We consider the laser gain as a figure of merit and define the instability criterion to satisfy stability conditions. Using the scattering times and calculating the dipole matrix elements, the gain coefficient g can be estimated for each structure as [108]:
|
| (6.7) |
where λ0 is the wavelength in vacuum, ϵ0 the vacuum dielectric constant, e the elementary charge, neff the effective refractive index of the mode, Lp the length of one period including active region and injector, and 2γ32 is the full width at half maximum value of the luminescence spectrum.
In fact, the optimization framework changes the geometrical parameters which affect the lifetimes and matrix elements. Simulation results indicate that the two key parameters to be optimized are the matrix element μ and the upper laser level lifetime τ3 (≈ T1 in Eq. 6.6). The gain coefficient increases with these two parameters (see Eq. 6.7). The reference design has a 3QW vertical active-region. The parameters of the reference design are mentioned in [221]. The layer sequence of the In0.52Al0.48As∕In0.53Ga0.47As heterostructure for the optimized design, starting with the injection barrier is: 3.74/2.36/1.50/7.45/1.63/5.63/2.15/4.43/3.86/3.85/ 3.15/3.18/4.40/2.79/2.67/2.46 nm, where the barrier layers are in bold and underlined layers are n-doped with Si at 2 × 1017cm-3.
Figure 6.2(a) shows the PSO results for different particles in the search space. Most of the points (samples) are located at the right-side of the vertical dashed-line which represents unstable laser operation. The samples at the left-side of the vertical dashed-line and above the horizontal dashed-line are the ones we are looking for.
The instability threshold behavior and gain spectrum for the reference design and two obtained optimized designs are exhibited in Fig. 6.2(b). The first structure is optimized at the same wavelength as the reference design and the second one is optimized at a higher wavelength but with a larger gain. Our developed framework has the advantage of optimizing the reference structure for a specific wavelength. We focus on the first optimized structure which has the same wavelength as the reference structure.
The conduction band diagram and the associated wave-functions of the reference design and optimized structure are shown in Fig. 6.3. The active region of the optimized structure is a three-well (3QW) vertical-transition design which provides high optical gain and concomitant robustness for laser action, see Fig. 6.3(b). The upper laser level for the optimized structure is delocalized which results in an increase of the upper laser level lifetime τ3 and consequently the laser gain, see Eq. 6.7.
| T1 | Gain recovery time | 0.46ps |
| T2 | Dephasing time | 0.06 ps |
| μ | Matrix element of lasing transition | 2.44 × 10-9 m |
| n | Refractive index | 3.25 |
| l0 | Linear cavity loss | 500 m-1 |
| L | Cavity length | 6 × 10-3 m |
| γ | Saturable absorber coefficient | 10-11m∕V2 |
The parameters extracted for the instability analysis of the optimized structure are shown in Table 6.1. Fig. 6.4(a) indicates that the instability threshold decreases uniformly with the SA coefficient, for example γ ≥ 3 m∕V2 triggers the instability for the optimized structure at a relatively low pumping strength (pf = 2). As reported in [126], a saturable absorber always favours a multimode regime to a single mode one, which explains why the instability threshold decreases with γ in the optimized design. As discussed later, this instability results in Rabi sidebands around the cw lasing frequency.
The parametric gain of the optimized structure at various pumping strengths is shown in Fig. 6.4(b). A larger pumping strength broadens the instability characteristics and decreases the instability threshold. The results are consistent with previous works [126].
Next, we present the optimization results for a reference QCL with a
superlattice (SL) active-region [105]. The layer sequence of the SL structure,
starting with the injection barrier, is as follows: 4.0/1.9/0.7/5.8/0.9/
5.7/0.9/5/2.2/3.4/1.4/3.3/ 1.3/3.2/1.5/
3.1/ 1.9/3.0/ 2.3/2.9/2.5/2.9 nm. The In0.52Al0.48As barrier layers are in bold
and In0.53Ga0.47As well layers are in roman. The underlined layers are n-doped
with Si at 4 × 1017cm-3.
The instability characteristics for the 3QW and SL active-regions are compared in Fig. 6.5(a). Because of the larger matrix element (μ) and longer upper laser state lifetime (τ3), which is approximately equal to the gain recovery time (T1), the SL active-region QCL indicates more stable operation and higher instability threshold, see Fig. 6.5(a). However, as we mentioned earlier, the matrix element and lifetimes of the lasing transition, which are the key parameters in linear stability analysis, are proportional to optical gain (see Eq. 6.7). As indicated in the inset of Fig. 6.5(b), the 3QW QCL exhibits larger optical gain at nearly the same wavelength. The optical gain of the 3QW structure is maximized by delocalizing the lasing states which increases the lifetimes τ3 and τ2. Because of the bound states in the SL active-region, there is no significant lifetime variation, however, due to the larger matrix element, a better instability condition is achieved.
Terahertz frequencies (1-10 THz, 30-300 μm) are among the least developed electromagnetic spectra even though they have wide ranging applications in spectroscopy, imaging, and remote sensing [222]. Since the report of quantum cascade lasers (QCLs) operating in the terahertz spectral region by Kšohler et al. [99] there has been significant progress concerning the available frequencies, the temperature performance, and the understanding of the dynamics [223].
Emitting radiation due to intersubband optical transitions in QCLs makes this possibility to generate terahertz radiation in these lasers. The terahertz radiation in QCLs could be engineered by electronic bandstructure. We select the designs that utilize strong electron-phonon interaction in the semiconductor as a means to establish population inversion for optical gain [2].
Based on the algorithm introduced in Sec. 6.1.2, we study the SA and pumping factor effects on instability characteristics of terahertz QCLs.
Here, a reference QCL operating at 3.44 THz, corresponding to λ = 87.2 μm, is employed [4]. The layer sequence of the reference structure, starting with the injection barrier, is as follows: 5.4/7.8/2.4/6.43.8/14.8/2.4/9.4 nm. The 14.8 nm well is doped at 1.9 × 1016cm-3. The operating temperature for reference design is 5 K up to 65 K, however, higher operating temperatures are achieved for optimized THz QCLs.
Figure 6.6 exhibits instability characteristics for two QCLs operating in infrared and terahertz spectral region. Figure 6.6(a) indicates that the instability threshold decreases faster with SA coefficient for mid infrared QCL sample while the terahertz QCL sample still operates below the instability threshold. SA can bring g(Ω) above zero, thereby triggering an instability which is more effective for mid infrared QCL. The parametric gain of the optimized structure at various pumping strengths is shown in Fig. 6.6(b). A larger pumping strength broadens the instability characteristics and decreases the instability threshold which is significant for mid infrared QCL. The results indicate more stability for QCLs operating in the terahertz spectral region.