Stabilizing nanolasers via polarization lifetime tuning

We investigate the emission dynamics of mutually coupled nanolasers and predict ways to optimize their stability, i.e., maximize their locking range. We find that tuning the cavity lifetime to the same order of magnitude as the dephasing time of the microscopic polarization yields optimal operation conditions, which allow for wider tuning ranges than usually observed in conventional semiconductor lasers. The lasers are modeled by Maxwell–Bloch type class-C equations. For our analysis, we analytically determine the steady state solutions, analyze the symmetries of the system and numerically characterize the emission dynamics via the underlying bifurcation structure. The polarization lifetime is found to be a crucial parameter, which impacts the observed dynamics in the parameter space spanned by frequency detuning, coupling strength and coupling phase.


Semi-classical laser model
Lasers generally consist of a cavity containing an active medium that is constantly being stimulated by an external pump current p supplying the laser system with energy. The resulting inversion N of the medium induces an electric field E, which retrospectively causes a polarization P of the active medium. The modelling is done with macroscopic Maxwell-Bloch equations, both for atomic and semiconductor lasers 20 . They can be derived by a quantum-mechanical treatment of the active medium and a classical treatment of the electric field. The resulting microscopic Maxwell-Bloch equations can be transformed to macroscopic equations 18 . To describe two coupled lasers as shown in Fig. 1a, six differential equations, comprising the complex electric field E, the carrier inversion N and the complex polarization P for both lasers, are used. For a single laser we use the model published in 12 . The equations read for j, k ∈ {1, 2} and j = k , where all time scales, such as the carrier lifetime T and the polarization lifetime T 2 are normalized to the photon lifetime in the cavity. To translate the timescales into SI-units, e.g., a photon lifetime of T ph = 1ps can be assumed 39 . The pump current is denoted by p. The parameter normalizes the system such that the first laser threshold is always at p (1) thr = 1/2 . Thus, it does not depend on the polarization lifetime T 2 , which allows for a fair comparison 12 . Throughout this paper, we choose a pumping of p = 2 , which corresponds to the lasers operating four times above threshold and below the second threshold 12 . The parameter �ω is the difference between the transition frequency of the two energy levels and the cavity mode. The system is further transformed into the reference frame of the average optical frequency ω 0 of the two (1a) where self-coupling is not considered. τ represents the propagation time between the lasers. We only consider a delay of τ = 10 which corresponds to a length of 800 µm on an integrated chip. K is the coupling strength.
In this manuscript we mainly choose a coupling of K = 0.1 , which refers to an intensity reflectivity of 20% . We consider superposition effects by back reflected light as small at this value and do not expect a significant impact on the observed dynamics 40 . The coupling phase θ is an additional phase shift acquired by the in-coupled light. The complex eigenvalues of the linearized Class-C equations for one solitary laser have been determined as a function of the polarization lifetime T 2 for the parameters discussed throughout the manuscript. Without the delayed coupling there are in total five eigenvalues out of which one is zero due to the phase symmetry 12 . The real and imaginary part of the (negative) eigenvalue which is undamped first (the one with the smallest absolute real part) are plotted in Fig. 2. The real part Re corresponds to the damping Ŵ RO of the relaxation oscillations, whereas the imaginary part Im yields the relaxation-oscillation frequency ν RO .
The class-C laser equations can be reduced to class-B equations by considering the limit T 2 → 0 , which mimicks the transition from a nanoscale to a mesoscale laser. By carrying out an adiabatic elimination of the polarization P and setting the derivative of the polarization to zero, we obtain the class-B laser equations 12,41 Note that the frequency difference between gain and cavity resonance �ω now has an effect equivalent to the commonly known line-width enhancement factor α 33,42,43 .

Symmetries
Our model for the coupled laser system given in Eqs. (1a)-(1c) contains a number of symmetries that facilitate the interpretation of the results or enable the analysis of steady state solutions and the investigation of their stability. Hence, before presenting the results, the relevant symmetries are discussed in the following. These have been introduced by 33 for two coupled class-B lasers and are reformulated and applied to class-C lasers here.
Reflection symmetry. For identical lasers, Eqs. (1a)-(1c) are invariant under an exchange of the lasers. For non-identical lasers, i.e., here for a detuning δω = 0 , the sign of the detuning has to be flipped 2πTranslational symmetry. The system is invariant under 2π-phase shifts of the coupling phase π-Translational symmetry. If the coupling phase θ is translated by π this translation can be cancelled out by a π-phase shift of one of the laser fields (3) (4) (E 1 , P 1 , N 1 , E 2 , P 2 , N 2 , δω) → (E 2 , P 2 , N 2 , E 1 , P 1 , N 1 , −δω).

Compound laser modes (CLMs)
The fundamental steady-state solutions describe the asymptotic behavior of the coupled laser system. Such solutions have been previously discussed in the context of class-B lasers 33 and are known as compound laser modes (CLMs). They describe fully synchronized, i.e., phase locked lasers. Due to the phase symmetry of the complex electric field and polarization we make the ansatz of constant light intensities |E j | 2 and constant rotation in the complex plane.
for j ∈ {1, 2} . The electric fields given by the optical frequency of the CLM are allowed to have a phase difference φ j,m for m ∈ {E, P} . Inserting the ansatz into Eqs. (1a)-(1c) yields a non-linear system of 10 equations that exhibits several branches of CLMs, characterized by different properties. In the following sections, we restrict the analysis to CLM branches that contain stable CLMs and CLM branches, which interact with the former.
For identical lasers ( δω = 0 ), the only CLMs found to be stable exhibit a phase difference between the lasers that is constant with respect to the coupling phase θ . Therefore such solutions correspond to phase-locking, i.e., synchronization, among the lasers. We refer to those steady states, also known as symmetric solutions 35 , as constant-phase CLMs in analogy to 33 , where a similar model for two coupled class-B lasers is discussed. We define the phase shift as The amplitudes of the two lasers are identical when working on a constant-phase CLM. Figure 3a shows the intensities |E j,0 | 2 of the CLMs as a function of the coupling phase θ . The color of the lines encodes the phase difference �φ between the two lasers and accordingly emphasizes two branches of constant phase CLMs, the in-and anti-phase CLMs (pink and yellow) with a phase difference �φ of 0 or π , respectively. The in-and anti-phase CLMs are connected by the π-translational symmetry: Given one full set of parameters, in particular θ = θ 0 , for which an in-phase CLM exists, a corresponding anti-phase CLM exists for θ = θ 0 + π . Figure 3a also depicts the intensity extrema found via numeric integration in dark green. It can be seen that the lasers either emits on the CLMs or they show periodic intensity oscillations that are born in Hopf bifurcations (the oscillations are identical for the two identical lasers). These periodic dynamics exist in regions where the CLMs are unstable and the oscillations form bridges between the in-and the anti-phase CLMs (these bridges are also found for the class-B limit with long delay 36 and for mutually delay-coupled semiconductor lasers in Ref. 38 where they are called symmetric two-color states).   33 , asymmetric solutions in Ref. 36 or out-of-phase solutions in Ref. 35 ) can be seen (curves that change their color) next to the constant phase CLMs. The symmetry broken CLMs are characterized by a phase difference �φ(θ) that varies with the coupling phase, and by non-equal amplitudes E j,0 33,36,38,44 . As a result the solutions for the two lasers that operate on a symmetry-broken CLM, do not parametrize the same curves in (�, N)-space. As can be seen in Fig. 3b, the symmetry-broken CLMs emerge in pitchfork bifurcations (black squares) from either in-or anti phase CLMs and vanish in the other.
In analogy to the external cavity modes of lasers subject to feedback (steady state solutions of the intensity) 12,30 , Fig. 4 shows the CLMs in a (�, N)-phase projection. Here a matrix of such projection-maps is shown, where the polarization lifetime increases from top to bottom (mimicking the transition from nano-to mesoscale lasers) and the detuning increases from left to right (i.e a change from identical to slightly different lasers). The in-and anti-phase CLMs can be found in pink and yellow in the left column for vanishing detuning. They lie on the same curves due to their identical amplitudes up to phase shifts of π . Stable solutions are indicated by black surroundings. Again the symmetry-broken CLMs (curves that change color in Fig. 4) connect the in-and antiphase CLMs via pitchfork bifurcations (squares). Within every column of the matrix the polarization lifetime T 2 ∈ {1, 2, 3} is varied from top to bottom. This change in T 2 decreases the width of the gain spectrum, which is proportional to 1 T 2 (see Fig. 1b). As a result, the solutions for the coupled laser system are found on a smaller range of CLM frequencies .
The consequences of non-vanishing detunings on the CLMs can be seen in the second and third column of Fig. 4. For each row, the polarization lifetime T 2 is constant and the detuning δω is increased. Apart from T 2 The deformed horseshoe-curves include stable CLMs, marked by black surrounding. The curves shrink, which indicates that the lasers can not stabilize for arbitrarily large detunings. When increasing the detuning δω we observe (not shown) that the horseshoe-formed curves pull together onto one point until they vanish completely and locking becomes impossible. As mentioned before, the CLM solutions plotted in Fig. 4 are obtained by continuously tuning the coupling phase θ . For the specific choice of θ = 0 , the CLMs are marked by white and black dots. By inspecting those dots it becomes obvious that, for the fixed parameter set, there is only one stable CLM solution (black dot). This solution determines the emission frequency of the synchronized state observed numerically. It is noted that the size of the CLM curves increases with the coupling strength K (not seen in the figure). Their shape is not influenced by K, although the number of solutions, found for a fixed coupling phase, changes.

Bifurcation scenarios
In order to analyze the synchronization behavior of the two coupled class-C lasers with respect to their coupling, we investigate the laser system with a focus on the coupling parameters θ , K and δω . Two dimensional bifurcation analyses are carried out by integrating Eqs. (1a)-(1c) numerically, using random initial conditions for each data point. Figure 5a shows a bifurcation diagram in the parameter space of coupling strength K and detuning δω . Every data point of the graph represents a time series of one of the lasers intensities |E| 2 , where the number of maxima of this time series is evaluated. For zero maxima, the lasers settle onto a CLM (see "Compound laser modes (CLMs)" section). In this case, instead of the number of maxima, the plots show the phase difference �φ of the electric fields. Figure 5b shows the difference |E max 1 | 2 − |E max 2 | 2 of the maxima of the two lasers. We concentrate on the period-one and steady state emission and thus areas with two or more maxima are masked in black in (b). The symmetry-breaking of the CLMs caused by non-zero detuning δω = 0 can be seen by the red and blueish colored regions. In Fig. 4 we saw this detuning induced symmetry breaking via the different closed loop of solutions (different intensity for each laser). The white regions in Fig. 5b indicate symmetric operation of both lasers, which interestingly also occurs for non-vanishing detuning within the regions of period-one oscillations (light blue regions in Fig. 5a).
The overall bifurcation structure for the small value of T 2 = 1 chosen in Fig. 5, is in agreement with existing results on delay coupled class-B lasers. As expected for lasers with weak damping of their relaxation oscillations the locking range of the lasers approximately increases linearly with the coupling strength K. Please see Fig. 2 for the values of relaxation-oscillation frequency ω RO and damping Ŵ RO of the solitary class-C laser used here. www.nature.com/scientificreports/ For an increasing detuning, the locking cone is bounded by saddle-node bifurcations, while it collapses in a Hopf-bifurcation for increasing K around K ≈ 0.2 35 . In Fig. 5 we can see the saddle-node characteristics as the dynamics changes abruptly from locked operation (yellow/pink) to complex dynamics (dark blue). Instead, at the Hopf bifurcation (vertical border of the locking range for increasing K) harmonic oscillations with one maximum (light blue colors in Fig. 5) are born (they can also be seen in the 1D-bifurcation scan in Fig. 3a). The bifurcation structure changes substantially with the coupling phase θ as can be seen in the animation (Supplementary Video S1), which shows graphs as in Fig. 5 with varying coupling phase θ , where also the periodicity in the coupling phase θ (translations of π or 2π ) can be nicely seen. In particular, the π-periodicity can directly be associated with the π-translational symmetry of the in-and anti-phase CLMs. For phase differences of π and 2π , complex dynamics are observed for detuning δω = 0 and small coupling strengths (Supplementary Video S1) which most likely emerge when the periodic bridges between in-and anti-phase dynamics form homoclinic connections 36 .
We note that the analysis of the coupled laser dynamics is performed for deterministic equations. Nevertheless, we checked if the dynamics is preserved when a Gaussian white noise term Dξ is added onto the field equation. Our results show that as long as the noise strength D is chosen below D ≈ 0.1 (corresponding to a relative standard deviation σ/�I� ≈ 0.8 for the solitary class-C laser), the locking structure is preserved. For D > 0.1 the noise dominates the dynamics.
The significant dependency of the locking behavior on the coupling phase θ suggests an analysis of the system with respect to this parameter. Within this projection we will also further investigate the impact of the polarization lifetime on the locking behavior of the lasers. Experimentally the coupling phase θ can easily be varied without crucially changing the other parameters. The normalized T 2 -time can be changed either by tuning the photon lifetime, i.e., the cavity, or by adjusting the active medium. Figure 6 shows two-dimensional bifurcation diagrams for which the coupling phase θ and the detuning δω are varied. If the lasers stabilize on a CLM (locking region), the diagrams show the phase difference �φ of the electric fields color coded with the yellow and pink color scale. Figure 6 a shows the results for a class-B laser  (Fig. 6 from top to bottom) and an optimal value of T 2 seems to exist. We investigate that in more detail in the next section. The numeric results in Fig. 6 also nicely highlight the symmetries of our coupled laser system. Due to the 2π -translational symmetry the results repeat and the system only needs to be investigated for θ ∈ [0, 2π) . The π -translational symmetry can be seen by focusing on the locked regions. If we perform a horizontal cut through the locking region along δω = 0 the change from in-phase (pink shading) to anti-phase CLMs (orange shadings) always occurs after a phase shift of π . For every in-phase CLM at coupling phase θ 0 , a corresponding anti-phase CLM exists at phase θ 0 + π solving Eqs. (1a)-(1c) equivalently. Therefore, also the stability of these in-and anti-phase CLMs coincides making the number of maxima or the locking regions π-periodic with respect to the coupling phase θ . At last, also the reflection symmetry can be found in Fig. 6. It appears as a symmetry of the locking regions at the horizontal axis along δω = 0.

Locking range
The goal of this section is to identify a polarization lifetime that is best suited to attain large locking ranges. Thus, we define the locking range δω lock (θ) . It is the maximum detuning δω > 0 for which the lasers lock onto a CLM for one specific coupling phase θ (see arrow in Fig. 6d for the definition). Negative detunings do not need to be considered since the locking regions are symmetrical. For some phases, the locking regions vanish completely, i.e., δω lock = 0 . As can be seen in Fig. 6, δω lock also strongly depends on the polarization lifetime T 2 . A monotonous correlation of the locking range to the polarization lifetime however can not be detected here, in contrast to a single class-C laser with feedback, which is stabilized by higher polarization lifetimes 12 .
To get a more comprehensive insight, Fig. 7 shows the locking range δω lock plotted color coded as a function of the coupling phase θ and the polarization lifetime T 2 . Every horizontal line of the diagram holds the same information about the locking range as one sub-figure in Fig. 6. Presenting the situation continuously with respect to T 2 shows that the locking range δω lock does not vary monotonously with T 2 for any given coupling phase θ . In fact, on the interval investigated here, there are at least two disjoint partial intervals for which the locking range δω lock vanishes.
We furthermore define the maximum locking range δω θ max as This maximally achievable locking range for the given coupling strength K = 0.1 is depicted as a function of the polarization lifetime T 2 in Fig. 8a for coupling strengths of K = 0.1 and K = 0.2 . For both coupling strengths, the maximum locking range δω θ max increases until it reaches its maximum at T 2 ≈ 1.06 for K = 0.1 or T 2 ≈ 1.39 for K = 0.2 . Afterwards, it decreases with δω θ max ∝ 1/T 2 for T 2 > 1.06 or for T 2 > 1.39 (see dashed lines in Fig. 8a). The decrease of the maximum locking range correlates with the decrease of the gain spectrum (as its width scales with 1/T 2 ) and consequently with a narrower frequency spectrum of possible CLMs (see "Compound laser modes (CLMs)" section). The increase of the maximum locking range for T 2 < 1.06 on the other hand can be explained by the rising dynamical impact of the polarization lifetime and the additional degree of freedom in phase space compared to the class-B case. This additional dynamical degree of freedom also leads to δω lock (θ, T 2 ). www.nature.com/scientificreports/ a maximum around T 2 = 1 for the damping of the relaxation oscillations of the solitary laser (see Fig. 2). Both effects combined nicely explain the occurrence of an optimal value for T 2 as seen in Fig. 8a. Further, the maximum locking range δω θ max increases with the coupling strength K as shown in the plot. In particular, we found that it increases linearly for small coupling strengths below K ≈ 0.15.
A second approach to characterize the locking behavior is by defining the relative stability µ δ , which represents the relative share of coupling phases on the interval [0, 2π) θ for which the locking range exceeds a certain threshold δ denotes the Heavyside function. The relative stability gives further insights on how stable a given polarization lifetime T 2 is regarding all coupling phases. Figure 8b shows the relative stability µ δ for several thresholds δ as a function of the polarization lifetime T 2 . Trivially, the relative stability increases if the threshold δ , that has to be exceeded, decreases. For all shown δ , the curves have maxima on the interval [1, 2] T 2 . Hence, the polarization lifetime T 2 that exhibits the optimal maximum locking range δω θ max approximately matches the polarization lifetime that shows the biggest share µ δ of of high locking ranges. Both quantities together provide a comprehensive view of the locking behavior of the coupled laser system and show that optimal locking is achieved for polarization lifetimes of the order of the photon lifetime.

Conclusion
We have investigated the locking and synchronization behavior of two coupled class-C lasers as a paradigmatic example of nanolasers with a small photon lifetime. An emphasis was put on the impact of the polarization lifetime, the frequency detuning, and the coupling phase. To approach the locking structure of the delay-coupled laser system, we have determined the compound laser modes, which are the fundamental steady-state solutions of the system, investigated the symmetries of the system, and performed a bifurcation analysis of the underlying dynamics. The polarization lifetime T 2 was found to decisively influence the locking range of the lasers. Interestingly a value of T 2 can be found where the locking range as well as the relative stability is optimal. The value depends on the photon lifetime within the cavity. Thus we can predict nanolasers to show best synchronization properties for the case where the photon lifetime is equal to the polarization lifetime for the case of weak coupling and about two third of the polarization lifetime for stronger coupling.  www.nature.com/scientificreports/