1. Introduction

Quasi-continuous-wave background (qCWB), also referred to as residual continuous-wave background in some literature [1,2], is an intriguing feature that emerges from mode-locked fiber lasers (MLFLs). In the time domain, the noisy qCWB spreads over the entire laser cavity and co-circulates with mode-locked pulses, whereas in the spectral domain, the qCWB appears as a sharp spectral peak on top of the broad optical spectrum of the mode-locked pulses. Since first reported in earlier studies on mode-locked erbium-doped fiber lasers [3], the qCWB has been observed in various regimes of mode-locked soliton pulses such as harmonic mode-locking [4], bound solitons and soliton molecules [2,5,6], soliton bunches [7,8], vector solitons [9], and soliton rains [10]. Previous studies suggest that the qCWB may arise from the gain inhomogeneity [11], the spontaneous emission noise from the laser gain medium [12], and the random beats of longitudinal cavity modes [10], although the origin of the qCWB has yet to be completely clarified.

While the qCWB has long been regarded as undesirable noises in MLFLs [12–14], it has recently attracted considerable attention as it can reportedly mediate long-range interactions among the soliton pulses [7–10,15–17], which in some cases leads to the emergence of nontrivial multi-pulse dynamics in MLFLs [18,19]. Thus, it is crucial to investigate the qCWB for unraveling the intricate multi-pulse dynamics, although such a study is not straightforward due to the complex and random nature of the qCWB. In this context, some attempts have been made to inject external continuous-wave light that plays the role of qCWB into the MLFL cavities to manipulate the state of mode-locked pulses [18–21]. However, to fully understand the multi-pulse dynamics based on the interplay between the qCWB and the mode-locked pulses, it is also essential to consider the influence of pulses on the qCWB. For instance, the gain depletion by mode-locked pulses and the gain recovery can induce the temporal variation of the qCWB, which gives rise to the drift of mode-locked pulses on the qCWB [16,17].

Besides the gain dynamics, it has been recently proposed that the transverse acoustic resonances (ARs) that are optomechanically excited in the fiber laser cavity by mode-locked pulses [22–25] can also act as a crucial physical element that directly affects the qCWB [26]. When a mode-locked pulse propagates along the fiber laser cavity, it creates a train of acoustic impulses in the fiber [27,28]. The acoustic impulses modulate impulsively the phase or polarization of the qCWB, which eventually creates a set of regularly spaced sharp intensity modulations on the qCWB landscape. It has been recently reported that such optomechanical qCWB modulations can dramatically alter the behaviors of non-equilibrium multi-pulse dynamics such as pulse drift, pulse pattern formation, and soliton rain in unconventional ways [26]. It would then be important to have the capability to adjust the properties of the qCWB modulations, such as their temporal period and magnitude and the type of the AR modes involved, for further studies and tailoring of the multi-pulse dynamics.

In this paper, we present a detailed experimental investigation on the optomechanically induced qCWB modulations in an MLFL. We confirm that the impulsive qCWB modulations indeed originate from the optomechanically excited ARs by experimentally verifying that the temporal period of the qCWB modulations is proportional to the fiber cladding diameter. Moreover, we demonstrate that we can further engineer the characteristics of optomechanical qCWB modulations by changing the fiber structure, e.g., the core parameters such as the core diameter and numerical aperture (NA), the fiber coating, as well as the fiber cladding diameter. In particular, we reveal that the fiber coating and the fiber core parameters determine the AR-mode-dependent mechanical dissipation and optomechanical overlap, respectively, which can determine which AR modes are predominantly apparent in the qCWB modulations. We also separately measure the forward stimulated intra-polarization (Raman-like) and inter-polarization Brillouin scattering spectra of the intracavity fibers to elucidate our experimental observations.

2. Results

Figure 1(a) shows our MLFL schematically. The passive laser mode-locking is achieved via nonlinear polarization rotation implemented with a fiber polarizer sandwiched by two polarization controllers (PCs). The laser cavity consists of standard single-mode fiber (SMF, group velocity dispersion β₂ = –22 ps²/km at 1550 nm) and dispersion compensating fiber (DCF, β₂ = +128 ps²/km at 1550 nm), together with a 78-cm-long section of erbium-doped fiber (absorption coefficient: 110 dB/m at 1530 nm, β₂ = +22 ps²/km at 1550 nm) that is used as a gain medium and pumped by a 976 nm laser diode through a wavelength division multiplexer with 117-cm-long pigtail fiber (β₂ = +20 ps²/km at 1550 nm). We change the lengths of the SMF and DCF to adjust the net cavity dispersion β_C, which yields the cavity round-trip time τ_C in the range of 130–160 ns that corresponds to the total cavity length of 26–32 m. The laser is operated in the anomalous net cavity dispersion regime. The laser output is obtained at the 10% port of a 90/10 fiber coupler, which is observed using a 12-GHz-bandwidth photodetector, whereas the optical spectrum is measured with a grating-based optical spectrum analyzer. We observe qCWB in the laser output that coexists with a mode-locked pulse train, as shown in the typical single-shot oscilloscope trace of Fig. 1(b), at the pump power of few hundreds of mW or higher, which is far above the laser mode-locking threshold (40–70 mW). Specifically, we first generate a multi-pulse state at such a high pump power and then carefully adjust the intracavity PCs to make a qCWB grow. In the process of PC adjustment, the nonlinear cavity loss is changed so that the intracavity pulse energy just exceeds the soliton upper limit, which leads to the formation of the CW component [14]. In the optical spectrum (Fig. 1(c)), the qCWB appears as a narrow spectral peak at 1561.5 nm that is superimposed on the broad pulse spectrum, which should be distinguished from the Kelly sidebands arising from a periodic perturbation on the mode-locked pulses [29]. In our experiments, we restrict the total cavity length within the range of 26–32 m, over which we can readily observe the qCWB, although the qCWB can be reportedly generated over a broad cavity length range, in some cases at cavity lengths even below 10 m [5] or longer than 100 m [7].

 

Fig. 1. (a) Schematic diagram of the mode-locked fiber laser. LD, laser diode; WDM, wavelength division multiplexer; EDF, erbium-doped fiber; FUT, fiber under test; PC, polarization controller. (b) Single-shot oscilloscope trace of a typical laser output obtained at a pump power of 300 mW, which displays a mode-locked pulse train at the fundamental repetition rate and the quasi-continuous-wave background (qCWB). The cavity round-trip time is τ_C = 151 ns, and the net cavity dispersion is β_C = –4.3 ps²/km (intracavity DCF length: 3.0 m). (c) Optical spectrum of the laser output. The spectral peak at 1561.5 nm corresponds to the qCWB.

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Zooming in the qCWB part of the oscilloscope trace in Fig. 1(b) (purple dashed box), we observe the detailed structure of the intensity envelope of the qCWB. We find out that two groups of almost regularly spaced sharp intensity modulations are routinely produced on the qCWB, as shown in Figs. 2(a) and 2(b). After a mode-locked pulse, a dominant set of intensity modulations appear nearly at integer multiples of 33 ns (purple vertical bars), and a relatively weak one nearly at integer multiples of 21 ns (dark yellow bars). These two temporal quasi-periods are very close to those of the electrostrictive impulse response to a single optical pulse in a 125-µm-thick-cladding silica glass fiber. In this process, the optical pulse creates a train of acoustic impulses composed of the superposition of several AR modes, where the AR amplitude is proportional to the pulse energy [28]. The resulting temporal interval between the adjacent acoustic impulses is τ_R = 21 ns or τ_T = 33 ns, depending on which type of AR modes form the acoustic impulses. τ_T = 33 ns is contributed by the torsional-radial TR_2m AR modes in the torsional branch, whereas τ_R = 21 ns by the radial R_0m AR modes and the TR_2m AR modes in the radial branch [28]. In the following description, we denote the former type of ARs by ‘T-AR’ and the latter by ‘R-AR.’ We check that the array of sharp intensity fluctuations disappears when we suppress the qCWB by adjusting the intracavity PCs. This observation, together with the laser optical spectrum in Fig. 1(c), indicates that the intensity modulations superimposed on the qCWB are not a part of the mode-locked pulses. Furthermore, we confirm that the two temporal quasi-periods, τ_T and τ_R, are almost constant over broad ranges of the pump power, cavity length, and net cavity dispersion, which indicates that the quasi-periods are not relevant to the purely optical properties of the laser cavity.

 

Fig. 2. Quasi-continuous-wave background (qCWB) modulations induced by optomechanical excitation of acoustic resonances (ARs) in the fiber laser cavity. (a) Zoomed-in oscilloscope trace corresponding to the purple dashed box in Fig. 1(b). (b) Oscilloscope trace obtained with 1000 times averaging. (c,d) Single-shot oscilloscope trace (c) and 1000-times-averaged one (d) obtained with the intracavity DCF length of 1.4 m, which results in (τ_C, β_C) = (141 ns, –11.6 ps²/km). (e,f) Single-shot oscilloscope trace (e) and 1000-times-averaged one (f) obtained with the intracavity DCF length of 1.0 m, which yields (τ_C, β_C) = (140 ns, –13.7 ps²/km). In (b), (d), and (f), the dark yellow and purple vertical bars indicate the theoretically predicted temporal locations at integer multiples of τ_R = 21 ns and τ_T = 33 ns, respectively. Note that the two temporal locations of 8τ_R and 5τ_T are almost the same, so they are hardly resolved. The solid bars correspond to the ARs driven by the mode-locked pulse at t = 0, whereas the dashed ones to the ARs by the mode-locked pulse at the previous round-trip (t = –τ_C). The pump power is fixed at 300 mW for all traces.

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The T-AR impulses induce local polarization modulations of qCWB with a temporal spacing of τ_T = 33 ns, which can be converted into intensity modulations (Figs. 2(a) and 2(b)) at the intracavity polarizer in the nonlinear-polarization-rotation (NPR)-based MLFL [26]. On the other hand, the R-AR impulses yield local phase modulations that cannot be directly converted into intensity modulations, whereas they have less influence on the polarization state of qCWB. In our previous work [26], another MLFL configuration employing a polarization-independent real saturable absorber (e.g., semiconductor saturable absorber mirror) as a laser mode-locker instead of the NPR was also observed to exhibit qCWB modulations spaced by τ_R = 21 ns similar to Figs. 2(c) and 2(d). We attribute such qCWB modulations to the transformation of the R-AR-induced local frequency chirps of qCWB into the deformation of the qCWB envelope through cavity dispersion [30], although a further theoretical investigation is necessary for a complete understanding of its mechanism.

Furthermore, we find out that we can make either set of qCWB modulations spaced by τ_R or τ_T predominant over the other by changing the intracavity fiber properties. First, we experimentally check that with sufficient intracavity DCF lengths around 3 m, strong qCWB intensity peaks separated by τ_T are routinely observed, whereas the qCWB modulations spaced by τ_R are very weak (Fig. 2(b)). On the other hand, as the intracavity DCF length is shortened, the dominant qCWB modulations at integer multiples of τ_T diminish, as shown in Figs. 2(c)–2(f). Even when the qCWB fluctuates so severely that regularly spaced qCWB modulations are hardly identified in single-shot oscilloscope traces (Figs. 2(c) and 2(e)), the averaged traces reveal the optomechanical qCWB modulations (Figs. 2(d) and 2(f)). In some cases, the qCWB modulation often appears dominantly as local intensity dips separated by τ_R, as shown in Fig. 2(d). Such dependence of optomechanical qCWB modulation on the intracavity DCF length indicates that the DCF length inside the MLFL critically affects the qCWB modulation properties, which we will discuss later in more detail.

Next, we examine the influence of fiber coating on the optomechanical qCWB modulation. As mentioned earlier, the T-AR-induced qCWB modulation is predominant when the intracavity DCF is sufficiently long. On the other hand, we find out that this tendency is strongly altered when the fiber coating of a strand of intracavity fiber is removed, the R-AR-induced qCWB modulation spaced by τ_R being significantly enhanced. We first set the intracavity DCF length as 3 m. In the absence of any uncoated fiber segment inside the laser cavity, the T-AR-induced qCWB peaks are seen dominantly (purple bars in Fig. 3(a)), similar to Fig. 2(a). We then remove the coating of the entire 3-m-long DCF without cutting and splicing of any intracavity fiber segment and suspend the DCF in the air using a few supports to minimize its contact with the optical table. We observe that a set of distinct intensity dips appear on the qCWB at integer multiples of τ_R (dark yellow bars in Fig. 3(b)), which is not observed when the DCF coating is not stripped, whereas the intensity peaks separated by τ_T are relatively weak. The qCWB intensity dips separated by τ_R also emerge when a much longer strand of standard SMF (20 m in the case of Fig. 3(c)) is coating-peeled and suspended in the air, instead of the 3-m-long DCF section. Removing the coating of both the 3-m-long DCF and the 20-m-long standard SMF yields more drastic R-AR-induced qCWB modulation, as shown in Fig. 3(d), where the intensity dips also become significant even at integer multiples of τ_R from the mode-locked pulse in the previous round-trip (dark yellow dashed bars in Fig. 3(d)). We note that in all cases such stripping of fiber coating does not cause any observable influence on the characteristics of mode-locked pulses.

 

Fig. 3. Optomechanical qCWB modulations in the case when the coating of a part of the intracavity fiber is removed. (a) No fiber coating is removed. The length of the intracavity DCF is 3 m. (b) The coating of the entire 3-m-long DCF is removed. (c) The coating of a 20-m-long SMF strand is removed, while that of the 3-m-long DCF is kept unstripped. (d) Both the 3-m-long DCF and the 20-m-long SMF strand are coating-removed. All the traces are obtained with 1000-times averaging. In (b)–(d), all coating-removed fiber segments are suspended in the air. In each plot, the dark yellow and purple vertical bars indicate the theoretically predicted temporal locations at integer multiples of τ_R = 21 ns and τ_T = 33 ns, respectively. Note that the two temporal locations of 8τ_R and 5τ_T are almost the same, so they are hardly resolved. The solid bars correspond to the ARs driven by the mode-locked pulse at t = 0, whereas the dashed ones to the ARs by the mode-locked pulse at the previous round-trip at t = –τ_C.

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These observations indicate that the predominance of the T-AR-induced qCWB modulation in Figs. 2(a) and 3(a) is assisted by the fiber coating, particularly the DCF coating. We explain this phenomenon in terms of the mode-selective acoustic dissipation in the fiber coating. The transverse AR modes generally have radial and/or azimuthal displacement components. The fiber coating surrounding the fiber cladding dissipates the radial displacement more strongly than the azimuthal one [31], which implies that the AR modes with larger radial displacements nearby the cladding-coating interface are damped more rapidly by the fiber coating. Since the R_0m AR modes have relatively large radial displacements compared to the TR_2m AR ones, the R_0m AR modes are dissipated more rapidly by the coating than the TR_2m AR ones. Furthermore, among the TR_2m AR modes, those belonging to the radial branch exhibit stronger radial displacements than those to the torsional branch [28]. Hence, removing a sufficiently long strand of intracavity fiber enhances the R-AR-induced qCWB modulation with the τ_R = 21 ns spacing, whereas the T-AR-induced one with the τ_T = 33 ns spacing can be predominant in the absence of the uncoated fiber strand. In addition, DCF has a smaller core than standard SMF, which leads to the excitation of a larger number of ARs in the DCF up to a few GHz, compared to the standard SMF. Therefore, the R-AR-induced qCWB modulation is more sensitive to removing the DCF coating than the standard SMF one.

To verify further our argument about the effect of fiber coating and fiber core parameters on the optomechanical qCWB modulation, we directly measure the forward stimulated Brillouin spectra of both the original and coating-peeled strands of the standard SMF and DCF we use in experiments. To this end, we employ the pump-probe measurement scheme [32], where the ARs are optically excited in each fiber sample by a sinusoidally intensity-modulated pump laser beam, and the resulting nonlinear phase shift or polarization rotation is detected with an actively stabilized Mach-Zehnder interferometer or a polarization analyzer, respectively. These measurements are performed over a range of pump modulation frequency, which yields the spectra of stimulated intra-polarization (Raman-like) and inter-polarization scattering, as summarized in Figs. 4 and 5. (Also see Figs. S1 and S2 in Supplemental Document.) Each scattering spectrum displays multiple Fano resonances that arise from the combination of the optomechanical and Kerr nonlinearities [32], where each Fano resonance corresponds to the respective AR mode. By fitting the scattering spectrum to the Fano resonance lineshape [32], we determine the resonant frequency, linewidth, mechanical quality factor, and Brillouin gain for each AR mode. We note that the AR linewidths and mechanical quality factors are governed by acoustic dissipation, whereas they are not affected by the optical power level in the fiber.

 

Fig. 4. Measured characteristics of forward stimulated Brillouin scattering in standard SMF (core diameter: 8.7 µm, NA: 0.12 at 1550 nm wavelength). (a)–(f) characterize the intra-polarization (Raman-like) scattering by the R_0m ARs, whereas (g)–(l) the inter-polarization scattering by the TR_2m ARs. (a,g) Scattering spectra for a 10-m-long strand of original (coated) SMF. (b,h) Scattering spectra for 10-m-long coating-removed SMF. By fitting each of the four scattering spectra to the Fano resonance lineshapes, the linewidths (c,i), AR quality factors Q (d,j), Brillouin gains G (e,k), and optomechanical overlaps G/Q (f,l) are determined for individual AR modes. In (c)–(f) and (i)–(l), the solid and open symbols correspond to the original and stripped fibers, respectively. In (i)–(l), the red circles and blue triangles stand for the radial and torsional branches, respectively, of the TR_2m ARs, while the violet squares indicate the points where two neighboring AR modes belonging to different branches are overlapped in the scattering spectra and thus cannot be resolved. (Also see Fig. S1 in Supplemental Document.) In (f), the dark red crosses (+) represent the theoretically predicted values. In (l), the dark red crosses (+) and dark blue ones (×) stand for the theoretically predicted values for the radial and torsional branches, respectively, of the TR_2m ARs.

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Fig. 5. Measured characteristics of forward stimulated Brillouin scattering in DCF (core diameter: 2.24 µm, NA: 0.33 at 1550 nm wavelength). (a)–(f) characterize the intra-polarization (Raman-like) scattering by the R_0m ARs, whereas (g)–(l) the inter-polarization scattering by the TR_2m ARs. (a,g) Scattering spectra for a 10-m-long strand of original (coated) DCF. (b,h) Scattering spectra for 10-m-long stripped DCF. Each symbol in (c)–(f) and (i)–(l) represents the same as that in Fig. 4. (Also see Fig. S2 in Supplemental Document.)

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The intra-polarization scattering spectra are contributed primarily by the R_0m AR modes, together with the weakly excited TR_2m AR modes. For original (coated) standard SMF, the AR linewidths are typically 4–5 MHz regardless of the acoustic frequency (Fig. 4(c)), where the fiber coating governs the acoustic dissipation. On the other hand, when the fiber coating is removed, the AR linewidths become as narrow as a few tens of kHz (Fig. 4(c)), corresponding to enhanced quality factors above 3000 (Fig. 4(d)), which is contributed primarily by the inherent viscosity of fused silica that tends to yield larger AR linewidths at higher acoustic frequencies [33]. Such enhancement of the AR quality factor through the removal of fiber coating also increases the Brillouin gain by one order of magnitude compared to that of the original coated fiber (Fig. 4(e)).

While the inter-polarization scattering spectra exhibit more complicated features, they provide more relevant information to our experimental observations of qCWB modulations, as they reveal the full characteristics of the TR_2m AR modes. The TR_2m ARs induce the local polarization modulations on the qCWB through optomechanical polarization coupling [26,33], which can be transformed into the sharp qCWB intensity modulations at the intracavity polarizer in the NPR-based MLFL. Figures 4(i)–4(k) show the AR dissipation and resulting decrease of the Brillouin gain in the presence of fiber coating similar to the case of the intra-polarization scattering in Figs. 4(c)–4(e). Here, we identify the two branches of the TR_2m AR modes, the torsional branch (blue triangles) and the radial one (red circles). For the ARs in the torsional branch, the torsional displacement is larger than the radial one, and the acoustic frequencies are spaced by ∼30 MHz in the 125-µm-thick fiber cladding, which yields the 33-ns-separated train of acoustic impulses. In contrast, for the ARs belonging to the radial branch, the radial displacement is dominant, and their acoustic frequency spacing of ∼48 MHz contributes to the acoustic impulses separated by 21 ns like the R_0m ARs. Figures 4(i) and 4(j) clearly show the ‘mode-selective’ dissipation of the TR_2m ARs by the fiber coating. When the fiber coating is removed, the torsional and radial branches exhibit similar linewidths and quality factors to each other. On the other hand, in the presence of the fiber coating, the linewidths of the ARs in the radial branch are a few MHz broader compared to those of the torsional branch (Fig. 4(i)), which leads to higher quality factors for the torsional branch than the radial branch (Fig. 4(j)). Furthermore, the difference in the resulting Brillouin gain between the two branches also becomes more apparent by the existence of the fiber coating, as shown in Fig. 4(k), the torsional branch having higher Brillouin gains.

We also theoretically estimate the optomechanical overlap, G/Q, the ratio between the Brillouin gain G and the quality factor Q for each AR mode and compare the resulting values with the experimentally determined ones, as shown in Figs. 4(f) and 4(l), to validate our measurements. G/Q is obtained from the full-vectorial calculation of the overlap integral between the AR displacement profile and the optical field distribution [34]. Since G/Q is independent of the AR linewidth [34], it is anticipated that the fiber coating does not significantly influence G/Q, unless the coating alters the AR displacement profile considerably. G/Q reaches a couple of 10⁻² W⁻¹km⁻¹ maximally at the acoustic frequencies of a few hundreds of MHz for both the intra-polarization and inter-polarization scattering. These behaviors and the modal dependence of G/Q show an excellent agreement with the theoretical predictions, verifying the reliability of our measurement and analysis, although there is a slight discrepancy in Fig. 4(f) due to the limited stability of our current interferometric measurement of very sharp spectral profiles, particularly for stripped SMF with very narrow AR linewidths. Furthermore, the AR frequency spacings in each family of AR modes (48 MHz or 30 MHz) are almost the same between the original and stripped fibers, their differences being only a few tens of kHz or smaller (below 0.1%). This result indicates that the ARs are confined almost entirely in the silica glass region, even for the case of coated fibers, so the fiber coating does not have a significant influence on the AR displacement profile. Accordingly, the two quasi-periods, τ_R and τ_T, are almost unaltered by the presence of fiber coating, as can be seen in Fig. 3.

All the tendencies described so far are also observed for DCF, as summarized in Fig. 5. The effect of fiber coating again appears as the AR linewidth broadening to the order of MHz (Figs. 5(c) and 5(i)) and the reduction of AR quality factors (Figs. 5(d) and 5(j)) and Brillouin gains (Figs. 5(e) and 5(k)). The acoustic dissipation in the absence of the fiber coating is determined primarily by the intrinsic viscosity of silica glass. The AR linewidths are larger at higher acoustic frequencies (Figs. 5(c) and 5(i)), and the AR quality factors are maintained on the order of 10⁻² over the acoustic frequency range (Figs. 5(d) and 5(j)). As a result, the influence of fiber coating on the AR linewidths and quality factors diminishes at high acoustic frequencies. We also observe the mode-selective dissipation of the TR_2m ARs by the fiber coating in the inter-polarization scattering (Figs. 5(i) and 5(j)), although this effect in DCF is not as drastic as that in standard SMF. For coated DCF, the AR linewidths are broader for the radial branch than the torsional one (Fig. 5(i)), leading to larger AR qualify factors and Brillouin gains for the torsional branch (Figs. 5(j) and 5(k)). The difference in the Brillouin gain between the two branches is enhanced by the fiber coating (Fig. 5(k)). Therefore, the insertion of coated DCF into the laser cavity makes the qCWB modulations with a period of τ_T = 33 ns predominant (Figs. 2(a) and 3(a)), whereas stripping its coating strongly enhances the qCWB modulations with a period of τ_R = 21 ns (Fig. 3(b)).

Apart from the fiber coating effects, it should be noticed that due to the small optical mode field diameter in DCF (core diameter: 2.24 µm, NA: 0.33 at 1550 nm wavelength), compared to that of the standard SMF (core diameter: 8.7 µm, NA: 0.12 at 1550 nm wavelength), the scattering spectra are extended to higher acoustic frequencies up to ∼2.5 GHz (Figs. 5(a), 5(b), 5(g), and 5(h)), in contrast to the case of standard SMF well below 1 GHz. The Brillouin gains also peak at higher acoustic frequencies compared to the standard SMF case (Figs. 5(e) and 5(k)). Such a broad acoustic spectral range that can be efficiently excited in the DCF accounts for our experimental observation that a couple of meters of DCF can affect the qCWB modulations significantly, whereas ∼20-m-long standard SMF needs to be stripped to obtain a similar degree of enhancement of qCWB modulations, as shown in Fig. 3. The optomechanical overlaps G/Q again exhibit similar values between coated and stripped DCF, indicating that the DCF coating does not significantly alter the AR displacement profile. Compared to the case of standard SMF, G/Q of DCF approaches higher values of 0.10 and 0.06 W⁻¹km⁻¹ for the intra-polarization and inter-polarization scattering, respectively. These experimental results on G/Q agree with theoretical predictions.

To further verify that the qCWB modulation that we observe indeed originates from the optically excited ARs, we examine the dependence of the temporal locations of the qCWB modulations on the fiber cladding diameter. Since the AR frequencies are inversely proportional to the cladding diameter [33], the optomechanical qCWB modulation periods, τ_R and τ_T, are linearly proportional to the cladding diameter d_clad as following.

 

Fig. 6. Optomechanical qCWB modulations in the case when a 20-m-long segment of air-suspended uncoated fiber with the cladding diameter less than 125 µm replaces the standard SMF of the same length in the laser cavity. (a,b) Optical micrographs of wet-etched standard SMF into the cladding diameters of 110 µm (a) and 103 µm (b). The white horizontal scale bars correspond to 100 µm. (c) Coating-removed commercial SMF with an 80-µm-thick cladding is used. (d–f) Standard SMF strands wet-etched into the cladding diameters of 110 µm (d) and 103 µm (e,f) are used. All the traces are obtained with 1000-times averaging. In each plot of (c)–(f), the upper row of vertical bars corresponds to the 125-µm-diameter cladding, and the lower row to the reduced cladding diameter. Dark yellow and purple indicate the theoretically predicted temporal locations at integer multiples of τ_R and τ_T, respectively. Note that the two temporal locations of 8τ_R and 5τ_T are almost the same regardless of the cladding diameter, so they are hardly distinguishable in any case.

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Figure 6(c) shows the qCWB modulation when an 80-µm-thick fiber strand is inserted into the laser cavity. Two major groups of temporal positions of impulsive qCWB modulations are observed. One group consists of a family of dip-like qCWB modulations with a temporal period of ∼14 ns (solid and dashed dark yellow bars), which coincides with the theoretically predicted value of τ_R = 13.4 ns from Eq. (1) with d_clad = 80 µm. The other group also forms dip-like qCWB modulations but with a different temporal period of ∼21 ns. One might think that the ∼21 ns period corresponds to the R-ARs in the pre-existing standard 125-µm-thick fiber that occupies about one-third (∼10 m) of the total cavity length (dotted dark yellow bars). However, Eq. (2) suggests that the cladding diameter of d_clad = 80 µm can also yield a set of T-AR-induced qCWB modulations at the integer multiples of 21.4 ns (solid and dashed purple bars). This indicates that it is accidentally ambiguous to distinguish between the T-AR-induced qCWB modulations from the 80-µm-thick cladding and the R-AR-induced ones from the 125-µm-thick cladding.

We repeat the same series of measurements with different cladding diameters to avoid the ambiguity. We prepare two 20-m-long standard SMF strands whose claddings are wet-etched down to respective thicknesses of 110 and 103 µm, using a 6:1 buffered oxide etch solution. When each etched SMF strand is inserted in the laser cavity, a new set of temporal locations of qCWB modulations is identified at the integer multiples of ∼19 ns (Fig. 6(d)) and ∼17 ns (Fig. 6(e)) for the cladding diameters of 110 and 103 µm, respectively. These periods agree with the theoretically predicted values of τ_R = 18.4 ns and τ_R = 17.2 ns, respectively, for the R-AR-induced qCWB modulations from Eq. (1). The qCWB modulations at the integer multiples of 21 ns are also seen again due to the pre-existing 125-µm-thick fiber. We also observe T-AR-induced qCWB modulations, as can be seen, for instance, in Fig. 6(f) for the case of d_clad = 103 µm (solid and dashed purple bars), where the observed temporal period of ∼27 ns matches the theoretically predicted value of τ_T = 27.5 ns from Eq. (2). However, they are relatively weak because removing the fiber coating enhances the R-AR-induced qCWB modulations primarily, as we explained previously.

3. Discussion and conclusion

In conclusion, the qCWB spreading over the entire MLFL cavity can be manipulated in the temporal domain using optomechanically excited ARs in the fiber. In this process, a train of acoustic impulses is produced from a family of simultaneously excited R_0m and TR_2m ARs by the mode-locked pulses. As a result, a set of impulsive intensity modulations are created on the qCWB landscape at well-defined temporal locations relative to the mode-locked pulses. The characteristics of the optomechanical qCWB modulations are governed by the properties of both the AR modes and optical modes, and such types of qCWB modulations appear over a wide range of laser cavity parameters (e.g., pump power, cavity length, and net cavity dispersion). Thus, the optomechanical qCWB modulations can be engineered over a broad range by changing the structure of the core and cladding that determine the optical and AR modes, respectively, and the fiber coating that acts as an AR-mode-selective acoustic attenuator.

We note that the co-existence of optical pulses and continuous-wave (CW) background is not a peculiar phenomenon limited to MLFLs only. An externally driven nonlinear fiber cavity that supports temporal cavity solitons always exhibits the CW background [35]. However, while it was reported that the ARs can be excited by the cavity solitons and mediate the long-range pulse-to-pulse interactions [36], the AR-mediated CW background modulation that we observe has not been reported so far in such passive fiber cavities. This may indicate that the qCWB modulations observed in our experiment could probably be related to some features of MLFLs, which should be clarified in a future investigation.

While the AR-induced phase modulation has been regarded as a primary mechanism of long-range pulse-to-pulse interactions [22–25], our study suggests another possibility that the optomechanical qCWB modulations facilitate the manipulation of the mode-locked pulse dynamics. In this case, the local intensity modulations in the qCWB envelope yield the phase modulations of the pulses via Kerr-induced cross-phase modulation, which significantly affects the generation and drift motion of multiple solitons in non-equilibrium multi-pulse dynamics (e.g., soliton drift and soliton rain) [26]. Our experimental demonstration of the engineering of optomechanical qCWB modulations through the adjustment of fiber structure thus indicates that the behaviors of multi-pulse dynamics in an MLFL can be further tailored via changing the fiber properties such as the fiber cladding diameter and the optical mode field diameter and controlling the AR dissipation with the aid of suitably designed fiber coatings.

Furthermore, mode-locked pulses exhibiting exotic dynamics may give rise to intriguing behaviors of AR impulses. For instance, it might be interesting to address a situation where breathing solitons are generated in an MLFL [37]. The typical breathing frequency of a breather soliton in an MLFL is on the order of 0.1 MHz [38,39], being much lower than the typical AR dissipation rate in silica glass fibers (order of MHz, according to our measurements in Figs. 4(c) and 4(i)). In this case, the amplitude of AR impulses excited by a breathing soliton would be simply modulated at the soliton’s breathing frequency. On the other hand, when the AR dissipation rate is comparable to or lower than the breathing frequency, e.g., by stripping the fiber coating (AR dissipation rate being on the order of 0.1 MHz or smaller, according to our measurements in Figs. 4(c) and 4(i)) or cooling fibers down to a very low temperature, the AR impulses would exhibit more complicated dynamics.

In these contexts, a more rigorous theoretical analysis of the optomechanical qCWB modulations and the resultant multi-pulse dynamics is an interesting future research direction. This study may require the numerical modeling of MLFLs, taking into account the generation of the noisy qCWB and its nonlinear interaction with the mode-locked pulses through the electrostrictively excited ARs, while such numerical studies have not been reported so far to the best of our knowledge. Since the time scales of the qCWB and acoustic impulses (>100 ns) are typically much longer than the nominal temporal width of mode-locked pulses (< 10 ps), we expect that precise numerical modeling would be challenging in view of the calculation time. Nevertheless, we believe that this study provides a useful means to better understand the underlying nature of the qCWB and access richer multi-pulse dynamics that emerge from MLFLs.