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Photonic generation of triangular-shaped waveform with variable symmetry based on dual-polarization modulation is proposed and demonstrated. Based on the external modulation method, a dual-polarization modulator is used to modulate the radio frequency signal to generate the needed optical signal. By setting the modulation index of the modulator and phase shift of phase shifters appropriately, the optical intensity of generated signal can equal approximately the first three terms of the Fourier series expansion of the ideal triangular-shaped waveform, so triangular-shaped waveforms with different symmetry factors can be generated. Most of previous triangular waveform generation schemes generate symmetrical triangular waveform or sawtooth waveform (sawtooth waveform can be regarded as an asymmetrical triangular waveform), and the symmetry factor is not tunable. The tunable range of symmetry factor of triangular-shaped waveform generated by this scheme can reach 0%–100%, which will greatly expand the application range of triangular-shaped waveforms. The root-mean-square error (RMSE) is introduced to measure the similarity between the generated waveform and the theoretical waveform. It can be found that theoretically the triangular-shaped waveform with a symmetry factor in a range from 14% to 86% has a good similarity to the ideal waveform (RMSE < 0.044), and the RMSE of the generated waveform in the simulation is also very close to the theoretical RMSE. Experimentally, the 4GHz triangular-shaped waveforms with different values of symmetry factor (20%–80%) are obtained by using 4GHz radio frequency signal.
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Keywords:
- triangular-shaped waveform /
- dual-polarization modulation /
- Fourier series expansion /
- root-mean-square error
[1] Bhamber R S, Latkin A I, Boscolo S, Turitsyn S K 2008 34th European Conference on Optical Communication, September 21–25, 2008 p1
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图 1 不同对称因子对应的b1, b2, b3值
Figure 1. Magnitude of b1, b2 and b3 versus symmetry factor δ.
图 2 可变对称三角波形生成方案的原理示意图. CW laser, 连续波激光器; PC, 偏振控制器; RF source, 射频源; EPS, 电功分器; EA, 电放大器; PS, 电移相器; OPS, 光功分器; MZM, 单驱动马赫曾德尔调制器; PBC, 偏振合束器; 90°PR, 90°偏振旋转器; PD, 光电探测器
Figure 2. Schematic diagram of the proposed scheme. CW laser, continuous wave laser; PC, polarization controller; RF source, radio frequency source; EPS, electrical power splitter; EA, electrical amplifier; PS, phase shifter; OPS, optical power splitter; MZM, single-drive Mach-Zehnder Modulator; PBC, polarization beam combiner; 90° PR, 90° polarization rotator; PD, photodiode.
图 3 仿真生成的对称因子为0%—100%的时域波形
Figure 3. Simulated generated temporal triangular-shaped waveforms with δ = 0%−100%.
图 4 仿真生成的对称因子为20%的三角波形 (a) 时域波形; (b) 对应的电谱
Figure 4. Simulated generated triangular-shaped waveform with δ = 20%: (a) Temporal waveform; (b) corresponding electrical spectrum.
图 5 对称因子为20%的三角波形的合成示意图
Figure 5. Synthesis schematic diagram of triangular-shaped waveform with δ = 20%.
图 6 (a) 三阶近似波形、五阶近似波形和十阶近似波形的理论RMSE; (b) 仿真生成波形的RMSE
Figure 6. (a) Theoretical RMSE for third-order approximate waveform, fifth-order approximate waveform and tenth-order approximate waveform; (b) the RMSE of simulated generated waveforms.
图 7 可变对称三角波形生成方案的实验平台. Dual-wavelength laser, 双波长激光器; Polarization controller, 偏振控制器; Signal generator, 信号发生器; Power splitter, 电功分器; Electrical delay line, 电延迟线; Electrical amplifier, 电放大器; 180° hybrid coupler, 180°电桥; DP-BPSK modulator, 双偏振BPSK调制器; Photodiode, 光电探测器; Oscilloscope, 示波器
Figure 7. Experimental platform for triangular-shaped waveform with variable symmetry generation scheme.
图 8 实验生成的对称因子为20%, 30%, 40%, 50%, 60%, 70%和80%的三角波形
Figure 8. Generated triangular-shaped waveforms with δ = 20%, 30%, 40%, 50%, 60%, 70% and 80% in the experiment.
图 9 实验生成的对称因子为20%的三角波形 (a) 时域波形; (b) 对应的电谱
Figure 9. Generated triangular-shaped waveform with δ = 20%: (a) Temporal waveform; (b) corresponding electrical spectrum.
表 1 不同对称因子三角波形的生成条件
Table 1. Generated conditions for triangular-shaped waveforms with different symmetry factor.
对称因子 相移1 相移2 相移3 调制系数1 调制系数2 δ θ1 θ2 θ3 m1 m2 0% 0 π/4 0 1.15 0.83 10% 0 π/4 0 1.1 0.81 20% 0 π/4 0 0.92 0.76 30% 0 π/4 0 0.49 0.53 40% π/2 π/2 3π/2 0.61 0.41 50% π/2 0 3π/2 0.76 0 60% π/2 0 3π/2 0.61 0.41 70% 0 3π/4 0 0.49 0.53 80% 0 3π/4 0 0.92 0.76 90% 0 3π/4 0 1.1 0.81 100% 0 3π/4 0 1.15 0.83 
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[1] Bhamber R S, Latkin A I, Boscolo S, Turitsyn S K 2008 34th European Conference on Optical Communication, September 21–25, 2008 p1
[2] Cho S W, Rassau A, Gornisiewicz W 2007 6th International Conference on Polymers and Adhesives in Microelectronics and Photonics, January 16–18, 2007 p217
[3] Xu S Z, Wang P, Dong Y G 2016 Sensors 16 576
Google Scholar
[4] Won Y S, Kim C H, Lee S G 2015 IEEE Sens. J. 15 7142
Google Scholar
[5] Verscheure N, Finot C 2011 Electron. Lett. 47 1194
Google Scholar
[6] Ye J, Yan L S, Pan W, Luo B, Zou X H, Yi A L, Yao S 2011 Opt. Lett. 36 1458
Google Scholar
[7] Jiang H Y, Yan L S, Sun Y F, Ye J, Pan W, Luo B, Zou X H 2013 Opt. Express 21 6488
Google Scholar
[8] Li W, Wang W Y, Sun W H, Wang W T, Liu J G, Zhu N H 2014 Opt. Lett. 39 4758
Google Scholar
[9] Dai B, Gao Z S, Wang X, Chen H W, Kataoka N, Wada N 2013 J. LightwaveTechnol. 31 145
Google Scholar
[10] Li J, Ning T G, Pei L, Zheng J J, Li Y Q, Yuan J, Wang Y Q, Zhang C, Chen H Y 2014 Chin. Opt. Lett. 12 120602
Google Scholar
[11] Gao Y S, Wen A J, Liu W Y, Zhang H X, Xiang S Y 2016 IEEE Photonics J. 8 7801609
Google Scholar
[12] Li W, Wang W T, Zhu N H 2014 IEEE Photonics J. 6 5500608
Google Scholar
[13] Liu X K, Pan W, Zou X H, Zheng D, Yan L S, Luo B, Lu B 2014 J. Lightwave Technol. 32 3797
Google Scholar
[14] Huang L, Chen D L, Wang P, Zhang T T, Xiang P, Zhang Y Y, Pu T, Chen X F 2015 IEEE Photonics Technol. Lett. 27 2500
Google Scholar
[15] Wang W Y, Li W, Sun W H, Wang W T, Liu J G, Zhu N H 2015 IEEE Photonics Technol. Lett. 27 522
Google Scholar
[16] Li Y Y, Wen A J, Zhang W, Wang Q, Li X R 2019 Opt. Commun. 445 231
Google Scholar
[17] He Y T, Jiang Y, Zi Y J, Bai G F, Tian J, Xia Y, Zhang X Y, Dong R Y, Luo H 2018 Opt. Express 26 7829
Google Scholar
[18] Wei C, Jiang Y, Luo H, Dong R Y, Tian J, Zi Y J, Liu H F, Wang R 2020 Opt. Express 28 8098
Google Scholar
[19] Xia Y, Jiang Y, Zi Y J, He Y T, Tian J, Zhang X Y, Luo H, Dong R Y 2018 Opt. Commun. 414 177
Google Scholar
[20] Bai G F, Hu L, Jiang Y, Tian J, Zi Y J, Wu T W, Huang F Q 2017 Opt. Commun. 396 134
Google Scholar
[21] Li J, Ning T G, Pei L, Zheng J J 2019 J. Mod. Opt. 66 1457
Google Scholar
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