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By using the truncated equations of the coupled wave, the expressions of the normalized electric field and the normalized intensity of the frequency doubling beam are derived in the cases with and without walk-off effect, caused by the off-axial vortex beam propagating through the negative uniaxial crystal. The influences of the off-axial magnitude, the weak walk-off angle and the crystal length on the output frequency doubling beam are mainly investigated. It is shown that while the walk-off angle is neglected, the two dark cores of the frequency doubling beam coincide with the point (0, 0); but while the weak walk-off angle is taken into account, the dark cores move along the direction where the walk-off effect is generated, and the two coincident dark cores separate in the direction perpendicular to the direction where the walk-off effect is generated on the cross-section. Especially, the distance the dark cores move is related to the off-axial magnitude, the weak walk-off angle and the crystal length. But the separation distance of the two dark cores is only related to the weak walk-off angle and the crystal length. The exact details show that when the off-axial magnitude increases, the distance the dark cores move along the direction where the off-axial magnitude is generated increases at the same time, but the separation distance has no connection with he off-axial magnitude. When the weak walk-off angle and the crystal length increase, the distance the dark cores move and the separation distance of the two dark cores increase. In addition, it can be found by comparison that when the crystal length reduces, the distance the dark cores move will decrease. And also the separation distance of the two dark cores will reduce, which is induced by the walk-off effect and the off-axial magnitude. Therefore, by reducing the crystal length, the output frequency doubling beam can be modified to a certain degree.
[1] Born M, Wolf E 1999 Principles of Optics (7th Ed.) (New York: Pergamon) pp554-632
[2] Pas'ko V A, Soskin M S, Vasnetsov M V 2001 Opt. Commun. 198 49
[3] Graham G, Johannes C, Miles J P 2004 Opt. Express 12 5448
[4] Shunichi S, Yasunori H, Yoshio W 1994 Opt. Lett. 19 1807
[5] Gahagan K T, Swartzlander Jr G A 1999 J. Opt. Soc. Am. B 16 533
[6] Dholakia K, Simpon N B, Padgett M J, Allen L 1996 Phys. Rev. A 54 3742
[7] Toda Y, Honda S, Morita R 2010 Opt. Express 18 17796
[8] Bovino F A, Braccini M, Bertolotti M, Sibilia C 2011 Opt. Commun. 284 2587
[9] Deng D M, Guo Q 2010 Appl. Phys. B 100 897
[10] Wu X, Deng D M, Guo Q 2011 Chin. Phys. B 20 084201
[11] Deng D M, Chen C D, Zhao X, Li H G 2013 Appl. Phys. B 110 433
[12] Zhou G Q 2014 Opt. Commun. 330 106
[13] Georgi M, Dragomir N N, Elitza P, Alexander D 2011 J. Opt. 13 064015
[14] Zhou Z Y, Ding D S, Jiang Y K, Li Y, Shi S, Wang X S, Shi B S 2014 Opt. Express 22 20298
[15] Zhou Z Y, Li Y, Ding D S, Zhang W, Shi S, Shi B S, Guo G C 2014 Opt. Express 22 23673
[16] Amold C L, Akturk S, Mysyrowjcz A, et al. 2015 J. Phys. B: Atom. Molecul. Opt. Phys. 48 1
[17] Chen B, Chen C D, Peng X, Deng D M 2015 J. Opt. Soc. Am. B 32 173
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[1] Born M, Wolf E 1999 Principles of Optics (7th Ed.) (New York: Pergamon) pp554-632
[2] Pas'ko V A, Soskin M S, Vasnetsov M V 2001 Opt. Commun. 198 49
[3] Graham G, Johannes C, Miles J P 2004 Opt. Express 12 5448
[4] Shunichi S, Yasunori H, Yoshio W 1994 Opt. Lett. 19 1807
[5] Gahagan K T, Swartzlander Jr G A 1999 J. Opt. Soc. Am. B 16 533
[6] Dholakia K, Simpon N B, Padgett M J, Allen L 1996 Phys. Rev. A 54 3742
[7] Toda Y, Honda S, Morita R 2010 Opt. Express 18 17796
[8] Bovino F A, Braccini M, Bertolotti M, Sibilia C 2011 Opt. Commun. 284 2587
[9] Deng D M, Guo Q 2010 Appl. Phys. B 100 897
[10] Wu X, Deng D M, Guo Q 2011 Chin. Phys. B 20 084201
[11] Deng D M, Chen C D, Zhao X, Li H G 2013 Appl. Phys. B 110 433
[12] Zhou G Q 2014 Opt. Commun. 330 106
[13] Georgi M, Dragomir N N, Elitza P, Alexander D 2011 J. Opt. 13 064015
[14] Zhou Z Y, Ding D S, Jiang Y K, Li Y, Shi S, Wang X S, Shi B S 2014 Opt. Express 22 20298
[15] Zhou Z Y, Li Y, Ding D S, Zhang W, Shi S, Shi B S, Guo G C 2014 Opt. Express 22 23673
[16] Amold C L, Akturk S, Mysyrowjcz A, et al. 2015 J. Phys. B: Atom. Molecul. Opt. Phys. 48 1
[17] Chen B, Chen C D, Peng X, Deng D M 2015 J. Opt. Soc. Am. B 32 173
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