## 1 Introduction
[PERSON] studied the impact of the LEF on the route to chaos of an LD subject to active external cavities [19]. He found that there is no difference in the route to chaos with minor changes in the LEF. Recently, [PERSON] et al. introduced an improved theoretical model of time delay, which is applicable through any OFB range and includes many round trips in the external cavity (EC) and optical phases due to OFB. This model has investigated the operation states and intensity noise of pumping LDs, which are used in fiber amplifier systems as a light source [26; 27]. Moreover, [PERSON] et al. studied the impacts of gain saturation, the radiative and non-radiative recombination lifetime, the LEF, and the combined optical phase with OFB strength on the dynamic states and RIN of LDs with short external cavities [28; 29]. On the other hand, the SEF specifies the ratio of the spontaneous field entered to the lasing mode, affecting the width of the laser linewidth. Since the emission rate is given in terms of \(f_{c}(1-f_{o})(S+1)-f_{o}(1-f_{c})S\), where \(f_{c},f_{o}\), and \(S\) represent the distribution functions of the electron in the conduction, valence bands, and photon density, respectively, accounting for the SEF and the stimulated emission coefficient for the laser. The SEF and the stimulated emission coefficient are identical in principle. [PERSON] investigated the impact of the SEF on the laser RIN and frequency noise levels [30]. He found that increasing the SEF and reducing the threshold current level gives low noise and a narrow linewidth when operating a micro-cavity laser at low power levels [30]. Many papers have studied the impacts of the SEF on the modulation and linewidth of the laser [31; 32; 33; 34]. However, a thorough investigation of the influences of the SEF and LEF on the dynamic states and RIN in the presence of OFB is still missing. Here, we propose for the first time, the impacts of the LEF and the SEF on the dynamics state and RIN of the LD with short EC, which is very important in designing the LDs, using the model mentioned in [26; 27]. This model can be used as the bedrock to cover a wide range of OFB, multiple round trips in the EC, and all the phase variations due to OFB, which leads to practically accurate results compared with the model in [25] used in many previous investigations. This work has novel features on the influences of the LEF, SEF, and OFB in LDs with short EC and their operation in optical communication systems, including fiber-optic communication networks. Moreover, LDs are used in optical data storage devices, such as CD and DVD players, for reading and writing data on optical discs. The main challenge for all these applications of LDs is the deformation of the signal due to OFB. This leads to difficulty in accurately positioning the laser spot on the application. Therefore, unnecessary OFB can cause data safety problems. During the writing or retrieval of data, any noise in the laser output due to OFB can result in the data being written or read with deformations. In this article, we study the LDs subject to external OFB operation, dynamics, states, and noise aiming to the roles played by the LEF and SEF in a self-contained manner. We used the model in Ref. [26], which counts multiple round trips to study the impact of the LEF and SEF on the states and to characterize the route to chaos in LDs. The model in [26] is used to run ferocious numerical calculations of a short EC760 nm AlGaAs/GaAs LD. We have automatically included the impact of linewidth narrowing due to feedback from an EC, which is often described by the so-called chirp reduction factor. The dynamics of the LD are classified using the peak value of the photon number BDs at each feedback value. Here, we obtain a general view of the longitudinal excess factor of the SEF and the LEF taking into attentiveness the gain nonlinear impacts, the effects of the Kerr effect, and the dispersion of the gain of such a structure.