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Based on the method of stationary phase and the theorem of the vectorial structure, the analytical expressions for the vectrorial terms, namely the TE and TM terms, of a linearly polarized Laguerre-Gauss beam are derived in the far-field. According to the far-field energy flux distributions of the TE and TM terms, the ratios of the powers of the TE and TM terms to the power of the Laguerre-Gauss beam are given. The analytical formulae of the far-field divergence angles of the Laguerre-Gauss beam and its TE and TM terms are presented, respectively. A relation among the far-field divergence angles of the TE term, the TM term, and the Laguerre-Gauss beam is also derived. The formulae obtained are applicable not only to the paraxial case, but also to the non-paraxial case. The dependences of the ratios of the powers of the TE and TM terms to the whole power onf-parameter and mode number are numerically examined. The effects of thef-parameter, the mode number, and the linearly polarized angle on the far-field divergence angle of the Laguerre-Gauss beam and its TE and TM terms are also analyzed. This research reveals the far-field divergent properties of the linearly polarized Laguerre-Gauss beam from the vectorial structure, and enriches the recognition of the propagation characteristics of the linearly polarized Laguerre-Gauss beam.
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Keywords:
- Laguerre-Gauss beam /
- far-field divergence angle /
- vectorial structure
[1] Kogelnik H, Li T 1966 Proc. IEEE 54 1312
[2] Tamm C 1988 Phys. Rev. A 38 5960
[3] He H, Heckenberg N R, Rubinsztein-Dunlop H 1995 J. Mod. Opt. 42 217
[4] Hasegawa T, Shimizu T 1999 Opt. Commun. 160 103
[5] Ishaaya A A, Davidson N, Friesem A A 2005 Opt. Express 13 4952
[6] Matsumoto N, Ando T, Inoue T, Ohtake Y, Fukuchi N, Hara T 2008 J. Opt. Soc. Am. A 25 1642
[7] Kuga T, Torii Y, Shiokawa N, Hirano T 1997 Phys. Rev. Lett. 78 4713
[8] Arlt J, Hitomi T, Dholakia K 2000 Appl. Phys. B 71 549
[9] Bradshaw D, Andrews D 2005 Opt. Lett. 30 3039
[10] Jarutis V, Paskauskas R, Stabinis A 2000 Opt. Commun. 184 105
[11] Simon R, Agarwal G S 2000 Opt. Lett. 25 1313
[12] Seshadri S R 2002 Opt. Lett. 27 1872
[13] Orlov S, Stabinis A 2003 Opt. Commun. 226 97
[14] Mei Z R, Zhao D M 2004 J. Opt. Soc. Am. A 21 2375
[15] Mei Z R, Zhao D M, Gu J G 2004 Opt. Commun. 240 337
[16] Mei Z R, Zhao D M 2004 J. Opt. A: Pure Appl. Opt. 6 1005
[17] Sheppard C J R 2009 Opt. Express 17 3690
[18] Takenaka T, Yokota M, Fukumitsu O1985 J. Opt. Soc. Am. A 2 826
[19] Duan K L, Wang B Z, Lü B D 2005 J. Opt. Soc. Am. A 22 1976
[20] Mei Z R, Zhao D M 2007 Opt. Express 15 11942
[21] Zhou G Q 2008 Opt. Laser Technol. 40 930
[22] Zhou G Q 2006 Opt. Lett. 31 2616
[23] Zhou G Q 2010 High Power Laser and Particle Beams 22 1187 (in Chinese)[周国泉 2010 强激光与粒子束 22 1187]
[24] Zhou G Q 2005 Acta Phys. Sin. 54 4710 ( in Chinese) [周国泉 2005 54 4710]
[25] Kang X P, Lü B D 2006 Acta Phys. Sin. 55 4564 (in Chinese) [康水平, 吕百达 2006 55 4564]
[26] Mart′?nez-Herrero R, Mej′ias P M, Bosch S, Carnicer A 2001 J. Opt. Soc. Am. A 18 1678
[27] Deng D M, Guo Q 2007 Opt. Lett. 32 2711
[28] Tang H Q, Li X G, Zhou G Q, Zhu K C 2009 Opt. Commun. 282 478
[29] Carter W H 1972 J. Opt. Soc. Am. 62 1195
[30] Gradshteyn I S, Ryzhik I M 1980 Table of integrals, series, and products (New York: Academic Press)
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[1] Kogelnik H, Li T 1966 Proc. IEEE 54 1312
[2] Tamm C 1988 Phys. Rev. A 38 5960
[3] He H, Heckenberg N R, Rubinsztein-Dunlop H 1995 J. Mod. Opt. 42 217
[4] Hasegawa T, Shimizu T 1999 Opt. Commun. 160 103
[5] Ishaaya A A, Davidson N, Friesem A A 2005 Opt. Express 13 4952
[6] Matsumoto N, Ando T, Inoue T, Ohtake Y, Fukuchi N, Hara T 2008 J. Opt. Soc. Am. A 25 1642
[7] Kuga T, Torii Y, Shiokawa N, Hirano T 1997 Phys. Rev. Lett. 78 4713
[8] Arlt J, Hitomi T, Dholakia K 2000 Appl. Phys. B 71 549
[9] Bradshaw D, Andrews D 2005 Opt. Lett. 30 3039
[10] Jarutis V, Paskauskas R, Stabinis A 2000 Opt. Commun. 184 105
[11] Simon R, Agarwal G S 2000 Opt. Lett. 25 1313
[12] Seshadri S R 2002 Opt. Lett. 27 1872
[13] Orlov S, Stabinis A 2003 Opt. Commun. 226 97
[14] Mei Z R, Zhao D M 2004 J. Opt. Soc. Am. A 21 2375
[15] Mei Z R, Zhao D M, Gu J G 2004 Opt. Commun. 240 337
[16] Mei Z R, Zhao D M 2004 J. Opt. A: Pure Appl. Opt. 6 1005
[17] Sheppard C J R 2009 Opt. Express 17 3690
[18] Takenaka T, Yokota M, Fukumitsu O1985 J. Opt. Soc. Am. A 2 826
[19] Duan K L, Wang B Z, Lü B D 2005 J. Opt. Soc. Am. A 22 1976
[20] Mei Z R, Zhao D M 2007 Opt. Express 15 11942
[21] Zhou G Q 2008 Opt. Laser Technol. 40 930
[22] Zhou G Q 2006 Opt. Lett. 31 2616
[23] Zhou G Q 2010 High Power Laser and Particle Beams 22 1187 (in Chinese)[周国泉 2010 强激光与粒子束 22 1187]
[24] Zhou G Q 2005 Acta Phys. Sin. 54 4710 ( in Chinese) [周国泉 2005 54 4710]
[25] Kang X P, Lü B D 2006 Acta Phys. Sin. 55 4564 (in Chinese) [康水平, 吕百达 2006 55 4564]
[26] Mart′?nez-Herrero R, Mej′ias P M, Bosch S, Carnicer A 2001 J. Opt. Soc. Am. A 18 1678
[27] Deng D M, Guo Q 2007 Opt. Lett. 32 2711
[28] Tang H Q, Li X G, Zhou G Q, Zhu K C 2009 Opt. Commun. 282 478
[29] Carter W H 1972 J. Opt. Soc. Am. 62 1195
[30] Gradshteyn I S, Ryzhik I M 1980 Table of integrals, series, and products (New York: Academic Press)
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