# Semiconductor Laser Linewidth Theory Revisited
## 1 Introduction  
Many applications of semiconductor lasers utilized in miniaturized contemporary photonic integrated devices for coherent optical communication, optical atomic clocks, atom interferometry, gravitational wave detection, space-based metrology, and optical quantum sensing impose strict requirements on the coherence of the light source, which can be expressed in terms of the spectral linewidth. The theoretical understanding of the governing factors limiting the linewidth is therefore of great practical relevance. After a first prediction by [PERSON] and [PERSON] [1], the corresponding theoretical framework was developed in the 1960s [2; 3; 4; 5; 6]. Later, in the 1980s and 1990s with the rapidly advancing technology of the fabrication of semiconductor lasers and their applications the theory of the spectral linewidth was further refined. The following five milestones can be distinguished:  
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2. The first milestone is the discovery of the enhancement of the fraction of the spontaneous emission going into the lasing mode in gain-guided lasers and the derivation of a corresponding excess factor (\(K\)-factor) by [PERSON] [7]. [PERSON] recognized this effect as a general property of non-Hermitian laser cavities [8]. Later his discussion of the power-nonorthogonality of the transversal modes and its consequences was extended to the case of the power-nonorthogonality of the longitudinal modesof laser cavities [9]. In Ref. [10] it was discovered, that the longitudinal modes can become degenerate for certain parameter configurations resulting in an infinite \(K\)-factor. The occurrence of such exceptional points is not restricted to lasers but is inherent to non-Hermitian systems, see [11] for a recent review. (ii) The second milestone is the discovery of a linewidth enhancement in semiconductor gain materials caused by refractive index fluctuations in response to fluctuations of the carrier density. Due to gain clamping, intensity fluctuations (which have negligible direct effect on the linewidth) can cause substantial refractive index changes, which in turn lead to fluctuations of the phase. The magnitude of this amplitude-phase coupling is quantified by the linewidth enhancement factor \(\alpha_{\text{H}}\) or \(\alpha\)-factor introduced by [PERSON] [12]. Later it was found that in DFB lasers the \(\alpha\)-factor has to be replaced by an effective factor \(\alpha_{\text{H,eff}}\)[13] and a general expression for \(\alpha_{\text{H,eff}}\) valid for distributed feedback (DFB), distributed Bragg reflector (DBR) and external cavity lasers was derived in [14]. (iii) The third milestone is the discovery of the possibility of linewidth reduction due to optical feedback from an external cavity by [PERSON] et al. [15]. Later a relation to the reduction of the frequency chirp was established [16; 17]. (iv) The forth milestone is related to the deterioration of the linewidth due to charge carriers injected into a phase tuning section within the cavity. [PERSON] and [PERSON] figured out that carrier noise is the origin resulting in an additional contribution to the linewidth, if the \(a\)-factors are different in the gain and phase tuning sections [18]. (v) The last milestone is the discovery of the enhancement of the linewidth due to fluctuations of the shape of the profile of the optical power in the cavity by [PERSON] and co-workers [14], which is particularly important in the vicinity of instabilities [19; 20]. The most sophisticated linewidth theory including fluctuations of the shape of the power profile was published in [21; 22]. In this paper, we will derive a semi-analytical expression of the spectral linewidth from a modern perspective step by step in a self-contained manner collecting all necessary ingredients that are otherwise found only scattered in the literature. The underlying theoretical approach is the classical Langevin formalism [23] where the deterministic equations for the optical field and the carrier density are supplemented by noise sources (Langevin forces).