Competing Turing and Faraday Instabilities in Longitudinally Modulated Passive Resonators

François Copie, Matteo Conforti, Alexandre Kudlinski, Arnaud Mussot, and Stefano Trillo

Phys. Rev. Lett. 116, 143901 – Published 7 April 2016

Abstract

We experimentally investigate the interplay of Turing (modulational) and Faraday (parametric) instabilities in a bistable passive nonlinear resonator. The Faraday branch is induced via parametric resonance owing to a periodic modulation of the resonator dispersion. We show that the bistable switching dynamics is dramatically affected by the competition between the two instability mechanisms, which dictates two completely novel scenarios. At low detunings from resonance, switching occurs between the stable stationary lower branch and the Faraday-unstable upper branch, whereas at high detunings we observe the crossover between the Turing and Faraday periodic structures. The results are well explained in terms of the universal Lugiato-Lefever model.

Received 17 December 2015

DOI:https://doi.org/10.1103/PhysRevLett.116.143901

Physics Subject Headings (PhySH)

Nonlinear optics

Cavity resonators

Atomic, Molecular & OpticalNonlinear Dynamics

Authors & Affiliations

François Copie^*, Matteo Conforti, Alexandre Kudlinski, and Arnaud Mussot

Univ. Lille, CNRS, UMR 8523—PhLAM—Physique des Lasers Atomes et Molécules, F-59000 Lille, France

Stefano Trillo

Department of Engineering, University of Ferrara, Via Saragat 1, 44122 Ferrara, Italy

^*francois.copie@univ-lille1.fr

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Vol. 116, Iss. 14 — 8 April 2016

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Images

Figure 1
(a) Schematic illustration of a nonuniform passive fiber ring cavity. (b) Piecewise constant dispersion map over one period of the GVD. (c) Normalized steady-state curves for detuning values $Δ = 1$ , 4 and 6.25. (d) Instability domains in a cavity with dispersion map as in (b) in the plane ( $Δ$ , $P_{u}$ ). The “Inaccesible” region corresponds to the negative-slope branch of the steady state. The green and blue domains represent the regions where Turing and Faraday modulated structures can be excited (at higher powers the Faraday structures destabilize giving rise to chaotic spatio-temporal evolutions). The bullets labeled 1, 2, 3, and 4 indicate the corresponding experimental results of Figs. 3 and 4. Parameters: $Λ_{a} = 0.97$ , $Λ_{b} = 0.03$ , $β_{a} = 1.5$ , $β_{b} = - 14$ , $α = 0.157$ , $θ^{2} = 0.1$ , $Λ = 1$ : the period of the GVD equals the length of the resonator.
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Figure 2
(a) Experimental setup [32]. (b) GVD map of the cavity over one round trip, centered on the $90 / 10$ SMF coupler. The gray horizontal line gives the average GVD $β_{2}^{a v}$ at pump wavelength 1550 nm (arrow on the right vertical axis, calibrated in terms of wavelengths). (c) Normalized transfer functions of the cavity for the control beam (dashed red) and the nonlinear beam in the linear regime (blue).
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Figure 3
(a) Bistable response of the cavity calculated for $Δ = 4$ ( $δ = π / 4.5 rad$ , $α = 0.175$ ); the hatched region is inaccessible. (b) Pseudocolor level plot of the gain spectrum as a function of the intracavity power calculated from Floquet analysis [28]. (c),(d) Comparison of experimental spectra (solid blue), spectra obtained from numerical integrations of the periodic LLE (1) (dashed red), and analytical estimates (vertical line, theory) from Eq. (3) with $m = 1$ for two different powers labeled 1 and 2 on the upper branch (see also Fig. 1): (c) $P_{i n} = 3.16 W$ ; (d) $P_{i n} = 1.97 W$ . Inset: close-up view of the sideband for the two powers.
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Figure 4
As in Fig. 3 for $Δ = 6.25$ ( $δ = π / 3.2 rad$ , $α = 0.157$ ). Here the two spectra in (c) and (d) are relative to points (labeled 4 and 3) on two different (upper and lower) branches corresponding to (c) $P_{i n} = 5.02 W$ (4.27 W in experiment); (d) $P_{i n} = 3.9 W$ (3.55 W in experiment). Estimated frequencies (theory) are from Eq. (3) with $m = 1$ in (c) and from Eq. (2) in (d).
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Figure 5
Experimental optical spectra at the cavity output vs input power for (a) $Δ = 4$ and (b) $Δ = 6.25$ . The numbered horizontal lines refer to the spectra shown in Figs. 3 and 4.
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