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Abstract
Nonlinear Fourier transform (NFT) is a powerful tool for characterizing optical soliton dynamics, which, however, suffers from fundamental limitations that ultra-wide bandwidth photodetectors and ultra-high sampling rate analog-to-digital converters should be used when accessing the full-field information of an ultrafast optical pulse. Herein, we report on the experimental demonstration of the linear optical sampling (LOS) enabled nonlinear frequency spectrum classification of ultrashort optical pulses, which could break this limitation. Instead of traditional coherent detection, the LOS overcomes the ultra-wide bandwidth constraint of commercially available optoelectrical devices. By finely adjusting the repetition rate difference between the soliton to be characterized and the sampling pulsed source, a 55.56-TSa/s equivalent sampling rate arising in the LOS can be secured, where only 400-MHz balanced photodetectors and 5-GSa/s analog-to-digital converter are used. Meanwhile, according to the nonlinear frequency spectrum calculated from the accurate full-field information, the promising concept of soliton distillation has been experimentally verified for the first time. The LOS-enabled NFT technique provides an alternative and efficient characterization tool for ultrafast fiber lasers, which facilities comprehensive insight into soliton dynamics.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The nonlinear Fourier transform (NFT), known as the inverse scattering transform in the mathematical literature, is a powerful method to analytically solve integrable nonlinear partial differential equations, especially in the theoretical field of optical solitons [1]. With the help of NFT, the optical signal can be transformed into the nonlinear frequency spectrum [2], including both continuous spectrum (non-soliton components) and discrete spectrum (soliton components), whose eigenvalues lie on the upper-half complex plane. Consequently, information can be encoded into the nonlinear spectrum of the signal, which can effectively address the nonlinear transmission impairments arising in a standard single mode fiber (SSMF) [3]. Recently, the NFT has shown its capability to characterize ultrashort pulses in the nonlinear frequency domain. Before calculating the nonlinear spectrum, it is essential to precisely acquire the full-field information of the optical signal. Simultaneous implementation of dispersive Fourier transform (DFT) and time-lens has been used to comprehensively characterize the spectral and temporal evolutions of an ultrashort dissipative soliton, where its dynamics before the stabilization can be investigated [4]. However, the DFT and time-lens will greatly complicate the experimental setup. In general, as for the eigenvalue distribution characterization for nonlinear spectrum classification, conventional coherent detection is a straightforward methodology to access the full-field information of optical pulses. To ensure the accuracy during the conventional coherent detection, the bandwidth of photodetectors (PDs) and analog-to-digital converters (ADCs) needs to satisfy the Nyquist sampling theorem. Recently, the laser output with a pulse width of about 200 ps has been successfully characterized with 50-GHz PDs, for the purpose of investigating its eigenvalue distribution [5]. In the nonlinear frequency domain, the soliton and the resonant continuous-wave (CW) background have different eigenvalue distributions. After filtering out the eigenvalues corresponding to the CW background in the nonlinear frequency domain, pure solitons are numerically recovered by the inverse nonlinear Fourier transform (INFT), which is also known as soliton distillation [6]. In our previous effort, the NFT-based soliton distillation was numerically implemented for various pulses generated in a fiber laser, including single pulse, period-doubling pulse, double pulses, and multiple pulses [7]. In addition, cavity-soliton distillation and high-order-soliton characterization using the NFT were both numerically carried out with high accuracy [8,9]. However, the ultimate requirements of balanced PD (BPD) with ultra-wide bandwidth (2.5 THz) and ADCs with ultra-high sampling rate (5 TSa/s) for characterizing a picosecond soliton considerably hinder the experimental verification. Frequency-resolved optical gating (FROG) involves obtaining simultaneous time- and frequency-resolved information regarding the ultrafast optical pulse [10]. FROG measures a two-dimensional (2-D) representation of the one-dimensional (1-D) field and consequently requires the collection of a relatively large amount of data [11]. Iterative algorithms are required for FROG, which consumes lots of time. Spectral phase interferometry for direct electric-field reconstruction (SPIDER) is a self-referencing interferometric method for measuring the time-dependent intensity and phase of ultrashort optical pulses [11]. SPIDER uses a Type II nonlinear crystal, and the corresponding characterization process is based on free space configuration, which is more complicated with huge loss.
Alternatively, the linear optical sampling (LOS) technique enables low-cost and high equivalent bandwidth characterization of optical pulses, serving as an effective solution to circumvent the bandwidth constraint of optoelectrical devices [12]. Moreover, the detection sensitivity of LOS can reach the shot-noise limit, owing to its inherent coherent detection [13]. Recently, LOS has been widely used to characterize various optical communication signals with distinct advanced modulation formats, including phase-modulated signals [14], 40-channel wavelength division multiplexed differential phase-shift keying (WDM-DPSK) signals [15], and 32-Gbaud polarization division multiplexed quadrature phase-shift keying (PDM-QPSK) signals [16–18]. Meanwhile, sub-THz-range linearly chirped signals have been recovered by the LOS, in order to secure the sub-millimeter resolution for optical sensing applications. Both 120-µm spatial resolution in optical fiber and 180-µm spatial resolution in free space have been realized [19]. Moreover, impulse responses of both in-phase and quadrature tributaries arising in the few-mode fiber (FMF) transmission have been measured by the LOS. A dynamic range of 80 dB and a time resolution of few picoseconds over tens of nanoseconds of differential mode group delay (DMGD) have been achieved [20]. Meanwhile, the impulse responses of a weakly-coupled homogeneous multicore fiber have been observed with the same LOS setup [21]. 6×6 spectral transfer functions arising in two non-degenerate LP modes transmission are simultaneously measured [22]. Benefitting from the high bandwidth characterization, LOS was used to evaluate the bandwidth of 11-cm long multimode polymer waveguides, featuring with an outstanding bandwidth enhancement of 241 GHz [23]. When operating at the steady-state regime, a passively mode-locked fiber laser (PMFL) normally generates pulse trains with a highly stable envelope [24]. Consequently, LOS can precisely access the full-field information of ultrashort pulse from a PMFL, facilitating the experimental verification of the NFT-based soliton distillation.
In this paper, we use the LOS technique to obtain the full-field information of ultrafast pulses from the fiber laser under test (FLUT). Two monitoring channels are used to track phase fluctuations of the FLUT and the local oscillator (LO). During the numerical simulation, the temporal profile, the optical spectrum, and the nonlinear frequency spectrum from the calculation of LOS data keep consistent with that of the original FLUT output, while the conventional coherent detection is uncapable when the pulsewidth becomes sub-picosecond and thus the severe bandwidth constraint of commercially available optoelectrical devices. Based on simulation results, we build up two PMFLs as the FLUT and the LO, respectively. An optical delay line is used to accurately adjust the repetition rate of FLUT, which can increase the equivalent sampling rate of LOS. Benefitting from the 55.56 TSa/s equivalent sampling rate, the full-field information of 1.71-ps optical pulse is correctly obtained. Finally, the NFT-based soliton distillation is experimentally verified for the first time, providing an effective methodology for the nonlinear frequency spectrum classification of ultrafast pulse.
2. Operation principle and simulation
In the LOS system, two PMFLs with slightly different repetition rates are used as the FLUT and the LO, respectively. With a 90-degree hybrid, the full-field information of both in-phase and quadrature tributaries are correctly characterized. Generally, pulses from the PMFL have stable envelopes. Meanwhile, there exist constant phase fluctuations between adjacent pulses for both the FLUT and the LO [25,26]. In the conventional coherent detection system, the phase information is directly acquired within one pulse duration. Therefore, phase fluctuations may not lead to an incorrect characterization. In contrast, only one point of the FLUT is acquired within one LO pulse cycle for the LOS. When sufficient sampling points are accumulated, the pulse from the FLUT can be reconstructed. In such a condition, phase fluctuations among pulses must be taken into account. Consequently, phase fluctuations monitoring and the corresponding mitigation are indispensable for the LOS-based soliton characterization and nonlinear frequency spectrum classification. The schematic LOS system is shown in Fig. 1(a). Three 90-degree hybrids are used for implementing three conventional coherent detections simultaneously. One is for the LOS implementation, while the others are necessary for the mitigation of phase fluctuations (MPFs). The complex amplitudes of the FLUT and the LO are
The mixing signal includes instantaneous complex electric-field of the FLUT and the relative phase fluctuation ΔϕS-ΔϕLO. As for two monitoring channels, parts of the FLUT and the LO are introduced into Hybrid 2 and Hybrid 3 by optical couplers (OCs), then optically mixed with the same CW laser, respectively. Since the spectral components of the FLUT at the operation wavelength of the CW can be effectively mixed with the CW laser, the phase fluctuation ΔϕS-ϕCW can be extracted, where ϕCW is the phase noise of the CW laser. Similarly, Hybrid 3 provides a phase fluctuation ΔϕLO-ϕCW of the LO. Finally, the relative phase fluctuation ΔϕS-ΔϕLO of the mixing signal in Eq. (3) is obtained. Then, the MPF can be implemented, with the help of digital signal processing (DSP) by multiplying the exponential term of the relative phase. Meanwhile, we can recover the mixing signal between the FLUT and the LO to obtain the full-field information of the pulse from the FLUT.
Fig. 1. (a) Linear optical sampling and (b) conventional coherent detection simulation setups and corresponding DSP flow, (c) temporal profile and (d) spectrum of the FLUT. 90° H: 2×4 90-degree hybrid, BPD: balanced photodetector, ADC: analog-to-digital converter, MPF: the mitigation of phase fluctuation, NFT: nonlinear Fourier transform, CW: continuous-wave.
We carry out numerical simulation for both the LOS and conventional coherent detection in Figs. 1(a) and 1(b). A PMFL is set as the FLUT [27]. The typical output is a hyperbolic secant pulse with a pulse width of ∼180 fs and 3-dB spectral bandwidth of 20 nm, as shown in Figs. 1(c) and 1(d). Since other fibers are SSMF with a total cavity length of 6 m, the repetition rate fFLUT of the FLUT is 33.33MHz. Another PMFL with a pulsewidth of 100 fs acts as the LO. The repetition rate fLO of the LO is 70-Hz smaller than that of the FLUT. The CW laser has a central wavelength of 1560 nm and a linewidth of 100 kHz. Noise sources include phase noise of laser source, shot noise, thermal noise, and the dark current of BPD. The corresponding parameters are chosen, according to common commercial devices. The mixing points move by Δf/(fFLUTfLO), where Δf is the repetition rate difference between the FLUT and LO. Since the equivalent time interval between adjacent sampling points is 63 fs, the corresponding equivalent sampling rate is up to 15.87 TSa/s. The data acquisition time for one FLUT pulse scanning is 1/Δf (∼14.3 ms). Ultrafast optical pulses can be broadened by the use of a narrow-band BPD. The minimum required bandwidth of BPD to guarantee that adjacent pulses do not interfere with each other, is approximately twice the repetition rate of local oscillator (LO). In the simulation, the optical-to-electrical conversion is realized by six BPDs with the 3-dB bandwidth of 400 MHz. Then, all electrical outputs are digitalized by a 6×5 GSa/s ADC. During the DSP flow, sampling points and two monitoring channels are synchronized. A peak extraction algorithm is used to extract sampling points.
Then, the MPF is implemented to ensure that the full-field information of the FLUT can be precisely obtained. Finally, the NFT methodology is applied to calculate the eigenvalue distribution of the FLUT. The pulses from the FLUT are processed with NFT and INFT algorithms, respectively, by the Boffetta-Osborne integration scheme and the Ablowitz-Ladik scheme [2]. Alternatively, the conventional coherent detection method is shown in Fig. 1(b). In order to comprehensively record the ultrashort optical pulse without distortion from the coherent detection, the operation bandwidths of both the BPD and the ADC need to satisfy the Nyquist sampling theorem. As shown in Fig. 1(d), since the 3-dB spectral width of the FLUT is 20 nm, the bandwidth of the used BPD is 1.25 THz. Moreover, the sampling rate of ADC is 5 TSa/s, which is far beyond the capability of commercial optoelectrical components.
Figure 2 shows the characterization results of the LOS and conventional coherent detection. Figure 2(a) indicates that, the temporal profile of the LOS method is almost the same as that of the original FLUT, regardless of the phase fluctuation, due to the fixed envelope of the mixing result. As for the conventional coherent detection, the recovered pulse width is broadened from 180 fs to 496 fs. As shown in Fig. 2(b), the LOS-based spectrum has a red-shift of 3.5 nm compared with the FLUT spectrum, which can be compensated by the MPF. Due to the bandwidth constraint, the spectral intensity of Kelly sidebands obtained by conventional coherent detection is 3 dB lower than that of the FLUT. In particular, the spectral intensities become much lower beyond the range of the BPD bandwidth. Next, the nonlinear frequency spectrum is calculated according to the full-field information of optical pulses, where the real and imaginary parts correspond to the frequency and amplitude, respectively. The eigenvalue corresponding to the soliton has a large imaginary part and a negligible zero real part, indicating that the soliton has zero velocity. Meanwhile, several eigenvalues corresponding to the resonant CW background have non-zero real parts and relatively small imaginary parts. As shown in Fig. 2(c), the LOS-based eigenvalues without the MPF have a shift of the real part compared with that of the FLUT, indicating a fixed frequency shift for both the soliton and the CW background. After the MPF, the LOS-based eigenvalue distribution agrees well with that of the FLUT. In contrast, the calculated eigenvalues from the conventional coherent detection cannot correctly characterize the FLUT, due to the loss of the full-field information. Therefore, the bandwidth-constrained coherent detection cannot accurately characterize the full-field information of the 180-fs optical pulse. Table 1 shows the 180-fs soliton characterization schemes and the parameters of corresponding optoelectrical components. With the help of optoelectrical components with the bandwidth far exceeding those of commercial devices, conventional balance detection is still unable to accurately characterize the pulse, while the LOS is capable of achieving this goal using low bandwidth devices. Furthermore, it is possible to implement the soliton distillation, according to different eigenvalue distributions. Consequently, both the temporal profiles and corresponding spectra of pure solitons can be reconstructed, as shown in Fig. 2(d) and (e), respectively.
Fig. 2. (a) temporal profiles, (b) spectra, (c) eigenvalue distributions, (d) temporal profiles after the soliton distillation and (e) spectra after the soliton distillation of the FLUT, LOS results with and without the MPF. MPF: the mitigation of phase fluctuation.
Table 1. 180-fs soliton characterization schemes and the corresponding characterization results
3. Experimental results and discussions
Motivated by the simulation results, we carry out the experimental verification with the setup shown in Fig. 3. Firstly, we build up two PMFLs as the FLUT and the LO, respectively. Then, the experimental implementation consists of the phase fluctuation characterization and the LOS.
Fig. 3. Experimental setup. CNT: carbon nanotube; EDF: Erbium-doped fiber; PC: polarization controller; OC: output coupler; LO: local oscillator; SESAM: semiconductor saturable absorber mirror; PD: photodetector; BPD: balanced photodetector; DSO: digital storage oscilloscope; FOE: frequency offset estimation; PM: polarization-maintained; Coated FC/PC: coated FC/PC fiber connector; CW: continuous-wave.
3.1 Fiber laser under test
As shown in Fig. 3(a), the FLUT is achieved by a PMFL based on the carbon nanotube (CNT) saturable absorber. The 10.26-m laser cavity consists of a hybrid optical component, a 1.4-m long erbium-doped fiber, an optical delay line, a fiber-based polarization controller, and several standard single mode fiber segments. It is worth noting that, the hybrid is composed of a 90:10 optical coupler, an isolator, and a 1550/980-nm wavelength division multiplexer, which can greatly reduce the cavity length. 10% of the optical power is extracted from the cavity as the laser output. The polarization controller is incorporated in the laser cavity for the optimization of the laser operation. By introducing an optical delay line with a tunable range of 300 ps and a temporal resolution of 0.05 ps into the cavity, the pulse repetition rate of the FLUT can be precisely tuned from 19.47 to 19.60 MHz with a 19-Hz tuning resolution. The pulsewidth is about 610 fs, which is quantified by a commercial autocorrelator.
3.2 Fiber laser as the LO
As shown in Fig. 3(b), the LO is realized by an all-polarization-maintaining PMFL based on a semiconductor saturable absorber mirror. The FC/PC fiber connector coated with a 20% transmission ratio mirror is used as the laser output port. The linear cavity is 1.025-m long, corresponding to a repetition rate of 97.476 MHz. Then, the laser output is amplified to satisfy the power requirement of the LO. The 980-nm pump diode is split by a 40:60 optical coupler to provide simultaneous pumping for the seed-laser and the optical amplifier. Finally, the output pulse has a pulse width of 360 fs and a 3-dB spectrum bandwidth of 8.43 nm.
3.3 Phase fluctuation characterization
We first characterize phase fluctuations of two PMFLs. Since adjacent pulses generated from the PMFL have a nearly fixed phase fluctuation, it is reasonable to use the slopes obtained by linear fitting of phases at pulse peaks as phase fluctuations between adjacent pulses. As shown in Fig. 3(c), the CW at the wavelength of 1560.12 nm from a semiconductor laser source acts as the LO, two PMFL outputs are chosen as the signal input, respectively. As shown in Fig. 4, we linearly fit the phase at pulse peaks. The phase fluctuations of the FLUT and LO obtained by the linear fitting are −120.07° and 74.9°, respectively.
Fig. 4. Peak phase and the corresponding linear fitting versus the pulse index, (a) the FLUT, and (b) the LO.
3.4 Linear optical sampling
When it comes to the LOS, as shown in Fig. 3(d), two PMFLs are split into two parts by two 50:50 optical couplers. One is for the LOS implementation, while the other is used for the direct detection, for the ease of tuning the equivalent sampling rate of LOS. It is worth noting that, the pulsewidth of the FLUT is broadened to ∼1.71 ps after the 17-m SSMF transmission, which is five times wider than the 360-fs pulsewidth of the LO. The coherent mixed optical signals are detected by 400-MHz BPDs (Thorlabs PDB470C) and captured by a real-time four-channel oscilloscope (Tektronix DPO 4104) with a sampling rate of 4×5 GSa/s. Here, the repetition rate of the FLUT is ∼19.495 MHz by precisely tuning the optical delay line. The equivalent sampling rate is 55.56 TSa/s, due to the equivalent repetition rate difference of 7 Hz [28], demonstrating the potential of linear optical sampling to characterize narrower optical pulses.
3.5 Digital signal processing
During the DSP process, the biased balance detection is used for the peak extraction. Then, frequency offset estimation is realized by the fast Fourier transform (FFT) [29]. Phase fluctuations are mitigated according to the fitting results in Fig. 4. Hence, the full-field information of an optical pulse from the FLUT can be correctly obtained. Figure 5 illustrates both the temporal profile and the calculated optical spectrum with solid lines. Due to the occurrence of resonant CW background, Kelly sideband appears on the optical spectrum, as well as noises on the temporal profile. Then, the NFT based eigenvalue calculation is implemented with the help of the correct full-field information of the ultrashort optical pulse. The complex eigenvalue distribution is shown in Fig. 5(c), where the real and imaginary parts correspond to the frequency and amplitude of the pulse, respectively. Then, the eigenvalue of the resonant CW background (circle) is filtered out and only soliton components (asterisks) remain. Finally, the pure soliton is recovered by utilizing the INFT. The temporal profile and optical spectrum of the soliton after the soliton distillation are shown in Figs. 5(a) and (b) with dashed lines. The experimental verification of soliton distillation provides a successful example of nonlinear frequency spectrum classification, which paves a new way for experimentally investigating solitons in nonlinear systems in the future. Moreover, when hardware synchronization and field programmable gate array (FPGA) based accelerated calculation are introduced, the effective time of linear optical sampling will be greatly reduced, close to the theoretical minimum time (1/Δf). We believe there occur three directions to further improve the characterization results. Firstly, under the condition of fully covering the FLUT spectrum, the LO spectrum needs to be narrowed down as much as possible, in order to improve the SNR of sampling signal. Secondly, improving the operation stability of laser is ideally desired, including temperature stabilization, phase-locking technique, polarization-maintaining resonant cavity. Finally, as long as the linear optical sampling system is relatively stable, data averaging is helpful to bring the SNR improvement [20].
Fig. 5. (a) temporal profile and (b) optical spectrum comparisons. (c) eigenvalue distribution of the fiber laser under test. SD: soliton distillation.
4. Conclusions
The LOS technique is proposed to precisely access the full-field information of ultrashort optical pulses from the PMFL and thus the experimental verification of soliton distillation is achieved. Two free-running PMFLs are used as the FLUT and the LO, respectively. Thanks to the 55.56-TSa/s ultra-high equivalent sampling rate of the LOS, we experimentally obtain the correct full-field information of the pulse from the FLUT after the MPF. Consequently, the nonlinear frequency spectrum classification leads to the experimental verification of soliton distillation concept. The proposed LOS-based pulse characterization together with the NFT technique are suitable for more complicated soliton dynamics, such as high-order soliton, soliton molecule, and soliton bunch, which will stimulate more soliton nonlinear frequency spectrum classifications, leading to the insight reinforcement of ultrafast fiber lasers.
Funding
Natural Science Foundation of Hubei Province (2020CFB440); National Natural Science Foundation of China (62005255); National Key Research and Development Program of China (2018YFB1801000).
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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