Observation of the Eckhaus Instability in Whispering-Gallery Mode Resonators

Gomila, D.; Parra-Rivas, P.; Colet, P.; Coillet, A.; Lin, G.; Chembo, Y.K.
submitted, (2021)

The Eckhaus instability is a secondary instability of nonlinear spatiotemporal patterns in which
high-wavenumber periodic solutions become unstable against long-wavelength perturbations. We
show in this letter that this instability can take place in Kerr combs generated with ultra-high
Q whispering-gallery mode resonators. In our experiment, sub-critical Turing patterns undergo
Eckhaus instabilities upon changes in the laser detuning leading to cracking patterns with long-
lived transients. In the spectral domain, this results in a metastable Kerr comb dynamics with
a timescale as large as few seconds. This ultra-slow timescale is at least six orders of magnitude
larger than the intracavity photon lifetime, and is in sharp contrast will all the transient behaviors
reported so far in dissipative nonlinear optics, that are typically only few photon lifetimes long
(microseconds). We show that this phenomenology is well explained by the Lugiato-Lefever model,
as the result of an Eckhaus instability. Our theoretical analysis is found to be in excellent agreement
with the experimental measurements.

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