Harnessing Dispersion in Soliton Microcombs to Mitigate Thermal Noise

We explore intrinsic thermal noise in soliton microcombs, revealing thermodynamic correlations induced by nonlinearity and group-velocity dispersion. A suitable dispersion design gives rise to control over thermal-noise transduction from the environment to a soliton microcomb. We present simulations with the Lugiato-Lefever equation (LLE), including temperature as a stochastic variable. By systematically tuning the dispersion, we suppress repetition-rate frequency fluctuations by up to 50 decibels for different LLE soliton solutions. In an experiment, we observe a measurement-system-limited 15-decibel reduction in the repetition-rate phase noise for various settings of the pump-laser frequency, and our measurements agree with a thermal-noise model. Finally, we compare two octave-spanning soliton microcombs with similar optical spectra and offset frequencies, but with designed differences in dispersion. Remarkably, their thermal-noise-limited carrier-envelope-offset frequency linewidths are 1 MHz and 100 Hz, which demonstrates an unprecedented potential to mitigate thermal noise. Our results guide future soliton-microcomb design for low-noise applications, and, more generally, they illuminate emergent properties of nonlinear, multimode optical systems subject to intrinsic fluctuations.

Figure 1

(a) The soliton velocity, $v_g$, depends on both its wavelength, ${\lambda }_{\mathrm{cs}}$, due to group-velocity dispersion (GVD), and the modal temperature, $T$, due to the thermo-optic effect. Hence, the pulse-to-pulse timing, $t_{\mathrm{rep}}=1/f_{\mathrm{rep}}$, is sensitive to $T$. (b) GVD curves with $T$ dependence. Dashed lines are lines of constant $n_g$. (c) Trajectory of $f_{\mathrm{rep}}$ for a change in $T$, $\mathrm{\Delta }T$. The GVD curve shifts due to the thermo-optic effect (vertical arrow), and the correlated change in ${\lambda }_{\mathrm{CS}}$ moves $f_{\mathrm{rep}}$ along its GVD curve (arrow parallel to curve). The change in $f_{\mathrm{rep}}$, $\mathrm{\Delta }{f}_{\mathrm{rep}}$, is the total vertical displacement.

Figure 2

LLE simulations of ${\eta }_T$ and soliton-microcomb thermal noise for various GVD settings. Parameters for the simulations are $D_2/2\pi =60\mathrm{MHz}$/mode and $F^2=10$. (a) Simulated $f_{\mathrm{rep}}$ versus $T$. As $D_3$ is increased from zero, ${\eta }_T$ decreases and eventually becomes negative. At the lowest values of $\alpha$, ripples in the data arise from small-amplitude breathing oscillations. (b) Optical spectra for various $D_3$. (c) Variance of $f_{\mathrm{rep}}$ frequency fluctuations, $⟨\delta f_{\mathrm{rep}}^2⟩$, calculated in two ways: From the slopes in (a) at $\alpha =7.65$ (gray data points) and by integrating $S_{\mathrm{rep}}$ (blue data points). (d) Simulated $S_{\mathrm{rep}}$ spectra for various $D_3$. Faded lines are LLE simulation results, and bold lines are fits to the data. The effective noise floor of our simulations is ${10}^{-4}{\mathrm{Hz}}^2/\mathrm{Hz}$.

Figure 3

Experimental evidence for thermal-noise mitigation by a balance of thermal shifts in GVD and $\mathrm{\Omega }$. (a) Optical spectrum used in the experiments. (b) Repetition frequency ($f_{\mathrm{rep}}$) detection by electro-optic modulation. The soliton-microcomb spectrum is shown both before electro-optic modulation (black) and after (blue). (c) Radio frequency spectrum (1 kHz resolution bandwidth) of the thermal-noise-limited $f_{\mathrm{rep}}$ signal. (d) $f_{\mathrm{rep}}$ versus ${\nu }_p$. These data are used to estimate ${\eta }_T$ and predict $S_{\mathrm{rep}}$. (e) $S_{\mathrm{rep}}$ phase-noise measurements of the 1-THz repetition rate for various settings of ${\nu }_p$. Dashed lines correspond to predictions from our thermal noise model using ${\eta }_T$ estimates from (d). (f) $S_{\mathrm{rep}}$ measurements at 10 kHz offset (blue circles) and predictions based on Eq. (7) and our thermal-noise model (orange stars). The dashed gray line corresponds to our measurement floor.

Figure 4

Simulations of thermal noise in octave-spanning soliton microcombs. (a) $D_{\mathrm{int}}$ for the two solitons analyzed in this study. Parameters for the dark blue traces (units omitted): $D_2/2\pi =20\times {10}^6$, $D_3/2\pi =1.5\times {10}^6$, $D_4/2\pi =-52\times {10}^3$, $D_5/2\pi =-4\times {10}^3$, $F^2=12$, $\alpha =9.3$. Parameters for the light blue traces: $D_2/2\pi =20\times {10}^6$, $D_3/2\pi =0$, $D_4/2\pi =-120\times {10}^3$, $D_5/2\pi =5.3\times {10}^3$, $D_6/2\pi =-150$, $F^2=12$, $\alpha =9.25$. (b) Optical spectra corresponding to the dispersion profiles in (a). (c) $f_{\mathrm{rep}}$ versus $T$. (d) $S_{\mathrm{rep}}$ spectra. (e) $S_{\mathrm{ceo}}$ spectra calculated from $m^2\times S_{\mathrm{rep}}$. The dashed line is the so-called beta line for understanding which Fourier frequencies contribute to the signal linewidth. (f) Simulated $f_{\mathrm{ceo}}$ beat note that indicates both coherence and signal-to-noise ratio are improved by mitigating thermal noise. The top (bottom) axis refers to the blue (cyan) trace.