Nyquist pulses, which are defined as impulse responses of a Nyquist filter, can be used to simultaneously achieve an ultrahigh data rate and spectral efficiency (SE). Coherent Nyquist optical time-division multiplexing transmission increases SE, but the optical signal-to-noise ratio (OSNR) is limited by the amplitude of the original CW beam. To further improve transmission performance, here we describe a new pulsed laser that can emit an optical Nyquist pulse train at a repetition rate of 40 GHz. The Nyquist laser is based on a regeneratively and harmonically mode-locked erbium fiber laser that has a special spectral filter to generate a Nyquist pulse as the output pulse. The pulse width was approximately 3 ps, and the oscillation wavelength was 1.55 μm. The spectral profile of the Nyquist pulse can be changed by changing the spectral curvature of the filter with a roll-off factor, , between 0 and 1. A Fabry–Perot etalon was also installed in the laser cavity to select longitudinal modes with a free spectral range of 40 GHz, resulting in the suppression of the mode hopping in the regenerative mode locking. A numerical analysis is also presented to explain the generation of a stable Nyquist pulse from the laser. The Nyquist laser is important not only for the direct generation of high-OSNR pulses but also for scientific advances, proving that pulse shapes that differ significantly from the conventional hyperbolic-secant and Gaussian pulse shapes can exist stably in a cavity.
© 2014 Optical Society of America
Intensive efforts have been made to increase spectral efficiency (SE) by adopting a coherent multilevel modulation format, such as quadrature amplitude modulation (QAM) or orthogonal frequency-division multiplexing [1,2]. We recently reported 1024 and 2048 QAM digital coherent transmission experiments and showed a potential SE of more than [3,4]. To increase the transmission speed in such a system, we recently proposed a Nyquist pulse that is given by the impulse response of a Nyquist filter . An optical Nyquist pulse was first generated by spectrally filtering a Gaussian pulse, and it was used for optical time-division multiplexing (OTDM) transmission [5,6]. The Nyquist pulse enables us to reduce bandwidth without causing intersymbol interference. The SE was improved from 0.5 to under differential phase shift keying transmission over 525 km, and we showed that the Nyquist pulse was robust against high-order dispersion and polarization-mode dispersion [6,7].
To further increase the SE, coherent Nyquist OTDM transmission was proposed, where a CW frequency-stabilized laser followed by a combination of sideband generation using a phase modulator and spectral filtering made it possible to generate a coherent Nyquist pulse . Thus, we successfully demonstrated , 64 QAM coherent Nyquist transmission over 150 km, where the SE was increased to . However, to improve the transmission performance of the coherent Nyquist transmission, it is important to increase the optical signal-to-noise ratio (OSNR) of the coherent Nyquist pulse, which is usually limited by the amplitude of the original CW beam.
In this paper, we propose a Nyquist laser that can directly emit a coherent Nyquist pulse with a high OSNR. To generate such a pulse train, we based our approach on our regeneratively and harmonically mode-locked erbium fiber laser at a wavelength of 1.55 μm . We suppressed mode hopping by installing a Fabry–Perot etalon to select longitudinal modes with a free spectral range (FSR) of 40 GHz. The shape of the Nyquist pulse was varied by changing the spectral curvature of the filter installed in the cavity, which depended on a roll-off factor of between 0 and 1. The installation of a special optical filter plays an important role in generating the Nyquist pulse, as it can control the amplitude and phase of each longitudinal mode.
2. CONFIGURATION OF THE NYQUIST LASER
Mode-locking techniques have generally been used to generate a transform-limited Gaussian or sech pulse train . In amplitude modulation (AM) mode locking, parabolic time-domain shaping near the top of a sinusoidal amplitude modulation gives rise to a stable Gaussian pulse. When we introduce an optical nonlinear effect such as self-phase modulation (SPM) into a laser cavity under anomalous dispersion, the laser starts to emit a sech pulse described by the nonlinear Schrödinger equation. This is called a soliton laser . Interesting schemes for mode locking such as interferential mode locking , additive pulse mode locking  and Kerr-lens mode locking  have also appeared. In FM mode locking, we employ a pulse formation mechanism combined with frequency chirping using a phase modulator and an optical bandpass filter. Here, the up or down linear chirping part of the sinusoidal phase modulation generates a linear chirping, and the two frequency components outside the optical bandpass filter can be removed. In other words, a combination of linear chirping and an optical bandpass filter in the cavity converts a CW wave into a pulse. Since this pulse-shaping mechanism exists with both up and down chirping, the FM mode locking is usually unstable unless cavity dispersion is introduced. For example, a stable FM mode-locked soliton laser uses up-chirping to counterbalance the anomalous dispersion during the course of soliton formation, and we can even accelerate up-chirping, which has the same chirping slope as SPM, for shorter soliton pulse generation. All these stable pulse formations in AM and FM mode locking were proven analytically in the 1970s, for example, by Kuizenga and Siegman [15,16], and Haus .
On the other hand, the Nyquist pulse is an entirely new pulse that has neatly repetitive ringing on the pulse tail. Therefore, we need a new mechanism to generate such ringing, which can be realized by installing a Nyquist optical filter as follows. The spectral profile of the raised-cosine Nyquist filter, , and its impulse response, , are given by18]. A Nyquist pulse, which depends on the value of , accompanies its specific ringing feature (or ripples) on the wing of the pulse. Therefore, to generate such a pulse, we need not only pulse shaping at the pulse peak (parabolic time-domain shaping by using an optical intensity modulator) but also shaping on the wing. Since it is not easy to introduce time-domain shaping on the wing of the pulse, we use spectral-domain shaping by incorporating spectrum manipulation based on a liquid-crystal-on-silicon (LCoS) spatial modulator , which can precisely control both spectral amplitude and its phase. Dark and bright pulses were generated in a passively mode-locked laser with LCoS in the cavity . Ultrashort pulses were also generated with an intracavity phase-shaping element . Furthermore, it is important to note that simply the installation of an ideal Nyquist filter does not produce an ideal Nyquist pulse as an output. That is, the output spectral profile (flattop spectral profile for the Nyquist pulse) is given by the product of a Gaussian-like convex spectral profile generated by parabolic shaping in the time domain and a concave spectral profile, which is newly introduced by, for example, an LCoS filter .
The experimental setup of the Nyquist laser is shown in Fig. 1(a), where we adopted regenerative and harmonic mode locking in a fiber laser with a total cavity length of 17 m. The fiber cavity consists of a 5 m long polarization-maintaining erbium-doped fiber (PM-EDF), a 30% output coupler, an optical etalon, a polarization-dependent isolator, a Mach–Zehnder intensity modulator, an LCoS filter, and a WDM coupler with fiber pig-tails. The LCoS filter can manipulate the spectrum with a frequency resolution of 1 GHz over the C band, and the intensity can be controlled over a range of 35 dB with a precision of 0.1 dB. All the fibers in the cavity were polarization-maintaining fibers to prevent polarization fluctuations. In addition, all the fibers and the LCoS filter had anomalous dispersions. The average cavity dispersion characteristic is shown in Fig. 1(b), where the laser cavity had an anomalous dispersion of at 1550 nm. The dispersion characteristic was obtained with a group delay time measurement by changing the peak oscillation wavelength and detecting the corresponding pulse delay under an ordinary regenerative mode-locking condition. The pumping source was a 1.48 μm InGaAsP laser diode. We installed an optical etalon with an FSR of 40 GHz and a finesse of 200 to suppress mode hopping. The insertion loss of the etalon was 3 dB, and the total cavity loss was approximately 15 dB, which was easily compensated for by the erbium-doped fiber amplifier (EDFA). The fundamental cavity mode spacing was 12 MHz, and we set the repetition rate in the harmonic mode locking at 40 GHz. To obtain a sinusoidal harmonic beat signal at 40 GHz between the longitudinal laser modes, part of the output beam was coupled into a clock extraction circuit consisting of a high-speed photodetector, a 40 GHz high- dielectric filter (), and a high-gain electrical amplifier. After adjusting the phase between the pulse and the modulation peak, the beat signal was amplified and fed back to the intensity modulator in the cavity, resulting in regenerative mode locking . The dispersion of the output pulse thus obtained was compensated for with appropriate opposite dispersion generated from a tunable grating-pair dispersion compensator. The output pulse was measured with an optical sampling scope with a resolution of 800 fs.
A Mach–Zehnder intensity modulator was used for parabolic intensity modulation. An LCoS spectral filter in the cavity plays an important role in changing the roll-off factor . The resolution of the spectral manipulation was 1 GHz, but in our experiment one spectral slot was set at a bandwidth of 40 GHz. That is, if the aim is to apply the Nyquist laser to coherent optical communication, no longitudinal mode of the laser should fluctuate with time. Therefore, we installed a Fabry–Perot etalon to suppress the mode-hopping effect in the harmonic mode locking . LCoS enabled us to modify the spectral curvature to obtain the Nyquist pulse. A phase-locked loop (PLL) operation was also adopted to keep the repetition rate constant at 40 GHz .
3. OUTPUT CHARACTERISTICS
The output power of the Nyquist laser as a function of the pump power is shown in Fig. 2(a), where the pump threshold was 30 mW and the slope efficiency was 8.1%. An output power of 15 mW was obtained at a pump power of 220 mW. The electrical spectrum of the output pulse train, which was detected with a high-speed photodetector, is shown in Fig. 2(b). Only one spectrum was observed at 39.813 GHz, with a noise suppression of 80 dB, which indicates that super-mode noise was well suppressed below by the installation of a Fabry–Perot etalon.
Figure 3 shows the output waveform characteristics of the Nyquist pulse when was set at zero. A rectangular spectral profile was obtained when the spectral filter profile generated by the LCoS was that given in Fig. 3(b). To obtain the rectangular profile, the transmittance on both edges of the optical filter was enhanced by approximately 5.3 dB. Thus, a sinc-like Nyquist pulse was obtained as shown in Fig. 3(c), where the FWHM of the Nyquist pulse was 3 ps. An output power of 15 mW with a 3 ps duration and a 40 GHz repetition rate gives a peak power of 125 mW, which is more than 10 times larger than that with a Nyquist pulse generated from a CW beam . By increasing the OSNR by 10 dB, the multiplicity of the QAM data can be increased fourfold. In Ref. , for example, a polarization-multiplexed , 64 QAM () transmission was demonstrated with an SE of . We therefore expect the multiplicity to be increased to 256 QAM and to achieve an SE of by using a Nyquist laser instead of the CW laser-based pulse source. It should be noted that when a Nyquist pulse is generated from a CW beam, the output power is limited by the maximum allowable input power of the optical modulator for optical comb generation. This inevitably results in OSNR limitation, even if a high-power CW laser is used. Therefore, a much better OSNR can be obtained with the Nyquist laser. Figure 4 shows the output waveform characteristics of the Nyquist pulse when was set at 0.2. When , the height of the shoulders in the spectral profile were decreased slightly to 1.7 dB and the spectral width was increased to 2.9 nm by controlling the LCoS filter as shown in Fig. 4(b). The obtained Nyquist waveform given in Fig. 4(c) agrees well with the theoretical waveform shown in red. The pulse width decreased slightly to 2.85 ps because of the larger filter width. When we set at 0.5, the spectral profile of the output pulse on both shoulders was further decreased by 2 dB by setting the LCoS filter as shown in Fig. 5(b). In addition, a wider loss band on the shoulders was introduced to make the Nyquist pulse. Ringing is still clearly observed on the wing of the pulse, as shown in Fig. 5(c), where the experimental result agrees well with the red theoretical curve.
Figure 6 shows the Nyquist output pulse when , where we broadened the LCoS passband from 2.9 () to 3.55 nm to obtain a broader spectral profile, as shown in Fig. 6(a). The spectral profile starts to be similar to the ordinary spectral profile of a conventional Gaussian or sech pulse. The ringing on the wing has almost disappeared, as shown in Fig. 6(c), where the pulse width was decreased to 2.6 ps. When we set at 1, we prepared a broader passband of 4.19 nm and a shoulder loss of 4.6 dB. Then we obtained the Nyquist spectral profile seen in Fig. 7(a). The pulse width was 2.4 ps, as shown in Fig. 7(c), and the pulses look similar to ordinary pulses. In part (a) of Figs. 3–7, dotted lines indicate theoretical spectral profiles for each . In addition, to obtain a better fitted Nyquist spectral profile, we changed the transmittance slightly for every 40 GHz band in the LCoS, as seen in part (b) of Figs. 4–7, which was effective in compensating for the small spectral amplitude fluctuation in the mode locking.
The mode locking occurred more easily when approached 1, since the Nyquist waveform became similar to ordinary pulses. An Nyquist pulse was the most difficult to obtain, because the top of the spectral profile had to be kept flat. All the experimental results described in Figs. 3–7 were obtained by AM mode locking with the use of a intensity modulator. We also generated the Nyquist pulses with FM mode locking using a phase modulator. We obtained the Nyquist pulses when exceeded 0.8. However, at an below 0.5, we were not able to obtain a stable Nyquist pulse, and a clock signal at 40 GHz for regenerative mode locking could not be extracted. This means that with FM mode locking, the phase of each longitudinal mode tends to be different because of the phase modulation given by the Bessel function, while the Nyquist pulse requires the same phase in each longitudinal mode. A flattop spectral profile under Nyquist mode locking can be realized with AM, because all longitudinal modes oscillate at the same phase due to the nature of AM.
4. NUMERICAL ANALYSIS OF THE NYQUIST LASER
A numerical model for the experimental setup of the Nyquist mode-locked laser is shown in Fig. 8, where intensity or phase modulation was applied with a sinusoidal function at 40 GHz. The split-step Fourier method was used for the pulse propagation in the fiber. However, since the laser cavity in the experiment was 17 m long with an average dispersion of , nonlinearity was not taken into account for the generation of a 3 ps pulse. This was realized by setting the nonlinear coefficient at zero in the nonlinear Schrödinger equation. We confirmed that the numerical result is identical even when including the nonlinear effects, which implies that the present laser operates in a linear regime and the nonlinearities do not play an important role because of the large dispersion. The gain in the EDFA, which compensates for the cavity loss, was calculated including gain saturation based on the rate equation between the pump power and signal power as follows:2) and (3) as
A. AM Mode Locking
Simulation results obtained for are shown in Fig. 9, where parts (a)–(d) correspond to the spectral filter shape, the evolution of the Nyquist pulse from ASE noise to steady-state oscillation, the steady-state Nyquist waveform, and its optical spectrum, respectively. The filter shape given in Fig. 9(a) was set so that it was similar to the experimental setup shown in Fig. 3(b). The frequency bandwidth of 320 GHz in Fig. 9(a) corresponds to a spectral width of 2.58 nm in Fig. 3(b). In this condition, a stable steady-state waveform with a pulse width of 2.9 ps was obtained and the waveform was completely fitted with the theoretical Nyquist waveform shown by the dashed line. The simulation results agree well with the experimental results. Therefore, judging from the obtained waveform and the spectral profile, we can say that the output waveform is a Nyquist pulse with .
Figure 10 shows the case for , where the filter shape given in Fig. 10(a) was set so that it was similar to our experimental setup shown in Fig. 5(b). A frequency bandwidth of 370 GHz corresponds to a bandwidth of 3 nm in Fig. 5(b). A 2.8 ps Nyquist pulse with a small ripple was clearly obtained and agrees well with the experiment. The simulation results for are shown in Fig. 11, where the filter bandwidth was set at 525 GHz, which corresponds to a spectral width of 4.2 nm. The pulse width was 2.5 ps. The simulation results also agree well with the experimental results shown in Fig. 7.
In the simulation, we confirmed that stable Nyquist pulses were also obtained when the cavity dispersion had an opposite GVD value of . This indicates that the Nyquist pulse can be generated in the linear pulse propagation regime.
B. FM Mode Locking
As we experienced in the FM mode-locked Nyquist laser experiments, there was a large difference as regards the AM mode locking. Here, we numerically investigate how the FM mode locking occurs at different values under different dispersion conditions. Figures 12–14 correspond to conditions of anomalous GVD, zero GVD, and normal GVD, respectively, where we chose values of 0.5, 0.8, and 1.0, respectively. In each figure, parts (a), (b), and (c) correspond to a waveform change toward a steady-state pulse, its steady-state waveform, and the corresponding spectral profile, respectively. For the anomalous dispersion shown in Fig. 12, the output pulse for was unstable and the spectral profile differed from the Nyquist shape, as we experienced in the experiment. For the smaller values, the oscillation was also unstable. When we increased to 0.8 and 1.0, we obtained stable pulse oscillations, and they could be fitted with the Nyquist profiles. Up-chirping in the phase modulation gave a stable oscillation to counterbalance the anomalous dispersion. These results indicate that it appears to be difficult to generate a sinc function Nyquist pulse with a flattop spectrum with FM mode locking, which is attributed to the fact that the phase in each longitudinal mode is different with phase modulation.
FM mode locking under zero GVD is shown in Fig. 13, where every case was unstable. This result is understandable, as the laser cavity has two possible oscillation conditions that occur every half-cycle due to up-chirping and down-chirping in the process of phase modulation. This causes unstable oscillation.
FM mode locking under a normal dispersion of is shown in Fig. 14. As with anomalous dispersion, stable Nyquist pulses were generated for values of 0.8 and 1.0, but a stable pulse was not obtained when was 0.5. It is important to note that the down-chirping of the phase modulation gave a stable pulse oscillation to counterbalance the normal dispersion.
We proposed and demonstrated a Nyquist laser that can directly emit an optical Nyquist pulse train. The laser was based on a 1.55 μm regeneratively and harmonically mode-locked erbium fiber laser, where a special optical filter called an LCoS was installed to generate a Nyquist pulse as the output pulse. The peak power reached 125 mW at a repetition rate of 40 GHz and a pulse width of 3 ps. Various types of Nyquist pulses were generated by changing the spectral curvature of the filter with a roll-off factor of between 0 and 1. A Fabry–Perot etalon was also installed in the laser cavity to select longitudinal modes with an FSR of 40 GHz, resulting in the suppression of mode hopping in the regenerative mode locking. Numerical analyses showed that there were stable Nyquist pulses in the laser cavity, and AM mode locking can generate better Nyquist pulses with smaller values than FM mode locking. The Nyquist laser makes it possible to realize coherent Nyquist OTDM transmission with a higher OSNR than conventional Nyquist pulse generation schemes, and thus is expected to constitute an attractive light source for ultrahigh-speed transmission with high-QAM multiplicity, which will lead to an ultrahigh SE of over even at a single-channel bit rate beyond .
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