# Multimode Physics in the Mode Locking of Semiconductor Quantum Dot Lasers
## 2 Multimode Mode-Locking Approach
### Progression to Mode-Locking Adler's Equation  
The mode-locking conditions in Equations (2) and (3) do not occur naturally because of group velocity dispersion experienced by the laser field. At a minimum, there are contributions from the passive waveguide material and from the carrier-induced refractive index change in the active region. A comparison of the mode spectra in Figure 2a shows the net effect of these contributions. In the upper spectrum with no dispersion, one has a frequency comb, i.e., an equal beat-note between adjacent modes. The shifts indicated in the lower spectrum with dispersion show differences in the beat-notes between adjacent modes. In this case, one has neither a frequency comb nor a mode-locked pulse train. A concise description of the path towards mode locking comes from Adler's equation [23]:  
\[\frac{d\psi}{dt}=a+bsint\psi \tag{4}\]  
which has been very useful in studies involving frequency locking, e.g., the synchronization of clocks, the locking of lasers (both mutually and via external injection), deadband in laser gyroscopes, etc. The beauty of Equation (4) is that it contains only two parameters: \(a\) for the obstacle and \(b\) for the fix. With mode locking, \(d\psi/dt\) is the difference between two beat-notes, \(a\) is the corresponding difference due to dispersion and \(b\) is the locking coefficient. To illustrate how Adler's equation gives a well-defined progression towards locking, we consider three adjacent modes \(n=-1\), \(0\), and \(1\) (see Figure 2a). When \(a>b\), numerically integrating Equation (4) gives \(d\psi/dt\approx a\), with modifications from \(bsint\psi\), as shown by the solid curve in Figure 2b. When \(a<b\), \(\psi\) initially increases according to \(\psi=at\). However, when \(bsint\psi=a\) (which is, of course, possible only when \(b\geq a\)), \(d\psi/dt=0\). This causes \(\psi\) to remain constant and, consequently, because of Equation (4), \(d\psi/dt\) stays as zero, i.e., the  
Figure 1: Mode-locked (**a**) frequency comb and (**b**) pulse train. system stays locked (dashed curve in Figure 2b). Without locking, the phase \(\psi\) increases constantly, as shown by the solid curve in Figure 2c. With locking, it assumes the value:  
\[\psi=sin^{-1}\left(\frac{a}{b}\right) \tag{5}\]  
as shown by the dashed curve in Figure 2c. Of course, the production of a frequency comb or pulse train involves many lasing modes. Then, one has many Adler's equations, each for the difference between two beat-notes. In practice, one tracks the phase of each lasing mode \(\psi_{n}\), which, from a semiclassical laser theory derivation, evolves according to \(\dot{\psi}_{n}=a_{n}+\sum\limits_{m}b_{nm}sin(\psi_{n}-\psi_{m})\)[21; 22]. One looks for steady state solutions where the difference \(\dot{\psi}_{n}-\dot{\psi}_{n-1}-\dot{\psi}_{m}+\dot{\psi}_{m-1}\) vanishes for all combinations of \(n\) and \(m\) for a frequency comb to occur. The steady-state phase difference \(\psi_{n}-\psi_{n-1}-\psi_{m}+\psi_{m-1}\) for all combinations of \(n\) and \(m\) determines the closeness to a transform-limited mode-locked pulse performance  
As an example, we can look at a simulation involving 11 lasing modes. For \(a_{n}\), we assume only dispersion from GaAs, so that the passive cavity frequency for the \(m\)th mode is \(\Omega_{m}=mc/(2 Ln_{GaAs})\), where \(m\) ranges from 1900 to 2100 for lasing around 1.315 \(\upmu\)m in a cavity of length 1.33 nm. For the refractive index \(n_{GaAs}(\lambda)\), we use the Sellmeier-type function with coefficients extracted from GaAs experiments [24]. With \(b\geq 5.0\) MHz, one obtains Equation (2) for \(-5\leq n\leq 5\). Figure 3a shows the modification to the beat-note, with the creation of a flat section (lock band) between \(n=-5\) and 5. A Fourier transformation of \(E(z,t)^{2}\) will show an infinitely narrow resonance at the radio frequency (RF) of 30.60 GHz, indicating a perfect frequency comb. However, a mode-locked pulse train may not result, as depicted in Figure 3b, because the condition specified by Equation (3) is unsatisfied.