1. Introduction

Periodic optical pulses generated by mode-locked lasers are of particular importance for a wide range of applications, including high speed optical communications [1], biomedical imaging [2], remote sensing [3], timing synchronization [4], optical signal processing [5], materials processing [6], spectroscopy [7,8], etc. Among all possible sources, semiconductor mode-locked lasers are very attractive devices because they generate stable periodic pulses at very high repetition rates while being potentially very compact, cheap, efficient, and robust [9,10]. Nanolasers in particular have attracted intense interest because they are expected to be more compact, integrable, and energy efficient than conventional lasers [11].

Recently, a new type of photonic crystal cavity, sustaining oscillation of Hermite-Gaussian (HG) modes in a parabolic potential, has been proposed and demonstrated [12–14]. These cavities are attractive not only because of their compactness, but also because they exhibit a comb of equally spaced modes. Based on this remarkable property, we proposed a new type of passively mode-locked laser, obtainable by incorporating saturable gain and absorber sections in such a cavity. In such a laser, the pulse repetition rate depends on the curvature of the potential instead of the cavity length [15]. In addition, different dynamical laser behaviors, including self-starting regimes, were investigated, such as Q-switching, Q-switched mode locking, continuous-wave mode locking, etc. [16].

However, potential practical implementation of such a mode-locked laser needs to take into account different types of defects that may affect the resonator. This is particularly true due to the fact that realization of such lasers relies on nano-fabrication techniques. For example, there might be some disorder in the size and position of the lattice holes [17–23], or simply some surface roughness or variation of the thickness of some layers [17,18,24–27]. Such general fabrication-related defects [19] may influence properties such as the band structure [18,27–33], the attenuation [21,28,31,33–37], or the coupling strength between cavities [38]. Usually such a disorder is undesirable except to obtain random laser operation [39–41], to observe Anderson localization [18,31,42–44], or to induce new functionalities [45,46]. Some new methods to probe disorder induced distortion in photonic devices have been developed [47,48]. Also, it has recently been shown that the intensity variations due to the disordered medium can be entirely suppressed by adding disorder-specific gain and loss components to the medium [49,50].

In the lasers that we consider, we can take most of these effects into account by introducing some distortion in the harmonic potential. Such a modification of the potential should be expected to lead to a modification of the Hermite-Gaussian modes and to a distortion of the cavity mode spectrum with respect to a perfect frequency comb. The question then arises to know whether the mode locking regime of operation will be robust with respect to such alterations of the cavity mode spectrum, which are unavoidable whatever the fabrication process. Thanks to the nonlinear nature of mode locking, in general, semiconductor mode-locked lasers are very robust. Even the locking of multi-mode fiber lasers is robust to disorder [51–54]. But this question needs to be addressed in the peculiar case of harmonic cavity nanolasers.

The aim of this paper is thus to investigate the robustness of mode locking of the Hermite-Gaussian modes in an harmonic cavity nanolaser from a theoretical and simulation standpoint. We focus on two different kinds of distortion of the harmonic potential. The first one is the non-parabolicity of the potential, which could for example be simply due to errors in the positioning of the elements of the structure. A fourth-order term is introduced in the potential to quantify this deformation. This distortion can strongly alter the Hermite-Gaussian modes and shift the initially equally spaced frequencies. The second type of potential distortion we consider consists in random errors in the shape of the potential, somewhat simulating disorder due to different kinds of structural imperfections. To this aim, a Gaussian random function with zero mean value is added to the potential. In both cases, we compare the mode spectrum of the “cold” cavity with the laser spectrum, in order to isolate the capability of the laser nonlinearities to restore mode-locked operation. We also take into account the existence of a non-zero Henry factor for the gain medium and the absorber.

The paper is organized as follows: Section 2 introduces the model and the two kinds of potential distortion. Section 3.1 focuses on the investigation of the effect of the non-parabolicity of the potential. Section 3.2 describes the laser behavior in the presence of disorder created using random errors in the shape of the potential. The evolution of the laser behavior is then statistically investigated, including the influence of the Henry factor. Section 4 gives the conclusions of the paper.

2. Harmonic cavity laser with imperfect potential

The starting point is the master equation initially developed in Refs. [15,16] to describe the evolution of light in a nanolaser based on a harmonic potential for light and containing a gain medium and a saturable absorber. The time evolution of the slowly varying amplitude $A(x,t)$ of the field inside the unidimensional cavity can be described by:

The three first terms in Eq. (1) lead to the existence of Hermite-Gaussian eigenmodes. The red full line in Fig. 1(a) shows the parabolic potential created by the term proportional to $x^2$. It also shows the spectrum of equally separated eigenfrequencies (the free spectral range $\Omega /2\pi$ is chosen equal to 100 GHz), together with the shapes of their associated Hermite-Gaussian (HG) modes.

 

Fig. 1. (a) Hermite-Gaussian modes in a parabolic potential, with three different values of the non-parabolicity coefficient $c_4$. Semi-transparent pink area: active media. (b-d) Evolution of the mode characteristics of the “cold” cavity versus $c_4$. (b) Frequency deviations $f_{n}-f_{n,0}$; (c) frequency separations $f_n-f_{n-1}$ between successive modes; (d) frequency differences $\Delta f_n = 2f_n-f_{n+1}-f_{n-1}$.

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The dissipative term $H_1$ describes the effects of the gain medium, the losses, and the saturable absorption:

The saturation of the gain and of the absorption is described by the following set of spatially local equations:

For a given field distribution $A(x,t)$, the slowly varying modal coefficient $C_n(t)$ can be calculated by projecting the field amplitude on $\Psi_n$:

2.1 Non-parabolicity of the potential

If the potential is not a perfect parabola, the HG modes in the cavity get distorted. The frequency difference between successive modes is then no longer uniform, and the cavity spectrum does no longer form a perfect frequency comb. For example, the mode frequencies should decrease (increase) if the potential is wider (narrower) than a perfect parabola.

To investigate the influence of such an anharmonicity on the mode locking, we introduce extra terms to the term proportional to $x^2$ in Eq. (1). Since an asymmetry of the cavity would play only a minor role, as shown in Supplement 1, we ignore here third-order terms. We thus add a fourth-order term to the parabolic potential in order to quantify the distortion. Since the size of the cavity is determined from the scaling factor $x_\Omega =\sqrt {\omega_{kk}/\Omega }$, the master equation (Eq. (1)) in the presence of distortion can be written in the following normalized form:

2.2 Potential disorder

We can follow a similar method to introduce the effect of random errors in the shape of the potential. To this aim, a term ${D}_{\textrm{max}}R(x)$ is added to the potential, where $R(x)$ is a standard centered Gaussian random variable with variance 1. The amplitude of the disorder is adjusted by varying $D_{\textrm{max}}$. The master equation can then be written as

3. Results and discussion

In this section we numerically simulate Eqs. (8) and (9) to predict the laser behavior in the presence of potential distortion. To this aim, like in Ref. [16], we use the split-step Fourier method. The spatial discretization period is equal to $0.13\;x_\Omega$. The time discretization period is equal to $10 \;\textrm{fs}$ for all simulations. We have checked that it is small enough to ensure a good accuracy for all the cases studied in this paper. As the gain width is fixed at $w=5x_\Omega$, the intensities of only the first six modes are significantly different from zero. Therefore, in the following, only modes labeled $n=0\cdots 5$ are taken into account.

In this section, we check whether the laser modes are locked using three different ways. The first one is to monitor the relative phase differences $\Delta \phi_n=2\phi_n-\phi_{n+1}-\phi_{n-1}$, which are obtained from the arguments of the $C_n(t)$’s in Eq. (7). The second one consists in calculating the frequency separations between the modes in the presence of distortion, in order to check whether the laser operation restores equally spaced frequencies. The last one consists of course in looking at the regularity of the generated pulse train.

3.1 Non-parabolicity of the potential

Three examples of distorted potentials with different values of $c_4$ are shown in Fig. 1(a). A positive value of $c_4$ leads to a narrower potential, and vice versa. The gain and absorber are supposed to spatially overlap in the middle of the cavity (transparent pink region of Fig. 1(a)).

Before studying the laser behavior in the presence of anharmonicity, we first calculate the eigenfrequencies of the “cold” cavity in the presence of distortion. To this aim, a random noise field is injected inside the cavity as initial value for the field. Then we compute the evolution of this field according to Eq. (9) with $\gamma_0=0$, $a_0=0$, and $g_0=0$, i. e. in the absence of any losses or gain. In these conditions, the “cold” cavity works as a spectral filter. The calculation is run during $30\;\textrm{ns}$. Then the spectrum $F(2\pi f)$ of the intracavity field is calculated using the following expression:

This equation is applied to the evolution of the field $A(x,t)$ during the last ${24}\;\textrm{ns}$ of the simulation. The eigenfrequencies $f_n$ of the cold cavity are then obtained by finding the peaks in the spectrum $F(2\pi f)$.

Figure 1(b) shows the result of such simulations for varying values of $c_4$. Each line corresponds to the evolution of $f_n-f_{n,0}$, for $n=0,\ldots ,5$, where $f_{n,0}=\omega_n/2\pi =(n+1/2)\Omega /2\pi$ is the eigenfrequency for $c_4=0$. One can see that these frequency shifts $f_n-f_{n,0}$ linearly increase as $c_4$ increases, and that the slope increases with the mode order $n$. By looking at the frequency difference $f_n-f_{n-1}$ between successive modes, as shown in Fig. 1(c), one notices that the potential distortion $c_4\neq 0$ destroys the periodicity of the frequency comb. The mode separation is larger than the initial value of $100\;\textrm{GHz}$ for $c_4>0$, and conversely. Moreover, for a given value of $c_4$, one can see that the frequency separation between successive modes increases linearly with $n$. This can be seen from Fig. 1(d), showing that $\Delta f_n = 2f_n-f_{n+1}-f_{n-1}= (f_n-f_{n-1})-(f_{n+1}-f_n)$ is independent of $n$.

The question then arises to know whether the nonlinearity of the active medium and of the saturable absorber is still sufficient to obtain mode locked operation, i. e. to restore the periodicity of the comb of modes, even for $c_4\neq 0$. We suppose here for example that the gain and absorber regions have a width $w=5x_\Omega$ and spatially overlap in the cavity center, as represented by the transparent pink region in Fig. 1(a). The values of the parameters are chosen as follows: intrinsic losses lead to a dissipation rate $\gamma_0 = 10^{10}\;\textrm{s}^{-1}$, the relative unsaturated gain coefficient is $r_{\textrm{g}}=g_0/\gamma_0=70$, and the relative unsaturated absorption coefficient is $r_{\textrm{a}}=a_0/\gamma_0=10$. The value of the gain lifetime is $\tau_g=1\;\;\textrm{ns}$, and the absorber lifetime is $\tau_a=10\;\textrm{ps}$. The ratio between the saturation energies of the gain and absorber media is $R_{\textrm{E}}=E_{\textrm{sat,g}}/E_{\textrm{sat,a}}=25$. In the following, three examples are investigated in details.

These three examples of the time evolution of the laser, corresponding to $c_4=-0.01$, $c_4=0.01$, and $c_4=0.02$, are plotted in Fig. 2. Let us first look at Fig. 2(a), which has been obtained for $c_4=-0.01$. The upper plots show the time evolution of the mode intensities $|C_n|^2$ and of the relative phase differences $\Delta \phi_n=2\phi_n-\phi_{n+1}-\phi_{n-1}$ between the modes. We can see that all the mode intensities are modulated. Furthermore, the phase difference $\Delta \phi_1$ varies in a monotonous manner versus time, corresponding to a beat frequency of the order of 300 MHz, while all other phase differences $\Delta \phi_n$ for $n=2,3,4$ are locked. This means that all the modes are phase locked except mode $n=0$. This unlocked mode is responsible for the modulation of the intensity $|A_+(x=0,t)|^2$ of the field propagating in the $+x$ direction at cavity center $x=0$, which is shown in the two plots of Fig. 2(b). This indicates that mode 0 is more easily influenced by the distortion of the potential when $c_4<0$ and then loses the locking.

 

Fig. 2. Steady-state time evolution of (a,c,e) the laser mode intensities and phase differences and (b,d,f) the laser intensity traveling in the $+x$ direction at cavity center ($x=0$) for (a,b) $c_4=-0.01$ (partial mode-locking), (c,d) $c_4=0.01$ (perfect mode-locking), (e,f) $c_4=0.02$ (partial mode-locking). Other parameter values: $r_{\textrm{g}}=g_0/\gamma_0=70$, $r_{\textrm{a}}=a_0/\gamma_0=10$, $w=5x_\Omega$, $\tau_{\textrm{g}}=1\;\textrm{ns}$, $\tau_{\textrm{a}} = 10\;\textrm{ps}$, $\gamma_0 = 10^{10}\;\textrm{s}^{-1}$, $\Omega /2\pi =100\;\textrm{GHz}$, $E_{\textrm{sat,g}}/E_{\textrm{sat,a}}=25$, $\alpha_{\textrm{g}}=\alpha_{\textrm{a}} =0$.

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On the contrary, with $c_4=0.01$ and all other parameters left unchanged, Figs. 2(c) and (d) shows that all the modes are locked. All modulations have disappeared in the mode intensities and relative phases, and the peak intensity of $|A_+(x=0,t)|^2$ remains stable with time, equal to $80\;I_{\textrm{sat,g}}$, as can be seen in Fig. 2(d).

However, if $c_4$ is further increased up to $c_4=0.02$, Figs. 2(e) and (f) show that some modes get unlocked, leading again to a modulation of the peak values of $|A_+(x=0,t)|^2$.

From the above results, it is clear that there exists a locking region between $c_4=-0.01$ and $c_4=0.02$, for which all modes can be locked. The same calculation is thus performed for varying values of $c_4$. The frequencies $f_n$ of the modes are obtained by calculating the spectrum of the field within the last $24\;\textrm{ns}$ of the simulation once the laser is in steady-state regime. The results are plotted in Fig. 3. Figure 3(a) shows the peak intensities observed at the center of the cavity for light traveling in the $+x$ direction, while Figs. 3(b)-(d) are presented in a similar manner as the results for the cold cavity shown in Figs. 1(b)-(d).

Figure 3(b) shows that the mode frequencies $f_n$ increase as $c_4$ increases. The shift of higher order modes is larger than that for lower order modes. But the comparison with Fig. 1(b) shows that the mode frequency shifts $f_n-f_{n,0}$ are clearly reduced by the nonlinearity of the active media, especially for $c_4>0$. It is clear that $f_0$ does no longer follow the same evolution when $c_4<-0.007$.

 

Fig. 3. Evolution of the steady-state laser characteristics versus $c_4$. (a) Peak intensities of the field propagating in the $+x$ direction at $x=0$. (b) Frequency deviations $f_{n}-f_{n,0}$; (c) frequency separations $f_n-f_{n-1}$ between successive modes; (d) frequency differences $\Delta f_n = 2f_n-f_{n+1}-f_{n-1}$. Same parameter values as in Fig. 2.

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Figure 3(c) shows that there exists an asymmetric locking region for $-0.007\leq c_4\leq 0.019$. With comparison with Fig. 2, one can see that the laser nonlinearity tends to reduce the frequency separation between the modes. It is thus more efficient to induce mode locking for positive values of $c_4$ than for negative values. In the locking region, for a given value of $c_4$, the mode separation is the same for all values of $n$ and linearly increases with $c_4$. In addition, one can notice that the mode frequency separation for $c_4=0$ is reduced to $99.8\;\textrm{GHz}$ due to the nonlinearity of active media. Outside this locking region, some modes are still locked, as can be seen from Fig. 3(d). Some of the modes still obey $\Delta f_n=0$, indicating that the corresponding mode frequencies are equally spaced. This is a big difference with respect to Fig. 1(d), induced by the nonlinearity of the amplifying and absorbing media.

Based on Figs. 2 and 3, we can see that there are several ways to observe the locking. The first one is to look at the phase difference $\Delta \phi_n$ between the modes, as can be seen by comparing Figs. 2(a) and 2(c). The second method consists in observing the frequency difference between successive modes, such as $f_n-f_{n-1}$ in Fig. 3(c) or $\Delta \phi_n$ in Fig. 3(d). These criteria are corroborated by observing the evolution of the peak intensities like in the three examples of Figs. 2(b),(d), and (f) and also summarized in Fig. 3(a). This plot shows all the peak intensities $|A_{\textrm{p+}}|^2$ of the field propagating into the $+x$ direction at $x=0$ detected during the last $10\;\textrm{ns}$ of each simulation. If all the modes are locked, this peak intensity has only one value as shown by the red dots in Fig. 3(a).

Outside the locking region, some modes are unlocked, leading to a slow modulation of the intensity, as is visible in the cases of Figs. 2(b) and (f). It also leads to several peak intensity values, as shown by the blue and green dots of Fig. 3(a). Unlocking is confirmed by the evolutions of the frequencies $f_n$ in Figs. 3(c) and (d).

Figure 4(a) shows the evolution of the locking range with the unsaturated gain and absorption rates $r_{\textrm{g}}=g_0/\gamma_0$ and $r_{\textrm{a}}=a_0/\gamma_0$. The two surfaces in this figure correspond to the two extremes values of $c_4$ between which all the laser modes are locked. In all cases, the upper limit for $c_4$ is larger than the lower limit, as already observed in Fig. 3(a). The locking range increases almost linearly with the saturable absorption rate $r_{\textrm{a}}$, except on the positive $c_4$ side where it decreases for large values of $r_{\textrm{a}}$. This corresponds to situations where the laser is close to the border to Q-switching regime [16].

 

Fig. 4. (a) Evolution of the mode-locking threshold values for $c_4$ versus $r_{\textrm{a}}$ and $r_{\textrm{g}}$. (b) Evolution of the locking region (red region) versus $\gamma_0$. $g_0/\gamma_0=120$ and $a_0/\gamma_0=10$ are fixed. Same parameter values as in Fig. 2.

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If the ratios $r_{\textrm{g}}=g_0/\gamma_0=120$ and $r_{\textrm{a}}=a_0/\gamma_0=10$ are fixed, varying the intrinsic losses $\gamma_0$ leads to the evolution of the locking region plotted in Fig. 4(b). The width of the region linearly increases with $\gamma_0$. In fact, this increase is due to the increase in saturable absorption $a_0$. At the same time, the increase in gain $g_0$ ensures that the laser is far away from the Q-switching region.

3.2 Potential disorder

The non-parabolicity of the potential can be regarded as a special kind of distortion. Here we extend the investigation of the robustness of mode locking from the influence of non-parabolicity to the influence of random errors in the shape of the potential. A term ${D}_{\textrm{max}}R(x)$ is thus added to the potential to describe this disorder, as given by Eq. (9). When the laser reaches its steady-state regime, the method of Eq. (10) is implemented to calculate the mode frequencies $f_n$.

To assess whether the laser remains locked in the presence of disorder, we define the free spectral range (FSR) relative root mean square (RMS) deviation $\delta$:

Before applying this criterion to the case of the disordered potential, let us first test its usefulness by applying it to the case of the non-parabolicity of the potential studied in the preceding subsection. We thus calculate $\delta$ according to Eq. (11) for the data of Figs. 1(b)-(d) and 3. We start with the “cold” cavity data of Fig. 1(b)-(d) and plot the corresponding FSR relative RMS deviations $\delta_{\textrm{C}}$ versus $c_4$ as open circles in Fig. 5(a). We then retrieve the linear evolution of the cold cavity FSR versus $c_4$, which was already observed in Figs. 1(c) and (d). The same treatment is applied to the laser data of Figs. 3(b)-(d), leading to the evolution of $\delta_{\textrm{L}}$ versus $c_4$ shown as triangles in Fig. 5(a). The locking region, in which $\delta_{\textrm{L}}$ remains equal to zero, is clearly visible in this figure. This shows that $\delta$ can be used as a criterion to recognize mode locked operation.

 

Fig. 5. (a) FSR relative RMS deviations $\delta_{\textrm{L}}$ and $\delta_{\textrm{c}}$ as a function of $c_4$. (b) Map of $\delta_{\textrm{L}}$ versus $\delta_{\textrm{C}}$ for the same values of $c_4$ as in (a). The blue (green) triangles correspond to unlocked solution with $c_4<0$ ($c_4>0$). The data for (a) and (b) are from Fig. 3 and Figs. 1(b)-(d). (c) Distribution of cold cavity FSR relative RMS deviation $\delta_{\textrm{C}}$ versus $D_{\textrm{max}}$. The calculation is run 50 times for each value of $D_{\textrm{max}}$. (d) Example of histogram of $\delta_{\textrm{C}}$ for $D_{\textrm{max}}=0.07$.

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It is worth observing the evolution of $\delta_{\textrm{L}}$ versus $\delta_{\textrm{C}}$, when $c_4$ is varied, as plotted in Fig. 5(b). It is clear that there exists a threshold value for $\delta_{\textrm{C}}$ below which locking is preserved.

The same method is implemented to calculate the FSR relative RMS deviation $\delta_{\textrm{C}}$ for the “cold” cavity for a randomly disordered potential. We run the simulation for $30\;\textrm{ns}$, removing the active media and losses ($\gamma_0=0$, $a_0=0$, $g_0=0$). The spectra are obtained from the last $24\;\textrm{ns}$ of each simulation, so that the frequency resolution is $0.042\;\textrm{GHz}$. The eigenfrequencies are extracted from the peaks of the field spectrum following Eq. (10) and used to calculate $\delta_{\textrm{C}}$ according to Eq. (11). This leads to the evolution of $\delta_{\textrm{C}}$ versus $D_{\textrm{max}}$, which is plotted in Fig. 5(c). Because of the random character of the potential disorder, the calculation is run 50 times for each value of $D_{\textrm{max}}$. Hence, in total this figure contains $1150$ different solutions for 23 different values of $D_{\textrm{max}}$. The general trend is that the range of values of $\delta_{\textrm{C}}$ increases linearly with $D_{\textrm{max}}$. One example of histogram of $\delta_{\textrm{C}}$ for $D_{\textrm{max}}=0.07$ is shown in Fig. 5(d).

To evaluate the robustness of mode locking to disorder, the RMS deviation $\delta_{\textrm{L}}$ is calculated for laser operation with the same parameters as in the preceding subsection and including the same potential disorder as in the case of the cold cavity. The results for $r_{\textrm{a}}=10$ and $r_{\textrm{g}}=150$ are shown in Figs. 6(a)-(d). Obtained from $1150$ solutions for 23 different values of $D_{\textrm{max}}$, the values of $\delta_{\textrm{L}}$ are plotted as a function of $D_{\textrm{max}}$ in Fig. 6(a). Some values of $\delta_{\textrm{L}}$ drop to $0$, contrary to what was happening in Fig. 5(c). Especially, for $D_{\textrm{max}}=0.005$, all the values of $\delta_{\textrm{L}}$ are close to $0$. This indicates that an equally spaced comb is formed by mode locking.

 

Fig. 6. (a,e) Distribution of laser FSR relative RMS deviation $\delta_{\textrm{L}}$ versus $D_{\textrm{max}}$. (b,f) Evolution of $\delta_{\textrm{L}}$ versus $\delta_{\textrm{C}}$. (c,g) Corresponding histograms of $\delta_{\textrm{L}}$ for all solutions (red) and solutions exhibiting mode locking (blue). (d,h) Same as (b,f) versus $\delta_{\textrm{C}}$. The fits are obtained from a Weibull distribution. (a-d) $r_{\textrm{a}}=10,\;r_{\textrm{g}}=150$; (e-h) $r_{\textrm{a}}=30,\;r_{\textrm{g}}=150$. All other parameter values are the same as in Fig. 2.

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Since we used exactly the same disordered potential in both cases, we can plot the value of $\delta_{\textrm{L}}$ versus $\delta_{\textrm{C}}$ , as shown in Fig. 6(b). We notice that there exists a threshold value $\delta_{\textrm{C,th}}$ of $\delta_{\textrm{C}}$ below which the laser remains locked. This means that the nonlinear locking effect can somewhat compensate the disorder in the potential.

To define and find this threshold value $\delta_{\textrm{C,th}}$, we first plot the histogram of $\delta_{\textrm{L}}$ for all the solutions (see Fig. 6(c)). It is clear that the histogram of $\delta_{\textrm{L}}$ can be divided into two different regions. The unlocked solutions distribute along a wide range of values of $\delta_{\textrm{L}}$ (marked by red bars). However, there are 153 solutions located in the first bin within the range $[0,1.51\times 10^{-4}]$ of values of $\delta_{\textrm{L}}$ (marked by blue bars). Figure 6(d) compares a histogram of the values of $\delta_{\textrm{C}}$ corresponding to mode locked operation, together with a histogram of all the values of $\delta_{\textrm{C}}$. One can notice that only a portion of the solutions with rather small values of $\delta_{\textrm{C}}$ are locked. From this figure, we can define $\delta_{\textrm{C,th}}$ as the threshold value above which less than 50% of simulations lead to mode locking.

One way to directly find $\delta_{\textrm{C,th}}$ would be to simply calculate the ratio between the number of locked solutions and the total number of simulations for each value of $\delta_{\textrm{C}}$. However, this strongly depends on the number and width of the bins in the histogram. To circumvent this problem, we fit the histograms using a Weibull distribution. The probability density function (PDF) of a Weibull random variable is expressed as

The results of these fits are shown as full lines in Fig. 6(d), where the fit parameters are $\lambda =5.61$ and $k=1.37$ for the ensemble of all solutions and $\lambda =0.98$ and $k=1.53$ for mode-locked solutions. This leads to the threshold value $\delta_{\textrm{C,th}}=1.2\times 10^{-3}$, evaluated as the value of $\delta_{\textrm{C}}$ for which the ratio of the two distributions is $0.5$.

The same calculations are performed for $r_{\textrm{a}}=30,\;r_{\textrm{g}}=150$. The corresponding results are plotted in Figs. 6(e)-(h). The fit parameters are unchanged for the ensemble of solutions and become $\lambda =2.7521$ and $k=1.395$ for the mode-locked solutions. We obtain in this case a larger threshold value $\delta_{\textrm{C,th}}=3.8\times 10^{-3}$ for mode-locking.

Using this method, we calculate the evolution of $\delta_{\textrm{C,th}}$ as a function of $r_{\textrm{a}}$, which is shown in Fig. 7(a) for different values of $r_{\textrm{g}}$. The threshold $\delta_{\textrm{C,th}}$ increases linearly with the saturable absorption rate $r_{\textrm{a}}$. However, the solutions finally approach the limit of Q-switching when $r_{\textrm{a}}$ is increased. This leads to a sharp decrease in $\delta_{\textrm{C,th}}$. On the contrary, increasing $r_{\textrm{g}}$ can keep away from that border so that higher values of $r_{\textrm{a}}$ can be allowed. This indicates that the robustness of locking to the disorder increases with the saturable absorption rate $r_{\textrm{a}}$.

 

Fig. 7. Evolution of the mode-locking threshold value of $\delta_{\textrm{C,th}}$ with different parameters: (a) $\delta_{\textrm{C}}$ versus $r_{\textrm{a}}$ for $\alpha_{\textrm{g}}=\alpha_{\textrm{a}}=0$ and several values of $r_{\textrm{g}}$; (b) $\delta_{\textrm{C}}$ versus $\alpha_{\textrm{g}}=\alpha_{\textrm{a}}$ for several values of $r_{\textrm{a}}$ and $r_{\textrm{g}}=150$; (c-d) $\delta_{\textrm{C}}$ versus $\alpha_{\textrm{g}}$ and $\alpha_{\textrm{a}}$ for $r_{\textrm{a}}=10$ and $r_{\textrm{g}}=150$. The dashed lines in each subfigure represent the solutions for the case where the gain region is located in the region $-2.5\;x_\Omega \leq x\leq 0$ and the absorber in the region $0\leq x\leq 2.5\;x_\Omega$. Other parameter values: $w=5x_\Omega$, $\tau_g=1\;\textrm{ns}$, $\tau_a=10\;\textrm{ps}$, $2\pi \gamma_0/\Omega =0.1$ and $E_{\textrm{sat,g}}/E_{\textrm{sat,a}}=25$.

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The same technique is used to investigate the influence of the Henry factors $\alpha_{\textrm{g}}$ and $\alpha_{\textrm{a}}$ of the gain medium and the absorber, respectively. For simplicity, we take $\alpha_{\textrm{g}}=\alpha_{\textrm{a}}$ and plot in Fig. 7(b) the evolution of $\delta_{\textrm{C,th}}$ as a function of $\alpha_{\textrm{g}}=\alpha_{\textrm{a}}$, with $r_{\textrm{g}}=150$ and for different values of $r_{\textrm{a}}$.

It is interesting to notice that the Henry factor first increases the threshold $\delta_{\textrm{C,th}}$. In other words, it increases the robustness of mode-locking to disorder, especially for small values of $r_{\textrm{a}}$. But, clearly, a large value of the Henry factors is detrimental to locking, as reported in several Refs. [16,55,56], so that the threshold $\delta_{\textrm{C,th}}$ eventually decreases.

We also investigate the robustness of mode locking to different values of the Henry factors for the gain medium and the absorber (see Fig. 7(c)). This result is reminiscent of previous works applied to standard semiconductor lasers [55–59], in which the difference between the Henry factors was found to be detrimental to mode locking. In our case, we find that this difference between the Henry factors reduces the robustness of the nanolaser mode locking to potential disorders.

In Ref. [16], we found that the dynamical behaviors of the lasers were similar in the situations where the gain and the absorber overlap or are separated inside the cavity. We thus also investigate here the robustness of the laser mode-locking to potential disorder in the situation where the gain and the absorber do no longer overlap. The values of the parameters as the same as before, except that the gain region is located in the region $-2.5\;x_\Omega \leq x\leq 0$ and the absorber in the region $0\leq x\leq 2.5\;x_\Omega$. The results are plotted as dashed lines in Figs. 7(a),(b), and (d). In Fig. 7(a), for separated gain and absorber regions, $\delta_{\textrm{C,th}}$ is found to increase with $r_{\textrm{a}}$ for $r_{\textrm{g}}=150$. One significant difference with respect to the case where the two media overlap is that $\delta_{\textrm{C,th}}$ is divided by a factor of 2. This is due to the fact that the pulse just stays only half of the time in the absorber region. The effective absorption for the whole field is then only one half of its value, leading to a reduction of $\delta_{\textrm{C,th}}$ to one half of its value when the absorber fills both sides of the resonator. The same results are observed for the investigation of the robustness to the Henry factors with $r_{\textrm{g}}=150$ and $r_{\textrm{g}}=10$ in Fig. 7(b) and Fig. 7(d).

4. Conclusion

In this paper, we have theoretically analyzed the robustness of the mode locked regime of oscillation in a harmonic cavity nanolaser, sustaining oscillation of Hermite-Gaussian modes, to imperfections of its harmonic potential. First, we considered the effect of the non-parabolicity of the potential. A fourth-order term was introduced to quantify the distortion of the potential, which leads to shifts of the eigenfrequencies. We found that locking the modes using a saturable absorber can to some extent compensate for this distortion.

Moreover, the more general case of random errors in the potential has been statistically analyzed by calculating the relative root mean square (RMS) deviation $\delta_{\textrm{C}}$ of the comb frequencies. By comparing the distribution of $\delta_C$ for the locked solutions with that for all the solutions, we could obtain the evolution of the threshold value $\delta_{\textrm{C,th}}$ with the laser parameters. The robustness of mode locking to disorder was found to increase with the saturable absorption rate on the condition that the gain is large enough to keep the laser away from the Q-switching regime. A well-chosen value for the Henry factors has been shown to increase the robustness of the mode-locked regime. Similar results are obtained for the robustness of the laser to the non-parabolicity of the potential and are reported in Supplement 1.