## 1 Introduction  
Recently, data traffic has been growing rapidly due to the increasing demand for high-definition video streaming, cloud computing, and a massive internet-of-things. Consequently, explosive bandwidth growth is expected in data centers and front haul deployments for fifth-generation wireless systems. To meet this need, high-speed optical transmitters that support data rates of 50, 100, and even 400 Gb/s are in great demand [1]. Compared with external modulators, the directly modulated semiconductor laser (DML) is an attractive candidate due to its compact size, cost-effectiveness, and high energy efficiency. To support the rapid increase of transmission rates over single lambda [2], further improving the modulation bandwidth of the DML is the primary challenge. One approach to achieving this is to increase the carrier-photon resonance (CPR) frequency by increasing the differential gain through optimizing multiple quantum well (MQW) structures, reducing the photon lifetime via shortening the cavity length [3; 4; 5; 6] or reducing parasitic parameters [7]. Another approach entails utilizing the photon-photon resonance (PPR) effect based on optical feedback [8; 9; 10] or optical injection-locking techniques [11; 12; 13]. The PPR effect is contributed by the cavity mode close to the main mode, with a suitable side-mode suppression ratio (SMSR), wherein a modulation sideband is resonantly enhanced by the cavity mode [10]. Consequently, a PPR peak that exceeds the usual CPR frequency is formed on the frequency response, which could contribute to further extending the modulation bandwidth. Utilizing optical feedback, several high-performance laser structures have been proposed, such as the passive feedback laser [9] and distributed reflector laser [10]. However, both are fabricated utilizing active-passive integration techniques and buried heterostructures, which increases their manufacturing costs and complications. For optical injection technology, one scheme involves employing a unidirectional optical injection-locking (OIL) system [13]. Several systematic studies on the injection-locking behaviors and modulation characteristics of OIL systems have been conducted [11; 12; 13; 14; 15; 16; 17], and an enhanced modulation bandwidth of 44 GHz has been achieved [18]. However, OIL systems are bulky and complex owing to their use of discrete optical components, especially when the isolator causes difficulty in implementing monolithic integration. A more practical OIL technology scheme has been proposed based on the optical mutual injection-locking (MIL) technique, which facilitates the implementation of monolithic integration by removing the optical isolator. Several integrated mutually coupled (IMC) lasers are employed to achieve MIL [19; 20; 21; 22; 23; 24; 25; 26]. The PPR effect on the IMC lasers is attributed to the red-shift cavity mode due to optical injection from one of the laser sections [27]. The research's detailed enhanced modulation performances are listed in Table 1, where f\({}_{\text{R}}\) is the resonance frequency and f\({}_{\text{3 dB}}\) is the 3-dB modulation bandwidth of the lasers. The enhancement of resonance frequency has also been reported with simultaneous improvements of other laser characteristics, including nonlinear distortion suppression [19], relative intensity noise reduction [23], chirp reduction [24], and linewidth narrowing [27]. Moreover, a modulation bandwidth enhancement of 37 GHz and an open eye of 36 Gb/s have been reported in mutually coupled vertical-cavity surface-emitting laser (VCSEL) arrays [25]. However, for edge-emitting lasers, most IMC lasers only achieve an increase of the resonance (or PPR) frequency without an enhanced modulation bandwidth, which is induced by the low-frequency roll-off effect and the deep gap between the CPR and PPR frequencies. This was caused by a long cavity length of directly modulated laser section [24], RC bandwidth roll-off effect in-package [28], and intrinsic bandwidth limitation induced by the first-order pole frequency [11].