- Email: [email protected]
Generation of pseudo-high-order polarization-locked vector solitons with super-Gaussian pulses
Optik - International Journal for Light and Electron Optics 194 (2019) 163132
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Generation of pseudo-high-order polarization-locked vector solitons with super-Gaussian pulses
T
⁎
Yan Zhoua, , Yuefeng Lia, Renli Zhangb, Tianxing Wangb, Wanjun Bib, Xia Lib, Peiwen Kuanb, Yongzheng Fangc, Meisong Liaob a
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 c School of Materials Science and Engineering, Shanghai Institute of Technology, Shanghai 201418, China b
A R T IC LE I N F O
ABS TRA CT
Keywords: Vector soliton Birefringence Fiber Polarization-locking Chirp Phase difference Time delay
In this paper, we simulate the generation of pseudo-high-order polarization-locked vector solitons (PLVSs) based on fundamental super-Gaussian PLVS. Through changing pulse chirp, pulse amplitude, projection angle, phase difference and time delay of incident pulses, different kinds of pseudo-high-order PLVS with the same central wavelength can be achieved in a polarization resolved fiber system.
1. Introduction Optical soliton is a kind of localized nonlinear wave, and it can be described by nonlinear Schrödinger equation (NLSE) [1–4]. Ultrafast fiber laser is a perfect platform to study the optical soliton. So far, several types of solitons have been theoretically and experimentally demonstrated in fiber lasers, such as conventional soliton [5,6], dark soliton [7,8], dispersion-managed soliton [9,10], dissipative soliton [11,12], dissipative soliton resonance [13,14] and so on [15,16]. Because of stress, fabrication process and other factors, single-mode-fiber (SMF) is not perfect isotropic, and it can support two orthogonal polarization modes. Due to the intrinsic birefringence of SMF, vector solitons (VSs) can be generated in SMFs and mode-locked fiber lasers. The coupled NLSEs can be employed to describe VSs. For VSs that generated in SMFs, the orthogonal polarization modes shift their own central frequency to compensate the linear birefringence. While in dissipative mode-locked fiber laser system, the generation of VSs is due to the balance between gain, loss, group-velocity dispersion (GVD), self-phase modulation (SPM), linear birefringence and so on. There are different kinds of VSs, such as group-velocity-locked vector soliton (GVLVS) [17–22], polarization-locked vector soliton (PLVS) [23–27], polarization rotation vector soliton (PRVS) [28–30] and so on. Among them, PLVS can be generated in relatively weak birefringence environment. The linear birefringence can be compensated with nonlinear birefringence (self-phase and cross-phase modulation). The orthogonal polarization modes of PLVS has the same central wavelength, the phase difference is fixed with ± π/2 and there is no energy transfer between them. The PLVS was firstly experimentally demonstrated in a fiber laser [23]. And “2 + 1” type high-order PLVS has also been demonstrated in a mode-locked Er-doped fiber laser in 2008 [26]. As for other VSs, for example, pseudo-highorder GVLVS with two humps in one polarization direction and one in orthogonal direction was simulated and experimentally
⁎
Corresponding author. E-mail address: [email protected] (Y. Zhou).
https://doi.org/10.1016/j.ijleo.2019.163132 Received 7 July 2019; Received in revised form 13 July 2019; Accepted 18 July 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 194 (2019) 163132
Y. Zhou, et al.
Fig. 1. Schematic diagram of polarization resolved optical fiber system.
demonstrated [19,20]. However, there are few report about the generation of pseudo-high-order PLVS. In this paper, we numerically generate pseudo-high-order PLVS through polarization controller (PC) and polarization beam splitter (PBS) in a polarization resolved fiber system. Pseudo-high-order PLVSs that consist more than two humps for orthogonal polarization modes can be obtained. The combined laser pulses along two principle axes of PBS are not locked together, so we call it “pseudo-high-order” PLVS. 2. Simulation The polarization resolved system is shown in Fig. 1, which is similar in Ref. [22]. A PLVS which has super-Gaussian pulse shape for horizontal and vertical polarization modes with the same central wavelength passes through a piece of SMF, and also goes through a PC and a PBS. The PC is to control and change intra-cavity birefringence, adjusting the phase difference between two orthogonal polarization components. The fiber collimator is for collimating the laser, which makes the laser incident perpendicularly on the PBS. The PBS acts as two important roles: one is changing the projection of PLVS along its principle axes, the other is to separate the orthogonal polarization modes for measurement. Two orthogonal polarization laser beams can be got with the PBS after the collimator. The incident orthogonally polarized wave A1(t) and A2(t) can be expressed as:
A1 (t ) = A1 e
−
A2 (t ) = A2 e
1.6652 (1 + iC1) t − ΔT /2 ⎛ ⎞ 2 ⎝ T1 ⎠
−
4
1.6652 (1 + iC2) t + ΔT /2 ⎛ ⎞ 2 ⎝ T2 ⎠
e
i 2πct λ1
(1)
i ( 2πct + Δφ) λ2
(2)
4
e
A1 and A2 are amplitudes of vertical and horizontal polarization modes. C1 and C2 are chirp parameters. t is time parameter and △T is time delay between two pulses. T1 and T2 are pulse width of incident pulses. c is light velocity in a vacuum. λ1 and λ2 are wavelengths of two pulses. △φ is the phase difference of orthogonal polarization modes. The orthogonal polarization projection after the PBS can be expressed:
Ahorizontal = A1 (t )cos θ + A2 (t )sin θ
(3)
Avertical = A1 (t )sin θ − A2 (t )cos θ
(4)
θ is the projection angle. In the simulation, we assume T1=T2 = 5 ps, λ1=λ2 = 1064 nm. Fig. 2 shows the change of temporal pulse waveform and optical spectrum of orthogonal polarization modes with different chirp values of 5、10、15、20 and 30. Other parameters are set as: A1=A2 = 1, θ = 0°, △φ=π/2, △T = 0.5 ps. We can see that orthogonal polarization modes have the same peak intensity and pulse width with different chirps. Also, the pulse width and pulse interval are unchanged. The optical spectrum for horizontal (red color) and vertical (black color) polarization mode is almost the same, with triangle shape and a small peak on the top. The 3-dB bandwidth is less than 1 nm with increased chirps. The corresponding peak intensity of the optical spectrum is -34.5 dB、-35.5 dB、-36.1 dB、-36.5 dB and -37.1 dB, respectively. Then, we consider the changes of temporal pulse waveform and optical spectrum with different amplitude ratio of incident fundamental super-Gaussian pulses, and the corresponding result is shown in Fig. 3 with amplitude ratio (A2/A1) of 2、4、6、8、10.
Fig. 2. The temporal pulse waveform (a–e) and optical spectrum (f–j) of horizontal and vertical polarization modes when chirp parameter (C1=C2) is set as 5、10、15、20 and 30. Other parameters are set as: A1=A2 = 1, θ = 0°, △φ=π/2, △T = 0.5 ps. 2
Optik - International Journal for Light and Electron Optics 194 (2019) 163132
Y. Zhou, et al.
Fig. 3. The temporal pulse waveform (a–e) and optical spectrum (f–j) of horizontal and vertical polarization modes when A1 = 1, while A2 is set as 2、4、6、8 and 10. Other parameters are set as: C1=C2 = 30, θ = 0°, △φ=π/2, △T = 0.5 ps.
Other parameters are set as: C1=C2 = 30, θ = 0°, △φ=π/2, △T = 0.5 ps. The amplitude of horizontal polarization mode (red color) is unchanged, but the vertical polarization mode (black color) increases with increased amplitude ratio because of the increased amplitude ratio. For optical spectrum, the intensity of vertical component is higher than horizontal component, and the peak intensity difference is 2.1 dB、4.1 dB、5.4 dB、6.2 dB and 6.9 dB respectively for increased amplitude ratio. After that, we consider the case when the projection angle θ between horizontal component and principle axis direction is changed. Other parameters are set as: A1=A2 = 1, C1=C2 = 30, △φ=π/2, △T = 0.5 ps. Fig. 4 shows the pulse waveform and optical spectrum of horizontal (red color) and vertical (black color) polarization component with different θ values. We can see the temporal pulse waveform is symmetric with respect to zero point, and the peak position for horizontal (or vertical) component is at the valley of vertical (or horizontal) component because of π/2 phase difference, while adjacent two peaks or valleys have π phase difference. Besides, more than two humps appear with specific θ value. For example, when θ = 80°, there are five humps in the orthogonal polarization directions, meaning “5 + 5” type pseudo-high-order PLVS is generated. The optical spectrum for orthogonal polarization modes both have triangle shape, but accompanied with multiple peaks or valleys, and a sharp peak at 1064 nm wavelength. These become obvious when θ = 45°, which may due to that the amplitude projection in orthogonal polarization direction is equal in this case. Fig. 5 gives the simulation result of orthogonal polarization modes’ pulse waveform when time delay between the incident two pulses is from 0 to 1.8 ps with 0.2 ps time interval. We can see the number of humps for horizontal or vertical polarization mode increases with increased time delay. There is only one hump for orthogonal polarization modes when △T = 0, but the hump number increases with increased △T. The peak intensity of main hump increases when the time delay reaches 0.6 ps, and becomes relatively stable in latter 1.2 ps. The time interval between adjacent main hump is 5.3 ps when △T = 0.2 ps, and decreases to 2.6 ps when △T = 1.8 ps. Fig. 6 gives the corresponding optical spectrum with different time delay ranging from 0 to 1.8 ps. The number of humps for two orthogonal polarization modes both increases with the time delay. The peak intensity is -37.1 dB when △T = 0, and for other cases, the peak intensity is in the range between -36 dB and-37 dB, meaning without obvious change for optical spectrum peak intensity with different time delay. At last, we consider the influence of phase difference △φ on pulse waveform and optical spectrum, △φ can be changed through rotating PC or bending SMF. The simulation result is shown in Fig. 7. We can see the pulse waveform is symmetric when △φ = 0 but becomes asymmetric when △φ ≠ 0. Horizontal mode always has six humps while vertical mode has five for different △φ values, which means “6 + 5” pseudo-high-order PLVS can be generated with appropriate parameters. The peak intensity difference decreases with increased △φ. In the optical spectrum of Fig. 7(f–j), the horizontal and vertical polarization mode has 5 and 6 obvious humps, respectively.
Fig. 4. The temporal pulse waveform (a–e) and optical spectrum (f–j) of horizontal and vertical polarization modes when projection angle θ is set as 10°、30°、45°、60° and 80°. Other parameters are set as: A1=A2 = 1, C1=C2 = 30, △φ=π/2, △T = 0.5 ps. 3
Optik - International Journal for Light and Electron Optics 194 (2019) 163132
Y. Zhou, et al.
Fig. 5. The temporal pulse waveform (a–j) of horizontal and vertical polarization modes when time delay △T is changed from 0 to 1.8 ps with 0.2 ps time interval. Other parameters are set as: A1=A2 = 1, C1=C2 = 30, θ = 45°, △φ=π/2.
Fig. 6. The temporal optical spectrum (a–j) of horizontal and vertical polarization modes when time delay △T is changed from 0 to 1.8 ps with 0.2 ps time interval. Other parameters are the same as in Fig. 5.
Fig. 7. The temporal pulse waveform (a–j) and optical spectrum (f–j) of horizontal and vertical polarization modes when △φ is 0°、20°、40°、60° and 80°. Other parameters are set as: A1=A2 = 1, C1=C2 = 30, θ = 45°, △T = 0.5 ps.
3. Conclusion In this paper, we simulate the generation of pseudo-high-order PLVS, in a polarization resolved fiber system. Through changing pulse chirp, amplitude ratio, projection angle, time delay and phase difference, different kinds of pseudo-high-order PLVS with 1064 nm central wavelength can be generated. Such as “3 + 3” type, “5 + 5” type and “6 + 5” type PLVSs. These simulations can provide beneficial conduct for latter experiment.
Acknowledgments Project supported by National Key Research and Development Program of China (2018YFB0504500), National Natural Science Foundation of China (NSFC) (51472162, 51672177, 61475171), the talent introduction research project of Shanghai Institute of Technology (YJ 2018-8).
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Optik - International Journal for Light and Electron Optics 194 (2019) 163132
Y. Zhou, et al.
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A 80 (4) (2009) 045803. [8] Y.Q. Ge, Z.M. Liang, Y.X. Chen, Y.F. Song, L. Li, H. Zhang, L.M. Zhao, D.Y. Fan, Characterization of dark soliton sidebands in all-normal-dispersion fiber lasers, IEEE J. Sel. Top. Quantum Electron. 24 (3) (2018) 0903407. [9] D.J. Lei, H. Dong, Generalized characteristics of soliton in dispersion-managed fiber lasers, Optik 124 (16) (2013) 2544–2548. [10] J.H. Shi, L. Chai, X.W. Zhao, J. Li, B.W. Liu, M.L. Hu, Y.F. Li, C.Y. Wang, Femtosecond pulse coupling dynamics between a dispersion-managed soliton oscillator and a nonlinear amplifier in an all-PCF-based laser system, Optik 145 (2017) 29–32. [11] P. Grelu, N. Akhmediev, Dissipative solitons for mode-locked fiber laser, Nat. Photonics 6 (2) (2012) 84–92. [12] L.N. Duan, L. Li, Y.G. Wang, X. Wang, All-fiber dissipative soliton laser based on single-walled carbon nanotube absorber in normal dispersion regime, Optik 137 (2017) 308–312. [13] W. Chang, N. Ankiewicz, J.M. Soto-Crespo, N. Akhmediev, Dissipative soliton resonances, Phys. Rev. A 78 (2) (2008) 033840. [14] Y.Y. Guo, Y. Xu, J.J. Zhang, Z.X. Zhang, High-power dissipative soliton resonance fiber laser with compact linear-cavity configuration, Optik 181 (2019) 13–17. [15] M. Gilles, P.Y. Bony, J. Garnier, A. Picozzi, M. Guasoni, J. Fatome, Polarization domain walls in optical fibres as topological bits for data transmission, Nat. Photonics 11 (2) (2017) 102–107. [16] J.S. Peng, H.P. Zeng, Dynamics of soliton molecules in a normal-dispersion fiber laser, Opt. Lett. 44 (11) (2019) 2899–2902. [17] C.R. Menyuk, Stability of solitons in birefringent optical fibers. I: equal propagation amplitudes, Opt. Lett. 12 (8) (1987) 614–616. [18] C.R. Menyuk, Stability of solitons in birefringent optical fibers. II. Arbitrary amplitudes, J. Opt. Soc. Am. B 5 (2) (1988) 392–402. [19] X.X. Jin, Z.C. Wu, Q. Zhang, L. Li, D.Y. Tang, D.Y. Shen, S.N. Fu, D.M. Liu, L.M. Zhao, Generation of high-order group-velocity-locked vector solitons, IEEE Photonics J. 7 (5) (2015) 7102206. [20] X.X. Jin, Z.C. Wu, L. Li, Q. Zhang, D.Y. Tang, D.Y. Shen, S.N. Fu, D.M. Liu, L.M. Zhao, Manipulation of group-velocity-locked vector solitons from fiber lasers, IEEE Photonics J. 8 (2) (2016) 1501206. [21] X. Wang, L. Li, Y. Geng, H.X. Wang, L. Su, L.M. Zhao, Decomposition of group-velocity-locked-vector-dissipative solitons and formation of the high-order soliton structure by the product of their recombination, Appl. Opt. 57 (4) (2018) 746–751. [22] S.N. Zhu, Z.C. Wu, S.N. Fu, L.M. Zhao, Manipulation of group-velocity-locked vector dissipative solitons and properties of the generated high-order vector soliton structure, Appl. Opt. 57 (9) (2018) 2064–2068. [23] S.T. Cundiff, B.C. Collings, N.N. Akhmediev, J.M. Soto-Crespo, K. Bergman, W.H. Knox, Observation of polarization-locked vector solitons in an optical fiber, Phys. Rev. Lett. 82 (20) (1999) 3988–3991. [24] B.C. Collings, S.T. Cundiff, N.N. Akhmediev, J.M. Soto-Crespo, K. Bergman, W.H. Knox, Polarization-locked temporal vector solitons in a fiber laser: experiment, J. Opt. Soc. Am. B 17 (3) (2000) 354–365. [25] J.M. Soto-Crespo, N.N. Akhmediev, B.C. Collings, S.T. Cundiff, K. Bergman, W.H. Knox, Polarization-locked temporal vector solitons in a fiber laser: theory, J. Opt. Soc. Am. B 17 (3) (2000) 366–372. [26] D.Y. Tang, H. Zhang, L.M. Zhao, X. Wu, Observation of high-order polarization-locked vector solitons in a fiber laser, Phys. Rev. Lett. 101 (15) (2008) 153904. [27] C. Mou, S. Sergeyev, A. Rozhin, S. Turistyn, All-fiber polarization locked vector soliton laser using carbon nanotubes, Opt. Lett. 36 (19) (2011) 3831–3833. [28] K.J. Blow, N.J. Doran, D. Wood, Polarization instabilities for solitons in birefringent fibers, Opt. Lett. 12 (3) (1987) 202–204. [29] V.V. Afanasjev, Soliton polarization rotation in fiber lasers, Opt. Lett. 20 (3) (1995) 270–272. [30] M. Liu, A.P. Luo, Z.C. Luo, W.C. Xu, Dynamic trapping of a polarization rotation vector soliton in a fiber laser, Opt. Lett. 42 (2) (2017) 330–333.
5
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Generation of pseudo-high-order polarization-locked vector solitons with super-Gaussian pulses
T
⁎
Yan Zhoua, , Yuefeng Lia, Renli Zhangb, Tianxing Wangb, Wanjun Bib, Xia Lib, Peiwen Kuanb, Yongzheng Fangc, Meisong Liaob a
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 c School of Materials Science and Engineering, Shanghai Institute of Technology, Shanghai 201418, China b
A R T IC LE I N F O
ABS TRA CT
Keywords: Vector soliton Birefringence Fiber Polarization-locking Chirp Phase difference Time delay
In this paper, we simulate the generation of pseudo-high-order polarization-locked vector solitons (PLVSs) based on fundamental super-Gaussian PLVS. Through changing pulse chirp, pulse amplitude, projection angle, phase difference and time delay of incident pulses, different kinds of pseudo-high-order PLVS with the same central wavelength can be achieved in a polarization resolved fiber system.
1. Introduction Optical soliton is a kind of localized nonlinear wave, and it can be described by nonlinear Schrödinger equation (NLSE) [1–4]. Ultrafast fiber laser is a perfect platform to study the optical soliton. So far, several types of solitons have been theoretically and experimentally demonstrated in fiber lasers, such as conventional soliton [5,6], dark soliton [7,8], dispersion-managed soliton [9,10], dissipative soliton [11,12], dissipative soliton resonance [13,14] and so on [15,16]. Because of stress, fabrication process and other factors, single-mode-fiber (SMF) is not perfect isotropic, and it can support two orthogonal polarization modes. Due to the intrinsic birefringence of SMF, vector solitons (VSs) can be generated in SMFs and mode-locked fiber lasers. The coupled NLSEs can be employed to describe VSs. For VSs that generated in SMFs, the orthogonal polarization modes shift their own central frequency to compensate the linear birefringence. While in dissipative mode-locked fiber laser system, the generation of VSs is due to the balance between gain, loss, group-velocity dispersion (GVD), self-phase modulation (SPM), linear birefringence and so on. There are different kinds of VSs, such as group-velocity-locked vector soliton (GVLVS) [17–22], polarization-locked vector soliton (PLVS) [23–27], polarization rotation vector soliton (PRVS) [28–30] and so on. Among them, PLVS can be generated in relatively weak birefringence environment. The linear birefringence can be compensated with nonlinear birefringence (self-phase and cross-phase modulation). The orthogonal polarization modes of PLVS has the same central wavelength, the phase difference is fixed with ± π/2 and there is no energy transfer between them. The PLVS was firstly experimentally demonstrated in a fiber laser [23]. And “2 + 1” type high-order PLVS has also been demonstrated in a mode-locked Er-doped fiber laser in 2008 [26]. As for other VSs, for example, pseudo-highorder GVLVS with two humps in one polarization direction and one in orthogonal direction was simulated and experimentally
⁎
Corresponding author. E-mail address: [email protected] (Y. Zhou).
https://doi.org/10.1016/j.ijleo.2019.163132 Received 7 July 2019; Received in revised form 13 July 2019; Accepted 18 July 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 194 (2019) 163132
Y. Zhou, et al.
Fig. 1. Schematic diagram of polarization resolved optical fiber system.
demonstrated [19,20]. However, there are few report about the generation of pseudo-high-order PLVS. In this paper, we numerically generate pseudo-high-order PLVS through polarization controller (PC) and polarization beam splitter (PBS) in a polarization resolved fiber system. Pseudo-high-order PLVSs that consist more than two humps for orthogonal polarization modes can be obtained. The combined laser pulses along two principle axes of PBS are not locked together, so we call it “pseudo-high-order” PLVS. 2. Simulation The polarization resolved system is shown in Fig. 1, which is similar in Ref. [22]. A PLVS which has super-Gaussian pulse shape for horizontal and vertical polarization modes with the same central wavelength passes through a piece of SMF, and also goes through a PC and a PBS. The PC is to control and change intra-cavity birefringence, adjusting the phase difference between two orthogonal polarization components. The fiber collimator is for collimating the laser, which makes the laser incident perpendicularly on the PBS. The PBS acts as two important roles: one is changing the projection of PLVS along its principle axes, the other is to separate the orthogonal polarization modes for measurement. Two orthogonal polarization laser beams can be got with the PBS after the collimator. The incident orthogonally polarized wave A1(t) and A2(t) can be expressed as:
A1 (t ) = A1 e
−
A2 (t ) = A2 e
1.6652 (1 + iC1) t − ΔT /2 ⎛ ⎞ 2 ⎝ T1 ⎠
−
4
1.6652 (1 + iC2) t + ΔT /2 ⎛ ⎞ 2 ⎝ T2 ⎠
e
i 2πct λ1
(1)
i ( 2πct + Δφ) λ2
(2)
4
e
A1 and A2 are amplitudes of vertical and horizontal polarization modes. C1 and C2 are chirp parameters. t is time parameter and △T is time delay between two pulses. T1 and T2 are pulse width of incident pulses. c is light velocity in a vacuum. λ1 and λ2 are wavelengths of two pulses. △φ is the phase difference of orthogonal polarization modes. The orthogonal polarization projection after the PBS can be expressed:
Ahorizontal = A1 (t )cos θ + A2 (t )sin θ
(3)
Avertical = A1 (t )sin θ − A2 (t )cos θ
(4)
θ is the projection angle. In the simulation, we assume T1=T2 = 5 ps, λ1=λ2 = 1064 nm. Fig. 2 shows the change of temporal pulse waveform and optical spectrum of orthogonal polarization modes with different chirp values of 5、10、15、20 and 30. Other parameters are set as: A1=A2 = 1, θ = 0°, △φ=π/2, △T = 0.5 ps. We can see that orthogonal polarization modes have the same peak intensity and pulse width with different chirps. Also, the pulse width and pulse interval are unchanged. The optical spectrum for horizontal (red color) and vertical (black color) polarization mode is almost the same, with triangle shape and a small peak on the top. The 3-dB bandwidth is less than 1 nm with increased chirps. The corresponding peak intensity of the optical spectrum is -34.5 dB、-35.5 dB、-36.1 dB、-36.5 dB and -37.1 dB, respectively. Then, we consider the changes of temporal pulse waveform and optical spectrum with different amplitude ratio of incident fundamental super-Gaussian pulses, and the corresponding result is shown in Fig. 3 with amplitude ratio (A2/A1) of 2、4、6、8、10.
Fig. 2. The temporal pulse waveform (a–e) and optical spectrum (f–j) of horizontal and vertical polarization modes when chirp parameter (C1=C2) is set as 5、10、15、20 and 30. Other parameters are set as: A1=A2 = 1, θ = 0°, △φ=π/2, △T = 0.5 ps. 2
Optik - International Journal for Light and Electron Optics 194 (2019) 163132
Y. Zhou, et al.
Fig. 3. The temporal pulse waveform (a–e) and optical spectrum (f–j) of horizontal and vertical polarization modes when A1 = 1, while A2 is set as 2、4、6、8 and 10. Other parameters are set as: C1=C2 = 30, θ = 0°, △φ=π/2, △T = 0.5 ps.
Other parameters are set as: C1=C2 = 30, θ = 0°, △φ=π/2, △T = 0.5 ps. The amplitude of horizontal polarization mode (red color) is unchanged, but the vertical polarization mode (black color) increases with increased amplitude ratio because of the increased amplitude ratio. For optical spectrum, the intensity of vertical component is higher than horizontal component, and the peak intensity difference is 2.1 dB、4.1 dB、5.4 dB、6.2 dB and 6.9 dB respectively for increased amplitude ratio. After that, we consider the case when the projection angle θ between horizontal component and principle axis direction is changed. Other parameters are set as: A1=A2 = 1, C1=C2 = 30, △φ=π/2, △T = 0.5 ps. Fig. 4 shows the pulse waveform and optical spectrum of horizontal (red color) and vertical (black color) polarization component with different θ values. We can see the temporal pulse waveform is symmetric with respect to zero point, and the peak position for horizontal (or vertical) component is at the valley of vertical (or horizontal) component because of π/2 phase difference, while adjacent two peaks or valleys have π phase difference. Besides, more than two humps appear with specific θ value. For example, when θ = 80°, there are five humps in the orthogonal polarization directions, meaning “5 + 5” type pseudo-high-order PLVS is generated. The optical spectrum for orthogonal polarization modes both have triangle shape, but accompanied with multiple peaks or valleys, and a sharp peak at 1064 nm wavelength. These become obvious when θ = 45°, which may due to that the amplitude projection in orthogonal polarization direction is equal in this case. Fig. 5 gives the simulation result of orthogonal polarization modes’ pulse waveform when time delay between the incident two pulses is from 0 to 1.8 ps with 0.2 ps time interval. We can see the number of humps for horizontal or vertical polarization mode increases with increased time delay. There is only one hump for orthogonal polarization modes when △T = 0, but the hump number increases with increased △T. The peak intensity of main hump increases when the time delay reaches 0.6 ps, and becomes relatively stable in latter 1.2 ps. The time interval between adjacent main hump is 5.3 ps when △T = 0.2 ps, and decreases to 2.6 ps when △T = 1.8 ps. Fig. 6 gives the corresponding optical spectrum with different time delay ranging from 0 to 1.8 ps. The number of humps for two orthogonal polarization modes both increases with the time delay. The peak intensity is -37.1 dB when △T = 0, and for other cases, the peak intensity is in the range between -36 dB and-37 dB, meaning without obvious change for optical spectrum peak intensity with different time delay. At last, we consider the influence of phase difference △φ on pulse waveform and optical spectrum, △φ can be changed through rotating PC or bending SMF. The simulation result is shown in Fig. 7. We can see the pulse waveform is symmetric when △φ = 0 but becomes asymmetric when △φ ≠ 0. Horizontal mode always has six humps while vertical mode has five for different △φ values, which means “6 + 5” pseudo-high-order PLVS can be generated with appropriate parameters. The peak intensity difference decreases with increased △φ. In the optical spectrum of Fig. 7(f–j), the horizontal and vertical polarization mode has 5 and 6 obvious humps, respectively.
Fig. 4. The temporal pulse waveform (a–e) and optical spectrum (f–j) of horizontal and vertical polarization modes when projection angle θ is set as 10°、30°、45°、60° and 80°. Other parameters are set as: A1=A2 = 1, C1=C2 = 30, △φ=π/2, △T = 0.5 ps. 3
Optik - International Journal for Light and Electron Optics 194 (2019) 163132
Y. Zhou, et al.
Fig. 5. The temporal pulse waveform (a–j) of horizontal and vertical polarization modes when time delay △T is changed from 0 to 1.8 ps with 0.2 ps time interval. Other parameters are set as: A1=A2 = 1, C1=C2 = 30, θ = 45°, △φ=π/2.
Fig. 6. The temporal optical spectrum (a–j) of horizontal and vertical polarization modes when time delay △T is changed from 0 to 1.8 ps with 0.2 ps time interval. Other parameters are the same as in Fig. 5.
Fig. 7. The temporal pulse waveform (a–j) and optical spectrum (f–j) of horizontal and vertical polarization modes when △φ is 0°、20°、40°、60° and 80°. Other parameters are set as: A1=A2 = 1, C1=C2 = 30, θ = 45°, △T = 0.5 ps.
3. Conclusion In this paper, we simulate the generation of pseudo-high-order PLVS, in a polarization resolved fiber system. Through changing pulse chirp, amplitude ratio, projection angle, time delay and phase difference, different kinds of pseudo-high-order PLVS with 1064 nm central wavelength can be generated. Such as “3 + 3” type, “5 + 5” type and “6 + 5” type PLVSs. These simulations can provide beneficial conduct for latter experiment.
Acknowledgments Project supported by National Key Research and Development Program of China (2018YFB0504500), National Natural Science Foundation of China (NSFC) (51472162, 51672177, 61475171), the talent introduction research project of Shanghai Institute of Technology (YJ 2018-8).
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Optik - International Journal for Light and Electron Optics 194 (2019) 163132
Y. Zhou, et al.
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