1. Introduction

Broadband optical frequency comb (OFC) generators delivering equidistantly spaced coherent laser lines is a rapidly growing area of modern nonlinear optics and laser physics [1]. Optical clocks, metrology, optical communication and microwave photonics are among their potential applications [2, 3]. Mode locked lasers emitting high repetition rate ultrashort pulse trains are commonly used as OFC sources. Short cavity lasers, like semiconductor disk lasers, are able to achieve OFC generation with frequency spacing of hundreds of GHz [4, 5]. Mode-locked soliton fiber lasers are among the most attractive sources for OFC generation [6], but to generate regular pulse trains at high frequencies the fiber laser should be harmonically mode-locked [7–9]. Harmonic mode-locking could be realized in fiber lasers employing a fast saturable absorber (e.g. nonlinear polarization rotation (NPR)) under conditions that uniform distribution of pulses along the cavity is established due to their mutual repulsion, in particular, provided by saturation and relaxation of the net gain [10, 11]. However, this regime is not stable for small interpulse distances, i.e. at high pulse frequencies [12, 13]. Another solution is dissipative four-wave-mixing (D-FWM) [14, 15]. The matter of the D-FWM mechanism is that only two longitudinal cavity modes in the cavity suffer from a positive net gain. The other cavity modes generated through FWM of initial modes are in the range where losses are higher than the gain and so they could acquire energy only through an efficient parametric interaction. As a result, all interacting modes get phase-locking leading to generation of pulses with the repetition rate equal to the frequency difference between two initial modes [16, 17]. To achieve this regime the desired high frequency should be somehow embedded into the fiber laser cavity. Highly selective frequency filters based on high-Q interferometers, like Fabry-Perot interferometer [15], Gires-Tournois interferometer [18] and ring microcavities [19] are commonly used for high repetition rate pulse generation. Unfortunately, these solutions enable laser operation with a fixed repetition rate and require rather complicate coupling of fiber and non-fiber elements.

The importance of ultrafast laser sources for industrial, scientific and medical applications continues to grow, making new demands for increasingly versatile and flexible all-fiber solutions [20, 21]. Recent progress in this field is associated with an all-fiber self-induced modulation instability (MI) laser comprising tunable Mach-Zehnder interferometer (MZI). The laser allows to control the repetition frequency of the pulse train generation by adjusting the optical path difference between the two arms of the MZ interferometer that makes it tunable in sub THz domain. Although this result has already been demonstrated experimentally [22], the reported laser operation seems to lack stability and reproducibility [23]. Moreover, theoretical understanding of the laser operation is still unclear. The key to the puzzle is a low-finesse of the two-arm interferometer that is not sufficient for effective mode selection. With the length of a standard fiber laser cavity of a few meters, its longitudinal mode density is so that one MZI spectral peak covers thousands of cavity modes. This prevents phase locking through D-FWM mechanism only. Besides, at low pulse frequencies too many cavity modes already selected by MZI occur in positive net gain region making their phase-locking through D-FWM (as reported in [22]) rather questionable.

In this paper, we use a simple approach to analyze potential physical mechanisms responsible for pulse train generation in a fiber laser with tunable MZI. In terms of the experimental observations [22, 23] we explore evolution of these mechanisms as the repetition rate changes from 0.01 to 1 THz. It is assumed that for each single round trip over the cavity the propagating light experiences cooperative action from three nonlinear mechanisms that are the saturating gain, NPR and MI followed by D-FWM. The pulse train repetition rate is set by a filter with the periodic spectral transfer function specified for MZI. To simplify interpretation of the results NPR mechanism is considered to be implemented in a passive fiber segment only. Such an approach allows to explore actions of each mechanism independently and to highlight key factors limiting its desired operation. Besides, for each mode-locking mechanism we evaluate zones of laser stability by putting on a map the range of gain saturation energies $E_g$ enabling a pulse train generation regime at different repetition rates. The crucial question is whether we can access all repetition-rate frequencies by adjusting the optical path difference between the two arms of the MZI.

2. Theoretical approach and computation algorithm

The considered experimental configuration of a fiber ring laser with an adjustable MZI is shown in Fig. 1(a). The laser cavity incorporates an erbium-doped fiber (EDF) used as the gain medium, a fiber-based MZ interferometer, whose time delay imbalance can be continuously adjusted by a manual optical delay line (OTDL), a length of passive single-mode fiber (SMF) and a polarization sensitive optical isolator (PS isolator) ensuring unidirectional laser operation and single polarization filtering. The polarization of light inside the cavity can be adjusted by two polarization controllers (PC). The net cavity dispersion is assumed to be slightly anomalous. The laser could operate in harmonic mode-locking regime generating a periodic pulse train with the period set by an optical path difference in a tunable MZI $\Delta L$:

 

Fig. 1 (a) The experimental configuration of the fiber laser with tunable repetition rate [22] and (b) an equivalent laser configuration used for numerical simulations.

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An equivalent configuration of the fiber ring laser used for numerical analysis is shown in Fig. 1(b). It comprises a gain fiber, a piece of passive single-mode fiber (SMF), two tunable polarizers, a polarization controller (PC), a MZI filter and an attenuator. We assume that the light propagating in the gain fiber is linearly polarized, while its propagation in SMF could possess an elliptical polarization. For this reason, a negligibly low fiber birefringence is assigned to the passive fiber and polarization-selective elements are replaced with two polarizers at the fiber input and output in accordance with the model [24, 25]. Such important parameters of the cavity as dispersion, nonlinearity and fiber gain factor are determined by their net cavity value and considered to be uniformly distributed over corresponding fiber segments.

The light propagation in the gain fiber is described by the Ginzburg-Landau equation

The light propagation in the passive fiber is described by two coupled nonlinear Schrödinger equations:

The polarization state of light inside the passive fiber is adjusted by setting of Polarizer 1 at the passive fiber input and described as $A_X=A\cos {\phi }_1,A_Y=A\sin {\phi }_1$, where ${\phi }_1$ is the input polarization angle. Adjusting of the polarization controller (PC) tunes the polarization angle as $A_Y=A_Y\exp i\theta$ and Polarizer 2 is assumed to restore the original state of polarization $A=A_X\cos {\phi }_2+A_Y\sin {\phi }_2$, where ${\phi }_2$ is the output polarization angle.

The spectrally selective filter is assumed to be a lump element with a linear transfer function corresponding to Mach-Zehnder interferometer (MZI) $T_{MZI}\equiv A^{\prime }\left(\Omega \right)/A\left(\Omega \right)=R-\left(1-R\right)\exp \left(-i\Omega /FSR\right)$, where, $A^{\prime }\left(\Omega \right)$ and $A\left(\Omega \right)$are the input and output optical field amplitudes in frequency domain, $R$ is the interferometer reflectivity (1/2 for an ideal MZI). For the desired laser operation, the central gain frequency ${\omega }_0$ is selected to be on the top of the fiber gain spectrum and exactly in the middle between two neighboring peaks of the MZI transfer function $\left|T_{MZI}\right|$. The optical frequency $\omega$ is determined in respect to this central frequency as $\omega =\Omega -{\omega }_0$. In this term, the modes at frequencies ${\omega }_k=\pm 2\pi \text{FSR}\left(1/2+k\right),k=0,1,\mathrm{2...}$ fit the peaks of the MZI transparency. Beating of two central modes with ${\omega }_{\pm 1}=\pm \pi \text{FSR}$ sets the period for the pulse train generation $t_{rep}=1/\text{FSR}$ and the phase difference between the nearest pulses of $\pm \pi$.

Total cavity losses induced by connectors and couplers are taken into account through a single lump attenuator described through its transfer function $T_{att}\equiv A^{\prime }\left(\Omega \right)/A\left(\Omega \right)=\text{const}<1$, where $A^{\prime }\left(\Omega \right)$ and $A\left(\Omega \right)$are the input and output optical field amplitudes in frequency domain.

Equations (2) and (4) with the boundary conditions corresponding to Fig. 1(b) have been numerically simulated employing the split step Fourier method [26]. Initial noise is introduced into the system as a superposition of 1000 low-amplitude central cavity modes possessing Gaussian statistics. Table 1 lists the cavity parameters used for calculations.

Table 1. The cavity parameters used for calculations.

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The anomalous GVD and nonlinearity are considered to be unchangeable over the fiber segments providing the net cavity values ${\beta }_{2\Sigma }=-0.015{\text{ps}}^2$ and ${\gamma }_{\Sigma }=0.025{\text{W}}^{-1}$, respectively. Values of the gain saturation energy $E_g$ determined by the pump power are altered in the range between $0.0025\text{nJ}$ and $0.2\text{nJ}$ to determinate zones of a stable laser operation at different FSR. The simulation time window of size ${\tau }_{win}=2^{15}\cdot 0.0125\text{ps=}409.6$ps has been used over all numerical experiments. For extension of the simulation results for different ring fiber laser configurations the following relations could be applied: ${\beta }_{2\Sigma }/a^2\to {{\beta }^{\prime }}_{2\Sigma },a\cdot FSR\to FS{R}^{\prime },a\cdot {\Omega }_g\to {{\Omega }^{\prime }}_g,a\cdot E_g\to {E^{\prime }}_g,$ where $a$ is the temporal scale factor and ${\gamma }_{\Sigma }{\left|A\right|}^2/{{\gamma }^{\prime }}_{\Sigma }\to {\left|A^{\prime }\right|}^2$ for power scaling. In particular, these relations could be used for qualitative comparison of the simulation results with the experimental observations [22, 23].

3. Results and discussion

We start analysis from a description of laser operation at low pulse frequencies ~10-100 GHz corresponding to $\text{FSR}=(4,\mathrm{..},40)/{\tau }_{win}$. Specific feature of this regime is that $\text{FSR}<<{\Omega }_g/2\pi$ and therefore a large number of cavity modes uniformly selected by MZI have quite equal conditions for their amplification in the laser cavity. It makes impossible preferable lasing of only a few cavity modes and prevents harmonic mode-locking governed by D-FWM mechanism. For successful mode-locking NPR mechanism should be taken into account.

To achieve the desired operation regime the polarization controller and polarizer shown in Fig. 1(b) have been assigned with the following angles ${\phi }_1=\pi /8,{\phi }_2=\pi /2$ and $\theta =3\pi /4$. The laser self-starts from a low level noise and after thousands of roundtrips comes to a stable generation of a soliton train (Figs. 2(a) and 2(b)). Typical feature of laser spectrum is a fringe structure with the period determined by FSR of MZI. Qualitatively, in this case, the laser behavior is similar to dynamics of the soliton laser with a fast saturable absorber [24]. The nonlinear element responsible for HML is the SMF bounded by two polarizers. Neglecting dispersion effects in Eq. (4) the intensity transmission function describing this element could be presented as [24, 25]:

 

Fig. 2 (a) Normalized spectrum and (b) a pulse train generated at $\text{FSR}=18/{\tau }_{win}\left(~45\text{GHz}\right),E_g=0.08\text{nJ}$ Laser spectrum is shown in comparison with the cavity net gain $G=\exp \left({\displaystyle {\int }_0^{L_g}g(z)dz}\right)$, losses and MZI transmission spectra $\left|T_{MZI}\right|$.

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With adjusted polarization elements the transmission coefficient ${\left|T_{SA}\right|}^2$ increases with the pulse peak power in a manner similar to that of a fast saturable absorber [24]. Similar to the soliton laser with a saturable absorber the generated pulses are not chirped and so almost transform-limited. The pulse duration is inversely proportional to the spectrum width and with the given repetition rate and saturation energy depends on the nonlinear cavity losses only (i.e. on the polarization angles). For the pulse train presented in Fig. 2 the pulse duration (FWHM) is ~0.5 ps.

Considering the laser dynamics at low FSR shown in Fig. 3(a) we should note that the spectrum evolution is a result of mode competition process mainly determined by the spectrum of MZI transmittance. These losses are the most significant part of all linear cavity losses responsible for mode filtering that seed the start of laser generation. As lasing is establishing, the nonlinear losses induced by NPR force the laser to operate in HML regime. NPR periodically opens the transparency gate for the most intensive pulses combined from the most successful lasing modes. As a result, phase-locking of all successful cavity modes occurs ensuring generation of the pulse train with the pulse frequency equal to FSR in accordance with Eq. (1). One can see that final intensities of central modes are almost equal highlighting equal conditions for their lasing at low FSR (Fig. 3(b)) and thus confirming a minor contribution of D-FWM mechanism to their formation. Lasing modes with initially uncorrelated phases are synchronized due to nonlinear filtering and the mutual phase differences are established to be multiples of $2\pi$ (Fig. 3(c)).

 

Fig. 3 Dynamics of laser mode-locking governed by NPR mechanism at $\text{FSR}=18/{\tau }_{win}\left(~45\text{GHz}\right),E_g=0.08\text{nJ}$. (a) Spectral density; (b) intensity and (с) phases of four modes centered around a MZI transmission $\left|T_{MZI}\right|$ peak.

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Figure 4(a) demonstrates the system parameters corresponding to successful laser mode-locking calculated for different FSR and levels of the saturation energy $E_g$. One can see that at a given FSR the laser operation remains stable in a range of the saturation energy above the laser threshold. This fact as well as an increase of the saturation energy with the repetition rate is explained by the matter of NPR mechanism that operates pulses with a certain level of the peak power. Hence, keeping the pulse peak power and duration unchangeable, the increase of the repetition rate should proportionally increase the average laser power and so the saturation energy. In this way, the maximum level of the saturation energy is achieved for the critical FSR values lying in the range ~70-75 GHz ($30/{\tau }_{win}-32/{\tau }_{win}$). For FSR above the critical values, the laser operation in harmonically mode-locking regime is not supported any more. It could be explained by two obstacles. First, the ability of the MZI to select individual lasing modes decreases with the increase of FSR. Second, action of NPR mechanism in the case of poor mode selection becomes rather unpredictable. It causes random phase-locking of different groups of modes leading to generation of rather irregular pulses. The last provokes unpredictable fluctuations of the population inversion gain in the cavity making individual mode selection by MZI even more difficult and stochastic. This result is in agreement with observations [23], where irregular pulsations above a critical FSR value have been observed.

 

Fig. 4 (a) A map of the saturation energies $E_g$ corresponding to successful mode-locking. (b) Pulse trains calculated for points A, B and C.

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Thus, above the critical FSR values the NPR mechanism makes rather negative contribution to laser dynamics preventing individual mode selection by MZI that is obligatory for harmonic mode-locking operation. Eliminating NPR mechanism in the laser operation at moderate FSR allows to remove the nonlinear losses providing better conditions for harmonic mode-locking achievable through D-FWM mechanism. So, now we focus on the laser operation at FSR ranging from 0.1 to 1 THz ($50/{\tau }_{win}$ - $400/{\tau }_{win}$). NPR mechanism is canceled by setting the polarization state angles to ${\phi }_1={\phi }_2=\pi /2$ and $\theta =0$. With such conditions Eqs. (4) are separated and light propagation in SMF is described by a single Schrödinger nonlinear equation. To explain a regular pulse train generation observed at this frequency range it is necessary to establish balance of dissipative factors considering low cavity mode selectivity provided by MZI. This result can be obtained by gain - loss control in spectral domain when most of cavity modes are removed from the range where net gain is higher than losses. It is the way how NPR mode-locking could be replaced with D-FWM mode-locking. A standard solution for precise control of the gain bandwidth is band-pass filter [17]. However, in our model such filtering can be realized by adjusting the gain saturation energy $E_g$ taking into account a finite gain spectrum width ${\Omega }_g$. It is clear that $E_g$ should be close to its threshold value for successful filtering.

At the repetition rates of about 0.25 - 0.75 THz ($50/{\tau }_{win}-300/{\tau }_{win}$) the laser starting from the noise level after several thousands of roundtrips comes into stable generation of high repetition pulse train with a typical optical frequency comb spectrum. In contrast to NPR driven mode-locking, to achieve lasing in D-FWM regime only two individual cavity modes should be in positive net gain region as shown in Fig. 5(a). Through FWM the energy of the central modes is redistributed to higher modes and compensated by dissipative effects. The dynamics of D-FWM driven mode-locking is shown in Fig. 6. In contrast to NPR mode-locking the intensities of neighboring fundamental modes differ by orders of magnitude. It causes generation of pulses under conditions when only small number of cavity modes is available for lasing. Due to D-FWM mechanism the phases of interacting modes get locking. Similar to NPR mode-locking, MZI supports anti-phase behavior $\Delta \varphi =\pi$ for neighboring pulses ensuring high repetition train stability. However, relatively broad pulses in the generated train [22] are explained by a small number of lasing modes contributing to mode-locking (see Fig. 4(b), point A).

 

Fig. 5 Normalized laser spectrum in comparison with the cavity net gain, losses and filter transmission spectra calculated for laser configurations with (a) MZI $\text{FSR}=200/{\tau }_{win},\left(~490\text{GHz}\right),E_g=0.008\text{nJ}$ – point A in Fig. 4(a); (b, c) DMZI $\text{FSR}=200/{\tau }_{win},$$\left(~490\text{GHz}\right),$$E_g=0.018\text{nJ}$ (b) and $\text{FSR}=400/{\tau }_{win},$ $\left(~980\text{GHz}\right),$$E_g=0.0025\text{nJ}$(c) – points B, C in Fig. 4 (a), respectively.

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Fig. 6 Dynamics of laser mode-locking governed by D-FWM mechanism at$\text{FSR}=200/{\tau }_{win}(~490\text{GHz}),E_g=0.008\text{nJ}$– point A in Fig. 4(a). (a) Spectral density, (b) intensity and (с) phases evolution of four first modes centered around the MZI transmission $T_{MZI}$ peak.

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Realization of these conditions at different FSR is determined by common ability of two filtering mechanisms that are the linear MZI filter and the gain spectrum line to select individual cavity modes. Selectivity of the first mechanism decreases with the FSR, while selectivity of the second increases. Indeed, in a scale of the cavity mode frequency spacing the transparency peak of the MZI filter is much narrower at low than at high FSR. So, its ability to select individual cavity modes is higher at low FSR. Similar, in a scale of the frequency spacing between the modes already selected by the MZI filter the fiber gain spectrum is much narrower at high than at low FSR. So, its ability to select these modes is stronger at high FSR.

In Fig. 4(a) the resulting effect of two filtering mechanisms on mode selection is evaluated through a map of the saturation energy values $E_g$ corresponding to successful D-FWM driven harmonic mode-locking. The maximal value of $E_g$ is achieved for moderate repetition rates in the range of 0.35 THz. Similar to NPR mode-locking the pulse duration is inversely proportional to the pulse spectrum width and for the given saturation energy increases with the repetition rate. At maximal available saturation energy the minimal pulse duration (FWHM) of ~1ps is achieved at 0.35 THz.

For higher repetition rates simulations show that successive mode filtration occurs near the threshold only, when the central modes suffer from very low net cavity gain just slightly exceeding the losses. The closer gain to the threshold the higher impact of selective losses caused by MZI. For the gain higher than the critical value, mode selectivity becomes weaker causing development of instabilities and noise generation. At the same time, an increase of the repetition rate requires better filtering in expense of decrease of the gain resulting in a low pulse train contrast (the pulse duration remains nearly unchangeable with FSR increase), fast recession of high harmonics and low peak power of generated pulses. Finally, at the repetition rates higher than 0.75 THz pulse train is transformed into continuous wave.

Thus, in order to achieve higher repetition rates and expand the FSR range of D-FWM phase-locking we have to enhance mode selection by a tunable filter. Considering just all-fiber solutions and keeping a tunable MZI as a basic system element, introduction of an additional filter into the cavity could be also the way to improve stability of the pulse train generation. The higher selectivity of the combined filtering, the higher gain can be introduced to the system. On other hand, adjustment of the additional filter synchronously with the MZI maintaining coherent superposition of transmission peaks requires tedious adjustment of cavity parameters [23]. The most obvious solution is the use of double-pass MZI configuration that has already demonstrated its capacity for a number of practical applications [27].

To demonstrate this idea all calculations have been repeated with $T_{MZI}$ replaced with a linear transfer function describing a double Mach-Zehnder interferometer (DMZI) $T_{DMZI}=T_{MZI}{}^2$ ($R=0.51$). One can see in Fig. 5(b) that an increase of the gain allows to enhance energy transfer from the central modes to side modes and thus to improve the quality of the pulse train generation (see Fig. 4(a), point B, Fig. 5(b)). The shortest achievable pulse duration decreases down to ~0.6 ps for DMZI configuration with a simultaneous increase of the pulse peak power. Besides, incorporating DMZI into the cavity expands the range of the laser tunability available through D-FWM mechanism enabling essentially higher repetition rates up to 1 THz (see Fig. 4(a), point C, Fig. 5(c)). For comparison, the pulse parameters obtained with MZI and DMZI configurations are listed in Table 2 for several given (FSR, saturation energy) points.

Table 2. Comparison of the pulses generated with MZI and DMZI configurations.

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4. Conclusion

In conclusion, we have analyzed potential mode-locking mechanisms responsible for high-repetition-rate pulse generation in fiber ring lasers with repetition frequency controlled by the delay imbalance of a MZ interferometer. Applying a simple laser model we have clearly demonstrated a presence of two pronounced ranges of the laser tunability. At low repetition rates the laser operates as a harmonically mode-locked soliton laser triggered by NPR mechanism. Our results qualitatively reproduce key features of the laser outputs recorded in the experiments [23], including observation of laser instabilities at upper FSR range limit. At high repetition rates the laser mode-locking occurs due to D-FWM seeded by MZI and gain spectrum filtering. For this range our results are in a qualitative agreement with the experimental data [22] presented at ~1 THz. In particular, relatively wide pulse is explained by a small number of phase-locked lasing modes provided by D-FWM. The model predicts a gap between two FSR ranges where pulse train generation is not supported. This result still needs a direct experimental verification, but it is not in contradiction with the reported experimental observations [22, 23]. We have demonstrated as well that the use of filters with higher Q-factors is the way to achieve higher repetition rates and expand the FSR range of D-FWM mode-locking. In particular, application of a double MZI instead of a single MZI could improve the laser stability and increases the range of powers available for laser operation.