{\mathrm{c}}\left( {1 - \frac{{\Delta n}}{{n_{\mathrm{{eff}}}}}} \right) - E_{\mathrm{x}}} \right)^2 +\, (\hbar {\mathrm{\Omega }})^2} } \right\} - E_{{\mathrm{LPB}}}^0$$ (6)

where E x , E c are the energies of the bare exciton and cavity modes respectively, and \(E_{\mathrm{{LPB}}}^0\) is the unperturbed energy of the ground polariton state in the limit of small excitation numbers (linear regime), \(\hbar {\mathrm{\Omega }}\) and Δn are the density-dependent Rabi splitting and the change of cavity refractive index, respectively.

In the case of a small saturation parameter ξ, namely \(\xi = \frac{{(n_{\mathrm{x}} + n_{\mathrm{p}})}}{{n_0}} \cong \frac{{n_{\mathrm{x}}}}{{n_0}} \ll 1\), we can significantly simplify the above equation for the polariton blueshift. First, we describe the change in refractive index Δn by means of parameter ξ as follows:

$$\Delta n = - \frac{{\mathrm{{ln}}10}}{{2\pi \sqrt \pi }} \cdot \xi \cdot \alpha \cdot F[d] \cong - \frac{1}{5}\xi \cdot \alpha \cdot F[d].$$ (7)

We replace the weighted sum over the Dawson functions from Eq. (4) to a single Dawson function with argument \(d = \frac{{\left\lceil \delta \right\rceil \cdot 2\sqrt {\mathrm{{ln}}2} }}{{\mathrm{{FWHM}}}}\), where δ = E c − E x is the detuning and the full-width at half-maximum (FWHM) of the main absorption peak which is attributed to the \(S_{0,0} \to S_{1,0}\) singlet optical transition. The scaling parameter α in Eq. (7) corresponds to the oscillator strength of the optical transition as it is proportional to the absorption maximum (Abs max ), \(\alpha = \frac{{\mathrm{{Abs}}_{\mathrm{{max}}} \cdot \lambda _{\mathrm{{max}}}}}{L}\), where L is the cavity thickness. Thus, we can now reformulate Eq. (6) for the total polariton blueshift in a more convenient way within the approximation of a small saturation parameter ξ:

$$\Delta E_{\mathrm{{LPB}}} = \Delta E_{\mathrm{{LPB}}}^{\mathrm{\Omega }} + \Delta E_{\mathrm{{LPB}}}^{\mathrm{c}} = \frac{\xi }{2}\frac{{s \cdot \hbar {\mathrm{\Omega }}_0}}{{\sqrt {1 + s^2} }} + \frac{\xi }{2}\frac{{E_{\mathrm{x}} - |\delta |}}{{5n_{\mathrm{{eff}}}}}F[d] \cdot \alpha \cdot \left( {1 + \frac{1}{{\sqrt {1 + s^2} }}} \right),$$ (8)

where \(s = \frac{{\hbar {\mathrm{\Omega }}_0}}{{|\delta |}}\) is a dimensionless parameter of strong coupling; we exploit the reasonable assumption of f c ≪ 1.

Both terms in Eq. (8) reflect the influence of the same physical process of saturation of the optical transition on the polariton energy, but rely on different subsets of molecules. The first term corresponds to the quenching of the vacuum Rabi splitting in strongly coupled molecules. The second term corresponds to the renormalization of the cavity mode energy due to the change of the intracavity effective refractive index from the excitation of weakly coupled molecules. Surprisingly, we find that the renormalization of the cavity mode energy dominates over the quenching of the Rabi splitting in the total polariton blueshift as \(\rho = \frac{{\Delta E_{\mathrm{{LPB}}}^{\mathrm{c}}}}{{\Delta E_{\mathrm{{LPB}}}^\Omega }} = \frac{{(E_{\mathrm{x}} - \left| \delta \right|) \cdot F[d] \cdot \alpha \cdot (\sqrt {1 + s^2} + 1)}}{{5n_{\mathrm{{eff}}} \cdot s \cdot \hbar {\mathrm{\Omega }}_0}} \,{> }\, 1\) for the whole range of exciton–photon detuning accessible in this study. Clearly, the ratio is invariant over the saturation parameter ξ, and it depends on \(\hbar {\mathrm{\Omega }}_0\), δ, the absorption of the optical transition, α, its linewidth FWHM and cavity thickness L. Figure 6b shows the ρ ratio versus the detuning. To the best of our knowledge, the significance of the renormalization of the cavity mode energy to the total energy shift of organic polariton condensates has not been considered to date. Although Eq. (8) ultimately describes the magnitude of blueshifts in strongly coupled organic microcavities, neither of the involved mechanisms nor their superposition can explain the ubiquitous step-like increase of the blueshift at condensation threshold, P th , but instead predicts a continuous increase of the blueshift with an increasing number of excitations in the system, characterized by parameter ξ.

Intermolecular energy transfer

To explain the pump power dependence of polariton blueshifts, we construct a model that distinguishes between molecules that have a non-zero projection of their optical dipole moment aligned parallel \((N_0^\parallel )\) and perpendicular \((N_0^ \bot )\) to the linear polarization of the excitation laser. We assume that upon non-resonant optical excitation, only parallel-aligned molecules are initially occupied. These molecules constitute an exciton reservoir \((N_{\mathrm{x}}^\parallel )\) whose population is then depleted through: (i) energetic relaxation to the ground polariton state having the same optical alignment \((N_{\mathrm{p}}^\parallel )\), (ii) intermolecular energy transfer to perpendicular-aligned molecules as well as to other uncoupled molecules having some out-of-plane projection of the dipole moment \(N_0^ \times\) and (iii) decay via other nonradiative channels (γ NR ). We propose that intermolecular energy transfer from exciton reservoir \(N_{\mathrm{x}}^\parallel\) populates exciton reservoirs \(N_{\mathrm{x}}^ \bot\) and \(N_{\mathrm{x}}^ \times\), whose populations are in turn depleted through the same energy relaxation channels with the \(N_{\mathrm{x}}^ \bot\) reservoir creating polaritons having an optical alignment that is perpendicular to the excitation laser \((N_{\mathrm{p}}^ \bot )\). Figure 7 shows a schematic of the involved molecular transitions and relaxation paths of excited states considered within our model.

Fig. 7 Schematic of the molecular transitions and relaxation paths of excited states considered within the model. Full size image

In densely packed organic films, intermolecular energy transfer is an efficient process that results in the ultrafast depolarization of fluorescence25,26. When such films are embedded in a strongly coupled microcavity, intermolecular energy transfer below condensation thresholds is evidenced by a near-zero degree of linear polarization, as shown in Fig. 2e. With increasing excitation density and upon condensation threshold, energy relaxation to the ground polariton state becomes stimulated, resulting in sub-picosecond relaxation times, i.e., stimulated relaxation becomes faster than intermolecular energy transfer. Thereby, polariton condensation occurs with optical alignment parallel to the excitation laser3,4,5,8. The interplay between stimulated relaxation to the ground polariton state and intermolecular energy transfer can qualitatively describe the step-like increase of the degree of linear polarization at the condensation threshold, experimentally observed here in Fig. 2e. The quenching of intermolecular energy transfer upon condensation threshold, effectively increases the occupation of \(N_0^\parallel\)-molecules, which in turn quenches the corresponding Rabi splitting, \(\hbar {\mathrm{\Omega }}^\parallel = \hbar {\mathrm{\Omega }}_0^\parallel \sqrt {1 - \frac{{2(N_{\mathrm{x}}^\parallel + N_{\mathrm{p}}^\parallel )}}{{N_0^\parallel }}}\), and blueshifts the ground polariton state accordingly, by \(\Delta E_{\mathrm{{LPB}}}^{\mathrm{\Omega }} = 1/2\cdot (E_{\mathrm{x}} + E_{\mathrm{c}} - \sqrt {(E_{\mathrm{c}} - E_{\mathrm{x}})^2 + (\hbar {\mathrm{\Omega }}^\parallel )^2} ) - E_{{\mathrm{LPB}}}^0\), where E x and E c are the energies of the bare exciton and cavity modes respectively, and \(E_{\mathrm{{LPB}}}^0\) is the energy of ground polariton state in the limit of small excitation numbers (linear regime). Analogously, the blueshift accompanied with renormalization of the cavity mode energy can be described by the density-dependent function \(\Delta E_{\mathrm{{LPB}}}^{\mathrm{c}} = 1/2\cdot \left\{ {E_{\mathrm{x}}\, + \, E_{\mathrm{c}}( {1 - \frac{{\Delta n}}{{n_{\mathrm{{eff}}}}}} )\, - \,\sqrt {( {E_{\mathrm{c}}( {1 - \frac{{\Delta n}}{{n_{\mathrm{{eff}}}}}} ) - E_{\mathrm{x}}} )^2 + (\hbar {\mathrm{\Omega }}_0^\parallel )^2} } \right\}\, - \, E_{{\mathrm{LPB}}}^0\), where Δn is defined by Eq. (7). In the case of small saturation, namely \(\frac{{\left( {N_{\mathrm{x}}^\parallel + N_{\mathrm{p}}^\parallel } \right)}}{{N_0^\parallel }} \ll 1\), the net polariton blueshift ΔE LPB is just a linear superposition of both contributions: \(\Delta E_{\mathrm{{LPB}}}^{\mathrm{\Omega }} + \Delta E_{\mathrm{{LPB}}}^{\mathrm{c}}\) that is described by Eq. (8). The competition between stimulated relaxation to the ground polariton state and intermolecular energy transfer qualitatively predicts the saturation of molecular optical transitions that are optically aligned with the excitation laser, and the concomitant step-like energy shift at condensation threshold, as shown in Fig. 2d.

To quantitatively describe the experimental dependence of the polariton emission intensity, energy shift, and degree of linear polarization with increasing excitation density, shown in Fig. 2c–e, we formulate the above model in terms of coupled rate equation (for more information see Supplementary Note 7):

$$\frac{{{\mathrm{d}}N_0^{\parallel , \bot , \times }\left( t \right)}}{{{\mathrm{d}}t}} = \, - P^{\parallel , \bot }\left( t \right)N_0^{\parallel , \bot }\left( t \right) + N_{\mathrm{p}}^{\parallel , \bot }\left( t \right){\gamma}_{\rm{p}} + N_{\mathrm{x}}^{\parallel , \bot , \times }\left( t \right){\gamma}_{\mathrm{NR}}\\ + N_{\mathrm{x}}^{\parallel , \bot , \times }\left( t \right)2{\gamma}_{\mathrm{xx}} - N_{\rm{x}}^{ \times ,\parallel , \bot }\left( t \right){\gamma}_{{\rm{xx}}} - N_{\rm{x}}^{ \bot , \times ,\parallel }\left( t \right){\gamma}_{{\rm{xx}}} \\ \frac{{{\mathrm{d}}N_{\mathrm{x}}^{\parallel , \bot , \times }\left( t \right)}}{{{\mathrm{d}}t}} = \,\, P^{\parallel , \bot }\left( t \right)N_0^{\parallel , \bot }\left( t \right) - N_{\rm{x}}^{\parallel , \bot }\left( t \right)\{ N_{\rm{p}}^{\parallel , \bot }\left( t \right) + 1\} {\gamma}_{xp}\\ - N_{\rm{x}}^{\parallel , \bot , \times }\left( t \right)2{\gamma}_{{\rm{xx}}} + N_{\rm{x}}^{ \times ,\parallel , \bot }\left( t \right){\gamma}_{{\rm{xx}}} + N_{\rm{x}}^{ \bot , \times ,\parallel }\\ \times \left( t \right){\gamma}_{\mathrm{xx}} - N_{\mathrm{x}}^{\parallel , \bot , \times }\left( t \right){\gamma}_{{\rm{NR}}} \\ {\frac{{{\mathrm{d}}N_{\mathrm{p}}^{\parallel , \bot }\left( t \right)}}{{{\mathrm{d}}t}}} = \,\, {N_{\mathrm{x}}^{\parallel , \bot }\left( t \right)\{ N_{\mathrm{p}}^{\parallel , \bot }\left( t \right) + 1\} {\gamma}_{\mathrm{xp}} - N_{\mathrm{p}}^{\parallel , \bot }\left( t \right){\gamma}_{\mathrm{p}}}$$ (9)