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We assess the scaling potential of high repetition rate, passively mode-locked erbium-doped soliton lasers. Our analysis focuses on three recently demonstrated lasers using saturable Bragg reflectors (SBR) as the mode-locking element. We use the soliton Area theorem to establish the limitations to increasing the repetition rate based on insufficient intracavity pulse energy, SBR properties, and dispersion engineering. Finally, we examine possible approaches to alleviate these limitations by changing the laser’s structure and composition.
©2008 Optical Society of America
1. Introduction
Sources of short optical pulses at GHz repetition rates are a key enabling technology for an array of applications. In fiber optic communication systems, optical time-division multiplexing schemes recently reached bit rates of 1.28 Tbps per channel [1]. Recent developments in high-speed photodetectors also permit the use of optical signal generators as replacements for high-speed radio-frequency sources. The development of photoconductive switches with ultrafast response times also offers the possibility of optically fed THz radiation generation [2]. Moreover, optical sampling requires both ultrashort pulses for measurement precision and high repetition rates for monitoring fast processes [3]. Optical comb generation also represents a fundamental pillar for ultraprecise clock signal synthesis and frequency metrology [4].
One of the fundamental approaches in designing sources able to provide ultrashort pulses at high repetition rates is mode-locking, a process known almost as long as the laser itself [4]. Fundamental mode-locking generates optical pulse trains with a period corresponding to the cavity roundtrip time, while harmonic mode-locking allows multiple regularly spaced pulses to circulate in the cavity, as long as their repetition rate corresponds to an integer multiple of the fundamental frequency. While the latter method allows higher repetition rates for a given cavity length, maintaining stable intrapulse spacing is often difficult, resulting in increased timing jitter, lack of coherence in between subsequent pulses, and noise. Mode-locking can be active, in which case an external signal source is used to generate the modulation window, or passive, which relies on an intrinsic intensity-dependent process which favours the build-up of pulses. Passive mode-locking offers substantial benefits of being simple, self-contained within the laser design and of avoiding the introduction of external noise and jitter. For reviews of mode-locking, the reader is referred to [6, 7]. Soliton operation in a mode-locked laser occurs when the combined effects of anomalous 2nd order dispersion and positive self-phase modulation dominate the pulse shaping process [8].
Mode-locked lasers employing soliton effects have already been extensively investigated on both the experimental and theoretical level [9-11]. As a result, a passively and fundamentally mode-locked soliton laser represents a very attractive design for ultrashort optical pulse sources exhibiting excellent jitter characteristics. Wavelength compatibility with fiber optic transmission, along with the fact that the long upper-state lifetime of erbium suppresses sub-millisecond pulse dynamics constitutes a major argument in favour of the 1.5 µm operating wavelength regime and the use of an erbium-doped gain medium. The use of saturable absorbers as mode-locking elements is preferred due to their low saturation energy fluence and compactness. The inherent simplicity of such a laser makes the design especially interesting for implementation in integrated optics. Nevertheless, it becomes increasingly difficult to scale the repetition rates of such lasers towards the high GHz regime. As the cavity (and thereby, the gain medium) becomes shorter and the pulse train period decreases, peak powers drop, resulting in smaller nonlinear phase shifts. To maintain soliton operation, this requires correspondingly smaller net cavity dispersion values. Another design issue originates from the fact that for shorter cavity lengths the dispersion introduced by the saturable absorber used for mode-locking may begin to dominate over the waveguide dispersion. Finally, one has to consider the evolution of not only the pulse peak power, but the intracavity average power as the absorber exhibits an energy saturation threshold, below which a sustained mode-locking is impossible to maintain. As a result, the two main issues to consider in scaling such lasers towards higher repetition rates are power level and dispersion engineering.
In this paper, we focus on determining simple, fundamental limitations to increasing the repetition rate of erbium-doped, soliton lasers passively modelocked by a saturable absorber. Our analysis considers three experimentally demonstrated lasers, differing by the dispersion engineering method used to achieve a net anomalous dispersion. In addition, we also examine the impact of SBR dispersion and reduced losses on the scaling potential. The theoretical extrapolation on the lasers’ behaviour for higher repetition rates is done through the soliton Area theorem that allows us to infer the evolution of pulse peak powers and widths, along with the required intracavity dispersion. In all cases, we demonstrate that reducing the cavity lengths eventually decreases the pulse energy along with increasing pulse widths beyond levels acceptable for soliton propagation. Further limitations originate from the SBR saturation limits and the required precision to tailor the intracavity dispersion. Nevertheless, we show that at least one of the experimentally demonstrated lasers has the potential to reach well into GHz repetition rates through simple dispersion engineering by relying on an anomalously dispersive gain medium combined with a positively dispersive SBR. In fact, such a configuration is established to be the most favourable for high repetition rate soliton lasers. We close our analysis by a brief discussion of some possibilities to overcome the aforementioned limitations.
2. Analytical approach
A simplified schematic of our general laser model along with the relevant parameters is shown in Fig. 1. below. We assume a cavity composed of a passive section as well as a distinct gain medium. Each section has its individual effective area, nonlinear index, and dispersion value. However, the reader will note that in two of three lasers subsequently considered in the analysis there is no intracavity passive waveguide, as the laser consists solely of a gain medium. As we focus our analysis on steady state soliton operation, the contribution of the SBR is reduced to a lumped second order dispersion. Losses are assumed to be lumped as well.
Tables 1-4;. List of parameters characterizing the laser
We first consider a laser’s steady-state operation from the average intracavity power (Pcaν) perspective. Pcaν is governed by the small signal gain per unit length g0 as well as the lumped losses αL:
where Psat is the gain medium saturation power, defined through the effective area AeffG, upper state lifetime τl, as well as emission and absorption cross sections σeff and σabs:
As we operate in the fundamental soliton regime, the pulse shape A(t) is given by (ignoring the nonlinear phase shift term):
Here, the pulse peak power Ppeak is easily obtained through:
where τ is the full width half maximum (measured) pulse width. For sech2 pulses, τ≈1.76 τ 0. We then use the fundamental soliton relation:
where λ is the central wavelength, Aeff the waveguide’s effective area, β2 the chromatic dispersion and δ the Kerr coefficient defined by δ=2π·n2·λ-1.
By relating the desired repetition rate (and thereby the cavity length) to the available intracavity power, as well as the resulting pulse width and required dispersion, the three above relations govern our subsequent analysis to assess the scaling potential of a laser. Finally, the above analysis will be done for different values of intracavity losses and SBR dispersion so as to reflect different designs. The latter is especially relevant insofar as saturable absorber mirrors incorporating dispersion compensation were recently demonstrated in the context of laser cavity dispersion engineering [12, 13].
3. Anomalously dispersive gain medium laser
3.1 Experimental setup and results
One possible approach consists of using an anomalously dispersive gain medium so as to alleviate the need for the SBR to provide the required anomalous dispersion. We recently demonstrated such a laser with a 491 MHz repetition rate [14]. The cavity was composed solely of a 20.7-cm long anomalously-dispersive erbium-doped fiber butt-coupled to a saturable absorber with a 6% modulation depth and a 50 µJ/cm2 saturation fluence. The optical spectrum indicates the possibility of 180 fs transform-limited pulses at an intracavity power of 120 mW. Figure 2 below depicts the reflectance and dispersion of the SBR used in the laser, while Tables 5-7 indicate relevant parameters.
Tables 5-7;. Estimated parameters of the 491 MHz soliton laser.
3.2 Scaling analysis and results
Using the approach described in Section 2 we project the evolution of both the pulse peak power and transform-limited pulse width as a function of the repetition rate (inversely proportional to the fiber length). As the dispersion curve of the SBR is varied between ±2000 fs2, in increments of 1000 fs2, we examine the predicted behavior of the laser’s peak pulse power and pulsewidth. The results are shown in Fig. 2. Next, for Fig. 3., we fix the SBR’s dispersion at -750 fs2 and sweep the losses between 5 and 25% in 5% increments.
Fig. 2. (a) Soliton peak power and (b) resulting pulse width as a function of the repetition rate for different SBR dispersion values. Dots indicate soliton modelocking failure points, while arrows indicate experimentally demonstrated values.
Fig. 3. (a) Soliton peak power and (b) resulting pulse width as a function of the repetition rate for different intracavity losses assuming a -750 fs2 dispersion contribution from the SBR.
Two distinct scaling limitations arise from the above analysis. The first one occurs when the lasing wavelength falls into the regime where the SBR exhibits positive dispersion (here, for example, +1000 fs2 and +2000 fs2 correspond to curves exhibiting an inflection point). Here, shortening the cavity length eventually results in the net intracavity dispersion (including the contribution from the SBR) shifting to positive values, where soliton operation is simply impossible. Interestingly, it is important to note the existence of a region occurring after the inflection point of the curves in Fig. 2. and prior to the soliton modelocking failure, where short, high peak power pulses can be achieved for high repetition rates. There, the laser cavity exhibits a small yet anomalous intracavity dispersion due to the positive SBR dispersion cancelling out almost all of the waveguide dispersion. As a result, short pulses could potentially be sustained despite a relatively low average power as long as the pulse energy exceeds the SBR saturation energy.
In the case where the SBR contributes anomalous dispersion to the cavity, soliton operation can in principle be sustained for any cavity length. Nevertheless, the limitation in this case to further increasing the repetition rate is due to an insufficient peak power due to excessively long pulses – a GHz regime for this laser would imply picosecond pulses with merely tens of watts of peak power. At this point, intracavity energy levels can simply be insufficient to saturate the SBR, and preventing the sustainability of a stable pulse train. More importantly, low power levels also imply that soliton pulse shaping becomes eventually negligible, in which case our analytical assumptions cease being valid.
Finally, Fig.3. depicts the impact of different cavity losses on the laser assuming a -750 fs2 dispersion contribution from the SBR. We can see that reducing the losses from 15% down to 5% would allow raising the repetition rate by a factor of 2 while keeping the same pulse peak power and width. However, even such a moderate reduction in losses would be difficult to implement, as it would require using an SBR with a lower modulation depth (in an SBR, the modulation depth and linear losses are of the same order). This in turn would reduce the discrimination of intracavity noise in favour of the pulses, potentially affecting the laser’s stability.
4. Anomalously dispersive high gain laser
4.1 Experimental setup and results
The analysis of the laser in Section 3 demonstrated that an anomalously dispersive gain medium with a positive dispersion contribution from the SBR yields an attractive configuration for a soliton laser. Such a combination makes it possible to reduce the net cavity dispersion via the SBR, thereby allowing for smaller nonlinear phase shifts. As such, it mitigates the drawbacks of the previous experimentally demonstrated laser. We recently demonstrated a configuration implementing this dispersion engineering approach [15]. The gain medium was a 3 cm-long highly doped erbium-ytterbium fiber (NP Photonics). The SBR was chosen so that the dispersion of the SBR is slightly positive with ~1500 fs2 to compensate part of the negative dispersion from the gain fiber. The SBR also has a 6% modulation depth, a 2 ps recovery time, and a saturation fluence of 50 µJ/cm2.
Tables 8-10;. Estimated parameters of the 3 GHz soliton laser
4.2 Scaling analysis and results
Here, we focus on the effect of different positive SBR dispersion values on the scaling potential of this laser. As seen in Section 3, this allows to operate in a region of short pulse solitons despite low intracavity power. Figure 5 depicts the pulse peak powers and widths for SBR dispersion values in between 0 fs2 and +1000 fs2, in 250 fs2 intervals.
Fig. 5. (a) Soliton peak power and (b) resulting pulse width as a function of the repetition rate for different SBR dispersion values. Dots indicate soliton modelocking failure points, while arrows indicate experimentally demonstrated values.
As in Section 4, one limitation in scaling this laser arises from the net intracavity dispersion eventually shifting towards positive values due to the SBR, manifesting through the soliton modelocking break points on the curves. Nevertheless, significantly higher repetition rates can be achieved as compared to the previously examined laser. Indeed, this laser exhibits a higher small signal gain due to a higher Er3+ concentration as well as stronger pumping. As a result, the repetition rate of the laser can be increased while still maintaining reasonable peak power levels. Moreover, the configuration demonstrated here implements the concept proposed in the previous section, namely using the SBR to cancel out a significant portion of the intracavity dispersion so as to allow for soliton operation at lower intracavity powers. While our laser reached a 3 GHz repetition rate, careful dispersion engineering resulting in a +250 fs2 contribution from the SBR would potentially allow operating above 20 GHz, while still generating sub-ps pulses.
5. Dispersion-engineered cavity with a normally dispersive gain medium
5.1 Experimental setup and results
One drawback of the previous configurations is their dependence on the SBR dispersion profile. The dispersive contribution of the SBR varies with the reflection wavelength, whereas the laser operating wavelength is determined by the lowest-loss point, itself a function of the erbium gain profile, pumping intensity, and SBR reflectivity. As such, it might be difficult to predict the lasing wavelength precisely enough to engineer an SBR with the appropriate characteristics for a specific target repetition rate. As a result, another possible approach consists of combining normally- and anomalously- dispersive waveguides within the laser cavity to achieve the desired dispersion. Recently, we demonstrated such a laser implemented in integrated silica waveguides and generated 440 fs pulses at 394 MHz [16]. The laser cavity consisted of a 5 cm section of erbium-doped alumino-silicate glass, normally-dispersive waveguide, and a 20 cm section of phosphorous-doped silica waveguide with anomalous dispersion was used to obtain a net anomalous intracavity dispersion to enable soliton mode-locking. Relevant laser parameters and measured pulse characteristics are summarized in Tables 11-14. A loop mirror was used at one end to provide 10% output coupling, while the other end is butt-coupled to an external SBR. The SBR was a commercial unit with 14% modulation depth, a 2 ps recovery time, and a saturation fluence of 25 µJ/cm2 (BATOP GmbH, model SAM-1550-23-x-2ps). The schematic of the laser setup is depicted in Fig. 6 with the inset depicting the dispersion and reflectance of the SBR. The SBR contributed an additional positive dispersion of approximately 1000 fs2.
Tables 11-14;. Estimted parameters of the waveguide soliton laser.
5.2. Scaling analysis and results
In order to simulate the behaviour of this laser for different repetition rates, we follow the same analytical method as in Section 3 and Section 4, namely extrapolating the intracavity power based on the gain medium length, and inferring the resulting pulse characteristic for different values of SBR dispersion. One important difference here is the additional ability to dispersion-engineer the cavity by adjusting the relative lengths of the active and passive waveguides. This, in turn, affects the length of the gain medium, and consequently the intracavity power. However, the soliton condition relates the intracavity power, the pulse width and the dispersion. The relation becomes therefore recursive and requires us to fix one parameter while finding the solution numerically. As we emphasize here the concept of dispersion engineering, we chose the pulse width to be 300 fs, and examine the resulting peak powers and required net roundtrip dispersion in the laser to sustain such pulses. The results are depicted in Fig. 7.
Fig. 7. (a) Soliton peak power and (b) total intracavity dispersion as a function of the repetition rate for different SBR dispersion values.
As in the previous configurations, the result clearly depicts the importance of the SBR’s dispersion on the laser. If the SBR’s dispersion is positive, a minimum length of the passive, anomalously dispersive waveguide is necessary to keep the overall roundtrip dispersion below zero. Nevertheless, to overcome the losses we then also need a section of erbium-doped waveguide whose normal dispersion needs to be compensated by an additional length of silica waveguide. In our case, the +1000 fs2 contribution from the SBR is nullified by 40 mm effective length of silica waveguide, yielding a theoretical 5 GHz maximum repetition rate – although as seen in Fig.7., the laser’s actual highest possible repetition rate is 3 GHz due to the aforementioned necessity of an additional erbium-doped section. Should the SBR be dispersionless or provide anomalous dispersion, this particular limitation disappears, as described in Section 3. The laser’s maximum repetition rate will be restrained by low power levels (Fig. 7(a).) which will eventually prevent the SBR from saturating, in addition to rendering soliton shaping negligible in comparison to other pulse shaping processes.
6. Possible alleviation of laser parameter constraints
In Sections 3 though 5 we analyzed the repetition rate scaling potential of our recently demonstrated soliton lasers. We chose to examine the impact of varying the SBR’s dispersion and losses, as these are arguably the two easiest experimental parameters to adjust. Nevertheless, a more general scaling potential study has to also address a range of other parameters based on the assumptions we initially established. First, we examine the losses and gains of the considered lasers. Experimentally, our configurations exhibited losses hovering around 10-25%, and we examined the impact of lowering them down to 5 % in Sections 3 and 4. In the integrated waveguide laser case the host material is low-loss silica, and sub-1% output couplers can be easily implemented. However, a substantial amount of the total loss originates from the saturable absorber. Usually, these have an unsaturated loss roughly equal to their modulation depth, and a few percent modulation is usually required to sustain mode-locking. As a result, unless a different mode-locking process is used, it is difficult to expect the total roundtrip losses to reach values below 1-5%.
Secondly, our lasers used erbium-doped gain media with small signal gains ranging from 0.5 to 4 dB/cm. As was apparent with the configuration examined in Section 4, a substantial improvement in the intracavity power can be achieved through an increase of the gain per unit length of the erbium-doped glass. Recently, D. Patel et al. demonstrated a 4.1 dB, 3-mm long Erbium-Ytterbium waveguide amplifier [17]; such a gain figure (13 dB/cm) combined with proper dispersion engineering will enable lasers to easily reach repetition rates in the GHz regime. Resonant pumping could also provide a significant increase in the intracavity energy. Arguably the most promising means to allow the design of ultrahigh repetition rate integrated soliton lasers would come through the use of heavily non-linear host glasses. Chalcogenide and silicon glasses have recently attracted a lot of interest due to their nonlinear index being orders of magnitude higher than silica, and most importantly they both have been successfully demonstrated to hold planar waveguide erbium-doped amplifiers [18, 19]. These recent results potentially offer the key towards novel, homogenous integrated pulsed sources. One problem for sustaining soliton propagation lies in the fact that, as opposed to silica, such glasses are strongly dispersive in the normal regime (β2>0). Nevertheless, waveguide dispersion engineering recently allowed shifting the net dispersion in both chalcogenide [20] and silicon [21, 22] waveguides towards anomalous values at 1.55 µm. Table 15 summarizes the linear and nonlinear refractive indices, material dispersion, and loss of these glasses along with a comparison to silica around 1.55 µm [20, 21].
Table 15. Optical properties of silica, arsenic trisulfide, and silicon glass waveguide around 1.55 um.
7. Discussion and conclusion
In this paper, we discussed the repetition rate scaling of three experimentally demonstrated passively mode-locked integrated soliton lasers. The considered configurations differed mainly through dispersion engineering due to either positive or negative contribution of the SBR. Moreover, an integrated laser was also discussed in which two sections of waveguides with opposite dispersion signs were used to tailor the net intracavity dispersion. In order to emulate experimentally-adjustable parameters, we considered different dispersive contributions from the SBR and examined the impact of lowering total roundtrip losses. We used the soliton Area theorem to assess the evolution of the peak intracavity pulse power pulse width, and overall net intracavity dispersion for increasing repetition rates.
Our analysis clearly shows that low intracavity power along with increasing pulse width represents the major limitation in scaling these lasers. In fact, picosecond long pulses with peak powers on the order of Watts simply do not acquire sufficient nonlinear phase shifts to be considered solitons in the first place. Moreover, a minimum amount of intracavity power is required to saturate the SBR in order to establish a steady state soliton regime. To date, the lowest demonstrated SBR saturation fluencies have been values of ~1 µJ/cm2 [23]. Nevertheless, we have established that precise dispersion engineering with the contribution of the SBR can be used to allow short, high peak power pulses despite limited intracavity energy. In this case, the main issues will come from controlling the overall dispersion profile to fs2 or sub-fs2 precision over the entire pulse bandwidth, thereby requiring a quasi-complete elimination of higher order dispersion terms.
Nevertheless, in each of the considered cases, it is clear that higher repetition rates, into the GHz regime, are readily achievable. Amongst the lasers presented here, the favoured configuration consists of a high gain, anomalously dispersive gain medium in conjunction with a normally dispersive mode-locking element so as to minimize the net intracavity dispersion while still allowing for soliton operation. Alternatively, one could use an equivalent normally dispersive gain medium, and rely on the SBR to provide net anomalous intracavity dispersion. Further improvements can be obtained by reducing the losses, using a higher gain medium, or considering a host material with a higher nonlinear index.
Acknowledgments
This work was supported by the Defense Advanced Research Projects Agency under grant HR0011-05-C-0155, and Air Force Office of Scientific Research under grant FA9550-07-1-0014. D. Pudo acknowledges support by the Natural Sciences and Engineering Research Council Canada.
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