Chirped-pulsed Kerr solitons in the Lugiato-Lefever equation with spectral filtering

Optical Kerr resonators support a variety of stable nonlinear phenomena in a simple and compact design. The generation of ultrashort pulses and frequency combs has been shown to benefit several applications, including spectroscopy and telecommunications. The most common anomalous dispersion Kerr resonators can be accurately described by a well-studied mean-field Lugiato-Lefever equation (LLE). Recently observed highly chirped pulses in normal dispersion resonators with a spectral filter, however, cannot. Here, we examine the LLE in the normal dispersion regime modified with a Gaussian spectral filter (LLE-F). In addition to solutions associated with the LLE, we find stable highly chirped pulses. Solutions are strongly dependent on the filter bandwidth. Because of the large changes per round trip, the validity of the LLE-F fails over a large range of experimentally relevant parameters. While the mean-field approach leads to accurate predictions with respect to the nonlinearity coefficient and the dispersion, the dependence of drive power on loss deviates significantly from an experimentally accurate model, which leads to opportunities for Kerr resonators including frequency comb generation from low-Q-factor cavities.

Figure 1

Location of stable nontrivial solutions in drive-detuning parameter space of normal dispersion cavity with (LLE-F) and without (LLE) spectral filtering. The gray region represents dark solitons of the LLE, the orange region represents dark solitons of the LLE-F with a 14.5 nm per unit length spectral filter, and the red region represents chirped-pulse solitons in the LLE-F with the same filter. Pulse and spectral profiles of four representative stable solutions are inset.

Figure 2

Comparison of chirped-pulse solutions in the LLE-F and Ikeda-F models. (a) Intracavity evolution of the spectral bandwidth, (c) the output spectrum, and (e) the output chirped (solid) and dechirped (dashed) pulse for chirped solitons of the LLE-F with $\mathrm{drive}=0.25\mathrm{W}/\mathrm{m}, \mathrm{detuning}=0.07$ rad/m, and with a 14.5- nm per unit length Gaussian spectral filter and (b) intracavity evolution of the spectral bandwidth, (d) the output spectrum, and (f) the output chirped (solid) and dechirped (dashed) pulse for chirped solitons of the Ikeda-F with $\mathrm{drive}=0.25$ W/m, $\mathrm{detuning}=0.1$ rad/m, and with a 14.5- nm per unit length Gaussian spectral filter.

Figure 3

Location of stable chirped pulses in drive-detuning parameter space for the LLE-F and Ikeda-F models. Stable chirped pulses are found in the LLE-F with 14.5 nm (20.3 nm) per unit length filter in the red (magenta) regions. Stable chirped pulses are found in the Ikeda-F with 14.5 nm (20.3 nm) per unit length filter in the blue (green) regions. Pulse and spectral profiles of four representative chirped pulses at a fixed drive power are inset.

Figure 4

The deviation of the required drive power of stable chirped pulses in the LLE-F and Ikeda-F models vs the total change of the field during one roundtrip in the cavity. Change per round trip is calculated as the ratio between the highest and the lowest intracavity root-mean-square (RMS) bandwidth.

Figure 5

Comparison of the chirped-pulse system and pulse parameters from the Ikeda-F model (points) with the scaling laws derived from the LLE-F (red lines). From Eq. (4), (a) the drive power varies with nonlinearity, and (b) the full- width at half maximum filter bandwidth varies with dispersion. From Eq. (5), (c) the peak power decreases, and (e) the pulse duration is constant with increasing nonlinearity, and (d) the peak power is constant, and (f) the pulse duration increases with increasing dispersion. For variations in nonlinearity and dispersion, the scaling laws agree well with the Ikeda-F model.

Figure 6

Location of stable chirped pulses in drive-detuning parameter space for the (a) LLE-F and (b) Ikeda-F models for different losses per unit length $\alpha$. The change in the required drive is much lower (1.3) for the Ikeda-F model than that predicted by the LLE-F model [8 from Eq. (4)].

Figure 7

Location of stable nontrivial solitons in drive-detuning parameter space with (LLE-F) and without (LLE) spectral filtering. Black points represent dark solitons of the LLE, orange points represent dark solitons of the LLE-F with a 14.5 nm per unit length spectral filter, and red dots represent chirped-pulse solitons in the LLE-F with the same filter. The shaded regions indicate the boundaries of each stable solution in parameter space. The dashed line and indicated points correspond to the solutions examined in Fig. 8.

Figure 8

Evolution to the converged solution for the Lugiato-Lefever equation with (LLE-F) and without (LLE) a spectral filter. The subpanels (a)–(h) correspond to the parameters with the same labels in Fig. 7. (a), (e), (d), and (h) At the highest and lowest drive powers, both systems converge to the trivial continuous wave (CW) solution. At the drive powers from (b) and (f), both systems converge to a dark soliton solution, with a smoother field supported by the LLE-F. At lower drive powers from (b) and (f), while without a filter, only the CW solution is found, but (g) with a filter, stable chirped pulses are stabilized.

Figure 9

Illustration of the deviation of the LLE-F from the Ikeda-F model. (a) Regions in the drive-detuning parameter space where chirped pulses are found in the LLE-F (red) and Ikeda-F (blue) models. Change in the required drive power between the LLE-F and Ikeda-F models as a function of (b) the ratio of the longest to the shortest pulse duration (full width at half maximum) and (c) the ratio of the highest to lowest peak power for five different detuning values with the other parameters fixed. The dashed line and indicated points in (a) correspond to the solutions examined in (b) and (c).