The calculations of the field distributions in the laser waveguide were performed with the finite element method using Photon Design Fimmwave software (
http://www.photond.com). We were looking for transversal modes with the highest confinement factor with the active region
\({\varGamma }\) and the lowest mode loss
\(\alpha\). The confinement factor is defined as
$$\varGamma \equiv \frac{\int ^{AR}{\left| \left\langle\mathbf {S}(r)\right\rangle\right| dr} }{\int ^{\infty }{\left| \left\langle\mathbf {S}(r)\right\rangle\right| dr} },$$
(1)
where
\(\langle\mathbf {S}(r)\rangle\) is the time-averaged Poynting vector, the numerator is the integral over the active region and the denominator is the integral over the whole structure. The mode loss were calculated with the assumption of zero gain in the active region as
$$\begin{aligned} \alpha =2{\text {Im}}(\beta ), \end{aligned}$$
(2)
where
\({\text {Im}}(\beta )\) is the imaginary part of the propagation constant
\(\beta\).
Threshold gain
\(g_{th}\) was calculated as
$$\begin{aligned} g_{th}=\frac{1}{{\varGamma }}\left( \alpha +\frac{1}{L}\log \left( \frac{1}{R}\right) \right) , \end{aligned}$$
(3)
where
L is the resonator length and
R is the mirrors reflectivity. We considered uncovered, identical laser mirrors with reflectivity
\(R=0.2858\).
Input material data (refractive indices
\(n_r\) and absorption coefficients
\(\alpha _r\)) for the simulations are listed in Table
1. These parameters for the semiconductors were calculated according to Drude–Lorentz model (Evans et al.
2012) with the following formula:
$$\begin{aligned} n_r+\iota \frac{\lambda }{4\pi } \alpha _r =\sqrt{\epsilon _{\infty }\left( 1 + \frac{\omega ^2_{LO}-\omega ^2_{TO}}{\omega ^2_{TO}-\frac{c}{\lambda }\left( \frac{c}{\lambda }+\iota \gamma _{phon}\right) } -\frac{{N_f q_e^2}}{\frac{c}{\lambda }\left( \frac{c}{\lambda }+\iota \frac{q_e}{m^{*}m\mu _e}\right) {\epsilon _0 \epsilon _{\infty }m^{*}m}} \right) }, \end{aligned}$$
(4)
where
\(N_f\) is the free-carrier concentration,
\(q_e\) is the electronic charge,
m is the electron rest mass,
c is the speed of light, and
\(\epsilon _0\) is the permittivity of free-space. For the binary materials (InAs, GaAs, AlAs and InP) the remaining parameters, i.e. high-frequency dielectric constant
\(\epsilon _\infty\), longitudinal optical phonon frequency
\(\omega _{LO}\), transversal optical phonon frequency
\(\omega _{TO}\), phonon damping frequency
\(\gamma _{phon}\) come from Evans et al. (
2012), electron effective mass
\(m^{*}\) comes from Vurgaftman et al. (
2001), and electron low-field mobility
\(\mu _e\) comes from Sotoodeh et al. (
2000). Parameters for the ternary materials were assumed as linear interpolation between the corresponding binary materials. The active region is considered as one bulk layer with zero absorption and with refractive index being weighted average of the refractive indices of the constituent materials with weights proportional to the thicknesses of the layers.
Table 1
Material parameters of the waveguide layers
\(\hbox {In}_{0.5273}\hbox {Ga}_{0.4727}\hbox {As}\)
|
\(8\times 10^{18}\)
| 2.5737 | 8627 | Drude–Lorentz model |
\(\hbox {In}_{0.5273}\hbox {Ga}_{0.4727}\hbox {As}\)
|
\(4\times 10^{16}\)
| 3.4028 | 7.7 | Drude–Lorentz model |
\(\hbox {In}_{0.6663}\hbox {Ga}_{0.3337}\hbox {As}\)
|
\(2\times 10^{17}\)
| 3.4145 | | Drude–Lorentz model |
\(\hbox {In}_{0.6663}\hbox {Ga}_{0.3337}\hbox {As}\)
| Undoped | 3.4345 | | Drude–Lorentz model |
\(\hbox {In}_{0.5226}\hbox {Al}_{0.4774}\hbox {As}\)
|
\(1\times 10^{17}\)
| 3.2224 | 64 | Drude–Lorentz model |
\(\hbox {In}_{0.3634}\hbox {Al}_{0.6366}\hbox {As}\)
|
\(2\times 10^{17}\)
| 3.1173 | | Drude–Lorentz model |
\(\hbox {In}_{0.3634}\hbox {Al}_{0.6366}\hbox {As}\)
| Undoped | 3.1385 | | Drude–Lorentz model |
InP substrate |
\(2\times 10^{17}\)
| 3.0747 | 78 | Drude–Lorentz model |
\(\hbox {Si}_{3}\hbox {N}_{4}\)
| | 2.3531 | 73 | |
Au | | 3.2022 | 792760 | |
Ti | | 3.392 | 233360 |
Mash and Motulevich ( 1973) |
Pt | | 3.8024 | 495230 | |