# Semiconductor Laser Linewidth Theory Revisited

Hans Wenzel  0000-0003-1726-0223<EMAIL_ADDRESS>Ferdinand–Braun–Institut gGmbH, Leibniz–Institut für Höchstfrequenztechnik,
Gustav–Kirchhoff–Str. 4, 12489 Berlin, Germany  Markus Kantner
0000-0003-4576-3135 Mindaugas Radziunas  0000-0003-0306-1266 Uwe Bandelow
0000-0003-3677-2347 Weierstrass Institute for Applied Analysis and Stochastics
Mohrenstr. 39, 10117 Berlin, Germany

###### Abstract

More and more applications require semiconductor lasers distinguished not only
by large modulation bandwidths or high output powers, but also by small
spectral linewidths. The theoretical understanding of the root causes limiting
the linewidth is therefore of great practical relevance. In this paper, we
derive a general expression for the calculation of the spectral linewidth step
by step in a self–contained manner. We build on the linewidth theory developed
in the 1980s and 1990s but look from a modern perspective, in the sense that
we choose as our starting points the time-dependent coupled–wave equations for
the forward and backward propagating fields and an expansion of the fields in
terms of the stationary longitudinal modes of the open cavity. As a result, we
obtain rather general expressions for the longitudinal excess factor of
spontaneous emission ($K$–factor) and the effective $\alpha$–factor including
the effects of nonlinear gain (gain compression) and refractive index (Kerr
effect), gain dispersion, and longitudinal spatial hole burning in
multi–section cavity structures. The effect of linewidth narrowing due to
feedback from an external cavity often described by the so–called chirp
reduction factor is also automatically included. We propose a new analytical
formula for the dependence of the spontaneous emission on the carrier density
avoiding the use of the population inversion factor. The presented theoretical
framework is applied to a numerical study of a two–section distributed Bragg
reflector laser.

semiconductor laser, spectral linewidth, coupled–wave equations, traveling
wave model, noise, Langevin equations, Henry factor, Petermann factor, chirp
reduction factor, population inversion factor

##  I 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 Schawlow and Townes [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 5 milestones can be
distinguished:

  1. (i)

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 Petermann [7].
Siegman 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 modes of 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.

  2. (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 $\alpha_{\mathrm{H}}$ or
$\alpha$–factor introduced by Henry [12]. Later it was found that in DFB
lasers the $\alpha$–factor has to be replaced by an effective factor
$\alpha_{\mathrm{H,eff}}$ [13] and a general expression for
$\alpha_{\mathrm{H,eff}}$ valid for distributed feedback (DFB), distributed
Bragg reflector (DBR) and external cavity lasers was derived in [14].

  3. (iii)

The third milestone is the discovery of the possibility of linewidth reduction
due to optical feedback from an external cavity by Patzak et al. [15]. Later a
relation to the reduction of the frequency chirp was established [16, 17].

  4. (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. Amann
and Schimpe figured out that carrier noise is the origin resulting in an
additional contribution to the linewidth, if the $\alpha$–factors are
different in the gain and phase tuning sections [18].

  5. (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
Tromborg 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). By means of these stochastic terms, quantum field theoretical
phenomena (in particular spontaneous emission), which are essential for the
laser linewidth, can be adequately treated within the framework of the semi-
classical theory. The correlation functions of the Langevin noise sources are
determined by the fluctuation-dissipation theorem, for which we refer to Refs.
[24, 25, 26].

While the starting point of most authors is the Helmholtz equation and an
expansion of its Green’s function [14, 27], it is more transparent to start
from the time-dependent coupled–wave equations and to expand the forward and
backward propagating fields in terms of the stationary longitudinal modes of
the open cavity [10] yielding simpler expressions. The linewidth expression
derived in such a manner includes automatically the findings described as
milestones (i)–(iii), i.e., longitudinal $K$–factor, effective
$\alpha$–factor, and chirp reduction factor. Additionally, we propose a new
analytical formula for the dependence of the spontaneous emission on the
carrier density, which is independent of the commonly employed _population
inversion factor_ , i.e., the ratio between the spontaneous rate of downward
band–to–band transitions and the stimulated rate of downward and upward
transitions [28]. The new formula avoids problems with the singularity of the
population inversion factor near the transparency density and is in good
agreement with microscopic calculations, see Sec. IX. The theory can be easily
extended to include carrier noise (iv), too, as sketched in the Outlook in
Sec. XI.

##  II Prerequisites and basic assumptions

Figure 1: Schematic view of a two–section DBR laser consisting of an
electrically biased gain section and a passive Bragg reflector section,
exemplifying an edge–emitting multi–section semiconductor laser.

We consider edge–emitting semiconductor lasers as sketched in Fig. 1
consisting of an arbitrary number of sections of different functionality. The
transverse cross section is uniform within each section, but may vary from
section to section. The preferred propagation direction of the optical field
under lasing conditions is along the cavity axis parallel to the longitudinal
coordinate $z$. Thus laser can be well described by the traveling–wave ansatz
for the main component of the electric field strength

$E(x,y,z,t)=\frac{e(x,y)}{\sqrt{2\varepsilon_{0}cn}}\left[\Psi^{+}(z,t)\mathrm{e}^{-i\beta_{0}z}+\Psi^{-}(z,t)\mathrm{e}^{i\beta_{0}z}\right]\mathrm{e}^{i\omega_{0}t}+\text{c.c.},$
(1)

employing a scalar approximation and neglecting the dynamics in the transverse
$(x,y)$–plane. This means that we assume _index-guiding_ in the transverse
(lateral–vertical) plane and lasing in a single transverse mode and exclude
gain–guiding, as appropriate for narrow–linewidth lasers. The real–valued
field distribution $e(x,y)$ is obtained as a solution of a real–valued
waveguide equation normalized to $\iint e^{2}(x,y)\,dxdy=1$ [29]. Moreover,
$\varepsilon_{0}$ is the vacuum permittivity, $c$ the vacuum speed of light,
$n$ the reference modal index, $\beta_{0}=2\pi n/\lambda_{0}$ the reference
propagation constant, $\lambda_{0}$ the reference wavelength and
$\omega_{0}=2\pi c/\lambda_{0}$ the reference frequency. The prefactor is
chosen such that $\|\Psi\|^{2}\equiv|\Psi^{+}|^{2}+|\Psi^{-}|^{2}$ is the
optical power $P$ with unit W. The left and right propagating fields
$\Psi^{\pm}$ vary slowly both in the longitudinal coordinate $z$ and time $t$
and fulfill the stochastic time–dependent coupled–wave equations [30, 31]

$\frac{n_{\mathrm{g}}(z)}{c}\frac{\partial}{\partial
t}\Psi(z,t)+\left[\sigma_{z}\frac{\partial}{\partial
z}+iM(z,N,\|\Psi\|^{2})+i\widehat{\cal
D}(z,t)\right]\Psi(z,t)=F_{\mathrm{sp}}(z,t,N)$ (2)

with

$\Psi(z,t)=\begin{bmatrix}\Psi^{+}(z,t)\\\
\Psi^{-}(z,t)\end{bmatrix},\qquad\sigma_{z}=\begin{bmatrix}1&0\\\
0&-1\end{bmatrix},$

and

$\displaystyle M(z,N,\|\Psi\|^{2})$
$\displaystyle=\begin{bmatrix}\Delta\beta(z,N,\|\Psi\|^{2})&\kappa^{+}(z)\\\
\kappa^{-}(z)&\Delta\beta(z,N,\|\Psi\|^{2})\end{bmatrix},$ (3a)
$\displaystyle\Delta\beta(z,N,\|\Psi\|^{2})$
$\displaystyle=\beta(z,N,\|\Psi\|^{2})-\beta_{0}=\frac{2\pi}{\lambda_{0}}\Delta
n(z)+\beta_{N}(z,N,\|\Psi\|^{2}),$ (3b)
$\displaystyle\beta_{N}(z,N,\|\Psi\|^{2})$
$\displaystyle=\frac{2\pi}{\lambda_{0}}\Delta
n_{N}(z,N,\|\Psi\|^{2})+\frac{i}{2}\left[g(z,N,\|\Psi\|^{2})-\alpha(z,N,\|\Psi\|^{2})\right].$
(3c)

The Langevin forces

$F_{\mathrm{sp}}(z,t,N)=\begin{bmatrix}F_{\mathrm{sp}}^{+}(z,t,N)\\\
F_{\mathrm{sp}}^{-}(z,t,N)\end{bmatrix},$

model the spontaneous emission into the forward and backward traveling waves
and are assumed to have zero mean

$\langle F_{\mathrm{sp}}\rangle=0.$

Here, $n_{\mathrm{g}}$ is the real–valued group index, $\beta$ is the
propagation factor and $\Delta n$ is the index detuning with respect to the
reference modal index. $\beta_{N}$ describes the dependence of the propagation
factor on the excess carrier density $N$, consisting of a real part
proportional to $\Delta n_{N}$ and an imaginary part with $g$ being the modal
gain and $\alpha$ the optical losses. $\kappa^{\pm}$ are complex coupling
coefficients describing a Bragg grating (coupling forward and backward
traveling waves) and $\widehat{\cal D}(z,t)$ is the dispersion operator. The
ensemble or temporal average is denoted by $\langle\cdot\rangle$. The
dependence of $\Delta n_{N}$, $g$ and $\alpha$ on the optical power
$\|\Psi\|^{2}$ is due to nonlinear effects like the Kerr effect, gain
compression, and two-photon absorption, which affect the stationary states
only slightly but can have a significant impact on the dynamic properties.

All quantities entering (2) are obtained by weighting the original quantities
entering Maxwell’s equations with the intensity distribution $e^{2}(x,y)$ of
the transverse mode. If we choose the origin of $z$ such that the grating
given by a complex dielectric function $\varepsilon(x,y,z)$ has the property
$\varepsilon(x,y,z)=\varepsilon(x,y,-z)$, then $\kappa^{+}=\kappa^{-}=\kappa$
holds. Eq. (2) has to be solved subject to the usual boundary conditions

$\Psi^{+}(0,t)=r_{0}\Psi^{-}(0,t)\quad\mathrm{and}\quad\Psi^{-}(L,t)=r_{L}\mathrm{e}^{-2i\beta_{0}L}\Psi^{+}(L,t),$
(4)

with $L$ being the total cavity length and $r_{0}$ and $r_{L}$ the
complex–valued reflection coefficients at the facets at $z=0$ and $z=L$,
respectively. Scattering matrices establish transition conditions on $\Psi$ at
the interfaces between different sections [22, 30].

The coupled–wave equations (2) must be supplemented with an equation for the
(excess) carrier density $N$. Although the theoretical treatment is quite
general, we will specifically treat diode lasers in this paper, where the
carrier dynamics is governed actually by both electrons and holes. We assume
charge neutrality and consider only the excess carriers in the active region
with density $N=n-n_{0}=p-p_{0}$ (where $n$ and $p$ are the electron and hole
densities and $n_{0}$, $p_{0}$ are the corresponding equilibrium densities),
neglecting any transport and capture effects. Thus, the excess carrier density
is assumed to obey the rate equation

$\frac{\partial N}{\partial
t}-\frac{j(z,N)}{qd}+R(z,N)+R_{\mathrm{st}}(z,N,\|\Psi\|^{2})=F_{N}(z,t,N)$
(5)

with the rate of spontaneous and non-radiative recombination $R$ and the rate
of stimulated recombination [30]111It can be derived by multiplying (2) from
the left with $[\Psi^{+\ast},\Psi^{-\ast}]$, the complex conjugate of (2) with
$[\Psi^{+},\Psi^{-}]$ and adding both equations to obtain a balance equation
for the electromagnetic energy in differential form (Poynting’s theorem).

$R_{\mathrm{st}}=\frac{\mathop{\mathrm{Re}}\left(\Psi^{\ast}\cdot[g-2i\widehat{\cal
D}]\Psi\right)}{dW\hbar\omega_{0}},$ (6)

where the dot product means
$\Psi_{1}\cdot\Psi_{2}=\Psi_{1}^{+}\Psi_{2}^{+}+\Psi_{1}^{-}\Psi_{2}^{-}$.
Here, $q$ is the elementary charge, $\hbar$ the reduced Planck’s constant and
$d$ and $W$ are the thickness and width, respectively, of the active region.
The current density can be related to the (sectional) applied voltage by means
of the Joyce model [32, 33]

$j(z,N)=\frac{U_{s}-U_{\mathrm{F}}(N)}{WL_{s}R_{s}}$ (7)

with sectional applied voltage $U_{s}$, resistance $R_{s}$, injection current
$I_{s}$, length $L_{s}$, and the Fermi voltage $U_{\mathrm{F}}$. The
carrier–density dependent injection current density given by (7) counteracts
longitudinal spatial hole burning, which substantially lowers the degradation
of the side mode suppression in strongly coupled DFB lasers compared to a
model assuming a constant current density [34]. The frequency modulation
response, albeit not topic of the present paper, is also affected as shown in
[35].

Dispersion, i.e., the frequency dependence of the dielectric function, enters
(2) first via the group index $n_{\mathrm{g}}$ in front of the time derivative
resulting from a linearization of the real part of the dielectric function
with respect to the frequency $\omega$. Second, the dispersion of the optical
gain (and associated refractive index) is described in the time domain by the
operator $\widehat{\cal D}(z,t)$, which can be obtained, e.g., by
approximating the gain spectrum by a Lorentzian (or a series of Lorentzians)
and a back-transformation into the time domain. This results in auxiliary
differential equations for polarization functions [36, 37], see Appendix A.
Due to the fact, that the spectral width of the gain in semiconductor lasers
is much larger than the width of the cavity resonances, $\widehat{\cal
D}(z,t)$ can be expanded again linearly in the frequency domain. In contrast
to [36], we neglect here the dependence of the dispersion operator on the
carrier density $N$.

Besides $\Psi$, we consider $N$ as the only further stochastic variable and
spontaneous emission as the only source of noise, neglecting all other noise
sources. The theory can be easily extended to include carrier noise as
sketched in the Outlook in Sec. XI. We assume that $\kappa$ does not depend on
$N$, i.e., we exclude gain–coupled lasers, and neglect fluctuations of the
shape of the power profile. We neglect the impact of non-lasing side modes on
the linewidth of the lasing mode and employ a single–mode approximation. Thus
the linewidth rebroadening caused by mode partitioning and poor side–mode
suppression, cf. [38, 39], can not be not accounted for.

Regarding the properties of the noise [40], we employ the usual assumptions of
ergodicity (ensemble or statistical average equals temporal average) and
stationarity (i.e., $\langle\xi(t)\xi(t^{\prime})\rangle$ depends only on the
time difference $t-t^{\prime}$) of any stochastic variable $\xi$. The
fluctuations are assumed to have Gaussian probability distributions and
delta–correlated covariance functions in time (Markov approximation) and
space, i.e.,
$\langle\xi(z,t)\zeta(z^{\prime},t^{\prime})\rangle=2D_{\xi\zeta}(z)\delta(t-t^{\prime})\delta(z-z^{\prime})$
with $D_{\xi\zeta}$ being called diffusion coefficient. In order to obtain a
Lorentzian shape of the optical spectrum around the lasing frequency, some
approximations have to be employed which are described in Sec. IV. The full
width at half maximum (FWHM) of the Lorentzian is commonly called _intrinsic_
linewidth. As we consider only spontaneous emission noise, we calculate the
quantum limit of the intrinsic linewidth. If non-Markovinan noise is
considered, which is beyond of the scope of the paper, the lineshape tends
towards a Gaussian in the vicinity of the lasing frequency with a
correspondingly larger FWHM [41, 42].

##  III Equation of the field amplitude

Throughout this work, we restrict ourselves to the analysis of the noise at
continuous wave (CW) emission, i.e., at steady-state lasing. Therefore, we
expand $\Psi(z,t)$ in terms of the stationary eigenmodes $\Phi_{m}(z)$
(longitudinal modes),

$\Psi(z,t)=\sum_{m}f_{m}(t)\Phi_{m}(z),$ (8)

satisfying the equation

$\left[\sigma_{z}\frac{\partial}{\partial
z}+iM\left(z,\overline{N},\overline{\|\Psi\|^{2}}\right)+i\beta_{\cal
D}(z,\Omega_{m})+i\frac{n_{\mathrm{g}}(z)}{c}\Omega_{m}\right]\Phi_{m}(z)=0$
(9)

with

$\displaystyle M\left(z,\overline{N},\overline{\|\Psi\|^{2}}\right)$
$\displaystyle=\begin{bmatrix}\Delta\beta\left(z,\overline{N},\overline{\|\Psi\|^{2}}\right)&\kappa(z)\\\
\kappa(z)&\Delta\beta\left(z,\overline{N},\overline{\|\Psi\|^{2}}\right)\end{bmatrix}\equiv\overline{M},$
(10)
$\displaystyle\Delta\beta\left(z,\overline{N},\overline{\|\Psi\|^{2}}\right)$
$\displaystyle=\beta\left(z,\overline{N},\overline{\|\Psi\|^{2}}\right)-\beta_{0}\equiv\overline{\beta}-\beta_{0}=\Delta\overline{\beta}.$
(11)

Here, $\overline{N}$ and $\overline{\|\Psi\|^{2}}$ are the steady-state
carrier density and optical power, respectively, at which the operator of the
eigenvalue problem (9) is expanded ($\overline{\beta}$, $\overline{M}$ are the
corresponding propagation factor and matrix $M$). The complex eigenvalue
$\Omega_{m}$ describes both, the frequency deviation
$\mathop{\mathrm{Re}}(\Omega_{m})=\omega_{m}-\omega_{0}$ (or the wavelength
deviation
$\Delta\lambda_{m}=\mathop{\mathrm{Re}}(\Omega_{m})\,d\lambda/d\omega|_{\lambda_{0}}$),
and the mode damping rate $\mathop{\mathrm{Im}}(\Omega_{m})$. The dispersion
$\beta_{\cal D}(z,\Omega_{m})$ is given by the operator $\mathcal{D}(z,t)$ in
the frequency domain. Eq. (9) has to be solved subject to the boundary
conditions

$\displaystyle\Phi_{m}^{+}(0)$ $\displaystyle=r_{0}\Phi_{m}^{-}(0)$ (12)
$\displaystyle\Phi_{m}^{-}(L)$
$\displaystyle=r_{L}\mathrm{e}^{-2i\beta_{0}L}\Phi_{m}^{+}(L)$

following from (4).

In the case of stationary eigenmodes considered here, the two-point boundary
value problem (9) together with the boundary conditions (12) is in fact a
quasi-linear eigenvalue problem. The point of expansion
$\big{(}\overline{\|\Psi\|^{2}},\overline{N}\big{)}$ of the nonlinear operator
must be chosen self-consistently, i.e., it is required to satisfy the
conditions (36) and (68) for the (mean) stationary state given below.
Furthermore, in the presence of dispersion $\beta_{\mathcal{D}}(\Omega_{m})$,
the eigenvalue problem is also nonlinear in the eigenvalue. The problem is
similar to the _self-consistent field method_ in electronic structure theory
[43] and can be treated in a similar way (e.g., linearization of the optical-
power and the frequency dependency of the gain dispersion and fixed-point
iteration, where the point of expansion is updated to the most recent
stationary state in each step). Alternatively, the system can be directly
propagated in the time-domain (as it has been done in Sec. X), until
convergence to a suitable CW state has been achieved.

Note that different modes (eigensolutions of (9)) fulfill the orthogonality
relation222It can be derived by multiplying (9) from the left by
$[\Phi^{-}_{n},\Phi^{+}_{n}]$, the corresponding equation for $\Phi_{n}$ by
$[\Phi^{-}_{m},\Phi^{+}_{m}]$, integrating along $z$, using (12) and
subtracting both equations.

$(\Phi_{m},\frac{\partial\beta}{\partial\omega}\Phi_{n})=0\quad\mathrm{for}\quad
m\neq n$ (13)

with the inner product (distinguished from a standard Hilbert space scalar
product)

$(\Phi,\Psi)=\int_{0}^{L}(\Phi^{+}\Psi^{-}+\Phi^{-}\Psi^{+})\,dz.$ (14)

At an exceptional point the expansion (8) includes also a generalized
eigenmode [10]. For $m=n$, the factor $\partial\beta/\partial\omega$, which
can be interpreted as an inverse complex–valued group velocity, is the exact
derivative

$\frac{\partial\beta}{\partial\omega}=\frac{n_{\mathrm{g}}}{c}+\frac{\partial\beta_{\cal
D}}{\partial\omega}\Big{|}_{\Omega_{m}}.$ (15)

In the general case of $m\neq n$ this holds only approximately:

$\frac{\partial\beta}{\partial\omega}=\frac{n_{\mathrm{g}}}{c}+\frac{\beta_{\cal
D}(\Omega_{m})-\beta_{\cal
D}(\Omega_{n})}{\Omega_{m}-\Omega_{n}}\approx\frac{n_{\mathrm{g}}}{c}+\frac{\partial\beta_{\cal
D}}{\partial\omega}\Big{|}_{\Omega_{m}}.$ (16)

A system of ordinary differential equations for the amplitudes $f_{m}(t)$ can
be obtained by inserting (8) into (2), using (9), multiplying from left with
$[\Phi^{-}_{n},\Phi^{+}_{n}]$, and integrating along $z$. In order to exploit
the orthogonality relation (13) with the approximation (16), we have again to
expand

$\beta_{\cal D}(\tilde{\omega})-\beta_{\cal
D}(\Omega_{m})=\frac{\partial\beta_{\cal
D}}{\partial\omega}\Big{|}_{\Omega_{m}}(\tilde{\omega}-\Omega_{m}),$ (17)

where $\tilde{\omega}=\omega-\omega_{0}$. Note that this is consistent with
the slowly varying amplitude approximation. Transforming back into the time
domain yields

$\int\left[\beta_{\cal D}(\tilde{\omega})-\beta_{\cal
D}(\Omega_{m})\right]f_{m}(\tilde{\omega})\mathrm{e}^{i\tilde{\omega}t}\,d\tilde{\omega}=\left[\widehat{\cal
D}-\beta_{\cal D}(\Omega_{m})\right]f_{m}(t)$ (18)

and

$\displaystyle\frac{\partial\beta_{\cal
D}}{\partial\omega}\Big{|}_{\Omega_{m}}\int(\tilde{\omega}-\Omega_{m})f_{m}(\tilde{\omega})\mathrm{e}^{i\tilde{\omega}t}\,d\tilde{\omega}$
$\displaystyle=-\frac{\partial\beta_{\cal
D}}{\partial\omega}\Big{|}_{\Omega_{m}}\left[\Omega_{m}f_{m}(t)+i\frac{\partial}{\partial
t}\int
f_{m}(\tilde{\omega})\mathrm{e}^{i\tilde{\omega}t}\,d\tilde{\omega}\right]$
(19) $\displaystyle=-\frac{\partial\beta_{\cal
D}}{\partial\omega}\Big{|}_{\Omega_{m}}\left[\Omega_{m}f_{m}(t)+i\frac{\partial
f_{m}}{\partial t}\right].$

Therefore the amplitudes $f_{m}$ fulfill

$(\Phi_{m},\frac{\partial\beta}{\partial\omega}\Phi_{m})\frac{\partial
f_{m}}{\partial
t}-i\Omega_{m}(\Phi_{m},\frac{\partial\beta}{\partial\omega}\Phi_{m})f_{m}+i\sum_{n}(\Phi_{m},\Delta
M\Phi_{n})f_{n}=(\Phi_{m},F_{\mathrm{sp}})$ (20)

with

$\Delta M=M-\overline{M}=(\beta-\overline{\beta})\begin{bmatrix}1&0\\\
0&1\end{bmatrix}.$ (21)

In what follows, we employ the single mode approximation and drop the
subscript $m$. Then Eq. (20) becomes a single differential equation for the
complex–valued amplitude $f$ of the lasing mode

$\frac{\partial f}{\partial t}-i\Omega f+i\frac{(\Phi,\Delta
M\Phi)}{(\Phi,\frac{\partial\beta}{\partial\omega}\Phi)}f=F_{f}.$ (22)

The new Langevin noise source entering (22) is given by

$F_{f}=\frac{(\Phi,F_{\mathrm{sp}})}{(\Phi,\frac{\partial\beta}{\partial\omega}\Phi)}.$
(23)

For an analysis above threshold, it is beneficial to derive equations for the
modulus $|f|$ and the phase $\varphi$ of $f$, because the fluctuations of
$|f|$ are damped but those of $\varphi$ are not. For the modulus squared
$|f|^{2}$ (called intensity in the rest of the paper) it follows

$\frac{\partial|f|^{2}}{\partial
t}+2\mathop{\mathrm{Im}}(\Omega)|f|^{2}-2\mathop{\mathrm{Im}}\frac{(\Phi,\Delta
M\Phi)}{(\Phi,\frac{\partial\beta}{\partial\omega}\Phi)}|f|^{2}=2\mathop{\mathrm{Re}}(F_{f}f^{\ast}).$
(24)

An equation for the phase can be determined from

$\mathop{\mathrm{Im}}\left(\frac{\partial f}{\partial
t}f^{\ast}\right)-\mathop{\mathrm{Re}}(\Omega)|f|^{2}+\mathop{\mathrm{Re}}\frac{(\Phi,\Delta
M\Phi)}{(\Phi,\frac{\partial\beta}{\partial\omega}\Phi)}|f|^{2}=\mathop{\mathrm{Im}}(F_{f}f^{\ast})$
(25)

and

$\frac{\partial f}{\partial
t}=\mathrm{e}^{i\varphi}\frac{\partial|f|}{\partial
t}+if\frac{\partial\varphi}{\partial
t}\quad\Longrightarrow\quad\mathop{\mathrm{Im}}\left(\frac{\partial
f}{\partial t}f^{\ast}\right)=|f|^{2}\frac{\partial\varphi}{\partial t}.$ (26)

The stochastic term on the right–hand side of (24) describes intensity noise,
which does not have a vanishing expectation value anymore: $\langle
F_{f}f^{\ast}\rangle\neq 0$. It is therefore convenient to rewrite the term by
distilling out the expectation value and introducing two new real-valued
Langevin forces $F_{|f|^{2}}$ and $F_{\varphi}$ with zero mean as

$2F_{f}f^{\ast}=2\langle
F_{f}f^{\ast}\rangle+F_{|f|^{2}}+2i|f|^{2}F_{\varphi}.$ (27)

It will be shown below that $\langle F_{f}f^{\ast}\rangle$ is real–valued, see
Eq. (66). Then, the final equations for the modulus squared and phase of $f$
read

$\frac{\partial|f|^{2}}{\partial
t}=-2\mathop{\mathrm{Im}}(\Omega)|f|^{2}+2\mathop{\mathrm{Im}}\frac{(\Phi,\Delta
M\Phi)}{(\Phi,\frac{\partial\beta}{\partial\omega}\Phi)}|f|^{2}+2\langle
F_{f}f^{\ast}\rangle+F_{|f|^{2}}$ (28)

and

$\frac{\partial\varphi}{\partial
t}=\mathop{\mathrm{Re}}(\Omega)-\mathop{\mathrm{Re}}\frac{(\Phi,\Delta
M\Phi)}{(\Phi,\frac{\partial\beta}{\partial\omega}\Phi)}+F_{\varphi}.$ (29)

Now we consider small fluctuations

$\displaystyle N(z,t)$ $\displaystyle=\langle N(z)\rangle+\delta N(z,t),$ (30)
$\displaystyle\varphi(t)$
$\displaystyle=\langle\varphi(t)\rangle+\delta\varphi(t),$ (31)
$\displaystyle|f(t)|^{2}$
$\displaystyle=\langle|f|^{2}\rangle+\delta|f(t)|^{2}$ (32)

and expand

$\beta=\beta(\langle
N\rangle,\langle|f|^{2}\rangle)+\frac{\partial\beta}{\partial
N}\bigg{|}_{\langle N\rangle,\langle|f|^{2}\rangle}\delta
N+\frac{\partial\beta}{\partial|f|^{2}}\bigg{|}_{\langle
N\rangle,\langle|f|^{2}\rangle}\delta|f|^{2}$ (33)

around the mean values

$\displaystyle\langle N(z)\rangle$ $\displaystyle=\overline{N}(z),$ (34)
$\displaystyle\langle\varphi(t)\rangle$
$\displaystyle=\mathop{\mathrm{Re}}(\Omega)t.$

The mean intensity $\langle|f|^{2}\rangle$ is required to satisfy

$\mathop{\mathrm{Im}}(\Omega)\langle|f|^{2}\rangle=\langle
F_{f}f^{\ast}\rangle.$ (35)

Due to the extreme smallness of the spontaneous emission $\langle
F_{f}f^{\ast}\rangle$ going into the lasing mode, (35) can be replaced by

$\mathop{\mathrm{Im}}(\Omega)=0$ (36)

above threshold $(\langle|f|^{2}\rangle>0)$, which will be exploited in what
follows (i.e., $\Omega=\mathop{\mathrm{Re}}(\Omega)$). The intensity and phase
fluctuations fulfill the equations

$\frac{\partial\delta|f|^{2}}{\partial
t}=2\mathop{\mathrm{Im}}\frac{(\Phi,\frac{\partial\beta}{\partial N}\delta
N\Phi)+(\Phi,\frac{\partial\beta}{\partial|f|^{2}}\Phi)\delta|f|^{2}}{(\Phi,\frac{\partial\beta}{\partial\omega}\Phi)}\langle|f|^{2}\rangle+F_{|f|^{2}}$
(37)

and

$\frac{\partial\delta\varphi}{\partial
t}=-\mathop{\mathrm{Re}}\frac{(\Phi,\frac{\partial\beta}{\partial N}\delta
N\Phi)+(\Phi,\frac{\partial\beta}{\partial|f|^{2}}\Phi)\delta|f|^{2}}{(\Phi,\frac{\partial\beta}{\partial\omega}\Phi)}+F_{\varphi},$
(38)

which are the basis for the derivation of the linewidth formula in the
following sections.

##  IV Lorentzian line shape

The power spectral density (PSD) of the optical field at $z=0$ can be
calculated utilizing the Wiener–Khinchin theorem as333We do not address
technical issues regarding the non-existence of Fourier integrals of
fluctuating quantities here and refer to Ref. [44].

$S_{E}(\omega)=\iint\int_{-\infty}^{+\infty}\left\langle
E^{\ast}(x,y,0,t)E(x,y,0,t+\tau)\right\rangle\mathrm{e}^{-i\omega\tau}\,d\tau\,dxdy$
(39)

with

$\langle
E^{\ast}(x,y,0,t)E(x,y,0,t+\tau)\rangle=\frac{1-|r_{0}|^{2}}{2\varepsilon_{0}cn}e^{2}(x,y)\langle
f^{\ast}(t)f(t+\tau)\rangle|\Phi^{-}(0)|^{2}\mathrm{e}^{i\omega_{0}\tau}+\text{c.c.}$
(40)

Hence we have to calculate the spectral density

$S_{f}(\omega)=\int_{-\infty}^{+\infty}\left\langle
f^{\ast}(t)f(t+\tau)\right\rangle\mathrm{e}^{-i(\omega-\omega_{0})\tau}\,d\tau$
(41)

with

$\langle
f^{\ast}(t)f(t+\tau)\rangle=\langle|f(t)||f(t+\tau)|\mathrm{e}^{i[\varphi(t+\tau)-\varphi(t)]}\rangle.$
(42)

Now we employ here a first approximation: We neglect in (42) the intensity
fluctuations because above threshold they are damped, in contrast to the phase
fluctuations, which can been seen by comparing (37) and (38), see the
discussion in [45]. Therefore, we obtain

$\langle
f^{\ast}(t)f(t+\tau)\rangle\approx\langle|f|^{2}\rangle\langle\mathrm{e}^{i\Delta_{\tau}\varphi}\rangle\mathrm{e}^{i\Omega\tau}$
(43)

with

$\Delta_{\tau}\varphi=\delta\varphi(t+\tau)-\delta\varphi(t)$ (44)

and $\Omega=\mathop{\mathrm{Re}}(\Omega)$. The random variable
$\Delta_{\tau}\varphi$ is assumed to be Gaussian distributed [46], such that
it holds

$\langle\mathrm{e}^{i\Delta_{\tau}\varphi}\rangle=\frac{1}{\sqrt{2\pi\mathop{\mathrm{Var}}(\Delta_{\tau}\varphi)}}\int_{-\infty}^{\infty}\mathrm{e}^{-\frac{(\Delta_{\tau}\varphi)^{2}}{2\mathop{\mathrm{Var}}(\Delta_{\tau}\varphi)}}\mathrm{e}^{i\Delta_{\tau}\varphi}d\Delta_{\tau}\varphi=\mathrm{e}^{-\frac{1}{2}\mathop{\mathrm{Var}}(\Delta_{\tau}\varphi)}.$
(45)

The variance of the phase fluctuations is

$\displaystyle\mathop{\mathrm{Var}}(\Delta_{\tau}\varphi)$
$\displaystyle=\langle\left(\Delta_{\tau}\varphi-\langle\Delta_{\tau}\varphi\rangle\right)^{2}\rangle$
(46) $\displaystyle=\langle(\Delta_{\tau}\varphi)^{2}\rangle$
$\displaystyle=\langle\left(\delta\varphi(t+\tau)\right)^{2}\rangle-2\langle\delta\varphi(t)\delta\varphi(t+\tau)\rangle+\langle\left(\delta\varphi(t)\right)^{2}\rangle$
$\displaystyle=2\left[\langle\left(\delta\varphi(0)\right)^{2}\rangle-\langle\delta\varphi(0)\delta\varphi(\tau)\rangle\right],$

where we exploited $\langle\Delta_{\tau}\varphi\rangle=0$ and the stationarity
property. We introduce the PSD of phase fluctuations444Note that the spectral
density of phase fluctuations is even in the frequency
$S_{\delta\varphi}(\tilde{\omega})=S_{\delta\varphi}(-\tilde{\omega})$.

$S_{\delta\varphi}(\tilde{\omega})=\int_{-\infty}^{+\infty}\left\langle\delta\varphi(t)\delta\varphi(t+\tau)\right\rangle\mathrm{e}^{-i\tilde{\omega}\tau}\,d\tau$
(47)

and the PSD of optical frequency fluctuations
$\delta\omega(\tilde{\omega})=i\tilde{\omega}\delta\varphi(\tilde{\omega})$

$S_{\delta\omega}(\tilde{\omega})=\int_{-\infty}^{+\infty}\left\langle\delta\omega(t)\delta\omega(t+\tau)\right\rangle\mathrm{e}^{-i\tilde{\omega}\tau}\,d\tau=\tilde{\omega}^{2}S_{\delta\varphi}(\tilde{\omega}).$
(48)

With these spectral densities, the variance of the phase fluctuations can be
written as

$\displaystyle\mathop{\mathrm{Var}}(\Delta_{\tau}\varphi)$
$\displaystyle=\frac{1}{\pi}\int_{-\infty}^{\infty}S_{\delta\varphi}(\tilde{\omega})\left(1-\mathrm{e}^{i\tilde{\omega}\tau}\right)\,d\tilde{\omega}$
(49)
$\displaystyle=\frac{1}{\pi}\int_{-\infty}^{\infty}S_{\delta\varphi}(\tilde{\omega})\left(2\sin^{2}\left(\frac{\tilde{\omega}\tau}{2}\right)-i\sin\left(\tilde{\omega}\tau\right)\right)\,d\tilde{\omega}$
$\displaystyle=\frac{2}{\pi}\int_{-\infty}^{\infty}S_{\delta\varphi}(\tilde{\omega})\sin^{2}\left(\frac{\tilde{\omega}\tau}{2}\right)\,d\tilde{\omega}$
$\displaystyle=\frac{\tau^{2}}{2\pi}\int_{-\infty}^{\infty}S_{\delta\omega}(\tilde{\omega})\left(\frac{\sin\left(\frac{\tilde{\omega}\tau}{2}\right)}{\frac{\tilde{\omega}\tau}{2}}\right)^{2}\,d\tilde{\omega}.$

We carry out a second approximation by noting that the function
$\sin^{2}\left(\frac{\tilde{\omega}\tau}{2}\right)/(\frac{\tilde{\omega}\tau}{2})^{2}$
peaks strongly at $\tilde{\omega}=0$ for large $|\tau|$, and obtain

$\mathop{\mathrm{Var}}(\Delta_{\tau}\varphi)\approx\frac{|\tau|}{\pi}S_{\delta\omega}(0)\int_{-\infty}^{\infty}\frac{\sin^{2}x}{x^{2}}\,dx=|\tau|\,S_{\delta\omega}(0).$
(50)

In this step, we have recovered a central property of Brownian motion, as we
can observe that the root mean square displacement of the phase fluctuation
grows as $\propto\sqrt{|\tau|}$. Inserting (50) into (45) and, finally, the
result into (43) yields

$\langle
f^{\ast}(t)f(t+\tau)\rangle\approx\langle|f|^{2}\rangle\mathrm{e}^{-\frac{1}{2}S_{\delta\omega}(0)|\tau|}\mathrm{e}^{i\Omega\tau}$
(51)

and

$\displaystyle S_{f}(\omega)$
$\displaystyle\approx\langle|f|^{2}\rangle\int_{-\infty}^{+\infty}\mathrm{e}^{-\frac{1}{2}S_{\delta\omega}(0)|\tau|+i(\omega_{0}+\Omega-\omega)\tau}\,d\tau$
(52)
$\displaystyle=\langle|f|^{2}\rangle\frac{S_{\delta\omega}(0)}{\left(\omega-(\omega_{0}+\Omega)\right)^{2}+\left(\frac{S_{\delta\omega}(0)}{2}\right)^{2}}.$

This is a Lorentzian centered at $\omega=\omega_{0}+\Omega$ with the FWHM

$\Delta\omega=S_{\delta\omega}(0)$ (53)

given by the PSD of the optical frequency fluctuations taken at zero Fourier
frequency.

In the following sections of this paper, we will focus on the static white
noise limit, where the frequency noise PSD $S_{\delta\omega}(\omega)$ is
effectively approximated by a constant, such that the second approximation
carried out on the step from Eq. (49) to (50) becomes in fact redundant.

##  V Correlation functions

The correlation function of the spontaneous emission noise $F_{f}$ defined in
(23) is

$\langle
F_{f}^{\ast}(t)F_{f}(t^{\prime})\rangle=\frac{\langle(\Phi^{\ast},F_{\mathrm{sp}}^{\ast}(z,t))(\Phi,F_{\mathrm{sp}}(z^{\prime},t^{\prime}))\rangle}{\left|(\Phi,\frac{\partial\beta}{\partial\omega}\Phi)\right|^{2}}.$
(54)

The numerator reads

$\displaystyle\langle(\Phi^{\ast},F_{\mathrm{sp}}^{\ast}(z,t))(\Phi,F_{\mathrm{sp}}(z^{\prime},t^{\prime}))\rangle$
(55)
$\displaystyle=\int_{0}^{L}\int_{0}^{L}\left\langle\left[\Phi^{+\ast}F_{\mathrm{sp}}^{-\ast}(z,t)+\Phi^{-\ast}F_{\mathrm{sp}}^{+\ast}(z,t)\right]\left[\Phi^{+}F_{\mathrm{sp}}^{-}(z^{\prime},t^{\prime})+\Phi^{-}F_{\mathrm{sp}}^{+}(z^{\prime},t^{\prime})\right]\right\rangle\,dzdz^{\prime}$
$\displaystyle=\int_{0}^{L}\int_{0}^{L}\left[|\Phi^{+}|^{2}\langle
F_{\mathrm{sp}}^{-\ast}(z,t)F_{\mathrm{sp}}^{-}(z^{\prime},t^{\prime})\rangle+|\Phi^{-}|^{2}\langle
F_{\mathrm{sp}}^{+\ast}(z,t)F_{\mathrm{sp}}^{+}(z^{\prime},t^{\prime})\rangle\right]\,dzdz^{\prime}$
$\displaystyle=2\int_{0}^{L}\|\Phi(z)\|^{2}D_{\mathrm{sp}}(z)\,dz\,\delta(t-t^{\prime})$

with

$\|\Phi\|^{2}=|\Phi^{+}|^{2}+|\Phi^{-}|^{2}$ (56)

taking into account the noise covariance functions [25, 26, 27] for index– and
absorption coupling555For gain–coupling see [47].

$\displaystyle\langle
F_{\mathrm{sp}}^{\pm\ast}(z,t)F_{\mathrm{sp}}^{\pm}(z^{\prime},t^{\prime})\rangle$
$\displaystyle=2D_{\mathrm{sp}}(z)\delta(t-t^{\prime})\delta(z-z^{\prime}),$
(57) $\displaystyle\langle
F_{\mathrm{sp}}^{\mp\ast}(z,t)F_{\mathrm{sp}}^{\pm}(z^{\prime},t^{\prime})\rangle$
$\displaystyle=0,$ $\displaystyle\langle
F_{\mathrm{sp}}^{\pm}(z,t)F_{\mathrm{sp}}^{\pm}(z^{\prime},t^{\prime})\rangle$
$\displaystyle=0,$ $\displaystyle\langle
F_{\mathrm{sp}}^{\mp}(z,t)F_{\mathrm{sp}}^{\pm}(z^{\prime},t^{\prime})\rangle$
$\displaystyle=0$

with

$2D_{\mathrm{sp}}(z,N,\|\Psi\|^{2})=\hbar\omega_{0}n_{\mathrm{sp}}(N)g(z,N,\|\Psi\|^{2}),$
(58)

where $n_{\mathrm{sp}}$ is the population inversion factor (a Bose–Einstein
distribution function multiplied by $-1$). Since $\Psi$ has the unit
$\sqrt{\text{W}}$, $D_{\mathrm{sp}}$ must have the unit Ws/m as it is. The
diffusion coefficient $D_{f^{\ast}f}$ defined by

$\langle
F_{f}^{\ast}(t)F_{f}(t^{\prime})\rangle=2D_{f^{\ast}f}\delta(t-t^{\prime}).$
(59)

is therefore

$D_{f^{\ast}f}=D_{f^{\ast}f}(N,\|\Psi\|^{2})=\frac{\int_{0}^{L}\|\Phi(z)\|^{2}D_{\mathrm{sp}}(z,N,\|\Psi\|^{2})\,dz}{\left|(\Phi,\frac{\partial\beta}{\partial\omega}\Phi)\right|^{2}}.$
(60)

The correlation functions of the Langevin forces entering (37) and (38) read

$\displaystyle\langle F_{|f|^{2}}(t)F_{|f|^{2}}(t^{\prime})\rangle$
$\displaystyle=2D_{|f|^{2}|f|^{2}}\delta(t-t^{\prime})=4D_{f^{\ast}f}\langle|f|^{2}\rangle\delta(t-t^{\prime}),$
(61) $\displaystyle\langle F_{\varphi}(t)F_{\varphi}(t^{\prime})\rangle$
$\displaystyle=2D_{\varphi\varphi}\delta(t-t^{\prime})=\frac{D_{f^{\ast}f}}{\langle|f|^{2}\rangle}\delta(t-t^{\prime}),$
(62) $\displaystyle\langle F_{|f|^{2}}(t)F_{\varphi}(t^{\prime})\rangle$
$\displaystyle=2D_{|f|^{2}\varphi}\delta(t-t^{\prime})=0,$ (63)

which can be derived using the transformation rules for the diffusion
coefficients [23]

$\displaystyle D_{|f|^{2}|f|^{2}}$
$\displaystyle=\left(\frac{\partial|f|^{2}}{\partial
f}\frac{\partial|f|^{2}}{\partial f^{\ast}}+\frac{\partial|f|^{2}}{\partial
f^{\ast}}\frac{\partial|f|^{2}}{\partial f}\right)D_{f^{\ast}f},$ (64)
$\displaystyle D_{\varphi\varphi}$
$\displaystyle=\left(\frac{\partial\varphi}{\partial
f}\frac{\partial\varphi}{\partial f^{\ast}}+\frac{\partial\varphi}{\partial
f^{\ast}}\frac{\partial\varphi}{\partial f}\right)D_{f^{\ast}f},$
$\displaystyle D_{|f|^{2}\varphi}$
$\displaystyle=\left(\frac{\partial|f|^{2}}{\partial
f}\frac{\partial\varphi}{\partial f^{\ast}}+\frac{\partial|f|^{2}}{\partial
f^{\ast}}\frac{\partial\varphi}{\partial f}\right)D_{f^{\ast}f},$

and the relations $|f|^{2}=f^{\ast}f$ and
$\varphi=-\frac{i}{2}\mathrm{ln}(f/f^{\ast})$. Following [48], the ensemble
average defined in (27) can be obtained using the formal solution of (22) in
the form of

$f\left(t\right)=f\left(t-\Delta t\right)+\int_{t-\Delta t}^{t}\partial
f^{*}(t^{\prime})/\partial t^{\prime}\,dt^{\prime},$ (65)

where the time step $0<\Delta t\ll\gamma_{R}^{-1}$ is assumed to be short in
comparison to the inverse relaxation rate
$\gamma_{R}=\mathop{\mathrm{Im}}\left(\Omega-\left(\Phi,\Delta
M\Phi\right)/\left(\Phi,\partial_{\omega}\beta\Phi\right)\right)$.
Substituting (65) into $\langle F_{f}\left(t\right)f^{*}\left(t\right)\rangle$
yields

$\displaystyle\langle F_{f}\left(t\right)f^{*}\left(t\right)\rangle$
$\displaystyle=\langle F_{f}\left(t\right)f^{*}\left(t-\Delta
t\right)\rangle+\int_{t-\Delta t}^{t}\langle F_{f}\left(t\right)\partial
f^{*}(t^{\prime})/\partial t^{\prime}\rangle\,dt^{\prime}$
$\displaystyle=\langle F_{f}\left(t\right)\rangle\langle f^{*}\left(t-\Delta
t\right)\rangle-i\int_{t-\Delta t}^{t}\bigg{(}\Omega-\frac{\left(\Phi,\Delta
M\Phi\right)}{\left(\Phi,\partial\beta/\partial\omega\,\Phi\right)}\bigg{)}^{*}\langle
F_{f}\left(t\right)f^{*}\left(t^{\prime}\right)\rangle\,dt^{\prime}$
$\displaystyle\phantom{=}+\int_{t-\Delta t}^{t}\langle
F_{f}\left(t\right)F_{f}^{*}\left(t^{\prime}\right)\rangle\,dt^{\prime},$

where we have used (22) in the second step. The first term vanishes because of
causality, since the fluctuations of $f^{*}\left(t-\Delta t\right)$ are not
correlated with the future noise $F_{f}\left(t\right)$. Consequently, the
correlation function factorizes and results in zero due to the zero-mean
property of the Langevin force. The second term can be dropped since it is
non-zero only over a set of measure zero (_i.e._ , at $t=t^{\prime}$).
Finally, we obtain with (59)

$\langle F_{f}\left(t\right)f^{*}\left(t\right)\rangle=\int_{t-\Delta
t}^{t}\langle
F_{f}\left(t\right)F_{f}^{*}\left(t^{\prime}\right)\rangle\,dt^{\prime}=D_{f^{*}f},$
(66)

where a factor 2 is canceled by 1/2 encountered in integrating only half of
the $\delta$–function.

##  VI Effective linewidth enhancement factor

Using the ansatz (8), we can approximate the rate of stimulated emission (6)
as

$R_{\mathrm{st}}=\frac{g+2\mathop{\mathrm{Im}}\left(\beta_{\cal
D}(\Omega)\right)}{dW\hbar\omega_{0}}|f|^{2}\|\Phi\|^{2}.$ (67)

The stationary mean value of the carrier density and its fluctuations satisfy

$\frac{j(\langle N\rangle)}{ed}=R(\langle N\rangle)+\frac{g(\langle
N\rangle,\langle|f|^{2}\rangle\|\Phi\|^{2})+2\mathop{\mathrm{Im}}\left(\beta_{\cal
D}(\Omega)\right)}{dW\hbar\omega_{0}}\langle|f|^{2}\rangle\|\Phi\|^{2}$ (68)

and

$\frac{\partial\delta N}{\partial t}+\frac{\delta
N}{\tau_{\mathrm{d}}}+\frac{\partial
R_{\mathrm{st}}}{\partial|f|^{2}}\delta|f|^{2}=F_{N},$ (69)

respectively, with the inverse differential carrier lifetime (taken at
$\langle N\rangle$ and $\langle|f|^{2}\rangle$)

$\frac{1}{\tau_{\mathrm{d}}}=-\frac{1}{qd}\frac{\partial j}{\partial
N}+\frac{\partial R}{\partial N}+\frac{\partial R_{\mathrm{st}}}{\partial N}.$
(70)

The FWHM of the Lorentzian line shape is given by the spectral density of the
frequency fluctuations at zero Fourier frequency, cf. (53). This is equivalent
to setting the time derivatives of the fluctuations equal to zero (static
limit)

$\frac{\partial\delta|f|^{2}}{\partial t}=0$ (71)

and

$\frac{\partial\delta N}{\partial t}=0.$ (72)

Furthermore, we omit carrier noise ($F_{N}=0$), considering only spontaneous
emission noise as mentioned in Sec. II. With these approximations, the carrier
density fluctuations are directly related to the intensity fluctuations by

$\delta N=-\tau_{\mathrm{d}}\frac{\partial
R_{\mathrm{st}}}{\partial|f|^{2}}\delta|f|^{2}.$ (73)

We substitute (73) into (37) and obtain

$\delta|f|^{2}=\frac{F_{|f|^{2}}}{2\langle|f|^{2}\rangle\mathop{\mathrm{Im}}(h_{\alpha})}$
(74)

with

$h_{\alpha}=\frac{(\Phi,\tau_{\mathrm{d}}\frac{\partial\beta}{\partial
N}\frac{\partial
R_{\mathrm{st}}}{\partial|f|^{2}}\Phi)-(\Phi,\frac{\partial\beta}{\partial|f|^{2}}\Phi)}{(\Phi,\frac{\partial\beta}{\partial\omega}\Phi)}.$
(75)

Inserting (74) back into (73) yields

$\delta N=-\tau_{\mathrm{d}}\frac{\partial
R_{\mathrm{st}}}{\partial|f|^{2}}\frac{F_{|f|^{2}}}{2\langle|f|^{2}\rangle\mathop{\mathrm{Im}}(h_{\alpha})}.$
(76)

Finally, substituting (76) into (38) results in

$\frac{\partial\delta\varphi}{\partial
t}=\frac{F_{|f|^{2}}}{2\langle|f|^{2}\rangle}\frac{\mathop{\mathrm{Re}}(h_{\alpha})}{\mathop{\mathrm{Im}}(h_{\alpha})}+F_{\varphi},$
(77)

which can be considered as a defining equation for the so–called effective
Henry’s $\alpha$-factor

$\alpha_{\mathrm{H,eff}}=-\frac{\mathop{\mathrm{Re}}(h_{\alpha})}{\mathop{\mathrm{Im}}(h_{\alpha})}.$
(78)

If $\alpha_{\mathrm{H,eff}}\neq 0$, the Langevin force resulting in
fluctuations of the intensity leads also to fluctuations of the phase. By
virtue of the different integrals in the numerator and denominator of (75),
the carrier dependence of the propagation factor, nonlinear gain (gain
compression), absorption (two–photon absorption) and index (Kerr effect) as
well as gain dispersion are accounted for. The impact of longitudinal spatial
hole burning and multi–section cavity structures (e.g., including Bragg
gratings, passive sections or external cavities) is also correctly described.
If all of these additional effects are neglected, for a Fabry–Pérot (FP) laser
the $\alpha$–factor is recovered

$\alpha_{\mathrm{H}}=-\frac{\mathop{\mathrm{Re}}\left(\frac{\partial\beta}{\partial
N}\right)}{\mathop{\mathrm{Im}}\left(\frac{\partial\beta}{\partial
N}\right)}=-\frac{4\pi}{\lambda_{0}}\frac{\frac{\partial\Delta n_{N}}{\partial
N}}{\frac{\partial g}{\partial N}-\frac{\partial\alpha}{\partial N}}.$ (79)

The second equality is obtained using (3c). The minus sign in (78), resulting
in a corresponding minus sign in (79), has been chosen in agreement with the
usual convention ensuring positive values of $\alpha_{\mathrm{H}}$ around the
gain peak [49]. The derivative $\partial\alpha/\partial N$ of the modal
absorption in (79) is often neglected, cf. [49]. One should keep in mind that
$\alpha_{\mathrm{H}}$ is not a constant but depends on the carrier density
(and the wavelength) because modal index and gain vary differently with
carrier density.

##  VII Spectral linewidth

Collecting the results of the previous sections, we can now calculate the
spectral linewidth. Rewriting (53) as

$\Delta\omega=\int_{-\infty}^{+\infty}\left\langle\frac{\partial\delta\varphi(t)}{\partial
t}\frac{\partial\delta\varphi(t+\tau)}{\partial t}\right\rangle\,d\tau$ (80)

and inserting the equation of the phase fluctuations (77), we obtain

$\Delta\omega=\frac{\alpha_{\mathrm{H,eff}}^{2}}{4\langle|f|^{2}\rangle^{2}}\int_{-\infty}^{\infty}\langle
F_{|f|^{2}}(t)F_{|f|^{2}}(t+\tau)\rangle\,d\tau+\int_{-\infty}^{\infty}\langle
F_{\varphi}(t)F_{\varphi}(t+\tau)\rangle\,d\tau,$ (81)

where we have taken (63) into account. Inserting (61) and (62) results in the
remarkably simple expression for the spectral linewidth

$\Delta\nu=\frac{\Delta\omega}{2\pi}=\frac{D_{f^{*}f}}{2\pi\langle|f|^{2}\rangle}\left(1+\alpha_{\mathrm{H,eff}}^{2}\right).$
(82)

In the following, we will rewrite this expression in terms of the intra-cavity
photon number

$I_{\mathrm{ph}}=\frac{1}{\hbar\omega_{0}}\int_{0}^{L}P\,\mathop{\mathrm{Re}}\left(\frac{\partial\beta}{\partial\omega}\right)\,dz=\frac{P_{0}}{\hbar\omega_{0}}\frac{\int_{0}^{L}\|\Phi\|^{2}\mathop{\mathrm{Re}}\left(\frac{\partial\beta}{\partial\omega}\right)\,dz}{(1-|r_{0}|^{2})|\Phi^{-}(0)|^{2}}$
(83)

where

$P(z)=\langle|f|^{2}\rangle\|\Phi(z)\|^{2}$ (84)

is the stationary optical power inside the cavity and

$P_{0}=(1-|r_{0}|^{2})\langle|f|^{2}\rangle|\Phi^{-}(0)|^{2}$ (85)

is the outcoupled power at $z=0$. Moreover, we define the rate of spontaneous
emission into the lasing mode

$R_{\mathrm{sp}}=\frac{\int_{0}^{L}\|\Phi\|^{2}n_{\mathrm{sp}}g\,dz}{\int_{0}^{L}\|\Phi\|^{2}\mathop{\mathrm{Re}}\left(\frac{\partial\beta}{\partial\omega}\right)\,dz}$
(86)

and the longitudinal excess factor of spontaneous emission

$K=\frac{\left(\int_{0}^{L}\|\Phi\|^{2}\mathop{\mathrm{Re}}\left(\frac{\partial\beta}{\partial\omega}\right)\,dz\right)^{2}}{\left|(\Phi,\frac{\partial\beta}{\partial\omega}\Phi)\right|^{2}}.$
(87)

Now, (82) can be written as

$\Delta\nu=\frac{KR_{\mathrm{sp}}}{4\pi
I_{\mathrm{ph}}}\left(1+\alpha_{\mathrm{H,eff}}^{2}\right)$ (88)

which is the standard form found in the literature [14, 46].

The mode profile $\Phi(z)$ entering (83), (86), and (87) is obtained by
solving (9), (36) and (68) self-consistently above threshold. Thus, $\Phi$ is
a mode of the active cavity affected by spatial hole burning. The expression
for the $K$–factor generalizes the one given in [10] for non-uniform and
complex–valued group velocity $(\partial\beta/\partial\omega)^{-1}$ and is in
basic correspondence with [14, 50, 51], but fundamentally different from the
modified $K$–factor introduced in [52].

At an exceptional point, the mode is orthogonal to itself, i.e.,
$(\Phi,\frac{\partial\beta}{\partial\omega}\Phi)=0$, and the $K$–factor
approaches infinity. However, one has to keep in mind that at an exceptional
point the single–mode approximation used here fails. Instead, for the
description of the dynamics in the vicinity of such a point one has use at
least one eigenmode and the corresponding generalized eigenmode [10].

##  VIII Impact of passive sections, chirp reduction factor and Fabry–Pérot
case

In this section, we apply the general theory developed above to special
configurations that allow for further simplifications (i.e., neglect of
spatial hole burning, nonlinear gain and index as well gain dispersion).
Thereby, it is shown that the general framework entails some well-established
formulas from the literature as special cases.

###  VIII.1 Linewidth of a laser consisting of a gain chip subject to feedback
from an external cavity

We consider a laser consisting of one active section for $z\in[0,l]$ with
length $l$ and an arbitrary number of passive sections (including an external
cavity) for $z\in[l,L]$ with total length $L-l$. The active section is of the
Fabry–Pérot (FP) type having no Bragg grating ($\kappa=0$ for $z\in[0,l])$.
The passive sections may contain Bragg gratings but no excess carriers ($N=0$,
$\partial\beta/\partial N=0$, $D_{\mathrm{sp}}=0$, $R_{\mathrm{st}}=0$ for
$z\in[l,L]$). The reference plane is located just inside the active section so
that a possible finite reflectivity at the interface between active and
passive sections belongs to the passive sections. Furthermore, we neglect
spatial hole burning (i.e., $\partial N/\partial z=0$), nonlinear gain and
index ($\partial\beta/\partial|f|^{2}=0$, $\partial
R_{\mathrm{st}}/\partial|f|^{2}=R_{\mathrm{st}}/|f|^{2}$) and gain dispersion
($\beta_{\cal D}=0$, $\partial\beta/\partial\omega=n_{\mathrm{g}}/c$). See
Fig. 2 for an illustration of the device.

Figure 2: Schematic illustration of a two-section DBR laser featuring an
active gain section and a passive Bragg grating as considered in Sec. VIII.

The functions $\Phi^{\pm}(z)$ as solution of (9) are defined on the whole
cavity $z\in[0,L]$. In the active section they are given by

$\Phi^{\pm}(z)|_{\mathrm{active}}=\Phi^{\pm}(l)\mathrm{e}^{\mp
i(\Delta\overline{\beta}+\frac{n_{\mathrm{g}}}{c}\Omega)(z-l)}$ (89)

and hence

$\Phi^{+}(z)\Phi^{-}(z)|_{\mathrm{active}}=\Phi^{+}(l)\Phi^{-}(l)$ (90)

holds. Furthermore we define the left and right reflectivities

$r^{+}=\frac{\Phi^{-}(l)}{\Phi^{+}(l)}\quad\mathrm{and}\quad
r^{-}=\frac{\Phi^{+}(l)}{\Phi^{-}(l)}\equiv
r_{0}\mathrm{e}^{-2il\left(\Delta\overline{\beta}+\frac{n_{\mathrm{g}}}{c}\Omega\right)},$
(91)

where we exploited the fact that the $\Phi^{\pm}(z)$ are continuous at $z=l$.
The derivative of $r^{+}$ can be expressed as (see Appendix B)

$\frac{\partial\mathrm{ln}(r^{+})}{\partial\omega}=-2i\frac{\int_{l}^{L}\Phi^{+}\Phi^{-}\frac{\partial\beta}{\partial\omega}\,dz}{\Phi^{+}(l)\Phi^{-}(l)}.$
(92)

Therefore,

$\displaystyle(\Phi,\frac{\partial\beta}{\partial\omega}\Phi)$
$\displaystyle=2\int_{0}^{l}\Phi^{+}\Phi^{-}\frac{\partial\beta}{\partial\omega}\,dz+2\int_{l}^{L}\Phi^{+}\Phi^{-}\frac{\partial\beta}{\partial\omega}\,dz$
(93)
$\displaystyle=2\Phi^{+}(l)\Phi^{-}(l)\left[l\frac{\partial\beta}{\partial\omega}\bigg{|}_{\mathrm{active}}+\frac{i}{2}\frac{\partial\mathrm{ln}(r^{+})}{\partial\omega}\right]$
$\displaystyle=2l\Phi^{+}(l)\Phi^{-}(l)\frac{\partial\beta}{\partial\omega}\bigg{|}_{\mathrm{active}}\chi$

where we introduced the parameter

$\chi=1+\frac{i\frac{\partial\mathrm{ln}(r^{+})}{\partial\omega}}{2l\frac{\partial\beta}{\partial\omega}}=1+i\frac{c}{2ln_{\mathrm{g}}}\frac{\partial\mathrm{ln}(r^{+})}{\partial\omega}$
(94)

as in [14]. We omit $|_{\mathrm{active}}$ in what follows. Furthermore

$\displaystyle(\Phi,\tau_{\mathrm{d}}\frac{\partial\beta}{\partial
N}\frac{\partial R_{\mathrm{st}}}{\partial|f|^{2}}\Phi)$
$\displaystyle=\frac{2}{|f|^{2}}\int_{0}^{l}\Phi^{+}\Phi^{-}\tau_{\mathrm{d}}\frac{\partial\beta}{\partial
N}R_{\mathrm{st}}\,dz$ (95)
$\displaystyle=\frac{2}{|f|^{2}}\Phi^{+}(l)\Phi^{-}(l)\frac{\partial\beta}{\partial
N}\int_{0}^{l}\tau_{\mathrm{d}}R_{\mathrm{st}}\,dz$

holds. Then from (75), (78) and (79)

$\alpha_{\mathrm{H,eff}}=-\frac{\mathrm{Re}\left(\frac{\frac{\partial\beta}{\partial
N}}{\frac{\partial\beta}{\partial\omega}}\chi^{\ast}\right)}{\mathrm{Im}\left(\frac{\frac{\partial\beta}{\partial
N}}{\frac{\partial\beta}{\partial\omega}}\chi^{\ast}\right)}=-\frac{\mathrm{Re}\left(\frac{\partial\beta}{\partial
N}\chi^{\ast}\right)}{\mathrm{Im}\left(\frac{\partial\beta}{\partial
N}\chi^{\ast}\right)}=\frac{\mathrm{Re}\left(\alpha_{\mathrm{H}}\chi+i\chi\right)}{\mathrm{Im}\left(\alpha_{\mathrm{H}}\chi+i\chi\right)}$
(96)

and

$1+\alpha_{\mathrm{H,eff}}^{2}=(1+\alpha_{\mathrm{H}}^{2})\frac{|\chi|^{2}}{\mathrm{Im}^{2}\left(\alpha_{\mathrm{H}}\chi+i\chi\right)}$
(97)

follow. Similarly, we obtain

$\displaystyle D_{f^{\ast}f}$
$\displaystyle=\frac{\int_{0}^{l}\|\Phi\|^{2}D_{\mathrm{sp}}\,dz}{\left|2\int_{0}^{l}\Phi^{+}\Phi^{-}\frac{\partial\beta}{\partial\omega}\,dz\right|^{2}}\frac{1}{\left|\chi\right|^{2}}.$
(98)

Hence, the product of (97) and (98) is given by

$D_{f^{\ast}f}\left(1+\alpha_{\mathrm{H,eff}}^{2}\right)=\frac{\int_{0}^{l}\|\Phi\|^{2}D_{\mathrm{sp}}\,dz}{\left|2\int_{0}^{l}\Phi^{+}\Phi^{-}\frac{\partial\beta}{\partial\omega}\,dz\right|^{2}}\frac{\left(1+\alpha_{\mathrm{H}}^{2}\right)}{\mathrm{Im}^{2}\left(\alpha_{\mathrm{H}}\chi+i\chi\right)}$
(99)

and the spectral linewidth can be written as

$\Delta\nu=\frac{\Delta\nu_{\mathrm{FP}}}{F^{2}}$ (100)

with the chirp reduction factor

$\displaystyle F$
$\displaystyle=\mathop{\mathrm{Im}}\left(\alpha_{\mathrm{H}}\chi+i\chi\right)$
(101)
$\displaystyle=1-\frac{c}{2n_{\mathrm{g}}l}\mathop{\mathrm{Im}}\left(\frac{\partial\mathrm{ln}(r^{+})}{\partial\omega}\right)+\alpha_{\mathrm{H}}\frac{c}{2n_{\mathrm{g}}l}\mathop{\mathrm{Re}}\left(\frac{\partial\mathrm{ln}(r^{+})}{\partial\omega}\right)$

and

$\Delta\nu_{\mathrm{FP}}=\frac{\int_{0}^{l}\|\Phi\|^{2}D_{\mathrm{sp}}\,dz}{2\pi\langle|f|^{2}\rangle\left|2\int_{0}^{l}\Phi^{+}\Phi^{-}\frac{\partial\beta}{\partial\omega}\,dz\right|^{2}}\left(1+\alpha_{\mathrm{H}}^{2}\right)$
(102)

being the linewidth of a FP laser having the cavity length $l$ and the
reflection coefficient $r^{+}(\Omega)$ at the rear facet. Eq. (101) agrees
with [53, 54] as well as [16, 55] if the differently chosen harmonic time
dependence ($\mathrm{e}^{i\omega_{0}t}$ as used here versus
$\mathrm{e}^{-i\omega_{0}t}$) is observed. The relation to the static
frequency chirp is given in Appendix C. Introducing an effective length of the
passive cavity

$l_{\mathrm{p}}=-\frac{c}{2n_{\mathrm{g,p}}}\mathop{\mathrm{Im}}\left(\frac{\partial\mathrm{ln}(r^{+})}{\partial\omega}\right),$
(103)

the chirp reduction factor can be written as

$F=1+\frac{n_{\mathrm{g,p}}l_{\mathrm{p}}}{n_{\mathrm{g}}l}+\alpha_{\mathrm{H}}\frac{c}{2n_{\mathrm{g}}l}\mathop{\mathrm{Re}}\left(\frac{\partial\mathrm{ln}(r^{+})}{\partial\omega}\right).$
(104)

Thus the second term on the right–hand side of (101) can be interpreted as the
ratio of the round-trip times in the passive and active sections. The chirp
reduction factor allows an estimation of the reduction of the linewidth by a
passive section or an external cavity for given group index, length and
Henry’s $\alpha$–factor of the active section.

###  VIII.2 Linewidth of the Fabry–Pérot laser cavity

We start from (88) and employ the same approximations resulting in (100) and
(101). The rate of spontaneous emission into the lasing mode (86) is

$R_{\mathrm{sp}}(N)=\frac{\int_{0}^{l}\|\Phi(z)\|^{2}n_{\mathrm{sp}}(N)g(N)\,dz}{\int_{0}^{l}\|\Phi(z)\|^{2}\mathop{\mathrm{Re}}\left(\frac{\partial\beta}{\partial\omega}\right)\,dz}=\frac{c}{n_{\mathrm{g}}}n_{\mathrm{sp}}(N)g(N).$
(105)

The expression (105) should be compared with

$\tilde{R}_{\mathrm{sp}}(N)=\beta_{\mathrm{sp}}VR_{\mathrm{rad}}(N)$ (106)

often used in rate equation based modeling [56], where $\beta_{\mathrm{sp}}$
is the dimensionless spontaneous emission factor (ratio between the
spontaneous emission going into the lasing mode and the spontaneous emission
into all modes), $V$ the volume of the active region and
$R_{\mathrm{rad}}=BN^{2}$ the rate of radiative spontaneous recombination ($B$
bimolecular recombination coefficient). Equalizing (105) and (106) at the
lasing threshold yields

$\beta_{\mathrm{sp}}=\frac{c}{n_{\mathrm{g}}}\frac{n_{\mathrm{sp}}(N_{\mathrm{th}})g(N_{\mathrm{th}})}{VR_{\mathrm{rad}}(N_{\mathrm{th}})},$
(107)

which is of the order $10^{-5}$ for typical edge–emitting lasers.

The photon number (83) can be written as

$\displaystyle I_{\mathrm{ph}}$
$\displaystyle=\frac{P_{0}}{\hbar\omega_{0}}\frac{\int_{0}^{l}\|\Phi(z)\|^{2}\mathop{\mathrm{Re}}\left(\frac{\partial\beta}{\partial\omega}\right)\,dz}{(1-|r_{0}|^{2})|\Phi^{-}(0)|^{2}}$
(108)
$\displaystyle=\frac{n_{\mathrm{g}}}{c}\frac{P_{0}}{\hbar\omega_{0}\left((1-|r_{0}|^{2})|\Phi^{-}(0)|^{2}\right)}\int_{0}^{l}\left[|\Phi^{+}(0)|^{2}\mathrm{e}^{2\mathop{\mathrm{Im}}(\overline{\beta})z}+|\Phi^{-}(0)|^{2}\mathrm{e}^{-2\mathop{\mathrm{Im}}(\overline{\beta})z}\right]\,dz$
$\displaystyle=\frac{n_{\mathrm{g}}}{c}\frac{P_{0}}{\hbar\omega_{0}}\frac{|r_{0}|^{2}\left(\mathrm{e}^{2\mathop{\mathrm{Im}}(\overline{\beta})l}-1\right)-\mathrm{e}^{-2\mathop{\mathrm{Im}}(\overline{\beta})l}+1}{2\mathop{\mathrm{Im}}(\overline{\beta})(1-|r_{0}|^{2})}.$

The complex mode frequencies of the FP cavity are the solutions 666They are
obtained from (89) applying the boundary conditions (12) at $z=0$ and $z=l$.
of

$\Omega_{m}=\frac{c}{n_{\mathrm{g}}}\left[\frac{m\pi}{l}-\overline{\beta}-\frac{i}{2l}\ln{\left(r_{0}r^{+}(\Omega_{m})\right)}\right]$
(109)

with $m\in\mathbb{Z}$. To ensure $\mathop{\mathrm{Im}}(\Omega)=0$ for the
lasing mode, (109) leads to the usual threshold condition

$\alpha_{\mathrm{out}}\equiv-\frac{1}{l}\ln(|r_{0}r^{+}(\mathop{\mathrm{Re}}(\Omega))|)=2\mathop{\mathrm{Im}}(\overline{\beta})\equiv
g-\alpha.$ (110)

In what follows, we set $r_{l}=r^{+}(\Omega)$ (reflectivity seen by the laser
at $z=l$). From (110) it follows

$I_{\mathrm{ph}}=\frac{n_{\mathrm{g}}}{c}\frac{P_{0}}{\hbar\omega_{0}}\frac{|r_{0}|^{2}\left(\frac{1}{|r_{0}r_{l}|}-1\right)-|r_{0}r_{l}|+1}{2\mathop{\mathrm{Im}}(\overline{\beta})(1-|r_{0}|^{2})}=\frac{n_{\mathrm{g}}}{c}\frac{P_{\mathrm{out}}}{\hbar\omega_{0}\alpha_{\mathrm{out}}},$
(111)

where we used the relation between outcoupled power $P_{0}$ and total output
power $P_{\mathrm{out}}$

$P_{0}=\frac{P_{\mathrm{out}}}{1+\frac{1-|r_{l}|^{2}}{1-|r_{0}|^{2}}\frac{|r_{0}|}{|r_{l}|}}.$
(112)

The final result is

$\Delta\nu_{\mathrm{FP}}=\frac{K_{\mathrm{FP}}}{4\pi}\frac{c^{2}}{n_{\mathrm{g}}^{2}}\frac{\hbar\omega_{0}n_{\mathrm{sp}}(\alpha+\alpha_{\mathrm{out}})\alpha_{\mathrm{out}}}{P_{\mathrm{out}}}\left(1+\alpha_{\mathrm{H}}^{2}\right),$
(113)

where Petermann’s $K$–factor

$K=\frac{\left(\int_{0}^{l}\|\Phi\|^{2}\mathop{\mathrm{Re}}\left(\frac{\partial\beta}{\partial\omega}\right)\,dz\right)^{2}}{\left|2\int_{0}^{l}\Phi^{+}\Phi^{-}\frac{\partial\beta}{\partial\omega}\,dz\right|^{2}}$
(114)

of the FP cavity can be similarly derived as

$K_{\mathrm{FP}}=\left[\frac{(|r_{0}|+|r_{l}|)(1-|r_{0}r_{l}|)}{2|r_{0}r_{l}|\ln(|r_{0}r_{l}|)}\right]^{2}$
(115)

in agreement with [27, 9]. Eq. (113) together with (112) was used by several
authors, e.g., [12, 57, 55]. However, one should be aware of the
approximations involved in the derivation.

The original Schawlow–Townes formula [1] is obtained by setting
$K_{\mathrm{FP}}=1$, $n_{\mathrm{sp}}=1$, $\alpha=0$, $\alpha_{\mathrm{H}}=0$,
and relating the outcoupling losses $\alpha_{\mathrm{out}}$ to the spectral
FWHM of a cavity resonance by
$\Delta\nu_{\mathrm{cav}}=c/n_{\mathrm{g}}\cdot\alpha_{\mathrm{out}}/2\pi$ to

$\Delta\nu_{\mathrm{ST}}=\frac{\pi\hbar\omega_{0}}{P_{\mathrm{out}}}(\Delta\nu_{\mathrm{cav}})^{2}.$
(116)

There are two reasons why this differs from the original formula by a factor
of $4$. First, in [1] the half widths instead of full widths are used (factor
of $2$). Second, in [1] the sub-threshold case is considered resulting in a
further factor of $2$, which can be seen as follows: Below threshold the
spectral density can be directly calculated from (22) for $\Delta M=0$ by
Fourier transformation via

$\langle f^{\ast}(\omega)f(\omega^{\prime})\rangle=2\pi
S_{f}(\omega)\delta(\omega-\omega^{\prime})$ (117)

with the result

$S_{f}(\omega)=\frac{2D_{f^{*}f}}{\left(\omega-\mathop{\mathrm{Re}}(\Omega)\right)^{2}+\left(\mathop{\mathrm{Im}}(\Omega)\right)^{2}}=\frac{2D_{f^{*}f}}{(\omega-\mathop{\mathrm{Re}}(\Omega))^{2}+\left(\frac{D_{f^{*}f}}{\langle|f|^{2}\rangle}\right)^{2}},$
(118)

where we have used (35) and (66). This is a Lorentzian with the FWHM

$\Delta\nu_{\mathrm{subthr}}=\frac{D_{f^{*}f}}{\pi\langle|f|^{2}\rangle},$
(119)

which is a factor of $2$ larger than (82) (for $\alpha_{\mathrm{H,eff}}=0$).
See [6] for a more thorough discussion.

Figure 3: (a) Modal gain (blue) and rate of spontaneous emission (green), left
axis, and population inversion factor (red), right axis, versus carrier
density obtained from a microscopic simulation (symbols) and the analytic
models (122), (123) and $n_{\mathrm{sp}}=r_{\mathrm{sp}}/g$ (solid lines) for
a fixed wavelength $\lambda_{0}=1064\,\text{nm}$. The dashed green line is the
spontaneous emission computed using a constant population inversion factor
$n_{\mathrm{sp}}=2$. See the Appendix A for the consideration of the gain at a
fixed wavelength within the framework of the present model. (b)
Microscopically computed gain spectra for carrier densities ranging from
$N=1\cdot 10^{18}\,\text{cm}^{-3}$ to $1\cdot 10^{19}\,\text{cm}^{-3}$. The
position of the fixed wavelength $\lambda_{0}$ is shown by a blue line, the
peak gain position is indicated by a red line.

##  IX Population inversion factor

The population inversion factor777Sometimes it is called ‘spontaneous emission
factor’ which could be misleading. introduced in (58) and used in (86) and
(113) is given by [28]

$n_{\mathrm{sp}}(N)=\frac{1}{1-\exp{\left(\frac{\hbar\omega_{0}-qU_{\mathrm{F}}(N)}{k_{\mathrm{B}}T}\right)}},$
(120)

where $U_{\mathrm{F}}$ is the Fermi voltage (spacing of quasi–Fermi potentials
of holes and electrons), $q$ the elementary charge, $k_{\mathrm{B}}$ the
Boltzmann constant and $T$ the temperature. It has a singularity at the
transparency density $N_{\mathrm{tr}}$, where
$\hbar\omega_{0}=U_{\mathrm{F}}(N_{\mathrm{tr}})$ and $g(N_{\mathrm{tr}})=0$.
Therefore, the replacement

$r_{\mathrm{sp}}(N)\equiv
n_{\mathrm{sp}}(N)g(N)=n_{\mathrm{sp}}(\alpha+\alpha_{\mathrm{out}})$ (121)

with constant $n_{\mathrm{sp}}$ often employed is critical because for
high–$Q$–cavities with low $\alpha$ and $\alpha_{\mathrm{out}}$ the
spontaneous emission is underestimated. We calculated the modal gain, the
spontaneous emission into the lasing mode and the inversion factor employing a
numerical simulation based on a $8\times 8$ $\mathbf{k\cdot p}$ band structure
calculation and a free carrier theory for the optical response functions with
phenomenological corrections for many-body effects such as band–gap
renormalization, Coulomb enhancement and transition broadening [58]. The
results for a 5 nm thick InGaAs quantum well emitting around $1064~{}$nm
embedded into an AlGaAs-based waveguide structure are compared with analytical
models in Fig. 3. The increase of the absorption (negative gain) at small
increasing carrier densities observed in the simulation is caused by band–gap
narrowing, shifting the absorption edge to longer wavelengths [59].

The gain at a fixed wavelength is modeled as

$g=g^{\prime}N_{\mathrm{tr}}\ln{\left[\frac{\mathrm{max}(N,N_{\mathrm{cl}})}{N_{\mathrm{tr}}}\right]}$
(122)

and the modal spontaneous emission as

$r_{\mathrm{sp}}=\frac{g^{\prime}N_{\mathrm{tr}}}{2}\ln{\left[1+\left(\frac{N}{N_{\mathrm{tr}}}\right)^{2}\right]}$
(123)

with differential gain $g^{\prime}=19\cdot 10^{-22}~{}\mathrm{m}^{2}$,
transparency density $N_{\mathrm{tr}}=1.7\cdot 10^{24}~{}\mathrm{m}^{-3}$ and
gain clamping density $N_{\mathrm{cl}}=7\cdot 10^{23}~{}\mathrm{m}^{-3}$. Eq.
(123) exhibits the correct asymptotic behavior $r_{\mathrm{sp}}\propto N^{2}$
for $N\ll N_{\mathrm{tr}}$ and $r_{\mathrm{sp}}=g$ for $N\to\infty$. The
agreement of $r_{\mathrm{sp}}$ between simulation and model is remarkably good
without the need of introducing any new parameters. The singularity of the
inversion factor at $N=N_{\mathrm{tr}}$ is clearly visible in Fig. 3. If a
constant value of $n_{\mathrm{sp}}=2$ is used, the spontaneous emission is
considerably under– or over–estimated (depending on the respective carrier
density).

##  X Numerical results for a DBR laser

parameter | symbol | unit | value | first used
---|---|---|---|---
reference wavelength | $\lambda_{0}$ | m | $1.064\cdot 10^{-6}$ |
front facet reflectivity | $|r_{0}|^{2}$ |  | $0.3$ | (4)
rear facet reflectivity | $|r_{L}|^{2}$ |  | $0$ | (4)
internal optical loss (both sections) | $\alpha$ | m-1 | $60$ | (3c)
group index (both sections) | $n_{\mathrm{g}}$ |  | 3.9 | (2)
built-in index detuning (both sections) | $\Delta n_{0}$ |  | 0 | (126)
_active section_
length | $l$ | m | $1\cdot 10^{-3}$ |
differential gain | $g^{\prime}$ | m2 | $18.62\cdot 10^{-22}$ | (122)
$\alpha$–factor | $\tilde{\alpha}_{\mathrm{H}}$ |  | $1$ | (79)
transparency carrier density | $N_{\mathrm{tr}}$ | m-3 | $1.7\cdot 10^{24}$ | (122)
self-heating induced index tuning | $\nu_{s}$ | A-1 | $9.16\cdot 10^{-3}$ | (126)
gain clamping density | $N_{\mathrm{cl}}$ | m-3 | $0.8\cdot 10^{24}$ | (122)
index clamping density | $N_{\mathrm{cl,i}}$ | m-3 | $0.01\cdot 10^{24}$ | (125)
gain saturation power | $P_{\mathrm{sat}}$ | W | $5.866$ | (124)
dispersion peak amplitude | $g_{\cal D}$ | m-1 | $50$ | (136)
dispersion peak frequency detuning | $\omega_{\cal D}$ | rad s-1 | $0$ | (136)
dispersion HWHM | $\gamma_{\cal D}$ | rad s-1 | $83.25\cdot 10^{12}$ | (136)
thickness of active region | $d$ | m | $4\cdot 10^{-6}$ | (5)
width of active region | $W$ | m | $10\cdot 10^{-9}$ | (6)
series resistance | $R_{s}$ | $\Omega$ | $1$ | (7)
Fermi voltage derivative | $\frac{dU_{\mathrm{F}}}{dN}$ | Vm3 | $3\cdot 10^{-26}$ | (128)
defect recombination coefficient | $A$ | s-1 | $4\cdot 10^{8}$ | (127)
bimolecular recombination coefficient | $B$ | m3 s-1 | $1\cdot 10^{-16}$ | (127)
Auger recombination coefficient | $C$ | m6 s-1 | $4\cdot 10^{-42}$ | (127)
injection current | $I$ | A | $[0,300]\cdot 10^{-3}$ | (7)
_passive section_
length | $L-l$ | m | $3\cdot 10^{-3}$ |
coupling coefficient | $\kappa$ | m-1 | $200$ | (10)
cross-heating induced index tuning | $\nu_{c}$ | A-1 | $1.83\cdot 10^{-3}$ | (126)
Table 1: Parameters of the simulated DBR laser.

In the simulations described below we study a simple two-section DBR laser as
sketched in Figs. 1 and 2. It consists of a $1$ mm long active gain section
(without Bragg grating) and a $3$ mm long passive Bragg reflector section
(without active layer) with a coupling coefficient of
$\kappa=2~{}\text{cm}^{-1}$. The usage of such DBR lasers having long
reflector sections with low coupling coefficients was suggested in Ref. [16]
for the first time. In fact, in Ref. [57] an intrinsic linewidth of $2$ kHz at
an output power of $180$ mW was reported. This device had a $3$ mm long gain
section and a $1$ mm long reflector section, where the active layer extended
over the whole cavity (i.e., including the reflector section). The device
studied here resembles the one presented in Ref. [60], where an intrinsic
linewidth of $4$ kHz at an output power of $73$ mW was reported.
Experimentally, the intrinsic linewidth is determined from the plateau of the
frequency noise spectrum, as in the theory [61].

Within the active section, we assume the logarithmic gain model (122)
supplemented with the nonlinear gain saturation factor

$g(z,N,\|\Psi\|^{2})=g^{\prime}(z)N_{\mathrm{tr}}(z)\ln{\left[\frac{\mathrm{max}(N,N_{\mathrm{cl}}(z))}{N_{\mathrm{tr}}(z)}\right]}\frac{1}{1+\|\Psi\|^{2}/P_{\mathrm{sat}}(z)},$
(124)

and the refractive index model with square–root–like dependency on the carrier
density

$\Delta
n_{N}=-\frac{\lambda_{0}\tilde{\alpha}_{\mathrm{H}}g^{\prime}N_{\mathrm{tr}}}{2\pi}\left(\sqrt{\frac{\max(N,N_{\mathrm{cl,i}})}{N_{\mathrm{tr}}}}-1\right).$
(125)

Note, that at the transparency carrier density $N_{\mathrm{tr}}$, the fixed
parameter $\tilde{\alpha}_{\mathrm{H}}$ agrees with the $\alpha$–factor
$\alpha_{\mathrm{H}}$ from (79), whereas $\Delta n_{N}$ vanishes. The
$\alpha$–factor correspondingly increases for $N>N_{\mathrm{tr}}$. The value
of $\tilde{\alpha}_{\mathrm{H}}=1$ was chosen in agreement with simulations
[58] and is in basic agreement with measurements, where $\alpha$–factors
between $1$ and $2$ for a highly-strained InGaAs quantum well were obtained,
depending on the detuning [62].

Figure 4: Simulated characteristics of the DBR laser as functions of the
upsweeped injection current. Dotted and dashed lines indicate mode jumps and
maxima of the DBR reflectivity, respectively.

Other changes of the refractive index, including self–heating–induced self-
and cross–heating contributions, are modeled by [30]

$\Delta n(z)=\Delta n_{0}(z)+\Delta n_{T}(z),\qquad\Delta
n_{T}(z)=\begin{cases}\nu_{s}I,&z\in[0,l]\cr\nu_{c}I,&z\in[l,L]\end{cases},$
(126)

where $I$ is the injection current into the active (gain) section. We employ
the commonly used cubic model for the rate of non-radiative and spontaneous
recombination of the carriers in the active section

$R(N)=AN+BN^{2}+CN^{3},$ (127)

is approximated by describing Shockley–Read–Hall recombination, direct band-
to-band recombination and Auger recombination. Finally, the current density
(7) in the gain section is approximated by

$j(z,N)=\frac{I}{Wl}-\frac{dU_{\mathrm{F}}}{dN}\;\frac{N-\frac{1}{l}\int_{0}^{l}N\,dz}{WlR_{s}},$
(128)

where the derivative of the Fermi voltage $dU_{\mathrm{F}}/dN$ is taken at a
fixed carrier density.

The numerical solution of the coupled–wave equations (2), (4), (5), (134) in
the time domain were performed with the software package LDSL-tool developed
at the Weierstrass Institute [63]. The set of parameters characterizing the
simulated DBR laser is listed in Table 1.

Several characteristics of the steady states obtained in the numerical
simulations of the DBR laser with an upsweep of bias current are shown in Fig.
4. The almost-periodic jumps towards the steady-state defined by the adjacent
longitudinal optical mode and corresponding changes of the state
characteristics are induced by the self–and cross–heating modeled according to
Eq. (126). While the self-heating of the active section is mainly responsible
for the fast shift of the lasing wavelength to longer values and periodic
jumps to the shorter-wavelength state, the cross–heating induces a reduced
shift of the peak wavelength of the DBR reflectivity and, therefore, the
corresponding shift of the mean lasing wavelength, see Fig. 4 (c). The maximum
emitted optical power, see Fig. 4 (a), and the minimal (averaged) carrier
density, see Fig. 4 (e), within each period coincide well with the maximum
field reflection provided by the Bragg grating shown in Fig. 4 (g).

The overall decay of the estimated spectral linewidth, see Fig. 4 (b),
reflects its inverse proportionality to the field intensity, see Eq. (82). The
decay of the linewidth within each period is consistent with the decrease of
the modulus of the effective linewidth enhancement factor
$\alpha_{\mathrm{H},\mathrm{eff}}$, see Fig. 4 (d), the increase of the chirp
reduction factor $F$ shown in Fig. 4 (f), and the behavior of Petermann’s
$K$–factor plotted in Fig. 4 (h). The narrowest linewidth within each period
is observed at the long–wavelength flank of the DBR reflectivity just before
the transition to the neighboring state. The dependence of $F$ and
$\alpha_{\mathrm{H}}$ on the deviation of the lasing wavelength from the peak
wavelength of the DBR reflectivity confirms earlier findings [16, 64, 65] that
$F$ increases and $\alpha_{\mathrm{H,eff}}$ decreases at the long–wavelength
flank of the reflection spectrum.

##  XI Outlook

The theory presented can be easily extended to take into account carrier
noise. Including the source $F_{N}\neq 0$, we obtain instead of (73) the
relation

$\delta N=\tau_{\mathrm{d}}F_{N}-\tau_{\mathrm{d}}\frac{\partial
R_{\mathrm{st}}}{\partial|f|^{2}}\delta|f|^{2}.$ (129)

Substituting this expression into (37) yields a simple modification of the
Langevin force

$F_{|f|^{2}}\to
F_{|f|^{2}}+2\mathrm{Im}\frac{(\Phi,\frac{\partial\beta}{\partial
N}\tau_{\mathrm{d}}F_{N}\Phi)}{(\Phi,\frac{\partial\beta}{\partial\omega}\Phi)}\langle|f|^{2}\rangle,$
(130)

leading to two additional additive contributions to the linewidth due to
carrier noise and the cross-correlation between carrier and intensity noise.
The same procedure can be applied to the calculation of the modulation of
amplitude and frequency in response to an external modulation of the current
injection term in (5).

In case of gain coupling, (21) has to be replaced by

$\Delta M=\begin{bmatrix}\overline{\beta}-\beta&\overline{\kappa}-\kappa\\\
\overline{\kappa}-\kappa&\overline{\beta}-\beta\end{bmatrix}$ (131)

and (33) has to be supplemented by

$\kappa=\kappa(\langle N\rangle)+\frac{\partial\kappa}{\partial N}\delta N.$
(132)

Note that for loss or gain coupling or for higher order gratings there are
additional contributions to the imaginary part of $\beta$. Furthermore, the
rate of stimulated recombination (6) has to be modified [66] and the diffusion
coefficient (60) is varied because of non–vanishing $\langle
F_{\mathrm{sp}}^{-\ast}(z,t)F_{\mathrm{sp}}^{+}(z^{\prime},t^{\prime})\rangle\neq
0$ and its complex conjugate [47, 22].

Fluctuations of the shape of the power profile can be accounted for by a
linearization of (2) and (5) around a steady state, performing a Fourier
transformation and solving for the fluctuations $\delta\Psi$, e.g., by the
Green’s function method [22] to calculate the noise spectral densities
$S_{\xi}(\tilde{\omega})$ from

$\langle\xi^{\ast}(\tilde{\omega})\xi(\tilde{\omega}^{\prime})\rangle=2\pi
S_{\xi}(\tilde{\omega})\delta(\tilde{\omega}-\tilde{\omega}^{\prime}),$ (133)

where $\xi$ is the variable of interest (e.g., phase or intensity).

##  XII Summary

We have derived in a self-contained manner the spectral linewidth of
edge–emitting multi–section semiconductor lasers starting from the time-
dependent coupled–wave equations with a Langevin noise source. For this, we
have expanded the forward and backward propagating fields into the
longitudinal modes of the open cavity and have obtained very general
expressions for the effective linewidth enhancement factor
$\alpha_{\mathrm{H}}$ and the longitudinal excess factor of spontaneous
emission ($K$–factor), including the effects of nonlinear gain (gain
compression) and index (Kerr effect), gain dispersion, longitudinal spatial
hole burning for multi–section cavity structures. We have shown that the
general linewidth expression contains the effect of linewidth narrowing due to
an external cavity by the chirp reduction factor $F$. Finally, we have
investigated the dependence of the population inversion factor on the carrier
density and proposed a new analytical formula. Based on the derived
expressions, the spectral linewidth as well as $\alpha_{\mathrm{H}}$, $K$, and
$F$ have been calculated for a two–section DBR laser as functions of the
injection current and the optical output power, taking into account the
thermal detuning between gain and reflector sections. The mode jumps appearing
with increasing current are accompanied by sudden rises of the spectral
linewidth due to the rise of the modulus of $\alpha_{\mathrm{H}}$ and the drop
of $F$.

###### Acknowledgements.

This research was partially funded by the Federal Ministry of Education and
Research (BMBF) under grant number 50RK1972 and by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy – The Berlin Mathematics Research Center MATH+
(EXC-2046/1, project ID: 390685689, grant AA2-13). The authors are indebted to
Hans-Jürgen Wünsche for a critical reading of the manuscript.

##  Appendix A Dispersion operator

The dispersion operator $\widehat{\mathcal{D}}(z,t)$ in Eq. (2) models the
frequency dependence of the gain and associated refractive index. The approach
followed here is based on approximating the material gain spectrum locally
around the gain peak by a Lorentzian and supplementing the time-domain
coupled–wave equations (2) with additional dynamical equations for _auxiliary_
polarizations [36, 37, 31]

$\frac{\partial}{\partial t}P(z,t)=\left(i\omega_{\cal D}(z)-\gamma_{\cal
D}(z)\right)P(z,t)+\gamma_{\cal D}(z)\Psi(z,t),$ (134)

where $\omega_{\mathcal{D}}$ is the detuning between the peak gain frequency
$\omega_{p}$ and the reference frequency $\omega_{0}$, $\gamma_{\mathcal{D}}$
is the half width at half maximum (HWHM) of the gain curve and

$P(z,t)=\begin{bmatrix}P^{+}(z,t)\\\ P^{-}(z,t)\end{bmatrix}.$

The corresponding dispersion operator in (2) reads

$\widehat{\cal D}(z,t)\Psi(z,t)=-\frac{i}{2}g_{\cal
D}(z)\left(\Psi(z,t)-P(z,t)\right),$ (135)

where $g_{\cal D}$ describes the difference between the maximum amplitude gain
$g(z,N,\|\Psi\|^{2})$, see Eq. (3c), and the gain at the reference frequency.
The parameters $\omega_{\mathcal{D}}$, $\gamma_{\mathcal{D}}$, and
$g_{\mathcal{D}}$ are obtained from a fit to a microscopically computed gain
spectrum or experimental data.

Figure 5: Illustration of the Lorentzian fitted (effective) gain spectrum and
labeling of the dispersion parameters $\omega_{\mathcal{D}}$,
$\gamma_{\mathcal{D}}$, and $g_{\mathcal{D}}$. The peak gain is denoted by
$g_{\text{eff}}\left(\Omega=\omega_{\mathcal{D}}\right)=g$ and
$g_{\text{eff}}\left(\Omega=0\right)$ is the effective gain at the reference
frequency $\omega_{0}$.

A Fourier transformation of (135) yields, together with the frequency domain
solution of the polarization equation (134), an expression for the dispersion
factor (complex Lorentzian)

$\beta_{\mathcal{D}}\left(z,\Omega\right)=\frac{g_{\mathcal{D}}(z)}{2}\frac{\Omega-\omega_{\mathcal{D}}\left(z\right)}{i\left(\Omega-\omega_{\mathcal{D}}\left(z\right)\right)+\gamma_{\mathcal{D}}\left(z\right)}.$
(136)

The associated _effective gain spectrum_ entering the rate of stimulated
emission (67) reads

$\displaystyle g_{\text{eff}}(z,N,\|\Psi\|^{2},\Omega)$
$\displaystyle=g(z,N,\|\Psi\|^{2})+2\mathop{\mathrm{Im}}\left(\beta_{\mathcal{D}}(z,\Omega)\right)$
(137)
$\displaystyle=g(z,N,\|\Psi\|^{2})-g_{\mathcal{D}}\frac{\left(\Omega-\omega_{\mathcal{D}}\left(z\right)\right)^{2}}{\left(\Omega-\omega_{\mathcal{D}}\left(z\right)\right)^{2}+\gamma_{\mathcal{D}}^{2}\left(z\right)},$

which is maximal at $\Omega=\omega_{\mathcal{D}}$, i.e., if the laser operates
at the peak gain frequency.

An alternative representation of Eq. (135) is obtained with the help of the
Green’s function of the polarization equations (134) as

$\widehat{\mathcal{D}}\left(z,t\right)\Psi\left(z,t\right)=-\frac{i}{2}g_{\mathcal{D}}(z)\left(\Psi\left(z,t\right)-\gamma_{\mathcal{D}}(z)\int_{0}^{t}\mathrm{e}^{\left(i\omega_{\mathcal{D}}(z)-\gamma_{\mathcal{D}}(z)\right)\tau}\Psi\left(z,t-\tau\right)\,d\tau\right),$
(138)

where dispersion is induced via the convolution of the memory kernel with the
time-delayed field amplitude (we omitted the quickly decaying initial value
term here).

##  Appendix B Calculation of derivative of $r^{+}$

To derive (92) we have to calculate

$\partial_{\omega}\ln(r^{+})=\frac{1}{r^{+}}\partial_{\omega}r^{+}=\frac{1}{\Phi^{-}}\partial_{\omega}\Phi^{-}-\frac{1}{\Phi^{+}}\partial_{\omega}\Phi^{+},$
(139)

where we abbreviated $\partial/\partial\omega\equiv\partial_{\omega}$. The
coupled–wave equations (9) together with their derivatives with respect to
frequency can be written as

$\displaystyle\frac{\partial}{\partial z}\Phi^{+}(z)$
$\displaystyle=-i\hat{\beta}(\Omega)\Phi^{+}(z)-i\kappa\Phi^{-}(z),$ (140)
$\displaystyle-\frac{\partial}{\partial z}\Phi^{-}(z)$
$\displaystyle=-i\hat{\beta}(\Omega)\Phi^{-}(z)-i\kappa\Phi^{+}(z),$
$\displaystyle\frac{\partial}{\partial z}\partial_{\omega}\Phi^{+}(z)$
$\displaystyle=-i\hat{\beta}(\Omega)\partial_{\omega}\Phi^{+}(z)-i\kappa\partial_{\omega}\Phi^{-}(z)-i\partial_{\omega}\beta(\Omega)\Phi^{+}(z),$
$\displaystyle-\frac{\partial}{\partial z}\partial_{\omega}\Phi^{-}(z)$
$\displaystyle=-i\hat{\beta}(\Omega)\partial_{\omega}\Phi^{-}(z)-i\kappa\partial_{\omega}\Phi^{+}(z)-i\partial_{\omega}\beta(\Omega)\Phi^{-}(z),$

where
$\hat{\beta}(\Omega)=\Delta\overline{\beta}(z)+\beta_{\mathcal{D}}(\Omega)+n_{\textrm{g}}\Omega/c$
and $\partial_{\omega}\beta(\Omega)$ is defined in (15). By multiplying the
left– and right–hand sides of these equations by $\partial_{\omega}\Phi^{-}$,
$\partial_{\omega}\Phi^{+}$, $-\Phi^{-}$, and $-\Phi^{+}$, respectively, we
get

$\displaystyle\partial_{\omega}\Phi^{-}\,\frac{\partial}{\partial z}\Phi^{+}$
$\displaystyle=-i\hat{\beta}(\Omega)\Phi^{+}\,\partial_{\omega}\Phi^{-}-i\kappa\Phi^{-}\,\partial_{\omega}\Phi^{-},$
(141) $\displaystyle-\partial_{\omega}\Phi^{+}\,\frac{\partial}{\partial
z}\Phi^{-}$
$\displaystyle=-i\hat{\beta}(\Omega)\Phi^{-}\,\partial_{\omega}\Phi^{+}-i\kappa\Phi^{+}\,\partial_{\omega}\Phi^{+},$
$\displaystyle-\Phi^{-}\,\frac{\partial}{\partial z}\partial_{\omega}\Phi^{+}$
$\displaystyle=i\hat{\beta}(\Omega)\Phi^{-}\,\partial_{\omega}\Phi^{+}+i\kappa\Phi^{-}\,\partial_{\omega}\Phi^{-}+i\partial_{\omega}\beta(\Omega)\Phi^{-}\,\Phi^{+},$
$\displaystyle\Phi^{+}\,\frac{\partial}{\partial z}\partial_{\omega}\Phi^{-}$
$\displaystyle=i\hat{\beta}(\Omega)\Phi^{+}\,\partial_{\omega}\Phi^{-}+i\kappa\Phi^{+}\,\partial_{\omega}\Phi^{+}+i\partial_{\omega}\beta(\Omega)\Phi^{+}\,\Phi^{-}.$

Adding all these equations implies

$\frac{\partial}{\partial
z}\left(\Phi^{+}\,\partial_{\omega}\Phi^{-}-\Phi^{-}\,\partial_{\omega}\Phi^{+}\right)=2i\partial_{\omega}\beta(\Omega)\Phi^{+}\,\Phi^{-}.$
(142)

An integration over the passive sections corresponding to the coordinate
interval $[l,L]$ yields

$\left[\Phi^{+}\,\partial_{\omega}\Phi^{-}-\Phi^{-}\,\partial_{\omega}\Phi^{+}\right]\big{|}_{l}^{L}=2i\int_{l}^{L}\partial_{\omega}\beta(\Omega)\Phi^{+}\,\Phi^{-}dz.$
(143)

Due to the reflecting boundary condition, the expression
$\left[\Phi^{+}\,\partial_{\omega}\Phi^{-}-\Phi^{-}\,\partial_{\omega}\Phi^{+}\right]$
vanishes at $z=L$ and

$-\Phi^{+}(l)\,\partial_{\omega}\Phi^{-}(l)+\Phi^{-}(l)\,\partial_{\omega}\Phi^{+}(l)=2i\int_{l}^{L}\partial_{\omega}\beta(z,\Omega)\Phi^{+}(z)\Phi^{-}(z)dz$
(144)

follows. Dividing by $-\Phi^{+}(l)\Phi^{-}(l)$ finally yields

$\partial_{\omega}\log(r^{+})=-2i\frac{\int_{l}^{L}\partial_{\omega}\beta(z,\Omega)\Phi^{+}(z)\Phi^{-}(z)dz}{\Phi^{+}(l)\Phi^{-}(l)}.$
(145)

##  Appendix C The relation to static frequency chirp

Here we justify the naming of the factor $F$ defined in (101). We start from
the roundtrip condition

$r^{+}(\tilde{\omega},g)r^{-}(\tilde{\omega},g)=1$ (146)

following from (91) which can be transformed into equations for the modulus

$\mathop{\mathrm{Re}}\left(\ln(r^{-}(\tilde{\omega},g))\right)+\mathop{\mathrm{Re}}\left(\ln(r^{+}(\tilde{\omega}))\right)=0$
(147)

and the phase

$h(\tilde{\omega},g)\equiv\mathrm{Im}\big{(}\ln(r^{-}(\tilde{\omega},g))\big{)}+\mathrm{Im}\big{(}\ln(r^{+}(\tilde{\omega}))\big{)}=2\pi
m$ (148)

with $\tilde{\omega}=\omega-\omega_{0}$ and $m\in\mathbb{Z}$. As a result of
the solution of (146) or, equivalently, (147) and (148), we obtain the
real–valued frequency $\Omega=\tilde{\omega}$ and the gain $g$ of the steady
lasing state.

Let the modal index in the active section vary due to a fluctuation of some
parameter such as carrier density or temperature. Eq. (148) establishes a
relation between the index fluctuation $\delta\Delta n$ and the frequency
fluctuation $\delta\omega$,

$\delta h=\frac{\partial h}{\partial\omega}\delta\omega+\frac{\partial
h}{\partial n}\delta n=0,$ (149)

where $\delta n\equiv\delta\Delta n$ and $1/\partial n\equiv 1/\partial\Delta
n$. The $\omega$–derivative of $h$ is

$\frac{\partial
h}{\partial\omega}=\mathop{\mathrm{Im}}\left(\frac{\partial\ln(r^{-})}{\partial\omega}\right)+\mathop{\mathrm{Im}}\left(\frac{\partial\ln(r^{+})}{\partial\omega}\right)+\mathop{\mathrm{Im}}\left(\frac{\partial\ln(r^{-})}{\partial
g}\right)\frac{\partial g}{\partial\omega}.$ (150)

The $\omega$–derivative of the lasing gain $g$ can be determined from (147),

$\frac{\partial
g}{\partial\omega}=-\frac{\mathop{\mathrm{Re}}\left(\frac{\partial\ln(r^{-})}{\partial\omega}\right)+\mathop{\mathrm{Re}}\left(\frac{\partial\ln(r^{+})}{\partial\omega}\right)}{\mathop{\mathrm{Re}}\left(\frac{\partial\ln(r^{-})}{\partial
g}\right)}=-\frac{1}{l}\mathop{\mathrm{Re}}\left(\frac{\partial\ln(r^{+})}{\partial\omega}\right)$
(151)

taking into account

$\frac{\partial\ln(r^{-})}{\partial\omega}=-2il\frac{n_{\mathrm{g}}}{c}$ (152)

and

$\frac{\partial\ln(r^{-})}{\partial g}=l(1+i\alpha_{\mathrm{H}}),$ (153)

following from (91) and (11), such that

$\frac{\partial
h}{\partial\omega}=-2l\frac{n_{\mathrm{g}}}{c}+\mathop{\mathrm{Im}}\left(\frac{\partial\ln(r^{+})}{\partial\omega}\right)-\alpha_{\mathrm{H}}\mathop{\mathrm{Re}}\left(\frac{\partial\ln(r^{+})}{\partial\omega}\right)$
(154)

is obtained. Furthermore, (91) and (11) imply

$\frac{\partial\ln(r^{-})}{\partial n}=-2il\frac{2\pi}{\lambda_{0}}$ (155)

and

$\frac{\partial h}{\partial n}=-\frac{4\pi l}{\lambda_{0}}.$ (156)

Therefore,

$\delta
h=-2l\frac{n_{\mathrm{g}}}{c}\left[1-\frac{c}{2n_{\mathrm{g}}l}\mathop{\mathrm{Im}}\left(\frac{\partial\ln(r^{+})}{\partial\omega}\right)+\alpha_{\mathrm{H}}\frac{c}{2n_{\mathrm{g}}l}\mathop{\mathrm{Re}}\left(\frac{\partial\ln(r^{+})}{\partial\omega}\right)\right]\delta\omega-\frac{4\pi
l}{\lambda_{0}}\delta n=0$ (157)

and

$\delta\omega=-\omega_{0}\frac{\delta n}{n_{\mathrm{g}}}F^{-1}$ (158)

is gained. For the solitary laser with $r^{+}(\tilde{\omega})$ taken at a
fixed $\tilde{\omega}=\Omega$, the same analysis results in

$\frac{\partial h}{\partial\omega}=-2l\frac{n_{\mathrm{g}}}{c}$ (159)

and

$\delta\omega_{\mathrm{s}}=-\omega_{0}\frac{\delta n}{n_{\mathrm{g}}}.$ (160)

Therefore in a laser with a passive section or an external cavity,
fluctuations of the frequency $\delta\omega$ due to a fluctuation of some
parameter in the active section are reduced (or enhanced) by the factor

$\frac{\delta\omega}{\delta\omega_{\mathrm{s}}}=\frac{1}{F}$ (161)

compared to a the solitary laser, as stated in, e.g., [16, 17]. If the active
section is not a single FP cavity, the chirp reduction factor can be
generalized to [67]

$F=1+\mathop{\mathrm{Re}}\left((1-i\alpha_{\mathrm{H}})\frac{\frac{\partial\mathrm{ln}(r^{+})}{\partial\omega}}{\frac{\partial\mathrm{ln}(r^{-})}{\partial\omega}}\right).$
(162)

## References

  * Schawlow and Townes [1958] A. L. Schawlow and C. H. Townes, Infrared and optical masers, Phys. Rev. 112, 1940 (1958).
  * Haken [1964] H. Haken, A nonlinear theory of laser noise and coherence. I, Zeitschrift für Physik 181, 96 (1964).
  * Lax [1967] M. Lax, Classical noise V: Noise in self-sustained oscillators, Phys. Rev. 160, 290 (1967).
  * Haug and Haken [1967] H. Haug and H. Haken, Theory of noise in semiconductor laser emission, Zeitschrift für Physik A 204, 262 (1967).
  * Scully and Lamb Jr [1967] M. O. Scully and W. E. Lamb Jr, Quantum theory of an optical maser. I. General theory, Phys. Rev. 159, 208 (1967).
  * Risken [1968] H. Risken, Zur Statistik des Laserlichts, Fortschritte der Physik 16, 261 (1968).
  * Petermann [1979] K. Petermann, Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding, IEEE J. Quantum Electron. 15, 566 (1979).
  * Siegman [1989] A. Siegman, Excess spontaneous emission in non-Hermitian optical systems. ii. Laser oscillators, Phys. Rev. A 39, 1264 (1989).
  * Hamel and Woerdman [1989] W. Hamel and J. Woerdman, Nonorthogonality of the longitudinal eigenmodes of a laser, Phys. Rev. A 40, 2785 (1989).
  * Wenzel  _et al._ [1996] H. Wenzel, U. Bandelow, H.-J. Wünsche, and J. Rehberg, Mechanisms of fast self pulsations in two-section DFB lasers, IEEE J. Quantum Electron. 32, 69 (1996).
  * Özdemir  _et al._ [2019] Ş. K. Özdemir, S. Rotter, F. Nori, and L. Yang, Parity–time symmetry and exceptional points in photonics, Nat. Mater. 18, 783 (2019).
  * Henry [1982] C. H. Henry, Theory of the linewidth of semiconductor lasers, IEEE J. Quantum Electron. 18, 259 (1982).
  * Amann [1990] M.-C. Amann, Linewidth enhancement in distributed-feedback semiconductor lasers, Electron. Lett. 26, 569 (1990).
  * Tromborg  _et al._ [1991] B. Tromborg, H. Olesen, and X. Pan, Theory of linewidth for multielectrode laser diodes with spatially distributed noise sources, IEEE J. Quantum Electron. 27, 178 (1991).
  * Patzak  _et al._ [1983] E. Patzak, A. Sugimura, S. Saito, T. Mukai, and H. Olesen, Semiconductor laser linewidth in optical feedback configurations, Electron. Lett. 19, 1026 (1983).
  * Kazarinov and Henry [1987] R. Kazarinov and C. H. Henry, The relation of line narrowing and chirp reduction resulting from the coupling of a semiconductor laser to passive resonator, IEEE J. Quantum Electron. 23, 1401 (1987).
  * Tromborg  _et al._ [1987] B. Tromborg, H. Olesen, X. Pan, and S. Saito, Transmission line description of optical feedback and injection locking for Fabry–Perot and DFB lasers, IEEE J. Quantum Electron. 23, 1875 (1987).
  * Amann and Schimpe [1990] M.-C. Amann and R. Schimpe, Excess linewidth broadening in wavelength-tunable laser diodes, Electron. Lett. 26, 279 (1990).
  * Olesen  _et al._ [1992] H. Olesen, B. Tromborg, H. Lassen, and X. Pan, Mode instability and linewidth rebroadening in DFB lasers, Electron. Lett. 28, 444 (1992).
  * Schatz [1992] R. Schatz, Longitudinal spatial instability in symmetric semiconductor lasers due to spatial hole burning, IEEE J. Quantum Electron. 28, 1443 (1992).
  * Olesen  _et al._ [1993] H. Olesen, B. Tromborg, X. Pan, and H. Lassen, Stability and dynamic properties of multi-electrode laser diodes using a Green’s function approach, IEEE J. Quantum Electron. 29, 2282 (1993).
  * Tromborg  _et al._ [1994] B. Tromborg, H. E. Lassen, and H. Olesen, Traveling wave analysis of semiconductor lasers: modulation responses, mode stability and quantum mechanical treatment of noise spectra, IEEE J. Quantum Electron. 30, 939 (1994).
  * Lax [1966a] M. Lax, Classical noise IV: Langevin methods, Rev. Mod. Phys 38, 541 (1966a).
  * Lax [1966b] M. Lax, Quantum noise. IV. Quantum theory of noise sources, Phys. Rev. 145, 110 (1966b).
  * Marani and Lax [1995] R. Marani and M. Lax, Spontaneous emission in non-Hermitian optical systems: Distributed-feedback semiconductor lasers, Phys. Rev. A 52, 2376 (1995).
  * Henry and Kazarinov [1996] C. H. Henry and R. F. Kazarinov, Quantum noise in photonics, Rev. Mod. Phys. 68, 801 (1996).
  * Henry [1986a] C. H. Henry, Theory of spontaneous emission noise in open resonators and its application to lasers and optical amplifiers, J. Lightwave Technol. 4, 288 (1986a).
  * Lasher and Stern [1964] G. Lasher and F. Stern, Spontaneous and stimulated recombination radiation in semiconductors, Phys. Rev. 133, A553 (1964).
  * Agrawal and Dutta [1993] G. P. Agrawal and N. K. Dutta,  _Semiconductor Lasers_, 2nd ed. (Kluwer Academic Publishers, Boston, 1993).
  * Radziunas [2017] M. Radziunas, Traveling wave modeling of nonlinear dynamics in multisection laser diodes, in  _Handbook of Optoelectronic Device Modeling & Simulation Volume II: Lasers, Modulators, Photodetectors, Solar Cells, and Numerical Methods_ (CRC Press, 2017) pp. 153–182.
  * Sieber  _et al._ [1998] J. Sieber, U. Bandelow, H. Wenzel, M. Wolfrum, and H.-J. Wünsche, Travelling wave equations for semiconductor lasers with gain dispersion, WIAS Preprint 459, 1 (1998).
  * Joyce [1980] W. B. Joyce, Current-crowded carrier confinement in double-heterostructure lasers, J. Appl. Phys. 51, 2394 (1980).
  * Zeghuzi  _et al._ [2018] A. Zeghuzi, M. Radziunas, H. Wenzel, H.-J. Wünsche, U. Bandelow, and A. Knigge, Modeling of current spreading in high-power broad-area lasers and its impact on the lateral far field divergence, Proc. SPIE 10526, 105261H (2018).
  * Bandelow  _et al._ [1992] U. Bandelow, H. Wenzel, and H.-J. Wünsche, Influence of inhomogeneous injection on sidemode suppression in strongly coupled DFB semiconductor lasers, Electron. Lett. 28, 1324 (1992).
  * Lassen  _et al._ [1993] H. E. Lassen, H. Wenzel, and B. Tromborg, Influence of series resistance on modulation responses of DFB lasers, Electron. Lett. 29, 1124 (1993).
  * Ning  _et al._ [1997] C.-Z. Ning, R. Indik, and J. Moloney, Effective Bloch equations for semiconductor lasers and amplifiers, IEEE J. Quantum Electron. 33, 1543 (1997).
  * Bandelow  _et al._ [2001] U. Bandelow, M. Radziunas, J. Sieber, and M. Wolfrum, Impact of gain dispersion on the spatio-temporal dynamics of multisection lasers, IEEE J. Quantum Electron. 37, 183 (2001).
  * Krüger and Petermann [1988] U. Krüger and K. Petermann, The semiconductor laser linewidth due to the presence of side modes, IEEE J. Quantum Electron. 24, 2355 (1988).
  * Pan  _et al._ [1991] X. Pan, B. Tromborg, and H. Olesen, Linewidth rebroadening in DFB lasers due to weak side modes, IEEE Photonic Tech. L. 3, 112 (1991).
  * Gardiner [2009] C. Gardiner,  _Stochastic methods: A Handbook for the Natural and Social Sciences_ (Springer, Berlin, 2009).
  * Agrawal and Roy [1988] G. P. Agrawal and R. Roy, Effect of injection-current fluctuations on the spectral linewidth of semiconductor lasers, Phys. Rev. A 37, 2495 (1988).
  * Di Domenico  _et al._ [2010] G. Di Domenico, S. Schilt, and P. Thomann, Simple approach to the relation between laser frequency noise and laser line shape, Appl. Opt. 49, 4801 (2010).
  * Woods  _et al._ [2019] N. D. Woods, M. C. Payne, and P. J. Hasnip, Computing the self-consistent field in Kohn–Sham density functional theory, J. Phys.: Condens. Matter 31, 453001 (2019).
  * Landau and Lifshitz [1980] L. D. Landau and E. M. Lifshitz,  _Course of Theoretical Physics Vol. 5: Statistical Physics (Part 1)_ (Butterworth–Heinemann, Amsterdam, 1980).
  * Vahala and Yariv [1983] K. Vahala and A. Yariv, Semiclassical theory of noise in semiconductor lasers - part II, IEEE J. Quantum Electron. 19, 1102 (1983).
  * Henry [1986b] C. H. Henry, Phase noise in semiconductor lasers, J. Lightwave Technol. 4, 298 (1986b).
  * Baets  _et al._ [1993] R. G. Baets, K. David, and G. Morthier, On the distinctive features of gain coupled DFB lasers and DFB lasers with second-order grating, IEEE J. Quantum Electron. 29, 1792 (1993).
  * Chow  _et al._ [1994] W. W. Chow, S. W. Koch, and M. I. Sargent,  _Semiconductor-Laser Physics_ (Springer Science & Business Media, Berlin, Heidelberg, 1994).
  * Osinski and Buus [1987] M. Osinski and J. Buus, Linewidth broadening factor in semiconductor lasers–an overview, IEEE J. Quantum Electron. 23, 9 (1987).
  * Champagne and McCarthy [1992] Y. Champagne and N. McCarthy, Global excess spontaneous emission factor of semiconductor lasers with axially varying characteristics, IEEE J. Quantum Electron. 28, 128 (1992).
  * Wenzel and Wünsche [1994] H. Wenzel and H.-J. Wünsche, An equation for the amplitudes of the modes in semiconductor lasers, IEEE J. Quantum Electron. 30, 2073 (1994).
  * Pick  _et al._ [2015] A. Pick, A. Cerjan, D. Liu, A. W. Rodriguez, A. D. Stone, Y. D. Chong, and S. G. Johnson, Ab initio multimode linewidth theory for arbitrary inhomogeneous laser cavities, Phys. Rev. A 91, 063806 (2015).
  * Komljenovic  _et al._ [2015] T. Komljenovic, S. Srinivasan, E. Norberg, M. Davenport, G. Fish, and J. E. Bowers, Widely tunable narrow-linewidth monolithically integrated external-cavity semiconductor lasers, IEEE J. Sel. Top. Quantum Electron. 21, 214 (2015).
  * Tran  _et al._ [2019] M. A. Tran, D. Huang, and J. E. Bowers, Tutorial on narrow linewidth tunable semiconductor lasers using Si/III-V heterogeneous integration, APL photonics 4, 111101 (2019).
  * Boller  _et al._ [2020] K.-J. Boller, A. van Rees, Y. Fan, J. Mak, R. E. Lammerink, C. A. Franken, P. J. van der Slot, D. A. Marpaung, C. Fallnich, J. P. Epping, R. M. Oldenbeuving, D. Geskus, R. Dekker, I. Visscher, R. Grootjans, C. G. H. Roeloffzen, M. Hoekman, E. J. Klein, A. Leinse, and R. G. Heideman, Hybrid integrated semiconductor lasers with silicon nitride feedback circuits, Photonics 7, 4 (2020).
  * Coldren  _et al._ [2012] L. A. Coldren, S. W. Corzine, and M. L. Mashanovitch,  _Diode lasers and photonic integrated circuits_ (John Wiley & Sons, Hoboken, New Jersey, 2012).
  * Spießberger  _et al._ [2011] S. Spießberger, M. Schiemangk, A. Wicht, H. Wenzel, G. Erbert, and G. Tränkle, DBR laser diodes emitting near 1064 nm with a narrow intrinsic linewidth of 2 kHz, Appl. Phys. B 104, 813 (2011).
  * Wenzel  _et al._ [1999] H. Wenzel, G. Erbert, and P. M. Enders, Improved theory of the refractive-index change in quantum-well lasers, IEEE J. Sel. Top. Quantum Electron. 5, 637 (1999).
  * Wenzel  _et al._ [2000] H. Wenzel, A. Klehr, G. Erbert, J. Sebastian, G. Tränkle, and M. Pereira Jr, Effect of band gap renormalization on threshold current and efficiency of a distributed Bragg reflector laser, Appl. Phys. Lett. 76, 2653 (2000).
  * Brox  _et al._ [2021] O. Brox, H. Wenzel, J. Fricke, P. Della Casa, A. Maaßdorf, M. Matalla, S. Wenzel, A. Wicht, and A. Knigge, Novel 1064 nm DBR lasers combining active layer removal and surface gratings, Electronics Letters 10.1049/ell2.12192 (2021).
  * Schiemangk  _et al._ [2014] M. Schiemangk, S. Spießberger, A. Wicht, G. Erbert, G. Tränkle, and A. Peters, Accurate frequency noise measurement of free-running lasers, Appl. Opt. 53, 7138 (2014).
  * Batrak  _et al._ [2005] D. V. Batrak, B. S. A, A. V. Borodaenko, A. E. Drakin, and A. P. Bogatov, Simulation of the material gain in quantum-well InGaAs layers used in 1.06-$\mu$m heterolasers, Quantum Electron. 35, 316 (2005).
  * [63] M. Radziunas, Longitudinal dynamics in multisection semiconductor lasers.
  * Patzak  _et al._ [1985] E. Patzak, P. Meissner, and D. Yevick, An analysis of the linewidth and spectral behavior of DBR lasers, IEEE J. Quantum Electron. 21, 1318 (1985).
  * Chaciński and Schatz [2010] M. Chaciński and R. Schatz, Impact of losses in the Bragg section on the dynamics of detuned loaded DBR lasers, IEEE J. Quantum Electron. 46, 1360 (2010).
  * Jonsson  _et al._ [1996] B. Jonsson, A. J. Lowery, H. Olesen, and B. Tromborg, Instabilities and nonlinear L-I characteristics in complex-coupled DFB lasers with antiphase gain and index gratings, IEEE J. Quantum Electron. 32, 839 (1996).
  * Tronciu  _et al._ [2017] V. Tronciu, H. Wenzel, and H. J. Wünsche, Instabilities and bifurcations of a DFB laser frequency-stabilized by a high-finesse resonator, IEEE J. Quantum Electron. 53, 1 (2017).