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In this paper, we investigate the propagation properties of radially polarized rectangular-symmetric cosine-Gaussian Schell-model (RCGSM) beam with multiple off-axis vortex phases by using Fourier transform and convolution method based on the unified theory of coherence and polarization. The results indicate that the radially polarized RCGSM beam has self-splitting properties and can be split into four identical lobes due to its unique spatial coherence structure. Furthermore, the modulation of multiple off-axis vortex phases can be acted on each lobe. For high coherence, the spot arrays with triangular or square hollow light intensity distribution can be generated in the focal plane by modulating the number of off-axis vortices (N0) and the beam order, and the corresponding state of polarization on each lobe presents an inverted triangular or oblique square elliptic distribution. However, for small coherence, the modulation effect of multiple off-axis vortex phases disappears and the light intensity of each lobe degenerates into a quasi-Gaussian distribution, whereas its state of polarization keeps invariant, which is independent of the beam order and coherence length. In addition, the beam still has a certain self-healing ability for one of the off-axis vortex phases partially blocked by an obstacle, but it will be destroyed for completely blocking, resulting in a notch on each lobe.
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Keywords:
- partially coherent beam /
- special spatial correlation structure /
- multiple off-axis vortex phases /
- propagation properties
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图 1 (a)源平面处第j个离轴涡旋中心与观察点的位置矢量关系; (b)—(d)携带3个离轴涡旋相位(拓扑荷数l = 1)的径向偏振RCGSM光束在源平面处的(b)离轴涡旋相位排列、(c)离轴涡旋相位分布和(d)光强分布
Figure 1. (a) An illustration of the position vector relationship between the j-th off-axis vortex phase core and the observation point in the initial plane; (b) arrangement of three off-axis vortex phases, (c) phase distribution, and (d) intensity distribution of a radially polarized RCGSM beam carrying three off-axis vortices in the source plane, respectively.
图 2 当光束阶数$ m = n = 5 $时, 携带3个离轴涡旋相位的径向偏振RCGSM光束通过薄透镜后的聚焦光强演化
Figure 2. Evolution of the focused intensity of a radially polarized RCGSM beam carrying three off-axis vortex phases through a thin lens when the beam order $ m = n = 5 $.
图 3 当光束阶数$ m = n = 5 $时, 携带4个离轴涡旋相位的径向偏振RCGSM光束的聚焦光强演化
Figure 3. Evolution of the focused intensity of a radially polarized RCGSM beam carrying four off-axis vortex phases through a thin lens when the beam order $ m = n = 5 $.
图 4 当光束阶数$ m = n = 5 $和相干长度$ {\delta _0} = 1.5{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\text{mm}} $时, 携带4个离轴涡旋相位的径向偏振RCGSM光束通过扇形障碍物的聚焦光强演化
Figure 4. Evolution of the focused intensity of a radially polarized RCGSM beam carrying four off-axis vortex phases after passing through a sector-shaped opaque obstacle when the beam order $ m = n = 5 $ and coherent length $ {\delta _0} = 1.5{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\text{mm}} $.
图 5 携带4个离轴涡旋相位的径向偏振RCGSM光束通过圆形障碍物后的聚焦光强演化($ m = n = 5 $, $ {\delta _0} = 1.5{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\text{mm}} $)
Figure 5. Evolution of the focused intensity of a radially polarized RCGSM beam carrying four off-axis vortex phases after passing through a circular obstacle when the beam order $ m = n = 5 $ and coherent length $ {\delta _0} = 1.5{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\text{mm}} $.
图 6 当光束阶数$ {m_0} = {n_0} = 5 $和相干长度$ {\delta _0} = 1.5{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\text{mm}} $时, 携带多个离轴涡旋相位的径向偏振RCGSM光束通过薄透镜后的偏振态演化情况
Figure 6. Evolution of the state of polarization of radially polarized RCGSM beams carrying multiple off-axis vortex phases through a thin lens when the beam order $ {m_0} = {n_0} = 5 $ and coherent length $ {\delta _0} = 1.5{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\text{mm}} $.
图 7 当相干长度$ {\delta _0} = 1.5{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\text{mm}} $时, 光束阶数$ {m_0}, {n_0} $对携带多个离轴涡旋相位的径向偏振RCGSM光束在焦平面处偏振态分布的影响
Figure 7. Influences of beam order $ {m_0}, {n_0} $ on the distribution of the state of polarization of radially polarized RCGSM beams carrying multiple off-axis vortex phases in the focal plane when the coherent length $ {\delta _0} = 1.5{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\text{mm}} $.
图 8 当光束阶数$ {m_0} = {n_0} = 2 $时, 相干长度$ {\delta _0} $对携带多个离轴涡旋相位的径向偏振RCGSM光束在焦平面处偏振态分布的影响
Figure 8. Influence of coherent length $ {\delta _0} $ on the distribution of the state of polarization of radially polarized RCGSM beams carrying multiple off-axis vortex phases in the focal plane when the beam order $ {m_0} = {n_0} = 2 $.
图 9 当光束阶数$ m = n = 2 $和相干长度$ {\delta _0} = 0.5{\text{mm}} $时, 携带多个离轴涡旋相位的径向偏振RCGSM光束通过扇形障碍物后在焦平面处的偏振态分布
Figure 9. Distribution of the state of polarization in the focal plane for the radially polarized RCGSM beams carrying multiple off-axis vortex phases through a sector-shaped opaque when the beam order $ m = n = 2 $ and the coherent length $ {\delta _0} = 0.5{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\text{mm}} $.
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[1] Ricklin J C, Davidson F M 2002 J. Opt. Soc. Am. A 19 1794
Google Scholar
[2] Cai Y J 2011 Proc. SPIE 7924 792402
Google Scholar
[3] 王飞, 余佳益, 刘显龙, 蔡阳健 2018 67 184203
Google Scholar
Wang F, Yu J Y, Liu X L, Cai Y J 2018 Acta Phys. Sin. 67 184203
Google Scholar
[4] 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
[5] Chen Y H, Wang F, Cai Y J 2022 Adv. Phys. X 7 2009742
Google Scholar
[6] 许文慧, 宁守琮, 张福才 2021 70 214201
Google Scholar
Xu W H, Ning S C, Zhang F C 2021 Acta Phys. Sin. 70 214201
Google Scholar
[7] Gori F, Santarsiero M 2007 Opt. Lett. 32 3531
Google Scholar
[8] Gori F, Ramírez-Sánchez V, Santarsiero M, Shirai T 2009 J. Opt. A: Pure Appl. Opt. 11 085706
Google Scholar
[9] Chen Y H, Gu J X, Wang F, Cai Y J 2015 Phys. Rev. A 91 013823
Google Scholar
[10] Mei Z R 2014 Opt. Lett. 39 347
Google Scholar
[11] Chen Y H, Cai Y J 2014 Opt. Lett. 39 2549
Google Scholar
[12] Mei Z R 2014 Opt. Express 22 13029
Google Scholar
[13] Liang C H, Wang F, Liu X L, Cai Y J, Korotkova O 2014 Opt. Lett. 39 769
Google Scholar
[14] Chen Y H, Wang F, Cai Y J 2015 物理学进展 35 51
Google Scholar
陈亚红, 王飞, 蔡阳健 2015 Prog. Phys. 35 51
Google Scholar
[15] 陈亚红, 蔡阳健 2016 光学学报 36 1026002
Google Scholar
Chen Y H, Cai Y J 2016 Acta Opt. Sin. 36 1026002
Google Scholar
[16] 曾军, 陈亚红, 刘显龙, 蔡阳健 2019 光学学报 39 1026004
Google Scholar
Zeng J, Chen Y H, Liu X L, Cai Y J 2019 Acta Opt. Sin. 39 1026004
Google Scholar
[17] Willner A E, Huang H, Yan Y, Ren Y, Ahmed N, Xie G 2015 Adv. Opt. Photonics 7 66
Google Scholar
[18] Ni J C, Wang C W, Zhang C C, Hu Y L, Yang L, Lao Z X, Xu B, Li J W, Wu D, Chu J R 2017 Light Sci. Appl. 6 e17011
Google Scholar
[19] Ginson G, Courtial J, Padgett M, Vasnetsov M, Pas'ko V, Barnett S, Franke-Arnold S 2004 Opt. Express 12 5448
Google Scholar
[20] 陈康, 马志远, 张明明, 窦建态, 胡友友 2022 71 014203
Google Scholar
Cheng K, Ma Z Y, Zhang M M, Dou J T, Hu Y Y 2022 Acta Phys. Sin. 71 014203
Google Scholar
[21] Zhang H, Zeng J, Lu X Y, Wang Z Y, Zhao C L, Cai Y J 2021 Nanophoto. 11 241
Google Scholar
[22] Ostrovsky A S, Rickenstorffparrao C, Arrizon V 2013 Opt. Lett. 38 534
Google Scholar
[23] 李阳月, 陈子阳, 刘辉, 蒲继雄 2010 59 1740
Google Scholar
Li Y Y, Chen Z Y, Liu H, Pu J X 2010 Acta Phys. Sin. 59 1740
Google Scholar
[24] Bekshaev A Y, Karamoch A I 2008 Opt. Commun. 281 3597
Google Scholar
[25] Wang X L, Zhu B W, Dong Y X, Wang S, Zhu Z Q, Bo F, Li X P 2017 Opt. Express 25 26844
Google Scholar
[26] Xu H F, Zhang R, Sheng Z Q, Qu J 2020 Opt. Express 28 28858
Google Scholar
[27] Mei Z R, Mao Y H, Wang J X, Shi X H 2023 Opt. Express 31 727
Google Scholar
[28] Zhan Q W 2009 Adv. Opt. Photon. 1 1
Google Scholar
[29] 刘森森, 宋华冬, 林伟强, 陈旭东, 蒲继雄 2019 68 074201
Google Scholar
Liu S S, Song H D, Lin W Q, Chen X D, Pu J X 2019 Acta Phys. Sin. 68 074201
Google Scholar
[30] Wolf E 2003 Phys. Lett. A 312 263
Google Scholar
[31] Mei Z R 2017 IEEE Photon. J. 9 6102306
Google Scholar
[32] Wu G F, Tao C Y 2018 Opt. Commun. 424 86
Google Scholar
[33] Zhou Y Q, Cui Z W, Han Y P 2022 Opt. Express 30 23448
Google Scholar
[34] Peng X F, Wang H Y, Liu L, Wang F, Popov S, Cai Y J 2020 Opt. Express 28 31510
Google Scholar
[35] Liu X L, Peng X F, Liu L, Wu G F, Zhao C L, Wang F, Cai Y J 2017 Appl. Phys. Lett. 110 181104
Google Scholar
[36] Zeng J, Liang C H, Wang H Y, Wang F, Zhao C L, Gbur G, Cai Y J 2020 Opt. Express 28 11493
Google Scholar
[37] Korotkova O, Wolf E 2005 Opt. Commun. 246 35
Google Scholar
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