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We present a detailed experimental and modeling study of fiber lasers passively mode locked using stimulated Raman scattering. We present experimental measurements of the mode locking behavior, and use the model to elucidate the nonlinear processes that create the intracavity pulses and determine the repetition rate and the order of fundamental and Raman pulses. We also present a simple method for harmonic mode locking Raman fiber lasers.
©2008 Optical Society of America
1. Introduction
Interest in pulsed fiber laser research remains strong because of the diverse range of applications including remote sensing, optical communications and material processing. Pulse widths from the femtosecond to the microsecond regime have been demonstrated. However, there is still a need for stable nanosecond pulsed fiber lasers designed to suit specific applications with tailored performance characteristics such as the pulse width and the average power (see for example [1]).
Nanosecond pulse durations from fiber lasers are typically produced using passive Q-switching, which relies on one or more of the phenomena of Rayleigh scattering and inelastic scattering, gain saturation, and saturable absorption [2–5]. For example, a passively Q-switched fiber laser that utilized distributed Rayleigh and Brillouin backscattering yielded pulses approximately 2 ns long [5]. These types of devices have the significant advantage of being essentially “all fiber” systems, but the pulse characteristics (e.g., the pulse duration, period and pulse shape) are too irregular for many applications since each pulse is seeded from stochastic processes [6]. More regular pulse characteristics may be demonstrated using active Q-switching methods but at the expense of added complexity, reduced robustness and the loss of all-fiber construction [7].
Passively mode locked fiber lasers, on the other hand, produce regular pulsed output using an “all fiber” design; the pulse duration and repetition rate are in general highly stable since these are defined by the basic fiber properties such as dispersion, gain and cavity length. Passive mode-locking methods include the use of nonlinear loop mirrors [8], saturable absorbers and polarization additive pulse mode locking techniques [9]. Most configurations require a figure-of-eight architecture with polarization controllers and optical diodes, and the output pulses are typically of sub-ps duration and at multi-GHz repetition rates [10].
Recently a simple fiber laser design was reported offering stable pulses at sub-MHz pulse rates and pulse durations of approximately 40 ns, i.e. pulse characteristics similar to Q-switched output [11]. The laser consisted of a 36 m section of Yb3+-doped silica fiber spliced to a 134 m section of Ge-doped fiber. Above a threshold level of pump power, the laser spontaneously emitted pulses at both the fundamental and first Stokes wavelengths with the Raman pulse preceding the leading edge of the fundamental pulse, and with an average Stokes output power of approximately 2W; the pulse repetition frequency was approximately equal to the inverse of cavity round trip time. Using a rate equation of model of the system, it was shown that the pulsing mechanism could be described as a form of mode locking resulting from the combined action of backwards stimulated Raman scattering and gain saturation [12].
In this paper, we report a detailed experimental and modeling study of the pulsing behavior, output spectra, and repetition rate of these Raman mode locked fiber lasers. Calculations from the numerical model display excellent agreement with the experimental results. We have used the model to investigate the contributions from each of the gain and loss processes within the fiber and show that backwards stimulated Raman scattering (SRS) is the major process that brings about mode locking of the fundamental cavity field and stabilizes the pulsing behavior. We also show that a fiber laser with an extended length exhibited harmonic mode locking with two pulses within the cavity at each wavelength. Finally, we discuss the limits on the output pulse duration and propose that a similar laser arrangement using non-fiber gain and Raman media may allow the generation of picosecond pulses.
2. Experiments
We investigated two fiber laser configurations shown schematically in Fig. 1. Each fiber laser was constructed from lengths of Yb3+-doped silica fiber, Ge-doped fiber, and standard SMF-28 fiber. The Yb3+-doped silica fiber was pumped using a continuous-wave (cw) 975 nm pump diode and provided broadband gain within the cavity. The Ge-fiber had a high cross-section for SRS, generating Stokes-shifted frequencies with a shift of 13.2 THz. The SMF-28 was used in the second laser arrangement to alter the overall length of the cavity. In each laser, a dichroic mirror was butted against the end of the Yb3+-doped fiber, transmitting 98% of the 975nm pump light, and reflecting over 99% of the light between 1040 and 1190 nm. The cavity was completed by the cleaved facet at the far end of the concatenated fiber chain, reflecting 3.75% of the radiation back into the cavity. The two cavity arrangements were designed to highlight features of the pulse forming mechanism, which depends on the fraction of the cavity that is occupied by the Raman gain fiber.
In order to accurately interpret the experimental results, we have used a numerical model similar to that presented in [12] to simulate the performance of each laser configuration. The physical processes in the model included: laser gain and seeding of the cavity fields by spontaneous emission in the Yb3+-doped silica fiber; forward- and backward-SRS that couple the fundamental, first-Stokes and second-Stokes fields in the Ge-doped silica fiber; and Rayleigh scattering and propagation loss in all fibers. The model in [12] was modified to enable incorporation of the SMF-28 fiber, and in particular to allow a different Rayleigh scattering coefficient, propagation loss coefficient, and group index for each type of fiber. As a result, a variety of cavity designs consisting of spliced lengths of Yb, Ge, and SMF28 fiber sections could be modeled by appropriately activating and deactivating laser-gain and Raman-gain terms for the different sections of the cavity, as well as selecting the appropriate scattering and loss coefficients. Appendix A describes the model in detail and lists the parameters used in the simulations.
Fig. 1. Schematics of the two laser configurations that are presented and discussed. Cavity I comprised a 14.6 m section of Yb gain fiber coupled to 100 m section of Ge Raman-shifting fiber. Cavity II included an additional 285 m length of passive fiber, so that the Raman-shifting fiber occupied only a quarter of the cavity. Both lasers were pumped from the left hand side through a dichroic mirror that was highly reflecting at all laser wavelengths and highly transmitting at the pump wavelength. The laser output was coupled out on the right hand side, the output coupler formed by a uncoated cleaved facet.
3. Raman mode locking
3.1. Experimental results
We used the configuration of Cavity I, which provided the most stable Raman mode locking, to investigate the mechanisms responsible for mode locking and to determine the detailed output characteristics. Figure 2(a) shows the output power and repetition rate of the laser as a function of absorbed pump power. For pump powers below the threshold for Stokes emission, the laser operated cw with a fundamental wavelength around 1100 nm and a spectral width of 6 nm. For pump powers greater than 4.9 W, Stokes laser emission centered at the wavelength 1150 nm was observed. The laser output consisted of sequential Stokes and fundamental pulses as shown in Fig. 3(a). For pump powers above 5.6 W, emission at the second Stokes wavelength around 1214 nm was observed. The output again was comprised of sequential pulses, with the second Stokes preceding the Stokes and fundamental wavelengths as shown in Fig. 3(b). The pulse durations of the Stokes outputs – the 1st Stokes was 100 ns and the 2nd Stokes was 60 ns – were markedly shorter than the fundamental pulse duration of 210 ns. The observed pulse repetition rate was 810 kHz at 4.9 W of pump power and decreased with pump power to 790 kHz at 7 W of pump power, as shown in Fig. 2(a). The repetition rate was always notably lower than the inverse round trip time 1/T of 876±9 kHz, calculated using the measured fiber lengths and group indices.
Fig. 2. (a). Output power of the laser at the fundamental (black), 1st Stokes (red) and 2nd Stokes (blue) wavelengths as a function of the absorbed pump power. The laser spontaneously mode locks above the Stokes threshold, with the repetition rate (grey, right axis) decreasing slightly with increasing pump power. (b) Spectral content of the laser output averaged over 1 ms, measured with an absorbed pump power of 6.3 W.
Fig. 3. Output pulse trains for Cavity I for a pump power of 5.6 W (a) and 6.3 W (b) at the fundamental (black), Stokes (red) and second Stokes (blue) wavelengths. The round trip time of the laser cavity was 1.142 microseconds.
The fundamental, 1st Stokes, and 2nd Stokes output spectra, measured using a Ocean Optics spectrometer integrating for 1 ms, showed that each was several tens of nanometers wide as shown in Fig. 2(b). The spectral width of the fundamental laser emission was 6 nm when the laser was operating below Stokes threshold. The width increased to 26 nm when operating above the threshold for second Stokes emission, with the FWHM of the 1st Stokes and 2nd Stokes emissions being 15 nm and 8 nm respectively. By observing the time behavior of different spectral components using a monochromator and a detector with a 1 ns response time, we deduced that each pulse contained the full spectral bandwidth.
3.2. Numerical modeling results
The numerical model was used to calculate a steady state solution for Cavity I for an absorbed pump power of 5.6W, corresponding to the experimental conditions for the results presented in Fig. 3(a) above. The parameters input to the model are listed in Table 1. A sample of the simulated pulse train is shown in Fig. 4. Excellent agreement was obtained between the model and experiment with respect to all the main features of output. The pulsing naturally emerged from the model, with the relative position of the fundamental and Stokes in agreement with the experiments. The calculated repetition rate was 846 kHz, which is 3.5% lower than 1/T. The good agreement between the model and the experiments indicates that the model incorporates the basic processes responsible for the gross laser features and in particular the pulsing behavior. We have previously published the results of similar simulations, including a movie showing how the pulses develop over several round trips [12].
Fig. 4. The output pulse train predicted by the numerical model, showing the fundamental (black) and the Stokes (red) predicted output pulse train. The round trip time of the laser cavity was 1.142 microseconds.
Fig. 5. Plots (a) and (b) show for two sequential round trips the values of the fundamental and Stokes cavity field at the output coupler. Plot (c) shows the gain for the fundamental field for each segment of the cavity field owing to the four processes acting on that field.
In order to further elucidate the processes that cause the pulsing behaviour, we have used the model to investigate how the gain and loss for the fundamental field depends on position relative to an intracavity Stokes pulse. To do this, the fundamental cavity field was split up into segments, and then each of those segments was tracked as it completed one round trip traveling at the group velocity. For each segment, the effects of laser gain, loss due to forwards-SRS, loss due to backwards-SRS, and passive losses were summed to obtain a small signal gain (or loss) associated with each process for one round trip time T. The results are shown in Fig. 5. In Fig. 5(a), the fields at the fundamental and Stokes wavelengths arriving at the output coupler of the laser cavity are plotted for one round trip; Fig. 5(b) shows the same for the subsequent round trip. For the intervening time, the small signal gain of each process was tracked for each segment of the fundamental cavity field as it traveled one round trip within the cavity at the group velocity. Figure 5(c) shows the values of gain for each process plotted against position in the cavity field: laser gain (red), loss due to forward (blue) and backwards (yellow) stimulated Raman scattering, and loss due to the output coupler and scattering (grey) are shown. When combined, the overall gain (black) for each part of the fundamental cavity field is of course that gain required to transform the magnitude of the field segment before the observed round trip (a) to its magnitude after (b).
The analysis shows that parts of the fundamental field that overlap with the Stokes pulse are very strongly attenuated by forward SRS, depleting the local fundamental field and transferring power to the Stokes field. However, for the fundamental field just trailing the Stokes pulse (i.e. further to the right in the plots of Fig. 5) the overall gain is enhanced due to a minimum in the loss by backwards SRS; the enhanced gain is maintained for many tens of nanoseconds before saturation of the laser gain causes the net gain fall back close to unity. This “window” of enhanced gain ensures that the fundamental field is continually and preferentially maintained just trailing the Stokes pulse. This leads to the stable situation in which the Stokes pulse is supported by the Raman conversion of the leading edge of the fundamental pulse, and the fundamental pulse is continually amplified behind the Stokes pulse. The pulsing emerged naturally from the numerical model, with the relaxation oscillations driven as the cw pump source was turned on sufficient to start the locking process.
The window of reduced loss due to backwards-SRS was first identified by Paschotta [13], and to understand this loss we must consider how a narrow Stokes pulse interacts by backwards-SRS with the whole fundamental cavity field. Each segment of the circulating fundamental cavity field collides with the counter-propagating Stokes pulse twice during each round trip – once in each half of the cavity; if the segment is close to the Stokes pulse, those two counter-propagating collisions happen close to each end mirror, whereas if the segment is approximately half a round trip away from the Stokes pulse, the two collisions happen close to the cavity midpoint. The two collisions per round trip are illustrated in diagrams I and II in Fig. 6(a), showing schematically a Stokes pulse and a fundamental pulse circulating within a cavity. The pulses collide once near the output coupler within the Ge fiber (I) and once close to the high-reflector in the Yb fiber (II).
It is the backwards-SRS interactions during these collisions that control the fundamental field and create the pulsing behavior. Only collisions that occur in the Raman-enhanced Ge fiber will contribute a significant interaction; provided however that the Ge fiber adjacent to the output coupler occupies at least half of the cavity, as in Cavity I, a single narrow Stokes pulse collides with every segment of the fundamental field at least once within the Ge fiber.
The fact that the Stokes pulse leads the fundamental can be explained by considering the effect of backwards SRS on the fundamental field just in front of and just behind the Stokes pulse, illustrated in Fig. 6(b). For a fundamental pulse F1 leading the Stokes pulse S, the backwards-SRS loss collision occurs at time A just before the Stokes pulse reaches the output coupler – the time at which the Stokes pulse has its maximum as it has just been amplified through the Ge fiber. On the other hand, for a fundamental pulse F2 trailing the Stokes pulse S, the pulses interact only after the Stokes pulse has reflected off the output coupler at time B – the time at which the Stokes pulse has its minimum intensity. The result is that the Stokes pulse is far weaker when it interacts with the trailing section of the fundamental field, and so that trailing section experiences reduced loss due to backwards-SRS.
Note that the fundamental pulse also undergoes a second interaction with the Stokes pulse located at the opposite end of the cavity near the high reflecting mirror on each round trip. If the Raman-active material extends over much more than half of the cavity length, this interaction may again partially occur within the Raman-active material, and may contribute to a preferred location either trailing or leading the Stokes pulse. Note, however, that at this end of the cavity, the magnitude of the Stokes pulse is not changing rapidly and the effect of this interaction on the laser dynamics is minor.
Fig. 6. Schematic diagram showing the interaction of a Stokes pulse (S) with a leading (F1) and a trailing (F2) fundamental pulse. (a) I and II show the backward interactions of these pulses at each end of the resonator. (b) Time A illustrates the collision between F1 and the counter-propagating S, just before S reaches the output coupler. Time B illustrates the collision between F2 and the counter-propagating S, just after S reaches the output coupler. It is during these collisions that the fundamental pulses experience loss due to backwards SRS; since S is far smaller at time B compared to time A, pulse F2 experiences a smaller loss than F1.
Enhanced gain for a favored part of the cavity field leading to the generation of a single intracavity pulse is characteristic of mode locking; we would therefore characterize this pulsing mechanism as a form of mode locking. Although mode locked lasers are often considered to have a repetition rate equal to 1/T, some discrepancy between the inter-pulse period and 1/T is expected: mode-locking theory (see e.g. [12]) only requires that the intracavity field is unchanged after a period od t+δt, where δt is the round trip time lag. For example, mode locking using a slow saturable absorber exhibits such a discrepancy, as the absorber continually erodes the leading edge of the pulse and the remainder of the pulse is preferentially amplified.
The fiber laser presented here was clearly only weakly mode locked. The pulse durations were as much as a twentieth of the round trip time, while the frequency content of the pulses was extremely broad, driven by the details of the SRS interaction. In the present case, the large round trip time lag δt means that the pulse peak is progressively shifted to a new location within the circulating cavity field. As a result, the intracavity fundamental pulse is not strongly sharpened. The numerical model indicates that the time lag δt and the durations of the pulses are strongly linked to the strength of Rayleigh scattering in the cavity. Simulations show that a ten-fold decrease in the scattering strength results in an approximately ten-fold decrease in the pulse durations. There are several avenues to further reduce the pulse durations. Reducing the doping levels of the Yb and Ge fibers would strongly decrease the scattering, reducing the present level of scattering of 10-6 m-1 towards the current lowest achievable value of 10-8 m-1 in SMF fiber. The model also shows that shortening the cavity length or changing the output coupling can lead to reduced pulse durations. An interesting approach may be to replace the Ge fibre with SMF, which is itself Raman active. Note however that the much lower Raman gain coefficient for SMF, and indeed any of the above changes to doping levels and the resonator design, will also influence other laser parameters such as Stokes threshold, power, efficiency and pulse rate. Finally, a laser operating at a longer fundamental wavelength may be promising for generating shorter pulses since the Rayleigh scattering decreases with the fourth power of the wavelength.
4. Harmonic mode locking
For arrangements in which the Raman-active Ge-doped fiber occupies only a quarter of the total cavity length the cavity dynamics were found to be notably different. Cavity II in Fig. 1 incorporated an additional 285 m section of passive SMF-28 fiber, with the result that the Raman-active Ge-doped fiber now accounted for only one quarter of the total cavity length. Measured output pulse trains at the fundamental and Stokes wavelengths are shown in Fig. 7(a). The mode locking remained self-starting, although it was noted that the pulse train was not as regular as for Cavity I in terms of pulse amplitude and shape. The pulse repetition rate was measured to be 485 kHz, which is slightly less than twice the inverse of the cavity round trip 1/T (254 kHz; as determined from measured fiber lengths and known group indices). We see then that for this cavity there were two intracavity pulses at each wavelength rather than just one, with the pulses approximately evenly spaced. This is indicative of harmonic mode locking.
Simulations of this cavity [Fig. 7(b)] are in again in good agreement with the experimental results with respect to pulse shape, pulse sequence, and repetition rate (the predicted repetition rate was 500 kHz). The simulated pulse train did not settle to an entirely stable pulse shape in this case, with the two intracavity pulses interacting with each other leading to a fluctuation in the pulse shape and peak power.
The origin of the two Stokes pulses each round trip is readily explained using a similar analysis to that used in the previous section. Due to the fact that the Raman section of fiber only occupies a quarter of the cavity, a single Stokes pulse can only interact via backwards-SRS with approximately half of the fundamental cavity field – the remaining half of the fundamental cavity field collides with the Stokes pulse only in the Yb and SMF fibers. Thus a second fundamental pulse can emerge in the remaining half of the cavity field, generating in turn a second Stokes pulse. Due to the finite widths of the Stokes pulses, each actually interacts with a little more than half of the fundamental cavity field – this creates an overlap between their domains of influence that along with the effects of gain saturation tends to stabilize their relative positions. These arguments apply equally to cavities with shorter and shorter lengths of Raman-shifting fiber, and it should be possible to achieve harmonic mode locking with larger numbers of pulses within the cavity.
The model also showed that a second mode of operation was supported with just a single fundamental and Stokes pulse within the cavity, but with greatly extended pulse durations, as shown in Fig. 7(c). Whether the simulation settled into the single or double pulsing mode depended on the exact details of how the simulation was started, such as the ramping rate of the pump power. No clear pattern was discerned, and small changes in the initial conditions could affect which mode came to dominate. The long-duration Stokes pulse interacts with the majority of the fundamental field despite the small length of Raman-active fiber with different parts of the Stokes pulse interacting with different parts of the fundamental field, and a stable pulsing regime is attained. This mode of operation was not observed experimentally; it may be that it is less stable against fluctuations (e.g. pump power variations) or is disturbed by the small discrete reflections off the splices between the different fibers within the cavity.
We note that harmonic locking was never observed in either simulations or experiments using Cavity I. With the Raman medium occupying more than half of the cavity, two sets of intracavity pulses would interact strongly, and the larger set would quickly suppress the smaller set.
Fig. 7. Experimental (a) and numerical simulation (b) of the output pulse trains for Cavity II for a pump power of 7 W at the fundamental (black) and Stokes (red) wavelengths. The round trip time of the laser cavity is 3.93 microseconds. There were now two pulses circulating in the cavity at each wavelength, with the inter-pulse period just over half the nominal round trip time. The numerical model also revealed a second stable mode of operation (c) with one lengthened pulse within the cavity at each wavelength.
5. Discussion and conclusions
We have presented a detailed set of experimental measurements of two cavities that demonstrate self mode locking. A cavity in which over half of the cavity is comprised of Raman-active fiber was shown to operate in a stable manner with Stokes and fundamental intracavity pulses generated at a repetition rate slightly lower than that expected from the optical length of the cavity. These observations were closely duplicated by a simple model, which shows that the mode locking is brought about by the action of backwards SRS on the fundamental field. A second cavity, extended so that the Raman-active fiber only comprised a quarter of the cavity, was observed to operate at a pulse repetition rate of approximately twice the round trip frequency. The numerical model again reproduced this behavior.
The results have implications for creating self mode locked Raman laser in bulk (non fiber) intracavity Raman lasers. The above results show that the fraction of the cavity filled by the Raman-active material is important: often in bulk lasers reported to date the Raman medium occupies only a small fraction of the cavity [15]. In other respects, they appear to meet all the criteria required for Raman mode locking. Indeed sub-ns pulses at a rate corresponding to ~1/T is often observed within a Q-switched pulse envelope (see for example reference [16]), it is often unclear whether this spiking behavior is a transient consequence of the rapid onset of Stokes generation, or if it is genuine mode locking. The recent emergence of continuous wave solid-state intracavity Raman lasers [17] provides an interesting platform for investigating the type of mode locking presented in this paper. The comparatively shorter cavity length of these systems along with the reduced impact of Rayleigh scattering offers significant potential for generating sub-picosecond pulse trains at GHz repetition rates. Designs that allow the Raman material to occupy approximately half or more of the cavity may lead to simple and efficient sources of picosecond pulse trains.
It is concluded that the Raman mode locking process is a simple method of generating pulsed output from an all-fiber laser, applicable to all forms of intracavity Raman lasers.
6. Appendix A – the laser model
The laser was modeled as a simple four level laser scheme with intracavity SRS, generating forward- and backward-travelling intracavity fields at a discrete fundamental wavelength of 1100 nm, and at the Stokes and second Stokes wavelengths of 1155 nm and 1215 nm. The model equations are shown in Eqs. (1–5). P ± f(z,t), P ± s(z,t), and P ± ss(z,t) are the powers of the intracavity fields as a function of position in the cavity z and time t: the subscripts f, s, and ss refer to the fundamental, Stokes, and second-Stokes fields respectively, and the ± superscript refers to the left- and right-travelling waves in the cavity. In the Yb fiber sections, the population inversion density N(z,t) is tracked. The pump power absorbed per unit length Pin(z) was set to simulate an exponential absorption of the 975 nm pump power along the Yb fiber, with 50% absorbed in total, consistent with that experimentally measured.
Equation (5) links the left- and right-travelling waves by simulating a 100% reflector at z=0 and a fraction reflection R at the far end of the cavity. Note that while the Stokes and second-Stokes fields are predominantly driven by stimulated Raman scattering, these wavelengths also experience gain in the Yb fiber sections owing to the very broad emission cross-section of Yb; the factors fs and fss quantify the relative gain for the Stokes and second-Stokes wavelengths with respect to the gain at the fundamental.
To solve the model equations, they were discretized into first-order-accurate difference equations, which were then solved with an adaptive-step-size Runge Kutta integrator to find the steady-state behaviour. Table 1 lists the definitions and values of the input parameters that appear in the model equations, where the subscripts Yb, Ge, and SMF refer to each type of fiber.
Table 1. Input parameters for the model.
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