1. Introduction

Since laser emission was first reported in erbium-doped fiber two decades ago, erbium-doped fiber lasers have proven very useful in a wide range of applications [1–2]. This rare-earth gain medium can serve as the basis for a wide variety of laser designs, including lasers operating in Q-switched, mode-locked, cw single-frequency, and quasi-cw multilongitudinal-mode regimes. The coherence length of the emitted radiation in these different regimes spans several orders of magnitude, ranging from a few tens of microns in the case of mode-locked lasers [2–3] to several tens of kilometers in the case of highly coherent single longitudinal mode lasers [4–5]. Between these two extremes there is an important regime of emission that is characterized by the simultaneous oscillation of multiple longitudinal modes, giving rise to a quasi-cw emission with a moderate coherence length (~ 0.01–1 m). However, the latter regime has not attracted as much interest as the others, and the mechanisms influencing its emission lineshape have not to date been clearly elucidated.

Experience has shown that a fiber laser readily oscillates on multiple longitudinal modes as a result of its long cavity [1,6–8]. Mode discrimination by the gain medium is not strong enough to restrict the oscillation to a single longitudinal mode, despite the fact that homogeneous broadening is known to dominate in erbium-doped fibers at room temperature [9–10]. Hence, one has to use one or more spectrally discriminating elements (i.e. interference filter, Bragg grating or Fabry-Perot etalon) or very short cavities to limit the number of modes over which the laser may effectively oscillate [8,11–12]. However, even doing so does not always prevent laser oscillation on more than one longitudinal mode [8,11].

Thus naturally arises the question of the mechanisms that affect the modal content of the emission spectrum in this particular regime. In this paper we report on the role played by the nonlinear refractive index of the optical fiber in the spectral distribution of the numerous lasing modes in an erbium-doped fiber ring laser. More specifically, we show that the nonlinear wave mixing that takes place between the closely-spaced longitudinal modes is responsible for the broadening of the emission spectrum. Similar effects have been reported in the nonlinear propagation of a partially coherent cw beam in a single-mode fiber [13] as well as in the spectral modeling of a Raman fiber laser [14–16]. In refs. [15–16], a semiempirical model based on rate equations for the multiple Stokes orders and taking into account four-wave mixing between individual longitudinal modes was developed in order to explain the broadening of the emission linewidth from a Raman fiber laser. In this paper, we proceed with the numerical modeling of the signal propagation in the laser cavity, with an emphasis on the signal buildup (see Section 2). The results of these simulations are found to be in good agreement with experimental measurements (see Section 3). The results presented herein provide an understanding of the processes that are responsible for the coherence properties of fiber lasers.

2. Theory

The model described in this section is designed to accurately represent the fiber laser used in the experiment. We consider a fiber ring cavity incorporating segments of erbium-doped fiber and passive single-mode fiber in addition to a bandpass filter with a bandwidth much narrower than that of the gain medium. The modeling is based on the nonlinear Schrödinger (NLS) equation that is commonly used to describe the nonlinear propagation of light in optical fibers [17]. Since the fiber ring cavity includes an active fiber to compensate for the cavity losses, we must therefore add a dissipative term to the NLS equation. The signal propagation in the laser cavity may then be modeled with the following equation:

where E(z,t) is the slowly-varying envelope of the optical field (as seen from the reference frame moving at the group velocity of light). In Eq. (1), g and α accounts for the gain and loss, respectively, while the real coefficient b represents the loss dispersion of the spectrally discriminating element inserted in the cavity. The average dispersion β ₂ of the laser cavity is given by β ₂ =(β _2,er L _er + β _2,smf L _smf)/(L _er + L _smf), where β _2,er and β _2,smf are, respectively, the group-velocity dispersions of the erbium-doped fiber (of length L _er) and of the single-mode fiber (of length L _smf). The nonlinear coefficient γ is related to the fiber nonlinear refractive index n ₂ by γ = n ₂ ω ₀ /cA_eff, where c is the speed of light, ω ₀ the carrier angular frequency and A_eff the effective area of the waveguide fundamental mode.

The gain medium is modeled as a collection of two-level systems in which homogeneous saturation dominates over other broadening mechanisms. Thus, the gain dynamics in the laser cavity is governed by the following equation [10]:

where g ₀ is the steady-state small-signal gain coefficient, P_sat the gain saturation power, T ₁ the gain relaxation time and $P_{\mathit{av}}\left(=\frac{1}{T}{\int }_0^T{\mid E\mid }^2\mathit{dt}\right)$ the power averaged over one cavity round-trip time T(= c/nL), where n and L are the refractive index and the length of the optical fiber forming the laser cavity. In writing Eq. (2), we assume the dipole phases to relax instantaneously, an approximation we find appropriate since the dynamics of the signal buildup proceeds on a radically longer time scale. As well, gain dispersion is absent from Eq. (2) since we assume the gain to be constant over the entire frequency content of the signal (as we discussed in the introduction). Furthermore, we assume the small-signal gain to be provided by a constant external pumping mechanism that we consider to be turned on instantaneously.

Equations (1–2) were integrated numerically using a standard split-step Fourier method [17]. The following values were used for the simulations (these values represent good estimates of the parameters involved in the experiment): ω ₀ = 1.22∙10¹⁵ rad/s, L = 24 m, α = 0.058 m^-1, b = 0.13 ps²/m , β ₂ = -0.003 ps²/m , γ = 0.003 W^-1m^-1, T ₁=102 s and P_sat = 0.6 mW . The value of g ₀ was varied to verify the broadening of the laser line as the signal power increases. The quantities E and g were sampled only once per cavity round trip. Such a procedure is justified since the last three terms on the right-hand-side of Eq. (1) have little effect over a single round trip. For instance, considering a signal with typical average power ~ 20 mW and frequency bandwidth ~ 5 GHz, we estimate these terms after one complete cavity round trip as b(ω - ω ₀)² L ~ 10^-3, β ₂(ω - ω ₀)² L ~ 10^-4 and γ|E|² L ~ 10^-3. Thus, starting from noise, the signal and the gain were computed iteratively until the laser had reached steady state. The relatively narrow frequency content of the laser emission allows for the sampling in the frequency domain to be matched with the actual longitudinal mode distribution. This, in turn, allows one to compute the optical field over a time interval that spans the entire cavity round-trip time.

 

Fig. 1. Simulated (a) waveform and (b) spectrum from an erbium-doped fiber ring laser oscillating on multiple longitudinal modes (g ₀ = 2.67 m^-1).

Download Full Size | PDF

Figure 1 illustrates the results of one such simulation. The laser signal consists of a modulated wave train (Fig. 1(a)) with a spectrum that comprises a large number of modes under an irregular envelope (Fig. 1(b)). The random phases of the multiple oscillating modes explain the irregular structure of the signal waveform. Indeed there is no mechanism in the laser cavity that may result in the ordering of the mode phases (such as a modulator or a saturable absorber). To facilitate the comparison with the experimental observations, the simulated spectra were convoluted with the response function of the optical spectrum analyzer we used in the experiment. For instance, a Gaussian transfer function with a spectral full-width at half maximum of 1.25 GHz (0.01 nm @ 1550.0 nm) was assumed for the subsequent calculations. As well, successive spectra were summed in order to account for the averaging inherent to the spectral measurement process. Examples of such spectra for different small-signal gain coefficients are shown in Fig. 2(a). The broadening of the laser line with the increase in the average signal power is evident from the spectra shown in that figure. This trend is confirmed by the theoretical curve displayed in Fig. 3, where the calculated linewidth (FWHM) is seen to follow a monotonous increase with the average signal power circulating in the cavity. In particular, the line broadening is observed to somewhat level off at higher signal powers. Other authors have reported a similar behavior in the nonlinear propagation of partially coherent cw signals in single-mode fibers [13] and in the spectral modeling of Raman fiber lasers [16].

 

Fig. 2. (a) Simulated and (b) measured spectra from an erbium-doped fiber ring laser oscillating on multiple longitudinal modes ((a): from bottom to top: g ₀ = 0.67, 1.75 and 2.92 m^-1 ; (b): from bottom to top: P_pump = 40, 115 and 185 mW , where the scale is the same as in (a)).

Download Full Size | PDF

 

Fig. 3. Linewidth (FWHM) as a function of the average signal power circulating in the laser cavity.

Download Full Size | PDF

3. Experiment

The predictions of the last section were verified experimentally. An erbium-doped fiber ring laser was built (see Fig. 4) and spectral measurements were performed in order to characterize the broadening of the laser linewidth. In our case, an interference bandpass filter with a full width at half-maximum of 1.2 nm was used as the spectrally discriminating element. The other components included in the laser cavity were a length of erbium-doped fiber (1500 ppm, L _er=16 m, β _2,er = 6.5 ps²/km), a polarization-sensitive Faraday isolator, a polarization controller, a directional coupler with a 75:25 coupling ratio and a WDM coupler to allow for the injection of the 980 nm pump laser. The remaining lengths of optical fiber consisted of single-mode fibers: a length of Corning SMF-28 (L _smf = 8 m , β _2,smf = -21.8 ps²/km) and a length of Corning Flexcor (L _Flexcor = 0.6 m, with negligible contribution to the overall cavity dispersion). The lasing wavelength was tuned to 1550.0 nm by tilting the bandpass filter. The polarization of the incident light at the optical isolator was adjusted with the polarization controller such that the power at the laser output port was maximized. The total cavity losses were estimated to be around 6 dB.

 

Fig. 4. Configuration of the erbium-doped ring laser used in the experiment (EDF: erbium-doped fiber, PC: polarization controller, P-ISO: polarization-sensitive isolator, F: bandpass filter, L: graded-index lens).

Download Full Size | PDF

The emission spectra for different levels of pump powers are shown in Fig. 2(b). Again, the broadening of the laser line with the increase in the average power is evident, since a higher pump power translates into a higher average signal power. In order to better understand the dependence of the laser linewidth on the signal power that propagates in the laser cavity, we have measured the FWHM linewidth for several levels of pump power using an optical spectrum analyzer with a nominal 0.01-nm resolution. Figure 3 shows the results of these measurements as a function of the average intracavity signal power. The latter quantity was inferred from the laser output power and the knowledge of the cavity losses. The average power was then assumed to equal the arithmetic mean of the circulating power at the input and output of the erbium-doped fiber.

4. Discussion

The observations reported in Section 3 are found to be in relatively good agreement with the predictions of Section 2, given the number of physical parameters relevant to this study. In particular, the simulated and measured line broadenings are seen to follow the same trend, although the leveling is less pronounced in the experimental data. As well, the predicted linewidths are slightly larger than the measured ones. Several factors may account for this small discrepancy. First and foremost, any deviations of the parameters used in the numerical simulations from the actual values are likely to cause small differences in the results. In addition, the sampling of the signal propagation in the fiber cavity as performed in the numerical calculations accounts for the distributed effects by replacing them with lumped average quantities. Though this process predicts a trend similar to that being observed, it may also result in small variations between the calculated and measured values. A more accurate calculation would have to consider the effect of the gain distribution on the self-phase modulation experienced by the laser signal upon its propagation along the laser cavity. Finally, the slit function used in the convolution of the predicted spectra may not fit perfectly well the actual response of the optical spectrum analyzer used in the experiment.

The broadening of the laser line theoretically predicted and experimentally described in the preceding sections results from the balance between two countervailing mechanisms: nonlinear wave mixing that takes place between the multiple longitudinal modes and spectral filtering. Nonlinear wave mixing spreads a fraction of the energy contained in the modes that fall near the center wavelength of the filter passband to the outer modes while spectral filtering restrains this flow. Despite the relatively low signal power that circulates in the laser cavity (about a few tens of mW, corresponding to a few tens of μW per lasing mode on average, assuming one thousand modes), the predictions and observations have proven the nonlinear wave mixing to be significant. In effect, we estimate the phase mismatch ΔkL between two adjacent longitudinal modes as β ₂Δω ² L ~ 10^-10. Such a small value, which results from the small mode spacing, clearly indicates that dispersion does not restrain the nonlinear wave mixing between the closely-spaced longitudinal modes. Even the estimate of the phase mismatch over the whole frequency content of the laser emission lineshape (as given in Section 2) confirms that dispersion has little effect in the case considered here.

Hence, the main factor that limits the linewidth broadening is the bandpass filter. Indeed, it was verified, both experimentally and numerically, that the use of an interference filter with a larger (smaller) bandwidth results in a broader (narrower) emission lineshape. However, the filter bandwidth is not the only factor that restricts/enhances the efficient exchange of energy between the multiple oscillating modes. In fact, any parameter entering the definition of the nonlinear parameter γ (see Section 2) is likely to have an influence as well. For instance, erbium-doped fibers usually have an effective mode area significantly smaller than ordinary single-mode fibers. As well, the nonlinear refractive index of optical fibers is known to vary according to the co-dopants used in the fabrication. Thus a particular active fiber is likely to result in an enhanced spectral broadening, depending upon its geometrical parameters and glass constituents, especially if one makes use of a long segment.

The results presented here may help to control the coherence in future designs of fiber lasers. In recent years, high-power fiber lasers have attracted a great deal of interest [18–19]. New designs, incorporating double-clad or microstructure ytterbium-doped fibers, now permit the output power to be scaled up by several orders of magnitude. The coherence of the emitted radiation is of great concern for some applications since the optical powers achieved in some cases far exceed the threshold for the onset of stimulated Brillouin scattering [20]. The results reported herein lead us to believe that nonlinear wave mixing is of paramount importance in such systems. Indeed, some authors have noticed the broadening of the laser line with an increase of the signal power [21–22]. Thus, the extension of our study to these systems would likely prove the optical fiber nonlinearity to impact the emission linewidth of high-power fiber lasers.

5. Conclusions

The influence of the nonlinear wave mixing on the lineshape of a multilongitudinal-mode erbium-doped fiber ring laser was investigated. The energy exchange between the closely-spaced longitudinal modes proceeds freely because of a negligible phase mismatch. In effect, an increase in the laser linewidth was observed for increased signal powers, both theoretically and experimentally. In practice, this spectral broadening is often restrained, in part, by one or more spectrally discriminating components (in our case, a bandpass filter). The results presented here will be useful for the control of the coherence in the design of future fiber lasers.