Cite this Article
Zhou Yan, Li Yuefeng, Li Xia, Liao Meisong, Hou Jingshan, Fang Yongzheng. Pulse shaping of bright-dark vector soliton pair. Chinese Physics B, 2020, 29(5): 054202  
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Pulse shaping of bright-dark vector soliton pair
Zhou Yan1, †, Li Yuefeng1, Li Xia2, Liao Meisong2, Hou Jingshan3, Fang Yongzheng3
School of Science, Shanghai Institute of Technology, Shanghai 201418, China
Key Laboratory of Materials for High Power Laser, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China
School of Materials Science and Engineering, Shanghai Institute of Technology, Shanghai 201418, China

 

† Corresponding author. E-mail: yzhou@sit.edu.cn

Project supported by the National Key Research and Development Program of China (Grant No. 2018YFB0504500) and the National Natural Science Foundation of China (Grant No. 51672177).

Abstract

We simulate pulse shaping of bright–dark vector soliton pair in an optical fiber system. Through changing input pulse parameters (amplitude ratio, projection angle, time delay, and phase difference), different kinds of pulse shapes and spectra can be generated. For input bright–dark vector soliton pair with the same central wavelength, “2+1”- and “2+2”-type pseudo-high-order bright–dark vector soliton pairs are achieved. While for the case of different central wavelengths, bright–dark vector soliton pairs with multiple pulse peaks/dips are demonstrated with appropriate pulse parameter setting.

PACS: ;42.25.Ja;;42.25.Lc;;42.55.Wd;;42.65.-k;
Keyword:;bright-dark vector soliton;;birefringence;;polarization-locking;;group-velocity-locking;
1. Introduction

Optical soliton is a kind of localized nonlinear wave in nonlinear optical systems, and it can propagate along long distances without pulse shape distortion. Optical solitons have been widely studied theoretically and experimentally. In theory, soliton solutions can be obtained by solving nonlinear Schrödinger equation (NLSE) and Ginzburg–Landau equation (GLE).[1] When the amplitude modulations (including gain, loss, spectral filtering, and saturable absorption) in GLE are negligible, GLE can be approximated with NLSE. While in experiments, optical solitons can be achieved in passively mode-locked solid-state and fiber lasers. Especially, solitons in ultrafast fiber laser field have been attracting much attention, because of the unique characteristics (including dispersion, nonlinearity, gain, and loss) offered by single-mode and multi-mode fibers. Currently, many researches are focused on optical solitons in single-mode fiber lasers, because of their compactness, stability, ease of maintenance, and so on. Until now, different kinds of optical solitons that exist in fiber lasers have been theoretically solved and experimentally demonstrated, such as traditional soliton,[2,3] dispersion-managed soliton,[4,5] self-similariton,[6,7] dissipative solion,[8,9] dual-wavelength domain wall soliton,[10] and so on.[11–14] They can be generated by appropriate design of intra-cavity parameters. Also, multi-state solitons can be emitted in bidirectional mode-locked fiber lasers.[15] As the development of detection techniques, soliton dynamics can be explored with time-stretch dispersive Fourier transform (TSDFT) technique,[16–19] which reveals the transient dynamics of soliton evolution and building processes.

In reality, single-mode-fiber is anisotropic even with circular symmetry, because of material imperfections and external environmental disturbance (such as stress and heat flow). So scalar approximation is not applicable and coupled NLSEs and GLEs should be considered for solving.[20] In this situation, various vector solitons could be generated, such as polarization/phase-locked vector soliton (PLVS),[21–23] group-velocity-locked vector soliton (GVLVS),[24–27] polarization rotation vector soliton (PRVS),[28,29] and so on, depending on the strength of intra-cavity linear birefringence and cross-phase modulation (XPM). Also, we can classify the vector solitons according to their pulse waveforms, such as bright–bright vector soliton, bright–dark vector soliton and dark–dark vector soliton. Bright soliton has a temporal intensity peak, while dark soliton has a temporal intensity dip with continuous light background. Vector solitons have potential applications in optical communications and optical data processing.[30,31]

In previous works, many different kinds of vector solitons have already been observed in mode-locked fiber lasers. For example, coherently and incoherently coupled bright–dark vector solitons in single-mode fiber lasers were experimentally shown.[32,33] Simulation about generating pseudo-high-order polarization-locked vector soliton was reported.[34] Manipulation of polarization- and group-velocity-locked dark vector solions was also theoretically simulated.[35] However, there are few reports about the pulse shaping of bright–dark vector soliton pair. In this paper, we simulate the pulse shaping of bright–dark vector solion pair in a polarization insensitive optical fiber system. Through changing input pulse parameters, different kinds of pulse shapes and optical spectra in orthogonal polarization directions can be generated. Our simulation result further explore the vector properties of optical soliton, and can provide beneficial conduct for latter experiments about pulse shaping of bright–dark vector soliton pair at different wavelengths.

2. Simulation

Figure 1 is the depiction of our theoretical model. It is a polarization-insensitive optical fiber system. “A2/A1 change” section is used for changing amplitude ratio of orthogonal modes and it can be adjusted by fiber amplifier or fiber attenuator; “ΔT change” section is for changing time delay between orthogonal modes and it can be changed with optical fiber time delay line; SMF is single-mode-fiber; PC (polarization controller) is for changing linear fiber birefringence; Col (collimator) is employed for collimating laser beam; PBS (polarization beam splitter) can separate orthogonally polarized electric fields and can also be rotated to change projection angle between the polarization direction and the projection axes. Bright–dark vector soliton pair passes through different stages one by one in order, and finally detected by photo-detector after passing through or being reflected by the PBS. Ahorizontal and Avertical stand for the amplitude projection along horizontal and vertical polarization directions.

Fig. 1. Picture of theoretical model.

Firstly, the temporal pulse amplitudes A1(t) and A2(t) of incident bright–dark vector solion pair in orthogonal polarization directions can be expressed as

In Eqs. (1) and (2), we make the assumption that horizontal mode is black soliton with “tanh”-type pulse shape, and vertical mode is bright soliton with “sech”-type pulse shape. A1 and A2 are amplitudes in vertical and horizontal polarization directions; t is time parameter; ΔT stands for time delay between orthogonal modes; T1 and T2 are pulse widths for vertical and horizontal mode; c is light velocity in a vacuum; Δφ is phase difference between orthogonal modes and it includes two parts: one is the phase difference of incident orthogonal polarization modes, the other is the linear birefringence-induced phase difference imposed in the optical fiber system; λ1 and λ2 are central wavelengths of orthogonal modes. We assume the projection angle is θ, then the projected amplitudes along horizontal and vertical polarization directions can be calculated according to

In the latter simulation, we let T1 = T2 = 5 ps and simulate the temporal pulse shapes and optical spectra of output horizontal and vertical polarization modes, through changing amplitude ratio, projection angle, time delay and phase difference for orthogonal modes. The simulation results include two parts: the first is considering orthogonal modes with the same central wavelength, and the second one considers those with different central wavelengths.

2.1. λ1 = λ2 = 1064 nm

In this section, we assume λ1 = λ2 = 1064 nm. The 1064-nm central wavelength is selected because it can be generated in Yb-doped mode-locked fiber laser experimentally. Firstly, we consider the case when input amplitude ratio of bright–dark soliton pair is changed, and the corresponding simulation result is shown in Fig. 2. Here, we simulate the changes of output pulse shapes and optical spectra when amplitude ratio (A2/A1) is decreased from 3/1 to 1/3. Other variables are set as: θ = 0°, ΔT = 0 ps, and Δφ = π/2. The phase difference is fixed at π/2, which is the typical characteristic of PLVS with ± π/2 phase difference in ytterbium fiber laser. The output pulse shapes in orthogonal polarization directions (see Figs. 2(a)–2(e)) shows the same tendency with amplitude ratio. While for optical spectra in Figs. 2(f)–2(j), both horizontal and vertical polarization modes have the same 3-dB bandwidth with different peak intensities. And the peak intensity is still different even though with the same input amplitude for orthogonal polarization modes. In all the simulated optical spectra, the optical spectrum resolution is ∼ 0.15 nm.

Fig. 2. Simulated output pulse shapes [(a)–(e)] and optical spectra [(f)–(j)] in orthogonal polarization directions when input amplitude ratio (A2/A1) is set as: 3/1, 2/1, 1/1, 1/2, and 1/3. Others are: θ = 0°, ΔT = 0 ps, and Δφ = π/2.

Then, we consider the change of projection angle θ. Other variables are: A1 = A2 = 1, ΔT = 3 ps, and Δφ = π/2. In Figs. 3(a)–3(e), we can see that as θ increases from 15° to 70°, the peak/dip pulse shape in horizontal/vertical direction will gradually evolve into dip/peak pulse shape. Besides, “2+2”-type pseudo-high-order bright–dark vector soliton pair occurs when θ = 44.5°. We call it “pseudo-high-order” rather than “high-order” because the phase of amplitude projection along two principle axes of PBS is not locked. It should be noted that the pulse shapes in orthogonal polarization directions will coincide with each other when θ = 45° because of symmetry projection. Besides, there is only one peak or dip in one polarization direction when ΔT = 0 ps. The peak/dip of bright/dark soliton is not equal to one/zero because of energy transfer between horizontal mode and vertical mode. For output optical spectra in Figs. 3(f)–3(j), both orthogonal polarization modes have different 3-dB bandwidth and slight peak intensity difference. Horizontal mode has narrower/broader bandwidth when θ is smaller/larger than 45°, but the critical point is different for optical spectrum peak intensity.

Fig. 3. Simulated output pulse shapes [(a)–(e)] and optical spectra [(f)–(j)] in orthogonal polarization directions when projection angle θ is set as: 15°, 30°, 44.5°, 60°, and 75°. Others are: A1 = A2 = 1, ΔT = 3 ps, and Δφ = π/2.

After that, we simulate the pulse shapes and optical spectra when time delay ΔT is changed from 1 ps to 10 ps. Others are: A1 = A2 = 1, θ = 10°, and Δφ = π/2. The corresponding simulation results are demonstrated in Figs. 4 and 5. In Fig. 4, the time separation of bright and dark solitons increases because the increasing ΔT value. Besides, the dark soliton in vertical polarization is gray soliton or black soliton, depending on ΔT. For optical spectra in Fig. 5, the peak intensity of orthogonal polarization modes is almost unchanged, accompanied with intensity difference ∼ 3 dB. The centering wavelength is 1064 nm and 3-dB bandwidth is smaller than 1 nm for both polarization modes, but the bandwidth is larger compared with Fig. 3.

Fig. 4. Simulated output pulse shapes [(a)–(j)] in orthogonal polarization directions when time delay ΔT increases from 1 ps to 10 ps (1-ps time interval). Others are: A1 = A2 = 1, θ=10°, and Δφ = π/2.
Fig. 5. Simulated output optical spectra [(a)–(j)] in orthogonal polarization directions when time delay ΔT increases from 1 ps to 10 ps (1-ps time interval). Others are the same as those in Fig. 4.

At last, we consider the influence of the phase difference Δφ on the output pulse properties. Figure 6 shows the simulation result when Δφ is increased from 0 to π. From pulse shapes of Figs. 6(a)–6(e), we can see that the peak intensity of horizontal bright soliton decreases with the increased Δφ. While for the vertical dark soliton, the dip intensity is almost unchanged, and another peak appears when Δφ reaches to 3π/4, which means that the “1+2”-type pseudo-high-order bright–dark vector soliton pair can be generated with appropriate Δφ value. Besides, the vertical dark soliton is always black soliton with all Δφ values. Figures 6(f)–6(j) are the corresponding optical spectra, and the figures show that the peak intensities of horizontal/vertical polarization modes have slight increase with the increased Δφ, and meanwhile, the peak intensity differences increase from 1.6 dB to 4.0 dB.

Fig. 6. Simulated output pulse shapes [(a)–(e)] and optical spectra [(f)–(j)] in orthogonal polarization directions when Δφ is set as 0, π/4, π/2, 3π/, and π. Others are: A1 = A2 = 1, θ=10°, and ΔT = 5 ps.
2.2. λ1 = 1063 nm, λ2 = 1065 nm

In this section, we simulate the output pulse shapes and optical spectra when λ1 ≠ λ2, which is the characteristic of GVLVS. Because for GVLVS, the two orthogonal polarization modes will shift their central wavelengths, combining with self-phase modulation (SPM) and XPM to compensate linear intra-cavity fiber birefringence. In later simulations, we set λ1 = 1063 nm and λ2 = 1065 nm, and the simulation procedure is similar to the former case when λ1 = λ2. Firstly, the amplitude ratio (A2/A1) of incident pulses is decreased from 3/1 to 1/3, and others are set as: θ = 0°, ΔT = 0 ps, and Δφ = 0. The simulation result is shown in Fig. 7. We can see that both pulse shape and optical spectrum amplitudes of orthogonal polarization modes have the same tendency with A2/A1 change. For output optical spectra, the horizontal mode with red color always has only one peak, while vertical mode always has two peaks, which is different from the case when λ1 = λ2.

Fig. 7. Simulated output pulse shapes [(a)–(e)] and optical spectra [(f)–(j)] in orthogonal polarization directions when amplitude ratio (A2/A1) is set as: 3/1, 2/1, 1/1, 1/2, and 1/3. Others are: θ=0°, ΔT = 0 ps, and Δφ = 0.

Then, we change projection angle θ and simulate the pulse shaping properties for incident group-velocity-locked bright–dark vector soliton pair. Figure 8 shows the corresponding simulation result when θ is increased from 15° to 75°, and other parameters are fixed at: A1 = A2 = 1, ΔT = 0 ps, and Δφ = 0. In Figs. 8(a)–8(e), it shows that both orthogonal modes have only one peak/dip with multiple side pulse oscillations when θ = 15°. When θ = 40°, side pulse oscillations become obvious and the main peak/dip is undistinguishable, chirp-like pulses appear in horizontal and vertical polarization directions. And the orthogonal polarization modes will have the same temporal oscilloscope and the pedestal value is equal to 0.5 when θ = 45°. For output optical spectra in Figs. 8(f)–8(j), both orthogonal modes have two peaks orientated at 1063 nm and 1067 nm. The red-shift of 1065 nm to 1067 nm may attribute to the energy transfer through amplitude projection along the two PBS axes. When θ is smaller than 45°, the horizontal mode has higher/lower peak intensity at 1063 nm/1067 nm wavelength position compared with vertical mode, and the result will become contrary when θ is larger than 45°. The orthogonal polarization modes have the same peak intensity at the above-mentioned two wavelengths when θ = 45° because of the projection symmetry.

Fig. 8. Simulated output pulse shapes [(a)–(e)] and optical spectra [(f)–(j)] in orthogonal polarization directions when projection angle θ is set as: 15°, 30°, 40°, 60°, and 75°. Others are: A1 = A2 = 1, ΔT = 0 ps, and Δφ = 0.

After that, the time delay ΔT is changed while with other variables fixed at appropriate values. ΔT increases from 1 ps to 10 ps while others are set as: A1 = A2 = 1, θ = 10°, and Δφ = 0. The simulation results are demonstrated in Fig. 9 and Fig. 10. In Fig. 9, we can see the horizontal mode is always bright soliton pulse with multiple side peaks across the whole temporal pulse shape. The peak intensity increases when ΔT is increased from 1 ps to 4 ps, and will have slight oscillation in latter 6 ps. For vertical mode, the dark soliton pulse is always black soliton, and as the increase of the time delay, chirp-like oscillated multiple-pulses appear at the pulse trailing edge and they become obvious when ΔT is larger than 2 ps. Figure 10 is the corresponding simulated output optical spectra. It shows that both orthogonal modes have two wavelength bands at 1063 nm and red-shifted 1067 nm, which is similar to the case with angle changes. Besides, the peak intensity of the horizontal mode is always higher/lower than the vertical mode at 1063 nm/1067 nm with different time delay values.

Fig. 9. Simulated output pulse shapes [(a)–(j)] in orthogonal polarization directions when time delay ΔT increases from 1 ps to 10 ps (1-ps time interval). Others are: A1 = A2 = 1, θ = 10°, and Δφ = 0.
Fig. 10. Simulated output optical spectra [(a)–(j)] in orthogonal polarization directions when time delay ΔT increases from 1 ps to 10 ps (1-ps time interval). Others are the same as those in Fig. 9.

At last, we simulate the pulse shaping through changing phase difference Δφ. Figure 11 is the simulation result when Δφ is changed from 0 to π, and other variables are set as: A1 = A2 = 1, θ = 10°, and ΔT = 5 ps. In Figs. 11(a)–11(e), we can see that horizontal polarization mode is always bright soliton with multiple side peaks, while vertical polarization mode is always black soliton with chirp-like oscillated pulses at pulse trailing edge position. Figures 11(f)–11(j) are the corresponding optical spectra, which show that the horizontal mode has higher/lower peak intensity at 1063 nm/1067 nm compared with the vertical mode, and the wavelength splits at 1067-nm position for both orthogonal polarization modes.

Fig. 11. Simulated output pulse shapes [(a)–(e)] and optical spectra [(f)–(j)] in orthogonal polarization directions when Δφ is set as 0, π/4, π/2, 3π/4, and π. Others are: A1 = A2 = 1, θ = 10°, and ΔT = 5 ps.

It should be noted that the simulation results in Figs. 9,10, and 11 are different. For pulse shapes in Fig. 9, the vertical dark soliton has side pulse oscillations at pulse leading edge when ΔT is not larger than 3 ps, while in Figs. 11(a)–11(e), the vertical dark soliton always has smooth pulse leading edge with different Δφ values. For optical spectra in Fig. 10, both orthogonal modes have multiple side peaks at 1067 nm and the number of peaks decreases with increased ΔT. While in Figs. 11(f)–11(j), the number of side peaks at 1067-nm position is almost unchanged.

3. Conclusion

We simulated the pulse shaping of bright–dark vector soliton pair at 1-μm wavelength region. Through changing input pulse pair parameters, various kinds of pulse shapes and optical spectra can be generated. When the orthogonal modes have the same central wavelength, “2+2”- and “1+2”-type pseudo-high-order polarization-locked bright–dark vector soliton pairs can be generated. While for different central wavelengths, group-velocity-locked bright–dark vector soliton pairs with chirp-like oscillated pulses at pulse trailing edge position can be generated. Our simulations can provide beneficial conduct for experiments about pulse shaping of bright–dark vector soliton pair with different wavelengths.

Reference
[1] Song Y F Shi X J Wu C F Tang D Y Zhang H 2019 Appl. Phys. Rev. 6 021313
[2] Luo Z C Liu M Luo A P Xu W C 2018 Chin. Phys. 27 094215
[3] Zhou Y Zhang R L Li X Kuan P W He D Y Hou J S Liu Y F Fang Y Z Liao M S 2019 Chin. Phys. 28 094203
[4] Chen S Xu Y Cai Y Shen J P Zhang Z X 2019 J. Opt. Soc. Am. 36 2688
[5] Yin K Li Y M Wang Y B Zheng X Jiang T 2019 Chin. Phys. 28 124203
[6] Wang Z Q Zhan L Fang X Luo H 2017 J. Opt. Soc. Am. 34 2325
[7] Chong A Wright L G Wise F W 2015 Rep. Prog. Phys. 78 113901
[8] Duan L N Wen J Fan W Wang W 2017 Chin. Phys. 26 104205
[9] Zhou Y Zhang R L Chen P Liu Y F Fang Y Z Wang T X Li X Kuan P W Liao M S 2019 Laser Phys. 29 055101
[10] Luo Z C Lin Z B Li J Y Zhu P F Ning Q Y Xing X B Luo A P Xu W C 2014 Chin. Phys. 23 064203
[11] Zheng Y Tian J R Dong Z K Xu R Q Li K X Song Y R 2017 Chin. Phys. 26 074212
[12] Wang H C Wei Y D Huang X Y Chen G H Ye H 2018 Chin. Phys. 27 044203
[13] Zhao J Q Zhou J Li L Zhao L M Tang D Y Shen D Y Su L 2019 Opt. Lett. 44 2414
[14] Xiong Z J Xu Q Ling L M 2019 Chin. Phys. 28 120201
[15] Liu B W Luo Y Y Xiang Y Xiao X P Sun Q Z Liu D M Shum P P 2018 Opt. Express 26 27461
[16] Liu X M Yao X K Cui Y D 2018 Phys. Rev. Lett. 121 023905
[17] Liu X M Pang M 2019 Laser Photon. Rev. 13 1800333
[18] Liu X M Cui Y D 2019 Adv. Photon. 1 016003
[19] Liu X M Popa D Akhmediev N 2019 Phys. Rev. Lett. 123 093901
[20] Balla P Agrawal G P 2018 J. Opt. Soc. Am. 35 2302
[21] Cundiff S T Collings B C Akhmediev N N Soto-Crespo J M Bergman K Knox W H 1999 Phys. Rev. Lett. 82 3988
[22] Collings B C Cundiff S T Akhmediev N N Soto-Crespo J M Bergman K Knox W H 2000 J. Opt. Soc. Am. 17 354
[23] Tang D Y Zhang H Zhao L M Wu X 2008 Phys. Rev. Lett. 101 153904
[24] Jin X X Wu Z C Li L Zhang Q Tang D Y Shen D Y Fu S N Liu D M Zhao L M 2016 IEEE Photon. J. 8 1501206
[25] Wang X Li L Geng Y Wang H X Su L Zhao L M 2018 Appl. Opt. 57 746
[26] Xiang Y Luo Y Y Liu B W Xia R Shum P P Tang X H Liu D M Sun Q Z 2019 J. Lightwave Technol. 37 5108
[27] Zhu S N Wu Z C Fu S N Zhao L M 2018 Appl. Opt. 57 2064
[28] Zhao L M Tang D Y Wu X 2008 Opt. Express 16 10053
[29] Liu M Luo A P Luo Z C Xu W C 2017 Opt. Lett. 42 330
[30] Willner A E Wang J Huang H 2012 Science 337 655
[31] Su Y L Feng H Hu H Wang W Duan T Wang Y S Si J H Xie X P Yang H N Huang X N 2019 Chin. Phys. 28 024216
[32] Ma J Shao G D Song Y F Zhao L M Xiang Y J Shen D Y Richardson M Tang D Y 2019 Opt. Lett. 44 2185
[33] Hu X Guo J Shao G D Song Y F Yoo S W Malomed B A Tang D Y 2019 Opt. Express 27 18311
[34] Zhou Y Li Y F Zhang R L Wang T X Bi W J Li X Kuan P W Fang Y Z Liao M S 2019 Optik 194 163132
[35] Zhou Y Li Y F Li X Zhao G Y Hou J S Zou J Fang Y Z Liao M S 2020 Optik 203 163925