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Phase-locking dynamics of a 2D VCSEL hexagonal array with an integrated Talbot cavity

Yanhong Ma, Tian Lan, Xiaofan Wang, Renjie Ruan, Yinhua Cao, Jingjing Dai, and Zhiyong Wang

Open Access Open Access

Abstract

In this paper, phase-locking dynamics of 2D VCSEL hexagonal array with an integrated Talbot cavity are numerically investigated based on rate equations aiming at achieving high brightness output. The processes of wavelength synchronization and phase locking under different fill factors ff and fractional Talbot cavity lengths L were addressed comprehensively. Different supermodes of phase-locked VCSEL array were then analyzed from both near-field and far-field pattern, and proved to be well matched with the results of coupled-mode theory. With appropriate configuration the Talbot-VCSEL system can operate in a full in-phase mode eventually, which is beneficial for determining the parameter interval corresponding to the most expected single narrow-lobe far-field pattern. Furthermore, the simulation results also indicate that, considering the parametric interactions the distribution of optical feedback from the fractional Talbot cavity should be consistent as much as possible to facilitate the realization of phase-locked state. Our study could provide a theoretical support to obtain the full in-phase coupled VCSEL array with high performance.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Vertical-cavity surface-emitting laser (VCSEL), as one of the most promising candidates for the next generation of high-power and high-brightness semiconductor laser source, has attracted great attention due to its intrinsic advantages of small size, narrow divergence, high efficiency, high thermal stability, free of COMD (catastrophic optical mirror damage) and especially its high scalability to high power by two-dimensional integration [1]. Therefore, high-brightness VCSEL arrays are widely used in many fields, including industrial processing, medical therapy, free-space communication [2,3], 3D sensing [4], etc. However, simply integrating hundreds of thousands of VCSELs to form a large 2D array could indeed produce the required output power level, but without obtaining high beam quality because of the independence of individual VCSEL and the incoherent nature, even if every VCSEL element works in its single fundamental mode by using a small aperture (<10 µm), thus resulting in a large divergent spot and poor brightness of the output beam. One of the commonly-used methods to achieve high brightness is coupling the output of 2D VCSEL array into the fiber through a focusing-lens system [5], but the coupling efficiency is rather low and the single-lobe far-field pattern could hardly be sustained. In principle, to achieve the high brightness output of 2D VCSEL array directly, synchronization of both lasing wavelength and phase among the whole array element is of crucial importance. In other words, it means the coherent optical coupling among VCSELs needs to be established to guarantee a naturally phase-locked 2D VCSEL array. Ideally speaking, if the entire array oscillates in the in-phase mode, the far-field pattern will evolve into a single narrow central lobe whose spot size changes inversely with the number of elements [6].

In previous work, several phase-locking approaches have already been proposed to obtain the coherently coupled VCSEL array. Yoo et al. [7] firstly demonstrated the validation of the concept of a 40×40 phase-locked VCSEL array with 100 nm etched gap between adjacent elements to enable evanescent coupling, but the emission beam was many times the diffraction limit. In contrast to the evanescent coupling, antiguided coupling, which was successfully developed in VCSEL array based on regrowth of high-index material or cavity-induced resonant shift structure, could achieve stable in-phase mode operation with specifically designed interelement spacing [8–10]. Another feasible mean of optical coupling between VCSELs is diffraction coupling due to the simplicity of its implementation, especially the following derived photonic crystal-based diffraction-coupled VCSEL array [11,12]. However, these above coupling methods suffer the difficulty and complexity of the fabrication process, preference to operate in the out-of-phase mode or limited number of elements involved in the coupling process, finally resulting in fragile character of coherent laser, small array scale and low emission power.

Embedding an external Talbot cavity into the VCSEL array [13,14] is another practical approach to achieve the passive phase locking based on the classical diffraction coupling theory when considering the advantage of monolithic integration of VCSEL array. Talbot effect is a well-known optical phenomenon that describes the self-imaging or lensless imaging of a periodic and coherent wavefield after propagating a certain distance (so-called Talbot distance zT) in free space [15]. Theoretically, it demonstrates a strong ability to accommodate the array scale to produce high power and high brightness output [16]. Thus, the coherent operations of both linear and orthogonal arrays have already been experimentally carried out [17–19], however, with limited scale and output power, mainly because the mutual injection process and phase-locking condition both remain unclear and puzzling. Additionally, the process of synchronizing emission wavelength, as one of the indispensable premises that guarantee the phase locking, is generally omitted. Consequently, the array structure, as well as the Talbot cavity, lacks specific designs corresponding to the phase locking.

In this paper, we demonstrate the numerical simulations of detailed phase-locking dynamics of hexagonally-arranged VCSEL array with fractional Talbot cavity based on the optical feedback rate equations. The dynamic behaviors and the mode characteristics after the phase locking of VCSEL array under different fill factors and Talbot cavity lengths are comprehensively analyzed to support the successful operation of wavelength synchronization and single pure in-phase supermode. The simulation results could offer a deeper understanding of fully-coupled phase locking of such VCSEL hexagonal array and help to obtain the monolithic integration configurations of the on-chip Talbot cavity with combined high output power with good beam quality.

2. Mathematical model

The conceptual structure of Talbot-VCSEL system, composed of a VCSEL array positioned inside an external cavity, is presented in Fig. 1(a). The VCSEL array consists of M emitters which are arranged into hexagonal geometry so as to achieve a wider in-phase operation range [20]. The emitters designed as circles with an aperture diameter of D have the periods of px and py in x and y dimensions, respectively, and py = √3px. Generally speaking, the external cavity is established by placing a flat mirror at fractional Talbot distance L = γzT from the front facet of the VCSEL array. However, in our model the length of external cavity is no more than that of a standard substrate so that the Talbot cavity could be achievable directly on-chip by using the substrate back surface as the mirror. As shown in Fig. 1(c), a proposed configuration of monolithic Talbot-VCSEL is a composite cavity structure of a standard VCSEL array with top and bottom DBRs and the substrate reflector. It is noteworthy that the Talbot distance zT of a hexagonal array is 3px2/2λ, which is different from the usual Talbot distance 2px2/λ in one-dimensional and two-dimensional orthogonal cases [21].

 figure: Fig. 1.

Fig. 1. (a) Conceptual structure of phase-locking Talbot-VCSEL system. (b) Arrangement of VCSEL array elements. (c) A proposed schematic of monolithic Talbot-VCSEL structure.

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On one hand, the Talbot cavity provides the optical feedback for the injection locking among the elements during the phase locking process. Specifically, two physical injection mechanisms are involved in our system, i.e. self injection and mutual injection, due to the optical feedback [22]. And the phase locking is the process of synchronizing the phase, that is, the phase difference is becoming constant by using the mutual injection between different elements to interfere with each other in frequency [23,24]. On the other hand, in Talbot cavity configuration, the reconstructed light by the Talbot effect can be efficiently coupled into the laser gain region. This not only has an enhanced effect on the phase locking process but also makes the phase-locked VCSEL array operate in the in-phase or out-of-phase mode eventually. In other words, the fractional Talbot cavity has selectivity to the array cavity mode (supermode) after the phase locking.

From the mathematical point of view, the phase locking process of the VCSEL array shown in Fig. 1 can be modeled by the optical feedback rate equations to investigate the temporal dynamics of each emitter. The complex optical field Em based on Lang-Kobayashi equations [25] is derived as follows:

$$\frac{{d{E_m}}}{{dt}} = \frac{1}{2}({1 + \textrm{i}\alpha } )\left( {\Gamma {G_m} - \frac{1}{{{\tau_\textrm{p}}}}} \right){E_m} + {E_{\textrm{sp,}m}} + \sum\limits_{n = 1}^M {\frac{{{\kappa _{mn}}}}{{{\tau _L}}}{E_n}} ,$$
where the subscripts m (= 1, …, M) and n (= 1, …, M, here, m = n denotes the case of self injection) refer to the mth emitter and nth emitter, respectively. Γ is the optical confinement factor. Gm, the optical gain, is dependent on the carrier density Nm and given by
$${G_m} = \frac{{{\nu _\textrm{g}}{g_0}({{N_m} - {N_{\textrm{tr}}}} )}}{{1 + \varepsilon {{|{{E_m}} |}^2}}},$$
where vg is the group velocity, g0 is the differential gain coefficient, and Ntr is the transparent carrier density. Besides, the gain saturation effect is also taken into account by introducing the gain suppression factor ɛ.

Moreover, the last term on the right-hand side of Eq. (1) indicates the sum of complex fields produced by the optical feedback at a frequency of 1/τL = c/2ηeffLd, where τL stands for the resonant cavity round-trip time. Considering the transmission and mode matching losses of the beams in Talbot cavity, the complex feedback coefficient κmn is then introduced to describe the relationship between the initial optical field En and the actual injected field from the nth to mth emitter. The parameter κmn is defined as [26]

$${\kappa _{mn}} = \frac{{1 - {R_2}}}{{\sqrt {{R_2}} }}\sqrt {{R_3}} {c_{mn}},$$
where R2 and R3 are the power reflectivities of the front facet and Talbot cavity mirror respectively, cmn represents the coupling coefficient of the feedback light into the active region, and can be obtained by calculating the overlapping integrals of optical field distribution [27]:
$${c_{mn}} = \frac{{\int\!\!\!\int {E_m^\ast ({x,y,0} ){E_n}({x,y,2L} )dxdy} }}{{{{\left[ {\int\!\!\!\int {{{|{{E_m}({x,y,0} )} |}^2}dxdy} } \right]}^{1/2}}{{\left[ {\int\!\!\!\int {{{|{{E_n}({x,y,2L} )} |}^2}dxdy} } \right]}^{1/2}}}}.$$

Here, it is assumed that each VCSEL element operates in its fundamental lateral mode. Em(n) (x,y,0) stands for the initial optical field with Gaussian distribution, and Em(n) (x,y,2L) stands for the optical field after propagation of distance 2L derived by Fresnel Kirchhoff integral.

Ignoring the factors including the time delay of optical feedback process, carrier diffusion, and Langevin noise F caused by spontaneous emission, we eventually established the rate equations model regarding the photon density Sm, the phase фm and the carrier density Nm based on the complex field rate equation mentioned above [28,29]

$$\frac{{d{S_m}}}{{dt}} = \Gamma {G_m}{S_m} - \frac{{{S_m}}}{{{\tau _\textrm{p}}}} + 2\sum\limits_{n = 1}^M {\frac{{|{{\kappa_{mn}}} |}}{{{\tau _L}}}\sqrt {{S_m}{S_n}} \cos ({{\phi_n} - {\phi_m} - {\theta_{mn}}} )} + \Gamma {R_{\textrm{sp}}},$$
$${\omega _m} = \frac{{d{\phi _m}}}{{dt}} = {\omega _{0m}} + \frac{\alpha }{2}\left[ {\Gamma {G_m} - \frac{1}{{{\tau_\textrm{p}}}}} \right] + \sum\limits_{n = 1}^M {\frac{{|{{\kappa_{mn}}} |}}{{{\tau _L}}}\sqrt {\frac{{{S_n}}}{{{S_m}}}} \sin ({{\phi_n} - {\phi_m} - {\theta_{mn}}} )} ,$$
$$\frac{{d{N_m}}}{{dt}} = \frac{I}{{eV}} - \frac{{{N_m}}}{{{\tau _n}}} - {G_m}{S_m},$$
where ω0m denotes the free-oscillating angular frequency of the mth emitter and the corresponding initial wavelength is λ0m = 2π/ω0m, θmn is the argument of the complex feedback coefficient κmn, indicating the phase shift of the optical field emitted from the nth emitter and reflected back into the mth emitter.

Theoretically, for such a nonlinear differential system composed of 3M equations, it is almost impossible to obtain the analytical solutions. However, we can study the temporal development of each state variable in this dynamical system by using the numerical approximation method. The phase locking problem is transformed into the question of whether there are stationary solutions to these differential equations. In this paper, the backward differentiation formulas (BDF) method in MATLAB program was used for numerical simulation, as we showed in Code 1, (Ref. [30]). For each element, the photon density, phase and carrier density in the steady state of the free-running VCSEL were set as the initial value of analysis. And in order to simulate the actual wavelength detuning, the initial wavelengths were set to distribute normally with center wavelength λ0 = 980 nm and standard deviation σ = 0.05 nm. Other relevant parameters are listed in Table 1 [31].

Tables Icon

Table 1. Parameters used in the simulation

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3. Results and discussions

Taking a VCSEL array with 19 elements in hexagonal arrangement as an example, the array period px was fixed at 30 µm. By changing the aperture diameter D, the temporal evolutions of wavelength and normalized output intensity with different fill factors ff = D/px for each laser were calculated, respectively. The simulation results are shown in Fig. 2 and Fig. 3, where the Talbot cavity length L is set to be 1/2 zT.

 figure: Fig. 2.

Fig. 2. The temporal evolutions of wavelength (left) and normalized output intensity (right) with different fill factors when L = 1/2 zT: (a) ff = 0.03, (b) ff = 0.1.

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 figure: Fig. 3.

Fig. 3. The temporal evolutions of wavelength (left) and normalized output intensity (right) with different fill factors when L = 1/2 zT: (a) ff = 0.2, (b) ff = 0.217.

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3.1 Dynamics of Phase Locking

In Fig. 2(a), it is found that the wavelength and output intensity of each element only slightly oscillate around their respective initial values when ff is lower than 0.05. The mutual interference among elements is too weak to be noticed. However, with ff increasing, the interference becomes stronger, as shown in Fig. 2(b). From 0 to 2 ns, the laser wavelengths and outputs fluctuate dramatically all the time, demonstrating a strong unpredictability.

Subsequently, when ff increases to 0.2, some of the wavelengths are synchronized gradually after mutual injection for 0.5 ns, as depicted in Fig. 3(a1). More specifically, the elements oscillating around 980.35 nm are treated to be locked due to their wavelength differences being less than 2×10−3 nm. And the locked wavelengths are red shifted compared with the initial center wavelength λ0 and constituted by two alternating groups of wavelengths that fluctuate periodically with an amplitude of about 0.025 nm. In comparison, the others below the locked wavelengths also have a regular co-oscillation but a larger deviation. Actually, the co-oscillation involving several VCSEL elements has already occurred when ff = 0.15. But it is unstable and easily damaged by the disturbance from other elements. With ff increasing larger than 0.195, the number of locked elements increases, and their wavelengths begin to present stable periodic oscillations. In addition, from the change of output intensity, it can also be seen that under this condition the VCSELs have transformed from the irregular, chaotic state with a low fill factor to the period-one oscillation state [32]. Thereafter, the dynamic behaviors of the VCSEL array are more susceptible to the change of fill factor. As shown in Fig. 3(b1), the wavelength synchronization of the whole VCSEL element has been realized with the differences being less than 2×10−4 nm when ff = 0.217, and the locked wavelength is red shifted to about 980.37 nm eventually. Besides, the output intensity of each element keeps relatively constant after the phase locking.

Furthermore, we simulated the case where the length of Talbot cavity is 1/4 zT. As shown in Fig. 4(a), the phase locking of the array has already been achieved when ff = 0.2. However, for even higher ff, the co-oscillation amplitudes of some wavelengths are instead enlarged and frequencies of the evolution speed up after the phase locking as shown in Fig. 4(b), resulting in an unstable phase-locked Talbot-VCSEL system. This phenomenon, which is not observed when L = 1/2 zT, is probably caused by the reduced injected light to other non-adjacent elements due to the short transmission distance.

 figure: Fig. 4.

Fig. 4. The temporal evolutions of wavelength with different fill factors when L = 1/4 zT: (a) ff = 0.2, (b) ff = 0.28.

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3.2 Supermode Analysis after Phase Locking

To have a more intuitive understanding of the phase locking, we then analyzed the near-field and far-field properties of VCSEL arrays under the conditions mentioned above, respectively. Figure 5(a1) ∼ (d2) show the array’s near-field intensity and phase distribution at around 2 ns. Before the mutual injection, the VCSEL arrays with different ffs and Ls share the same initial conditions, namely all the elements lase with uniform intensity and randomly distributed phases ranging from – π to π.

 figure: Fig. 5.

Fig. 5. The near-field intensity and phase distribution of the array and the corresponding far-field intensity distribution and the profile along the vertical direction when (a) ff = 0.1, L = 1/2 zT, (b) ff = 0.2, L = 1/2 zT, (c) ff = 0.217, L = 1/2 zT, (d) ff = 0.2, L = 1/4 zT.

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In Fig. 5(a1) and (a2), the device is still in chaotic state after the optical injection of 2 ns according to Fig. 2(b), and no distinct regularities of both near-field intensity and phase distribution are observed. When ff is increased to have realized the partial phase locking, we find that the near-field intensities corresponding to wavelength-locked elements are obviously larger than those of the unlocked ones, and the phase difference of adjacent locked elements keeps π, as shown in Fig. 5(b1) and (b2) respectively. This accords with the fundamental out-of-phase coupling characteristics, although the near-field intensities are not uniformly distributed due to the likely influence of finite array and wavelength detuning [14]. From the corresponding far-field pattern in Fig. 5(b3), it is found that the light intensity on propagation axis is zero, and the principal maximums are located around the axis. The beam energy is comparatively dispersed, thus resulting in a poor output brightness.

After all the wavelengths are entirely locked, the cases of L = 1/2 zT and L = 1/4 zT were both investigated. In Fig. 5(c2), the constant phase difference between two arbitrary elements once again proves the realization of complete phase locking, however, the phase difference of adjacent elements lies between 0 and π, which does not belong to the characteristics of any basic array mode. It is considered to be the result of co-existence of multiple supermodes. And after changing the cavity length L to 1/4 zT, as shown in Fig. 5(d), the simulated near-field intensity gradually weakens from the center to edge, and the phase difference between adjacent elements remains zero. The far-field pattern in Fig. 5(d3) also shows that the optical energy is mainly concentrated in the on-axis central lobe surrounded by six weaker side lobes, clearly indicating that the VCSEL array operates in the in-phase mode with high brightness.

3.3 Feedback Coefficient κmn

We then obtained the locking range of the system by solving the stationary solutions of the rate equations given in Eq. (5–7). After neglecting a small term representing spontaneous emission, the formula is simplified as below:

$$- \sqrt {1 + {\alpha ^2}} \sum\limits_{n = 1}^M {\frac{{|{{\kappa_{mn}}} |}}{{{\tau _L}}}} \sqrt {\frac{{S{^{\prime}_n}}}{{S{^{\prime}_m}}}} \le \varDelta \omega {^{\prime}_m} \le \sqrt {1 + {\alpha ^2}} \sum\limits_{n = 1}^M {\frac{{|{{\kappa_{mn}}} |}}{{{\tau _L}}}} \sqrt {\frac{{S{^{\prime}_n}}}{{S{^{\prime}_m}}}} ,$$
where Δω'm = ω'm-ω0m is the frequency detuning of the mth laser with respect to the locked frequency ω'm, and S'm(n) is the photon density in the steady state. It can be known that in the phase-locked arrays with both self injection and mutual injection, the locking range is substantially symmetric concerning the frequency detuning, as well as the wavelength detuning. Moreover, a higher injection power could result in a larger locking range. Besides increasing the array size [24], the injection power is also associated with the feedback coefficient |κmn| which can be used as an intermediate quantity to describe the influence of key structural parameters, such as fill factor ff and Talbot cavity length L, on the device’s phase-locking dynamic state and the mode after the phase locking.

Considering that after the emergent light of each element is reflected by the Talbot cavity, the self feedback intensity is always the strongest, then followed by the feedback to other elements which is exponentially weakened as the distance between two elements increasing, so we only enumerate the cases of self feedback and feedback to adjacent elements here. As shown in Fig. 6, the solid line and dotted line represent the relationship between the feedback coefficient |κmn| and the fill factor ff when L = 1/2 zT and L = 1/4 zT, respectively.

 figure: Fig. 6.

Fig. 6. (a) The curves of feedback coefficient |κmn| as a function of fill factor ff, where m = n denotes the self feedback coefficient and m ≠ n denotes the mutual feedback coefficient to adjacent elements. (b) The curves of feedback coefficient ratio |κmn(m ≠ n)|/|κmn(m = n)| as a function of fill factor ff. The solid line and dotted line represent the cases of L = 1/2 zT and L = 1/4 zT, respectively.

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Overall, with the ff increasing, the amount of light feedback into itself and the other elements increases due to the increase of aperture diameter D of each element. And the optical coupling among the elements is gradually strengthened so that the Talbot-VCSEL system changes from the state, in which each element works independently, with incoherent properties to the coherent phase-locked state. Compared with the case of L = 1/2 zT, the transmission distance of emergent light decreases when L = 1/4 zT, leading to that the self feedback is strengthened throughout, while the light feedback into adjacent elements increases slightly at the beginning (ff < 0.25) as depicted in Fig. 6(a). So under the same ff, it is easier for the case where L = 1/4 zT to realize the phase locking. However, the optical coupling with other elements, especially the non-adjacent elements, is instead weakened as ff increases continuously. From the curve of feedback coefficient ratio in Fig. 6(b), we can also see that the array element is dominated by the self feedback under this condition, which could disturb and restrain the mutual coupling among the elements, resulting in the destruction of the phase-locked state. Hence, the optimized fill factor to achieve a phase-locked state for our Talbot-VCSEL structure when L = 688 µm (1/2 zT) and L = 344 µm (1/4 zT) with a fixed array spacing px of 30 µm are >0.217 and 0.18∼0.28, respectively. Of course, the fill factor should not be designed too high to avoid the appearance of high-order transverse modes due to the large aperture.

In addition, the array mode after the phase locking is also affected by the fill factor ff and Talbot cavity length L. In general, the array with M emitters allows for oscillation of M supermodes. Using the coupling matrix composed of the coupling coefficients cmn, we solved the thresholds of all supermodes according to the following formula [25,33]

$${g_j} ={-} \frac{1}{{{L_d}}}\ln \left( {\left|{\sqrt {{R_2}} + (1 - {R_2})\sqrt {{R_{ext}}} {\beta_j}} \right|} \right),$$
where βj (j = 1, 2, …, M) is the eigenvalue of coupling matrix, as shown in Fig. 7. The two insets depict the corresponding far-field intensity profiles calculated by the coupling matrix’s eigenvectors.

 figure: Fig. 7.

Fig. 7. The threshold gains of all supermodes under different ffs and Ls. The insets depict the corresponding far-field intensity profiles of the lowest order (j = 1) and highest order (j = M) supermodes.

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According to the calculation results, the mode with the lowest threshold is out-of-phase mode when L = 1/2 zT and ff = 0.2, so the first locked elements show the relevant out-of-phase characteristics as discussed above, and their far-field intensity distribution is also matched with that of the out-of-phase mode calculated by the coupled-mode theory. Moreover, the output intensity of each element is redistributed after the appearance of such mode characteristics, which subsequently accelerates the locking of those elements with low output according to Eq. (8). Furthermore, the increase of fill factor reduces the threshold gain of the supermode, which would lead to the multiple modes oscillating simultaneously as shown in Fig. 5(c). When L becomes 1/4 zT, the threshold gain of in-phase mode is the lowest. This explains the function of L on mode switching under the same ff. Hence, considering the refractive index of GaAs of 3.527, the Talbot cavity length L, or thickness of substrate, can be designed around 100 µm to support the in-phase operation.

4. Conclusions

In conclusion, this paper has presented a detailed theoretical analysis concerning the phase-locking dynamics of 2D VCSEL hexagonal arrays based on the rate equations with external optical feedback. The dynamic behaviors of Talbot-VCSEL system are quite sensitive to the parametric interaction such as fill factor ff and fractional Talbot cavity length L through the numerical calculations. And the distribution of optical feedback among the elements on the array need be consistent as much as possible when optimizing the structural parameters of the VCSEL array, since the competition between self injection and mutual injection by optical feedback could significantly control the phase locking. Additionally, we analyzed the mode of phase-locked VCSEL arrays by the research on temporal dynamics, which is in good agreement with the results of coupled-mode theory. And the numerical results also reveal that the promoting effect of Talbot cavity on phase locking is implemented by the redistribution of near-field intensity due to the newly-generated supermode characteristics shown in partially coherent elements.

Furthermore, this model can be easily scaled to much larger 2D VCSEL arrays by a feasible design to achieve higher output power with much better coherent properties, and also be well applied to investigate the influence of other array arrangements, structure parameters (such as reflectivity, active volume etc.) on phase locking. Thus study on the mechanism of Talbot cavity feedback can provide a theoretical support to design and fabricate the in-phase coupled monolithic VCSEL arrays with high performance.

Funding

Beijing Postdoctoral Science Foundation (2020-Z2-043).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Supplementary Material (1)

NameDescription
Code 1       The Matlab code used in numerical calculation that solving the rate equations.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (7)
Fig. 1.
Fig. 1. (a) Conceptual structure of phase-locking Talbot-VCSEL system. (b) Arrangement of VCSEL array elements. (c) A proposed schematic of monolithic Talbot-VCSEL structure.

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Fig. 2.
Fig. 2. The temporal evolutions of wavelength (left) and normalized output intensity (right) with different fill factors when L = 1/2 zT: (a) ff = 0.03, (b) ff = 0.1.

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Fig. 3.
Fig. 3. The temporal evolutions of wavelength (left) and normalized output intensity (right) with different fill factors when L = 1/2 zT: (a) ff = 0.2, (b) ff = 0.217.

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Fig. 4.
Fig. 4. The temporal evolutions of wavelength with different fill factors when L = 1/4 zT: (a) ff = 0.2, (b) ff = 0.28.

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Fig. 5.
Fig. 5. The near-field intensity and phase distribution of the array and the corresponding far-field intensity distribution and the profile along the vertical direction when (a) ff = 0.1, L = 1/2 zT, (b) ff = 0.2, L = 1/2 zT, (c) ff = 0.217, L = 1/2 zT, (d) ff = 0.2, L = 1/4 zT.

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Fig. 6.
Fig. 6. (a) The curves of feedback coefficient |κmn| as a function of fill factor ff, where m = n denotes the self feedback coefficient and m ≠ n denotes the mutual feedback coefficient to adjacent elements. (b) The curves of feedback coefficient ratio |κmn(m ≠ n)|/|κmn(m = n)| as a function of fill factor ff. The solid line and dotted line represent the cases of L = 1/2 zT and L = 1/4 zT, respectively.

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Fig. 7.
Fig. 7. The threshold gains of all supermodes under different ffs and Ls. The insets depict the corresponding far-field intensity profiles of the lowest order (j = 1) and highest order (j = M) supermodes.

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Tables (1)

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Table 1. Parameters used in the simulation

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Equations (9)

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d E m d t = 1 2 ( 1 + i α ) ( Γ G m − 1 τ p ) E m + E sp, m + ∑ n = 1 M κ m n τ L E n ,
G m = ν g g 0 ( N m − N tr ) 1 + ε | E m | 2 ,
κ m n = 1 − R 2 R 2 R 3 c m n ,
c m n = ∫ ∫ E m ∗ ( x , y , 0 ) E n ( x , y , 2 L ) d x d y [ ∫ ∫ | E m ( x , y , 0 ) | 2 d x d y ] 1 / 2 [ ∫ ∫ | E n ( x , y , 2 L ) | 2 d x d y ] 1 / 2 .
d S m d t = Γ G m S m − S m τ p + 2 ∑ n = 1 M | κ m n | τ L S m S n cos ⁡ ( ϕ n − ϕ m − θ m n ) + Γ R sp ,
ω m = d ϕ m d t = ω 0 m + α 2 [ Γ G m − 1 τ p ] + ∑ n = 1 M | κ m n | τ L S n S m sin ⁡ ( ϕ n − ϕ m − θ m n ) ,
d N m d t = I e V − N m τ n − G m S m ,
− 1 + α 2 ∑ n = 1 M | κ m n | τ L S n ′ S m ′ ≤ Δ ω m ′ ≤ 1 + α 2 ∑ n = 1 M | κ m n | τ L S n ′ S m ′ ,
g j = − 1 L d ln ⁡ ( | R 2 + ( 1 − R 2 ) R e x t β j | ) ,
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