A particular category of structured light beams, called optical vortices (OVs), was discovered a few decades ago, and since then has been considered for many applications. Such OV beams exhibit a phase singularity at the center, around which the phase accumulated by the field is an integer multiple of $2\pi$. This integer number is known as the topological charge (TC)/orbital angular momentum (OAM) of OV beams. The annular intensity distribution of OV has been found useful for optical trapping [1–3], micromotoring [4,5], microscopy [6], nano-structuring [7], and optical data transfer [8–11].

In many of these applications, it was shown that improved performance with additional control could be achieved by introducing intensity and/or phase asymmetry in the beam, which then becomes an asymmetric optical vortex (AOV) beam. Particularly, the asymmetry has been exploited to form flexible optical tweezers, which are used in many areas of physics [12–14]. For example, the rate of microparticle motion was shown to increase linearly with the asymmetry of vortex-carrying Bessel- or Laguerre-Gaussian beams [12,13]. The thermal damage of the live cells was also reported to be better managed when the symmetry of OV is broken [12]. The asymmetry has also been exploited in generating a pair of entangled photons with OAM using spontaneous parametric down-conversion [15].

For some particular applications, this maximization of the vortex beam asymmetry may be detrimental to some extent for parameters. For example, an experiment involving particle trapping would become less efficient in terms of particle conservation if the involved OV is asymmetric. A trade-off must also be found when one considers OAM-based optical data transfer [8,9]. The vortex-based information storage using the topological charge can be improved using some additional information encoded in the non-symmetrical intensity profile [16]. A large asymmetry can allow the beam to increase its storage capacity. However, a strong deformation of the OV can make the information carried by the phase profile more difficult to read. The asymmetric aberration laser beams have also been demonstrated for obtaining additional control on generating high-energy density at desired spatial locations [17]. Other types of asymmetric beams have also been investigated such as Kummer laser beams with transverse complex shifts [18], asymmetric Gaussian optical vortices [19], and asymmetric Bessel modes [20]. Consequently, the flexibility and adaptive control of the beam profile are critical for all these applications.

The most common solutions for AOV beam generation are the optical system misalignment, digital micromirror device (DMD), spatial light modulator (SLM), diffractive optical elements (DOEs), micrograting arrays, astigmatic mode converters, and pinhole plate [17,21–30]. In most of these works, the AOV beams are generated with a single laser and thus can pose power limitations for various applications. Further, the generation of AOV beams external to the laser source using DMD and SLM can further reduce the output power due to filtration of higher diffraction orders containing considerable power. To overcome these limitations as well as to get high-purity beams, the most promising approach is adaptive beam-shaping integrated directly into the laser source [31]. To this aim, the dynamics of OV generation in degenerate cavity solid-state lasers has been broadly studied. In particular, the self-healing properties of OVs generated by phase-locked laser arrays as well as the probability of generating different topological charges have been thoroughly investigated [31–35].

The probability to generate OV with certain TC was predicted to be also quite large in degenerate cavity lasers based on semiconductor gain chips, namely, VECSELs (vertical external cavity surface emitting lasers). Moreover, this probability was shown to depend on the value of the Henry factor in such gain structures [36]. Additionally, these lasers are known for their smooth class-A synchronization dynamics [36] and very low-noise operation [37]. The gain chip production technology allows to choose the best resonant wavelength in the near-infrared range for any particular application. Thus degenerate VECSELs are now considered to be promising solutions for all of the above-mentioned applications.

The aim of the present paper is thus to theoretically explore the dynamics of non-symmetrical OV generation using a ring array of lasers formed in a degenerate VECSEL. The rotational symmetry of the system is broken by altering the geometry of the amplitude-mask, which forms a ring array of lasers. Thus one originality of the approach discussed here is to investigate different kinds of asymmetric masks, namely, point defects, gradient in diameter, and random hole diameters. The difference of such masks with respect to uniform hole parameters is expected to affect the coupling coefficient between the lasers and thus affect the dynamics of the system and the choice of the system steady state. Moreover, another original point addressed here is to assess the influence of the Henry factor, which is particular to semiconductor gain media [36], on the probability to achieve phase-locking of the laser array with non-symmetrical OV phase differences, which is studied in detail.

We investigate AOV generation using a ring array of lasers based on a degenerate cavity VECSEL. The scheme of such a laser system is shown in Fig. 1. The semiconductor gain chip typically consists of a Bragg reflector and a few optically pumped quantum wells, thus acting as an amplifying planar mirror. The cavity is closed by a second planar mirror, and its self-imaging properties are obtained by inserting two lenses forming a telescope. An amplitude mask is inserted close to the output coupler (OC), i.e., close to the self-imaging point of the intracavity telescope. In this configuration, every hole in the mask defines an individual laser. The gap between the mask and output coupler has a length of $z$ and influences the amount of laser light injected into neighboring apertures due to diffraction at the mask’s edges.

For the modeling, we rely on the values of the parameters of our recent experiments [38] in which $f_1=5\mathrm{cm}$, $f_2=20\mathrm{cm}$, and the holes in the mask form a regular circular pattern with diameter of 200 μm and center-to-center separation between them of 250 μm.

A mask pattern in which the holes have identical diameters will be called a “uniform” mask geometry. In the present paper, we investigate the generation of AOV beams by introducing different kinds of asymmetries in the array, called “gradient”, “random”, and “point defect” masks, as presented in Fig. 2. In the point defect case, one of the holes has a smaller or bigger diameter than the others. In the gradient case, the hole diameters increase gradually from the first to the last one. Finally, in the random configuration, the hole diameters are randomly chosen around some average value.

The hole diameters in Fig. 2 are not to scale with the array geometry. In agreement with experiments [36], the standard hole diameter is taken to be 200 μm and the center-to-center separation between two successive holes is $d=250\mathrm{\mu m}$. In the following, we consider arrays of 20 lasers. The details of the different masks are given in Table 1.Table 1.

Hole Diameters (in µm) in Different Chosen Configurations of 20 Lasers^a

At every round-trip inside the cavity, each laser undergoes diffraction by its mask hole, and after reflection on the planar mirror, a fraction of the diffracted light is injected into its neighbors. This process is schematically shown in Fig. 3. The corresponding coupling parameter is obtained by calculating the overlap between the diffracted laser field and the neighboring laser mode profiles. To calculate the wavefront diffracted by the laser labeled by $i$, we suppose that the incident field on the circular hole has a uniform amplitude $E_{0,i}$. Then, after propagating along the distance $2z$, i.e., twice the distance between the mask and the output coupler, the diffracted beam profile is obtained by the following Huygens-Fresnel equation in cylindrical coordinates as [39] $$\begin{gathered} E_{s,i}({\rho }^{\prime })=\frac{i\pi E_{0,i}}{z\lambda }{\int }_0^{{\sigma }_i/2}\rho \exp \left[\frac{-i\pi ({\rho }^2+{\rho }^{\prime 2})}{2z\lambda }\right] \\ \times J_0\left(\frac{\pi \rho {\rho }^{\prime }}{z\lambda }\right)\mathrm{d}\rho , \end{gathered}$$(1)where $E_{s,i}$ is the field diffracted at a distance $2z$ from the $i$th hole with diameter ${\sigma }_i$, and where $\rho$ and ${\rho }^{\prime }$ are the radial coordinates in the planes of the incident and diffracted fields, respectively. The VECSEL wavelength is taken to be $\lambda =1\mathrm{\mu m}$.

The diffracted waveform exhibits strong oscillations, whose characteristics mainly depend on the hole diameter ${\sigma }_i$ and on the propagation distance $z$. The fraction of the field from laser number $i$ injected by diffraction to laser number $i\pm 1$ at each round trip is a complex coupling coefficient, whose value can be calculated by projecting the field diffracted by hole $i$ on laser $i\pm 1$ as $${\kappa }_{i\to i\pm 1}=\frac{{\int }_0^{2\pi }\mathrm{d}{\varphi }_{i\pm 1}{\int }_0^{{\sigma }_{i\pm 1}/2}{\rho }_{i\pm 1}\mathrm{d}{\rho }_{i\pm 1}{E}_{0,i\pm 1}^{*}({\rho }_{i\pm 1})E_{s,i}\left(\sqrt{d^2+{\rho }_{i\pm 1}^2+2d{\rho }_{i\pm 1}\cos {\varphi }_{i\pm 1}}\right)}{\sqrt{{\int }_0^{2\pi }\mathrm{d}{\varphi }_i{\int }_0^{{\sigma }_i/2}{\rho }_i\mathrm{d}{\rho }_i{|E_{0,i}({\rho }_i)|}^2{\int }_0^{2\pi }\mathrm{d}{\varphi }_{i\pm 1}{\int }_0^{{\sigma }_{i\pm 1}/2}{\rho }_{i\pm 1}\mathrm{d}{\rho }_{i\pm 1}{|E_{0,i\pm 1}({\rho }_{i\pm 1})|}^2}}.$$(2)In Eq. (2), $({\rho }_i,{\varphi }_i)$ and $({\rho }_{i\pm 1},{\varphi }_{i\pm 1})$ are the cylindrical coordinates centered on aperture numbers $i$ and $i\pm 1$, respectively. We have also supposed that all the laser modes have a cylindrical symmetry and thus their fields depend only on their radial coordinates ${\rho }_i$ and ${\rho }_{i\pm 1}$. We treat the incident mode profiles on the mask holes as top hat profiles with a width equal to ${\sigma }_i$ and in a range given by Table 1. Thus, the denominator of Eq. (2) simply becomes $\pi {\sigma }_i{\sigma }_{i\pm 1}|E_{0,i}{E}_{0,i\pm 1}|/4$. The calculations of the overlap with more realistic field profiles would add some scaling factor to its modulus value, but would not affect the coupling argument.

Figure 4(a) shows the modulus of the field $E_{s,i}$ diffracted from laser number $i$, calculated with different distances between the mask and OC, namely, $z=500\mathrm{\mu m},1000\mathrm{\mu m}$, and 2000 μm. The edges of the uniform mask are marked as black circles. The period of the field oscillations increases with an increase in the distance $z$ between mask and OC. It is easy to understand then that the argument of the complex coupling can be modified by the variation of the mask hole dimensions.

As shown in Fig. 4(a), the modulus of the coupling coefficient between neighboring lasers generally increases with distance $z$. Moreover, Fig. 4(b) shows the dependence of the argument ${\theta }_{i\to i\pm 1}$ of the field coupling coefficient ${\kappa }_{i\to i\pm 1}$ on the diffraction path length $z$, i.e., the distance between mask and OC. Both holes labeled $i$ and $i\pm 1$ have diameters of 200 μm. The argument of the overlap varies very fast for small distances $z$. It is thus expected to be hard to control the phase-locking in this range, because of the easy break of symmetry of the laser array through the tilts of the mask, and hole diameter variations. A very precise control of parameters would be needed in this case. The distances $z$ longer than 1 mm are characterized by much slower variations of the coupling argument, and thus robust phase-locking conditions even with significant asymmetries of the laser array. Due to these reasons, we study the effect of the gradient and random masks with a diffraction range $z$ larger than 1 mm.

We then calculate the coupling coefficient for different coupling geometries and different distances between the mask and output coupler. Figure 5 shows the computed values of the modulus and argument of the coupling coefficients ${\kappa }_{i\to i+1}$ (solid lines) and ${\kappa }_{i\to i-1}$ (dashed lines) versus laser index $i$ for the different schemes shown in Fig. 2, with the values of the hole diameters given in Table 1 and distance between two neighboring holes (center-to-center) $d=250\mathrm{\mu m}$.

In the case of the uniform mask [see Figs. 5(a) and 5(b)], the coupling coefficient is independent of $i$. The “big” and “small” defect masks respectively lead to an increase or a decrease of the coupling of the first hole with its neighbors.

As expected, the random mask [see Figs. 5(c)–5(f)] leads to random variations of the coupling coefficient across the different lasers, while the gradient mask leads to a progressive variation of the coupling with $i$. Figures 5(b), 5(d), and 5(f) underline how the variations of the hole diameters affect not only the modulus of the coupling but also its phase. We believe that these variations of ${\theta }_{i\to i\pm 1}$ will favor the solutions exhibiting a non-zero topological charge, as we investigate in the following sections.

To describe the dynamics of the VECSEL array and analyze its phase-locking behavior, we use a standard rate equation approach. Thanks to the class-A nature of the VECSEL, one can eliminate adiabatically the population inversion dynamics [36]. In the system of $n$ lasers, the laser labeled by $i\in [1,n]$ is described by a complex field amplitude $E_i=A_i(t){\mathrm{e}}^{\mathrm{i}{\varphi }_i(t)}$, where $A_i(t)$ and ${\varphi }_i(t)$ are its real-valued amplitude and phase, respectively. In the ring array configuration (Fig. 2), each laser field obeys the following set of coupled equations: $$\begin{gathered} \frac{\mathrm{d}{A}_i}{\mathrm{d}t}=-\frac{A_i}{2{\tau }_{\mathrm{cav}}}(1-\frac{r_i}{1+A_i^2/F_{\mathrm{sat}}}) \\ +\frac{c}{2L_{\mathrm{cav}}}[|{\kappa }_{i+1\to i}|A_{i+1}\cos ({\varphi }_{i+1}-{\varphi }_i+{\theta }_{i+1\to i})+|{\kappa }_{i-1\to i}|A_{i-1}\cos ({\varphi }_{i-1}-{\varphi }_i+{\theta }_{i-1,i})], \\ \frac{\mathrm{d}{\varphi }_i}{\mathrm{d}t}={\mathrm{\Omega }}_i+\frac{\alpha }{2{\tau }_{\mathrm{cav}}}\frac{r_i}{1+A_i^2/F_{\mathrm{sat}}} \\ +\frac{c}{2L_{\mathrm{cav}}}[|{\kappa }_{i+1\to i}|\frac{A_{i+1}}{A_i}\sin ({\varphi }_{i+1}-{\varphi }_i+{\theta }_{i+1\to i})+|{\kappa }_{i-1\to i}|\frac{A_{i-1}}{A_i}\sin ({\varphi }_{i-1}-{\varphi }_i+{\theta }_{i-1\to i})]. \end{gathered}$$(3)In these $2n$ coupled differential equations, ${\tau }_{\mathrm{cav}}$ is the photon lifetime, $r_i$ the excitation ratio of laser $i$, ${\mathrm{\Omega }}_i$ the frequency detuning between lasers, $c$ the speed of light, and $L_{\mathrm{cav}}=4(f_1+f_2)$ the cavity round-trip length. The real amplitude $A_i$ is normalized in such a way that its square corresponds to the photon number in the corresponding laser. Here, $\alpha$ is the Henry factor and $F_{\mathrm{sat}}$ is the saturation photon number, whose exact value serves as a scaling factor for the laser power and does not affect the laser dynamics. For simplicity, we consider the system with zero detuning (same value of ${\mathrm{\Omega }}_i$ for all lasers). In the experimental implementation, this happens in a perfectly degenerate system, thanks to identical cavity lengths and other laser array parameters. This simplification corresponds to a critical coupling value equal to zero [40]. Thus, the phase-locking can be observed for small values of the coupling coefficients ${\kappa }_{i\to i\pm 1}$.

In the numerical simulations described below, we have considered the parameters from our recent experiment [38], with a cavity length $L_{\mathrm{cav}}=0.5\mathrm{m}$ in which the photon lifetime is equal to ${\tau }_{\mathrm{cav}}=30\mathrm{ns}$. The steady-state solution of such a system of differential equations is known for exhibiting a multistability of solutions [36]. Following the same approach as given in our previous works [30,32], we can analytically derive the steady-state solutions for a simplified case, such as a mask with uniform holes (Fig. 2). Identical coupling parameters lead to phase-locking with the same amplitude all over the array and $A_i=A_{\mathrm{st}}$ for every $i$ derived in Appendix B and given by Eq. (4) below. The phase-differences derived in Appendix B are also identical (${\psi }_i={\psi }_{\mathrm{st}}$) and given by Eq. (5): $$A_{\mathrm{st}}=\sqrt{F_{\mathrm{sat}}}\sqrt{\frac{r}{1-2\eta \cos {\psi }_{\mathrm{st}}\cos \theta }-1},$$(4)$${\psi }_{\mathrm{st}}=\frac{2\pi q}{n},$$(5)where $q\in \mathbb{Z}$ is often referred to as the topological charge and $\eta =|\kappa |\frac{{\tau }_{\mathrm{cav}C}}{L_{\mathrm{cav}}}$. Phase-locking with $|q|=1$ corresponds to the simplest case of optical vortex, with $2\pi$ phase-shift accumulated around the center of the beam.

To analyze the ring laser array in various mask configurations (Fig. 2), we have numerically solved the laser rate equations using Wolfram Mathematica software (with the Adams method of the NDSolve). Since the system exhibits multistability, the steady-state solution and the associated topological charge that the system will eventually reach strongly depend on the initial conditions. It is possible to obtain a vortex solution for all of the mask configurations within several runs (realizations) of the rate equation with different initial conditions. For example, Fig. 6 shows the values of the phase differences ${\psi }_i={\varphi }_{i+1}-{\varphi }_i$ between successive lasers for two different sets of initial conditions, leading to $q=0$, without a vortex [see Fig. 6(a)], and $q=1$, which is a vortex solution [see Fig. 6(b)]. Indeed, we can see that the numerically calculated ${\psi }_i$ are close to zero in Fig. 6(a) and to $2\pi /20$ in Fig. 6(b), as expected from the analytical solutions of Eq. (5) with $q=0$ and 1.

It is important to mention that we define the topological charge as $q={\sum }_{i=1}^n\arg (E_i^{*}{E}_{i+1})/2\pi$ [32]. For all kinds of non-uniform masks, we can see that the steady-state phase differences no longer satisfy Eq. (5). It is particularly interesting to have a closer look at what happens with the point defect masks [see Figs. 6(a) and 6(b)]. We can see that a variation of the diameter of a single hole in the mask results in a redistribution of the perturbation all over the laser array even though only the nearest neighbors are coupled to that particular hole. Figures 6(c) and 6(d) show the same redistribution effect for the gradient and random masks. One notices that a gradual variation of the hole sizes leads to smooth variations of the relative phase from hole to hole, while random variations do not.

By utilizing the amplitudes and phase differences of the numerically determined steady-state solutions, we can visualize the far-field patterns of the VECSEL array as shown in Fig. 7. Corresponding to various ring array masks (Fig. 2), the total field is calculated by treating each laser as the fundamental Gaussian distribution, with the amplitude given by the rate equations $A_{\mathrm{st},i}$ and the phase differences according to ${\varphi }_{\mathrm{st},i}$. The arrangement of lasers in the ring arrays (corresponding to masks) with the amplitude and phase distributions obtained numerically by the laser rate equations serves as the near-field distributions [34]. The corresponding far-field intensity distributions are obtained by taking the Fourier transform of the near-field distributions. Different ring array masks (Fig. 2) possess different hole configurations and hence different coupling configurations, which leads to the different steady-state values of amplitude and phase distributions of lasers (Fig. 6), and therefore results in different far-field intensity distributions (as shown in Fig. 7).

The doughnut shape and phase singularity are preserved with the non-uniform masks but with a strongly varying amount of distortion. By comparison with the case of the uniform mask of Fig. 7(a), it is easy to see that the vortices shown in Figs. 7(c), 7(f), and 7(g) have a less distorted phase profile than the other ones. These cases correspond to a mean coupling argument ${\theta }_{i\to i\pm 1}$ closer to $2\pi /n$ than the other cases.

The patterns in Figs. 7(b), 7(d), and 7(e) show more asymmetric beam structures with complicated phase profiles. In particular, their singularity point is shifted from the center of the beam because of the strong variations of the coupling argument around its mean value ${\theta }_{i\to i\pm 1}$. For example, the strongest deviation of the coupling strength is found for the “big” defect mask between the lasers labeled $i=20$, 1, and 2, where the defect hole corresponds to $i=1$. This leads to a large spatial shift of the topological defect from the center of the beam, as can be seen in Fig. 7(b).

Each intensity pattern presented in Figs. 7(b)–7(g) can be useful as an asymmetric optical vortex. As these AOVs are obtained by phase-locking a ring array of lasers, the output power of the system can be larger than conventional AOVs obtained by a single laser. Thus, these high-power AOVs can be potentially useful in various applications as mentioned earlier. For example, the rate of micro-particle motion is shown to increase linearly with the asymmetry of vortex-carrying Bessel- or Laguerre-Gaussian beams [12,13]. The thermal damage of the live cells is also reported to be better managed when the OV symmetry is broken [12]. Another example of application lies in the fact that vortex-based information storage using the topological charge can be improved by encoding additional information in the non-symmetrical intensity profile [16].

The system has a chance to end up phase-locked in a vortex solution by itself, depending on the initial conditions. However, stable phase-locking cannot be observed for every set of parameters of the system. Among these parameters, a peculiarity of semiconductor lasers is the presence of a relatively large Henry factor. In preceding works, it was established that this factor limits the probability to obtain a vortex solution [36]. In the case of weak real positive coupling, it was shown that the probability for the laser array to exhibit a vortex solution with $|q|=1$ becomes negligible if the Henry factor increases above limiting value ${\alpha }_{\lim }=2/\tan {\psi }_q$. However, this investigation was limited to real-valued coupling coefficients, and thus needs to be re-considered for the complex coupling coefficients that we consider here.

We thus follow the same approach to obtain a more general expression for the complex coupling case. First of all, we linearize the rate equations given by Eq. (3) around the steady-state solutions of Eqs. (4) and (5). In this way, we obtain the Jacobian of the system, from which the stability of the steady-state solution can be analyzed based on its trace and its determinant. A given steady-state solution is stable if the trace is negative and the determinant positive.

The details of the calculations are given in the Appendix A. They show that the condition on the trace is always fulfilled for our system. Furthermore, the condition on the determinant leads to the following stability criterion: stable phase-locking with ${\psi }_q$ is possible only when the $\alpha$ factor of the gain chip is smaller than a limit value given by $${\alpha }_{\lim }=\frac{2\cos \theta \cos {\psi }_q}{\sin ({\psi }_q-\theta )}.$$(6)The limitations are different for different topological charges and vary with the number of lasers. Additionally, we can see a clear dependency of the limiting Henry factor on the coupling argument.

A graphical illustration of Eq. (6) is given in Fig. 8(a) for $|q|=1$, 2, and 3. The horizontal dashed lines correspond to the limiting values of the Henry factor predicted in Ref. [36] for the real coupling factor, i.e., $\theta =0$, as given by the blue squares. We then compare the analytical formula of Eq. (6) with numerical simulations in the case of a uniform mask with different values of the coupling argument ($\theta =-\pi /2,-\pi /10,-\pi /20,\pi /20,\pi /10$, and $\pi /2$) and with $|{\kappa }_{i\to i\pm 1}|=0.28\times {10}^{-3}$. We calculate the number of outcomes phase-locked with $|q|=1$ among the 500 runs of the rate equations of Eq. (3) with different initial conditions. From this, we obtain the value of $\alpha$ above which the probability to obtain a steady-state solution with $|q|=1$ becomes negligible. These simulations lead to the red dots in Fig. 8(a), which perfectly confirm the analytical result of Eq. (6).

Figure 5(a) has shown that the uniform mask leads to a coupling argument $\theta =\pi /10$ when $z=500\mathrm{\mu m}$. We thus take this value to compute the probabilities of the different solutions shown in Fig. 8(b). As expected from Fig. 8(a), no limit value for the Henry factor was observed in this case. Indeed, the analytically predicted value for this case exceeds 10. This shows that if one is able to control the value of the argument of the coupling coefficient, then steady-state solutions carrying a non-zero topological charge can be obtained with significant probability even with large values of the Henry factor.

In this section, we investigate whether the probabilities to generate phase-locked vortex solutions with $|q|=1$ and 2 can be increased by using different non-uniform masks. The probabilities obtained for the “big” and “small” point defect masks with the coupling coefficients presented in Figs. 5(a) and 5(b) are shown in Fig. 9. They are calculated with 500 random initial conditions for each value of $\alpha$. The probabilities for the uniform [see Fig. 8(b)] and “big” defect [see Fig. 9(a)] masks are very close. In both cases, the argument of the coupling coefficient has a positive sign for each pair of lasers in the array. The limit value for the $\alpha$ factor is very large in both cases.

A very different result is obtained for the “small” defect mask, as can be seen in Fig. 9(b). In this case, the limit value of the Henry factor is approximately ${\alpha }_{\lim }=3.8$, which is very small compared to the preceding cases. Only the $q=0$ solution appears to be stable in the range $3.8\le \alpha \le 5.3$. There is an interesting effect close to $\alpha =2$, where 100% of the phase-locked cases lead to a non-zero topological charge. The width of this region, in which the $q=0$ experiences large losses, increases with a decrease of the coupling strength $|\eta |$, but the physical explanation of this effect is not fully clear. None of the masks investigated in Fig. 9 leads to an asymmetry between solutions corresponding to positive and negative values of $q$ and the probabilities of finding OV with positive and negative TCs remain equal.

The behavior of the laser array with the gradient and random masks, calculated for three different values of the distance $z$, is shown in Fig. 10. The first two rows of the figure are calculated for $z=1050\mathrm{\mu m}$ and 1100 μm, a situation in which the signs of ${\theta }_{i\to i\pm 1}$ can be either positive or negative depending on the considered pair of lasers. This makes the average of these arguments over the array be equal approximately to $-\pi /60$ and $\pi /25$ in Figs. 10(a) and 10(b), respectively. Thus the phase-locking ranges decrease or increase with respect to the preceding case, as expected from Eq. (6). The arguments of the coupling coefficients are all positive in the cases of Figs. 10(e) and 10(f), which have been calculated for $z=1200\mathrm{\mu m}$.

Interestingly, the results for the gradient mask exhibit a break of symmetry (unequal probability) between positive and negative values of $q$ for $2.5\le \alpha \le 2.8$ in Fig. 10(a) and $5.8\le \alpha \le 8$ in Fig. 10(c). This is a promising result for the generation of anti-vortices. Also, a small range of values of $\alpha$ around 0.4 exhibit destabilization of the $q=1$ solution, leading consequently to a symmetry breaking between positive and negative vortices in Fig. 10(c). This breaking of the symmetry just gives an advantage of a few percent for the probability $P_{q=1}$ over $P_{q=-1}$.

The effect of the random mask is less spectacular. Let us first focus on the comparison of the data presented in Figs. 10(a) and 10(b), which respectively correspond to the gradient and random masks. Interestingly, the last one exhibits a narrow range of values of $\alpha$ ($3.8\le \alpha \le 4$), in which $P_{q=1}>P_{q=-1}$. The dominant solution is still $q=0$, but the degeneracy between positive and negative topological charges is lifted. At the same time, the probability of phase-locking of the laser array decreases dramatically in this range.

The distribution of the hole diameters in the random mask averages the non-uniformity of the mask parameters, thus explaining the fact that the symmetry-breaking range is significantly smaller in this case.

Comparing Figs. 10(e) and 10(f) shows that only the gradient mask leads to a predominance of the vortex solution over the anti-vortex one, however with a probability difference of only a few percents. Overall, the most promising mask geometry is the gradient one. The best positioning strategy is to first consider the limiting $\alpha$ factor effect within the $\alpha$ range of the used gain chip. The average of ${\theta }_{i\pm 1\to i}$ can then be obtained using Eq. (6), and the distance $z$ can be determined from Fig. 4(b) or from the calculation based on the specific mask being used.

In the next step, we consider the case of a virtual mask that adds phase shifts ${\theta }_{i\to i+1}=2\pi /n$ and ${\theta }_{i\to i-1}=-2\pi /n$ between lasers to force the system to phase lock according to the anti-vortex solution $q=-1$. The result is shown in Fig. 11. Figure 11(a) corresponds to the case where ${\theta }_{i\to i+1}=2\pi /n$ and ${\theta }_{i\to i-1}=-2\pi /n$ are valid only for $i=1$, while for the other lasers, this argument keeps the value obtained for the uniform mask. In Fig. 11(b) this condition is valid for $i=1$ and $i=2$. It is extended to half of the lasers, i.e., for $1\le i\le 10$ [see Fig. 11(c)], and to the entire array [see Fig. 11(d)].

The results of Fig. 11 show that such masks would indeed allow to increase the value of $P_{q=-1}$, and to lift the degeneracy between positive and negative values of the topological charge. Interestingly, this increase of $P_{q=-1}$ occurs thanks to a decrease of $P_{q=0}$. Moreover, when one increases the number of lasers in the array for which ${\theta }_{i\to i\pm 1}=\pm 2\pi /n$, the probability $P_{q=-1}$ increases, as it can be seen by comparing the successive plots in Fig. 11. In particular, when the number of lasers for which ${\theta }_{i\to i+1}$ is equal to $\pm 2\pi /n$ exceeds $n/2$, one obtains $P_{q=-1}\ge P_{q=0}$. In this case, the solution with non-zero topological charge becomes dominant, as can be seen, for example, in Fig. 11(d).

Finally, let us mention that the generation of asymmetric OVs with higher values of $q$ would probably be possible by increasing the number of lasers $n$.

We present a new approach for asymmetric vortex beam generation by phase-locking a ring array of lasers in a VECSEL. Particularly, we explore the impact of non-uniform masks on the phase-locking of a ring array of lasers, taking into account the complex nature of the coupling coefficients and the existence of a non-zero Henry factor. We have shown that the phase-locking of laser arrays and the predominance of the solution in which all the lasers are in phase are relatively robust to variations of the hole diameters with respect to their average values. However, we have also observed that strong deviations of the hole diameters are detrimental to the phase-locked operation of the laser array. For example, in the case of an amplitude mask containing holes with random diameters (random mask), the standard deviation in hole diameters of average diameter equal to 200 μm should not exceed approximately 2 μm at a distance $z=1\mathrm{mm}$ and 5 μm for $z=2\mathrm{mm}$. This also makes the fabrication of the mask by standard machining techniques a bit challenging.

We have observed that the in-phase solution always dominates irrespective of non-uniform masks. Moreover, we have found that the quality of the vortex beam, and its asymmetries, are strongly dependent on the complex coupling argument.

Further, we have predicted a possibility to make one particular vortex solution dominant over other possible steady-state solutions. This solution consists of imprinting the necessary phase shift among neighboring lasers in the argument of their coupling coefficients, namely, ${\theta }_{i,i+1}=2\pi /n$ and ${\theta }_{i+1,i}=-2\pi /n$. In particular, a larger number of holes satisfying this condition are favorable for the establishment of a given vortex solution. However, the design of a mask allowing to fulfill this condition for a significant number of holes in the array remains to be found.

We believe that this method can prove useful to select one desired vortex phase-locked solution in the laser array, in conjunction, if necessary, with far-field Fourier filtering. However, this can be possible only through further studies of “diffraction engineering”, to some extent related to what is required for laser neuron network training and control [41], which we plan to explore in future work.

Acknowledgment. VP acknowledges the funding from the Science and Engineering Research Board.