Optical properties of one-dimensional photonic crystals based on multiple-quantum-well structures

A general approach to the analysis of optical properties of photonic crystals based on multiple-quantum-well structures is developed. The effect of the polarization state and a nonperpendicular incidence of the electromagnetic wave is taken into account by introduction of an effective excitonic susceptibility and an effective optical width of the quantum wells. This approach is applied to consideration of optical properties of structures with a pre-engineered break of the translational symmetry. It is shown, in particular, that a layer with different exciton frequency placed at the middle of an MQW structure leads to appearance of a resonance suppression of the reflection.

Figure 1

A scheme of the angular dependence of the Fresnel coefficients ${\rho }_s$ and ${\rho }_p$ on the angle ${\theta }_w$ is shown on the complex plane. At normal incidence the coefficients have values shown by small filled circles (the same for both polarizations). When the angle of incidence increases, the coefficients follow the arrows on the lines. ${\rho }_p$ passes through zero at the Brewster’s angle, both coefficients reach the unit circle at the angle of total internal reflection. When the angle increases further, the Fresnel coefficients become complex with the unit modulus and increasing argument.

Figure 2

Change of the effective radiative decay rates ${\Gamma }_s$ (solid line) and ${\Gamma }_p$ (dotted line) with the angle of incidence for a single quantum well.

Figure 3

Dependence of the amplitude reflection coefficient ${\mid r_N\mid }^2$ on the frequency. The main plot shows the reflection of structure satisfying the modified Bragg condition with $\rho =0.005$, the radiative decay rate ${\Gamma }_0=67\mu \mathrm{eV}$, the exciton frequency ${\omega }_0=1.491\mathrm{eV}$, the homogeneous broadening $\gamma =500\mu \mathrm{eV}$ and the length $N=10$, 25, 100 (dotted, dashed, solid lines, respectively). On the inset reflection spectra of structures that satisfy the standard Bragg condition are shown. To make the correspondence with the results of Ref. 35 clearer we chose $\rho =-0.005$.

Figure 4

The change of the reflection spectrum with the angle of incidence. The parameters of the structure are the same as those used in Fig. 3 except $\gamma =25\mu \mathrm{eV}$, $\rho =0.01$. The dotted line shows the reflection at normal incidence. The solid and dashed lines show the reflection at ${\theta }_b=\pi ∕18$ of $s$- and $p$-polarized waves, respectively. The main plot corresponds to the structure that is tuned to the Bragg resonance at normal incidence. The inset shows the reflection of a structure that is tuned to the Bragg resonance at ${\theta }_b=\pi ∕18$.

Figure 5

The reflection spectrum of $s$- and $p$-polarized waves (solid and dashed lines, respectively) of a structure that is tuned to the Bragg resonance at ${\theta }_b=\pi ∕4$. The Bragg condition is met for $s$-polarized waves.

Figure 6

The reflection (dotted line, right scale), transmission and absorption (solid and dashed lines, respectively, left scale) are shown in a vicinity of ${\omega }_R$ for a Bragg 5-1-5 structure. The parameters of the quantum wells are the same as those used in Fig. 3 except $\gamma =25\mu \mathrm{eV}$, ${\omega }_h=1.491\mathrm{eV}$, and ${\omega }_d=1.495\mathrm{eV}$. To emphasize the resonant character of the change of the reflection it is plotted in the log scale. It should be noted that because of the large (in comparison with $\gamma$) separation between ${\omega }_R$ and ${\omega }_d$ the drop of the reflection is not accompanied with a resonant absorption.