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Vortex beam (VB) is a structured light beam with a helical wavefront and carrying orbital angular momentum (OAM). Compared with Gaussian beam, the VB possesses the rotational Doppler effect (RDE), which is anticipated to compensate for the shortcoming of traditional detection methods in the spin motion of the target object. However, in practical applications, the rotational speed measurement technology based on the VB is facing some challenges, such as weak echo signal intensity due to low vortex beam light power and OAM spectrum expansion caused by off-axis incidence of the vortex beam. These above-mentioned problems directly limit the accuracy and application range of rotational speed measurement. To expand the application range of detection scheme based on the VB, we study the measurement scheme of the target rotational speed based on the combined vortex beam (CVB), which is on the basis of the experimental device for rotational speed measurement with CVB generated by fibre laser arrays. Firstly, the OAM spectra of the off-axis incidence situation are simulated. According to the simulation results, we derive a general model of the peak distribution of echo signals under the off-axis incidence, and propose a rotational speed measurement scheme based on the frequency interval between adjacent spectral peaks. Secondly, we carry out the target rotational speed measurement experiment in off-axis incidence case, and the difference in frequency between two adjacent spectral peaks is obtained from the spectrum map of the echo signal to measure the rotational speed of the target object. The results show that the target rotational speed can be accurately measured regardless of the lateral displacement and angular deflection in the case of off-axis incidence, which confirms the validity of the universal model for rotational speed measurement. The rotational speed measurement scheme proposed in this study takes into consideration the off-axis incidence prevalent in practical application, thereby improving the applicability in the target object rotational speed measurement, and providing technical reference for remote sensing detection application based on the VB.
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Keywords:
- rotational Doppler effect /
- vortex beam /
- off-axis incident /
- rotational speed measurement
[1] Coullet P, Gil L, Rocca F 1989 Opt. Commun. 73 403
Google Scholar
[2] Allen L, Beijersbergen M W, Spreeuw R, Woerdman J 1992 Phys. Rev. A 45 8185
Google Scholar
[3] Lavery M P, Speirits F C, Barnett S M, Padgett M J 2013 Science 341 537
Google Scholar
[4] Liu K, Cheng Y Q, Li X, Gao Y 2019 IEEE Veh. Technol. Mag. 14 112
Google Scholar
[5] Fang L, Padgett M J, Wang J 2017 Laser Photon. Rev. 11 1700183
Google Scholar
[6] Zhu X Y, Qiu S, Liu T, Ding Y, Tang R Y, Liu Z L, Chen X C, Ren Y 2023 Nanophotonics 12 2157
Google Scholar
[7] Tang R Y, Li X, Qiu S, Zhu X Y, Liu T, Liu Z L, Chen X C, Ren Y 2023 Opt. Express 31 39995
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[8] Zhai Y W, Fu S Y, Yin C, Zhou H, Gao C Q 2019 Opt. Express 27 15518
Google Scholar
[9] Zhai Y W, Fu S Y, Zhang J, Lü Y, Zhou H, Gao C Q 2020 Appl. Phys. Express 13 022012
Google Scholar
[10] Qiu S, Liu T, Li Z, Wang C, Ren Y, Shao Q L, Xing C Y 2019 Appl. Opt. 58 2650
Google Scholar
[11] Zhou H L, Fu D Z, Dong J J, Zhang P, Zhang X L 2016 Opt. Express 24 10050
Google Scholar
[12] Belmonte A, Torres J P 2011 Opt. Lett 36 4437
Google Scholar
[13] Li K F, Deng J H, Liu X, Li G 2018 Laser Photon. Rev. 12 1700204
Google Scholar
[14] Phillips D, Lee M, Speirits F, Barnett S M, Simpson S H, Lavery M P, Padgett M J, Gibson G M 2014 Phys. Rev. A 90 011801
Google Scholar
[15] Qiu S, Liu T, Ren Y, Li Z M, Wang C, Shao Q L 2019 Opt. Express 27 24781
Google Scholar
[16] Zhi D, Hou T, Ma P, Ma Y, Zhou P, Tao R, Wang X, Si L 2019 High Power Laser Sci. Eng. 7 e33
Google Scholar
[17] Long J, Chang H, Zhang Y, Hou T, Chang Q, Su R, Ma Y, Ma P, Zhou P 2022 Opt. Laser Technol 148 107775
Google Scholar
[18] Yu T, Xia H, Xie Q, Qin G W, Zhao Y F, Xie W K 2022 Opt. Express 30 39294
Google Scholar
[19] 于涛, 夏辉, 樊志华, 谢文科, 张盼, 刘俊胜, 陈欣 2018 物理学报 67 134203
Google Scholar
Yu T, Xia H, Fan Z H, Xie W K, Zhang P, Liu J S, Chen X 2018 Acta Phys. Sin. 67 134203
Google Scholar
[20] Yu T, Xia H, Xie W K, Xiao G Z, Li H J 2020 Res. Phys. 16 102872
Google Scholar
[21] Ding Y, Ren Y, Liu T, Qiu S, Wang C, Li Z M, Liu Z L 2021 Opt. Express 29 15288
Google Scholar
[22] 谢巧 2023 硕士学位论文 (长沙: 中南大学)
Xie Q 2023 M. S. Thesis (Changsha: Central South University
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图 2 传输距离z = 1 km时的±6阶相干合成涡旋光束 (a)光场强度分布; (b) OAM谱分布
Figure 2. Coherent synthesis of the ±6th order vortex beam at a transmission distance of z = 1 km: (a) Distribution of light field intensity; (b) distribution of OAM spectrum.
图 3 涡旋光离轴入射示意图, 偏心距离为d, 倾斜角度为γ
Figure 3. Schematic of off-axis incident vortex beams, and the lateral displacement is d, the angle of inclination is γ.
图 4 离轴入射时的OAM谱分布 (a)—(f)倾斜角度$ \gamma =10°$不变时, 以偏心距离分别为d = 1.25, 1.75, 2.25, 2.75, 3.25, 3.75 cm入射
Figure 4. Spectrum of OAM modes at off-axis incidence: (a)–(f) For different lateral displacements incident at constant inclination γ = 10°, transverse displacements were d = 1.25, 1.75, 2.25, 2.75, 3.25, 3.75 cm.
图 5 (a)实验装置图; (b)相位控制后的±6阶涡旋光束
Figure 5. (a) Experimental setup; (b) phase-controlled ±6th-order vortex beams.
图 6 离轴入射时, 转速为Ω = 73.304 rad/s的目标散射场时域信号和旋转多普勒频率信号 (a), (b) γ = 10°, d = 1.25 cm; (c), (d) γ = 10°, d = 3.75 cm; (e), (f) γ = 15°, d = 1.25 cm
Figure 6. At off-axis incidence, time domain signal and rotational Doppler frequency signal of the target scattered light field with a rotational speed of Ω = 73.304 rad/s: (a), (b) γ = 10°, d = 1.25 cm; (c), (d) γ = 10°, d = 3.75 cm; (e), (f) γ = 15°, d = 1.25 cm.
图 7 探测光离轴入射时, 不同转速下的相对误差比较曲线 (a)倾斜角度γ = 10°不变, 偏心距离分别为d = 1.25, 1.75, 2.25, 2.75, 3.25, 3.75 cm; (b)偏心距离d = 1.25 cm不变, 倾斜角度分别为γ = 10°, 15°, 20°, 25°, 30°, 35°, 40°, 45°
Figure 7. Detecting light at off-axis incidence, relative error comparison curves at different RPMs: (a) Angle of inclination γ = 10° is constant and the lateral displacements are d = 1.25, 1.75, 2.25, 2.75, 3.25, 3.75 cm; (b) the lateral displacement d = 1.25 cm is constant and the angles of inclination are γ = 10°, 15°, 20°, 25°, 30°, 35°, 40°, 45°
表 1 在73.304 rad/s的转速下, 探测光以不同偏心距离入射的误差结果
Table 1. Error results for beams incident at different transverse displacements at a speed of 73.304 rad/s.
d/cm $ {\varOmega }_{m} $
/(rad·s–1)$ \Delta \varOmega ={\varOmega }_{m}-\varOmega $
/(rad·s–1)$ \eta = \dfrac{|\Delta \varOmega |}{\varOmega} \times 100{\mathrm{{\text{%}}}} $ 1.25 73.633 0.329 0.4% 1.75 73.370 0.066 0.09% 2.25 73.679 0.375 0.5% 2.75 73.588 0.284 0.4% 3.25 73.631 0.327 0.4% 3.75 74.901 1.597 2.2% 
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表 2 在73.304 rad/s的转速下, 以不同倾斜角度入射的误差结果
Table 2. Error results for beams incident at different inclination angles at a speed of 73.304 rad/s.
γ/(°) $ {\varOmega }_{m} $
/(rad·s–1)$ \Delta \varOmega ={\varOmega }_{m}-\varOmega $
/(rad·s–1)$ \eta = \dfrac{|\Delta \varOmega |}{\varOmega} \times 100{\mathrm{{\text{%}}}} $ 10 73.633 0.329 0.4% 15 73.545 0.241 0.3% 20 73.594 0.291 0.4% 25 73.504 0.201 0.3% 30 73.714 0.410 0.6% 35 73.597 0.293 0.4% 40 73.589 0.285 0.4% 45 73.528 0.224 0.3%
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[1] Coullet P, Gil L, Rocca F 1989 Opt. Commun. 73 403
Google Scholar
[2] Allen L, Beijersbergen M W, Spreeuw R, Woerdman J 1992 Phys. Rev. A 45 8185
Google Scholar
[3] Lavery M P, Speirits F C, Barnett S M, Padgett M J 2013 Science 341 537
Google Scholar
[4] Liu K, Cheng Y Q, Li X, Gao Y 2019 IEEE Veh. Technol. Mag. 14 112
Google Scholar
[5] Fang L, Padgett M J, Wang J 2017 Laser Photon. Rev. 11 1700183
Google Scholar
[6] Zhu X Y, Qiu S, Liu T, Ding Y, Tang R Y, Liu Z L, Chen X C, Ren Y 2023 Nanophotonics 12 2157
Google Scholar
[7] Tang R Y, Li X, Qiu S, Zhu X Y, Liu T, Liu Z L, Chen X C, Ren Y 2023 Opt. Express 31 39995
Google Scholar
[8] Zhai Y W, Fu S Y, Yin C, Zhou H, Gao C Q 2019 Opt. Express 27 15518
Google Scholar
[9] Zhai Y W, Fu S Y, Zhang J, Lü Y, Zhou H, Gao C Q 2020 Appl. Phys. Express 13 022012
Google Scholar
[10] Qiu S, Liu T, Li Z, Wang C, Ren Y, Shao Q L, Xing C Y 2019 Appl. Opt. 58 2650
Google Scholar
[11] Zhou H L, Fu D Z, Dong J J, Zhang P, Zhang X L 2016 Opt. Express 24 10050
Google Scholar
[12] Belmonte A, Torres J P 2011 Opt. Lett 36 4437
Google Scholar
[13] Li K F, Deng J H, Liu X, Li G 2018 Laser Photon. Rev. 12 1700204
Google Scholar
[14] Phillips D, Lee M, Speirits F, Barnett S M, Simpson S H, Lavery M P, Padgett M J, Gibson G M 2014 Phys. Rev. A 90 011801
Google Scholar
[15] Qiu S, Liu T, Ren Y, Li Z M, Wang C, Shao Q L 2019 Opt. Express 27 24781
Google Scholar
[16] Zhi D, Hou T, Ma P, Ma Y, Zhou P, Tao R, Wang X, Si L 2019 High Power Laser Sci. Eng. 7 e33
Google Scholar
[17] Long J, Chang H, Zhang Y, Hou T, Chang Q, Su R, Ma Y, Ma P, Zhou P 2022 Opt. Laser Technol 148 107775
Google Scholar
[18] Yu T, Xia H, Xie Q, Qin G W, Zhao Y F, Xie W K 2022 Opt. Express 30 39294
Google Scholar
[19] 于涛, 夏辉, 樊志华, 谢文科, 张盼, 刘俊胜, 陈欣 2018 物理学报 67 134203
Google Scholar
Yu T, Xia H, Fan Z H, Xie W K, Zhang P, Liu J S, Chen X 2018 Acta Phys. Sin. 67 134203
Google Scholar
[20] Yu T, Xia H, Xie W K, Xiao G Z, Li H J 2020 Res. Phys. 16 102872
Google Scholar
[21] Ding Y, Ren Y, Liu T, Qiu S, Wang C, Li Z M, Liu Z L 2021 Opt. Express 29 15288
Google Scholar
[22] 谢巧 2023 硕士学位论文 (长沙: 中南大学)
Xie Q 2023 M. S. Thesis (Changsha: Central South University
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