## 1 Introduction  
Since the advent of the information age, laser communication and laser sensing technologies have played pivotal roles in driving societal development and progress. The ITU-C band, positioned within both the optical fiber and atmospheric transmission windows, offers significant advantages, such as the low relative intensity noise (RIN), high gain, and high saturation amplification achievable with optical fiber amplifiers [1, 2, 3]. These characteristics make it one of the most widely used bands in various optical communication and sensing systems. Semiconductor lasers, with their compact size, light weight, high electro-optic efficiency, and ease of integration, are among the most utilized lasers in modern applications [4, 5]. Laser linewidth is a critical parameter of semiconductor lasers, as it reflects the intensity of phase noise within the laser's optical field [6, 7, 8, 9, 10]. Therefore, narrow-linewidth semiconductor lasers are frequently used in systems sensitive to source phase noise. For instance, in coherent optical communication, the laser linewidth influences the bit error rate and other performance metrics of phase modulation formats [9]. In laser sensing systems, the linewidth directly impacts the detection accuracy and sensitivity of the sensors [10]. Consequently, narrow-linewidth semiconductor lasers have become indispensable in modern information technologies and have been the focus of extensive research. Narrow-linewidth external cavity lasers, owing to the ultra-high quality factor (often exceeding 10\({}^{6}\)) of their external cavities [11], offer significant advantages in terms of linewidth reduction. For instance, external cavity lasers with a Littman configuration can achieve linewidths on the order of tens of kHz or even a few kHz, which are substantially narrower than the linewidths of DFB lasers, which are typically in the range of hundreds of kHz [12; 13; 14; 15; 16]. As a result, external cavity lasers have become the preferred solution for achieving narrow-linewidth performance. The RSOA, with its broad 3 dB spectral bandwidth and flexible wavelength selection, has found widespread application in narrow-linewidth external cavity lasers that require specific wavelength selection and tunability [8]. For example, semiconductor narrow-linewidth lasers based on an RSOA, combined with the Vernier effect achieved using dual micro-ring resonators (MRRs), have achieved a remarkable tuning range of 172 nm in the C + L band [17]. This broad tuning capability is enabled by the 3 dB spectral bandwidth of the RSOA. Therefore, further advancements in RSOA performance are crucial for the continued development of high-performance narrow-linewidth lasers. The RSOA is a critical component in narrow-linewidth external cavity lasers. As the laser's active medium, it enables lasing under electrical pumping, and thus plays a pivotal role, not only in determining the wavelength tuning range of the external cavity laser but also in influencing its output power and linewidth [18]. The ripple of the RSOA is a key factor in determining the linewidth of the narrow-linewidth laser [18]. Ripple, which represents the flatness index of the amplified spontaneous emission (ASE) spectrum of the RSOA, reflects the resonance strength of the FP cavity formed by the front and rear facets of the RSOA [19]. For external cavity lasers, the laser linewidth ideally should be governed by the external cavity, given the high quality factor, while any internal resonances from other cavities should be minimized [11]. This means that the ripple of the RSOA should be as low as possible, so that the optimal laser linewidth can be obtained. Extensive research has been conducted on RSOAs [20; 21; 22; 23; 24; 25], with the lowest reported ripple being 1 dB; the lowest ripple for C-band RSOAs is typically around 2 dB. This limitation imposes constraints on further optimizing the linewidth of external cavity lasers.