1. Introduction

Generation of laser pulses with picosecond and femtosecond durations using the technique of mode locking is a vital technology for observing fast events, generating ultra-high-intensity laser pulses for numerous applications, as well as special applications such as frequency metrology. There are many specific techniques used to mode lock lasers [1] including Kerr lens mode locking, additive pulse mode locking, and using semiconductor saturable absorbers, non-linear mirrors [2, 3], and slow saturable absorbers.

Trains of short pulses are also frequently observed from multi-wavelength lasers that employ stimulated Raman scattering (SRS) [4–7]. All of these devices are pulse-pumped rather than continuous wave (cw) lasers, with the duration of the output pulse train of the same order as the oscillation build-up time and the cavity lifetime. Since the turn-on behaviour of lasers, particularly in very high gain systems such as these, is often characterized by output with strong fluctuations on a time-scale that corresponds to the round trip time [8], it is not clear whether these fluctuations would eventually dissipate if under continuous pumping, or are evidence for a genuine form of mode-locking.

It at first seems counterintuitive that a cw intracavity Raman laser could be mode locked. While an intra-cavity pulse in the fundamental field will quickly lead to an overlapping pulse in the co-propagating Stokes field, it seems then that the Stokes pulse will then rapidly deplete the driving fundamental pulse and then decay with the cavity lifetime. A peak in the fundamental field thus appears to be suppressed, rather than favoured as required for mode locking.

Recent studies by Zhao and Jackson of a cw-pumped fibre laser incorporating a section of Raman-shifting fibre have demonstrated the generation of a stable train of pulses with a period close to the round-trip time [9]. Two suggestions for the mechanism underlying this behaviour have been proposed - passive Q-switching [9], and a partially-random pulsing behaviour [10] - but neither can fully account for the observed behavior [11].

In this paper we present an explanation of this behavior as a new type of passive mode locking. We present a physical description of the mode locking mechanism, along with the results of a numerical model that support our interpretation. We show that stable mode locking of both the fundamental and Stokes fields occurs through a dynamic equilibrium in which the fundamental and Stokes pulses are only partially overlapping. Significant reshaping of the fundamental and Stokes pulses during each round trip leads to a repetition rate that is not equal to 1/T , the inverse of the round trip time of the laser cavity. This method of mode locking can be applied to other laser architectures such as crystalline intracavity Raman lasers, and may be a new route for generating picosecond pulses.

2. Description of the numerical model

The numerical model developed here aims to simulate the performance of the cw-pumped fibre Raman laser of Zhao and Jackson [9]. That laser consisted of a 36 m length of Yb-doped fibre providing laser gain spliced to a 134 m length of Ge-doped fibre that has an enhanced SRS cross-section. A highly reflecting mirror was placed at the input to the Yb fibre, with the other cavity mirror provided solely by the Fresnel reflection from the cleaved end-face of the Ge fibre. The Yb-doped fibre was pumped by a cw diode laser at 975 nm. The laser generated output at up to the three wavelengths: 1112 nm (fundamental), 1168 nm (first Stokes) and 1232 nm (second Stokes).

For low pump powers, cw output was observed at the fundamental only. For higher pump powers above the threshold for Stokes oscillation, the output was pulsed with a repetition rate close to the round-trip frequency of the cavity, with the Stokes pulse slightly leading the fundamental pulse. The pulse train was uniform and indefinitely stable [11]. The repetition rate was initially reported as being equal to reciprocal of the cavity round trip time; however, a careful measurement of the fibre lengths and refractive indices revealed that the repetition rate was actually up to 15% less that 1/T , with the discrepancy increasing with increasing pump power [12]. This discrepancy in the repetition rate is crucial in understanding the mechanism of this behavior.

The numerical model is based on Eqs. (1) to (5), the rate equations describing a simple four level laser system and the generation of the fundamental and first- and second-Stokes waves in the cavity. In these equations A_L and A_R are the effective mode areas of the Yb fibre and the Ge fibre, σ_L is the Yb stimulated emission cross-section for the fundamental wavelength, σ_R is the stimulated Raman scattering cross-section (assumed equal for all wavelengths) and τ is the lifetime of the upper laser level. N(z,t) is the population inversion density as a function of the position in the cavity z and time t. In the following parameters the subscripts f, s, and ss refer to the fundamental, Stokes and second Stokes radiation respectively: P ^±(z,t) are powers of the right (+) and left (-) traveling waves in the cavity, ω is the angular frequency of the radiation, and v is the group velocity. The parameters f_s and f_ss are the strength of the Yb stimulated emission cross-sections for the Stokes and the second-Stokes wavelengths relative to the fundamental. The magnitude of spontaneous emission into the cavity mode is determined by γ, β is the strength of backward Rayleigh scattering, and α is the (non-backscattered) propagation loss coefficient. The pump power absorbed per unit length of fibre P_in (z) is in these simulations set to simulate an exponential absorption of the incident pump power along the Yb fibre with a 75% total absorption.

Equation (1) describes the time derivative of the population inversion density in the Yb fibre. Equations (2–4) describe the evolution in time and space of the forward- and backward-traveling intracavity waves at the fundamental and first- and second-Stokes wavelengths. The model includes laser gain in the Yb fibre for fundamental and Stokes-shifted wavelengths [8], forward- and backward-SRS for the Stokes and second-Stokes waves and depletion of the associated pump waves [13]. Terms are included for spontaneous emission into the cavity mode and for coupling of the forward- and backward-traveling waves by backward Rayleigh scattering, as well as a distributed round-trip loss for all wavelengths. Equation (5) determines the boundary conditions for all fields at the ends of the cavity, simulating a 100% reflector at the cavity entrance and a reflectance R at the far end of the cavity of length L. Spontaneous Raman scattering is neglected, since it is extremely small compared to the strength of spontaneous emission coupled into the cavity within the bandwidth of the Stokes and second Stokes fields [13].

These equations are transformed into a set of one-dimensional difference equations by discretizing the z-direction into m steps and creating variables that are a function of t only, replacing the ∂/∂z derivatives with a finite forward difference approximation. Simulating a cavity that is composed of a length of Yb gain fibre spliced to a length of Ge Raman-shifting fibre, appropriate terms are activated or deactivated at different z-positions along the simulation grid. The resulting set of differential equations is solved using an adaptive-step-size Runge-Kutta integrator with the initial conditions at t = 0 that all variables are zero. This method of solution is accurate to first order only, and is prone to numerical dispersion; for a sufficiently fine discretization (m > 10⁴) the effects of numerical dispersion become insignificant.

Table 1. Input parameters for the model.

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3. Results

Simulations were carried out for the input parameters listed in Table 1, derived from the data presented in Ref [9]. The calculated threshold for Stokes generation is 5 W of incident pump power, in reasonable agreement with that determined experimentally (~7W). Above Stokes threshold the fundamental and Stokes fields are found to spontaneously mode lock, with the output consisting of a stable and regular train of identical pulses. Figure 1(a) shows the predicted form of the stable pulses at the fundamental and Stokes wavelengths for an incident pump power of 5.5 W. The model accurately reproduces the experimental observations that the Stokes pulse leads the fundamental pulse, and that the Stokes pulse is significantly narrower. Moreover, the predicted inter-pulse period is 3.6% lower than 1/T , consistent with the experimentally-observed discrepancy. Figure 1(b) shows the laser output for a pump power of 9 W, which is above the threshold for oscillation at the second-Stokes wavelength. The second-Stokes intracavity pulse is positioned on leading edge of the Stokes pulse and the inter-pulse period is now 6.5% lower than 1/T , again matching experimental observations.

 

Fig. 1. Predicted output pulses at the fundamental, first-Stokes and second-Stokes wavelengths for incident pump powers of 5.5 W (a) and 9 W (b).

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The development of the intracavity fields on each round trip can be appreciated in the movie Figure 2(a). This is an animated time sequence showing the evolution of intracavity fields during three complete round trips as a function of position in the resonator. During a complete round trip, we see that first the pulses are both amplified during the transit through the Yb fibre, with a higher gain for the fundamental field. Propagating through the Ge fibre on the way towards the output coupler, the Stokes pulse on the leading edge of the fundamental pulse is strongly amplified, depleting that leading edge of the fundamental pulse. Only 3.75% of the power is reflected at the output coupler, and there is little development of the fields during the transit back through the Ge fibre. When the residual fundamental pulse is again amplified in the Yb fibre, the pulse is reformed in a slightly retarded position compared to the previous round trip owing to the erosion of its leading edge. After each round trip the pulse shape of the fundamental and Stokes pulses are entirely unchanged apart from the small retardation relative to the group velocity. The retardation is more clearly visible in Fig. 2(b), in which the fundamental and Stokes cavity fields are plotted in a moving coordinate frame traveling at the group velocity.

 

Fig. 2. Movies showing the development of the fundamental and Stokes intracavity fields during three round trips, for a pump power of 5.5 W. Movie (a) shows an animation of the total cavity field at each wavelength vs. position (3.4 MB) [Media 1]. Movie (b) shows the cavity fields (unwrapped so the left-traveling cavity field is plotted on the right of the right-traveling field) viewed in a moving frame that travels around the cavity at the group velocity (2.5 MB). [Media 2]

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4. Discussion

The excellent agreement between the model and the experimental data of [9] confirms that the important physical processes are included in the numerical model. The conclusions that the laser generates stable pulses at a uniform rate, with a smooth evolution of the pulses in the cavity to ensure a controlled phase relationship between successive pulses, lead us to assert that the laser is mode locked. Note, as discussed in more detail below, the discrepancy between the repetition rate and 1/T is not inconsistent with mode locking.

We identify backward-SRS as the mode locking mechanism for this laser. Consider a situation where there is a narrow spike at one point in the Stokes cavity field. This spike interacts with the fundamental cavity field through both forward- and backward-SRS. Forward-SRS creates a strong loss for the part of the fundamental field that overlaps with the Stokes pulse. The effect of backward-SRS is more subtle. Consider the simplest case where the Raman medium fills exactly half of the laser cavity and is adjacent to the output coupler. Each part of the fundamental cavity field collides with the Stokes pulse within the Raman medium exactly once per round trip; the loss due to backward-SRS depends then on the magnitude of the Stokes pulse at the time of that one collision. The portion of the fundamental cavity field that just precedes the Stokes pulse collides just before the Stokes pulse reaches the output coupler - the point in the resonator that the Stokes pulse is at its largest. On the other hand, a point in the fundamental cavity field just trailing the Stokes pulse collides just after the Stokes pulse has reflected off the output coupler - the point that the Stokes pulse is at its smallest. Thus the fundamental cavity field immediately trailing the Stokes pulse sees reduced loss due to backwards SRS compared to the remainder of the cavity field. The window of enhanced round-trip gain behind the Stokes pulse that is opened by the sudden decrease in loss due to backward-SRS is closed by a combination of saturation of the gain by the fundamental pulse and increasing backward-SRS as the Stokes pulse is amplified on its way back through the Raman material. This action provides the pressure required to mode lock the fundamental field, creating a fundamental pulse just trailing the Stokes pulse.

The Stokes pulse creating this mode locking pressure is itself maintained against the round trip losses as its trailing edge extracts power from the leading edge of the fundamental pulse, as well as the whole Stokes pulse being amplified by backward-SRS. The Stokes pulse continually leads the fundamental pulse that is being continuously generated behind it. Net loss at the front of the fundamental pulse followed by net gain at the back leads to a dynamic pulse shaping the causes both pulses to slip backwards relative the group velocity, leading to a repetition rate that is slightly less than 1/T. Such a decrease in repetition rate owing to pulse shaping is common to mode locked lasers in general, with mode locking theories generally requiring only that the pulse shape is unchanged after a period T + δT; see for example [1]. Most mode locked lasers have relatively low gain and low output coupling, and so the intracavity pulse evolves only slightly during a single round trip and δT is small. In the case of the fibre laser discussed here, the high gain and the dynamics of the locking process lead to substantial pulse shaping on each round trip.

This mechanism creating a favoured position in the cavity for the fundamental field behind a Stokes pulse was previously recognized by Paschotta [10], although in that work it was suggested that a new fundamental pulse was created randomly behind the original each round trip while the original was depleted. That process, however, is incompatible with the subsequent experimental measurements confirming that the pulse spacing was stable and uniform [11]. The mechanism we describe here is similar to that proposed by Paschotta, with the important difference that the fundamental pulse is itself reshaped during each round trip with a slight lag with respect to the round trip time. This reshaping has no random component and is repeated exactly during each round trip to produce a uniform mode locked pulse train.

We note that the described mode locking mechanism only provides for a single stable intracavity pulse at each wavelength if the Raman material fills at least half of the cavity. If the Raman material occupies less than half of the cavity, a single Stokes pulse in the cavity interacts though backward-SRS only with part of the fundamental cavity field. This leaves the possibility of more than one pulse at each wavelength coexisting in the cavity; the dynamics and stability of this situation will be investigated in a future work.

We find that the output pulse duration is presently determined by β, the coefficient for backward Rayleigh scattering (BRS). For a ten-fold decrease in the scattering strength, the predicted pulse widths also decrease by a factor of approximately ten, suggesting that shorter pulses may be generated by experimental systems with reduced BRS. Inclusion of different group velocities for each of the cavity fields in accordance with the estimated group velocity dispersion of the fibers does not affect the results.

The mode locking mechanism is not specific to fibre Raman lasers, and may have been at least in part responsible for the pulsing seen by researchers investigating crystalline Raman lasers using pulsed pumping [4–7]. Recent demonstrations of cw crystalline intracavity Raman lasers [14, 15] offer the exciting new prospect of simple, compact oscillators generating picosecond pulses at repetition rates up to several GHz.

5. Conclusion

We have described a new Raman mode locking mechanism that accounts for the observed pulse shape, pulse sequence and pulse repetition rate of a previously-published self-pulsed fiber Raman laser. The model reveals that the origin of the mode locking pressure is due to the backward-SRS. The results have implications for realizing compact and simple mode locked oscillators.