Mode-Locked Topological Insulator Laser Utilizing Synthetic Dimensions

We propose a system that exploits the fundamental features of topological photonics and synthetic dimensions to force many semiconductor laser resonators to synchronize, mutually lock, and under suitable modulation emit a train of transform-limited mode-locked pulses. These lasers exploit the Floquet topological edge states in a 1D array of ring resonators, which corresponds to a 2D topological system with one spatial dimension and one synthetic frequency dimension. We show that the lasing state of the multielement laser system possesses the distinct characteristics of spatial topological edge states while exhibiting topologically protected transport. The topological synthetic-space edge mode imposes a constant-phase difference between the multifrequency modes on the edges, and together with modulation of the individual elements forces the ensemble of resonators to mode lock and emit short pulses, robust to disorder in the multiresonator system. Our results offer a proof-of-concept mechanism to actively mode lock a laser diode array of many lasing elements, which is otherwise extremely difficult due to the presence of many spatial modes of the array. The topological synthetic-space concepts proposed here offer an avenue to overcome this major technological challenge and open new opportunities in laser physics.

Popular Summary

For decades, researchers have tried to use arrays of coupled laser diodes to produce a high-power laser source with semiconductor technology. However, arrays of semiconductor lasers tend to lase in many modes simultaneously, and they quickly become mutually incoherent as the size of the array increases to more than ten emitters. Consequently, after almost 40 years of research, laser diode arrays are currently used only as an incoherent source to pump solid-state lasers. For the same reasons, all attempts to make arrays of semiconductor lasers emit mode-locked pulses have technologically failed, because it is extremely difficult to synchronize all the elements in an array. Here, we take a totally different approach. We exploit the unique properties of topological insulators—materials that are insulators in the bulk but conduct robust current on their surface—to design a topological photonic platform that forces many semiconductor laser resonators to synchronize, mutually lock, and emit a train of mode-locked pulses.

Our proposed system works in a synthetic 2D space: one spatial dimension (the physical array of laser resonators) and frequency dimension (the modes of the resonators). We show that the lasing state possesses the distinct characteristics of spatial topological edge states while exhibiting topologically protected transport and immunity to disorder. The topological synthetic-space edge mode imposes a constant-phase difference between the multifrequency modes, forcing the resonators to mode lock and emit short pulses, robust to disorder in the multiresonator system.

Our results offer a proof-of-concept mechanism to mode lock a laser diode array of many lasing elements, which is otherwise extremely difficult to achieve. The topological synthetic-space concepts proposed here offer an avenue to overcome this major technological challenge and open new opportunities in laser physics.

Figure 1

Schematic of one module of the topological insulator laser in synthetic dimensions. (a) The photonic array consists of modulated rings, in the form of two independent rows connected by two side resonators. The blue and white staggered pattern indicates the different pumping schemes. The shaded regions show the synthetic space including one spatial and one synthetic mode dimensions. The blue (gray) circles indicate the gain (loss), where the gain is provided only to the upper (lower) mode in the upper (lower) row, and to all modes in the side rings. (b) The typical topological lasing state extending over the artificial perimeter of the 2D synthetic space. The color bar indicates the intensity of light. The unitless parameters for simulation are $N=18$, $M=20$, $t_0=t_f=1$, $\Omega =100$, $I_{\mathrm{sat}}=1$, and $\gamma =0.3$. The topological system has band gap size of $1.6t_0$.

Figure 2

Laser output versus (in arbitrary units, arb. units) pumping intensity and the spectra of the topologically nontrivial and trivial lasers. (a) The total output power versus pumping intensity. The slope efficiency of the topological laser is about twice that of the trivial laser. (b) Slope efficiency versus disorder strength. The on-site disorder is composed of independent random variations in the resonance frequencies, uniformly distributed in $[-W,W]$, where $W$ is in the units of $t_0$. This panel presents the mean of 100 realizations of disorder for every point of disorder strength. (c),(d) Frequency spectra of the topologically nontrivial (c) and trivial (d) lasers, as emitted from the connecting side resonator on the right-hand side; the panels on the right show the enlarged spectra around $9.5\Omega$. The topological system emits a comb of narrow single frequencies, whereas the trivial system has multiple modes around each central frequency. The original resonances in each ring are set to be $-(M-1/2)\mathrm{\Omega },\dots ,(M-1/2)\mathrm{\Omega }$ centered at ${\omega }_0$ ($\mathrm{\Omega }=100$ is the free spectral range, $M=20$ is the number of modes in each ring; ${\omega }_0$ is set to 0 for simplicity). The topological landscape causes a constant shift to all the resonance frequencies by a value of $0.019\mathrm{\Omega }$, as shown by the shift from $9.5\mathrm{\Omega }$ in the enlarged spectrum in (c) (see explanation in Sec. A in Supplemental Material [59]). The blue and gray curves represent the topological and trivial lasers, respectively. The blue shaded region on the right in (c) indicates the topological band gap.

Figure 3

Temporal dynamics of the phase difference between adjacent resonant modes in the side rings. The color bar indicates the absolute value squared of the spatial Fourier transform of the field inside the side-ring. (a) The topological laser exhibits a fixed phase difference between the modes in the right-hand side ring, which gives rise to the mode locking. (b) The trivial laser displays multivalued phase differences between the modes of the same ring, which therefore cannot be synchronized to yield mode-locked operation. The time axis is in units of $\tau =2\pi /\mathrm{\Omega }$.

Figure 4

Temporal dynamics of the laser output power (in arbitrary units, arb. units) emitted from the side resonators. The topological module emits mode-locked pulses (a) and maintains clean mode-locked operation even under considerable disorder with $W=0.5$ (b). On the other hand, the trivial laser system ($\varphi =0$) displays multiple bursts of power, which broaden and fluctuate in the absence (c) and presence (d) of on-site disorder which introduces frequency mismatch among the resonators. The blue and gray curves indicate the results of topological and trivial lasers, respectively. The time axis is in units of $\tau =2\pi /\mathrm{\Omega }$. Note that in the simulations, background noise is present at all times. The background noise is distributed in $[-\delta ,\delta ]$ and $\delta =0.2$.

Figure 5

Topological mode-locked laser array. (a) Scheme showing a laser array of $n=5$ modules [each of which is described by Fig. 1]. The side resonators in the laser array configuration connect different chains of ring resonators. The lower panel shows an enlarged view of the blue box in (a). The laser light is extracted through output couplers. (b) The gain profile in synthetic dimensions. The ring resonators correspond to the ones within the blue box in (a). (c),(e) The topological laser array generates a train of mode-locked pulses (c) which are emitted from the rings with homogenous gain [top or bottom rings in (b)], and remain clean mode-locked pulses even in the presence of considerable on-site disorder with $W=0.3$ (which introduces frequency mismatch among the resonators) (e). (d),(f) The trivial laser array emits fluctuating pulses with much less output power (in arbitrary units, arb. units) even in the absence (d) and presence (f) of disorder. (g) Peak power scales with the number of emitters quadratically for topological laser (blue) and linearly for trivial laser (gray). The simulation parameters we used are the same as those in Fig. 1. The time axis is in units of $\tau =2\pi /\mathrm{\Omega }$. In panels (c)–(f) the laser output power is calculated from the sum of the electric fields of side rings in the laser array.