Crossover of exciton-photon coupled modes in a finite system

An exciton and a photon behave as a polariton in macroscopic materials while they can be treated almost independently in nanostructured ones. We have theoretically investigated the crossover of exciton-photon coupled modes, each of which is characterized with a resonance frequency and a radiative decay rate, in a semiconductor film by continuously increasing the thickness. The resonance frequency and the radiative decay rate are calculated from poles of exciton correlation functions renormalizing the exciton-photon interaction in the film, and we also introduce an intuitive calculation method with considering additional boundary conditions in order to derive the crossover condition. The general properties of the coupled modes are analytically discussed by the intuitive method.

Figure 1

[(a) and (b)] Radiative decay rates $\gamma$ and [(c) and (d)] resonance frequency ${\omega }_{\text{res}}$ of the exciton-photon coupled modes in a CuCl film are plotted as a function of the film thickness. (a) and (c) are, respectively, enlarged views of (b) and (d) for small thickness region. While 200 states of exciton center-of-mass motion are considered in the numerical calculation, for the sake of easy view, the values of the lowest ten and 45 modes are plotted in (a) and (c), and (b) and (d), respectively. The dielectric constants of outside media are ${\epsilon }_L={\epsilon }_R={\epsilon }_{\text{bg}}$.

Figure 2

Schematic view of polariton and photon fields to calculate the reflection coefficients from inside of the film. $A_{1/2}$ and $B_{1/2}$ are polariton fields with wave number ${\tilde{k}}_{1/2}$ inside the film, $C_{L/R}$ is an outgoing field to left/right-hand side, and $F_1$ is an incident field with wave number $k_1$ from inside of the film.

Figure 3

(a) Wave number $k$, (b) radiative decay rate $\gamma$, (c) factor $\alpha$ of effective thickness $d/\alpha =-\text{Im}{\left[\tilde{k}\right]}^{-1}$, and (d) apparent propagation speed $\gamma d_{\text{eff}}$ are plotted as a function of resonance frequency ${\omega }_{\text{res}}$ of exciton-photon coupled modes in a CuCl film with thicknesses of 50, 200, and 500 nm. The lines are calculated by solving Eqs. (25, 32) for a given real wave number $k$ and the symbols are calculated by the correlation function method. The dielectric constants of outside media are ${\epsilon }_L={\epsilon }_R={\epsilon }_{\text{bg}}$.

Figure 4

(a) Wave number $k$, (b) radiative decay rate $\gamma$, (c) factor $\alpha$ of effective thickness $d/\alpha =-\text{Im}{\left[\tilde{k}\right]}^{-1}$, and (d) apparent propagation speed $\gamma d_{\text{eff}}$ are plotted as a function of resonance frequency ${\omega }_{\text{res}}$ of exciton-photon coupled modes in a CuCl film with thicknesses of 1.6, 2, and $3\mu \text{m}$. The lines are calculated by solving Eqs. (25, 32) for a given real wave number $k$, and the symbols are calculated by the correlation function method, where 400 states of exciton center-of-mass motion are considered in the numerical calculation. The dielectric constants of outside media are ${\epsilon }_L={\epsilon }_R={\epsilon }_{\text{bg}}$.

Figure 5

Thickness dependence of apparent propagation speed $\gamma d_{\text{eff}}$. Dashed lines are calculated by the intuitive method with ABC and solid lines are by the correlation function method. 200 exciton states are considered in the latter calculation and the small deviation at large thicknesses comes from this finiteness. The dielectric constants of outside media are ${\epsilon }_L={\epsilon }_R={\epsilon }_{\text{bg}}$.

Figure 6

Apparent propagation speeds $\gamma d_{\text{eff}}$ of exciton-photon coupled modes are plotted as a function of their resonance frequencies ${\omega }_{\text{res}}$ by changing the film thickness. The data is calculated by the intuitive method with ABC and plotted (a) in linear scale and (b) in log scale. The dielectric constants of outside media are ${\epsilon }_L={\epsilon }_R={\epsilon }_{\text{bg}}$.

Figure 7

(a) Apparent propagation speed $\gamma d_{\text{eff}}$ and (b) radiative decay rate of exciton-photon coupled modes in a CuCl film are plotted as a function of the film thickness. The dielectric constants of outside media are ${\epsilon }_L={\epsilon }_R=1$. (a) is calculated by the intuitive method with ABC and (b) is by the correlation function method.

Figure 8

Apparent propagation speeds $\gamma d_{\text{eff}}$ of exciton-photon coupled modes are plotted as a function of their resonance frequencies ${\omega }_{\text{res}}$ by changing the film thickness. The data is calculated by the intuitive method with ABC and plotted (a) in linear scale and (b) in log scale. The dielectric constants of outside media are ${\epsilon }_L={\epsilon }_R=1$.