# Multimode Physics in the Mode Locking of Semiconductor Quantum Dot Lasers
## 3 Towards Transform-Limited Performance
### Linewidth Enhancement Factor  
Amplitude phase coupling, namely, the dynamic change of the active medium refractive index in relation to changes in the gain, commonly characterized by the linewidth enhancement factor (\(\alpha_{H}\) factor) reflects a basic characteristics of semiconductor lasers. It is associated with several basic characteristics of the QD laser, such as linewidth, conversion efficiency of four-wave mixing and other nonlinear properties [33]. Let us stress that in our model, described in the previous sections, the contributions to the alpha factor, i.e., the real and imaginary parts of the active medium's susceptibility, come out of the perturbation analysis in the form of the frequency pulling and linear gain, respectively. There are several ways to measure the \(\alpha_{H}\)-factor. Typically, the \(\alpha_{H}\)-factor is extracted using peak shift and intensity change to reflect the dynamics of the refractive index and gain change, which we call the amplified spontaneous emission (ASE) method [34]. While this method can extract the spectral dependence of the \(\alpha_{H}\)-factor, it is fundamentally restricted to the ASE regime and cannot be used to extract the \(\alpha_{H}\)-factor above the threshold current. Alternatively, an interferometric method relying on a Mach-Zehnder interferometer and optical injection locking can be used to extract an \(\alpha_{H}\)-factor above the threshold current, but both can only be applied to single-mode lasers [34]. The effective extraction of an \(\alpha_{H}\)-factor above the threshold current of a multimode laser helps to analyze the nonlinear dynamics with the current or wavelength. In this study, the phase modulation (PM) method is used to obtain information on the longitudinal modes of a multimode laser in order to extract an \(\alpha_{H}\)-factor above the threshold value [35]. Recently, the \(\alpha_{H}\)-factor of the quantum cascade laser was also extracted using this method [36]. In what follows, the effects of different modulation frequencies on the \(\alpha_{H}\)-factor are measured on a silicon-based FP QD laser. In order to confirm the advantages of such lasers, we also compare the \(\alpha_{H}\)-factor results with those obtained from quantum well lasers heterogeneously grown on silicon. The experimental set-up for optical phase modulation is shown in Figure 9. The QD laser was kept at an operating temperature of 30 \({}^{\circ}\)C during the test. The laser maintains a bias current of twice the threshold value. The two RF signals (\(f_{m}\)) are split by the RF power divider (RFPS) and then modulate both the QD laser and the phase modulator (PM). In order to operate the laser in a small signal modulation range, the sinusoidal signal applied to the QD laser needs to be controlled by an RF attenuator. A bias tee (BT) channels the direct current pumping and RF signal transmission. Once the laser is running, the beam travels through an optical delay line (ODL) and is fed to a phase modulator to control the delay between the optical and electrical signals. With this method, an optical spectrum analyzer (OSA) records spectra at four different delays (\(1/(4f_{m})\)) to extract the \(\alpha_{m}\)-factor, which is related to the modulation frequency. These are the normalized spectra for four different delays at a modulation frequency of 8.5 GHz, as shown in Figure 10a. Figure 9: Schematics of the experimental set-up of optical phase modulation method. RFPS, radio-frequency power divider; BT, bias tee; ODL, optical delay line; PM, phase modulator; OSA, optical spectrum analyzer. The \(\alpha_{m}\)-factor can be obtained by extracting the intensity of the side modes using the phase modulation method [7]. The relationship between the \(\alpha_{H}\)-factor and the corner frequency (\(f_{c}\)), modulation frequency (\(f_{m}\)) and \(\alpha_{m}\)-factor is \(\alpha_{m}=\alpha_{H}[1+(f_{c}f_{m})^{2}]^{1/2}\)[34]. Thus, we obtain the spectral dependence \(\alpha_{m}\) at different modulation frequencies from 6.5 GHz to 8.5 GHz.