# Multimode Physics in the Mode Locking of Semiconductor Quantum Dot Lasers
## 2 Multimode Mode-Locking Approach
### Physics of the Locking Mechanism  
To investigate the physical origin of the locking coefficient \(b\), we generalize Adler's equation with the help of multimode semiclassical laser theory to obtain [22]:  
\[\dot{\psi}_{n}=\Omega_{n}+\sigma_{n}-\tau_{n}\epsilon_{n}^{2}-\sum_{m\
eq n} \tau_{n,m}\epsilon_{m}^{2}\]  
\[-\sum_{\{n^{\prime},m,m^{\prime}\}}Im\Big{[}\vartheta_{m^{\prime}mm^{\prime}} e^{-i\phi_{m^{\prime}m^{\prime}m^{\prime}}}\Big{]}\epsilon_{n^{\prime}}(t) \epsilon_{m}(t)\epsilon_{m^{\prime}}(t)\epsilon_{n}^{-1}(t) \tag{6}\]  
where the right-hand side of the first line corresponds to \(a\) in Equation (4) and the locking coefficient \(b\) is contained within the expression in the second line. The uncluttered appearance of Equation (6) is obtained by defining a dimensionless electric field amplitude \(\varepsilon_{n}=\wp E_{n}/2\hbar\gamma\), where \(\gamma\) is the dephasing rate, \(\wp\) is the electron-hole dipole matrix element, \(E_{n}\) is the modal electric field amplitude in Equation (1) and \(\hbar\) is the Planck constant divide by \(2\pi\). In the equation, \(\Omega_{n}\) is the passive cavity frequency including waveguide dispersion, and the remaining terms describe the carrier-induced refractive index change and its correction due to saturation, according to the coefficients \(\sigma_{n}\) and \(\tau_{n}\), respectively. [21,22]. In the second line, the bracket \(\{\}\) denotes summing up only the combinations of \(n^{\prime},m,m^{\prime}\) not included in the terms in the first line. The phase difference,  
\[\psi_{nn^{\prime}mm^{\prime}}=\psi_{n}-\psi_{n^{\prime}}+\psi_{m}-\psi_{m^{ \prime}} \tag{7}\]  
and associated coefficient \(\vartheta_{nn^{\prime}mm^{\prime}}\) come from the relative phase angle contributions, either from the saturable absorber in a two-section laser or from four-wave mixing in the gain medium in a single-section laser. The general equations for \(\vartheta_{nn^{\prime}mm^{\prime}}\) for both homogeneously broadened lasers and Doppler-broadened lasers may be found in early laser physics papers and textbooks [21]. The equation, specifically for an inhomogeneously broadened QD gain medium, is given in one of our recent papers [22]. The equation of motion for \(\psi_{nn^{\prime}mm^{\prime}}\) for a given set of indices has the form of Adler's equation in Equation (4). In the numerical model, we track directly the set of Equation (6) and look for steady-state solutions where \(d\psi_{n.n+1,m,m+1}/dt\) vanishes for a combination of indices. Figure 3: (**a**) Beat-note showing a lock band for modes \(-5\leq n\leq 5\) with locking coefficient \(b=5.0\) MHz and dispersion from GaAs [24]. The dashed curve shows the beat-notes with \(b=0\). (**b**) Intensity (\(|\sum_{n}E_{n}cos[\psi_{n}(t)]|^{2}\)) versus time showing progression towards transform-limited pulse train (blue dotted curves) with \(b\) increasing from 5 MHz to (**c**) 6 MHz and (**d**) 7 MHz. According to multimode laser theory, \(\vartheta_{nn^{\prime}mm^{\prime}}\) includes spatial and spectral hole burning, with input parameters involving electronic structure, material constants and carrier relaxation rates. An examination of the expression for \(\vartheta_{nn^{\prime}mm^{\prime}}\) indicates that spatial hole burning dominates in a two-section laser with a saturable absorber, while spectral hole burning is important in determining the strength of self-mode locking in a single-section laser. Lastly, unlike Adler's equation, the locking term in Equation (6) contains the electric field amplitudes, giving rise to rich dynamical behaviors. To allow the treatment of these behaviors, the model has additional equations of motion that are solved simultaneously with that for the laser phase.