## 1 Introduction  
Recent technology has witnessed the prominent role played by LDs in different disciplines, including optical communications, laser pumping, optoelectronics, and spectroscopy. The detailed and precise theoretical understanding of the roots of the analytical and numerical calculations is the bedrock for establishing highly efficient and practical solid lasers. Modulation bandwidths, enhanced output powers, and control spectral linewidths are fundamentally important for practical relevance. The spatial coherences of high-power semiconductor lasers, such as quantum dots or quantum wells, are significantly manipulated by the LEF. This factor plays a vigorous role as a critical parameter in manipulating the linewidth targeting high-frequency diode lasers [1, 2]. The emission linewidth and the carrier dynamics are the two vital characteristics of lasers, which can be enhanced by engineering the coupling of the refractive index and gain medium to the carrier density. Potentially, in 1982, [PERSON] introduced such a factor through the expression \(\alpha=-\frac{4\pi}{\lambda}\frac{du/dN}{d\xi/dN}\) to enumerate the effect of the refractive index of the medium with the carrier density on the behaviors of LDs. Here, \(n\) is the refractive index, \(g\) is the gain per unit length, \(\lambda\) is the excitation wavelength, and \(N\) is the carrier density [3]. The insertion of the \(\alpha\)-factor into the compacted set rate equations of laser dynamics has significantly explained the experimentally observed phenomenon related to broad linewidth and the onset of the coherent revival and collapse induced by OFB in many fields of LDs and amplifiers. This parameter also plays a vital role in the spectral features of the LD and its dynamic response and estimates the sensitivity of the LD to OFB [4]. The robust laser holds promising and fascinating applications for coherent communication and sensing devices, which has led to enormous efforts toward engineering narrow linewidth LDs to boost their performance beyond the conventional resonators. Many methodologies have been proposed, such as external cavities, phase shifted, chirped grating, fiber lasers, gain-loss interplay in non-Hermitian systems, and so on, with the aim of determining and reducing the spectral linewidth. A comprehensive related review can be found in [5; 6]. One type of laser is called a spaser or plasmonic laser, which is a device that confines light at a sub-wavelength scale and enables localized or propagation light with high efficiency far below Rayleigh's diffraction limit [7]. The spaser, surface plasmon amplification by stimulated emission of radiation, has been recently predicted and experimentally realized as an active element in nanostructures for generating coherent local fields [8]. If the spaser can be manipulated as an optical counterpart of the MOSFET, greater progress will be achieved for the coming promising devices. However, establishing a spaser for feature amplification is still challenging due to the intrinsic feedback amplification, causing the generation of nanolocalized plasmonic modes and saturation and the consequent elimination of the net gain. Alternatively, topological insulator lasers have been demonstrated theoretically and experimentally for crafting robust and topologically protected lasers [9; 10]. Particularly, non-Hermitian systems with inherent gain and losses enable the occurrence of exceptional points (EPs), which leads to highly efficient lasers that are robust to disorder and defects, reaching single-mode lasing even high above the laser threshold. In fact, the LEF, in several application aspects of LDs, similar to the linewidth, affects the chirp under current modulation, the mode stability, and the circumstance of filamentation in broad-area bias [3; 11]. Many articles have studied the influences of the operation states and dynamics of time-delayed LD by the LEF and OFB [12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25].