# Optimizing the Cavity-Arm Ratio of V-Shaped Semiconductor Disk Lasers

Stefan Meinecke<EMAIL_ADDRESS>Institut für Theoretische Physik,
Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany  Kathy
Lüdge<EMAIL_ADDRESS>Technische Universität Ilmenau, Institut für
Physik, Weimarer Straße 25, 98693 Ilmenau, Germany

(September 3, 2024)

###### Abstract

Passively mode-locked semiconductor disk lasers have received tremendous
attention from both science and industry. Their relatively inexpensive
production combined with excellent pulse performance and great emission
wavelength flexibility make them suitable laser candidates for applications
ranging from frequency comb tomography to spectroscopy. However, due to the
interaction of the active medium dynamics and the device geometry, emission
instabilities occur at high pump powers and thereby limit their performance
potential. Hence, understanding those instabilities becomes critical for an
optimal laser design. Using a delay-differential equation model, we are able
to detect, understand, and classify three distinct instabilities that limit
the maximum achievable pump power for the fundamental mode-locking state and
link them to characteristic positive net-gain windows. We furthermore derive a
simple analytic approximation in order to quantitatively describe the
stability boundary. Our results enable us to predict the optimal laser cavity
configuration with respect to positive net-gain instabilities and are
therefore of great relevance for the future development of passively mode-
locking semiconductor disk lasers.

Semiconductor laser, mode-locking, frequency comb

##  I Introduction

Passively mode-locked semiconductor disk lasers generate regular sequences of
short optical pulses without requiring any external modulation [1]. The
research on and the engineering of such lasers has been subject to an
outstanding progress for the past decades [2, 3, 4]. Nowadays, such lasers
provide very competitive pulse performances, while offering flexible emission
wavelengths and repetition rates due to their modular laser cavity design and
the modern semiconductor band-gap engineering technologies [2]. Pulse peak
powers on the order of a few kilowatts [5], pulse widths as low as
$100\,\mathrm{fs}$ [6], and repetition rates ranging from $85\,\mathrm{MHz}$
[7] to $190\,\mathrm{GHz}$ [8, 9] have been reported. Moreover, the mature
semiconductor fabrication processes enable a relatively inexpensive production
of semiconductor disk lasers, which makes them a popular choice for
applications in frequency comb generation for metrology and spectroscopy [10,
11, 12, 13, 14, 15], super-continuum generation [16, 17, 3, 18], and two-
photon microscopy [19, 20]. Since all those applications benefit from ultra
short and high peak power pulses, the further optimization of passively mode-
locked semiconductor disk lasers is highly desirable and an ongoing quest in
both science and industry.

Typically, the achievable pulse peak power is limited by multi-pulse
instabilities that arise at larger pump powers due to the relatively short
upper state lifetimes of semiconductor gain media [21]. One approach to
overcome this issue is to construct elaborate multi-pass geometries [22, 7,
23, 24], which prevent the built-up of an excess inversion in the gain chip
and thus the formation of additional pulses. The complex interplay between the
gain medium dynamics and the laser cavity setup yields a fundamental mode-
locking stability boundary, which characteristically depends on the geometric
cavity configuration. Understanding this dependency therefore becomes crucial
for an intelligent device design, which maximizes the laser performance.

In this manuscript, we consider the optimization of a V-shaped external
cavity, which represents one of the simplest multi-pass geometries. As
sketched in Fig. 1 (a), the output-coupler, gain chip, and absorber chip are
arranged in V-shape, where the gain chip sits in the middle. Along one round
trip through the cavity, the electric field therefore interacts twice with the
gain chip and only once with the other two components. The passive free space
in between the optical elements results in the field propagation time
$\tau_{1}$ between the output coupler and the gain chip and the field
propagation time $\tau_{2}$ between the gain chip and the absorber chip. Such
setups have been proven popular with semiconductor gain media and have thus
been realized experimentally many times with examples given in Refs. [25, 6,
5, 3, 26, 27]. The particular device investigated in this manuscript is based
on quantum-well active media and has been presented and characterized in
detail in Refs. [3, 4].

We investigate the impact of the geometric cavity configuration, i.e., the
cavity-arm ratio, by the means of a minimalistic delay-differential equation
model. The direct integration of the equations reveals the characteristic
stability boundary of the, for applications relevant, fundamental mode-locking
state. Moreover, the analysis of the temporal dynamics yields three distinct
destabilizing mechanisms, which each can be fully understood in terms of the
net-gain dynamics. We then use this insight to derive a simple analytic
approximation for the net gain, which enables us to predict the stability
boundary and thus to study the dependency on the other laser parameters. Our
results allow us to predict the optimal cavity-arm ratio for passively mode-
locked V-shaped lasers with respect to positive net-gain instabilities in
terms of the maximum achievable pump power. Lastly, we discuss how our
approach can also be applied to more complex gain media.

Figure 1: (a) Sketch of the V-shaped external cavity laser. The setup contains
three optical components: A high reflectivity output coupler, a semiconductor
gain chip and a semiconductor saturable absorber chip. The two latter are
mounted on top of high reflectivity DBRs. The active region of gain chip is
indicated in green and the active region of the absorber in red. The lengths
of the cavity arms are denoted by their propagation times $\tau_{1}$ and
$\tau_{2}$. (b) Pulse shape and (c) pseudo space-time plot of the fundamental
mode-locking state.

##  II Delay-Differential Equation Model

To self-consistently describe the laser at the time scales required for a net-
gain stability analysis, we use a multiple-delay differential equation model,
which we have presented and derived in Refs. [4, 28, 29]. The model is
obtained from a traveling-wave framework, which takes the forward and backward
propagating light beams and the carrier dynamics inside the gain and absorber
chips into account. Integrating those quantities within the co-moving frame of
the field propagation and applying a Lorentzian filter to model the gain
spectrum yields delay-differential equations for the electric field $E$ at the
output coupler and the integrated inversion densities $G$ and $Q$ for the gain
and absorber chip [28, 29]:

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}{t}}E(t)=$ $\displaystyle-\gamma
E(t)+\gamma\sqrt{r_{\mathrm{OC}}}E(t-T)$ $\displaystyle\times
e^{\frac{1-i\alpha_{\mathrm{G}}}{2}[G(t-\tau_{1})+G(t-\tau_{1}-2\tau_{2})]}$
$\displaystyle\times e^{(1-i\alpha_{\mathrm{Q}})Q(t-\tau_{1}-\tau_{2})}$
$\displaystyle+\sqrt{D_{\mathrm{sp}}}\xi(t)$ (1)
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}{t}}G(t)=$
$\displaystyle-\gamma_{\mathrm{G}}G(t)+J_{\mathrm{G}}-\left(e^{G(t)}-1\right)$
$\displaystyle\times\bigg{[}\left|E(t-\tau_{1})\right|^{2}+\left|E(t-\tau_{1}-2\tau_{2})\right|^{2}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\times
e^{2Q(t-\tau_{2})}e^{G(t-2\tau_{2})}\bigg{]}$ (2)
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}{t}}Q(t)=$
$\displaystyle\gamma_{\mathrm{Q}}(Q_{0}-Q(t))$
$\displaystyle-s\left(e^{Q(t)}-1\right)\left|E(t-\tau_{1}-\tau_{2})\right|^{2}e^{G(t-\tau_{2})}.$
(3)

The geometric cavity configuration, i.e., the length of the cavity arms, is
then encoded via the propagation times $\tau_{1}$ and $\tau_{2}$ (s. Fig. 1),
which appear in the time-delayed terms. Within the system equations, $\gamma$
describes the bandwidth of the gain medium, $r_{\mathrm{OC}}$ the output-
coupler intensity reflectivity, $T$ the cold-cavity round-trip time,
$\alpha_{\mathrm{G}}$ and $\alpha_{\mathrm{Q}}$ the amplitude-phase coupling
coefficients of the gain and absorber chip, $\gamma_{G}$ the gain recovery
rate, $J_{\mathrm{G}}$ the gain-chip pump current, $\gamma_{Q}$ the absorber
recovery rate, $Q_{0}$ the unsaturated absorption and $s$ the ratio between
the gain and absorber chip differential gain coefficients. Spontaneous noise
is phenomenologically taken into account by a stochastic Langevin term in Eq.
(1), where $D_{\mathrm{sp}}$ describes the noise strength and $\xi(t)$ delta-
correlation Gaussian white noise.

Table 1: Simulation parameters normalized to the cold-cavity round-trip time $T=625\,\mathrm{ps}$ unless stated otherwise. Symbol | Value | Parameter
---|---|---
$\gamma$ | 10000 | gain bandwidth
$r_{\mathrm{OC}}$ | 0.99 | output-coupler reflectivity
$\alpha_{\mathrm{G,Q}}$ | 0.0 | amplitude-phase coupling coefficient
$\tau_{1}$ | 0.25 | output-coupler - gain propagation time
$\tau_{2}$ | 0.25 | gain - absorber propagation time
$D_{\mathrm{sp}}$ | 0.1 | spontaneous emission noise strength
$J_{\mathrm{G}}$ | 0.04 | pump current
$\gamma_{\mathrm{G}}$ | 0.625 | gain recovery rate
$Q_{0}$ | -0.03 | unsaturated absorption
$\gamma_{\mathrm{Q}}$ | 200.0 | absorber recovery rate
$s$ | 2.0 | differential gain ratio

The model parameters are chosen to describe the laser investigated in [3, 4]
and are given in Tab. 1. However, we deliberately chose a narrower gain
bandwidth $\gamma$ to stay within the slowly-varying envelope approximation
and to enable fast simulation times. The resulting pulse widths are of the
order of $\sim 1\,\mathrm{ps}$ and thus still much shorter than the round-trip
time $T$. Note that we set the amplitude-phase coupling coefficients to be
zero, since the underlying approximations have been shown to break down at
fast and high-power excitations [30, 31, 32], while the simulation of mode-
locked lasers has been shown to be successful with vanishing amplitude-phase
coupling [33, 34, 35]. Nonetheless, we point out that our model at the chosen
parameters does not include the phase dynamics of the electric field and
subsequently can not describe any phase related instabilities. To investigate
those, one should resort to more elaborate models [36, 37, 38, 39, 40, 41],
which, however, come at the cost of increased computational demands.

To integrate the system equations, we utilize an explicit fourth-order Runge-
Kutta algorithm [42] with cubic Hermite interpolation for the delay terms. The
stochastic term in Eq. (1) is small at the chosen parameters and can thus be
integrated alongside the deterministic equations via a simple Euler method.

##  III Results

###  III.1 Net-Gain Instabilities of the Fundamental Mode-Locking Regime

Figure 2: Map of the emission states in the pump current and gain-chip
position parameter space $(J_{\mathrm{G}},\tau_{1})$. (a) Numerically obtained
emission state states with color-codes indicated on top. FML and HML denote
fundamental and harmonic mode-locking and the subscripts LEI and PS denote
leading-edge instability and pulse switching. Corresponding to the three
instabilities, purple, dark green, and light green lines indicate the critical
pump current $J_{\mathrm{crit}}$ obtained from the net-gain approximation Eq.
(9) with the effective pulse width $\Delta_{\mathrm{pw}}=0.03$. Black circles
indicate the parameter combinations used in Fig. 3 and red squares indicate
the parameter combinations used in Fig. 4. (b) Branches of the corresponding
gain-recovery intervals $\Delta_{1},\Delta_{2}$ (solid and dashed,
respectively). Horizontal black dotted lines indicate $\tau_{1}=T/6$ and
$\tau_{1}=2T/6$.

The V-shaped semiconductors disk laser’s underlying bifurcation structure of
the relevant lasing solutions, albeit at broader pulses and larger gain
parameters, has been published in Ref. [28]. The results reveal that the
fundamental mode-locking state, which is the focus of this work, is born from
a Andronov-Hopf bifurcation of the cw lasing state, similar to the cases of
ring cavities [43, 44]. This bifurcation occurs very close to the lasing
threshold, which can be determined analytically and reads

$\displaystyle
J_{\mathrm{th}}=\gamma_{\mathrm{G}}\left[\frac{1}{2}\ln{\left(\frac{\omega^{2}}{\gamma^{2}}+1\right)}-\frac{1}{2}\ln{\left(\kappa\right)}-Q_{0}\right],$
(4)

where $\omega$ denotes the maximum gain mode [28]. Note that the threshold
does not depend on the cavity configuration $\tau_{1},\tau_{2}$. The dynamics
of the out-coupled field in the fundamental mode-locking state are exemplified
in Fig. 1, where (b) shows the pulse shape and (c) a color-coded pseudo space-
time plot. The smooth pulse shape remains stable from round trip to round
trip, which demonstrates the stability of the chosen operation point.

To further investigate the impact of the cavity configuration on the
fundamental mode-locking state, we numerically integrate the laser equations
in the parameter space of the pump current and the cavity configuration. On
that account, the overall cavity length, i.e., the cold-cavity round-trip time
$T$, is kept constant while the ratio between the two cavity arms is changed.
The geometric configuration is then specified in terms of the propagation time
$\tau_{1}$, while the propagation time $\tau_{2}$ follows from the condition

$\displaystyle 2\tau_{1}+2\tau_{2}=T,$ (5)

where $T$ remains fixed as aforementioned. The propagation times thus take
values $\tau_{1},\tau_{2}\in[0.0T,0.5T]$, where on the extremes, the gain chip
is either located at the output coupler or at the absorber chip. These cases
are not directly experimentally realizable with a V-shaped external cavity
since the optical components take up some space by themselves. The
intermediate configurations, however, can be easily achieved on an optical
table by positioning the components accordingly.

The results are presented in Fig. 2 (a), where the various lasing states are
color-coded with the labels indicated on the top. While higher order states
also exist, we want to focus our analysis in this manuscript onto the
fundamental state and its instabilities. Fundamental mode-locking (FML, blue)
can be observed for all cavity configurations. As anticipated via the
analytical threshold condition Eq. (4), the gain-chip position neither affects
the lasing threshold nor the lower FML boundary. The upper pump current
stability boundary, on the other hand, critically depends on the cavity
configuration $\tau_{1}$. With the gain-chip located at the output coupler,
i.e., $\tau_{1}=0.0$, the upper FML stability boundary is found at
$J_{\mathrm{G}}\approx 0.048$ and pulse-switching unstable fundamental mode-
locking (FMLPS, dark purple) is found beyond the boundary. For an increasing
gain-chip position $\tau_{1}$, the upper FML stability boundary shifts towards
larger pump currents until it reaches a local maximum at
$J_{\mathrm{G}}\approx 0.088$ for the configuration $\tau_{1}=1/6$. From there
on, the upper stability boundary marginally reduces to the local minimum
$J_{\mathrm{G}}\approx 0.086$ at the symmetric configuration $\tau_{1}=0.25$.
For cavity configurations $1/6<\tau_{1}<0.25$, leading-edge unstable
fundamental mode-locking (FMLLEI, dark green) is observed beyond the stability
boundary instead of FMLPS. That is with the exception of the symmetric cavity
configuration $\tau_{1}=0.25$, where leading-edge unstable second-order
harmonic mode-locking (HML2LEI, light green) can be found beyond the stability
boundary. The remaining cavity configurations $\tau_{1}\in\,]0.25,0.5]$
exhibit an apparent reflection symmetry with respect to $\tau_{1}=0.25$, which
indicates that the output coupler and the absorber chip have roughly similar
effects on the pulse shaping in fundamental mode-locking operation.

In conclusion, the cavity configuration has profound consequences on the
possible fundamental mode-locking operation conditions. At the configurations
$\tau_{1}=1/6$ and $\tau_{1}=2/6$, the range of available pump currents is
$\approx 3.2$ larger than it is at $\tau_{1}=0.0$ and $\tau_{1}=0.5$, which
moreover translates into maximally achievable pulse energies, which are
$\approx 4.8$ times larger.

Figure 3: Pseudo space-time plots of the normalized intensity $|E|^{2}$ (top
row) and the net-gain $\mathcal{G}$ dynamics (bottom row). From left to right:
pulse-switching unstable fundamental mode-locking ((a), FMLPS,
$J_{\mathrm{G}}=0.07$, $\tau_{1}=0.1$), leading-edge unstable fundamental
mode-locking ((b), FMLLEI, $J_{\mathrm{G}}=0.095$, $\tau_{1}=0.2$), and
leading-edge unstable second-order harmonic mode-locking ((c), HML2LEI,
$J_{G}=0.095$, $\tau_{1}=0.25$). The three parameter combinations are
indicated by black circles in Fig. 2.

To unravel the mechanism, which leads to the characteristic fundamental mode-
locking region, we take a closer look at the three different instabilities
that can be observed beyond the stability boundary. Those are illustrated in
Fig. 3, which shows color-coded pseudo space-time plots of the normalized
intensity $|E|^{2}$ (top row) and the net gain $\mathcal{G}$ (bottom row). The
columns from left to right present pulse-switching unstable fundamental mode-
locking (FMLPS), leading-edge unstable fundamental mode-locking (FMLLEI), and
leading-edge unstable second-order harmonic mode-locking (HML2LEI).

The net gain [45] measures the intensity amplification or attenuation, which a
small perturbation experiences per round trip. Evaluated for the V-shape laser
model, the net gain contains the individual contributions of the various
optical elements, which are picked up along the propagation and thus reads

$\displaystyle\mathcal{G}(t)=$ $\displaystyle
G(t-\tau_{1})+G(t-\tau_{1}-2\tau_{2})$
$\displaystyle+2Q(t-\tau_{1}-\tau_{2})+\mathrm{ln}(\kappa).$ (6)

Positive net-gain, which does not coincide with the emission of a pulse,
implies that perturbations can grow and thus may destabilize a regular mode-
locking state. Note, however, that the net gain only represents a small signal
measure that describes the short-term evolution of a perturbation that sits on
top of existing lasing dynamics. Non-linear effects, e.g., mediated by the
group-velocity dispersion, may allow mode-locking states to be stable despite
positive net-gain windows [43, 46]. Nevertheless, the net gain will turn out
to be the critical quantity to describe the observed instabilities of the
fundamental mode-locking state.

Pulse-switching unstable fundamental mode-locking (FMLPS) is characterized by
pulse trains which only exist for a finite number of round trips before they
vanish and a new pulse train is born at a different position within the
cavity. As can be seen in Fig. 3 (a2), the birth of a new pulse train is
preceded by a broad positive net-gain window (white areas with a slight red
tint). The new pulse then emerges at the trailing edge of the net-gain window.
As soon as the new pulse train has achieved sufficient power to effectively
bleach the gain and absorber chip, it causes the positive net-gain window to
shrink and only overlap with the pulse. Shortly after the new pulse train is
established, a new broad net-gain window establishes at different cavity
position as well. Hence, the switching process regularly repeats and thereby
prevents stable fundamental mode-locking emission.

Leading-edge unstable fundamental mode-locking (FMLLEI) is characterized by a
single pulse train that exhibits a broad positive net-gain window at the
leading edge of the pulse (s. Fig. 3 (b2)). Hence, noise induced perturbations
at the leading edge can be amplified and thus destabilize the pulse train.
This produces pronounced fluctuations of the pulse peak intensities and of the
pulse positions.

Lastly, leading-edge unstable second-order harmonic mode locking (HML2LEI) is
characterized by two unstable and equidistant pulse trains. As shown in Fig. 3
(c2), both pulses are accompanied by a large leading positive net-gain window,
which continuously causes noise induced perturbations to grow and to
destabilize the existing pulse trains. Being observed for a symmetric cavity
configuration, the HML2LEI state suffers from an additional destabilizing
mechanism: The equidistant pulses collide in the gain chip, which tightly
couples them via the interaction with the gain and thereby transfers
instabilities from one pulse to the other.

In conclusion, the three different destabilizing mechanisms of the fundamental
mode-locking state, which appear for different cavity configurations, are
caused by characteristic positive net-gain windows. Those allow spontaneous
emission noise to be amplified and consequently destabilize the regular mode-
locking dynamics. We will see in the next section that this insight allows for
an analytic treatment.

###  III.2 Analytic Net-Gain Approximation

To understand the shape of the stability boundary in the
($\tau_{1},J_{\mathrm{G}}$) parameter plane (s. Fig. 2), we make further use
of the net gain and derive an analytic approximation for the critical pump
current of the three different instabilities. The net gain $\mathcal{G}$ (s.
Eq. (6)) dynamically depends on the integrated gain $G$ and absorption $Q$. In
the absence of optical pulses, both quantities, however, exhibit rather simple
exponential relaxations. This occurs very quickly in the fast absorber chip,
such that $Q(t)$ can be approximated by its unsaturated equilibrium value
$Q_{0}$. Assuming a complete saturation of the gain by the optical pulses,
i.e., $G\approx 0.0$, the gain recovery can be analytically integrated and the
result reads

$\displaystyle
G(\Delta)=\frac{J_{\mathrm{G}}}{\gamma_{\mathrm{G}}}\left(1-\exp(-\gamma_{\mathrm{G}}\Delta)\right),$
(7)

where $\Delta$ denotes the time interval starting from the interaction of the
pulse and the gain. The net gain, in the absence of pulse interactions with
the gain and absorber chip can then be written as

$\displaystyle\mathcal{G}(\Delta_{1},\Delta_{2})=G(\Delta_{1})+G(\Delta_{2})+2Q_{0}+\mathrm{ln}(r_{\mathrm{OC}}),$
(8)

where $\Delta_{1}$ and $\Delta_{2}$ denote the available gain-recovery time-
intervals, which result from the last gain-chip interaction with the
respective forward and backward travelling high-power pulse. It is important
to note that $\Delta_{1}$ and $\Delta_{2}$ depend on the emissions state and
the cavity geometry as encoded by the gain-chip position $\tau_{1}$. The
previously introduced instabilities occur due to the appearance of distinct
positive net-gain windows. This implies that our net-gain approximation in the
absence of optical pulses Eq. (8) transitions to positive values for recovery
intervals $\Delta_{1},\Delta_{2}$, which are characteristic for the
instability and the cavity configuration. Hence, Eq. (8) is set to zero and
solved for the critical pump current, which yields the expression

$\displaystyle
J_{G}^{\mathrm{crit}}=\frac{\gamma_{\mathrm{G}}\left(-\mathrm{ln}(r_{\mathrm{OC}})-2Q_{0}\right)}{2-e^{-\gamma_{\mathrm{G}}[\Delta_{1}-\Delta_{\mathrm{pw}}]}-e^{-\gamma_{\mathrm{G}}[\Delta_{2}-\Delta_{\mathrm{pw}}]}},$
(9)

where the recovery intervals $\Delta_{1},\Delta_{2}$ are yet to be determined.
Additionally, the recovery times have been adjusted for the finite pulse width
$\Delta_{\mathrm{pw}}$, during which the electric field is not negligible and
thus no free gain recovery occurs.

Figure 4: Intensity and carrier dynamics of the fundamental mode-locking state
close to the (a) pulse-switching instability ($J_{\mathrm{G}}=0.0675$,
$\tau_{1}=0.1$), (b) leading-edge instability ($J_{\mathrm{G}}=0.0905$,
$\tau_{1}=0.2$), and (c) second-order harmonic mode-locking leading-edge
instability ($J_{\mathrm{G}}=0.0880$, $\tau_{1}=0.25$). Subfigures (a1) to
(c1) show the temporal evolution of the intensity (blue lines), the gain
(green lines), and the absorption (red lines), which have been normalized to
$0.1$. Their respective zero positions (black dotted lines) have been shifted
proportional to their cavity positions. The propagation of the pulses is
illustrated by blue dashed lines and the propagation of the semistable
perturbations by brown dashed lines. (a2) to (c2) plot the respective net gain
$\mathcal{G}$. Black circles indicate semistable ($\mathcal{G}\approx 0$) net
gain windows. The three parameter combinations are indicated by red squares in
Fig. 2.

Each of the three instabilities produces an independent branch of recovery
intervals $(\Delta_{1},\Delta_{2})$ that lead to positive net-gain windows. To
identify the respective recovery intervals, the three different instabilities
at their respective critical pump current $J_{\mathrm{G}}^{\mathrm{crit}}$ are
further illustrated in Fig. 4. The columns from left to right show the pulse-
switching instability (a), the leading-edge instability (b), and the second-
order harmonic mode-locking leading-edge instability (c). The top and bottom
row present the normalized intensity $|E|^{2}$ (blue), the gain $G$ (green),
and the absorber $Q$ (red) dynamics ((a1) to (c1)) and the corresponding net-
gain $\mathcal{G}$ dynamics ((a2) to (c2)). The zero baselines of the gain and
absorption have been shifted proportional to their cavity position with
respect to the output coupler and are indicated via horizontal black dotted
lines. The field propagation through the laser cavity is illustrated via blue
dashed lines. The intersections with the black dotted lines thus represent the
interactions with the gain and absorber chips. Additionally, black circles in
(a2) to (c2) indicate semistable ($\mathcal{G}\approx 0$) net-gain windows for
which perturbations are neither amplified nor attenuated. The propagation of
those perturbations is further indicated by brown dashed lines in (a1) to
(c1). Note that the perturbations are considered to be small and thus leave no
characteristic footprint on the gain dynamics, i.e., do not affect the gain
recovery. On the other hand, the gain (green lines) is fully bleached by the
optical pulses.

The recovery intervals $(\Delta_{1},\Delta_{2})$ can be identified as the time
differences between a gain-chip interaction with the perturbation and the
preceding high-power pulse. Crucially, the considered noise induced
perturbations are small when they are generated, such that they can
potentially be amplified twice by the same gain per round trip without causing
any meaningful gain saturation.

The pulse-switching instability (FMLPS) directly benefits from this mechanism.
As Fig. 4 (a1) exemplifies, the backward-moving perturbation first passes the
gain chip, is reflected at the output coupler, and then passes the gain chip
in the forward direction right before the existing backward-moving pulse
bleaches the gain. The associated recovery intervals read

$\displaystyle\Delta_{1}=4\left|\tau_{1}-\frac{1}{4}\right|,\,\,\,\,\,\Delta_{2}=\frac{1}{2}+2\left|\tau_{1}-\frac{1}{4}\right|.$
(10)

Note that the perturbation is amplified twice by the long gain-recovery, which
leads to $\Delta_{1},\Delta_{2}\approx 1$ for configurations with
$\tau_{1}\approx 0.0$ and $\tau_{1}\approx 0.5$.

The leading-edge instability (FMLLEI), as illustrated in Fig. 4 (b1), is
produced by a perturbation that travels right in front of the existing pulse.
Hence, the characteristic recovery intervals correspond to the propagation
times that determine the pulse amplification in Eq. (1) and read

$\displaystyle\Delta_{1,2}=\frac{1}{2}\pm 2\left|\tau_{1}-\frac{1}{4}\right|$
(11)

Lastly, the second-order leading-edge instability (HML2LEI) is generated by a
perturbation that reaches the gain chip in the backward direction right before
the existing forward-moving pulse saturates the gain, as shown in Fig. 4 (c1).
This produces identical recovery intervals

$\displaystyle\Delta_{1,2}$
$\displaystyle=\frac{1}{2}-2\left|\tau_{1}-\frac{1}{4}\right|.$ (12)

Note that both recovery intervals drop to zero for the configurations
$\tau_{1}=0.0$ and $\tau_{1}=0.5$, which causes Eq. (9) to diverge.

The three branches of gain-recovery intervals $\Delta_{1,2}$ are plotted as
functions of the cavity configuration $\tau_{1}$ in Fig. 2 (b), where solid
lines denote $\Delta_{1}$ and dashed lines denote $\Delta_{2}$. Purple
represents the pulse-switching instability (FMLPS), dark green the leading-
edge instability (FMLLEI), and light green the second-order leading-edge
instability (HML2LEI). The corresponding critical pump currents
$J_{\mathrm{G}}^{\mathrm{crit}}$ are computed via Eq. (9) and plotted in Fig.
2 (a).

As demonstrated by the comparison to the underlying numerically obtained
emission states, the analytic approximation Eq. (9) excellently predicts the
fundamental mode-locking upper stability boundary as well as the respective
destabilizing mechanism that appears beyond the critical pump current. On that
account, fundamental mode-locking stability is determined by the lowest
critical pump current $J_{\mathrm{G}}^{\mathrm{crit}}$ among the three
instabilities. For the rather asymmetric configurations, i.e., $\tau_{1}<1/6$
and $\tau_{1}>2/6$, this turns out to be the pulse-switching instability
(purple line). According to Fig. 2 (b), this can be attributed to the longest
available recovery intervals $\Delta_{1}$ and $\Delta_{2}$, which already
enable smaller pump currents to achieve positive net-gain windows.

For rather symmetric configurations $1/6<\tau_{1}<2/6$, the leading-edge
instability (dark green lines in Fig. 2) provides the longest recovery
intervals $(\Delta_{1},\Delta_{2})$ and thus first destabilizes the
fundamental mode-locking state. The pump current stability boundary maxima at
$\tau_{1}=1/6$ and $\tau_{1}=2/6$ are generated by the intersection of the
pulse-switching (purple line) and leading-edge instability (dark green line).
At those points, the device geometry yields identical recovery intervals
$\Delta_{1,2}$ for both instabilities (s. Fig. 2 (b)). Moreover, the leading-
edge instability exhibits a global minimum for the symmetric cavity
$\tau_{1}=0.25$, despite the constant sum $\Delta_{1}+\Delta_{2}=1$ of the two
recovery intervals. This feature is caused by the concave downward recovery of
the gain $G$, which is more efficient, i.e., faster, for shorter recovery
intervals and thus requires a smaller critical pump current to produce
positive net-gain windows.

Lastly, the second-order leading-edge instability (light green lines) becomes
only relevant for the symmetric cavity configuration $\tau_{1}=0.25$, where
the recovery intervals $(\Delta_{1},\Delta_{2})$ are identical to the leading-
edge instability. For all other configurations, the critical pump current
quickly grows and diverges towards $\tau_{1}=0.0$ and $\tau_{1}=0.5$. Hence,
the leading-edge unstable second-order harmonic mode-locking state HML2LEI can
only be found right beyond the stability boundary for the symmetric
configuration as well as in a small cone, which grows in the
$(J_{\mathrm{G}},\tau_{1})$ parameter space for further increasing pump
currents $J_{\mathrm{G}}$.

Figure 5: Critical pump current $J_{\mathrm{G}}^{\mathrm{crit}}$ (upper
stability boundary) of the fundamental mode-locking state in the pump current
and gain-chip position parameter space $(J_{\mathrm{G}},\tau_{1})$. Subfigure
(a) plots $J_{\mathrm{G}}^{\mathrm{crit}}$ for a geometrically increasing
gain-chip recovery rate $\gamma_{\mathrm{G}}\in\\{0.185,\ldots,2.11\\}$.
Subfigure (b) plots $J_{\mathrm{G}}^{\mathrm{crit}}$ for a linearly increasing
pulse width $\Delta_{\mathrm{pw}}\in\\{0.0,\ldots,0.12\\}$.

In summary, the presented results demonstrate that the stability boundaries of
the fundamental mode-locking state can be entirely understood in terms of the
net-gain dynamics and can be analytically predicted via a simple gain-recovery
approximation. Lastly, we now utilize this approximation to study the relevant
parameter dependencies of the upper FML stability boundary. The critical pump
current $J_{G}^{\mathrm{crit}}$ given in Eq. (9) linearly scales with the
logarithm of the out-coupling losses $r_{\mathrm{OC}}$ and the unsaturated
absorption $Q_{0}$ and nonlinearly with the gain recovery rate
$\gamma_{\mathrm{G}}$ and the effective pulse width $\Delta_{\mathrm{pw}}$.
The influence of the two latter in the pump current and gain-chip position
parameter space $(J_{\mathrm{G}},\tau_{1})$ is presented in Fig. 5. The
critical pump current $J_{G}^{\mathrm{crit}}$ is determined as the minimum
among the three destabilizing mechanisms and seven representative stability
boundaries are plotted, respectively.

The gain-recovery rate $\gamma_{\mathrm{G}}$, shown in Fig. 5 (a), is
increased geometrically (constant multiplication factor) from
$\gamma_{\mathrm{G}}=0.185$ to $\gamma_{\mathrm{G}}=2.11$. The increasing
recovery rate most dominantly shifts the critical pump current to larger
values due to the linear factor of $\gamma_{\mathrm{G}}$ in the numerator of
Eq. (9). Secondly, a fast (large) gain-recovery rate produces an effectively
’more concave’ gain recovery via the contributions in the exponentials in the
denominator of Eq. (9). This leads to more pronounced local maxima and a
stronger curvature of the stability boundary at fast recovery rates. At small
rates, the recovery is slow and effectively linear at the available recovery
intervals $\Delta_{1},\Delta_{2}$, which leads to a leading-edge instability
boundary that is almost constant between gain-chip positions
$1/6\leq\tau_{1}\leq 2/6$.

The effective pulse width $\Delta_{\mathrm{pw}}$, shown in Fig. 5 (b), is
increased linearly with equidistant steps from $\Delta_{\mathrm{pw}}=0.0$ to
$\Delta_{\mathrm{pw}}=0.12$. The effective pulse width only appears in the
exponentials in Eq. (9) and reduces the effective gain recovery intervals
$\Delta_{1}$ and $\Delta_{2}$. Hence, it increases the pump current
$J_{\mathrm{G}}$, which is needed to achieve positive net gain. The effect is
nonlinear and, similarly to the gain-recovery rate, leads to a more pronounced
curvature and stronger local maxima at larger pulse widths (light green curve
to the right in (b)).

In conclusion, the nonlinear dependencies of the critical pump current
determine the curvature of the stability boundary. Nonetheless, the local
maxima are always located at $\tau_{1}\approx 1/6$ and $\tau_{1}\approx 2/6$.
However, their relative advantages in terms of the pump current tuning range
with respect to neighboring cavity configurations increases for a stronger
nonlinearity of the gain-recovery process. Note that the relative nonlinearity
of the gain recovery can also be controlled via the scaling of the cavity
size, since the presented discussion was performed with parameters normalized
to the cold-cavity round-trip time.

##  IV Discussion

We have demonstrated that the stability of the fundamental mode-locking state
of a V-shaped semiconductor disk laser can be qualitatively and quantitatively
understood in terms of its net-gain dynamics. In particular, the maximum
achievable pump current is limited by positive net-gain instabilities, which,
depending on the geometric cavity configuration, manifests either as a pulse
switching, leading edge, or second-order leading-edge instability. The
corresponding stability boundaries of those three destabilizing mechanisms can
be analytically predicted by solving the gain recovery in the absence of
optical pulses and using the results to construct the net gain as a function
of the available gain-recovery intervals.

We have associated the three instabilities each with a branch of gain-recovery
times $(\Delta_{1},\Delta_{2})$. Those depend on the cavity-arm ratio and
thereby determine, which instability occurs at the lowest pump current and
thus limits the maximum pump current. The branches $(\Delta_{1},\Delta_{2})$
intersect at characteristic cavity configurations, which in turn causes the
corresponding stability boundaries to also intersect. This produces the
characteristic region of stable FML emission in the pump current and cavity
configuration parameter space, which exhibits the optimal pump current tuning
range for the cavity-arm ratios $\tau_{1}/\tau_{2}=1/2$ and
$\tau_{1}/\tau_{2}=2/1$. We have furthermore observed that the region of
stable FML emission retains its qualitative features, i.e., its shape, for all
changes of the relevant laser parameters. We attribute those results to the
concave downward gain-recovery process and the cavity configuration, which
determines the available gain-recovery intervals.

For that reason, the qualitative shape of the stable fundamental mode-locking
region should generalize to any gain-chip active medium, which exhibits a
concave downward recovery process. Hence, we predict that the largest pump
current tuning range should always be observed at the characteristic cavity-
arm ratios $\tau_{1}/\tau_{2}=1/2$ and $\tau_{1}/\tau_{2}=2/1$. However, for
more complex active media, the gain dynamics can likely not be solved
analytically, which would prevent a simple approximation of the stability
boundaries. Nevertheless, to obtain quantitative predictions, the stability
boundary condition ($\mathcal{G}\approx 0$) could be formulated as an
ordinary-differential equation boundary problem. The numerical solution of
this problem should be always much cheaper than the simulation of the complete
spatio-temporal laser dynamics. This approach could be easily implemented to
calculate the stability boundaries for gain-chips with quantum-dot [47, 48,
49, 50], submonolayer quantum-dot [32, 51, 52], and multi-level quantum-well
active media [53, 4].

##  V Achknowledgements

We thank Jan Hausen for fruitful discussions and pioneering the bifurcation
analysis of the V-shaped laser model.

##  VI Code Availability

The simulation code is available on GitHub under a MIT license
(https://github.com/stmeinecke/VShape)

## References

  * Keller [2003] U. Keller, Recent developments in compact ultrafast lasers, Nature 424, 831 (2003).
  * Tilma  _et al._ [2015] B. W. Tilma, M. Mangold, C. A. Zaugg, S. M. Link, D. Waldburger, A. Klenner, A. S. Mayer, E. Gini, M. Golling, and U. Keller, Recent advances in ultrafast semiconductor disk lasers, Light Sci. Appl. 4, https://doi.org/10.1038/lsa.2015.83 (2015).
  * Waldburger  _et al._ [2016] D. Waldburger, S. M. Link, M. Mangold, C. G. E. Alfieri, E. Gini, M. Golling, B. W. Tilma, and U. Keller, High-power 100 fs semiconductor disk lasers, Optica 3, 844 (2016).
  * Waldburger  _et al._ [2018] D. Waldburger, C. G. E. Alfieri, S. M. Link, S. Meinecke, L. C. Jaurigue, K. Lüdge, and U. Keller, Multipulse instabilities of a femtosecond SESAM-modelocked VECSEL, Opt. Express 26, 21872 (2018).
  * Wilcox  _et al._ [2013] K. G. Wilcox, A. C. Tropper, H. E. Beere, D. A. Ritchie, B. Kunert, B. Heinen, and W. Stolz, 4.35 kW peak power femtosecond pulse mode-locked VECSEL for supercontinuum generation, Opt. Express 21, 1599 (2013).
  * Klopp  _et al._ [2011] P. Klopp, U. Griebner, M. Zorn, and M. Weyers, Pulse repetition rate up to 92 GHz or pulse duration shorter than 110 fs from a mode-locked semiconductor disk laser, Appl. Phys. Lett. 98, 071103 (2011), https://doi.org/10.1063/1.3554751 .
  * Butkus  _et al._ [2013] M. Butkus, E. A. Viktorov, T. Erneux, G. Maker, G. P. A. Malcolm, and E. U. Rafailov, 85.7 MHz repetition rate mode-locked semiconductor disk laser: fundamental and soliton bound states, Opt. Express 21, 25526 (2013).
  * Mangold  _et al._ [2014] M. Mangold, S. M. Link, A. Klenner, C. A. Zaugg, M. Golling, B. W. Tilma, and U. Keller, Amplitude noise and timing jitter characterization of a high-power mode-locked integrated external-cavity surface emitting laser, IEEE Photonics J. 6, 1500309 (2014).
  * Gaafar  _et al._ [2016] M. A. Gaafar, A. Rahimi-Iman, K. A. Fedorova, W. Stolz, E. U. Rafailov, and M. Koch, Mode-locked semiconductor disk lasers, Adv. Opt. Photonics 8, 370 (2016).
  * Udem  _et al._ [2002] T. Udem, R. Holzwarth, and T. W. Hänsch, Optical frequency metrology, Nature 416, 233 (2002).
  * Keller and Tropper [2006] U. Keller and A. C. Tropper, Passively modelocked surface-emitting semiconductor lasers, Phys. Rep. 429, 67 (2006).
  * Klenner  _et al._ [2014] A. Klenner, S. Schilt, T. Südmeyer, and U. Keller, Gigahertz frequency comb from a diode-pumped solid-state laser, Opt. Express 22, 31008 (2014).
  * Coddington  _et al._ [2016] I. Coddington, N. Newbury, and W. Swann, Dual-comb spectroscopy, Optica 3, 414 (2016).
  * Link  _et al._ [2017a] S. M. Link, D. J. H. C. Maas, D. Waldburger, and U. Keller, Dual-comb spectroscopy of water vapor with a free-running semiconductor disk laser, Science 356, 1164 (2017a).
  * Diddams  _et al._ [2020] S. A. Diddams, K. Vahala, and T. Udem, Optical frequency combs: Coherently uniting the electromagnetic spectrum, Science 369, eaay3676 (2020).
  * Mayer  _et al._ [2015] A. S. Mayer, A. Klenner, A. R. Johnson, K. Luke, M. R. E. Lamont, Y. Okawachi, M. Lipson, A. L. Gaeta, and U. Keller, Frequency comb offset detection using supercontinuum generation in silicon nitride waveguides, Opt. Express 23, 15440 (2015).
  * Klenner  _et al._ [2016] A. Klenner, A. S. Mayer, A. R. Johnson, K. Luke, M. R. E. Lamont, Y. Okawachi, M. Lipson, A. L. Gaeta, and U. Keller, Gigahertz frequency comb offset stabilization based on supercontinuum generation in silicon nitride waveguides, Opt. Express 24, 11043 (2016).
  * Waldburger  _et al._ [2019] D. Waldburger, A. S. Mayer, C. G. E. Alfieri, J. Nürnberg, A. R. Johnson, X. Ji, A. Klenner, Y. Okawachi, M. Lipson, A. L. Gaeta, and U. Keller, Tightly locked optical frequency comb from a semiconductor disk laser, Opt. Express 27, 1786 (2019).
  * Aviles-Espinosa  _et al._ [2011] R. Aviles-Espinosa, G. Filippidis, C. Hamilton, G. Malcolm, K. J. Weingarten, T. Südmeyer, Y. Barbarin, U. Keller, S. I. C. O. Santos, D. Artigas, and P. Loza-Alvarez, Compact ultrafast semiconductor disk laser: targeting gfp based nonlinear applications in living organisms, Biomed. Opt. Express 2, 739 (2011).
  * Voigt  _et al._ [2017] F. F. Voigt, F. Emaury, P. Bethge, D. Waldburger, S. M. Link, S. Carta, A. van der Bourg, F. Helmchen, and U. Keller, Multiphoton in vivo imaging with a femtosecond semiconductor disk laser, Biomed. Opt. Express 8, 3213 (2017).
  * Saraceno  _et al._ [2014] C. J. Saraceno, F. Emaury, C. Schriber, M. Hoffmann, M. Golling, T. Südmeyer, and U. Keller, Ultrafast thin-disk laser with 80 $\mu$J pulse energy and 242 W of average power, Opt. Lett. 39, 9 (2014).
  * Zaugg  _et al._ [2013] C. A. Zaugg, A. Klenner, O. D. Sieber, M. Golling, B. W. Tilma, and U. Keller, Sub-100 mhz passively modelocked vecsel, in  _CLEO: 2013_ , IEEE (IEEE, 2013) pp. 1–2, paper CW1G.6.
  * Link  _et al._ [2017b] S. M. Link, D. Waldburger, C. G. E. Alfieri, M. Golling, and U. Keller, Coherent beam combining and noise analysis of a colliding pulse modelocked VECSEL, Opt. Express 25, 19281 (2017b).
  * Avrutin and Panajotov [2019] E. A. Avrutin and K. Panajotov, Delay-differential-equation modeling of mode-locked vertical-external-cavity surface-emitting lasers in different cavity configurations, Materials 12, 3224 (2019).
  * Hoogland  _et al._ [2000] S. Hoogland, S. Dhanjal, A. C. Tropper, J. S. Roberts, R. Häring, R. Paschotta, F. Morier-Gemoud, and U. Keller, Passively mode-locked diode-pumped surface-emitting semiconductor laser, IEEE Photon. Technol. Lett. 12, 1135 (2000).
  * Großmann  _et al._ [2020] M. Großmann, R. Bek, M. Jetter, and P. Michler, Stable fundamental and dual-pulse mode locking of red-emitting VECSELs, Laser Phys. Lett. 17, 065001 (2020).
  * Meyer  _et al._ [2020] J. T. Meyer, M. L. Lukowski, C. Hessenius, E. M. Wright, and M. Fallahi, High peak power, sub-ps green emission in a passively mode locked W-cavity VECSEL, Opt. Express 28, 5794 (2020).
  * Hausen  _et al._ [2019] J. Hausen, S. Meinecke, B. Lingnau, and K. Lüdge, Pulse cluster dynamics in passively mode-locked semiconductor vertical-external-cavity surface-emitting lasers, Phys. Rev. Appl. 11, 044055 (2019).
  * Meinecke [2022] S. Meinecke,  _Spatio-Temporal Modeling and Device Optimization of Passively Mode-Locked Semiconductor Lasers_, Springer Theses (Springer, Cham, 2022).
  * Agrawal and Dutta [1993] G. P. Agrawal and N. K. Dutta,  _Semiconductor Lasers_ (Van Nostrand Reinhold, New York, 1993).
  * Wang  _et al._ [2007] J. Wang, A. Maitra, C. G. Poulton, W. Freude, and J. Leuthold, Temporal dynamics of the alpha factor in semiconductor optical amplifiers, J. Light. Technol. 25, 891 (2007).
  * Herzog  _et al._ [2016] B. Herzog, B. Lingnau, M. Kolarczik, Y. Kaptan, D. Bimberg, A. Maaßdorf, U. W. Pohl, R. Rosales, J. H. Schulze, A. Strittmatter, M. Weyers, U. Woggon, K. Lüdge, and N. Owschimikow, Strong amplitude-phase coupling in submonolayer quantum dots, Appl. Phys. Lett. 109, 201102 (2016).
  * Pimenov  _et al._ [2014] A. S. Pimenov, E. A. Viktorov, S. P. Hegarty, T. Habruseva, G. Huyet, D. Rachinskii, and A. G. Vladimirov, Bistability and hysteresis in an optically injected two-section semiconductor laser, Phys. Rev. E 89, 052903 (2014).
  * Jaurigue  _et al._ [2016] L. C. Jaurigue, O. Nikiforov, E. Schöll, S. Breuer, and K. Lüdge, Dynamics of a passively mode-locked semiconductor laser subject to dual-cavity optical feedback, Phys. Rev. E 93, 022205 (2016).
  * Nikiforov  _et al._ [2016] O. Nikiforov, L. C. Jaurigue, L. Drzewietzki, K. Lüdge, and S. Breuer, Experimental demonstration of change of dynamical properties of a passively mode-locked semiconductor laser subject to dual optical feedback by dual full delay-range tuning, Opt. Express 24, 14301 (2016).
  * Kilen  _et al._ [2014] I. Kilen, J. Hader, J. V. Moloney, and S. W. Koch, Ultrafast nonequilibrium carrier dynamics in semiconductor laser mode locking, Optica 1, 192 (2014).
  * Kilen  _et al._ [2016] I. Kilen, S. W. Koch, J. Hader, and J. V. Moloney, Fully microscopic modeling of mode locking in microcavity lasers, J. Opt. Soc. Am. B 33, 75 (2016).
  * Pimenov  _et al._ [2017] A. S. Pimenov, S. Slepneva, G. Huyet, and A. G. Vladimirov, Dispersive time-delay dynamical systems, Phys. Rev. Lett. 118, 193901 (2017).
  * Pimenov and Vladimirov [2019] A. S. Pimenov and A. G. Vladimirov, Dynamics of an inhomogeneously broadened passively mode-locked laser, Eur. Phys. J. B 92, 114 (2019).
  * Hausen  _et al._ [2020] J. Hausen, K. Lüdge, S. V. Gurevich, and J. Javaloyes, How carrier memory enters the haus master equation of mode-locking, Opt. Lett. 45, 6210 (2020), arXiv:2008.09058 [physics.optics] .
  * McLaren  _et al._ [2020] S. McLaren, I. Kilen, and J. V. Moloney, Microscopic modeling of transverse mode instabilities in mode-locked vertical external-cavity surface-emitting lasers, Appl. Phys. Lett. 116, 031102 (2020), https://doi.org/10.1063/1.5134070 .
  * Press  _et al._ [2007] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vettering,  _Numerical Recipes (3rd ed.)_ (Cambridge University Press, Cambridge, 2007).
  * Vladimirov and Turaev [2005] A. G. Vladimirov and D. V. Turaev, Model for passive mode locking in semiconductor lasers, Phys. Rev. A 72, 033808 (2005).
  * Vladimirov  _et al._ [2010] A. G. Vladimirov, U. Bandelow, G. Fiol, D. Arsenijević, M. Kleinert, D. Bimberg, A. S. Pimenov, and D. Rachinskii, Dynamical regimes in a monolithic passively mode-locked quantum dot laser, J. Opt. Soc. Am. B 27, 2102 (2010).
  * New [1974] G. New, Pulse evolution in mode-locked quasi-continuous lasers, IEEE J. Quantum Electron. 10, 115 (1974).
  * Vladimirov  _et al._ [2011] A. G. Vladimirov, D. Rachinskii, and M. Wolfrum, Modeling of passively mode-locked semiconductor lasers, in  _Nonlinear Laser Dynamics - From Quantum Dots to Cryptography_ , Reviews in Nonlinear Dynamics and Complexity, edited by K. Lüdge (Wiley-VCH, Weinheim, 2011) Chap. 8, pp. 183–213.
  * Lüdge [2012] K. Lüdge,  _Nonlinear Laser Dynamics - From Quantum Dots to Cryptography_ (Wiley-VCH, Weinheim, 2012).
  * Hoffmann  _et al._ [2011] M. Hoffmann, O. D. Sieber, V. J. Wittwer, I. L. Krestnikov, D. A. Livshits, Y. Barbarin, T. Südmeyer, and U. Keller, Femtosecond high-power quantum dot vertical external cavity surface emitting laser, Opt. Express 19, 8108 (2011).
  * Lingnau  _et al._ [2014] B. Lingnau, W. W. Chow, and K. Lüdge, Amplitude-phase coupling and chirp in quantum-dot lasers: influence of charge carrier scattering dynamics, Opt. Express 22, 4867 (2014).
  * Meinecke  _et al._ [2019] S. Meinecke, L. Drzewietzki, C. Weber, B. Lingnau, S. Breuer, and K. Lüdge, Ultra-short pulse generation in a three section tapered passively mode-locked quantum-dot semiconductor laser, Sci. Rep. 9, 1783 (2019).
  * Alfieri  _et al._ [2018] C. G. E. Alfieri, D. Waldburger, R. Nürnberg, M. Golling, L. C. Jaurigue, K. Lüdge, and U. Keller, Modelocking instabilities for high-gain semiconductor disk lasers based on active submonolayer quantum dots, Phys. Rev. Appl. 10, 044015 (2018).
  * Hausen  _et al._ [2021] J. Hausen, B. Herzog, A. Nelde, S. Meinecke, N. Owschimikow, and K. Lüdge, Feedback induced locking in semiconductor lasers with strong amplitude-phase coupling, Phys. Rev. A 103, 043511 (2021), arXiv:2101.01623, arXiv:2101.01623 [physics.optics] .
  * Alfieri  _et al._ [2017] C. G. E. Alfieri, D. Waldburger, S. M. Link, E. Gini, M. Golling, G. Eisenstein, and U. Keller, Optical efficiency and gain dynamics of modelocked semiconductor disk lasers, Opt. Express 25, 6402 (2017).