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We introduce a class of specular and antispecular twisted Gaussian Schell-model beams, which are generated by inserting a twisted Gaussian Schell-model beam into a wavefront folding interferometer (WFI). The analytical expression for the cross-spectral density function of the beam propagating in free space is derived, and the statistical properties of the optical field are investigated in detail. The results show that the twisted effect is still maintained after the transformation, and the spectral density of the light field always rotates to 90 degrees around the axis during propagation. Furthermore, with appropriate optical field adjustment, the twist effect of the spectral degree of coherence (DOC) can be observed, but in opposite directions to the irradiance profile. We also find that the twisted phase not only controls the rotation of the field, but also effectively modulates the overall spot contour. For the far-field spectral density distribution, a larger twist effect will induce a smaller ellipticity of the beam spot. However, the intensity pattern in the central area is mainly determined by the phase difference of WFI. To be specific, the specular twisted field always has a sharp central peak during propagation, and in the antispecular case it has a central dip. Besides, the DOC distribution can be flexibly adjusted by the source coherence, the twisted phase and the phase difference of the WFI. The results of our work have important applications in the fields of free-space beam communication and particle trapping.
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Keywords:
- partially coherent beams /
- specular and antispecular /
- twisted phase /
- propagation
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Google Scholar
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Google Scholar
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图 1 波前折叠干涉仪(WFI), 其中S为光源, BS为非偏振分束器, P1和P2为直角棱镜, D为探测器
Figure 1. Wavefront-folding interferometer (WFI). S is the source, BS is a non-polarizing beam splitter, P1 and P2 are right-angle prisms, and D is a detector.
图 2 WFI输出平面上的归一化光谱密度$ {{S\left( {x', y'} \right)} {/ } {{S_{\max }}}} $ (a) $\phi = 0$; (b) $ \phi = {{\text{π}} {/ } {2}} $; (c) $\phi = {\text{π}}$
Figure 2. Normalized spectral density $ {{S\left( {x', y'} \right)} {/ } {{S_{\max }}}} $ in the WFI output plane: (a) $\phi = 0$; (b) $ \phi = {{\text{π}} {/ } {2}} $; (c) $\phi = {\text{π}}$.
图 3 WFI输出平面上的归一化光谱相干度$ \mu \left( {{x'_1}, {0}, {x'_2}, {0}} \right) $
Figure 3. Spectral degree of coherence $ \mu \left( {{x'_1}, {0}, {x'_2}, {0}} \right) $ in the WFI output plane.
图 4 变换扭曲光场在自由空间传输过程中的归一化光谱密度$S\left( {x, y, z} \right)/{S_{\max }}$的演变规律
Figure 4. Evolution of the normalized spectral density $S\left( {x, y, z} \right)/{S_{\max }}$ of the transformed twisted field on propagation.
图 5 扭曲因子对镜像扭曲光场的归一化光谱密度$S\left( {x, y, z} \right)/{S_{\max }}$的影响 (a) z = 0 mm; (b) z = 200 mm; (c) z = 4000 mm
Figure 5. Influence of the twist factor on the normalized spectral density $S\left( {x, y, z} \right)/{S_{\max }}$ of the specular twisted field: (a) z = 0 mm; (b) z = 200 mm; (c) z = 4000 mm.
图 6 两个对称点之间的光谱相干度$\mu \left( {x/2, y/2, - x/2, - y/2, z} \right)$在传输距离z = 400 mm处沿$ x $轴的二维分布 (a) wx = 0.5 mm, $ {\delta _x} = {\delta _y} = 0.2 {\text{ mm}} $; (b) $\phi = {{\text{π}} {/ } {4}}$, $ {\delta _x} = {\delta _y} = 0.2 {\text{ mm}} $; (c) $\phi = {{\text{π}} {/ } {4}}$, wx = 0.5 mm
Figure 6. Spectral degree of coherence $\mu \left( {x/2, y/2, - x/2, - y/2, z} \right)$ between two symmetrical points at the propagation distance z = 400 mm along $ x $ axis: (a) wx = 0.5 mm, $ {\delta _x} = {\delta _y} = 0.2 {\text{ mm}} $; (b) $\phi = {{\text{π}} {/ } {4}}$, $ {\delta _x} = {\delta _y} = 0.2 {\text{ mm}} $; (c) $\phi = {{\text{π}} {/ } {4}}$, wx = 0.5 mm.
图 7 两个对称点之间的光谱相干度$\mu \left( {x/2, y/2, - x/2, - y/2, z} \right)$随干涉仪两光路相位差$ \phi $的分布情况 (a) wx = 0.5 mm, $ {\delta _x} = {\delta _y} = 0.2 {\text{ mm}} $; (b) z = 400 mm, $ {\delta _x} = {\delta _y} = 0.2 {\text{ mm}} $
Figure 7. Spectral degree of coherence $\mu \left( {x/2, y/2, - x/2, - y/2, z} \right)$ along $ \phi $: (a) wx = 0.5 mm, $ {\delta _x} = {\delta _y} = 0.2 {\text{ mm}} $; (b) z = 400 mm, $ {\delta _x} = {\delta _y} = 0.2 {\text{ mm}} $.
图 8 两个对称点之间的光谱相干度$\mu ( {x/2, y/2, - x/2, - y/2, z} )$在传输过程中的演化规律, 其中$ \phi = {\text{π}}/4 $
Figure 8. Evolution of the spectral degree of coherence $\mu ( {x/2, y/2, - x/2, - y/2, z} )$ between two symmetrical points on propagation, and $ \phi = {\text{π}}/4 $.
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[1] Simon R, Sudarshan E, Mukunda N 1985 Phys. Rev. A 31 2419
Google Scholar
[2] Cai Y J, Korotkova O 2009 Appl. Phys. B 96 499
Google Scholar
[3] Tong Z S, Korotkova O 2012 Opt. Lett. 37 2595
Google Scholar
[4] Cui Y, Wang F, Cai Y J 2014 Opt. Commun. 324 108
Google Scholar
[5] Cai Y J, Lin Q, Korotkova O 2009 Opt. Express 17 2453
Google Scholar
[6] Mao Y H, Mei Z R, Wang Y Y, Zhou G Q, Qiu P Z 2020 Opt. Commun. 477 126321
Google Scholar
[7] Simon R, Mukunda N 1993 J. Opt. Soc. Am. A 10 95
Google Scholar
[8] Friberg A T, Tervonen E, Turunen J 1994 J. Opt. Soc. Am. A 11 1818
Google Scholar
[9] Borghi R, Gori F, Guattari G, Santarsiero M 2015 Opt. Lett. 40 4504
Google Scholar
[10] Borghi R 2018 Opt. Lett. 43 1627
Google Scholar
[11] Mei Z R, Korotkova O 2017 Opt. Lett. 42 255
Google Scholar
[12] Gori F, Santarsiero M 2018 Opt. Lett. 43 595
Google Scholar
[13] Peng X F, Liu L, Wang F, Popov S, Cai Y J 2018 Opt. Express 26 33956
Google Scholar
[14] Santarsiero M, Gori F, Alonzo M 2019 Opt. Express 27 8554
Google Scholar
[15] Mei Z, Korotkova O 2018 Opt. Lett. 43 3905
Google Scholar
[16] Tian C, Zhu S J, Huang H K, Cai Y J, Li Z H 2020 Opt. Lett. 45 5880
Google Scholar
[17] Wang H Y, Peng X F, Zhang H, Liu L, Chen Y H, Wang F, Cai Y J 2022 Nanophotonics-Berlin 11 689
Google Scholar
[18] Dong S J, Yang Y Z, Zhou Y J, Li X Z, Tang M M 2024 J. Opt. 26 065608
Google Scholar
[19] Ponomarenko S A 2001 Phys. Rev. E. 64 036618
Google Scholar
[20] Wu G F 2016 J. Opt. Soc. Am. A 33 345
Google Scholar
[21] Zhang C, Zhou Z L, Xu H F, Zhou Z X, Han Y S, Yuan Y S, Qu J 2022 Opt. Express 30 4071
Google Scholar
[22] Zhang C Y, Fu W Y 2024 Opt. Appl. 54 15
Google Scholar
[23] Wan L P, Zhao D M 2019 Opt. Lett. 44 735
Google Scholar
[24] Cai Y J, Lin Q, Ge D 2002 J. Opt. Soc. Am. A Opt. Image Sci. Vis. 19 2036
Google Scholar
[25] Gori F, Guattari G, Palma C, Padovani C 1988 Opt. Commun. 68 239
Google Scholar
[26] Partanen H, Sharmin N, Tervo J, Turunen J 2015 Opt. Express 23 28718
Google Scholar
[27] Guo M W, Zhao D M 2016 Opt. Express 24 6115
Google Scholar
[28] Zhou Z T, Guo M W, Zhao D M 2016 Appl. Opt. 55 6757
Google Scholar
[29] Zhou Z T, Guo M W, Zhao D M 2017 Opt. Commun. 383 287
Google Scholar
[30] Das D, Halder A, Partanen H, Koivurova M, Turunen J 2022 Opt. Express 30 5709
Google Scholar
[31] Tang M M, Dong S J, Yang Y Z, Zhou Y J, Guo M W, Li X Z 2024 J. Opt. 26 065601
Google Scholar
[32] Guo M W, Zhao D M 2018 Opt. Express 26 8581
Google Scholar
[33] Tang M M, Feng X X, Liu S Y, Li H H, Li X Z 2021 J. Opt. 23 045605
Google Scholar
[34] Li C Q, Zhang H Y, Wang T F, Liu L S, Guo J 2013 Acta Phys. Sin. 62 224203
Google Scholar
[35] 徐华锋, 张兴宇, 王仁杰 2024 物理学报 73 034201
Google Scholar
Xu H F, Zhang X Y, Wang R J 2024 Acta Phys. Sin. 73 034201
Google Scholar
[36] 王飞, 余佳益, 刘显龙, 蔡阳健 2018 物理学报 67 184203
Google Scholar
Wang F, Yu J Y, Liu X L, Cai Y J 2018 Acta Phys. Sin. 67 184203
Google Scholar
[37] Liu Y L, Dong Z, Zhu Y M, Wang H Y, Wang F, Chen Y H, Cai Y J 2024 PhotoniX 5 8
Google Scholar
[38] Yu J Y, Zhu X L, Wang F, Chen Y H, Cai Y J 2023 Prog. Quant. Electron. 91-92 100486
Google Scholar
[39] Chen Y H, Wang F, Cai Y J 2022 Adv. Phys-X 7 2009742
Google Scholar
[40] Peng D M, Huang Z F, Liu Y L, Chen Y H, Wang F, Ponomarenko S A, Cai Y J 2021 PhotoniX 2 6
Google Scholar
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