1. Introduction

From the observation of the phenomenon of supercontinumm (SC) generation in optical fibers, questions related to the nonlinear propagation of one-dimensional incoherent light waves have attracted a great deal of interest [1–3]. Those questions are not restricted to the only process of SC generation : they have also been examined for partially coherent cw fiber lasers such as Raman fiber lasers (RFLs) [4–7]. RFLs are now seen as light sources having an optical power spectrum that is determined by turbulentlike weak interactions among their multiple (typically ∼ 10⁵–10⁸) cavity modes [4, 5, 7].

Although RFLs are cavity-based sources that differ conceptually from single-pass sources emitting SC, analogous features have been experimentally observed in both systems. In particular an extreme-type statistics has been observed both in SC generation and in RFLs [8,9]. From the theoretical point of view, wave turbulence theory has been used to describe the phenomenon of spectral broadening occurring both in SC generation and in RFLs [4, 10].

Apart from experimental and theoretical approaches, numerical modelling of spectral, temporal and statistical properties of SC or RFLs is also currently a very active field of research [11]. In the specific case of RFLs, Turitsyna et al have used numerical simulations to show that the sign of the second-order dispersion parameter β₂ drastically changes the spectral shape together with the statistics of the laser radiation [5]. In addition, numerical simulations made in ref. [5] and [7] predict a phenomenom of spectral condensate for RFLs operating in the normal dispersion regime (β₂ > 0). The spectral condensate consists of a set of a few modes persisting over a time depending on the value of β₂ and on the total number of modes used in the numerical calculation. It persists typically over hundreds of round trips after the laser turn-on and the RFL power is found to be quite constant during the condensate lifetime. The condensate destruction is manifested by a sharp transition to a lower mean power and to a wider optical spectrum corresponding to strong Stokes intensity fluctuations [5, 7].

Up to now, the spectral condensate has not been observed in experiments and it remains a pure numerical prediction. Beyond an experimental tracking of this phenomenon, questions related to the transient buildup of the optical power spectrum in RFLs have not been investigated from experiments. To date only questions related to the shape of the mean optical power spectrum at steady state have been considered together with issues concerning the influence of some laser parameters (see e.g. [4,12]). However note that the switching dynamics of cascaded RFLs has been studied both from experiments and numerical simulations in ref. [13] and [14]. Let us emphasize that these works [13, 14] only provide information on the transient evolution of the Stokes power but no information on the transient buildup of the optical power spectrum.

In this paper, we demonstrate an experimental technique enabling to record the mean optical power spectrum of a RFL during the transient regime in which the intracavity Stokes power builds up consecutively to a sudden switch-on of the pump power. The proposed setup permits to question the way through which the Stokes optical power spectrum broadens before reaching its shape at steady-state. In Sec. 2, we detail the design of our experimental setup. In Sec. 3, we explore the broadening of the RFL optical power spectrum in the transient regime associated with the buildup of Stokes emission from noise to steady state.

2. Experimental setup

Our experimental setup schematically shown in Fig. 1 is basically similar to the one already used in ref. [6,9,12]. The RFL is pumped by a linearly-polarized Yb-doped fiber laser operating at λ_p = 1100 nm. It is made with a 500-m long polarization-maintaining fiber (PMF) having a measured Raman gain of 12.3 dB/km/W at the Stokes wavelength λ_s ≃ 1159 nm. The pump wave undergoes a single pass inside the fiber. The fiber losses are of 0.9 dB/km and 0.8 dB/km at λ_p and λ_s, respectively. At λ_s, the second-order dispersion coefficient of the fiber is β₂(λ_s)= +15.7 ps²/km. The laser cavity is made with two uniform (unchirped) fiber Bragg gratings with a bandwidth of ∼ 0.5 nm and peak reflectivities of R₁ ≃ 99% and R₂ ≃ 80%. The pump power available after the first cavity mirror FBG1 can be measured from a fiber coupler (FC) labeled FC1. In this work, we consider the spectrum of the Stokes light incident on the output cavity mirror FBG2. It can be recorded by using another FC (FC2) and a wavelength-dense multiplexer (WDM) separating Stokes light from pump light. The single-pass insertion losses of FC1 (resp. FC2) are 14% (resp.10%) at λ_s. The laser power threshold P_Th measured at FC1 is close to 350 mW. The round trip time τ_R of light inside the laser cavity is close to 5μs.

 

Fig. 1 Experimental setup. HWP: Half-wave plate. PC: Polarization controller. AOM: acousto-optic modulator. OSA: Optical spectrum analyzer. Depending on the experiments, the power modulator can be either a chopper wheel or an AOM.

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The experimental investigation of the transient buildup of the Stokes power spectrum requires to abruptly switch on the pump power and to subsequently record the Stokes optical power spectrum at some precise steps in the transient regime associated with the buildup of Stokes emission from noise to steady state. The duration of this transitory regime is much shorter than the time required to record the optical power spectrum by using an optical spectrum analyzer (OSA). Therefore, in our setup, the pump power is periodically switched on and off by using a power modulator. In our experiments, we have used either an acousto-optic modulator (AOM) or a chopper wheel to modulate the pump power. The AOM presents the advantage of a short pump-power rise time (∼ 10 ns) but the disadvantages of relatively high insertion losses (∼ 4 dB) and low damage threshold (∼ 5 Watt) that do not permit to push the incident pump power P_in above ∼ 3P_Th. With the chopper wheel, the incident pump power P_in can be increased up to ∼ 11P_Th but the minimum pump power rise time is around ∼ 2 μs. The power modulator (AOM or chopper wheel) is driven by a waveform generator that delivers a square waveform with a period T typically around 1 ms (i.e. T ≫ τ_R)

To measure the Stokes optical power spectrum at some precise steps in the RFL transitory response, we need an optical gate that can slice short time windows in the intracavity Stokes signal growing from noise to steady-state. The key element to achieve this time slicing is a second fiber AOM that is placed after FC2 and WDM2, as shown in Fig. 1. This fiber AOM is periodically driven by a short electrical pulse that has a duration τ_p much shorter than τ_R (τ_p is either taken to 1 μs or to 0.125 μs). The pulsed signal driving the fiber AOM has a period T and it is synchronously generated by the waveform generator already driving the power modulator. Since the time delay τ_d between the leading edge of the square signal and the pulsed signal can be continuously changed, we can slice short time windows at some arbitrary positions in the transient buildup of the Stokes power (see Fig. 2 and 3).

 

Fig. 2 (a), (b) (d) Dynamics of the input pump power (black line), of the intracavity Stokes power (red line) and of the Stokes signal (blue line) launched in the OSA. (b) is a magnified view of (a) for P_in ∼ 3P_Th with the AOM as power modulator. (d) is recorded for P_in ∼ 11P_Th with the chopper wheel as power modulator. (c) Comparison between the intracavity Stokes power spectra recorded at steady state with (red line) and without (green line) switching the input pump power (inset: laser spectra in vertical logarithmic scale).

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Fig. 3 (a), (c) Transient buildup of the intracavity Stokes power and (b), (d) corresponding buildup of the Stokes optical power spectrum when the incident pump power is abruptly switched on between zero and P_in. In (a), (b) P_in ∼ 3P_Th. In (c), (d), P_in ∼ 11P_Th. The normalized spectra (1), (2), (3) in the right-column are measured during the time windows (1), (2), (3) shown in the left-column by using the experimental technique described in the text. The normalized spectra plotted in red in (b) and (d) are the laser spectra at steady-state. The insets in (b), (d) are the laser spectra plotted in vertical logarithmic scale.

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The pulsed periodic Stokes signal found at the output of the fiber AOM is finally launched in an OSA (Advantest Q8384), as shown in Fig. 1. The Stokes optical power spectrum at some particular point inside the transient buildup is acquired in a typical time of ∼ 30 s. This relatively long acquisition time is necessary to record a mean spectrum from a signal with a weak duty cycle τ_p/T of only ∼ 10⁻³. Note that we have checked that the power of the RFL does not significantly drift over the acquisition time of the OSA.

3. Transient buildup of the intracavity Stokes optical power spectrum

Figure 2(a) and 2(b) show the response of the RFL to periodic and abrupt switches of the pump power. Figure 2(a) and 2(b) have been recorded with the AOM as power modulator. The incident pump power P_in reaches a maximum and constant value of ∼ 1 Watt (i.e. P_in ∼ 3P_Th) in a short duration of 10 ns. In these conditions, the RFL switches on after a delay of ∼ 60 μs. As illustrated in Fig. 2(d), this delay is reduced to ∼ 12 μs when the incident pump power is increased to ∼ 4 Watt (i.e. P_in ∼ 11P_Th). For a relatively low pump power (P_in ∼ 3P_Th), the intracavity Stokes power builds up in several discrete steps, each of them having a duration τ_R of one cavity round trip (∼ 5μs) (see Fig. 2(b)). On the other hand, the duration of the RFL switching transient is reduced to ∼ 5μs at higher pump power (P_in ∼ 11P_Th) (see Fig. 2(d)). Because of the use of the chopper wheel at high pump power, the rise time for the pump power (∼ 2μs) in Fig. 2(d) is close to the duration of the laser switching transient, thus possibly smoothing discrete steps such as the ones reported in Fig. 2(b).

Preliminary experiments have been performed in order to evaluate the performances of the time-resolved measurement of the Stokes optical power spectrum described in Sec. 2. To this end, Stokes optical spectra at steady-state have been compared with and without activating the periodic modulation of the input pump power. When the pump power is not modulated, we simply record the mean Stokes optical power spectrum at steady state for a given intracavity Stokes power P_s. In these simple conditions, the time-resolved measurement of the spectrum is obviously not activated and the corresponding spectrum is plotted in green lines in Fig. 2(c). When the pump power is periodically switched on and off, we use the optical gating technique described in Sec. 2 and we record the Stokes optical power spectrum at steady state, well after the transient evolution of the Stokes power (see Fig. 2(a), 2(b) showing that a delay τ_d ∼ 190μs is used to record spectra plotted in red lines in Fig. 2(c)).

Figure 2(c) shows that there is only a small difference between the spectra measured with the two methods at the same intracavity Stokes power P_s. Secondary experiments have shown that this small difference arises from thermal expansion of the FBGs, a phenomenon that has already been pointed out in ref. [15]. Switching on and off the incident pump power divide in half the average pump power seen by the FBGs, thus slightly reducing the thermal expansion of the FBGs and consequently the weak thermal shift (∼ 0.02 nm) of the Stokes central wavelength.

Considering that we can now confidently assume that our time-resolved spectral measurement technique is valid, we have measured the intracavity Stokes spectra at some discrete positions in the transient buildup of the intracavity Stokes power, as illustrated in Fig. 3(a), 3(c). As shown in Fig. 3(b) and 3(d), experiments reveal a monotonic broadening of the intracavity Stokes optical power spectrum. Note that the spectral condensate predicted in ref. [5, 7] has not been observed with the parameters of our RFL. To further investigate the buildup of the Stokes optical power spectrum, we have studied the evolution of its full width at half maximum (FWHM) as a function of the intracavity Stokes power. The black squares and green triangles plotted in Fig. 4 are obtained in the transient regime from the experiments above described. On the other hand, the blue cross plotted in Fig. 4 represent the FWHM of the intracavity Stokes spectrum at steady state. Figure 4 shows that the RFL spectrum at steady state broadens with the intracavity Stokes power according to a square-root law already evidenced in ref. [4]. Regarding the transient evolution of the RFL spectrum, our experiments strikingly show that it follows a square-root broadening close to the one observed at steady state.

 

Fig. 4 Square-root broadening of the intracavity Stokes optical power spectrum. The blue cross represent experimental values of the FWHM of Stokes spectrum at steady state and the red line is a square-root fit of these data. The green triangles and the black squares are measurements of the FWHM made in the transient build up of Stokes emission. The green triangles have been measured with experimental conditions close to those of Fig. 3(a), 3(b) (P_in ∼ 3P_Th) and for three values of τ_d (τ_d = 65, 70, 75μs). The black squares have been measured in three different runs with experimental conditions close to those of Fig. 3(c), 3(d) (P_in ∼ 11P_Th) and for three values of τ_d (τ_d ∼ 13.1, 15.5, 16.2μs).

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Let us emphasize that Fig. 3 and Fig. 4 evidence features of qualitatively different natures: Fig. 3 shows that transient broadening of the spectrum to its steady-state is monotonic whereas Fig. 4 shows the existence of a square-root law connecting the FWHM of the Stokes spectrum and the intracavity Stokes power both in transient and in steady-state regimes.

From the square-root law found in our experiments, we conclude that the transient process through which the Stokes power spectrum builds up is physically essentially governed by the intracavity Stokes power. For RFLs operating in cavities made with hundred-meters long fibers, we can infer that the optical power spectrum at some step in the transient buildup is essentially determined by the instantaneous value taken by the intracavity Stokes power.

4. Conclusion

We have implemented an experimental technique enabling to record the mean optical power spectrum of a RFL during the transient regime in which the laser switches on consecutively to an abrupt change of the pump power. This setup permits to question the way through which the Stokes optical power spectrum broadens before reaching its shape at steady-state. In the transient regime associated with the buildup of the Stokes emission, our experiments show that the FWHM of the RFL spectrum monotonically broadens with the intracavity Stokes power.

Our study made with a RFL operating in a typical configuration commonly found proves that the square-root broadening evidenced in ref. [4] is a robust law not only valid at steady state but also in the transient regime. For the RFL used in our experiment, experiments did not reveal a signature showing the existence of a narrow Stokes optical power spectrum possibly associated with a spectral condensate. This is consistent with numerical simulations showing that the existence of the condensate dramatically depends on some laser parameters such as the number of interacting modes [5, 7]. The experimental tracking of the phenomenon of spectral condensate requires the design of a specific RFL from numerical simulations and the setup presented in this paper could be useful for future experimental researches on this phenomenon.