Multiple-quantum-well-based photonic crystals with simple and compound elementary supercells

Exciton polaritons in one-dimensional photonic crystals based on multiple quantum well structures are investigated. The effects due to interplay between resonant interaction of light with quantum well excitons, and light scattering from well-barrier interface, are elucidated. Polariton dispersion equations and reflection spectra in structures with two wells in an elementary supercell of the periodic structure are studied. Several examples of different compound elementary supercells are considered. Special attention is paid to structures with the period or the distance between quantum wells satisfying the resonance Bragg condition. Such structures are characterized by a presence of a larger-than-usual polariton stop band. It is shown that in structures with a complex elementary supercell, the width of such a stop band can be significantly enhanced in comparison to that in simple structures.

Figure 1

Allowed minibands (white) and forbidden minigaps (gray) for propagation of light through MQW structures with a contrast of refraction indices in dependence on relative detuning of the characteristic frequency $\overline{\omega }$ from the exciton resonance frequency ${\omega }_0$. The boundary curves 1–4 describe the exciton-polariton frequencies (10), (11) at the Brillouin-zone edge and center. The following parameters are taken in the calculation: ${\Gamma }_0∕{\omega }_0=7\times {10}^{-5}$, $n_a∕n_b=1.1$, and $a∕b=0.05$.

Figure 2

The periodic structure with two quantum wells (dark rectangulars) in the elementary supercell. Indices $m$ and $j$ enumerate the supercells and quantum wells inside one supercell.

Figure 3

The dependence of the position and the width of the forbidden minigaps upon the distance $d_1$ between the quantum wells in the elementary supercell of the periodic structure with the period satisfying doubled resonant Bragg condition $({\omega }_0∕c)n_b(d_1+d_2)=2\pi$. Bright and dark regions correspond to the bands and forbidden gaps, respectively. The third forbidden band $\mid \omega -{\omega }_0\mid ⩽{\Gamma }_0\mid \sin (2\pi d_1∕d)\mid$ is indistinguishable on this scale since ${\Gamma }_0⪡\Delta$.

Figure 4

The evolution of the reflection spectrum with increasing the number of supercells in the structure with $d_1∕d=0.1$ and the period satisfying the doubled Bragg condition. Curves 1, 2, 3 are calculated for $N=10$, 40, and $\infty$ correspondingly.

Figure 5

The effect of the difference between the exciton frequencies ${\omega }_{01}$ and ${\omega }_{02}$ on the reflection spectrum of the structure with 10 pairs of the quantum wells with equal both radiative and nonradiative decay rates ${\Gamma }_{01}={\Gamma }_{02}={\Gamma }_1={\Gamma }_2=7\times {10}^{-5}{\omega }_{01}$. The quantum wells are equidistant and the period meets the Bragg condition at the frequency ${\omega }_{01}$. The curves 1–5 have been obtained for $({\omega }_{02}-{\omega }_{01})∕{\omega }_{01}=0$, 0.002, 0.005, 0.01, and 0.015 correspondingly.