The generation regimes of an all-fiber passively mode-locked ytterbium laser with intra-cavity photonic crystal fiber have been studied with the aim to provide recipes for obtaining chirp-free sub-picosecond pulses directly from the cavity. Small-beam area photonic-crystal fiber is used for dispersion compensation of the intra-cavity normal dispersion of Yb-doped and single-mode fibers as well as for spectrum expanding due to enhanced nonlinearity. Regions of the gain and fiber parameters near the generation threshold were found in both cases of normal and anomalous net intra-cavity dispersion, which provide a stable generation of ultra-short sub-picosecond pulses directly from the cavity. Laser parameters of a transition to the multi-pulsed generation regimes were also found.
©2006 Optical Society of America
Yb-doped fiber (YDF) lasers have attracted great interest, mainly by virtue of their applications in a variety of spectroscopic applications in material science, biology and medicine [1, 2, 3]. Apart from their broader gain bandwidth as compared to Er-doped fibers, they offer higher output power and better pump power conversion efficiency. Many of the complications which are well-known from Er-doped amplifiers are avoided, such as excited state absorption and concentration quenching and Yb-doped fibers can potentially be utilized for development of short-length, high-power and short-pulse fiber lasers. There is also a wide range of possible pumpwavelengths (from 850 to 1050 nm), allowing a variety of pumping schemes including the use of diode lasers. Yb-doped fiber lasers also offer practical advantages over ultra-short solid-state lasers which provide to date the shortest optical pulses from the laser cavity, including small size, weaker sensitivity to misalignments, low cost.
Significant progress was achieved recently in the development of passively mode-locked Yb-doped fiber lasers [2, 3, 4, 5, 6, 7, 8] as well as in the theoretical analysis of their generation regimes [9, 10]. The main emphasis was placed on the generation dynamics, and the well-known soliton , self-similar pulse  and stretched-pulse generation regimes  were studied. However, the challenge remains of studying the generated pulse characteristics themselves in dependence on laser parameters. It is especially important to find ranges of the parameters which provide the generation of the shortest possible pulses directly from the cavity without additional extra-cavity bulk optical elements for chirp compensation.
In terms of the shortest pulse generation, for both fiber and solid-state passive mode-locked lasers [15, 16], the pulse duration is defined by a complicated balance of gain, gain bandwidth, gain saturation, passive modulator efficiency, intra-cavity nonlinearity and dispersion. For a fixed gain bandwidth and gain saturation energy, the higher the sensitivity of the passive modulator to instantaneous changing of the pulse intensity, the shorter the pulse duration that could be achieved. The nonlinear polarization modulator has a high modulation sensitivity and permits to generate the shortest to date pulses from a Yb fiber laser when an additional external pulse compression is used [7, 8, 13, 14]. In spite of this fact, for passive mode-locking we have chosen a semiconductor saturable modulator (saturable Bragg reflector, SBR) which has relatively slow responce to pulse intensity variations, but is less sensitive to the environmental variations and adjustment [5, 6, 10].
Ultra-short and high-intensity intra-cavity pulses necessarily cause an onset of pulse spectrum broadening due to Kerr nonlinearity and nonlinear spectral self-phase modulation. Additionally, chromatic dispersion of the intra-cavity fibers necessarily causes a spectral phase modulation. Both substantially affect the generated pulse duration. Self-phase modulation and dispersion of the typical silica fibers, as YDF and single-mode fibers (SMF) result in normal group-velocity dispersion of the pulse in the near-IR. For a laser system without additional bulk optical elements used for extra-cavity pulse compression, this modulation should be compensated, possibly in a complete way, to ensure the ultimately shortest duration of the pulses generated directly from the cavity for a given gain bandwidth and modulator efficiency. Intra-cavity fiber-based dispersion compensation is possible by a photonic crystal fiber, the only known type of silica fibers with anomalous dispersion in the near-IR . We have chosen photonic crystal fiber to serve as a dispersion-compensating fiber (DCF) similar to the laser system developed in Refs. [4, 8].
Hence, we have considered a Yb-doped fiber laser with fiber-based semiconductor saturable modulator for the passive mode-locking and with intra-cavity photonic crystal fiber for the dispersion compensation. This system presents a reliable, environmentally-insensitive all-fiber source of ultra-short pulses at 1050 nm suitable for a number of important applications. The motivation of the simulations was to study a fragile balance of numerous factors of pulse shape formation on the basis of a round-trip model of passive mode-locked generation [15, 16, 18] and to reveal the ranges of YDF gain and DCF length which provide cw generation of the shortest possible pulses ~100 fs for the currently available YDF, SMF and SBR. Pulse parameters as a function of the laser parameters were studied in both cases of normal and anomalous intra-cavity dispersion. Intra-cavity fiber parameters and gain were also found which provide the transition to the multi-pulsed generation regime.
2. Round-trip laser model and laser elements characterization
The laser system under consideration includes Yb-doped fiber, single-mode fiber, photonic crystal fiber as a dispersion-compensating fiber, passive modulator, and output coupler (Fig. 1). In the round-trip model of the passive mode-locked laser the generation pulse is formed as a result of multiple uni-directional passes of a seed noise pulse through all intra-cavity elements.
The propagation of the generation pulse in the YDF is described by the nonlinear Ginzburg-Landau equation which includes the effects of dispersion, Kerr nonlinearity, linear loss and saturated gain of the finite bandwidth :
where E(z,t) is the slowly-varying pulse field, β 2 is the group-velocity dispersion parameter, γ is the nonlinear parameter, Γ is the linear loss, z is the distance along the fiber, t is the pulse local time. The Yb-doped fiber is characterized also by the finite gain bandwidth Δλ g with the corresponding temporal parameter τg=2π/(ck 2Δλg) and by the saturated dynamic gain
where g 0 is the small-signal gain defined by the pump power level, 𝓔 sat is the saturation energy, 𝓔 0(z) is the current total pulse energy, k=ω 0/c is the wavenumber, ω 0 is the carrier frequency, c is the velocity of light. The propagation within the SMF and DCF is modeled by the nonlinear Schrödinger equation , which includes the effects of dispersion, Kerr nonlinearity and linear loss (first three terms in the rhs part of Eq. (1)):
As we expect generation of ~100-fs pulses we can ignore higher-order dispersion terms in Eqs. (1) and (3). This is in contrast to the case of Yb lasers using nonlinear polarization evolution for mode locking [7, 8, 13, 14] which generate broader spectra corresponding to sub-50-fs pulses.
In our laser system we use the active fiber with the parameters of commercially available Yb-doped fiber from INO  and single-mode fiber FLEXCOR 1060. The parameters β 2, γ and Γ are different for each of the intra-cavity fibers and are given in Table 1. One should especially note the negative group-velocity dispersion parameter β 2 of the DCF at the carrier wavelength λ0=1050 nm (Fig. 2(a)) and the order-of-magnitude smaller effective area of the fundamental mode A eff (Fig. 2(b)). Therefore, varying the length of DCF one can balance the normal dispersion of the two other intra-cavity fibers and their normal self-phase modulation. Additionally, the DCF has a higher nonlinear parameter γ=kn 2/A eff and contributes substantially into the nonlinear broadening of the generating pulse, as is shown below. Here n 2=2.6×10-20 m2/W is the nonlinear refractive index which we suppose to be the same for all intra-cavity silica fibers.
The saturated Bragg reflector consists of multiple AlGaAs/GaAs layers grown by molecular beam epitaxy on a thick GaAs substrate, which provide high broadband reflection at ~1050 nm, and InGaAs quantum wells, which provide the nonlinear response of the reflection to the intensity variations of the reflected pulse with fast and slow temporal responses. The fast component of the reflection response arises from the thermalization of the bandedge-excited carriers and is faster than ~100 fs. The slow component with characteristic time constant τ slow=10 ps is caused by the nonradiative recombination of the excited carriers.
The combined SBR response is implemented into the round-trip model by the relative changing of the electric field of the pulse field as follows :
Here 𝓔(t) and 𝓔 0 are the instantaneous and total pulse energy, respectively, E peak and t peak are the pulse field peak and its temporal position, and H(t) is the Heaviside function. The parameters σlin, σfast and σslow in Eq. (4) characterize the portion of the SBR reflectivity due to the linear loss, fast and slow saturation, respectively, and can be found by fitting the modulator reflection measured in the pump-probe configuration. For the SBR from BATOP  we have found that low-signal losses related to these effects are equal to 0.086, 0.112 and 0.112 dB, respectively. Finally, we assume that the output loss of the laser system is equal to 0.087 dB, so that the generation threshold of the laser is approximately g 0 L YDF=0.42 dB.
We have modeled the generation of a passive mode-locked fiber laser by multiple solutions successively of Eq. (1) for YDF and corresponding Eqs. (3) for SMF and DCF in the described order (Fig. 1), starting from the initial noise pulse. This was accompanied by field transformation according to Eq. (4) and output losses at each cavity round-trip. We have proceeded for many 1000s of round-trips till steady state was reached with the pulse shape and spectrum unchanging during several 100s of round-trips. Steady-state pulse and all its properties such as shape, spectrum, instantaneous frequency and spectral delay were monitored at different intra-cavity positions.
For the chosen group-velocity dispersion parameters of the fibers (Table 1) the net intra-cavity dispersion parameter LYDF + LSMF + L DCF is zero for the specific DCF length L*DCF ≈ 86.9 cm. We study the laser dynamics in the vicinity of L*DCF for both signs of the net intra-cavity dispersion. Typically when there is a large excess of normal or anomalous dispersion in the cavity the model developed does not give a stable ultra-short pulse generation.
3. Generated pulse parameters of the passively mode-locked fiber laser
Our aim is to explore the role of the photonic crystal fiber with its anomalous dispersion in intra-cavity dispersion compensation and in the spectrum broadening owing to its enhanced nonlinearity. Because of a large number of laser parameters and a large variety of generation regimes, let us first consider the oversimplified case where the nonlinearity in all the fibers is neglected. Then the role of the SBR becomes more transparent.
Figure 3 presents steady-state pulse spectra for the case of the absence of Kerr nonlinearity in all intra-cavity fibers (all nonlinear parameters were supposed to be equal to zero in Eqs. (1–3)). It is seen that for the DCF length close to L*DCF a compensation of intra-cavity dispersion is achieved. When the net intra-cavity dispersion is normal (anomalous) the pulse spectrum has a red (blue) shift from the carrier frequency (inset). The shift is caused by the pulse transformation due to combined fast and slow response of the SBR. According to Eq. (4) the pulse leading front gets the enhancement by the SBR, where red or blue part of the spectrum is located in dependence on the sign of the intra-cavity dispersion. This results in the pulse spectrum shift to the corresponding direction from the carrier frequency which is ultimately restricted by the finite gain bandwidth at the steady-state. For the DCF length close to L*DCF of zero net dispersion the laser generates a steady-state pulse with an extremely broad spectrum that is comparable to the gain linewidth, negligible phase modulation and minimal pulse duration (Fig. 3(b)). This duration is close to the duration of the corresponding transform-limited pulse obtained by a complete elimination of nonlinear spectral phase dependence on the frequency.
Next we focus on the role of the nonlinearity of the DCF. Figure 4 illustrates the cases when the nonlinearities in the YDF and SMF are taken into account but the nonlinearity in the DCF is neglected or is also taken into account (curves 1 and 2, respectively). We note that the former case (curves 1) can be considered as related to a laser that implements the intra-cavity dispersion compensation through a hollow-core photonic crystal fiber with a negligible nonlinearity and anomalous dispersion . When the enhanced nonlinearity of the photonic-crystal fiber is included into consideration, the spectrum of the generating pulse becomes much broader as compared to the case without the nonlinearity. It is especially obvious for the case of normal net intra-cavity dispersion with L DCF<L*DCF (Fig. 4, top row), whereas for the anomalous net dispersion the spectrum broadening owing to the presence of the DCF nonlinearity is smaller. For the latter case the pulse is shorter, as the phase modulation is partially compensated due to the nonlinearity and the dispersion similar to the soliton-pulse regime of the generation . Overall, in both cases the group delay slope corresponds to the sign of the net dispersion. For example, in the case of normal net dispersion the blue part of the pulse spectrum has smaller delay and comes first whereas for anomalous dispersion the red part comes first (rhs column, green lines). In Fig. 4 (top row) we observe the typical spectrum profiles of self-similar pulses [9, 10] with a flat parabolic-like top and steep edges which result from the parabolic gain profile and normal net intra-cavity dispersion. In turn, when the net dispersion is anomalous, the laser demonstrates the stretched-pulse generation regime  with sign changing of the pulse chirp after propagation through different fibers and with complete chirp compensation at some positions in the cavity for a definite choice of the laser parameters (see below Fig. 7).
The central result of the laser dynamics simulations is presented in Fig. 5 where the generated pulse duration versus the gain g 0 is shown for different DCF fiber length, when the net intra-cavity dispersion is normal (a) and anomalous (b) and when the nonlinearity in all the fibers is taken into account. The curves are shown at limited gain parameter ranges where the stable pulsed generation takes place. They start at the threshold gain values and end at the gain parameters which give the onset of multi-pulsed generation. The ranges of the stable ultra-short pulse generation are wider in the case of the normal net intra-cavity dispersion. For this case the shorter the DCF length, the larger the gain parameter interval where the stable pulsed generation occurs. However, for decreasing DCF length, beginning from the value of zero net intra-cavity dispersion, the pulse becomes longer and possesses larger phase modulation. For the DCF length closest to that of zero net intra-cavity dispersion the stable generation is obtained in the restricted interval of the gain parameter but with sub-ps duration generated directly from the cavity without additional chirp compensating elements. The duration of the corresponding transform-limited pulses is well below 100 fs.
Generation of sub-ps directly from the cavity takes place also for the DCF lengths which result in anomalous net intra-cavity dispersion, with even shorter pulse duration ~300 fs (Fig. 5(b)). The gain parameter ranges of the stable pulsed generation are much narrower in this case. As the mutual compensation of the phase modulation due to the self-phase modulation and anomalous dispersion partially occurs, the durations of the pulse and the corresponding transform-limited pulse differ from each other less dramatically. The pulse is shorter in the case of anomalous net dispersion as compared to the case of normal dispersion (cf. red lines in Fig. 5(a) and (b)). For the anomalous dispersion the stretched-pulse generation regime takes place with different signs of the phase modulation in different cavity locations, therefore the phase compensation could be even more complete at some positions inside the cavity and the difference between the pulse and its transform-limited counterpart could be even smaller there.
We note that according to Fig. 4 the dispersion of the intra-cavity fibers plays the dominant role in the steady-state pulse phase modulation over the finite gain bandwidth and passive modulator efficiency. However, the pulse duration itself is a sensitive function of all parameters of the laser such as nonlinearity, dispersion, gain, gain bandwidth, passive modulator parameters. The generation of the shortest possible pulses directly from the cavity and without additional chirp compensation requires a fragile adjustment of all these factors (Fig. 5).
The round-trip model of the passive mode-locked fiber laser incorporates self-consistently the interrelation of both phase effects (typically referred to as a soliton pulse formation by the balance of the dispersion and the nonlinearity) and amplitude effects (owing to the passive mode locking in competition with the gain of finite bandwidth). Varying the amount of the intra-cavity dispersion by DCF fiber length, we have revealed different generation regimes of the laser system. In soliton lasers  a pulse is generated which maintains an almost unchanged temporal shape during one round-trip. A laser in the self-similar regime under normal intra-cavity dispersion [9, 10, 6] generates pulses with spectra of a parabolic top and steep edges, and with linear chirp, increasing during the propagation in the fibers. A stretched-pulse laser [12, 13, 14] has a cavity with fiber sections of normal and anomalous dispersion, causing the phase modulation to change the sign during one round trip.
Actually when one desires to obtain a shortest possible pulse directly from the cavity, the features of these different mechanisms should be combined with the advantage of the pulse shortening due to the passive modulator and the restrictions from the finite gain bandwidth. A precise balance of all effects should be found for this purpose. The shortest pulses are generated at the gain slightly exceeding the threshold (Fig. 5). When the gain exceeds the threshold further, the nonlinearity in the fibers contributes substantially into the pulse phase modulation giving the onset of the multi-pulsed regime and generation instability. A stable cw generation of a single pulse per round-trip is hindered with the gain growth stronger at the stretched-pulse regime due to complexification of the self-phase modulation and inevitable increasing of the stretching ratio at different intra-cavity fibers.
For the shortest pulse generated from the cavity one also has to take into account how the pulse transforms along the cavity. For normal net intra-cavity dispersion the shortest pulse is after propagation through the DCF (Fig. 6). The group delay interval becomes successively smaller after passes of each of the fibers and preserves the positive slope corresponding to the dispersion sign of the fibers (Fig. 6(b)). For anomalous net dispersion (Fig. 7) the pulse chirp could be of opposite signs after propagation of SMF and DCF and for some sets of fibers and gain parameters a complete chirp compensation with an almost transform-limited pulse could occur at some intra-cavity locations (curves 1 in Fig. 7). As a result, for the chosen parameters and the order of uni-directional passes through all three intra-cavity fibers (section 2), a preferable place to output the shortest generated pulse is before and after the YDF for normal and anomalous net dispersion, respectively.
In conclusion, we have modeled the mode-locking pulsed dynamics in an all-fiber Yb-doped laser with a photonic crystal fiber and a saturable Bragg reflector. The simulations were performed for both cases of normal and anomalous net intra-cavity dispersion. We have demonstrated the features of the self-similar pulse and stretched pulse generation as a function of the amount of anomalous group-velocity dispersion introduced by the photonic crystal fiber. It is shown that with a balance of all factors of pulse formation, such as intra-cavity dispersion, nonlinearity, saturated gain and gain bandwidth, and modulation efficiency of the passive modulator, the range of laser parameters could be found which provide cw generation of the pulses as short as 100–200 fs directly from the laser cavity. Thereby the possibility of sub-ps all-optical fiber ytterbium laser without additional extra-cavity chirp-compensating elements is shown. The ranges of laser parameters were also revealed which provide transition to the multi-pulsed generation regime.
This work was supported by the Canadian Institute of Photonic Innovation, Natural Science and Engineering Research Council of Canada, and Research Chair Program. One of the authors (V.P.K.) thanks Dr. Y. Logvin for valuable discussions and suggestions.
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