Manipulation of polarization- and group-velocity-locked dark vector solitons

Journal Pre-proof Manipulation of polarization- and group-velocity-locked dark vector solitons Yan Zhou, Yuefeng Li, Xia Li, Guoying Zhao, Jingshan Hou, Jun Zou, Yongzheng Fang, Meisong Liao

PII:

S0030-4026(19)31823-6

DOI:

https://doi.org/10.1016/j.ijleo.2019.163925

Reference:

IJLEO 163925

To appear in:

Optik

Received Date:

11 November 2019

Revised Date:

28 November 2019

Accepted Date:

29 November 2019

Please cite this article as: Zhou Y, Li Y, Li X, Zhao G, Hou J, Zou J, Fang Y, Liao M, Manipulation of polarization- and group-velocity-locked dark vector solitons, Optik (2019), doi: https://doi.org/10.1016/j.ijleo.2019.163925

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Manipulation of polarization- and group-velocity-locked dark vector solitons Yan Zhou,1,* Yuefeng Li,1 Xia Li,2 Guoying Zhao,3 Jingshan Hou,3 Jun Zou,1 Yongzheng Fang,3 and Meisong Liao2 1School

of Science, Shanghai Institute of Technology, Shanghai 201418, China Laboratory of Materials for High Power Laser, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China 3 School of Materials Science and Engineering, Shanghai Institute of Technology, Shanghai 201418, China *Corresponding author: [email protected]

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Abstract: In this manuscript, we simulate the manipulation of polarization- and group-velocity-locked dark vector solitons. Through changing pulse parameters of incident dark vector solitons, different kinds of pulse shapes could be generated. For incident fundamental dark polarization-locked vector soliton (PLVS), “2+2” pseudo-high-order dark PLVS and bright-dark PLVS could be generated. While for incident fundamental dark group-velocity-locked vector soliton (GVLVS), dark vector solitons with multiple side dips could be achieved depending on input pulse parameters.

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1. Introduction Optical soliton is widely studied by many research groups. The soliton that exist in single-mode fiber (SMF) can be theoretically solved with nonlinear Schrödinger equation [1,2]. While in fiber lasers, Ginzburg-Landau equation can be employed to study the soliton dynamics [3,4]. So far, different kinds of solitons have been explored in ultrafast mode-locked fiber lasers, such as traditional soliton [5-8], dispersion-managed soliton [9,10], dark soliton [11,12], dissipative soliton [13,14], dissipative soliton resonance [15,16], soliton molecular (or bound soliton) [17,18] and so on [19]. Among them, dark soliton is a kind of soliton that occurs in normal-dispersion regime in optical fiber. In theory, inverse scattering method can be used to obtain solutions of dark soliton. In the experiment, Zhang et al. firstly observed dark pulse train in 2009 [11]. After that, Tang et al. experimentally demonstrated the details of dark solitons that formed in fiber lasers [12]. Compared with bright soliton, the intensity profile of dark soliton has a dip (or hole) in a continuous background. If the dip center of dark soliton can fall to zero, we call it black soliton. Otherwise, it is called gray soliton. Recently, time-stretch dispersive Fourier transform (TS DFT) technique has been proved to be an efficient method to investigate the dynamic behavior of optical soliton in mode-locked fiber lasers [20,21]. It opens new opportunities for exploring ultrafast transient process in ultrafast mode-locked lasers and the dynamics of complex nonlinear systems. Because of material defects and environmental perturbation, SMF will become anisotropic and exhibit linear birefringence, so scalar approximation is not suitable. In this

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case, vector solitons (VSs) with orthogonally polarized electrical field modes (or components) can be supported. There is a variety of VSs that have been reported, such as polarization-locked vector soliton (PLVS) [22-26], group-velocity-locked vector soliton (GVLVS) [27-30], polarization rotation vector soliton (PRVS) [31,32] and so on. Among these VSs, GVLVS exists in large linear birefringence environment. It has elliptical polarization with rotational polarization axis. The two orthogonal modes will change their center wavelengths and combined with fiber nonlinearity (including self-phase modulation/SPM and cross-phase modulation/XPM), to compensate linear fiber birefringence. While for PLVS, it exists in the environment with weak birefringence, so SPM and XPM can completely compensate the linear fiber birefringence. For orthogonal modes of PLVS, they have the same center frequency (or wavelength) with fixed phase difference of ±π/2, and without energy transfer between them. PLVS has fixed polarization (linear or elliptical) compared with GVLVS. For another type of VS, PRVS, the polarization rotation period is equal to or multiple of the cavity round-trip time. The total soliton intensity is uniform, but the intensity of the orthogonal polarization modes varies among two or several values. The dynamic trapping property of PRVS was explored in ultrafast Er-doped fiber laser in Ref. [32]. Until now, there have been many research works about VSs, however, there are few reports about the manipulation of dark PLVS and dark GVLVS. In this manuscript, we theoretically simulate the manipulation of dark PLVS and dark GVLVS in a non-polarization-maintaining (non-PM) optical fiber system. For incident fundamental dark PLVS, pseudo-high-order dark PLVS and bright-dark PLVS could be generated. The projection of laser beams along the directions of two principle axes of polarization beam splitter (PBS) are un-locked, so we call the output soliton “pseudo-high-order” PLVS. While for incident fundamental dark GVLVS, dark vector solitons with multiple side dips could be achieved depending on input pulse parameters. The organizational structure of this paper is as follows. Firstly, we describe the theoretical model and give the corresponding equations that are used in the simulation. Then, we simulate the manipulation of dark PLVS at 1-μm wavelength region through changing input pulse parameters, and analyze the simulation result. After that, we consider the case of GVLVS with different central wavelengths and simulate the manipulation of it. Also, we compare the simulation result of PLVS and GVLVS. At last, we conclude the whole simulation result.

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Fig. 1. Schematic picture of the theoretical model. The theoretical model in our simulation is described in Fig. 1, and it is similar to that in Ref. [26,29]. A dark PLVS/GVLVS goes through a section of SMF without

polarization-maintaining, a polarization controller (PC), an optical collimator (Col), and at last a PBS, successively. The PC can impose external stress on SMF, and stress induced linear birefringence will produce phase difference between horizontal mode and vertical mode. The Col at the end of SMF is used for collimating (or aligning) the laser beam, making the laser beam perpendicularly passes through the end-face of PBS. Through rotating the PBS, we can change the projected intensity of dark PLVS/GVLVS along the two principle axes, so pulse shapes or optical spectra of orthogonal polarization modes can be changed. Two orthogonally polarized laser beams (Ahorizontal and Avertical, labelled in Fig. 1) can be got with the PBS at different directions. In the simulation, we assume both polarization modes have “tanh” temporal pulse shapes, meaning incident vector soliton is black soliton, so the amplitude of vertical or horizontal polarized waves A1(t) and A2(t) at time t can be described as below: 2 ct

1.763(t  T / 2)  i ( A2 (t )  A2 tanh  e T2  

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Ahorizontal  A1 (t ) cos   A2 (t )sin  ,

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In above equations, A1 and A2 are the maximum amplitudes of orthogonal polarization modes. t is temporal parameter, △T is time delay of orthogonal polarization modes. T1 and T2 are temporal pulse widths (full width at half maximum/FWHM) of initial orthogonal polarization modes. c represents speed of light in a vacuum. λ1 and λ2 are center wavelengths of vertical and horizontal mode, respectively. △φ is corresponding phase difference. The amplitude in horizontal and vertical directions after PBS can be calculated:

Avertical  A1 (t )sin   A2 (t ) cos  .

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In equations (3-4), 𝜃 is the projection angle and can be changed through rotating PBS. In later simulation, we assume incident horizontal mode and vertical mode of the VS have the same FWHM: T1=T2=5 ps.

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A. With the same central wavelength Firstly, we assume that horizontal mode and vertical mode have the same center wavelength (λ1=λ2=1064 nm), which is the typical characteristic of PLVS that exist in Yb-doped fiber laser. Fig. 2 gives simulation result of temporal pulse shapes and optical spectra of corresponding orthogonal modes, with increased amplitude ratios (A2/A1=2、3、4、5、6, A1=1) of incident fundamental dark vector pulses. Other variables are set as: 𝜃=0°, △T=0 ps, △φ=π/2. In pulse shapes of Fig. 2(a-e), the horizontal (with red color, the same in latter cases) polarization mode’s amplitude is fixed at a constant, however, the vertical (black color, the same in latter cases) polarization mode’s amplitude will increase, when amplitude ratio A2/A1 of incident dark vector pulses is increased. Because the projection angle is equal to zero, the output intensity of vertical mode increases with increased input amplitude (or

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intensity) ratio. Both horizontal and vertical modes are black solitons. In optical spectra of Fig. 2(f-j), we can see that the vertical mode has higher intensity than horizontal mode, and the corresponding optical spectrum peak intensity difference is 4.2 dB、6.6 dB、8.3 dB、9.6 dB and 10.8 dB respectively for increased input amplitude ratio. Then, we consider the case when 𝜃 is changed. Other variables are: A1=A2=1, △T=6 ps, △φ=π/2. Fig. 3 shows the corresponding simulation result. In Fig. 3(a-e), we can see that for all cases, the temporal pulse shapes are always be symmetric about original (or zero) point, also, both orthogonal polarization modes have two dips, meaning “2+2” pseudo-high-order dark PLVS could be got with appropriate projection angle setting. The amplitude ratio of the two dips varies with different angle values, and the two dips have the same amplitude (or dip depth) when 𝜃=45°, because under this condition, the amplitude projection of orthogonal polarization modes along the principle axes are equal. Besides, orthogonal polarization modes are gray solitons when 𝜃 is not equal to zero, and the dip depth decreases when 𝜃 is approaching 45°. In optical spectra of Fig. 3(f-j), both orthogonal polarization modes have one sharp peak (3-dB optical spectrum bandwidth is smaller than 1 nm) at 1064 nm wavelength position. Besides, there appear one or two obvious side peaks for orthogonal polarization modes.

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After that, we consider the influence of time delay △T on the properties of output orthogonal polarization modes. Fig. 4 gives the corresponding simulation result when △T is increased from 1 ps to 10 ps with time spacing of 1 ps. Other variables are set: A1=A2=1, 𝜃=43°, △φ=π/2. It shows both horizontal mode and vertical mode have only one dip when △T increases from 1 ps to 3 ps. However, when △T is larger than 4 ps, two obvious dip appear for both orthogonal modes. The time interval between the two dips is 4.3 ps、5.7 ps、 6.8 ps、8.0 ps、9.0 ps and 9.9 ps when △T increases from 5 ps to 10 ps. Besides, the dip depth of output pulses decreases with increased △T in the first 5 ps time delay range, then it becomes stable in the later 5 ps time delay. Fig. 5 gives the corresponding optical spectra under the condition of different time delay values. It shows that the peak intensity and 3-dB bandwidth of horizontal mode and vertical mode are the same, and the peak intensity is almost unchanged with increased △T. Finally, we change the phase difference △φ and study its influence on output pulse shapes and optical spectra. In the experiment, △φ can be changed by rotating the PC or bending/stretching SMF. The corresponding result is given in Fig. 6. From Fig. 6(a-e), we can see that when △φ=0 or π, bright-dark PLVS could be generated. The horizontal (or vertical) polarization mode is dark (or bright) when △φ=0, and it becomes contrary when △φ=π. Also, bright-dark PLVS can be transformed into “2+2” pseudo-high-order dark PLVS when △φ=π/2. Besides, the orthogonal polarization modes are black or gray solitons depending on phase difference. In Fig. 6(f-j), the orthogonal polarization modes’ optical spectrum peak intensity are the same, but the signal-to-noise ratio (SNR) is different, depending on △φ. When △φ=0 or π, the SNR difference is as large as 41.4 dB, while SNR difference is 0 when △φ=π/2.

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B. With different central wavelengths We now consider the case when the central wavelengths λ1 and λ2 are not equal, which is the typical characteristic of GVLVS. We assume λ1=1063 nm and λ2=1065 nm in later simulation. And the simulation procedure is similar to former case for PLVS.

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Firstly, we consider the change of amplitude ratio. Fig. 7 shows the simulation result with different amplitude ratios (A2/A1=2、3、4、5、6, A1=1) of incident fundamental dark vector pulses. Other variables are: 𝜃=0°, △T=0 ps, △φ=0. Similar to Fig. 2, in Fig. 7(a-e), we can see horizontal (red color, the same in later cases) polarization mode’s amplitude has no change, but for vertical (black color, the same in later cases) polarization mode, the dip intensity will increase with increased A2/A1 for incident dark vector pulses, because the projection angle is zero. While in optical spectra (see Fig. 7(f-j), zero point is 1064 nm), there are two peaks for horizontal mode with equal peak intensity, but only one peak in vertical mode, which means optical spectrum splits into two components for horizontal mode. The vertical mode’s optical spectrum peak intensity is higher than horizontal mode, and the corresponding intensity difference is 5.1 dB、7.5 dB、9.2 dB、10.6 dB and 11.6 dB respectively for increased amplitude ratio.

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changed. Other parameters are: A1=A2=1, △T=0 ps, △φ=0. Fig. 8 shows the corresponding result. We can see the pulse shape (see Fig. 8(a-e)) always have one dip shape with different 𝜃 values. But there appear multiple side dips for dark GVLVS pulses. The intensity of side dips increases when 𝜃 value is approaching 45°, and it becomes the largest and almost equal to the amplitude of main dip when 𝜃=45°. The optical spectra (see Fig. 8(f-j)) for orthogonal polarization modes both have two main peak bands, with two input wavelengths of 1063 nm and 1065 nm. The multiple peaks in the optical spectra correspond to multiple side dips of temporal pulse shapes. The intensity of two peak bands is different, but becomes equal when 𝜃=45°, because the projection on the principle axes is equal in this situation. However, the optical spectra are not symmetric about 1064 nm wavelength position in this case. (a)

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After that, we consider the influence of time delay △T on the properties of output orthogonal polarization modes. Fig. 9 shows output orthogonal polarization modes’ pulse shapes, when time delay is increased from 1 ps to 10 ps with time spacing of 1 ps. Other variables are: A1=A2=1, 𝜃=1°, △φ=0. It shows that both horizontal mode and vertical mode are black solions with only one dip, with increased △T, and the side dip intensity is suppressed because 𝜃 is approaching zero. Besides, the time separation of two dips increases with increased △T. Fig. 10 shows output optical spectra when time delay increases. The peak intensity for orthogonal modes are almost the same, accompanied with two peaks at two wavelengths’ position, and the occurrence of two peaks may due to the influence of temporal side dips. Besides, horizontal (or vertical) polarization mode is centered at 1063 (or 1065) nm wavelength band, meaning one wavelength band is suppressed for orthogonal polarization modes. Because the projection angle is approximate to zero, the orthogonally polarized modes will be fewly influenced by the other and transmit the PBS nearly independently .

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Finally, we change the phase difference △φ of orthogonal modes, and study output pulse shapes and optical spectra, other variables are: A1=A2=1, 𝜃=2°, △T=5 ps. Fig. 11 gives the corresponding simulation result. In Fig. 11(a-e), it shows that the horizontal mode and vertical mode always have one main dip with equal amplitude under the condition of different △φ values. Besides, the orthogonal polarization modes are always black solitons, which is different from PLVS when phase difference is changed. In Fig. 11(f-j), we can see that the orthogonal modes have almost the same peak intensity, but accompanied with one rather than two peak bands, meaning one wavelength band is suppressed for orthogonal modes, which is similar to former case when time delay is changed.

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3. Conclusion In this manuscript, we theoretically simulated the manipulation of dark PLVS and dark GVLVS at 1-μm wavelength region, in a non-PM optical fiber system. Through changing pulse parameters of incident dark vector pulses, different kinds of PLVSs and GVLVSs at 1-μm wavelength region could be generated, such as “2+2” pseudo-high-order PLVS, bright-dark PLVS and dark GVLVS with multiple side dips. The output dark vector soliton could be black or gray depending on input pulse parameters. The simulations have beneficial guide for later experiments.

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Declaration of interests

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement This work is supported by the Talent Introduction Research Project of Shanghai Institute of Technology (YJ 2018-8).

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