1. Introduction

The use of mode-locked fiber lasers as sources of ultrashort pulses of coherent light is now well established. They are ubiquitous in biomedical applications, chemical applications, imaging systems, spectroscopy, optical communication, material processing and metrology. The repetition rate at which the laser emits pulses is often an important parameter for different reasons. In telecom and imaging applications, the repetition rate determines the rate at which the information is transmitted or generated. In material processing applications requiring a given pulse energy, a higher repetition rate leads to a higher average power which could accelerate the process. A higher repetition rate is associated with a larger frequency separation between the longitudinal modes of the laser; this feature is advantageous in many applications of frequency combs such as spectroscopic studies and metrology [1]. On a more practical level, many complex optical systems developed in the past to be used with solid-state lasers are now shifting toward fiber lasers as the primary source of light pulses; for compatibility reasons, it is thus desirable to produce pulses with repetition rates similar to their solid-state counterparts which are typically in the range of 80 to 100 MHz. Consequently, during the last decade or so, different groups came up with fiber laser designs at 1550 nm with repetition rates of 80 MHz or more.

In 2005, Ruehl et al. [2] introduced a 108-MHz laser producing 64-fs 6.2-nJ pulses. This laser was operated in the strong normal dispersion regime and was based on a nonlinear polarization evolution (NPE) mode-locking (ML) mechanism implemented via a free-space section that included a polarization beam splitter and two wave plates. Shortly after, a series of articles on high-repetition rate fiber lasers were published. These lasers were all based on an NPE ML mechanism and they all included a free-space section. Chen at al [3]. presented a 194-MHz, 167-fs and 0.15-nJ soliton laser. Wilken et al. [1] introduced a 250-MHz, 70-fs and 0.12-nJ laser; few details about the oscillator were provided since their pulses were further amplified in another system. Jing et al. [4] reported a 102-MHz, 105-fs and 0.34-nJ soliton laser that involved a free-space section. Zhao et al. [5] operated a laser in the strong normal dispersion regime to obtain 230-MHz, 286-fs and 0.11-nJ pulses. To reach this dispersion regime, they used a highly-doped bismuth-based erbium-doped fiber. Peng et al. [6] came up with a 300-MHz, 93-fs and 0.20-nJ laser. Ma et al. [7] introduced a 225-MHz and 0.31-nJ laser with an impressive pulse duration of 37.4 fs. More recently, in 2013, Krempzek et al. [8] presented a sub-100 fs stretched-pulse erbium-doped all-fiber laser with a repetition rate of 205 MHz followed by a booster amplifier providing 70 mW of average power at the output, which means approximately 0.34-nJ of energy per pulse. In some instances, different ML mechanisms were used. In 2009, Morse at al [9]. obtained 108-fs 0.20-nJ pulses at a repetition rate of 301 MHz using a saturable Bragg reflector and Byun et al. [10] obtained 187-fs, 0.03-nJ pulses at an impressive 967-MHz fundamental repetition rate using the same ML mechanism in an extremely short linear cavity. More recently, repetition rates of several GHz were obtained based on harmonic mode-locking in fiber lasers using different ML mechanisms such as a topological insulator saturable absorber [11] or nonlinear polarization rotation [12, 13]. The quest for extremely high repetition rates is an important one for applications such as frequency combs. However, in other applications, a repetition rate comparable to that of solid-state lasers is sufficient but the pulse energy is more important. This is the case, for instance, for terahertz wave generation. The laser design proposed in this article aims towards this latter type of applications.

Recent conceptual advances in our understanding of mode-locked fiber lasers opened the door to a significant increase of their performances in terms of pulse energy. Since the pionneering work of Zhao et al. [14, 15] and Chong et al. [16], it is known that fiber lasers operated in normal dispersion can support pulses with higher energies before wavebreaking occurs in the so-called dissipative soliton lasers. A good review of this topic is given in [17]. The concept of self-similar evolution in normal-dispersion fibers also led to more energetic pulses [18–20]. The basic idea is that a parabolic pulse acquires a linear chirp during its nonlinear propagation in such a fiber and can thus avoid wavebreaking. The similariton lasers are based on this property. The pulses generated by such lasers have picosecond durations but can be compressed externally down to sub-100-fs durations by compensating their linear chirp with a linear or a nonlinear dispersive delay line. An excellent review of ultrafast fiber lasers based on self-similar pulse evolution was recently published by Chong et al. [21]. In order to reach higher pulse energy in our high-repetition-rate fiber laser, we decided to use this approach. It is important to note that the laser introduced by Ruehl et al. in 2005 [2] was based on self-similar pulse evolution that probably occured within their gain fiber which had normal dispersion. As such, it was the first amplifier similariton erbium-doped fiber laser. At 108 MHz, it was also a high-repetition-rate fiber laser. Inspired by their pioneering work, we introduce in this article a novel design of high-repetition-rate fiber laser based on self-similar pulse propagation and built entirely from standard fibers and fiber-pigtailed optical components. Technological advances in the areas of erbium-doped fibers and semiconductor lasers in the last decade have made our simple cavity design possible. Effectively, the use of a shorter active fiber without affecting the overall gain of the laser is feasible because of the commercial availability of single-mode erbium-doped fibers (EDF) with high concentrations of erbium in combination with high-power single-mode semiconductor lasers at 980 nm used for optical pumping. As a bonus, these fibers have a relatively large normal dispersion at 1550 nm allowing the self-similar behavior of the pulse in the gain fiber.

Our goal was to develop a highly-reliable and easy-to-operate high-repetition rate fiber laser at 1550 nm capable of generating pulses at a 100-MHz repetition rate. Maximizing the pulse energy while keeping the pulse duration below 100 fs was a must. The result of our research is an all-fiber dispersion-mapped amplifier similariton (DMAS) laser. This kind of laser was first introduced and implemented in an ytterbium fiber laser by Renninger et al. [22]. Our erbium-doped laser is passively mode-locked via nonlinear polarization evolution. It is entirely made up of relatively low cost standard commercial optical fiber components. The laser generates 76-fs pulses with 1.17 nJ of energy at 1550 nm at a repetition rate of 100 MHz. Our laser thus produces ultrashort pulses with an energy at least a factor of three larger than all the high-repetition-rate erbium-doped fiber lasers discussed in our introduction with the exception of the laser presented by Ruehl et al. [2] which involved a relatively complex pumping system and a free-space section. This article presents the general configuration of the laser and the characterization of its experimental performance. Numerical simulations confirm that it is indeed a DMAS laser operating in the large normal-dispersion regime. Moreover, we introduce the concept of vector similariton in fiber lasers. A vector similariton is a pulse for which the x- and y- polarization components, that are coupled through nonlinear polarization evolution, have a linear chirp and a combined power profile that evolves self-similarly when the nonlinear asymptotic regime is reached in the gain fiber. This vector similariton behavior is in qualitative agreement with the theoretical analysis of two-component parabolic pulses in optical amplifiers presented by Kruglov et al. in 2008 [23] and we believe it is the first experimental demonstration of such a behavior.

2. General configuration of the cavity

To reach a high fundamental repetition rate, a short cavity is required. The first challenge encountered in this situation is the high number of components required to build a mode-locked fiber laser cavity. One way to combine these components without over-increasing the cavity length is using free-space components in combination with the gain fiber. As mentioned above, this method was used by several groups to build fiber lasers with repetition rates of 100 MHz or more. However, due to the presence of free-space optics, re-alignment is occasionnally required. Taking that into account, it becomes obvious that an all-fiber design is best suited because, once aligned, the all-fiber laser does not require further alignment. This is especially appropriate for a commercial fiber laser design. Then, joining together several fiber-pigtailed optical components when building the cavity becomes the challenge. Effectively, fusion-splicing components requires a certain extra length of fiber (approximately 10 to 20 cm of fiber for each component). This could become restrictive for high-repetition rate cavities. For instance, the length of a 100-MHz cavity such as the one presented here must be 2 m. Combining 5 or 6 optical components could then become difficult. It is important to note however that in recent years many miniature components and hybrid components such as wavelenght division multiplexers, couplers and isolators appeared in the market, making it easier to reach 100-MHz or even 200-MHz fundamental repetition rates in all-fiber designs.

The next challenge is related to pulse duration and energy. The use of a short cavity could become a major concern if the laser is to emit sub-100 fs pulses with energies in the nJ range. The dynamics of such short pulses is influenced by many physical processes among which the most important are the gain associated with the EDF, the dispersive effects and the nonlinear effects occurring in all fibers. As mentionned in the introduction, commercially available highly-doped erbium fibers and semi-conductor laser pumps provide a sufficient gain with a short length of gain fiber. These fibers allow the average power to reach sufficient levels to get 1 nJ pulses with a repetition rate of 100 MHz while keeping the length of gain fiber near 50 cm or so. The effect of dispersion is not so easy to predict. In the 1990’s, it was commonly accepted that a laser cavity should present a net dispersion close to zero to generate short pulses. This was the case in solid-state lasers with intracavity prisms and grating systems controlling this parameter. In the realm of fiber lasers, the stretched-pulse laser introduced by Tamura et al. [24] was the first fiber laser emitting sub-100 fs pulses using dispersion management that provided a net dispersion close to zero. In 2004 and later on, the concept of self-similar pulse propagation was exploited to generate ultrashort pulses in fiber lasers [18–21]. In this case the asymptotic evolution of the pulse in a normal-dispersion fiber governs the dynamics of the laser cavity such that the net dispersion does not play an important role [22]. In 2006, Zhao et al. [14] obtained ultrashort pulses from a cavity in which the dynamics seemed to be controlled mostly by the gain bandwidth in combination with nonlinear effects in the cavity, a phenomenon they dubbed gain-guided solitons. Chong et al. [16] then introduced the all-normal dispersion laser in which a filter can play a major role to generate ultrashort pulses even if the net dispersion is largely normal. This concept was then extended to dissipative soliton lasers in which, more generally, a balance between different effects such as filtering, self-phase and self-amplitude modulation and gain can lead to the formation of ultrashort pulses [17]. Our high-repetition rate fiber laser thus operates in a regime where the net dispersion is largely normal. This regime was achievable mostly because the EDF being used presents a relatively large normal dispersion. Consequently, it was possible to reach a net dispersion largely normal even if most of the other fibers in the cavity had an anomalous dispersion. At first, the expectation that a short cavity would not lead to nonlinear effects as important as would a longer cavity was a major concern, since our design is based on nonlinear polarization evolution to achieve ML [25]. Reduced nonlinearity could have prevented the formation of pulses from noise when the laser is turned on. But, thanks to its small mode-field diameter at 1550 nm, it turned out that the nonlinear parameter of the EDF was high enough to initiate and sustain ML. On the other hand, the fact that the overall nonlinearity of the cavity is low becomes an advantage since the NPE ML mechanism saturates at higher power, leading to larger pulse energy and allowing for the generation of nJ pulses. Finally, and most importantly, it was not a priori obvious that the pulse evolution in such a short gain fiber would reach the asymptotic self-similar behavior. Usually, amplifier similariton lasers make use of long segments of normal-dispersion gain fiber in order to ensure it occurs [21]. As shown in sections 3 and 4, our laser design is able to produce similaritons.

Based on all these considerations, we designed an all-fiber ring cavity which is shown in Fig. 1. The cavity components are, along the CW direction: an EDF pumped by two 980-nm laser diodes injected through wavelength-division multiplexers (WDMs) at both ends, an isolator incorporated in the second WDM, a 50-50 output coupler, a polarizer followed by a Yao-type polarization controller. All these components are commercially available and are fiber-pigtailed. During a roundtrip, the signal travels through lengths of 77 cm of SMF-28 fiber (β₂ = −22 ps²/km, γ = 1.4 × 10⁻³ W⁻¹m⁻¹), 40 cm of Corning HI1060 fiber (β₂ = −5.8 ps²/km, γ = 4.0 × 10⁻³ W⁻¹m⁻¹) and 65 cm of erbium-doped fiber OFS EDF-150 (β₂ = 59 ps²/km, γ = 6.2 × 10⁻³ W⁻¹m⁻¹) that provides gain. The net cavity dispersion is thus 0.020 ps² and the laser operates in the large normal dispersion regime. The ML mechanism is based on NPE within the cavity and is implemented via a polarizer and a manual Yao-type polarization controller. The isolator-WDM hybrid component forces propagation of the 1550-nm signal in the CW direction while injecting a counter-propagating pump signal at 980 nm. The EDF is pumped at both ends to make sure population inversion is achieved throughout the entire length of the highly-doped fiber. The choice of a 50-50 coupler nearly optimizes the performance of our laser as was checked through numerical simulations. It turns out that a higher external coupling prevents lasing because the available non-saturated gain is limited. On the other hand, a lower coupling leads to wavebreaking of the pulse due to the saturation of the ML NPE mechanism when we work at the highest experimental pump power available.

 

Fig. 1 The laser cavity. The different fibers are identified by their colors: Corning SMF-28 (black), OFS EDF-150 (red) and Corning HI1060 (blue).

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The design was targeted to operate in the amplifier similariton regime. The main reason is that the soliton regime and the stretched-pulse regime are known to limit the energy to about a few tenths of nJ. On the other hand, similariton lasers usually produce pulses with nJ energies or more while permitting sub-100-fs durations after dechirping. The dissipative-soliton regime has also been considered but a filter is generally required in this regime. Inserting this component would have strongly impaired the possibility of having an all-fiber laser with a cavity short enough to reach a 100-MHz repetition rate. The NPE ML mechanism was selected for three reasons: simplicity, low cost and effectiveness.

3. Simulations

In order to guarantee that the laser will operate correctly and perform adequately, other aspects must be considered. Self-starting from noise to form the expected pulses is critical in order to facilitate its operation. Moreover, even if most of the cavity parameters are fixed once the cavity is built, some of them can still be varied. We thus have to check if the small-signal gain, the gain saturation energy and the adjustment of the PC will have an impact on the formation or the characteristics of the pulses. We thus ran several numerical simulations to understand the dynamics of the proposed laser.

The simplified model represented in Fig. 2 was used for these simulations. The model considers the effects of gain, group-velocity dispersion, nonlinear self- and cross-phase modulation and the nonlinear Raman effect through a self-frequency shift term. It is similar to the one presented by Olivier et al. in 2006 [26]. Propagation of the slowly-varying field envelopes components A_x(t) and A_y(t) in the CW direction in the fibers is modeled by two coupled generalized nonlinear Schrödinger equations that are solved with the split-step Fourier method. The dispersive and nonlinear parameters associated with the different fibers at 1550 nm were given in the previous section. For the erbium-doped fiber, we considered a lorentzian gain profile with a 5-THz FWHM bandwidth, a small-signal gain g₀ = 3.0 m⁻¹ and a saturation energy E_sat that could be varied to represent the effect of changing the pump power. The polarizer and the PC were represented by a polarizer followed by a quarter-wave plate and a half-wave plate making angles θ_Q and θ_H with respect to the polarizer's axis. A Jones matrix acting on both components of the field was used to calculate the effect of this system.

 

Fig. 2 The simplified model of the cavity used for the simulations. The same color code as in Fig. 1 is used here.

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The output coupler was represented by a multiplication of the field by a scalar. The Raman parameter was fixed to T_R = 3.0 fs [27]. Simulations were started from white noise, i.e. a signal represented by constant spectral amplitude and an arbitrary spectral phase. Convergence was achieved within less than 100 roundtrips in each case. Figure 3 shows the mode-locked pulses obtained directly after the output coupler as E_sat is varied from 1.15 nJ up to 1.60 nJ and the PC angles are kept fixed with values θ_Q = 0.20 rad and θ_H = −0.64 rad. The energy of the pulses varies from 0.79 nJ up to 1.46 nJ, the latter being the highest achievable value since further increasing the saturation energy leads to an unstable behavior. The FWHM duration of the pulses varies from 1.04 ps up to 1.66 ps. At the output of the coupler, the pulse spectra are approximately centered on the gain peak and their width varies from 6 THz to 8 THz. In each case, the steepness of the pulse spectrum wings and the absence of a pronounced dip in its middle tend to confirm that the laser is operating in the similariton regime and not in the soliton, the stretched-pulse nor the dissipative-soliton regimes.

 

Fig. 3 The simulated pulses profiles (top) and spectra (bottom) directly at the output of the 50-50 coupler for different saturation energies. From blue to red Esat = 1.15, 1.30, 1.45 and 1.60 nJ and the energies of the pulses are respectively 0.79, 0.96, 1.17 and 1.46 nJ.

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The fact that the pulses are largely chirped within the cavity allows to reach higher energies in similariton lasers. In such lasers, a necessary procedure to reduce the pulse duration is to compress the pulses externally using a linear dispersive delay line (such as a grating pair) or propagation in a nonlinear dispersive fiber. Figure 4 shows the compressed pulse profiles and spectra obtained using both compression schemes in the case where the pulse energy is 1.17 nJ. As discussed later, this case corresponds to the experimental situation in which the available pump power was limited. In the linear compression scheme, the pulse spectrum remains unchanged and the combined pulse FWHM duration that can be achieved is 120 fs. The two polarization components have comparable energies. In the nonlinear compression scheme, the pulse spectrum develops important peaks at ± 3.6 THz. These features are mainly due to self-phase modulation occurring during the solitonic compression which leads to an FWHM duration of 65 fs. The nonlinear polarization evolution occurring during propagation transfers most of the energy of the pulse in the x polarization component. Finally, close to the point of maximum compression, the nonlinear Raman effect transfers some of the energy from the high frequencies toward the low frequencies, explaining the asymmetry developed by the spectrum.

 

Fig. 4 Pulse profiles and spectra after external compression in the case where the pulse energy is 1.17 nJ. The left column shows the results for compression in a linear dispersive delay line while the right column shows the results for nonlinear compression in a 132-cm length of SMF-28 fiber. On each graph, the x component of polarization is shown in blue, the y polarization component in green and the combined signal in red.

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Figure 5 shows the intracavity dynamics of the pulse for the case of the 1.17-nJ pulse. The behavior of each polarization component of the field is shown. An energy exchange between the polarization components occurs in all the fiber sections but mostly during and after amplification. This exchange is due to the nonlinear polarization evolution that forces the rotation of the polarization state and thus a redistribution of the energy between the polarization components. The pulses are positively chirped throughout the cavity which confirms that the laser is not operating in the stretched-pulse regime. The pulse duration and spectral width of the y-component, that transports most of the energy, increase simultaneously during the amplification in the EDF. This reveals that self-similar behavior is reached. The chirp of the pulse evolves asymptotically toward a constant value of 1.6 THz/ps in the EDF. As explained in the article by Renninger et al. [22], this indicates that the EDF acts as a strong nonlinear attractor and controls most of the pulse evolution that becomes decoupled from the net cavity dispersion. We would thus have a DMAS laser operated in the large normal dispersion regime [22].

 

Fig. 5 The evolution of the energy (E), the FWHM duration (Δt), the chirp and the spectral FWHM (Δf) of the pulse vs the position in the cavity (measured from the output coupler in the clockwise direction). The different fibers along the cavity are represented at the bottom with the same color code as in Fig. 1. The first purple line is the polarizer - waveplates system and the second one is the output coupler. On the graphs, the x- and y-polarization components are shown in blue and green. Red corresponds to their sum (when applicable). Chirp is defined as the slope of the instantaneous frequency at the pulse peak.

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There is no narrowband filter in the cavity. As discussed by Renninger et al. [22], the role of the filter is to stabilize the pulse evolution in DMAS lasers. In our case, this role is played by the NPE ML mechanism. Effectively, one can see in Fig. 5 that when the y-polarized pulse goes through the polarizer - waveplates system, its duration and its spectral width are both reduced significantly. In fact, the effect of this system is to almost completely annihilate the input y-pulse and to transform the input x-pulse into an identical y-pulse at its output.

Another important remark concerns the shape of the pulses that are obtained. As seen in Fig. 3, the temporal profile of the pulses is not exactly parabolic as it is commonly expected for a self-similar behavior. In fact, it has been shown in recent years that similaritons with non-parabolic pulse profiles can exist [28], especially in situations where other physical phenomena such as gain saturation play a role [29]. The gain bandwidth also perturbs the self-similar behavior [20]. The evolution of the pulse and its polarization components in the EDF are shown in Fig. 6. The x-component has a closely triangular shape and the y-component has a more complex shape which looks like a rectangular pulse [28] combined with a triangular pulse on top. Both undergo an increase of their duration and amplitude during propagation. The two polarization components have an almost identical linear chirp that reaches a fixed value of 1.6 THz/ps near the end of the fiber as discussed before. The energy exchange between the x- and y-components renders the comparison with the standard scalar similaritons more difficult to realize. The combined power profile shows self-similar behavior during propagation in the last half of the EDF. Even though the combined profile is not purely parabolic, this behavior is in qualitative agreement with the results presented by Kruglov et al. [23] when they studied the propagation of two-component pulses in optical amplifiers with positive group-velocity dispersion. They found different regimes of similaritons in which the two pulse components have different non-parabolic power profile but their combined power profile is parabolic and evolves self-similarly. Both components have relatively identical linear or quasi-linear chirps. This is in agreement with the behavior of the pulses obtained in our case if we assume that other effects such as gain saturation [29] and gain bandwidth [20], not considered in their model, might have a certain impact on the combined pulse shape and modify it from a purely-parabolic profile. In fact, when the lower-energy cases presented before (E_sat = 1.15 or 1.30 nJ) are analyzed, we find that the combined power profile looks more and more parabolic as E_sat is reduced. It does appear that the amount of nonlinear coupling occurring between the polarization components plays a role in determining the combined power profile. The purely-parabolic pulse presented in the article by Kruglov et al. existed only for a certain range of this coupling parameter. We also note that the unstable behavior obtained when the saturation energy was set higher than 1.60 nJ in our laser was probably due to the polarization instability discussed by Kruglov et al. We thus conclude that the pulse obtained is a vector similariton, a term by which we mean that the coupled x- and y- polarizations of the pulse have an almost identical linear chirp while exchanging some energy and preserving a combined pulse profile which evolves self-similarly as the nonlinear asymptotic regime is reached in the gain fiber.

 

Fig. 6 Evolution of the power and chirp profiles in the EDF: the x-polarized component (blue hue), the y-polarized component (green hue) and the combined power profile (red hue) are shown. In each case, the hue goes from darker to lighter colors as the profile are shown at 25 cm, 35 cm, 45 cm, 55 cm and 65 cm of propagation in the EDF.

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The concept of a stationary pulse with coupled field components evolving in a nonlinear medium has been known for a while in optics. In fact, the general term vector soliton refers to a multidimensional entity that propagates in an invariant or periodic manner in an otherwise destructive environment [30, 31]. Such structures can exist in nonlinear, dispersive and birefringent optical fibers. In a fiber with large birefringence, a temporal pulse composed of both polarization components can propagate as a single vector soliton unit. This is due to the fact that the non-coherent nonlinear coupling between the components can modify their group velocities in such a way as to compensate exactly their group velocity difference associated with the birefringence [32]. This type of pulse is called a group-velocity-locked vector soliton (GVLVS). The polarization state of such a pulse is not fixed, it evolves according to its birefringent environment. In a fiber with small birefringence and in which the group velocity difference of the polarization components can be ignored, the coherent nonlinear coupling between the polarization components can lead to a compensation of the phase velocity difference associated with the birefringence. We then have a stationary pulse solution composed of two orthogonally polarized components that are locked in phase. Such a pulse is called a polarization-locked vector soliton (PLVS) [33]. For such a pulse, the polarizationstate profile is not uniform across the pulse. However, this polarization profile remains invariant during propagation. PLVS have been observed in an anomalous-dispersion fiber laser modelocked with a SESAM [31]. It seems that vector solitons can also exist in the presence of bandwidth-limited gain and other dissipative effects, and also if the group-velocity dispersion is normal. We refer to these pulses as dissipative vector solitons (DVS). Such pulses were observed in a fiber laser with a large net normal dispersion modelocked with a SESAM by Zhang et al. [34]. In this situation, there is no polarizing component inside the laser cavity since a SESAM is used to force modelocking instead of the nonlinear polarization rotation technique. The coupled polarization components of the pulse can thus evolve “freely” and follow the theoretical behavior described in the lines above. By measuring the output polarization state of the pulse over several roundtrips, the authors were able to conclude that a dissipative vector soliton existed in their cavity. Depending on the cavity birefringence, which was adjusted with a polarization controller inside the laser cavity, they were able to find both polarization-rotating and polarization-locked vector solitons. Numerical simulations based on coupled Ginzburg-Landau equations confirmed this behavior and also showed that the existence of DVS in such a cavity depended strongly on the gain bandwidth. Finally, group-velocity-locked dissipative vector solitons were also found in a normal dispersion fiber laser modelocked with a SESAM [35]. In our case, the situation is different. First of all, the presence of a polarizer in the cavity prevents the existence of a DVS that would survive over several roundtrips. The polarization state of the pulse is reset each time it goes through the polarizer and we cannot expect that the polarization state of the pulse will change from one roundtrip to the next. In fact, the vector similariton must be reshaped almost from scratch at each roundtrip. Secondly, the laser is operated in the similariton regime. As explained before, in this regime the evolution of the pulse is controlled mainly by the normal-dispersion gain fiber in which a nonlinear attractor fixes the asymptotic evolution of the pulse. The evolution is thus decoupled from the average properties of the cavity as is the case for dissipative solitons. Thirdly, our numerical simulations showed that the pulse forming in the gain fiber behaves self-similarly in the last portion of the fiber if we consider the combined power profile of the x- and y- polarization components. Each component does not individually behave self-similarly. The polarization state of the pulse thus varies across the pulse and is not invariant during the propagation. These features clearly distinguish the vector similariton presented here from the DVS discussed previously in the literature. The closest theoretical description that we found was the one given by Kruglov et al. [23] concerning parabolic and quasi-parabolic two-component coupled propagating regimes in optical amplifiers as discussed in the previous paragraph.

The results obtained in this section show that the proposed cavity design could yield pulses with an energy of 1 nJ or more and a sub-100 fs dechirped duration at a repetition rate of 100 MHz. It was also established that an amplifier vector similariton forms in the EDF section of this dispersion-mapped fiber laser in the strong normal dispersion regime.

4. Experimental results

A laser cavity was built according to the design shown in Fig. 1. For a combined pump power of 1500 mW and a given adjustment of the PC, stable self-starting ML was obtained. The length of the external SMF-28 fiber was adjusted to 30 cm to characterize the output of the laser before any significant compression occurs, thus limiting the nonlinear effects. It was also adjusted to 132 cm in order to compress the pulses as much as possible. An angle-polished connector (APC) was used at this end of the fiber in order to avoid any back reflection that could destabilize the laser. The output average power was measured to be 117 mW. Characterization of the signal is shown in Fig. 7. The numerical results obtained in the previous sections for the case of a 1.17-nJ pulse after propagation in a 132-cm external SMF-28 fiber are also shown for comparison and are consistent with the experimental results.

 

Fig. 7 Experimental results (blue) in comparison with simulations results (red) for the case of the 1.17-nJ pulse. On top, the pulse train measured with a 3 GHz photodiode. The left column shows the autocorrelation trace and the optical spectrum (log scale covering from 10^-5 up to 10²) after a propagation of 30 cm in the external SMF-28 fiber, i.e. almost directly at the output of the laser. The right column shows the autocorrelation trace and the optical spectrum (log scale covering from 10^-5 up to 10²) after 132 cm of propagation in the same fiber, i.e. near the point of maximum compression.

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The pulse train has a repetition rate of 100 MHz corresponding to the fundamental repetition rate of the cavity. Near the output of the laser, the autocorrelation trace has an FWHM of 1.58 ps which means a pulse with an FWHM of 1.12 ps (assuming a gaussian profile). This is in good agreement with the simulations results as can be seen by comparing the red curve with the blue one. At the point of maximum compression, the experimental autocorrelation trace has an FWHM of 107 fs. Assuming a Gaussian pulse profile, this value leads to a pulse FWHM duration of 76 fs. Again, the simulation result are in relative agreement with the experimental results. The pedestal accompanying the main peak is formed during the compression of the pulse. It is associated with the presence of much smaller secondary pulses forming on both sides of the main pulse as was seen in the simulations results in Fig. 4. As the mean pulse undergoes a significant Raman self-frequency shift, it acquires a different group velocity near the optimal compression point and starts to move relative to the secondary pulses. The non-symmetric positions of the secondary pulses with respect to the main pulse lead to a flattened pedestal instead of two distinct sidelobes as would be the case in the absence of the Raman effect. Since the AC intensity of the pedestal is about 5 times smaller than the main peak, its presence should not have a significant impact on most applications. The autocorrelation trace was also measured for a full range of 100 ps and no other pulse was detected. We conclude that a single pulse exists in the cavity and its energy is about 1.17 nJ after the output.

Finally, the spectrum after propagation in 30 cm of SMF-28 fiber has an FWHM of 53 nm. The spectrum looks like a similariton spectrum but it has an important asymmetry similar to the one presented in Ruehl et al. [36]. As mentionned in their paper, this asymmetry is probably due to intrapulse Raman scattering which transfers energy from the short to the long wavelenghts. Our simulations were not able to reproduce this feature adequately because they only included a self-frequency shift term [27]. In our laser cavity, a picosecond pulse with a large chirp and a wide spectrum is propagating so an exact treatment of the Raman scattering as presented in [37], for instance, would be required but would make the simulations much more complex. After propagation in 132 cm of SMF-28 fiber, the spectrum shows a central peak around 1555 nm with important peaks at 1528 and 1603 nm, similar but not exactly like the ones obtained and discussed in the context of simulations. Again, the asymmetry of the spectrum is more pronounced in the experimental case. We believe this is related to the original asymmetry of the pulse at the output of the laser which is enhanced by the Raman scattering occurring near the point of maximum compression. Probably other perturbations occuring in the cavity or during external propagation such as third-order dispersion, parasitic filtering component or polarizing component not included in the simulations have also a certain role. Once generated, this ML regime is stable. The pulse train and pulse characteristics remain unaltered for several hours. The laser can also be turned on and off as often as desired and the same emission regime is always obtained. Finally, in this experiment, the energy of the pulses was limited by the available pump power. No sign of wavebreaking was observed. The simulations show that the pulse energy could probably be increased by about 25% would the appropriate pump power be available. The agreement between the simulations and the experimental results shown in this section support our interpretation of the existence of a vector similariton in this laser cavity. However, a deeper investigation of vector similaritons to further confirm their existence at the experimental level will be required because the evidence shown here is rather indirect.

5. Conclusion

To sum up, a self-starting passively mode-locked amplifier similariton all-fiber laser yielding energetic ultrashort pulses at a high repetition rate was developed at 1550 nm. This laser emits 76-fs pulses with an energy of 1.17 nJ at a repetition rate of 100 MHz. It is easy and relatively inexpensive to implement since it is built from standard optical components. Its all-fiber design makes it robust and easy to use since no alignment of optical components is required once the cavity is built. Its performance makes it a valuable potential tool for different applications. The fact that an amplifier similariton could be observed in such a short length of erbium-doped fiber (65 cm) is remarkable since usually fibers of at least a few meters are required for the pulse to reach this asymptotic regime in fiber lasers. We have also emphasized that to understand the self-similar behavior of the pulse in this laser one must consider both of its polarization components which are coupled through nonlinear polarization rotation and we have thus introduced the concept of vector similariton in fiber lasers.

Acknowledgments

Funding was provided by the Canadian Institute for Photonic Innovations (CIPI – TEN grant), the Fonds de recherche du Québec - Nature et technologies (FRQNT - 177503), the Natural Sciences and Engineering Research Council of Canada (NSERC – RGPIN-2015-04753) and TeTechS Inc., Waterloo, Canada. The authors would like to thank Mrs. Nariné Tovmasyan, Mr. Pascal Hogan-Lamarre and Mr. Maxime Hardy for their involvement in different aspects related to this project and Pr. Christophe Finot for providing useful informations.