Analysis of Kerr comb generation in silicon microresonators under the influence of two-photon absorption and fast free-carrier dynamics

Kerr frequency comb generation relies on dedicated waveguide platforms that are optimized toward ultralow loss while offering comparatively limited functionality restricted to passive building blocks. In contrast to that, the silicon-photonic platform offers a highly developed portfolio of high-performance devices, but suffers from strong two-photon absorption (TPA) and subsequent free-carrier absorption (FCA) at near-infrared telecommunication wavelengths, thereby rendering Kerr comb generation a challenge. Here we present a model to investigate the impact of TPA and FCA on Kerr comb formation. Our model combines a modified version of the Lugiato-Lefever equation with a refined relation to precisely describe the fast space and time dependence of the free-carrier concentration along the circumference of the microresonator. Using this refined model, we derive conditions for modulation instability, in particular for necessary pump powers depending on TPA parameters and free-carrier lifetimes. We validate our analytical predictions by time integration and study the impact of fast free-carrier dynamics on Kerr comb formation. We find that silicon microresonators may be suitable for Kerr comb generation in the NIR, provided that the dwell time of the TPA-generated free carriers in the waveguide core is reduced by a reverse-biased $p\text{−}i\text{−}n$-junction and that the pump parameters are chosen appropriately.

Figure 1

Silicon-photonic microresonator as an example of a device suffering from both TPA and FCA. The bus waveguide as well as the resonator ring waveguide are connected to $p^{+}$-doped (red) and $n^{-}$-doped silicon (Si) regions (blue) that form a p-i-n-junction. This junction allows the reduction of the dwell time of free carriers by applying a reverse-bias voltage U through contact pads and vias. At the coupling section between the bus and the ring waveguide, the doping is locally inverted from an $n^{-}$ doping to a $p^{+}$ doping next to the bus waveguide to ensure maximum free-carrier removal in the microresonator [28]. The geometrical dimensions of the waveguide cross section can be chosen such that the waveguide features anomalous group-velocity dispersion (GVD) at wavelengths near $1550\mathrm{nm}$, see, e.g., [29]. Further, the microresonator diameter can be chosen to achieve a specific free spectral range of, e.g., $f_{\mathrm{FSR}}=100\mathrm{GHz}$, corresponding to a round-trip time $t_{\mathrm{RT}}=f_{\mathrm{FSR}}^{-1}=10\mathrm{ps}.$ For details on the geometrical dimensions used in our simulations, see Sec. 5.

Figure 2

Threshold forcing needed to achieve MI as a function of $r$ and ${\sigma }^{\prime }{t}_{\mathrm{eff}}^{\prime }$. Modulation instability and comb formation can be observed only to the left of the vertical dashed line $r=3^{-1/2}$. The gray area indicates the parameter space for which modulation instability does not occur. Magenta crosses mark real data points reported for silicon-photonic waveguides [20, 22, 38, 39, 40, 41], assuming waveguide losses of ${\alpha }_i=46{\mathrm{m}}^{-1}$ ($2\mathrm{dB}{\mathrm{cm}}^{-1}$) [20]. Reference [20] reports on silicon-photonic waveguides in which free carriers are actively removed by a reverse-biased p-i-n-junction, leading to a dwell time of $t_{\mathrm{eff}}^{\prime }=12\mathrm{ps}$. In all other cases, no free-carrier removal was used, leading to dwell times of the order of $1-5\mathrm{ns}$ according to [16, 20, 21, 22, 23]. Specifically, a value of $3\mathrm{ns}$ was used for the data points related to Refs. [38, 39, 40, 41], which do not specify values for $t_{\mathrm{eff}}^{\prime }$. The value for the FCA cross section $\sigma =1.45\times {10}^{-21}{\mathrm{m}}^2$ is consistently found in various publications [20, 22, 39]. The operating wavelength is $1550\mathrm{nm}$.

Figure 3

Results of time integration of the LLE for different resonator designs and pump parameters. Row R1: Simulation parameters. Row R2: Dispersion profile for the central 101 modes. Row R3: Computed normalized gain rates at the beginning of the simulation (slow time $t^{\prime }=0$). Row R4: Evolution of power spectra with normalized slow time $t^{\prime }$. Row R5: Spectra at stationary state for $t^{\prime }=150$. Row R6: Color-coded intracavity (IC) power $|\underline{\mathit{a}}\left(t^{\prime },{\tau }^{\prime }\right){|}^2$ evolving during comb formation over $t^{\prime }$ as a function of the normalized fast time ${\tau }^{\prime }$. We observe a temporal shift of the IC pulse pattern within the retarded time frame of the fast time axis ${\tau }^{\prime }$, which can, e.g., be caused by FC dispersion or avoided mode crossings. Besides this shift along the ${\tau }^{\prime }$ axis, the IC power distribution evolves into a “quasistationary” final state in all cases. Row R7: Quasistationary IC power $|\underline{\mathit{a}}\left(t^{\prime }=150,{\tau }^{\prime }\right){|}^2$. Row R8: Free-carrier density $N_{\mathrm{c}}^{\prime }\left(t^{\prime }=150,{\tau }^{\prime }-150v^{\prime }\right)$ at final state. Column (a): Modulation instability inhibited by long free-carrier dwell time $t_{\mathrm{eff}}$, which corresponds to five times the cavity round-trip time $t_{\mathrm{RT}}$. Column (b): Modulation instability with sufficiently reduced carrier dwell time $t_{\mathrm{eff}}=1.22\times t_{\mathrm{RT}}$. Column (c): Single-soliton formation for dynamically increasing detuning. Column (d): Comb formation in a normal-dispersion resonator with avoided mode crossing.

Figure 4

(a) On-chip threshold pump power for the onset of modulation instability as a function of TPA coefficient ${\beta }_{\mathrm{TPA}}$ for different FC dwell times $t_{\mathrm{eff}}$. We assume a silicon microresonator with an $f_{\mathrm{FSR}}=100\mathrm{GHz}$ at $1550\mathrm{nm}$; see Sec. 5 for details. The microresonator is assumed to have $2\mathrm{dB}{\mathrm{cm}}^{-1}$ waveguide loss, anomalous dispersion, and critical coupling to the bus waveguide. Reported values of ${\beta }_{\mathrm{TPA}}$ for silicon range from $0.45$ to $1.5\mathrm{cm}{\mathrm{GW}}^{-1}$ as indicated by the vertical dashed lines. (b) Soliton comb spectrum obtained in the bus waveguide after the microresonator. The resonator is identical to the one considered in (a). We assume ${\beta }_{\mathrm{TPA}}=0.7\mathrm{cm}{\mathrm{GW}}^{-1}$, $t_{\mathrm{eff}}=12\mathrm{ps}$, and $P=50\mathrm{mW}$. The spectrum is derived from the normalized intracavity comb spectrum indicated in Fig. 3, row R5.

Figure 5

Threshold pump power needed to achieve modulation instability. (a) Approximate analytical evaluation; see also Fig. 2 of the main text. (b) Corresponding numerical calculation. (c) Relative color-coded difference in threshold pump powers when comparing the analytical and the numerical method. The deviation is smaller than $1\%$ in the whole region.

Figure 6

Normalized modulus squared of the total electric field of the quasi-TE mode at $1550\mathrm{nm}$. We assume a waveguide bent to the left with a bend radius of $115\mu \mathrm{m}$. The white outline marks the undoped silicon part of the waveguide.

Figure 7

Results of time integration of the LLE and the free-carrier rate equation for different integration formalisms. Row R1: Simulation parameters. Row R2: Color-coded intracavity (IC) power $|\underline{\mathit{a}}\left(t^{\prime },{\tau }^{\prime }\right){|}^2$ as a function of fast time ${\tau }^{\prime }$, evolving over slow time $t^{\prime }$. Row R3: Color-coded free-carrier density (FCD) $N_{\mathrm{c}}^{\prime }\left(t^{\prime },{\tau }^{\prime }-t^{\prime }{v}^{\prime }\right)$ as a function of fast time ${\tau }^{\prime }$, evolving over slow time $t^{\prime }$. Row R4: Mean IC power $\langle |\underline{\mathit{a}}\left(t^{\prime },{\tau }^{\prime }\right){|}^2{\rangle }_{{\tau }^{\prime }}$ as a function of the slow time $t^{\prime }$. Vertical dashed lines indicate the temporal positions of maximum intracavity power. Row R5: Mean free-carrier density $\langle N_{\mathrm{c}}^{\prime }\left(t^{\prime },{\tau }^{\prime }-t^{\prime }{v}^{\prime }\right){\rangle }_{{\tau }^{\prime }}$ as a function of the slow time $t^{\prime }$. Column (a): Free-carrier oscillations simulated using the full model for the description of FC dynamics, Eqs. (E1) and (E2). Column (b): Free-carrier oscillations modeled using a mean free-carrier density. Both models give essentially the same results.

Figure 8

Results of time integration of the LLE and the free-carrier rate equation for different integration formalisms. Row R1: Simulation parameters. Row R2: Color-coded intracavity (IC) power $|\underline{\mathit{a}}\left(t^{\prime },{\tau }^{\prime }\right){|}^2$ as a function of fast time ${\tau }^{\prime }$, evolving over slow time $t^{\prime }$. Row R3: IC power at the end of the simulation, $|\underline{\mathit{a}}\left(t^{\prime }=100,{\tau }^{\prime }\right){|}^2$. Row R4: Color-coded free-carrier density (FCD) $N_{\mathrm{c}}^{\prime }\left(t^{\prime },{\tau }^{\prime }-t^{\prime }{v}^{\prime }\right)$ as a function of fast time ${\tau }^{\prime }$ evolving over slow time $t^{\prime }$. Row R5: Free-carrier density at the end of the simulation, $N_{\mathrm{c}}^{\prime }\left(t^{\prime }=100,{\tau }^{\prime }-100v^{\prime }\right)$. Column (a): Free-carrier oscillations simulated using the full model for the description of FC dynamics, Eqs. (E1) and (E2). Column (b): Free-carrier oscillations modeled using a mean free-carrier density. Correct representation of the fast-time free-carrier dynamics results in a drift of the soliton pulse along the fast time axis ${\tau }^{\prime }$, which cannot be described by the mean-value approximation of the intracavity free-carrier distribution.