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We precisely generate dark solitons using an optical pulse synthesizer (OPS) at a repetition rate of 25 GHz and experimentally investigate soliton transmission through a normal-dispersion fiber. Because of their particular waveform, there are not many experimental studies. The OPS provides frequency-domain line-by-line modulation and produces arbitrary pulse waveforms. The soliton waveform has an intensity contrast greater than 20 dB. At certain input peak power, the pulse exhibits soliton transmission and maintains its initial waveform. The power agrees with soliton transmission theory. We confirm that the π phase shift at the center of the dark soliton is maintained after transmission through the fiber. We also investigate the influence of stimulated Brillouin scattering for long-distance transmission.
© 2013 Optical Society of America
1. Introduction
Short pulses propagating through optical fibers experience dispersion in the fiber, which changes their waveforms. The interplay of dispersion and self-phase modulation can support certain initial waveforms, namely, optical solitons. The nonlinear Schrodinger equation has two solutions: the bright soliton in the anomalous dispersion regime and the dark soliton in the normal dispersion regime. The bright soliton has been widely studied for diverse applications, including optical communication, mode-locked fiber lasers, and ultrashort pulse generation [1–3]. In contrast, although dark solitons are an interesting phenomenon in both science and engineering, there are not many experimental studies because of their specific waveform, which complicates their generation. Dark solitons consist of a dip in a constant-intensity background. The electric field waveform of a dark soliton is expressed as
Progress in frequency-domain pulse shaping, which uses spatial light modulators [4] and fiber Bragg gratings [5,6], makes it possible to generate precise dark solitons, and some experimental reports are now available. Experimentally generated dark solitons are mainly realized by a narrow dip centered in a bright, broad finite pulse [4,7]. A method that colliding two chirped-pulses generated multiple dark solitons in a single broad bright pulse at a high repetition rate, over 60 GHz [8]. Other generation methods such as nonlinear generation and direct modulation of continuous-wave light have been proposed [9,10]. Although these methods successfully generated the precision of the waveforms is insufficient for transmission studies. Furthermore, the frequency-domain pulse shaping method has difficulty in the generation because the waveform is very particular, namely, an intensity waveform constituting a temporal dip in a uniform background with a π phase shift.We have developed an optical pulse synthesizer (OPS) that can generate arbitrary optical waveforms based on frequency-domain pulse shaping [11,12]. The OPS can manipulate a spectrum line-by-line with a spacing of 12.5 GHz. We have experimentally investigated the characteristics of the synthesized pulses after transmission through various types of fibers, including bright solitons and parabolic pulses [13–15]. The theory describing these pulses is in agreement with the experimental results, and the agreement indicates that the pulses are synthesized with precision. One of the greatest limitations is that the OPS can only generate repetitive waveforms; and the OPS has difficulty in generating dark solitons because of the π phase shift.
To overcome this limitation, we propose the synthesis of two dark solitons in a single repetition period so that the two π phase shifts at the two pulse centers can constitute a repetitive electric field waveform by a total phase shift of 2π. In this study, we synthesize dark solitons at a repetition rate of 25 GHz and experimentally investigate their transmission through a normal-dispersion fiber (NDF). Because the dark solitons were generated in a precise manner, their soliton transmission intensity agreed with the calculated intensity. We confirmed that the intensity waveform and phase waveform were maintained after soliton transmission.
2. Experimental setup and procedure for dark soliton synthesis
Figure 1 shows the experimental setup for dark soliton synthesis and transmission through optical fibers. By using the OPS, dark solitons were generated from an optical frequency comb (OFC). The OFC, which served to seed the solitons, was generated by the phase modulation of single-frequency light. The dark solitons were phase modulated to broaden their linewidth, thereby suppressing stimulated Brillouin scattering (SBS). Next, they were transmitted through an NDF after amplification by an erbium-doped fiber amplifier (EDFA). We measured their waveforms and spectra for different input power at the input and output of the fiber. We used a delayed interferometer to confirm that the phase waveforms had the π phase shift at the pulse center.
An OFC was generated by modulation of single frequency light at 1540.7 nm from a tunable laser diode. Two-phase modulators driven at 12.5 and 25.0 GHz produced the OFC with a bandwidth of approximately 350 GHz from the light. The OPS is fabricated in silica-based technology and can generate an intensity and phase modulation of 12.5 GHz spacing for 30 channels by current injection to heaters installed on the modulators. The modulation depths of intensity and phase modulators were greater 20 dB and 2π, respectively. Required current for π phase shift is approximately 20 mA for the modulators. The output was amplified by the EDFA, following which a band path filter eliminated the noise that is generated by the amplifier. Power spectra were adjusted to match the target spectra calculated from the Fourier transform of the target waveform, and the reference spectra were measured with an optical spectrum analyzer (OSA). To minimize differences between the generated and target pulse waveforms, phase spectra were manipulated through genetic-algorithm-based feedback control. The waveforms were measured by an optical sampling oscilloscope (OSO) with a bandwidth greater than 500 GHz.
After synthesizing the pulses, they were transmitted through a 1-km-long NDF, whose propagation loss, dispersion, and nonlinear coefficient were 0.82 dB/km, −21.0 ps/(nm·km), and 17.4 (W·km)−1, respectively, at the wavelength of 1550 nm. Because the Brillouin threshold of the fiber was below the power required for soliton propagation, the spectral linewidth was broadened to suppress SBS by phase dithering using an LiNbO3 (LN) phase modulator driven by an 80 MHz signal at the OPS output. The modulator was not driven during the pulse synthesis processes. The transmitted pulse was measured by the OSO and OSA.
The spectral control of the comb was performed as follows: The power spectrum was directly measured with the OSA so that the comb power spectrum could be controlled to match the target spectrum. First, to widen frequency spacing of the OFC from 12.5 to 25.0 GHz, we filtered out the every other comb lines by driving the corresponding intensity modulators at the highest extinction ratio bias, 20 mA. Next, the remained comb lines were controlled to be the target powers. The phase spectra were difficult to measure directly and were controlled by the feedback control of the OPS. We controlled the waveform to match the target waveform by using a genetic algorithm (GA) [16–18]. A computer-based digital signal processor acquired the intensity waveforms from the OSO, calculated the fitness values, and fed the signal back to the controller of the synthesizer. The fitness function was root mean square difference between the intensity waveform and the target one. We controlled the phase spectrum to minimize the fitness by GA-based feedback control. This approach compensated signal crosstalk between the modulators for precise phase spectrum control. Although we did not measure the phase waveforms during feedback control, we obtained the desired dark soliton waveform including the phase waveform. We confirmed that the pulses were π shifted at the pulse center by measuring the delayed interferometer output. The delay time of the interferometer was 45.6 ps.
3. Dark soliton synthesis
When we defined a target waveform as a single dark soliton in a single repetition period [Fig. 2(a)], the waveform intensity was continuous between the neighboring periods, but the phase was not. The phase shift resulted in the sharp dip at the period edge, as shown in Fig. 3. The phase shift without intensity variation cannot be realized under the limited controllable bandwidth of the OPS. Dark solitons were experimentally characterized by using the waveform shown in Fig. 2(b).
Fig. 2 Target waveform: (a) Single dark soliton in a single repetition period. (b) Two dark solitons in a single repetition period.
In contrast, when the target waveform contained two dark solitons in a single period [Fig. 2(b)], the two phase shifts at the pulse centers completed the 2π phase shift, and the electric field waveform was continuous at the edges of the repetition periods. In this case, no waveform dip was generated. We generated this type of dark soliton with pulse widths of 8 and 10 ps. Figure 4 shows the waveform and spectrum of a synthesized dark soliton with a pulse width of 10 ps. There were two pulses in a single repetition period of 80 ps; hence, the repetition rate was 25 GHz. The measured waveform and spectrum were compatible with the target waveform and spectrum, respectively, and the power contrast was 22.7 dB at the pulse center. This pulse was transmitted through the NDF with different peak powers, where peak power is defined as the maximum of the waveform.
We also confirmed the π phase shift by measuring the delayed interferometer output. Figure 5 shows the output waveform from the delayed interferometer with a phase difference of π between the two arms in the interferometer. The ideal waveform, calculated from the ideal dark soliton waveform with a pulse width of 10 ps, is also shown. The two waveforms were compatible, which shows that the pulse has a π phase shift at the pulse center, as expected for the waveform defined in Eq. (1).
Fig. 5 Delayed interferometer output waveform for the waveform shown in Fig. 4(a) with a π phase difference.
4. Dark soliton transmission through a normal-dispersion fiber
The theoretical peak power P0 for supporting dark solitons in normally dispersive fibers is
where β2, γ, and TFWHM are the second-order dispersion, nonlinear coefficient, and pulse width, respectively [19].Figure 6(a) shows the propagated waveforms along with the initial pulse waveform. To measure the fundamental soliton peak power, we measured the waveforms of pulses transmitted through the 1-km-long NDF for different input powers. The results indicated that the peak power for supporting 10-ps-wide solitons is 16.8 dBm. At the peak power, the pulse maintained its waveform through the fiber and experienced soliton transmission. Below the peak power, dispersion broadened the pulse. Conversely, the pulse was compressed at a higher peak power because of higher-order soliton compression. The output spectra are shown in Fig. 6(b). In the low-power regime, the spectrum maintained the input profile. The high-power pulse underwent nonlinear spectral broadening, and the normal dispersion of the fiber caused higher-order soliton compression in the time domain.
Fig. 6 (a) Intensity waveforms and (b) spectra of dark solitons with different input peak powers at the NDF output. Peak power was 8.2 dBm (black solid curve), 16.8 dBm (red solid curve), and 24.0 dBm (green solid curve). The blue dashed curve indicates the initial waveform.
Figure 7(a) shows the dependence of output pulse width on input peak power. The theoretical soliton peak power for 8- and 10-ps-wide dark solitons, calculated from Eq. (2), is also shown. We assumed the same parameters for the fiber as were used in the experiment. Figure 7(b) shows the dependence of soliton peak power on dark soliton width. From the figures, the experimental and theoretical soliton peak powers were compatible.
Fig. 7 (a) Dependence of output pulse width on input peak power, (b) Soliton peak power as a function of dark soliton pulse width.
Finally, we investigated the phase waveform of the solitons at the output of the NDF. We inserted a delayed interferometer with a delay time of 45.6 ps and with a phase difference of π between the two arms. Figures 8(a) and 8(b) show the experimental results with the ideal waveforms for the 10- and 8-ps-wide solitons, respectively. The input peak power was the soliton peak power for each soliton: 16.7 dBm and 18.6 dBm [Figs. 8(a) and 8(b), respectively]. The ideal waveforms were calculated from the intensity waveforms directly measured by the OSO at the NDF output by assuming a phase difference of π between the two arms of the delayed interferometer. In both cases, the experimental results matched the ideal value well, indicating that the π phase shift at the pulse center is maintained even after transmission through the fiber. Conversely, higher-peak power distorted the waveform. Figure 9 shows the delayed interferometer output for a 10-ps-wide transmitted pulse at an input peak power of 24.0 dBm. The ideal waveform for the pulse according to the abovementioned method is also shown. The two waveforms were compatible near the pulse center but not in the pulse tails. The difference shows that the nonlinear effect distorts the phase waveform, especially the π phase shift at the pulse center.
Fig. 8 Delayed interferometer output waveform: (a) 10 ps width, 16.7 dBm peak power; (b) 8 ps width, 18.6 dBm peak power.
5. Discussion
The previous section presents the experimental results of dark soliton transmission by a high-repetition-rate (25 GHz) pulse train. However, the high-repetition-rate dark soliton has two strong frequency components that efficiently induce SBS, as shown in Fig. 4(b). The SBS threshold power is determined by the intensities of each frequency component. When the dark soliton propagates through long fibers or highly nonlinear fibers, the two components exceed the threshold power, and they experience high loss. Thus, to measure the transmission characteristics, we phase modulated the pulse to suppress SBS.
The SBS threshold power Pth is expressed as
where gB, Aeff, and Leff are the peak value of the Brillouin gain, the effective core area, and the effective interaction length, respectively. The effective interaction length Leff is related to the physical length of the fiber L according towhere α is the fiber-loss parameter. From Eqs. (3) and (4), the SBS threshold power for the NDF is 48.4 mW, where we used the NDF Aeff of 10.5 μm2 and the standard gB of 5 × 10−11 m/W for silica fibers.The optical power of dark solitons is concentrated in the two center-frequency components. For example, one center-frequency component of the 8- and 10-ps-wide dark solitons contains 47.3% and 48.2% of the total power, respectively. The two center-frequency components together contain over 90% of the total power. The values were numerically calculated by dividing the center-frequency component power by total power of the spectra. The spectra were Fourier-transoform of the single repetitive period of the waveforms. Figure 10 shows the dependence of power concentration on pulse width for a repetition rate of 25 GHz. The spectrum of the shorter pulse was wider, and the optical power was distributed over a wider spectral range. Although the concentration ratio decreased for the shorter pulse, each center-frequency component contained over 40% of the power. This concentrated power efficiently excites SBS, which distorts the waveforms. This power concentration is a distinct feature of bright solitons.
Fig. 10 Optical power concentration ratio of one of the spectral peaks around the center wavelength.
Figure 11 shows the intensity waveforms and spectra of a 10-ps-wide dark soliton at the NDF output without linewidth-broadening modulation and for different input powers. The waveform and spectrum of the input pulse are also shown. At the soliton power (16.7 dBm peak power), no frequency components exceeded the SBS threshold, and soliton transmission was observed. At a higher input power (24.0 dBm peak power), the two center-frequency components experienced higher loss caused by SBS, as shown in Fig. 11(b). Because the two components mainly form the background part in the time domain, the output waveforms in Fig. 11(a) contained dips in the background. The other frequency components, which form the dark soliton part, had the same spectral envelopes as the input components, and SBS was not excited. By calculating spectrograms of dark solitons, we have confirmed that the background part is formed by the center-frequency components (not shown). To suppress SBS, we introduced phase modulation by an 80 MHz signal, which caused linewidth broadening. The modulation sufficiently suppressed SBS, and the pulse width was reduced by higher-order soliton compression, as shown in Fig. 6 in the previous section. To transmit the dark solitons over long distances, such as in an optical fiber communication system, SBS excitation is a significant drawback, because it causes high loss and waveform distortion. Considering data transmission, data modulation generates spectral sidebands so that the power concentration can be relaxed to raise the SBS threshold. In long haul transmission, isolators included in fiber amplifiers prevent backward propagation, and the SBS thresholds are determined by the transmission fibers between the amplifiers, not by the total transmission length. If spacing between the neighboring sidebands is greater than the bandwidth of the SBS gain, about 15 MHz for standard silica-fibers, the each frequency component excites the SBS independently. For dark solitons modulated by random data sequence at a repetition rate of 25 GHz in the limit of afore-mentioned condition, the SBS threshold can be increased by 3000 times [20]. Therefore, SBS is not a significant drawback for data-modulated dark-soliton transmission.
Fig. 11 (a) Intensity waveforms and (b) spectra of NDF-transmitted dark solitons without linewidth broadening.
6. Summary
We synthesized dark solitons at a repetition rate of 25 GHz using an OPS and experimentally investigated soliton transmission through a 1-km-long NDF. By making the waveform continuous at the period boundary, we synthesized two dark solitons in a single repetition period to generate the pulses in a precise manner. The generated pulses exhibited soliton transmission, and the theoretical and experimental results were compatible. We confirmed that the intensity waveform and phase waveform were maintained after transmission through the NDF. Finally, we investigated the influence of SBS on long-distance transmission. The power concentration in the two spectral peaks at the center frequency excited SBS. The pulse waveform was distorted by SBS, especially, the background part. In a communications system, data modulation generates sidebands and suppresses SBS, as does phase modulation for linewidth-broadening modulation for long-range transmission.
Acknowledgments
This study was financially supported by a Grant-in-Aid for Scientific Research (B) Grant Number 25286065 from the Japan Society for the Promotion of Science (JSPS) and R&D through Fostering Young ICT Researchers as part of the Strategic Information and Communications R&D Promotion Programme (SCOPE) from the Ministry of Internal Affairs and Communications (MIC), Japan.
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