Coupling strength as a function of the number of absorbers

In the samples of thedorgseries presented here, the number of absorbers is changed at constant concentration, i.e., in a neat film, by changing the thicknessd of the organic film (N∝d).

Figure 4.9 shows the data of three exemplary sets of samples of thed_orgseries, one without any C545T (a), one containing 15 nm of C545T (b) and one containing 40 nm of C545T (c). Reflectance spectra (second row) show that cavities of positive and negative detuning were

Figure 4.9:Characterisation of thedorgseries from reflectance spectra. Each column shows a set of samples containing a C545T film of different thickness. (a) left: no C545T, (b) centre: 15 nm C545T, (c) right: 40 nm C545T. The top row shows an illustration of the microcavities and the second row the experimental reflectance spectra for cavities with different cavity thick- ness (optical thicknessdoptindicated in legend). The bottom row compares the mode position of the experimental spectra (red symbols) to TMM calculations (grey background). Dotted, dashed and solid lines represent modes calculated from the CO model. Arrows indicate the position and the extent of the Rabi splitting.

produced (EX ≈ 2.5 eV). By comparison to TMM calculations, the optical thickness of the microcavities was determined. In order to determine the Rabi splitting for each of the cavities, the reflectance of corresponding structures was calculated using the TMM. Additionally, TMM calculations were performed for a continuous variation of the thickness of the SiO₂ films, see the background in the third row of Figure 4.9. The mode positions extracted from the experimental spectra are overlaid (red symbols) and agree well with the TMM calculations.

Moreover, green, solid lines from CO calculations demonstrate that the shape of the modes follows that expected for a strongly coupled system. In agreement with expectations, the plots show an increased Rabi splitting for larger thicknesses of the organic film inside the microcavity. Note, however, that in order to obtain the agreement between TMM and CO

calculations, not only the coupling constant gwas changed. Instead, the energy of the cavity mode (grey, dashed line) had to be changed slightly via the background refractive index of the microcavity and the exciton energy (grey, dotted line) also had to be adjusted between different sets. The small adjustments ofneffmight be due to misestimations of the background refractive index of SiO₂ and/or C545T. The changes in EX are more noticeable; the best agreement for the samples with 15 nm C545T is found for EX = (2.52±0.05) eV, while samples with 40 nm C545T require EX = (2.62±0.05) eV. Even though the change in EX is not drastic, the observed increase of EX for increasing thickness of C545T was continuous. The small values of E_X, which are observed for small Rabi splittings, roughly correspond to the energy of the 0–0 transition (see Figure 4.8). At large Rabi splittings, by contrast, EX ® 2.67 eV is found, which is similar to the position of the peak value of the entire absorption band (at 2.7 eV). This shift inEXcould indicate an asymmetry in the splitting, as would be expected for the ultra-strong coupling regime.103 Since the simple CO model used here does not take into account any counter-rotating terms, an asymmetry might be misinterpreted as a shift in exciton energy (with symmetric splitting). Indeed, TMM calculations show that ford_C545T ¦20 nm,

ħ

hΩ ¦0.5 eV, which would correspond to ħhΩ ≈0.2EX and thus limit the applicability of the rotating-wave approximation on which the CO model relies. Since we only care about the extent of the splitting here, the assumed spectral position of the exciton does not matter and we only accept this discrepancy.

The Rabi splitting was determined from the TMM calculations as the energy of the minimal splitting between two modes. These values agreed well (±0.04 eV) with the values of the splitting obtained from the CO model. Figure 4.10a shows the squared Rabi splitting (symbols) determined in this way as a function of the thickness of the C545T film, dC545T, which is proportional to the number of excitons,N. In agreement with expectations, the Rabi splitting increases with increasing dC545T. However, even though theory predicts that g ∝

p

d, the linear fit (green line) fails to describe the data well; a systematic deviation of the data from the fit is visible. This systematic nature of the deviation is confirmed by obtaining very different fit parameters when only fitting a subset of the values. This points to a hidden variable influencing the coupling strength, which is identified as the electric field. Since the electric field forms a standing wave inside the microcavity with nodes at its edges, films of different thickness will not be exposed to the same average field, A(see middle panel of Figure 4.10). This means that for the d_orgseries, g is not only affected by the change in N, but also by changes inA.

Figure 4.10: Squared Rabi splitting of the dorg series as a function of different parameters: (a) as a function of the physical thickness of the organic film, d_C545T, and (b) as a function of d_C545T ·A2, considering the average electric field overlapping with the organic film (see blue lines in central panel). Each symbol represents one set of microcavities with samples of constant thickness of the organic film. The experimental data are fitted with a linear fit for the entire range where a finite splitting was observed (green, solid lines). R2values for the fits are indicated in the legend. The central panel illustrates the difference between the x-axes of (a) and (b).

Assuming the correctness of Equations 2.24 and 2.27, this can be accounted for by considering both variables in the fit:

ħhΩ2=4g2 − (γX− γC)2 =a_N·dC545T·A

2

−b_N.

(4.2)

Hence, a linear fit witha_N andb_Nas fit parameters can be used to test the validity of Equation 4.2 withx =dC545T·A

2

as the dependent variable.

While plottingħhΩas a function ofdC545Tis straightforward, the calculation ofAdeserves a few comments. In order to determineA(C545T₎,A(z)was calculated for an empty microcavity withEC≈EX. The average electric fieldAwas calculated from the knowledge of the position (centre, z₀) and optical thickness of the film of C545T (d_C545T· n, where n ₌ 1.8). This value was then normalised to the maximum value of the average electric fieldAmax that was measured in any of the subsets:

A= z0+dC545T/2 z0−dC545T/2 A(z)dz·dC545T·n·Amax ₋1 . (4.3)

This might seem an oversimplified method, given that the electric field inside the microcavity is certainly changed by introducing a strongly absorbing material into the cavity. Not only is

the refractive index no longer constant, but the mode positions change. Hence, the electric field inside the strongly coupled microcavities is negligible at EC. This is, however, also the reason why it is difficult to compare the electric field in different cavity structures directly from corresponding TMM calculations. The described method, which was also used by Wanget al.

[42], Schouwinket al.[71], and Hobsonet al.[72], was thus chosen as an estimate ofA. The normalised values forAranged from 0.81 for the thickest film (104 nm) to 1 for the thinnest film (3 nm) of C545T.

In order to test Equation 4.2, the Rabi splitting in Figure 4.10b is plotted as a function of d_C545T·A2. This corresponds to a rescaling of the physical film thickness for the average electric field overlapping with the film. A linear fit was performed on all sample sets for which a finite Rabi splitting was detected, i.e., for d_C545T > 3 nm (see green lines in Figure 4.10). The coefficients of determination for each fit are indicated in the legend. Clearly, not only the coefficient of determination improves when changing taking the overlap with the electric field into account (R2 =0.96_→R2 =0.997). Also, no systematic deviation of the data from the fit is observed; the remaining deviations between data and fit are of a random nature. The experiment thus confirms that g ∝ pN. However, a comparison between x = dC545T and x = dC545T·A

2

also shows that in the studied samples, the effect of the electric field is small compared to the changes induced by changingdC545T. Subtle deviations from a g∝A dependence would thus not necessarily have been obvious in the previously discussed results. Therefore, the next section will look at the effect of the electric field in detail.