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Structured beam plays an important role in optical communication, microscopy and particle manipulations. Traditionally, structured beam can be obtained by solving Helmholtz wave equation. This method involves complex mathematical procedures, and the properties of solved light beam are obscure. It is worth noting that the structured beam can also be constructed by ray-optical Poincaré sphere method: this method is a rather intuitive and convenient for designing the structured beam with novel properties. This method also provides a ray-based way to study the propagation properties of structured beam. In this paper, the ray-optical Poincaré sphere method combined with plum-blossom curve is used to build a family of structured beams. The optical field distributions on beam waist, including intensity and phase, are calculated by the ray-optical Poincaré sphere method. The shape of inner and outer caustics of optical field are also detailed in order to demonstrate the self-healing or non-diffraction features of beams. By using angular spectrum diffraction, the free space evolutions of such structured beams are demonstrated. The results show that the structured beam turns to be the well-known Laguerre-Gaussian beam when the leaf number of plum-blossom curve is 0. While the leaf number equals 1, the structured beam has non-diffraction property, for its inner caustic concentrates onto two points. In geometrical optics sight, all light rays are tangent to the inner caustic, and the optical fields carried by rays interfere near the caustic, leading the beam to possess a self-healing capacity. The self-healing property is demonstrated in terms of rays. With the beam's propagating, rays which launch from the inner side of beam gradually reach the outer side of beam. On the contrary, the rays launching from the inner side of beam arrive at the outer side of beam. When the center of beam is blocked, the inner rays are also blocked. After propagating, outer side rays will reach the inner side, fill up the hole of beam, and recover the injury of optical field. Furthermore, we demonstrate the structured beam with a 5leave plum-blossom curve. In this case, the inner caustic of this beam turns into a decagonal star structure; our simulation results show that this beam has relatively strong self-healing capability. Theoretically, one can simply change the parameters of plum-blossom curve or choose other kind of Poincaré sphere curve to create more complex structured beams.
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Keywords:
- structured light beam /
- Poincaré sphere /
- geometrical optics /
- ray tracing
[1] Simpson N B, Dholakia K, Allen L, Padgett M J 1997 Opt. Lett. 22 52
[2] Gutiérrez-Vega J C, Bandres M A 2005 J. Opt. Soc. Am. A 22 289
[3] Siviloglou G A, Broky J, Dogariu A, Christodoulides D N 2007 Phys. Rev. Lett. 99 213901
[4] Penciu R S, Paltoglou V, Efremidis N K 2015 Opt. Lett. 40 1444
[5] Bandres M A, Gutiérrez-Vega J C 2004 Opt. Lett. 29 144
[6] Bandres M A, Gutiérrez-Vega J C 2004 J. Opt. Soc. Am. A 21 873
[7] Wang J 2016 Photon. Res. 4 B14
[8] Fahrbach F O, Simon P, Rohrbach A 2010 Nat. Photon. 4 780
[9] Lei M, Zumbusch A 2010 Opt. Express 18 19232
[10] Baumgartl J, Mazilu M, Dholakia K 2008 Nat. Photon. 2 675
[11] Woerdemann M, Alpmann C, Esseling M, Denz C 2013 Laser Photon. Rev. 7 839
[12] Dholakia K, Čižmár T 2011 Nat. Photon. 5 335
[13] Dietrich M 1972 Light Transmission Optics (New York: van Nostrand Reinhold) pp230-238
[14] Vainshtein L A 1964 Sov. Phys. Jetp. 18 471
[15] Chen Y Q, Wang J H 2004 Laser Principle (Hangzhou: Zhejiang Universir publisher) pp55-159 [陈钰清, 王静环 2004 激光原理 (杭州: 浙江大学出版社) 第55–159页]
[16] Alonso M A, Dennis M R 2017 Optica 4 476
[17] Alonso M A, Forbes G W 2002 Opt. Express 10 728
[18] Goodman J W 1968 Introduction to Fourier Optics (New York: McGraw-Hill) pp55-61
[19] Li M 2006 M. S. Thesis (Chengdu: University of Electronic Science and Technology) (in Chinese) [黎茂 2006 硕士学位论文 (成都: 电子科技大学)]
[20] Anguiano-Morales M, Martínez A, Iturbe-Castillo M D, Chávez-Cerda S, Alcalá-Ochoa N 2007 Appl. Opt. 46 8284
[21] Born M, Wolf E 1999 Principles of Optics (Cambridge: Cambridge University Press) pp349-352
[22] Vaveliuk P, Martínez-Matos ó, Ren Y X, Lu R D 2017 Phys. Rev. A 95 063838
[23] Zhang S H, Zhou J H, Gong L 2018 Opt. Express 26 3381
[24] Zhang S H, Liang Z, Zhou J H 2017 Acta Phys. Sin. 66 048701 (in Chinese) [张书赫, 梁振, 周金华 2017 物理学报 66 048701]
[25] McNamara D A, Pistorius C W I, Malherbe J A G 1990 Introduction to the Uniform Geometrical Theory of Diffraction (Norwood: Artech House) pp263-288
[26] Alonso M A 2013 J. Opt. Soc. Am. A 30 1223
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[1] Simpson N B, Dholakia K, Allen L, Padgett M J 1997 Opt. Lett. 22 52
[2] Gutiérrez-Vega J C, Bandres M A 2005 J. Opt. Soc. Am. A 22 289
[3] Siviloglou G A, Broky J, Dogariu A, Christodoulides D N 2007 Phys. Rev. Lett. 99 213901
[4] Penciu R S, Paltoglou V, Efremidis N K 2015 Opt. Lett. 40 1444
[5] Bandres M A, Gutiérrez-Vega J C 2004 Opt. Lett. 29 144
[6] Bandres M A, Gutiérrez-Vega J C 2004 J. Opt. Soc. Am. A 21 873
[7] Wang J 2016 Photon. Res. 4 B14
[8] Fahrbach F O, Simon P, Rohrbach A 2010 Nat. Photon. 4 780
[9] Lei M, Zumbusch A 2010 Opt. Express 18 19232
[10] Baumgartl J, Mazilu M, Dholakia K 2008 Nat. Photon. 2 675
[11] Woerdemann M, Alpmann C, Esseling M, Denz C 2013 Laser Photon. Rev. 7 839
[12] Dholakia K, Čižmár T 2011 Nat. Photon. 5 335
[13] Dietrich M 1972 Light Transmission Optics (New York: van Nostrand Reinhold) pp230-238
[14] Vainshtein L A 1964 Sov. Phys. Jetp. 18 471
[15] Chen Y Q, Wang J H 2004 Laser Principle (Hangzhou: Zhejiang Universir publisher) pp55-159 [陈钰清, 王静环 2004 激光原理 (杭州: 浙江大学出版社) 第55–159页]
[16] Alonso M A, Dennis M R 2017 Optica 4 476
[17] Alonso M A, Forbes G W 2002 Opt. Express 10 728
[18] Goodman J W 1968 Introduction to Fourier Optics (New York: McGraw-Hill) pp55-61
[19] Li M 2006 M. S. Thesis (Chengdu: University of Electronic Science and Technology) (in Chinese) [黎茂 2006 硕士学位论文 (成都: 电子科技大学)]
[20] Anguiano-Morales M, Martínez A, Iturbe-Castillo M D, Chávez-Cerda S, Alcalá-Ochoa N 2007 Appl. Opt. 46 8284
[21] Born M, Wolf E 1999 Principles of Optics (Cambridge: Cambridge University Press) pp349-352
[22] Vaveliuk P, Martínez-Matos ó, Ren Y X, Lu R D 2017 Phys. Rev. A 95 063838
[23] Zhang S H, Zhou J H, Gong L 2018 Opt. Express 26 3381
[24] Zhang S H, Liang Z, Zhou J H 2017 Acta Phys. Sin. 66 048701 (in Chinese) [张书赫, 梁振, 周金华 2017 物理学报 66 048701]
[25] McNamara D A, Pistorius C W I, Malherbe J A G 1990 Introduction to the Uniform Geometrical Theory of Diffraction (Norwood: Artech House) pp263-288
[26] Alonso M A 2013 J. Opt. Soc. Am. A 30 1223
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