Synchronization of complex spatio-temporal dynamics with lasers

We experimentally study unidirectional optical injection of one BA-VCSEL into another BA-VCSEL of the same model and analyze the dynamical response of the slave laser to the signal of the master laser. A simplified schematic of the experimental setup is shown in Fig. 1a, and a detailed description is provided in Supplementary Section 1. We begin with a summary of the properties of the master laser (see also refs. ^30,31), and a comparison to the slave laser is presented in Supplementary Section 2.

The free-running BA-VCSELs lase in several transverse modes and two linear orthogonal polarization states, denoted by u and v. The dominant polarization depends on the pump current and changes at several polarization-switching points (PSPs, see Supplementary Fig. S2)³⁰. The birefringence splitting between u– and v– polarization modes is around Δν_b ≈ 9 GHz (see Supplementary Section 2), and we refer to “red” (u) and “blue” (v) polarization according to their relative position in the optical spectrum.

Figure 1b shows spatio-spectral images measured with an imaging spectrometer (see Supplementary Section 1) for different pump currents. With increasing pump current, more transverse modes are progressively excited. A transverse mode with mode indices (m, n) has 2m intensity maxima in azimuthal direction and n maxima in radial direction. Furthermore, for m > 0, two distinct spatial orientations are possible, which we denote with + ( − ) when they are (anti-) symmetric with respect to a symmetry axis as shown in Fig. 1b₂. In the following we denote the modes with X(m, n, ± ) where X ∈ {M, S} refers to the master and slave laser, respectively. It should be noted that modes (m, n, + ) and (m, n, − ) are not necessarily degenerate³⁷, but can exhibit a splitting Δν_O of the order of several GHz (see Supplementary Section 2 A d). The actual splitting seems to depend on the transverse mode. Typically, only one of the orientations is excited for a given polarization, and the corresponding mode in the other polarization has the opposite orientation due to gain competition^30,38,39,40. This is exemplified in Fig. 1b₂ with the modes M_v(3, 1, + ) and M_u(3, 1, − ). However, in some cases both orientations of a transverse mode (m, n) lase in the same polarizations as shown in Supplementary Fig. S5.

The competition of lasing modes with different spatial profiles and polarizations results in rich nonlinear dynamics, as shown in Fig. 1c. As the pump current increases, the system undergoes a sequence of bifurcations that progressively enrich the temporal behavior^30,31: initially periodic oscillations transition to quasi-periodic regimes, and eventually to chaotic dynamics (see Supplementary Fig. S6). These bifurcations are usually accompanied by a redistribution of power between different transverse modes and in some cases polarization switching points³⁰. We restrict our analysis to parameter regions in which the master laser features chaotic dynamics for a focused investigation of synchronization in the presence of intrinsic multimode chaos.

The lasing dynamics features different time scales: the dominant frequencies are around the birefringence Δν_b [see Fig. 1c]. However, in some current regimes like around 6 mA, this fast dynamics coexists with a slower polarization-hopping dynamics with frequencies of the order of 100 MHz [see Fig. 1c₁ and Supplementary Fig. S4], similar to observations for single-mode VCSELs^41,42,43,44.

Next, we implement optical injection between two BA-VCSELs to investigate synchronization between them. Although the two lasers are of the same model, some differences in their spectral and dynamical properties inevitably remain. The comparison of their optical spectra and dynamics in Supplementary Section 2 B shows that spectral alignment can be achieved by adjusting the detuning via the temperature or the pump current of the slave (see Supplementary Section 2 C). The two distinct dynamical regimes in Fig. 1c are injected into the slave laser in order to analyze the synchronization quality for both high frequency chaos and slower mode-hopping dynamics. Different synchronization mechanisms emerge depending on the properties of the injected dynamics. The response of the slave laser strongly depends on the temporal and spectral structure of the injected signal, highlighting the richness of coupling phenomena in chaotic multimode systems.

The first case we study is when the master laser operates at I_M = 6.17 mA and exhibits a dynamics characterized by both high-frequency components and slow polarization-hopping (typically around 100 MHz). The master laser current and temperature are fixed to keep its dynamical properties constant. The slave laser current is varied between I_S = 2 mA and 10 mA at a fixed temperature of 20^∘C. The slave laser modes are red-shifted due to Joule heating as its current increases [see Supplementary Section 2 C]. Furthermore, its output power increases, thereby reducing the optical injection ratio (see Supplementary Section 3 A). We study parallel injection, meaning that the u-polarization component of the master laser, which is its dominant polarization for this pump current, is injected into the u-polarization of the slave laser using a HWP. The slave laser has several PSPs in this current range, but its red polarization (u) remains mostly dominant (see Supplementary Fig. S2).

Figure 2b₁ shows the correlation between the time traces (see Methods) of u-polarized emissions of master and slave laser as function of the frequency detuning Δν between the two lasers, which is defined here as the frequency of mode M_u(0, 1) minus the frequency of S_u(0, 1) as illustrated in Fig. 2a₁. We observe several cases of significant positive and negative values of correlation, corresponding to synchronization and inverse synchronization, respectively, which are discussed in the following.

The positive correlation peaks are related to spectral alignment between transverse modes of the master and the slave laser with high power. For Δν = 0, the spatio-spectral image in Fig. 2a₁ shows that both the S_u(0, 1) and the S_u(3, 1) modes emit strongly and are spectrally aligned with the M_u(0, 1) and M_u(3, 1) modes, respectively. We originally expected this to yield the best synchronization since theoretically each transverse mode of the master laser would be aligned with its counterpart of the slave, but in practice perfect matching of all transverse mode frequencies is not achievable due to small differences in modal spacings between the two lasers. We also observe that for Δν = 0, when the S_u(3, 1) mode aligns with the M_u(3, 1) mode, its power significantly increases while neighboring modes become weaker, as discussed in Supplementary Fig. S10. This highlights the strong interaction of these two modes as their frequencies align.

Similar observations are made for the other positive correlation peaks. At Δν ≈ 30 GHz, a weak synchronization peak appears, associated with the alignment of M_u(0, 1) with S_u(1, 1). The peak at Δν ≈ 80 GHz corresponds to the alignment between M_u(0, 1) and S_u(2, 1) [see Fig. 2a₂ and Supplementary Fig. S10]. Similarly, at Δν ≈ 133 GHz, another peak emerges from the alignment between modes M_u(0, 1) and S_u(3, 1) [see Fig. 2a₂]. These examples demonstrate that synchronization can be achieved when a strong transverse mode of the master laser spectrally aligns with a transverse mode of the slave, however, it need not be the same transverse modes: spectral alignment appears to be more important than spatial alignment for successful synchronization. It should also be emphasized that the other transverse modes of the slave laser remain active and are different from those of the master laser [see Fig. 2a], meaning that synchronizing the temporal dynamics of the lasers does not require synchronizing the optical spectrum or the spatial intensity distribution.

We also observe inverse synchronization, that is negative correlation between the time traces of the u-polarized emission of master and slave, with the most significant examples at Δν = 1.7 GHz and at Δν = 180 GHz. Synchronization in antiphase for coupled oscillators has been known since early studies of coupled pendula and has also been observed in coupled lasers with feedback, for example in the low-frequency fluctuation regime⁴⁵. Polarization dynamics were identified as a key factor in the emergence of inverse synchronization in refs. ^46,47.

The first example at Δν = 1.7 GHz happens right after the alignment of the M_u(3, 1) mode with the S_u(3, 1) mode when the former approaches the frequency of the S_v(3, 1) mode, as shown by the time traces of the master and slave, low-pass filtered at 0.1 GHz cutoff, in Fig. 2c. Indeed, Supplementary Fig. S8a shows that the S_v(3, 1) mode is strongly excited at Δν = 1.7 GHz. We surmise that when injecting light into the u-polarization at a frequency near a v-polarized mode of the slave, it can create a synchronization of the v-polarized emission with the mode-hopping dynamics of the u-polarized injection signal from the master, though this is evidently not always the case. Since the emission in u– and v-polarization is highly anticorrelated in the polarization-hopping regime [see Supplementary Fig. S4], the u-polarized emission of the slave becomes anti-correlated to the u-polarized master signal since the latter is synchronized with the v-polarized emission of the slave. At 180 GHz detuning, we observe that the S_v(1, 2) mode is enhanced when its frequency comes close to the M_v(0, 1) mode.

These examples demonstrate that coupling of two multimode VCSELs can create different types of synchronization depending on system parameters, and that moreover, strong dynamic correlations can be established between transverse modes of very different order.

Finally, we analyze the master-slave correlation across different timescales by applying spectral filtering (see Methods). Figure 2d shows the evolution of the correlation as function of the low-pass filter cutoff frequency for three examples. The correlation significantly improves with correlations of up to 90% for 100 MHz cutoff at Δν = 180 GHz. Furthermore, Fig. 2b₂ shows the correlation of the low-pass filtered time traces with 1 GHz cutoff as a function of detuning. A global increase is observed across all correlation regions. These low-frequency components represent the relatively slow polarization-hopping dynamics which appears to synchronize much better than the high-frequency components of the dynamics as demonstrated by the low-pass filtered time traces of master and slave laser at Δν = 0 and 1.7 GHz in Fig. 2c. This demonstrates that it is the polarization-hopping dynamics which is synchronized, and that one can achieve very high synchronization quality even though the BA-VCSELs are not perfectly identical.

For the second case of injection that we study, the master laser is set to I_M = 8.8 mA and T_M = 20^∘C, the slave laser is operated at a fixed current of I_S = 4.8 mA, and the detuning is varied via the temperature of the slave. As before, the master laser emits predominantly along the u-polarization, and parallel injection of the u-polarization is performed. These experimental parameters lead to three differences to the first case. First, the injection ratio is higher though it remains relatively weak (see Supplementary Section 3 A). Second, while the master laser has higher power, it is distributed over a larger number of transverse modes [see Fig. 3a]. Third, the master laser operates in a chaotic state with high frequency components (see Fig. 1c and Ref. ³¹) without polarization-hopping. In the following we discuss how these changes affect the synchronization.

The detuning Δν is controlled by the temperature of the slave laser, with a measured tuning coefficient of Δν/ΔT = 20.7GHz/K. We define here that Δν = 0 GHz when the mode S_u(2, 1) is aligned with the mode M_u(6, 1), which is the dominant mode of the master and plays an important role in the following. The spatio-spectral images in Fig. 3 reveal that this mode lases in both orientations, M_u(6, 1, + ) and M_u(6, 1, − ), at the same time, and with a negligible splitting Δν_O. This property of the master strongly influences the synchronization behavior.

Figure 3c shows several regions of positive correlation. The region around Δν ≈ 0 GHz stems from synchronization of the M_u(6, 1, ± ) modes with the S_u(2, 1, ± ) modes, which are very weak in free-running operation but can be strongly excited by the injection. A closer look shows that there are actually three successive correlation peaks: S_u(2, 1, − ) is excited at Δν = − 3.5 GHz, both S_u(2, 1, + ) and S_u(2, 1, − ) are excited at Δν = 0 GHz, and S_u(2, 1, + ) is excited at Δν = 4.14 GHz as shown in Fig. 3a. We attribute this behavior to the splitting between the S_u(2, 1, − ) and S_u(2, 1, + ) modes, which seems to be about Δν_O ≈ 7.5 GHz (see also Supplementary Section 2 A d): these two modes are excited alone when the M_u(6, 1, ± ) modes aligns spectrally with them, and their superposition is created when M_u(6, 1, ± ) is in between the two orientations of S_u(2, 1). The same behavior is found when the M_u(6, 1, ± ) modes come close to the S_u(3, 1, ± ) modes [see Fig. 3b and the magnification in Fig. 3c]. We associate this scenario to the M_u(6, 1, ± ) modes lasing simultaneously, which creates an azimuthally uniform intensity distribution that is able to excite both orientations of the slave laser modes equally well.

However, spectral alignment alone does not guarantee strong synchronization. For instance, around Δν ≈ 115 GHz only a low correlation around 9% is observed despite the spectral matching between the M_u(6, 1, ± ) modes and the S_u(4, 1) mode. The third region of positive correlation shows a somewhat different scenario: at Δν ≈ 76 GHz, the M_u(5, 1) mode of the master, which lases only in a single orientation, excites a superposition of the S_u(2, 1, + ) and S_u(2, 1, − ) modes. However, the correlation is low, probably because the M_u(5, 1) has less power than the M_u(6, 1, ± ) modes.

This second injection experiment confirms our earlier observation that spectral alignment between a strong mode of the master and a transverse mode of the slave laser is an important prerequisite for synchronization. While the transverse modes of the master and slave lasers that are coupled can be different, the spatial structure of the master laser mode can also play an important role: simultaneous lasing of the master laser mode in both orientations favors the emergence of three correlation peaks, in which different orientations of the same transverse mode of the slave laser are excited. Furthermore, we find that a higher injection ratio does not necessarily lead to higher correlations, which may be due to the master power being spread across more modes, while power concentration in a single mode seems to play a key role.

Applying a low-pass filter improves the correlations to some extent, typically reaching peak values between 20% and 30% with cutoff frequencies around 0.5 to 1 GHz [see Fig. 3c], but these values are below those found for the first case [see Fig. 2]. We believe this is related to the type the master laser dynamics, which is chaotic with high frequency components up to 20 GHz [see Fig. 1c], in contrast to the slower polarization-hopping dynamics in the first case. Whereas the slow polarization-hopping dynamics creates strong synchronization which is revealed by low-pass filtering, the chaotic dynamics in the second case relies on high-frequency components, so filtering does not help much. It seems that fast chaotic dynamics is harder to synchronize. The absence of inverse synchronization in the second case is explained by the absence of polarization-hopping dynamics to which we attribute the inverse synchronization in the first case. In summary, the second injection experiment confirms the importance of spectral alignment for synchronization [see also Supplementary Fig. S9]. However, we also find significant differences between the two cases, demonstrating the diversity of synchronization scenarios and their dependency on the dynamical and spatial properties of the master laser.