1. Introduction

Passively mode-locked fiber lasers have the advantage of being potentially compact and portable. These lasers are generally constructed with some method of intra-cavity dispersion compensation. For an all-fiber configuration, standard optical fibers that provide anomalous dispersion to counter the normal dispersion of the gain media are available in the 1550-nm wavelength region [1] but not in the 1060-nm wavelength region. Photonic crystal fibers [2], prisms [3], and free-space gratings [4] have been used for dispersion management over a wide range of wavelengths. With intra-cavity dispersion compensation, net normal-dispersion regime lasers have previously generated hyperbolic secant [1], self-similar (parabolic) [5], Gaussian [6], as well as dark soliton pulse shapes [7]. Buckley, et al, have observed an unpredicted (non-parabolic) mode of operation in their net-normal, dispersion-compensated cavity for repetition rates greater than 15 MHz [8]. For ease of construction and use, as well as to capitalize on the portability of all-fiber configurations, it is desirable to construct a fiber laser without any dispersion compensation. Unfortunately, there has been little experimental work on uncompensated fiber lasers and their performance is not well characterized or understood.

The existence of solitons for straight fiber propagation with gain and spectral filtering has been proposed in both the anomalous and normal dispersion regimes [e.g., 9]. Hyperbolic-secant functions, in contrast to parabolic amplitude profiles, produce solitary pulse solutions to the nonlinear Schrödinger equation. The phase profile of a hyperbolic-secant pulse solution is ϕ = C ln[cosh(t/τ)], where C is the chirp parameter and τ is the normalized pulse width. More precisely, the solitary pulse is represented as V ₀ sech(t/τ) exp(iϕ), where V₀ is the amplitude. Analytical evaluation of this functional form for the normal dispersion regime indicates a spectrum with extremely steep edges and an increasingly “square top” for increasingly large chirp parameters [10]. In the well-known anomalous-dispersion mode-locked soliton lasers, self-phase-modulation compensates for the net dispersion in the cavity. Walton, et al, have numerically shown that a passively-modelocked all-normal-dispersion laser is also possible by using an active nonlinear directional coupler, which acts like a nonlinear polarization rotation (NPR) mode-locker [11]. However, it has not been experimentally shown until very recently that a normal-dispersion solitary pulse could be mode-locked in a laser cavity without any dispersion compensation [12,13].

In this paper, we present further concrete evidence as to the solitary pulse nature of normal-dispersion laser outputs. Our laser has a longer cavity length and hence higher dispersion by more than a factor of 2 over previously demonstrated lasers, allowing the output pulse spectra to develop the expected steep spectral edges. Our results clearly show the steep edges and we measure the chirp parameter based on the optical spectrum to characterize this stable operating point. We also characterize both the amplitude and phase of the mode-locked pulses as well as those of the externally compressed pulses, demonstrating good pulse compression to near the transform limit. Additionally, we observe energy quantization in an all-normal-dispersion solitary pulse laser for the first time to our knowledge. Other passively -modelocked soliton lasers of net-anomalous and net-normal dispersions have exhibited energy quantization in the form of multi-pulsing. That is, if the intra-cavity energy is high enough to support more than one fundamental soliton, then the laser will tend to operate with multiple pulses simultaneously in the cavity with discrete jumps in power for each additional pulse [14]. This is in contrast to so-called “parabolic pulses” where the spectrum assumes a parabolic shape and the power of a single-pulse increases as opposed to being shed from the pulse or generating multiple-pulsing [15].

2. Experimental setup

To study the formation of solitary pulses in a laser with no dispersion compensation, we have built an 8.3-MHz tunable Yb-doped fiber laser with only normal-dispersion fibers. The low repetition rate of the laser is advantageous for achieving higher pulse energies in chirped pulse amplified applications, as long as the initially chirped pulses can be compressed. A frequency-resolved optical gating (FROG) setup is constructed to fully characterize the amplitude and phase of the output pulses [16]. Using the phase information, we can numerically compress the pulses to determine the minimum achievable pulse width. We also experimentally demonstrate the compressibility of the laser pulses using free-space gratings. We find excellent agreement between the experimental, numerical, and theoretical minimum pulse widths.

Figure 1 illustrates the configuration of our laser. Mode-locking is achieved by nonlinear polarization rotation [1]. Unlike most other normal-dispersion fiber lasers, it does not have any anomalous dispersion region. We use 4 meters of Yb-doped fiber (INO Yb118), pumped by a 976-nm laser diode as our gain stage. The gain fiber output is sent through a 76-mm free-space optical bench that includes a 10-nm tunable bandpass filter, an isolator, and a set of waveplates (λ/4, λ/2, and λ/4) for polarization control. The laser output is then extracted with the 34% output of a fiber coupler and the remaining 66% is sent back into the gain fiber. An NPR port output is also accessible with the addition of a polarization beam splitter before the isolator. A total additional length of ~20.5 m of Corning HI-1060 fiber contribute to the normal dispersion of the laser cavity. Spectral filtering allows soliton formation in much the same way as in a gain-guided laser [12] and has been shown to be necessary for obtaining stable pulses in this dispersion regime [17]. By angle-tuning the filter, we can select the center wavelength (λ_c) from 1050 to 1080 nm. Qualitative behavior of the laser is similar for all the wavelengths and thus, for these demonstrations we use a fixed λ_c≈1080 nm. Mode-locking is attained by increasing the pump power to above the threshold (>160 mW) and adjusting the polarization controllers. Once the laser is mode-locked, the pump power can be reduced to eliminate any multiple pulsing or continuous-wave (cw) background. Even without active temperature control, the laser is highly stable and once the polarization is set, the laser is completely self-starting, even after powering off for several days.

 

Fig.1. Schematic diagram of 8.3-MHz soliton Yb-doped fiber laser. WDM = wavelength-division multiplexer. BPF = band-pass filter. PBS = polarizing beam splitter. Iso = isolator. AC = autocorrelator. OSA = optical spectrum analyzer.

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In order to show that these pulses meet the criteria of solitons as predicted in Ref. 9, we characterize both the laser output and the NPR port output. We find that the NPR port output is similar to the laser output and thus, focus our reported findings on the laser output. All widths in this paper are reported as the full width at half maximum. Initial autocorrelator and optical spectrum analyzer (OSA) measurements indicate widths of the laser output are τ_AC≈20.8 ps and Δλ=6.2 nm. The combination of such large time and spectral widths suggest a nonlinear phase profile. To characterize both the phase and amplitude profiles of these pulses, we have used FROG with type-II phase-matching as described in Ref. 18. Large chirps are supported by using 1024 × 1024 FROG matrices, and we typically achieve FROG errors < 0.005.

3. Pulse characterization results

Because the normal dispersion solitary pulse solution is always chirped, the “classic” (anomalous dispersion soliton) definitions of soliton parameters are not necessarily applicable. From our experimental results, we find that the normalized intensity [8] is related to the number of pulses in the cavity. This is analogous to the anomalous-dispersion soliton laser cavity where power levels corresponding to an Nth order soliton result in N simultaneous pulses in the cavity. Using the following equations and values for the lowest power pulse that we measured, N = -n₂ k|V₀|² τ ²/(2/β ₂), |V₀|² = P ₀/A_eff, n ₂ = 3.2×10^-20 m ²/W , β ₂ = 6.1×10^-27 s ²/m (experimentally derived using the sideband method [19,20]), P ₀ = 16.3W (peak power), A_eff = 15.5μm ², and τ = 204fs (based on a numerical fit of the FROG measured spectrum to the analytical solution, Eqn. 2 of Ref. 21), the result for this laser is |N|≈0.66.

We compare the measured average pulse output power and number of pulses per round trip with the expected soliton “orders”, as a function of the pump power (shown in Fig. 2).

 

Fig. 2. Average laser pulse output power (left y-axis) and calculated normalized intensity |N| (right y-axis) vs. pump power.

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Fig. 3. Output power vs. pump power. The shaded regions indicate the difference between the total and the pulse power. The output spectra over a 60-dB intensity range for 3 different pump levels are shown.

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Note the discrete jumps in average power of the pulses (i.e., total pulse energy) as additional pulses are allowed to exist simultaneously in the laser cavity. These steps agree well with integer values of the calculated normalized intensity |N|. The total output power always increased with increasing pump. Figure 3 shows the difference between the total and the pulse output power. The inset examples of the spectrum show the presence of the cw background and its absence when the laser generates additional pulses. Between ~130 and 260 mW of pump, the total power continues to increase linearly due to the growth of a cw background, while the pulse power stays relatively constant. A second pulse appears only after the total power exceeds that needed for two |N|≈0.95 pulses (i.e., ~5.4mW). A third pulse occurs at a measured total pulse power of ~7.4 mW (approximately three |N|≈0.87 pulses). For pump powers that produce >3 simultaneous pulses, the mode-locking became unstable. Note that a hysteretic effect [14] is observed as the pump power is decreased. When the total power drops below ~3.9 mW (i.e., |N|≈1.34), or rather, when a pair of pulses drops below |N|≈0.67 for each pulse, a single pulse regime is re-established. Such hysteretic multi-pulsing behavior is expected of solitons.

The intensity and phase of the output pulses from typical FROG measurements (shown in Fig. 4) indicate a pulse width Δτ=13.9 ps and a spectral width Δλ=6.1 nm. This corresponds to a time-bandwidth product (ΔτΔv) of 21.8. Figure 5 shows that the autocorrelation derived from the FROG measurement matches well with that measured by a commercial autocorrelator. Due to the high frequency chirp, the actual time-domain pulse shape evolves to look like the spectral shape leading to a structured and asymmetric appearance. The laser output spectrum measured by an OSA shows good agreement with the FROG spectral data in Fig. 4(b). The spectrum has steep edges as expected and fitting the spectrum to Eqn. 2 of Ref. 21, we find that the chirp parameter, C, is approximately 0.95.

 

Fig. 4. FROG measurements of the laser output in (a) Time domain and (b) Spectral domain. The spectrum, measured by an OSA, is shown as a solid line. x = Intensity. o = Phase.

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Fig. 5. Comparison of pulse autocorrelation from the FROG measurement and from a commerical autocorrelator.

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4. Minimum pulse width

The laser output can be reasonably compressed to a hyperbolic-secant pulse shape with a pair of 1500 lines/mm free-space gratings. Physical space limitations prevent the gratings from being placed at the optimal separation distance for pulse compression immediately after the laser output. Thus, we use ~22 m of Corning HI-1060 fiber to first stretch the pulses by ~5 ps. Then, the pair of gratings in a Treacy configuration compresses the output pulses to ~430 fs with Δλ=5.4 nm (ΔτΔv=0.60), as measured by FROG and shown in Fig. 6. Numerical calculations based on these results suggest that with optimization of the linear chirp compensation, the laser output could be compressed to ~410 fs. This value is close to the theoretical minimum pulse width of 360 fs, calculated by deriving the normalized pulse width τ as previously described and then multiplying by the factor 1.763 for hyperbolic-secant pulse width. The fit for a hyperbolic-secant pulse (R=0.9985) is also shown in Fig. 6(a). These results show that the minimum chirp-free pulses have relatively flat phases as would be expected for solitons while the time-bandwidth product is higher than that of the transform-limited anomalous-dispersion soliton (0.32).

 

Fig. 6. After compression, the laser output measured by FROG system (a) Time domain and (b) Spectral domain. Hyperbolic-secant pulse shape fit is shown as a solid line in (a). The spectrum, measured by an OSA, is shown as a solid line in (b). x = Intensity. o = Phase.

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5. Conclusion

In conclusion, we have demonstrated a long-cavity soliton Yb-doped fiber laser operating entirely in the normal-dispersion regime with no intra-cavity dispersion compensation. Soliton energy quantization of up to 3 simultaneous pulses in the laser cavity is observed. The chirp parameter is measured to be ~0.95. Finally, using a FROG setup, we show that the pulse is consistent with the defining equation V ₀ = sech (t/τ) exp{iC ln[cosh(t/τ)]}and can be compressed to 430 fs, near its calculated transform-limit of 410 fs. The fundamental output pulses have a chirped hyperbolic-secant profile with a pulse width of ~14 ps and an average output power of ~1.2 mW at 8.3 MHz. The spectral width is ~6.1 nm with steep band edges, which is expected of a solitary pulse of this form.

More detailed studies of this laser cavity can be carried out by testing a range of gains, dispersions, etc. to better understand and optimize the laser performance. Although the compressed pulse widths presently are not as short as the minimum pulse widths obtainable from some dispersion compensated lasers, the achievable pulse widths are still much less than 1 ps. These lasers can be particularly useful for applications where ultrashort pulse widths and high energies are not critical but long cavity lengths (low repetition rates) that have cumulatively high dispersion, which can be difficult to compensate compactly, are required. Additionally, a chirped pulse amplification system can be easily constructed by including a fiber amplifier between the stretching fiber and the compressor gratings, thus boosting the output power levels.