1. Introduction

The need to compensate group-velocity dispersion (GVD) is ubiquitous in femtosecond pulse generation and propagation. Prisms [1], diffraction gratings [2], and chirped mirrors [3] have all been used to compensate or control GVD. Reliable femtosecond lasers had to await the development of a low-loss means of introducing controllable GVD [1]. Pulse formation in modern femtosecond lasers is dominated by the interplay between nonlinearity and dispersion [4,5]. In all cases of practical interest, a positive (self-focusing) nonlinearity is balanced by anomalous GVD. The need to compensate normal GVD in the laser, along with the balance of nonlinearity in soliton-like pulse shaping, underlies the presence of anomalous GVD in femtosecond lasers.

Most femtosecond lasers have segments of normal and anomalous GVD, so the cavity consists of a dispersion map, and the net or path-averaged cavity dispersion can be normal or anomalous. With large anomalous GVD, soliton-like pulse shaping produces short pulses with little chirp. Some amplitude modulation is required to stabilize the pulse against the periodic perturbations of the laser resonator. Pulse formation and pulse evolution become more complex as the cavity GVD approaches zero, and then becomes normal. The master-equation treatment of solid-state lasers, based on the assumption of small changes of the pulse as it traverses cavity elements, shows that stable pulses can be formed with net normal GVD [5]. Nonlinear phase accumulation, coupled with normal GVD, chirps the pulse. The resulting spectral broadening is balanced by gain-narrowing. By cutting off the wings of the spectrum, gain dispersion shapes the temporal profile of the chirped pulse. Proctor et al showed that the resulting pulses are long and highly-chirped [6], as predicted by the analytic theory [5]. Stable pulse trains can even be produced without dispersion compensation, but the output pulses are picoseconds in duration and deviate substantially from the Fourier-transform limited duration, even after dechirping with anomalous GVD external to the cavity.

Fiber lasers can be constructed entirely of fiber with anomalous GVD, to generate solitons as short as ~200 fs in duration. However, the pulse energy is restricted by the soliton area theorem and spectral sidebands [7] to ~0.1 nJ. Much higher energies are obtained when the laser has segments of normal and anomalous GVD. In general, the pulse breathes (i.e., the pulse duration varies periodically) as it traverses the cavity. Dispersion-managed solitons are observed as the net GVD varies from small and anomalous to small and normal [8], and self-similar [9] and wave-breaking-free [10] pulses are observed with larger normal GVD. The large changes in the pulse as it traverses the laser preclude an accurate analytical treatment, so numerical simulations are employed to study these modes. Among fiber lasers, Yb-based lasers have produced the highest femtosecond-pulse energies, recently reaching 15–20 nJ [11]. The normal GVD of single-mode fiber (SMF) around 1 µm wavelength has been compensated by diffraction gratings, which detract from the benefits of the waveguide medium.

With the goal of building integrated fiber lasers, microstructure fibers [12,13] and fiber Bragg gratings [14] have been implemented to compensate dispersion at 1 µm. However, performance is sacrificed compared to lasers that employ diffraction gratings. From a practical point of view, it would be highly desirable to design femtosecond-pulse fiber lasers without compensation of the GVD of several meters of fiber. However, to our knowledge there is no prior report of any laser that generates ~100-fs pulses without elements that provide anomalous GVD in the cavity.

Recently, Buckley et al. showed that the introduction of a frequency filter stabilizes modelocked operation of a Yb-doped fiber laser with normal cavity GVD (~0.015 ps 2), which allows the routine generation of 15-nJ pulses as short as 55 fs [15]. The frequency filter produces self-amplitude modulation, which allows nonlinear polarization evolution (NPE) to be biased for higher pulse energies. By altering the laser cavity to operate at large normal GVD (0.04–0.10 ps ²), the frequency filter was found to stabilize modelocked operation characterized by highly chirped, nearly static pulses as predicted by the theory of self-similar lasers [9]. Although Buckley et al. succeeded in enhancing the stability of modelocking at large normal GVD, the laser still required some dispersion compensation with a grating pair.

Here we describe a femtosecond fiber laser with a cavity consisting only of elements with normal GVD. By increasing the nonlinear phase shift accumulated by the pulse and inserting a spectral filter in the cavity, self-amplitude modulation via spectral filtering is enhanced. The laser generates chirped picosecond pulses, which are dechirped to 170 fs outside the laser. These results are remarkable considering that the cavity consists of ~10 characteristic dispersion lengths of fiber with respect to the dechirped pulse, yet no dispersion control is provided. The pulse energy is 1–3 nJ, and the laser is stable and self-starting. The laser is thus a first step in a new approach to modelocking. Systematic understanding of the pulse-shaping and evolution will be interesting scientifically, and the freedom from anomalous dispersion offers significant practical advantages.

2. Design rationale and numerical simulations

The design of a femtosecond fiber laser without dispersion control in the cavity exploits the understanding gained by the recent work of Buckley et al. [15]. The master-equation analysis does not apply quantitatively to fiber lasers, but we are guided qualitatively and intuitively by its predictions. The key elements of such a laser (Fig. 1(a)) are a fairly long segment of SMF, a short segment of gain fiber, a segment of SMF after the gain fiber, and components that produce self-amplitude modulation. A significant nonlinear phase shift is impressed on the pulse in the SMF that follows the gain, and NPE converts the differential phase shift to amplitude modulation. Numerical simulations show that stable solutions do exist in such a laser, for a reasonable range of parameters. The gain bandwidth has a major influence on the pulse evolution. With large gain bandwidth (>~30 nm), approximately parabolic pulses evolve as in a self-similar laser [9]. As the bandwidth is reduced to ~10 nm, the spectrum develops sharp peaks on its edges, and for narrower bandwidths the solutions do not converge.

 

Fig. 1. Numerical simulation result: a) schematic diagram of the laser. A ring cavity is assumed, so the pulse enters the first SMF after the NPE. Results of numerical simulations are shown on the bottom. Power spectrum (b) and temporal intensity profile (c) after the second SMF.

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Results of simulations with 10-nm gain bandwidth and 2-nJ pulse energy are shown in Fig. 1. The pulse duration increases monotonically in the SMF, and then decreases abruptly in the gain fiber. In the second segment of SMF the pulse duration increases slightly, before dropping again owing to the NPE. The spectrum (Fig. 1(b)) exhibits a characteristic shape, with sharp peaks near its steep edges. The pulse is highly-chirped throughout the cavity, with the duration varying from ~10 to ~20 times the transform limit (Fig. 1(c)).

The simulations show that spectral filtering of a strongly phase-modulated pulse can produce substantial amplitude modulation under realistic conditions. With additional amplitude modulation from NPE, stable solutions exist. The pulse is highly-chirped inside the cavity, but the phase is roughly parabolic near the peak of the pulse, so the pulse can be dechirped outside the laser.

3. Experimental results

The numerical simulations offer a guide to the construction of a laser without anomalous dispersion. The laser (shown schematically in Fig. 2) is similar to the Yb fiber laser of Lim et al. [16], but without the grating pair that provides anomalous GVD in earlier designs. The fiber section consists of ~3 m of SMF and 20 cm of highly-doped Yb gain fiber, followed by another ~1 m of SMF. Gain fiber with a 4-µm core diameter (which is smaller than the 6-µm core of SMF) was chosen to increase self-phase modulation (SPM) in the gain fiber. A 980-nm laser diode delivers ~350mW into the core of the gain fiber. NPE is implemented with quarter-waveplates, a half-waveplate, and a polarizing beamsplitter. The output of laser is taken directly from the NPE ejection port.

 

Fig. 2. Schematic of all-normal-dispersion fiber laser: QWP: quarter-waveplate; HWP: half-waveplate; PBS: polarizing beam splitter; WDM: wavelength-division multiplexer.

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In contrast to the simulations, it is not possible to vary the gain bandwidth easily. An interference filter centered at 1030 nm, with 10 nm bandwidth, is employed. The optimum location for the filter is not clear. Placing it after the gain or second SMF segment would maximize the amplitude modulation from spectral filtering and correspond most closely to the simulations described above. However, we also want to output the broadest spectrum and the largest pulse energy, to achieve the shortest and most intense pulse. Considering these factors, we placed the filter after the beam splitter. This location also allows as much of the laser to be spliced together as possible. The total cavity dispersion is ~0.1 ps ².

The threshold pump power for modelocking is ~300 mW. Self-starting modelocked operation is achieved by adjustment of the waveplates. The laser produces a stable pulse train with 45 MHz repetition rate. Although the continuous-wave output power can be as high as ~200 mW, in modelocked operation the power is limited to 120 mW, which corresponds to a pulse energy of ~3 nJ. Stable single-pulsing is verified with a fast detector down to 500 ps, and by monitoring the interferometric autocorrelation out to delays of ~100 ps. Also, the spectrum is carefully monitored for any modulation that would be consistent with multiple pulses in the cavity. Remarkably, there is no evidence of multi-pulsing at any available pump power. However, with a single pump diode the pump power only exceeds the modelocking threshold by ~20%.

 

Fig. 3. Output of the laser: a) spectrum, b) interferometric autocorrelation of the output, c) interferometric autocorrelation of dechirped pulse and the interferometric autocorrelation of zero-phase Fourier-transform of the spectrum (inset), d) intensity autocorrelation of the dechirped pulse.

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Typical results for the output of the laser are shown in Fig. 3. The spectrum (Fig. 3(a)) is qualitatively similar to the simulated spectrum (Fig. 1(b)) and is consistent with significant SPM within the cavity. The laser generates ~1.4-ps chirped pulses (Fig. 3(b)), which are dechirped to 170 fs (Fig. 3(c and d)) with a pair of diffraction gratings outside the laser. The dechirped pulse duration is within ~16% of the Fourier-transform limit (Fig. 3(c) inset). The interferometric autocorrelation shows noticeable side-lobes, which arise from the steep sides and structure of the spectrum. Nevertheless, these amount to only ~10% of the pulse energy. The output pulse energy is ~2.7 nJ, and after dechirping with lossy gratings the pulse energy is ~1 nJ. Pulse energies of 2 nJ could be obtained by dechirping with high-efficiency gratings or photonic-bandgap fiber. The laser is stable and self-starting. In addition to verifying as carefully as possible that the laser is not multi-pulsing, we compared the pulse peak power to that of a fully-characterized femtosecond laser available in our lab. Within the experimental uncertainties, the two-photon photocurrent induced by the all-normal-dispersion laser scales correctly with the nominal peak power, which is ~5 kW.

Detailed understanding of pulse formation and evolution in this laser will require more experimental work and theoretical analysis. Because the simulated laser is not identical to the experimental version, it is not appropriate to compare the calculated and measured performance in detail. However, qualitative and even semi-quantitative observations of the laser properties are consistent with the intended pulse-shaping through spectral filtering. The behavior of the laser depends critically on the spectral filter: without it, stable pulse trains are not generated. By rotating the spectral filter to vary the center wavelength, either of the sharp spectral features can be suppressed, which may slightly improve the pulse quality. When the spectrum changes, the magnitude of the chirp on the output pulse can change substantially: the pulse duration varies from approximately 1 to 2 ps. With standard femtosecond Yb-doped fiber lasers, mechanical perturbation of the fiber extinguishes modelocking. In the laser described here, we find that it is possible to touch and move the fiber without disrupting modelocking, which indicates that NPE plays a reduced role in pulse-shaping. The simulations (e.g., Fig. 1) show that the role of NPE is reduced compared to a laser with a dispersion map, but it is still crucial to the generation of stable pulses.

4. Conclusion

In conclusion, we have demonstrated a fiber laser that generates reasonably high-quality femtosecond pulses without the use of intracavity dispersion control. The behavior and performance of the laser agree qualitatively with numerical simulations that illustrate the intended pulse-shaping mechanism by enhanced spectral filtering of chirped pulses in the cavity. Nevertheless, our picture of this modelocking process is rudimentary, and more work will be required to obtain a systematic understanding. Improved performance should accompany better understanding of this modelocking process.