Turing patterns in a fiber laser with a nested microresonator: Robust and controllable microcomb generation

Microcombs based on Turing patterns have been extensively studied in configurations that can be modeled by the Lugiato-Lefever equation. Typically, such schemes are implemented experimentally by resonant coupling of a continuous wave laser to a Kerr microcavity in order to generate highly coherent and robust waves. Here, we study the formation of such patterns in a system composed of a microresonator nested in an amplifying laser cavity, a scheme recently used to demonstrate laser cavity solitons with high optical efficiency and easy repetition rate control. Utilizing this concept, we study different regimes of Turing patterns, unveiling their formation dynamics and demonstrating their controllability and robustness. By conducting a comprehensive modulational instability study with a mean-field model of the system, we explain the pattern formation in terms of its evolution from background noise, paving the way towards complete self-starting operation. Our theoretical and experimental paper provides a clear pathway for repetition rate control of these waves over both fine (Megahertz) and large (Gigahertz) scales, featuring a fractional frequency nonuniformity better than $7\times {10}^{-14}$ with a 100-ms time gate and without the need for active stabilization.

Figure 1

(a) Depiction of the laser operation. The Kerr microresonator (green ring) is nested into an amplifying fiber loop (yellow). A Turing pattern waveform (light green) is excited in the microcavity and is sustained by a leading-order (red) and a first-order (blue) “supermode” waveforms from the amplifying loop. These supermodes are periodical with the microcavity length, highlighted by a line segment $L_{\mathrm{a}}$. (b) Spectral distribution of the modes in the cold-cavity condition. The microcavity resonances (green) have a FSR denoted $F_{\mathrm{a}}$ whereas the FSR of the amplifying main-cavity modes (black) is denoted $F_{\mathrm{b}}$. The leading-order and first-order supermodes are plotted in red and blue, respectively. (c)–(e) Zoom of the $m\mathrm{th}$, central, and $-m\mathrm{th}$ resonances. Δ is the normalized frequency offset between the central frequency of the leading supermode and the microcavity resonance. $v$ is the normalized FSR detuning, appearing when the two cavities are not commensurate.

Figure 2

MI maps for the stationary states. For the calculations: ${\zeta }_a=1.7\times {10}^{-4},{\zeta }_b=3.5\times {10}^{-4},\sigma =2.5\times {10}^{-4}$, and $\sqrt{\kappa }=2.5$, which are compatible with the experimental regime discussed in the sections below. Here, we used $N=7$, corresponding to a total of 15 supermodes. (a) MI map for the trivial solution (TS) showing the regions of instability as a function of the cavity gain $g$ and detuning Δ with $\nu =0$. (b) MI gain of the zero solution with $g=0.06$ (indicated by a white dashed line) and $\nu =0$. (c) MI gain of the low-energy cw state $I_0^{+}.$ (d) Zoom out of (c). (e)–(h) are the same as (a)–(d) with $\nu =1/3$.

Figure 3

Self-starting Turing pattern states. The zero solution evolves into state $I_{+}^{\left(0\right)}$, which is also unstable and gives rise to a Turing pattern. The simulation parameters are: ${\zeta }_a=1.7\times {10}^{-4},{\zeta }_b=3.5\times {10}^{-4},\sigma =2.5\times {10}^{-4},\mathrm{\Delta }=-0.17$, $\sqrt{\kappa }=2.5$, $g=0.0365$, $\nu =-0.3342$, $N=7$. (a) Evolution of the temporal profile of the field in the microcavity with the $y$ axis normalized to the microcavity length. (b) Maximum intensity of the evolution in the microcavity. (c) Low-energy stationary states. (d) Output time profile of the fields in the micro- and main cavities. (e) Output spectral profiles of the fields within the micro- and the main cavities with power spectral density (PSD) as a function of the normalized relative frequency.

Figure 4

(a) Experimental setup of the micro-cavity-filtered laser. Erbium-ytterbium-doped fiber amplifier: EYDFA; translation delay stage: TD; bandpass filter: BPF; output coupler: OC. (b) Diagnostic setup for the frequency comb-assisted spectroscopy. Photodiode: PD; MZI; radio-frequency bandpass filter: rf-BPF. (c) Diagnostic setup for the frequency uniformity measurement. The beat notes B1–B3 among the three desired comb lines W1–W3 and their adjacent reference frequency comb lines are selected using a dense wavelength division multiplexer (DWDM). They are then mixed and counted by a high-resolution frequency counter.

Figure 5

(a), (d), and (g) Experimental optical power spectral density (PSD) for the microcomb modes. The insets depict the autocorrelation (AC) and the intracavity spectrum of the oscillating spectral mode with the highest power. The red line here shows the position of detuning Δ used in the respective simulations. The color shadings in (d) are in reference to the resonance profiles presented in Fig. 6. (b), (e), and (h) Simulated optical spectra calculated at $\nu =1/2,1/3,1/4,$ respectively, with corresponding gain settings and cavity detunings of (b) $g=0.0400$, $\mathrm{\Delta }=-0.36$, (e) $g=0.0320$, $\mathrm{\Delta }=-0.17$, and (h) $g=0.0501$, $\mathrm{\Delta }=-0.54$. (c), (f), and (i) MI gain spectrum of the cw states. In the calculations: $\sigma =2.5\times {10}^{-4}$, ${\zeta }_a=1.7\times {10}^{-4}{\zeta }_b=3.5\times {10}^{-4}$, $\sqrt{\kappa }=2.5$, and $N=7$

Figure 6

Experimental resonance profiles and beat note signals obtained with the probe laser scanning oscillating comb lines. (a)–(e) Descending rows correspond to a relative change in the main cavity length of 0, −100, −200, −300, and −400 µm. The four sequential color-shaded plots correspond to different comb line wavelengths, 1547.13, 1549.47, 1550.64, and 1553.01 nm, respectively, from the left to the right panels as indicated by the color shading in Fig. 5. The dashed red line is a Lorentzian fit to the profile of the hot resonance whereas the vertical dashed-blue lines indicate the relative shift of the oscillating comb lines within their respective resonances.

Figure 7

(a) and (b) Numerical results showing the evolution of the (a) temporal and the (b) spectral profile of the field. The temporal evolution has a corresponding zoom of the stable operation to illustrate more clearly the pattern. For the simulations: ${\zeta }_a=1.7\times {10}^{-4}$, ${\zeta }_b=3.5\times {10}^{-4}$, $\sigma =2.5\times {10}^{-4}$, $\mathrm{\Delta }=-0.17$, $\sqrt{\kappa }=2.5$, $g=0.0365$, $\nu =-0.3942$, and $N=7$. (c) Optical spectrum of the stable pattern state. (d) Differential phase evolution of three comb modes indicated by colored shading in (c). The differential phase is calculated as the derivative of the absolute phase ${\phi }_{\mathrm{n}}$ over the temporal evolution, and it is indicative of the stability intrinsic to the repetition rate of the comb.

Figure 8

(a) The average beat note frequencies between two selected microcomb lines and their adjacent reference frequency comb teeth (recorded by an oscilloscope) as a function of the decreasing main-cavity length. The raw frequency measurement, in megahertz, relative to the frequency for 0-µm delay, is shown in blue. (b) Distribution of the microcomb output spectrum stemming from an equidistant mode spacing, including a Gaussian fit (dashed red line). Here, we consider the −300-µm delay case with a frequency counter gate time of 100 ms.