T-shaped waveguides for quantum-wire intersubband lasers

We present a T-shaped waveguide for a quantum-wire intersubband laser in the $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{Ga}\mathrm{As}$ heterosystem for the midinfrared spectral region around $8\mu \mathrm{m}$. Such a device is fabricated using the cleaved edge overgrowth (CEO) technique. The suggested waveguide structure is compatible to this growth technique. The confinement factor and the waveguide losses due to free carrier absorption are calculated solving the scalar wave equation and compared to the results for a conventional quantum cascade laser (QCL) in the $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{Ga}\mathrm{As}$ material system based on coupled quantum-wells. We show that the optical mode can be confined near the active region of a quantum-wire cascade structure with the presented T-shaped waveguide design. Simultaneously the waveguide losses can be reduced compared to a conventional QCL waveguide design. However, the confinement factor is substantially reduced as a result of a small optically active region in the quantum-wire cascade device.

Figure 1

Sample structure of the quantum-wire cascade structure considered for the presented waveguide calculations. This structure is realized employing the cleaved edge overgrowth (CEO) technique (Ref. 12). The relevant crystal directions are given. The extension of the quantum wires is normal to the drawing plane. The sketch is not to scale.

Figure 2

Contour plots of the fundamental modes and the first order mode (upper left contour plot) of the quantum-wire cascade structure of Fig. 1 for several GaAs waveguide layer thicknesses $d_{wg}$ and a ridge width of $d_{rw}=30\mu \mathrm{m}$. The corresponding calculated confinement factors $\Gamma$ of the optical modes are also given. The confinement factor has a maximum for a thickness of the GaAs waveguide layer of about $5.3\mu \mathrm{m}$ (see also Fig. 3).

Figure 3

Confinement factor $\Gamma$ of the fundamental mode (closed symbols) in dependence of the thickness of the GaAs waveguide layer along the [110] direction for different ridge widths. For wider ridges the GaAs waveguide layer has to be thicker to achieve best confinement. For smaller GaAs waveguide layer thicknesses the first order mode (open symbols) has a better confinement factor than the corresponding fundamental mode.

Figure 4

Waveguide loss of the fundamental mode (closed symbols) in dependence of the thickness of the GaAs waveguide layer along the [110] direction for different ridge widths. The losses decrease with increasing GaAs waveguide layer thickness along the second growth direction, due to a decreased overlap of the fundamental mode with the lossy $n^{++}$-contact layers. For small GaAs waveguide layer thicknesses the first order mode (open symbols) has lower waveguide losses than the corresponding fundamental mode. The different dependence of the waveguide losses on the waveguide layer thickness along the [110] direction is explained by the splitting of the first order mode.

Figure 5

Figure of merit of the fundamental mode (closed symbols) in dependence of the thickness of the GaAs waveguide layer along the [110] direction for different ridge widths. For small GaAs waveguide layer thicknesses the figure of merit of the first order mode (open symbols) is higher than the figure of merit of the corresponding fundamental mode. Because of the higher waveguide losses, the maximum value of the figure of merit for the first order mode is smaller than the maximum value for the fundamental mode for all calculated ridge widths.