Exciton-polaritonic quasicrystalline and aperiodic structures

We have theoretically studied propagation of exciton-polaritons in deterministic aperiodic multiple-quantum-well structures, particularly, in the Fibonacci and Thue-Morse chains. The attention is concentrated on the structures tuned to the resonant Bragg condition with two-dimensional quantum-well exciton. Depending on the number of wells, the super-radiant either photonic-quasicrystal regimes are realized in these aperiodic structures. For moderate values of the exciton nonradiative damping rate $\Gamma$, the developed theory based on the two-wave approximation allows one to perceive and describe analytically the exact transfer-matrix computations for transmittance and reflectance spectra in the whole frequency range except for a narrow region near the exciton resonance ${\omega }_0$. In this region the optical spectra and the exciton-polariton dispersion demonstrate scaling invariance and self-similarity which can be interpreted in terms of the “band-edge” cycle of the trace map, in the case of Fibonacci structures, and in terms of zero reflection frequencies, in the case of Thue-Morse structures. With decreasing $\Gamma$, in the whole allowed polariton band the two-wave approximation stops to be valid, and a transition occurs from Bloch-like to localized states, with modes closer to ${\omega }_0$ becoming localized first.

Figure 1

(Color online) Reflection spectra calculated for four QW structures, each containing $N=50$ wells and tuned to the resonant Bragg condition $2\overline{d}=\lambda \left({\omega }_0\right)$: periodic structure, with $a\equiv b\equiv \overline{d}$ (dashed); Fibonacci chain, with $a/b=\tau$ (solid curve); Thue-Morse sequence with $a/b=3/2$ (dotted); and weakly disordered periodic MQWs with ${\sigma }_z={\lambda }_0/20$ (dashed-and-dotted). Note the break on the abscissa axis in panel (a) around $\omega ={\omega }_0$. Panels (b)–(d) show the same spectra in larger scale of the variable $\left(\omega -{\omega }_0\right)/{\Gamma }_0$. Calculated for $\hslash {\Gamma }_0=50\mu \text{eV}$, $\hslash {\omega }_0=1.533\text{eV}$, and $\Gamma =0.1{\Gamma }_0$.

Figure 2

(Color online) Conventional and differential reflection spectra calculated for Fibonacci QW structures containing $N=50$ wells. Panel (a) shows reflection spectrum $R\left(\omega \right)$ calculated for $\Gamma =0.1{\Gamma }_0$ (thin curve) and $\Gamma =2{\Gamma }_0$ (thick curve). Panels (b) and (c) demonstrate the first- and second-order differential spectra $R^{\prime }\left(\omega \right)$ and $R^{″}\left(\omega \right)$ in arbitrary units for $\Gamma =2{\Gamma }_0$. Other parameters are the same as in Fig. 1.

Figure 3

(Color online) Reflection spectra calculated for a Fibonacci QW structure containing 1000 wells. The parameters used are the same as those in Fig. 1 except for the nonradiative decay which now is $\Gamma =0.2{\Gamma }_0$. Upper and lower panels correspond, respectively, to the exact calculation and calculation in the two-wave approximation. Vertical arrow at $\omega ={\omega }_0$ in panel (a) indicates the narrow reflectivity stop band, which cannot be described by the two-wave approximation.

Figure 4

(Color online) Reflection spectra calculated for Thue-Morse and periodic QW structures. Thin solid curve corresponds to the exact calculation for $N=1000$ Thue-Morse QWs, thick solid curve is calculated in the two-wave approximation for $N\to \infty$. Dotted curve represents the spectrum for periodic MQWs with $N=1000$ wells and the period $d=\lambda \left({\omega }_0\right)/2$. The parameters used are the same as those in Fig. 3.

Figure 5

(Color online) Complex eigenfrequencies of exciton polaritons in Fibonacci 56-QW structure calculated exactly (filled symbols) and in the two-wave approximation (empty symbols). Note the break in the ordinate axis. Vertical arrows indicate the edges of inner stop band. Calculated for $\Gamma =0$ and other parameters same as in Fig. 1.

Figure 6

(Color online) (a). Exciton-polariton allowed (thin lines) and forbidden (thick stripes) bands in periodically repeated Fibonacci sequences of the order $j=1\dots 13$. (b) Bands for $j=11$ ad $j=13$ in the spectral range around the frequency $\omega ={\omega }_0$. (c) and (d) Bands for $j=11$ and $j=13$, respectively, in a large scales near the frequency $\omega ={\omega }_0+0.935{\Gamma }_0$ indicated by the vertical line. Calculated for $\Gamma =0$ and other parameters same as in Fig. 1.

Figure 7

(Color online) Trace scaling for the Fibonacci structures. The panels (a)–(c) show the half-trace $x_j\left(\omega \right)$ for $j=11,13,15$, respectively. The filled ribbons indicate the regions of polariton band gaps, where $\left|x_j\left(\omega \right)\right|>1$. Calculated for $\Gamma =0$ and other parameters same as in Fig. 1.

Figure 8

(Color online) Transmission spectra of the Fibonacci quantum-well sequences of the order $j=11$ (a) and $j=13$ (b) and (c), containing 89 and 233 QWs, respectively. Vertical lines indicate the real parts of complex eigenfrequencies of the structures. Calculated for $\Gamma =0$ and other parameters same as in Fig. 1.

Figure 9

(Color online) Fine structure of the transmission spectra calculated for Thue-Morse quantum-well sequences of order $j=4,5,6,7$ $\left(N=16\dots 128\right)$. Calculated for $\Gamma =0$ and other parameters the same as in Fig. 1.