# Multimode Physics in the Mode Locking of Semiconductor Quantum Dot Lasers
## 1 Introduction  
In recent years, the market for data center servers has been boosted by a huge increase in demand from the digitalization of data in commercial and home applications, including media and social networks. Optical interconnects play a crucial role in supporting the enormous volume of data that propagates in short- and long-haul fiber-optic networks. Presently, optical interconnect bandwidth and power consumption contribute to a bottleneck limiting further increase in transmission capacity. Dense wavelength-division multiplexing (DWDM) technologies provide a solution to increase link capacity [1]. Silicon photonic integrated circuits (PICs) based on a mature complementary metal-oxide-semiconductor (CMOS) industry is paving the way with low-cost and energy-efficient solutions [2]. DWDM requires high-performance on-chip laser sources. Mode-locked (ML) lasers producing optical frequency combs are ideal for DWDM systems owing to their small footprint and low power consumption [3, 4]. Pulsed ML lasers are also highly desired for applications including optical time-division multiplexing [5], metrology [6], LIDAR systems for self-driving automobiles [7] and optical clocks for distribution and recovery in future computer processors [8]. A quantum dot (QD) medium is regarded as an important option for monolithically integrated light sources and photodiodes in silicon photonic PICs due to its improved tolerance for dislocations arising during the epitaxial growth of III-V on silicon [9; 10; 11]. In addition, the three-dimensional charge carrier confinement in QDs contributes to high material gain [12], low noise [13] and remarkable temperature stability [10]. Additionally, the inherently low linewidth enhancement factor is important, which results in a high degree of tolerance for chip-scale back reflections [14; 15]. In terms of mode locking, a recent study revealed that QDs are beneficial for generating ultra-short ML pulses and supporting a high transmission rate over 4 Tbits/s [16]. Moreover, rich QD optical nonlinearities, specifically a high \(\chi^{(3)}\) coefficient from four-wave mixing (FWM), is found to be sufficiently strong to overcome group velocity dispersion and result in self-mode locking. A self-ML, single-section laser has a significantly smaller footprint and less electronic complexity than a typical two-section ML laser with forward- and reverse-bias sections [17]. In the near future, QD FWM may generate quantum states of light for quantum photonic integrated circuits (QPICs) [18]. On the basis of the above arguments, our experimental and theoretical studies focus on passively ML QD lasers epitaxially grown on silicon. Mode locking in both single- and multi-section diode lasers is typically described in the time domain or traveling wave domain, because it has the advantage of giving a physical picture of pulse train formation [19]. We have, instead, returned to the earlier complimentary frequency domain description [20]. There are several motivations, including a direct connection of optical nonlinearities to electronic structure via multimode semiclassical laser theory, where the quantum mechanical electron-hole polarization treats the locking mechanism, gain saturation, mode competition and carrier-induced refractive index on an equal footing [21; 22]. Equally important and useful from using the frequency-based approach is that it allows a precise and physically intuitive description of the mode locking process to be made from the Adler equation perspective [23]. Section 2 describes the frequency domain approach. Three important advantages are discussed. Section 2.1 describes a precise quantitative description of the conditions for frequency comb and pulse train formation. Section 2.2 presents a concise explanation of the progression to mode locking via Adler's equation. Section 2.3 shows the equation for the mode-locking strength derived from multimode laser theory. It allows the computation of the mode-locking coefficient with band structure properties as input parameters.