Theoretical study of a passively mode-locked integrated laser based on transition-metal dichalcogenides and graphene

We assess the theoretical performance of a mode-locked silicon-rich-nitride ring laser, which is partly overlayed with a ${\mathrm{MoS}}_2/{\mathrm{WSe}}_2$ bilayer and partly with a graphene monolayer. Through an external vertical optical pump at 740 nm, the transition-metal dichalcogenide (TMD) bilayer can be inverted and provide gain at 1128 nm while the graphene monolayer acts as the fast broadband saturable absorber. We show that under modest pumping conditions, we can reach a stable mode-locked regime that can deliver on-chip pulsed output with milliwatt peak power and down to 400 fs in duration. The ring laser is studied by rigorously modeling the TMD bilayer as a semiclassical three-level system and incorporating the resulting resonant polarization to the nonlinear Schrödinger equation (NLSE). Within the NLSE formalism we are able to also incorporate material and waveguide dispersion as well as the significant Kerr-type nonlinearity of the silicon-rich-nitride waveguide. Furthermore, we discuss all the necessary physical and mathematical approximations needed to numerically solve the propagation problem, based on an implementation of the split-step Fourier method with the atomic polarization term. Our mathematical treatment is very general and can be adapted to any two-dimensional-material-enhanced traveling wave source. Finally, this work shows the feasibility and potential of TMD bilayers as well as graphene for the development of efficient integrated pulsed on-chip sources.

Figure 1

(a) Top-down view of the SRN ring resonator overlaid with a ${\mathrm{MoSe}}_2/{\mathrm{WSe}}_2$ bilayer and a graphene monolayer (not to scale). The arrow shows the slice of the cross section. (b) SRN wire waveguide cross section with vertical external optical pump direction (plane wave). (c) Parallel to the 2D material electric-field distribution $|{\mathbf{E}}_{\parallel }{|}^2$ of the TE-supported mode.

Figure 2

(a) Blue curve: $|{\mathbf{e}}_{\parallel }{|}^2$ distribution on the TMD layer, evaluated by the FEM mode solver. Red curve: Piecewise approximation of $|{\mathbf{e}}_{\parallel }{|}^2$, where each segment is calculated as the mean of the respective true field distribution. Inset: Cross-sectional $|{\mathbf{e}}_{\parallel }{|}^2$ in the waveguide. The position of the TMD bilayer is shown with a dashed black line. (b) Circulating power of the stable Gaussian-like pulse. The inset shows the initial random noise seed. (c) Spectral power density of the circulating pulse. The red curve is a reference Lorentzian distribution normalized to the maximum spectral density of the signal and with the same bandwidth as that of the signal transition (20 Trad/s).

Figure 3

(a) Circulating pulse peak power in mW, (b) temporal (ps) and spectral (THz) FWHM width, and (c) time-bandwidth product of the stable pulse formed after around 600 round trips versus the normalized pumping rate $W_p{\tau }_{10}$.

Figure 4

2D evolution diagrams of the circulating laser intensity for various normalized pump rates $W_p{\tau }_{10}$. The corresponding pump rate is noted in each diagram in white. The vertical axis corresponds to the round-trip number while the horizontal axis corresponds to the fast time $T$. The latter is the same as the moving time-frame $T$ of the NLSE, which is moving with the group velocity $v_g$. Panel (a) shows no pulse formations, while panels (b)–(e) correspond to the formation and propagation of stable pulses. The random noise seed that simulates spontaneous emission can be seen in all panels around $T=0$ and is explicitly denoted in panel (e) with the white dashed rectangle. (f) The pulse shows significant changes from round trip to round trip and thus cannot be characterized as stable. Note that the label and units for the horizontal axis are common between all panels and are given below them.