1. Introduction

Optical solitons not only are fascinating physical phenomena but also possess important practical applications. Mode-locked lasers have been one of most important platforms to explore various soliton effects. As this kind of solitary wave is generated from, in essence, a non-conservative system involving constant exchanging of energy, such lasers’ output is often characterized as dissipative solitons (DSs). Unlike optical solitons in the conservative systems under equilibrium, their characteristics and dynamics can be remarkably different, depending on the balance between not only dispersion and Kerr effect but also other gain/loss and linear/nonlinear effects [1, 2]. Traditionally, most mode-locked lasers are operated and modeled under the condition where one pulse or multiple identical pulses is emitted during one roundtrip time of the cavity [3]. In soliton lasers, the pulse spectrum would slightly vary under different pump powers. By further ramping up the pump, a bifurcation to a double-pulse stationary state with identical pulses of a slightly narrower bandwidth [4] would occur. This kind of multi-pulsing dynamics have been observed in a variety of laser configurations based on nonlinear polarization rotation (NPR) or saturable absorber (SA) mode-lockers.

Now, there are increasing interests in exploring the simultaneously emission of multiple diverse and asynchronous pulses from one laser cavity. It had been demonstrated that soliton fiber lasers can generate two or more ultrashort pulses circulating in the same direction in the cavity with different center wavelengths or states of polarization [5, 6]. While traveling along the same physical path, chromatic or polarization-mode dispersion results in a small different roundtrip time between each pulse train. Moreover, multi-pulse lasing had also been observed in bidirectional oscillating cavities [7, 8] or a unidirectional one with slightly different beam paths [9]. Recently, asynchronous pulse generation from a bidirectional oscillating Ti:sapphire laser had been demonstrated, likely due to the difference in the nonlinear phase associated the beam focusing configuration [10]. Such lasers could be an alternative to the traditional dual-laser, dual-comb source in frequency metrology [11], high-precision ranging [12] and spectroscopy [13]. It had been demonstrated that they could have good stability and inherent coherence for dual-comb application including pump-probe, ranging, terahertz measurement and optical spectroscopy [14–17].

In this paper, we demonstrate a dual-comb laser emitting multiple unidirectional oscillating pulses due to the existence of different pulse formation mechanisms. Picosecond and femtosecond pulses with four-fold difference in pulsewidths and very small repetition rate difference are observed to co-exist in one cavity. In contrast to the previous schemes, they share the same light path, state of polarization, and have completely overlapped spectra.

2. Experimental setup

The laser cavity setup for unidirectional dual-comb mode-locking shown in Fig. 1 is ~9.0 m, which consists of a 2.1-meter-long piece of Erbium-doped fiber (EDF, Type 1022) forward pumped by a 980 nm diode, an in-line polarizer aligned roughly along the fast axis of two 1-meter-long polarization maintaining fiber (PMF) pigtails. A polarization-independent isolator (ISO) and a fiber-squeezer-type polarization controller (PC) are used to ensure the unidirectional operation and control the intracavity polarization evolution, respectively. An 80/20 optical coupler (OC) is used. A single-wall carbon nanotube (SWNT) modelocker is fabricated on an FC/APC ferrule using the optical deposition method [18, 19] with an insertion loss of about 1.5 dB at 1550 nm. With its relatively fast picosecond nonlinear saturable absorption [20], SWNT helps to initiate the mode-locking process. The total GVD is ~0.11 ps/nm, and the laser could operate in the soliton regime.

 

Fig. 1 Configuration of the unidirectional dual-comb mode-locked fiber laser.

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Besides these well-known effects in soliton fiber lasers, the introduction of the in-line polarizer with birefringent PMF pigtails in this seemingly simple design bring into plays additional mode-locking and pulse formation mechanisms [21]. In contrast to the traditional hybrid mode-locking lasers based on a mode-locker and the NPR effect, instead of waveplates and a polarizer, the PMF of the in-line polarizer introduces relatively large birefringence into the cavity that results in an ostensibly small yet critical difference in the pulse formation process. When combined with the polarization-dependent loss from the polarizer, spectral filtering effect would emerge for the imperfectly aligned fiber polarizer. For its 1-meter-long PMF pigtails with the modal birefringence of 4.8 × 10⁻⁴, the in-line polarizer could introduce spectral filtering with a free spectral range (FSR) of ~5 nm at 1560 nm [22]. While NPR is to further narrow the pulses as an ultrafast ‘modelocker’, the filtering would curtail the spectral bandwidth of pulses and, thus, may broaden the pulses. Since the shape of a mode-locked laser output is affected by the delicate balance between the pulse formation mechanisms in the cavity, the simultaneous existence of multiple intracavity linear or nonlinear, pulse narrowing and broadening mechanisms could enable emission of different pulses from the same platform.

3. Experimental results and discussions

Mode-locking under the traditional soliton regime When PC is adjusted to a ‘zero’ position so that the polarization state of light passing into the polarizer remains the same after one round trip in the cavity, the polarization-related filtering and NPR pulse narrowing effect as well as the intracavity loss would be minimized. Mode-locking can self-start at a fundamental repetition rate f_rep of ~23 MHz. The optical spectrum and RF spectrum of the outputs are shown in Fig. 2. It shows a typical soliton shape with a 3-dB spectral bandwidth of 2.6 nm, mostly determined by the intracavity dispersion, fiber nonlinearity and the mode-locker characteristics. When compared with that obtained using a 2-m-long piece of SMF in the place of the in-line polarizer, it bears strong similarities. This further suggests that the polarization-related mechanisms are not affecting the pulse forming process in this case.

 

Fig. 2 (a) Optical spectra and (b) RF spectra of the laser under the traditional soliton regime with and without a polarizer.

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Mode-locking under the additional NPR and filtering effects By adjusting the polarization controller away from the ‘zero’ position, spectral interferometric filtering in the lasing spectrum is observed. Periodic spectral modulation whose period is determined by the amount of birefringence could limit the pulse bandwidth. Therefore, when the pump power is relatively low, the bandwidth of pulses with relative low energy is reduced from the above level. On the other hand, at higher pulse energy when the laser is pumped harder, the NPR effect may kick in under the correct polarization ‘bias’ positions, and it reduces the linear polarization-dependent loss for pulses with higher optical intensity. This pulse narrowing mechanism could balance and eventually overcome the filtering effect. Therefore, even shorter pulses with a broader spectrum can be formed. As shown in Fig. 3(a), by adjusting PC, a gain peak at ~1562.8 nm can be observed. The laser starts to emit CW light at that wavelength, when the pump power reaches the lasing threshold. Once the pump power further reaches 14.5 mW, a pulse with a spectrum bandwidth of 1.8 nm self-starts to mode-lock at the fundamental cavity frequency. As discussed above, this pulse’s narrower spectrum is attributed to the spectral filtering in the absence of the additional hybrid mode-locking effect.

 

Fig. 3 (a) Optical spectra evolution and (b) the corresponding temporal oscilloscope waveforms, when increasing the pump power; (c) the temporal waveforms of the ‘1 fs + 1 ps’ case captured at different time.

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At a larger pump power while keeping the PC state unchanged, the pulse would accumulate additional energy so that the nonlinear phase shifts it experiences also increase. The pulse’s bandwidth broadens to 4 nm at a pump power of 16.9 mW. Its bandwidth indicates the existence of the hybrid mode-locking effect in the presence of NPR, as it is significantly wider than that under the soliton regime alone. Higher height of the pulse train when observed on the oscilloscope shown in the top row of Fig. 3(b) also indicates the increased pulse energy. The center wavelength of this shorter pulse shifts slightly to the shorter wavelength and is 0.8 nm away from the CW peak, which is almost unchanged. Significant Kelly sidebands are observed at about 11.4 nm away from the center wavelength, indicating the presence of relatively strong, quasi-CW dispersive wave in the cavity [23, 24].

If further increasing the pump power without other adjustment, an intriguing phenomenon would occur, as shown in Fig. 3(c). An extra ‘bump’ shows up on top of the spectrum of the shorter pulse. Its position is close to the CW peak, and its shape bears a strong resemblance to that of the wider pulse first appeared. The pulse trace shows an additional pulse (denoted as ‘ps’) besides the taller short pulse (denoted as ‘fs’), while the relative temporal position between the ‘fs’ and ‘ps’ pulses slightly shifts after each round trip. This indicates that a bifurcation from a single-pulse state to a dual-pulse state has occurred with quite different spectral characteristics, and their repetition rates are also slightly different.

As the pump power is further increased, more pulses are observed in each period of the round trip time. As observed in Fig. 3(b), up to three smaller pulses could co-exist with a taller one, before additional taller pulse would emerge while one smaller pulse ‘disappears’. This could be understood as the smaller pulse may have accumulated enough energy so that it transforms into a taller pulse with a broader spectrum.

Dual-comb lasing When gradually decreasing the pump power from the above states, the CW peak on top of the optical spectrum could be removed by carefully decreasing the pump power. Due to the well-known hysteresis effect, the laser would remain mode-locked at a lower power than its starting threshold. The laser can be adjusted into a state emitting just one ‘fs’ pulse and one ‘ps’ one. Additional measurements and analyses are carried out to characterize this dual-pulse output. Its optical spectrum at a pump power of 14.7 mW is shown in Fig. 4(a). Despite of its special shape, it’s shown that it can be fitted with two standard solitons’ spectra of different bandwidths. The sum of a 4.4-nm-bandwidth spectrum (blue dashed line) and a 1.3 nm one (red dashed line) separated by 0.8 nm close matches the measured data in Fig. 4(a). The inset shows an excellent agreement between the fitted narrow-bandwidth spectrum and that at a pump power of 9.2 mW.

 

Fig. 4 (a) Experimental and sech² fitted optical spectra; (b) experimental autocorrelation trace and calculated ones based on spectral information; (c) RF frequency spectrum of the laser output when Δf = 45 Hz.

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Intensity autocorrelation measurement is done by sending both pulses into an autocorrelator. The measured autocorrelation trace matches the sum of two individual autocorrelation functions of two sech² pulses with 570 fs and 2.0 ps pulsewidth, which are estimated based on the fitted optical spectra respectively in Fig. 4(a) and assuming transformed-limited solitons. In this case, the ‘fs’ pulse’s peak power is estimated to be one order of magnitude larger than that of the ‘ps’ pulse. Obviously, the peak of autocorrelation trace is mostly dominated by the ‘fs’ pulse and the broader tails illustrates the presence of the longer and weaker ‘ps’ pulse as observed in Fig. 4(b). From the RF spectrum shown in Fig. 4(c), the existence of two repetition rates closely spaced to each other with the difference Δf of 45 Hz further proves the simultaneous stable lasing of one femtosecond and one picosecond pulses in the cavity. This value of Δf corresponds to a round-trip delay difference of 0.085 ps between the two pulses with a center wavelength separation of 0.8 nm, and is in good agreement with the total cavity dispersion ~0.11 ps/nm. The smaller peaks are beat notes generated by the photodetector [6, 9]. These observations confirm that two pulse trains with overlapped spectra and pulsewidths different by a factor of >3 are simultaneously generated.

By slightly tuning the state of polarization, it is observed that the center wavelength of the ‘ps’ pulse can be adjusted, as the maxima of spectral interference shift. Also, when stronger polarization-dependent loss is introduced, the ‘fs’ pulse’s spectrum under hybrid mechanism becomes wider and further shifts to shorter wavelengths due to the changed gain tilt. In the tuning process, center wavelength separations can be change between 0.5 nm and 1.4 nm. Accordingly, the repetition rate difference is tunable over a range of about 30 to 80 Hz. Figure 5 shows a case where the spectra of the two pulses are 5.2 and 0.8 nm wide, respectively. A center wavelength separation of 1.4 nm leads to a Δf of 78 Hz. Shown in the linear plot of the inset of Fig. 5 (a), the large narrower spectral peak and the lower, much broader one with two small Kelly sidebands corrspongding to the ‘ps’ and ‘fs’ pulses, respectively, are clearly distinguishable.

 

Fig. 5 (a) Experimental and sech² fitted optical spectra; (b) RF frequency spectrum and (c) waveform of the laser output when Δf = 78 Hz.

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Asynchronous coherent sampling and interferogram Two coherent pulse trains with a stable Δf could asynchronously sample each other [12–15]. To further verify the characteristics of the dual-comb output, the laser output is split into two paths and then recombined by using 2*2 50:50 optical couplers (see Fig. 6(a)). Two outputs of the coupler are sent to a balanced photodetector (BPD, Thorlabs PDB420C). The setup is similar to the commonly used dual-comb spectroscopy setup [17]. As an ideally BPD only yields an output when the electrical field of pulses from the two arms arriving at the coupler are temporally overlapped and can interfere, it gives the background-free interferogram trace corresponding to the field cross-correlation of the pulses. As both arms have two asynchronous pulses and the optical path lengths are not balanced, there would be two interferograms in each period of 1/Δf as indicated in Fig. 6(a). An NI-5122, 14-bit ADC at a sampling rate of 100 MS/s is used to digitize the signal. At f_rep = 22.975 MHz and Δf = 63 Hz, the following measurements are made with a corresponding sampling step of 119 fs. The BPD output, shown in Fig. 6(b), as expected, is similar to the cross-correlation scaled by a factor of f_rep/Δf. Its envelope can be fitted with a 1.08 μs-wide sech² shape, corresponding to an actual temporal width of 2.96 ps. Optical spectral shape in Fig. 6(c) is obtained by taking the averaged Fourier transform of 34 temporal interferograms as in the dual-comb spectroscopy measurement [17]. Good signal quality indicates high coherence between the comb lines of the dual-comb output. Its shape matches the calculated product of the dual-comb spectra, mostly determined by the ‘ps’ pulse, which is 1.2 nm (i.e. 150 GHz) wide during this measurement.

 

Fig. 6 (a) Experimental setup for asynchronous sampling; (b) measured interferogram trace and (c) its Fourier transform.

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4. Conclusion

We demonstrate a unidirectional dual-comb laser that can asynchronously emit multiple ultrashort pulses with quite different pulsewidths. It is believed that the additional linear and nonlinear pulse shaping mechanisms introduced into the cavity besides the SA lead to this unusual output bifurcation at elevated pump powers. Good spectral coherence is demonstrated due to the completely common-path cavity, which is evident from the asynchronous sampling and corresponding dual-comb spectroscopy result.