We present a theoretical and experimental study of the modulation instability process in a dispersion oscillating passive fiber-ring resonator in the low dispersion region. Generally, the modulation of the dispersion along the cavity length is responsible for the emergence of a regime characterised by multiple parametric resonances (or Faraday instabilities). We show that, under weak dispersion conditions, a huge number of Faraday sidebands can grow under the influence of fourth order dispersion. We specifically designed a piecewise uniform fiber-ring cavity and report on experiments that confirm our theoretical predictions. We recorded the dynamics of this system revealing strong interactions between the different sidebands in agreement with numerical simulations.
© 2017 Optical Society of America
8 May 2017: Corrections were made to Eqs. (3) and (6).
Nonlinear fiber cavities have been widely studied during the last 30 years. The inclusion of fibers’ Kerr nonlinearity in a passive system subject to optical feedback opens the way to a large variety of instabilities  among which modulation instability (MI) plays a significant role [2,3]. Arising from the interplay between losses and driving on one side and nonlinearity and dispersion on the other side, MI in nonlinear cavities has attracted a lot of attention as it allows the generation of ultra-fast train of pulses from a continuous or quasi-continuous wave excitation . In this context MI is usually referred to as Turing instability by analogy with the mechanism of pattern formation in the context of chemical reactions .
The group velocity dispersion (GVD) is one of the most important parameters of the cavity . To date, most studies of intracavity MI have addressed the case of rather large group velocity dispersion in both the anomalous [2, 4] and normal  regimes. In these cases, it is legitimate to take into account only the second order dispersion term β2, but when investigating MI under weak dispersion the impact of higher order dispersion (HOD) becomes significant. Indeed, it is well known that in fibers (in the single-pass configuration) with small β2, the MI process is affected by HOD terms. For example, the fourth order dispersion (FOD, β4) can extend the range of existence of MI to the normal dispersion regime [8–10]. In periodic dispersion fibers the impact of β4 is also significant. Indeed, it can either produce new sidebands in the spectrum  or suppress the MI process . HOD terms have been found to be at the origin of numerous effects in cavities as well. Recent studies have shown, for example, that the first HOD term (β3) is responsible for the emergence of convective and absolute instabilities , the asymmetry observed in intracavity MI spectrum  or short pulses spectrum . β3 also impacts the stability of Kerr frequency combs triggered by MI in both fiber cavities and microresonators . It was also predicted that the FOD might give rise to a new unstable frequency in low dispersion photonic crystal fiber cavities .
In previous works we found that a longitudinal step-like dispersion profile along the length of the cavity gives rise to a new regime of instability originating from parametric resonance that we refer to as Faraday regime, which coexists with the well established Turing regime [18–21]. Such a Faraday mechanism is attracting a lot of attention as a general mechanism related to parametric driving in fiber cavities . In our passive cavity, the Faraday regime of instability results in high frequency parametric sidebands in the optical spectrum. In the first experimental investigations of this kind of resonator, composed of two sections of fibers spliced together, the local dispersion encountered by light was rather high so that the impact of HOD was not significant [20, 21].
In this work, we study a dispersion modulated fiber-ring cavity pumped close to the zero-dispersion wavelength (ZDW) of one fiber section. As a consequence, a huge number, up to 125, of quasi-phase-matched Faraday sidebands are observed at the output of the cavity thanks to the contribution of FOD. The paper is organized as follows: in Section 2 we extend our previous model  to higher order dispersion terms. We especially show that β4 drastically changes the shape of the parametric gain spectrum when the cavity is pumped in the vicinity of a ZDW with the formation of high frequency sideband clusters. In Section 3 we report on the experimental observation of such spectra in excellent agreement with theoretical predictions. The round-trip-to-round-trip evolution of the instability spectrum is also investigated and features strong multiple wave mixing leading to a stationary regime in agreement with numerical simulations. Conclusions are drown in Section 4 and some perspectives are given.
In passive fiber-ring cavities with a step-like GVD profile an additional regime of parametric instability appears, that we called Faraday regime [18, 20, 21] in addition to the well-known Turing regime. This latter can be fully interpreted by considering average parameters of the cavity contrary to the Faraday regime which only exists thanks to the modulated GVD [18–20]. In the previous experiments, the values of dispersion of each fiber were relatively high and thus the behaviour of the system was dominated by the second order dispersion β2.
In this section, we extend the previous theoretical model to higher order dispersion terms which become significant in regions of low dispersion. To do so, we consider the following generalized mean field Lugiato-Lefever (LL) equation [3, 23] which describes light propagation inside the ring cavity:3, 20, 21, 24].
We perform the stability analysis of this equation relative to the ansatz , where Pu is the intracavity power and u, v are small real perturbations. By introducing the Fourier transform in time [ and ], the linearised problem reduces to the following set of coupled linear differential equations:
Starting from Eqs. (2) and (3), we follow the same approach discussed in detail in the supplemental of Ref. . We consider a ring cavity composed of two different pieces of uniform fiber labelled fiber a and fiber b in Fig. 1 and we limit the expansion up to the fourth order dispersion term β4 (i.e., N = 4 in Eq. (1)) since higher orders do not significantly impact the behaviour of the system. Under the assumption that losses and odd order dispersion terms do not vary inside the cavity, the following parametric resonance conditions are derived:
From this equation it appears that an infinite number of resonances exists each one related to an order m. The parametric gain G(ω) can be calculated according to the following set of expressions:
At this point, we recall the expression of the frequencies of the MI sidelobes in cavities with modulated dispersion without the contribution of β4 :3]:
We computed the parametric gain from Eq. (6) for a cavity composed of two uniform fibers of equal length (i. e. La = Lb = L/2). We consider the system to be in the bistable regime (Δ = 4 ) with steady state set on the upper branch of the cavity response as illustrated in Fig. 2(b) (Pu = 7.8 W). For uniform cavities, this branch is modulationally unstable in the average anomalous dispersion regime with the characteristic frequency of Eq. (8) and stable in the average normal dispersion regime .
Let us first consider the case β4 = 0 (Fig. 2). The evolution of the GVD of each fiber composing the cavity is represented in Fig. 2(a) as a function of the wavelength and shows each fiber zero dispersion wavelength (ZDW) and respectively. The average dispersion of the cavity is plotted in black and lies in between the two previous curves and the resulting average zero dispersion wavelength is . Figure 2(c) displays a 2D color plot of the parametric gain spectrum (y axis) as a function of the pump wavelength (x axis) calculated from Eqs. (6). For the average dispersion is normal and the spectrum exhibits only two weak sidebands. When increasing the wavelength the number of bands grows, is maximum for , and then decreases until no sidebands exists in the vicinity of . The position of each sideband corresponds to a solution of Eq. (7) for m ≠ 0. For example, the calculated position for the m = 1 condition is superimposed (see dashed gray lines). For longer wavelengths the average dispersion becomes anomalous. Again, several pairs of sidebands emerge and when a large number of bands appear in a fashion similar to the case . Then, less and less sidebands exist as the wavelength is further increased above . The main difference between the average normal and average anomalous region is the pair of broad, low frequency and high gain bands that can be seen in the anomalous regime. These bands are associated to the well-known MI in the anomalous dispersion region of uniform cavities  as it can be inferred from the agreement with the predictions of Eq.(8) (superimposed in dashed green lines) which gives the position of the MI sidelobes in the case of constant dispersion.
In order to get a clearer insight of what occurs when λP lies in the vicinity of one of the ZDW of the cavity, a closer view of the region around (rotated by 90 degrees anticlockwise for convenience) is plotted in Fig. 2(d). The gain spectrum exhibits more and more sidebands as λP approaches as highlighted in Figs. 2(e–g) which show three specific spectra. For , an infinite number of sidebands are destabilized with almost the same gain (Fig. 3(e)).
This latter observation results from the exact cancellation of the dispersion operator for that specific pump wavelength and is thus not physically meaningful. Indeed, in regions of low dispersion (i. e. in the vicinity of a ZDW) HOD terms become significant and cannot be neglected anymore. This realistic configuration is represented in Fig. 3 which keeps the same structure as Fig. 2 except that we accounted for the FOD. In this exemple β4 is negative as it is the case in most optical fibers, and constant all along the cavity length. Note that β3 and more generally odd order dispersion terms do not play any role in the parametric gain. Figure 3(a) represent the dispersion of each fiber as a function of the wavelength and Fig. 3(b) the bistable response of the cavity. It appears from Fig. 3(c) that the parametric gain spectrum is influenced by β4 mostly in the vicinity of the ring’s ZDWs (i. e. when the dispersion is locally low) and remains the same otherwise. It can also be pointed out that the predicted positions of the sidebands by Eq. (5) (in dashed grey lines) is essentially the same as when β4 = 0. Fig. 3(d) shows a close-up view of the region around as in Fig. 2(d) and clearly illustrates the strong modification of the gain spectrum under the influence of β4. Indeed, the branches that tend toward infinite frequencies in Fig. 2(d) are now curved toward lower wavelengths. At the ZDW of fiber the gain spectrum is now limited to a rather narrow range of frequencies and the number of sidebands becomes finite (Fig. 3(e)). For shorter pump wavelengths the gain spectrum is also significantly modified by the presence of β4. Indeed, new higher frequency clusters of sidebands appear. For wavelength just slightly shorter than (Fig. 3(f)) these clusters are still connected to the lowest frequency sideband pairs (which are nearly unaffected by β4) and form a rather broad spectrum. But for even shorter wavelengths (Fig. 3(g)) they eventually form detached high frequency clusters. Note that the same observation stands around the ZDW of fiber in the average anomalous GVD region (see Fig. 3(c)).
The existence of the high frequency clusters is thus due to the FOD and their spectral positions can be calculated by noting that it is solely driven by the even dispersion coefficients (β2 and β4) of the locally low dispersion fiber section (fiber a in the case of Fig. 3(c–f)). It simply follows by dropping the contribution of the dispersion modulation in Eq. (5). This leads to the following expression for the position of the β4 induced clusters:
This frequency is reported in Fig. 3(d) in dashed black lines and coincide with the maximum of the envelope of the spectrum.
We designed a ring cavity similar to the one depicted in Fig. 1. It is composed of 22.6 m of dispersion shifted fiber (DSF) with a ZDW at 1551.05 nm (fiber a) and 20.5 m of dispersion compensating fiber (DCF) (fiber b). The evolution of the GVD as a function of the wavelength is represented in Fig. 4(a) for the two fibers. The range of pump wavelengths accessible in the experiment is shaded and corresponds to a small region below . In this region, all the predicted sidebands exhibit a parametric gain of the same order of magnitude (see Fig. 3(d)) which facilitate their observation as opposed to the region around where the first pair of sidebands has a parametric gain six times larger than the others. Figures 4(b, c) display the evolution over three round-trips of β2 and β4 respectively. We can see that, for the range of pump wavelength investigated, both fibers exhibit normal dispersion with a nearly constant difference (≈ 10 ps2/km). Furthermore, fiber a exhibits a very low GVD in this region which will emphasize the effect of β4. The two fibers are characterized by fourth order dispersions of respectively and and the nonlinearity is estimated to be γ = 2.5/W/km for both fibers. We measured a finesse of 18 and thus α ≈ 0.17. For all the following experiments the detuning was fixed to π/4.5 rad which leads to a normalized detuning , that is to say the cavity was driven in the bistable regime . It is worth noting that for this set of parameters, only the upper branch of the bistable cycle exhibits Faraday MI since the system cannot exhibit stable Turing patterns on the lower branch unless Δ > 4.25 .
The setup used for the experiments is depicted in Fig. 5(a) and is similar to the one used in our previous studies ([20, 21]). A continuous wave laser with very low spectral width (< 1 kHz) and tunable wavelength (between 1549.6 nm and 1550.3 nm) is chopped by an electro-optic modulator (EOM 1) to produce a train of 400 ps square pulses at a repetition rate of 4.74 MHz which corresponds to the round-trip time of the 43.14 m long cavity. Those pulses are amplified to peak powers up to a few watts thanks to an erbium doped fiber amplifier (EDFA) and the pump is filterd out by a narrow tunable band-pass filter (BPF) to remove amplified spontaneous emission (ASE) in excess. A second polarization dependant EOM (EOM 2) works as a variable attenuator to control the input pump power and is also used to tailor the train of pulses into periodic bursts (see Fig. 5(b)). Input pulses are directed to the input 90/10 coupler of the cavity through a circulator which allows us to actively stabilise the cavity length against acousto-mechanical perturbation with a precision of the order of λ0/200 using a counter propagating reference beam. The output of the cavity can be analysed simultaneously using two different detection schemes: (i) An optical spectrum analyser (OSA) displays the full spectrum averaged over thousands of round-trips typically with a resolution of 4 GHz; (ii) A setup based on the real-time dispersive Fourier transformation (rt-DFT) technique allows the recording of round-trip-to-round-trip spectra but offers a lower resolution (≈ 30 GHz) and dynamics .
A particular attention has to be paid to the driving intensity of the cavity. Indeed, the regime of instability we investigate is characterized by tens of spectral sidebands. It is important to emphasize here that the theoretical spectra shown so far in Fig. 2 and 3 are gain spectra which do not take into account wave mixing between the different MI gain bands. The consequence of such a potential parasitic wave mixing is that the output spectra when a steady state is reached is not likely to contain every predicted gain bands. It is thus necessary to circumvent this detrimental wave mixing to allow the recording of all the sidebands by means of an OSA. One way to do so is to reduce the power of the quasi-phase-matched (QPM) sidebands as it is known that the wave mixing is less effective when the power of the frequency components is low. To achieve this we propose a pumping technique that we labelled short burst pumping which is illustrated in Fig. 5(b). It consists of periodically turning on and off the coherent driving of the cavity so that the QPM sidebands do not reach a power level for which wave mixing is detrimental. When the driving field is on, the power of the sidebands grows (see the dashed red line in Fig. 5(b)) and when it is turned off, the power decays exponentially as the cavity empties itself. For comparison, the evolution of the signal power if the driving field is not turned off is sketched as a dashed black line. The on and off durations were adjusted experimentally: ΔT1 is the largest value for which no evidence of spurious wave mixing is seen in the output spectrum, thus maximising the signal power while circumventing the wave mixing issue. The dead time ΔT0 was set just long enough to ensure that each burst acts as an independent run of the experiment. It should be noted that it typically takes a few seconds to record an OSA trace which corresponds to thousands of consecutive bursts in our case. Such recordings are therefore average representations of the dynamics of the cavity at the early stage of the parametric amplification as it will be discussed in the last section.
3.2. Observation of complex parametric instability spectra
We applied the “short bursts pumping” method to observe a very large number of sidebands when the pump wavelength lies in the vicinity of the ZDW of one fiber (Figure 3(f) typically).
Firstly we confirm the dependence of the instability spectrum on the pump wavelength. Fig. 6(a) displays a 2D color plot of the experimental output spectrum for pump wavelengths varying from 1549.6 to 1550.3 nm, constant input peak power of 4 W and normalized detuning Δ = 4. ΔT1 = 100 μs (475 pulses) and ΔT0 = 10 μs. Under these conditions the system evolves on the higher transmission branch of the bistable cycle. As predicted by our theoretical study, multiple sideband pairs are observed with symmetric location around the pump, as shown in Fig. 6(a). In particular, clusters of narrow sidebands which position strongly depends on the pump wavelength appear at high frequency. As the pump wavelength is increased closer to the ZDW of fiber a (1551.05 nm), the clusters drift toward lower frequency until the spectrum exhibits a quasi-continuum of sidebands spanning more than 16 T Hz (130 nm) for λP = 1550.3 nm. The position of the clusters is in excellent agreement with the predictions of Eq. (9) (superimposed in dashed black lines) which correspond to a uniform cavity with FOD. Figure 6(b) shows eight spectra picked along Fig. 6(a) in order to better appreciate these results, which qualitatively agree with the theoretical behaviour described in Fig. 3(d–g). The position of each pair of sidebands corresponds to a fulfilled parametric resonance condition (Eq. (4)). By noting that the frequency of the band grows with the value of the integer m (Eq. (5)), we can identify each peak up to the order m = 125 in the spectrum at the bottom.
A quantitative comparison between theory and experiments is displayed in Fig. 7 for the largest pump wavelength (λP = 1550.3 nm) which is the one producing the largest number of sidebands in the spectrum. The experimental spectrum (Fig. 7(a)) shows a very strong resemblance in its shape with the analytical gain spectrum (Fig. 7(b)). Notice the reduced visibility of the bands in the clusters around ±7 T Hz in experiments that are likely due to the limited spectral resolution of the OSA (4 GHz). Fig. 7(c) displays the position of each sideband as a function of its order m for both the experiment and the analytics. It highlights that the bands get closer to each other as m increases as expected from Eq. (5). As can be seen, a perfect agreement between theory and experiments is obtained for the positions of the 87 pairs of sidebands.
3.3. Competition between sidebands
As described before, parasitic wave mixing between the different parametric sidebands is expected to be increasingly significant as the power of the sidebands grow (and thus as the number of round-trips increases). This is why we limited the burst duration to 475 round-trips so far. To investigate the competition between the different sidebands we need to access the round-trip-to-round-trip dynamics of the spectrum which is not possible using an OSA due to its large recording time compared to the round-trip time of the cavity. We thus employ the DFT method described previously and in Ref.  to unveil the dynamical behaviour of our system. Indeed, this method allows us to record the spectrum of each pulse exiting the cavity. Then the dynamics of the spectrum is obtained by stacking the successive recordings. We use the same pumping method as before, though setting the duration of the bursts to ΔT1 = 1.5 ms (≈ 7000 pulses) to let the system reach a stationary state. We use a highly dispersive fiber, characterized by the following parameters: β2L = 870 ps2 and β3L = −5.5 ps3, to realise the frequency-to-time conversion. The spectrum is then retrieved from the time traces thanks to the mapping T (f) = 2π2 β3L(f − f0)2 + 2π β2L(f − f0), where f0 = c/λ0 is the optical frequency of the pump laser.
Figure 8(b) showcases the experimental evolution of the instability spectrum at the output of the cavity for 1500 consecutive round-trips after turning on the driving input field for the same pump wavelength as before (λP = 1550.3 nm). The plot of the spectrum is restricted to one side of the pump for the sake of clarity. The pump, which is supposed to appear at null frequency shift, has been filtered out before detection to help detecting weak parametric sidebands without saturating the detector.
It appears from this 2D color plot that the system goes through a transient regime, which lasts nearly 800 round-trips. First, multiple sidebands appear around the round-trip 150 and we can identify the low frequency components (range 0 − 4 T Hz) and the cluster of sidebands (range 6 – 8 T Hz). Note that the rather large noise level of the DFT setup prevent us from revealing the weakest sidebands which are expected to appear between 4 and 6 THz (see Fig. 7(a)). After the 500-th round-trip the cluster has completely vanished and some low frequency sidebands persist until the 800-th round-trip. After that, only 4 sidebands (corresponding to m = 1, 2, 5 and 6) “survive” and persist as long as the driving field is sustained (although we recorded this steady spectrum for more than 6000 round-trips, we reported a truncated version for the sake of clarity). Two snapshots at round-trips 1500 and 400 are plotted in Fig. 8(a) and (c), respectively.
We carried out numerical simulations of the LL model (Eq. (1)) with dispersion truncated up to the FOD term, and the results are plotted in Fig. 8(d–f) for comparison. The main features are well reproduced and a stable spectrum arises after a transitory regime of a few hundred round-trips. These results constitute a clear evidence of a competition scenario driven by the parametric gain.
A perfect quantitative agreement is not achieved in Fig. 8 because MI is a noise driven process where the random initial conditions play a critical role both in experiments and simulations [25, 26]. We illustrate this fact in Fig. 9 by plotting side-by-side the output spectrum of three different realizations of both experiments (Fig. 9(a)) and simulations (Fig. 9(b)) after 400 round-trips. Those realizations only differ from each other by the random initial conditions. It clearly appears that, at this stage, of the experiment/simulation, a large variety of results can be obtained. This is further illustrated in Visualization 1, which shows the roundtrip-to-roundtrip evolution of the output spectrum during the first 400 round-trips for 40 realizations of the experiment using the short burst pumping and the DFT technique. Each realization leads to a different result.
In conclusion, we demonstrated that, under low dispersion conditions, an unprecedented large number of quasi-phase-matched frequency sidebands can appear in dispersion oscillating passive fiber-ring cavities pumped in the low dispersion regime. For the relevant case of cavities with piecewise constant parameters, we developed a theoretical model based on an extended mean-field Lugiato-Lefever model that allows us to compute the parametric gain by taking into account higher order dispersion terms. In particular, we show that the fourth order dispersion β4 strongly affects the gain spectrum when the cavity is pumped close to a zero dispersion wavelength, and leads to high frequency clusters of sidebands. To confirm this experimentally, we built a fiber-ring cavity with a specially tailored longitudinal dispersion profile. By periodically turning on and off the driving field, we managed to record a remarkably large number of sideband pairs, up to 87, in a single spectrum, in perfect agreement with theoretical predictions. The dependence of the instability spectrum on the pump wavelength is also clearly observed. Finally, we proceed to the recording of the dynamical evolution of the instability spectrum disclosing the birth of the sidebands and their interactions until a stationary state is reached. These results, in agreement with numerical simulations of the mean-field model, provide an example of the dynamics of gain competition in a parametric process. Such an investigation of the real-time evolution of the system was made possible thanks to the very low free spectral range of macroscopic fiber cavities compared to microresonators. Also, this study is clearly of interest in the domain of microresonators where a similar azimuthal modulation of the cavity dispersion leads to Faraday instability which can trigger the formation of Kerr frequency combs .
Agence Nationale de la Recherche NoAWE (ANR-14- ACHN-0014) and TOPWAVE (ANR-13-JS04-0004) projects; LABEX CEMPI (ANR-11-LABX-0007); Equipex Flux (ANR-11-EQPX-0017); Ministry of Higher Education and Research, Nord-Pas de Calais Regional Council; European Regional Development Fund (ERDF) through the Contrat de Projets Etat-Region (CPER Photonics for Society P4S); IRCICA, USR 3380, CNRS-Univ, F-59000 Lille, France (http://www.ircica.univ-lille1.fr); PRIN (2012BFNWZ2).
We thank Laure Lago and Julie Beaucé for providing us with the fiber Bragg gratings.
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