Abstract

The numbers of the focal spots and the dominant field (i.e., whether the axial field or the transverse fields play dominant role in the focusing field) have significant effects on various applications. In this paper, we have derived the universal imaging model of the composite vector beam (CVB) composed of two orthogonally linearly polarized beams with inhomogeneous polarization modulation, which is also suitable for various polarized beams, such as linearly, circularly, radially, azimuthally, and vortex polarized beams. Moreover, the sin&cos amplitude modulation with arbitrary orders and the pupil filters with cylindrical symmetry are also involved in this imaging model. On the basis of this imaging model, the regulars to control the focal numbers and the dominant field are drawn. For the various applications, some important conclusions and constructive advices are given.

© 2011 OSA

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    [CrossRef] [PubMed]
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    [CrossRef]
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2011 (3)

2010 (1)

Y. Shao, J. Qu, H. Li, Y. Wang, J. Qi, G. Xu, and H. Niu, “High-speed spectrally resolved multifocal multiphoton microscopy,” Appl. Phys. B 99(4), 633–637 (2010).
[CrossRef]

2009 (4)

2008 (1)

K. A. Serrels, E. Ramsay, R. J. Warburton, and D. T. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nat. Photonics 2(5), 311–314 (2008).
[CrossRef]

2007 (2)

X. Dong, Z. Zhao, and X. Duan, “Micronanofabrication of assembled three-dimensional microstructures by designable multiple beams multiphoton processing,” Appl. Phys. Lett. 91(12), 124103 (2007).
[CrossRef]

K. J. Moh, X. C. Yuan, J. Bu, R. E. Burge, and B. Z. Gao, “Generating radial or azimuthal polarization by axial sampling of circularly polarized vortex beams,” Appl. Opt. 46(30), 7544–7551 (2007).
[CrossRef] [PubMed]

2006 (1)

2004 (1)

2003 (2)

2000 (2)

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000).
[CrossRef] [PubMed]

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[CrossRef]

1997 (1)

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[CrossRef]

Antolini, R.

Arisaka, K.

A. Cheng, J. T. Gonçalves, P. Golshani, K. Arisaka, and C. Portera-Cailliau, “Simultaneous two-photon calcium imaging at different depths with spatiotemporal multiplexing,” Nat. Methods 8(2), 139–142 (2011).
[CrossRef] [PubMed]

Armstrong, D. J.

Balla, N. K.

C. J. R. Sheppard, S. Rehman, N. K. Balla, E. Y. S. Yew, and T. W. Teng, “Besssel beams: Effects of polarization,” Opt. Commun. 282(24), 4647–4656 (2009).
[CrossRef]

Blit, S.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[CrossRef]

Bomzon, Z.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[CrossRef]

Brown, T. G.

Bu, J.

Burge, R. E.

Cao, G. W.

Chen, J.

Cheng, A.

A. Cheng, J. T. Gonçalves, P. Golshani, K. Arisaka, and C. Portera-Cailliau, “Simultaneous two-photon calcium imaging at different depths with spatiotemporal multiplexing,” Nat. Methods 8(2), 139–142 (2011).
[CrossRef] [PubMed]

Cheong, W. C.

Choudhury, A.

Davidson, N.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[CrossRef]

De Koninck, Y.

Dehez, H.

Dong, X.

H. Guo, X. Dong, X. Weng, G. Sui, N. Yang, and S. Zhuang, “Multifocus with small size, uniform intensity, and nearly circular symmetry,” Opt. Lett. 36(12), 2200–2202 (2011).
[CrossRef] [PubMed]

X. Dong, Z. Zhao, and X. Duan, “Micronanofabrication of assembled three-dimensional microstructures by designable multiple beams multiphoton processing,” Appl. Phys. Lett. 91(12), 124103 (2007).
[CrossRef]

Duan, X.

X. Dong, Z. Zhao, and X. Duan, “Micronanofabrication of assembled three-dimensional microstructures by designable multiple beams multiphoton processing,” Appl. Phys. Lett. 91(12), 124103 (2007).
[CrossRef]

Friesem, A. A.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[CrossRef]

Froner, E.

Gao, B. Z.

Golshani, P.

A. Cheng, J. T. Gonçalves, P. Golshani, K. Arisaka, and C. Portera-Cailliau, “Simultaneous two-photon calcium imaging at different depths with spatiotemporal multiplexing,” Nat. Methods 8(2), 139–142 (2011).
[CrossRef] [PubMed]

Gonçalves, J. T.

A. Cheng, J. T. Gonçalves, P. Golshani, K. Arisaka, and C. Portera-Cailliau, “Simultaneous two-photon calcium imaging at different depths with spatiotemporal multiplexing,” Nat. Methods 8(2), 139–142 (2011).
[CrossRef] [PubMed]

Guo, H.

Hasman, E.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[CrossRef]

Huang, K.

Kano, H.

Kim, W. C.

Lee, W. M.

Li, H.

Y. Shao, J. Qu, H. Li, Y. Wang, J. Qi, G. Xu, and H. Niu, “High-speed spectrally resolved multifocal multiphoton microscopy,” Appl. Phys. B 99(4), 633–637 (2010).
[CrossRef]

Li, K.

Li, Y. P.

Moh, K. J.

Niu, H.

Y. Shao, J. Qu, H. Li, Y. Wang, J. Qi, G. Xu, and H. Niu, “High-speed spectrally resolved multifocal multiphoton microscopy,” Appl. Phys. B 99(4), 633–637 (2010).
[CrossRef]

Oron, R.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[CrossRef]

Park, K. S.

Park, N. C.

Park, Y. P.

Pavone, F. S.

Piché, M.

Portera-Cailliau, C.

A. Cheng, J. T. Gonçalves, P. Golshani, K. Arisaka, and C. Portera-Cailliau, “Simultaneous two-photon calcium imaging at different depths with spatiotemporal multiplexing,” Nat. Methods 8(2), 139–142 (2011).
[CrossRef] [PubMed]

Qi, J.

Y. Shao, J. Qu, H. Li, Y. Wang, J. Qi, G. Xu, and H. Niu, “High-speed spectrally resolved multifocal multiphoton microscopy,” Appl. Phys. B 99(4), 633–637 (2010).
[CrossRef]

Qu, J.

Y. Shao, J. Qu, H. Li, Y. Wang, J. Qi, G. Xu, and H. Niu, “High-speed spectrally resolved multifocal multiphoton microscopy,” Appl. Phys. B 99(4), 633–637 (2010).
[CrossRef]

Ramsay, E.

K. A. Serrels, E. Ramsay, R. J. Warburton, and D. T. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nat. Photonics 2(5), 311–314 (2008).
[CrossRef]

Rehman, S.

C. J. R. Sheppard, S. Rehman, N. K. Balla, E. Y. S. Yew, and T. W. Teng, “Besssel beams: Effects of polarization,” Opt. Commun. 282(24), 4647–4656 (2009).
[CrossRef]

Reid, D. T.

K. A. Serrels, E. Ramsay, R. J. Warburton, and D. T. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nat. Photonics 2(5), 311–314 (2008).
[CrossRef]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[CrossRef]

Sacconi, L.

Serrels, K. A.

K. A. Serrels, E. Ramsay, R. J. Warburton, and D. T. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nat. Photonics 2(5), 311–314 (2008).
[CrossRef]

Shao, Y.

Y. Shao, J. Qu, H. Li, Y. Wang, J. Qi, G. Xu, and H. Niu, “High-speed spectrally resolved multifocal multiphoton microscopy,” Appl. Phys. B 99(4), 633–637 (2010).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard, S. Rehman, N. K. Balla, E. Y. S. Yew, and T. W. Teng, “Besssel beams: Effects of polarization,” Opt. Commun. 282(24), 4647–4656 (2009).
[CrossRef]

Shi, P.

Smith, A. V.

Sui, G.

Taghizadeh, M. R.

Teng, T. W.

C. J. R. Sheppard, S. Rehman, N. K. Balla, E. Y. S. Yew, and T. W. Teng, “Besssel beams: Effects of polarization,” Opt. Commun. 282(24), 4647–4656 (2009).
[CrossRef]

Terakado, G.

Török, P.

Varga, P.

Wang, Y.

Y. Shao, J. Qu, H. Li, Y. Wang, J. Qi, G. Xu, and H. Niu, “High-speed spectrally resolved multifocal multiphoton microscopy,” Appl. Phys. B 99(4), 633–637 (2010).
[CrossRef]

Warburton, R. J.

K. A. Serrels, E. Ramsay, R. J. Warburton, and D. T. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nat. Photonics 2(5), 311–314 (2008).
[CrossRef]

Watanabe, K.

Weng, X.

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[CrossRef]

Xu, G.

Y. Shao, J. Qu, H. Li, Y. Wang, J. Qi, G. Xu, and H. Niu, “High-speed spectrally resolved multifocal multiphoton microscopy,” Appl. Phys. B 99(4), 633–637 (2010).
[CrossRef]

Yang, N.

Yew, E. Y. S.

C. J. R. Sheppard, S. Rehman, N. K. Balla, E. Y. S. Yew, and T. W. Teng, “Besssel beams: Effects of polarization,” Opt. Commun. 282(24), 4647–4656 (2009).
[CrossRef]

Yoon, Y. J.

Youngworth, K. S.

Yuan, X. C.

Zhang, X. B.

Zhao, Z.

X. Dong, Z. Zhao, and X. Duan, “Micronanofabrication of assembled three-dimensional microstructures by designable multiple beams multiphoton processing,” Appl. Phys. Lett. 91(12), 124103 (2007).
[CrossRef]

Zhuang, S.

Appl. Opt. (3)

Appl. Phys. B (1)

Y. Shao, J. Qu, H. Li, Y. Wang, J. Qi, G. Xu, and H. Niu, “High-speed spectrally resolved multifocal multiphoton microscopy,” Appl. Phys. B 99(4), 633–637 (2010).
[CrossRef]

Appl. Phys. Lett. (2)

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[CrossRef]

X. Dong, Z. Zhao, and X. Duan, “Micronanofabrication of assembled three-dimensional microstructures by designable multiple beams multiphoton processing,” Appl. Phys. Lett. 91(12), 124103 (2007).
[CrossRef]

J. Opt. Soc. Am. A (1)

Nat. Methods (1)

A. Cheng, J. T. Gonçalves, P. Golshani, K. Arisaka, and C. Portera-Cailliau, “Simultaneous two-photon calcium imaging at different depths with spatiotemporal multiplexing,” Nat. Methods 8(2), 139–142 (2011).
[CrossRef] [PubMed]

Nat. Photonics (1)

K. A. Serrels, E. Ramsay, R. J. Warburton, and D. T. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nat. Photonics 2(5), 311–314 (2008).
[CrossRef]

Opt. Commun. (1)

C. J. R. Sheppard, S. Rehman, N. K. Balla, E. Y. S. Yew, and T. W. Teng, “Besssel beams: Effects of polarization,” Opt. Commun. 282(24), 4647–4656 (2009).
[CrossRef]

Opt. Express (2)

Opt. Lett. (6)

Proc. R. Soc. Lond. A Math. Phys. Sci. (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Geometry of imaging of an aplanatic system.

Fig. 2
Fig. 2

(Color online) Light intensities of the transverse [(a) and (d)] and axial [(b) and (e)]components and the total light intensity [(c) and (f)] at the focal plane for the case of n = 1, m = 2, e x = e y =1 , and n 1 = n 2 =1 , where Ω=0.5 [(a)~(c)] and Ω=0.8 [(d)~(f)].

Fig. 3
Fig. 3

(Color online) The total focusing light intensity at the focal plane for the case of m = 1, e x = e y =1 , n 1 = n 2 =1 , Ω=0.8 , (a) n = 4, (b) n = 6, and (c) n = 8.

Fig. 4
Fig. 4

(Color online) Normalized intensities of the total focusing light intensity at the focal plane for the case of n = 4, m = 1, e x = e y =1 , n 1 = n 2 =1 , and Ω=0.8 (solid curve), 0.7 (dashed curve), and 0.6 (dotted curve) along the 45° direction with the x axis.

Fig. 5
Fig. 5

(Color online) Light intensities of the (a) transverse and (b) axial components and (c) the total light intensity at the focal plane for the case of n = 3, m = 2, e x =0 , e y =1 , n 1 =1 and n 2 =1.5 .

Fig. 6
Fig. 6

(Color online) Normalized intensities of the left focus in the first row of Fig. 5(c) (dashed and solid curves) and the focus of the linearly polarized beam (dotted and dashed dotted curves) along the x and y axes.

Fig. 7
Fig. 7

(Color online) The total focusing light intensity at the focal plane for the case of m = 2, e x =0 , e y =1 , and n 1 =1 . (a) n = 3 and n 2 =1 . (b) n = 1 and n 2 =1.5 .

Equations (14)

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E o =[x e x cos n (mφ+ φ 0 )+y e y sin n (mφ+ φ 0 )]B(h) A o (h),
cos n (mφ+ φ 0 )= 1 2 2 n ˜ k=0 n ˜ C n k cos[(2 n ˜ 2k+1)(mφ+ φ 0 )],
sin n (mφ+ φ 0 )= 1 2 2 n ˜ k=0 n ˜ (1) n ˜ +k C n k sin[(2 n ˜ 2k+1)(mφ+ φ 0 )],
cos n (mφ+ φ 0 )= 1 2 2 n ˜ 1 { k=0 n ˜ 1 C n k cos[(2 n ˜ 2k)(mφ+ φ 0 )]+ 1 2 C n n ˜ },
sin n (mφ+ φ 0 )= 1 2 2 n ˜ 1 { k=0 n ˜ 1 (1) n ˜ +k C n k cos[(2 n ˜ 2k)(mφ+ φ 0 )]+ 1 2 C n n ˜ }.
E x ( s i )=jA k=0 n ˜ j C n k [ e x (2 A cos Θ 0+ + D +2 cos Θ 2+ + D 2 cos Θ 2 ) e y (1) n ˜ +k ( D +2 cos Θ 2+ D 2 cos Θ 2 )],
E y ( s i )=jA k=0 n ˜ j C n k [ e x ( D +2 sin Θ 2+ D 2 sin Θ 2 ) + e y (1) n ˜ +k (2 A sin Θ 0+ D +2 sin Θ 2+ D 2 sin Θ 2 )],
E z ( s i )=2A k=0 n ˜ j C n k [ e x ( B +1 cos Θ 1+ B 1 cos Θ 1 ) + e y (1) n ˜ +k1 ( B +1 cos Θ 1+ + B 1 cos Θ 1 )],
A = 0 Φ cos 1 2 θ 1 sin θ 1 ( T + T cos θ 1 ) d θ 1 ,
B = 0 Φ cos 3 2 θ 1 sin θ 1 cos 1 θ 2 sin θ 2 T d θ 1 ,
D = 0 Φ cos 1 2 θ 1 sin θ 1 ( T T cos θ 1 )d θ 1 ,
E x ( s i )=jA C n n ˜ [ e x ( A 0 + D 2 cos2ϕ)+ e y D 2 sin2ϕ] jA k=0 n ˜ 1 (1) ( n ˜ k)m C n k [ e x (2 A cos Θ 0+ + D 2( n ˜ k)m+2 cos Θ 2+ + D 2 cos Θ 2 ) + (1) n ˜ +k e y ( D +2 sin Θ 2+ D 2 sin Θ 2 )],
E ˜ y ( s i )=jA C n n ˜ [ e x D 2 sin2ϕ+ e y ( A 0 D 2 cos2ϕ)] jA k=0 n ˜ 1 (1) ( n ˜ k)m C n k [ e x ( D +2 sin Θ 2+ D 2 sin Θ 2 ) + (1) n ˜ +k e y (2 A cos Θ 0+ D +2 cos Θ 2+ D 2 cos Θ 2 )],
E ˜ z ( s i )=2A C n n ˜ B 1 ( e x cosϕ+ e y sinϕ) +2A k=0 n ˜ 1 (1) ( n ˜ k)m C n k [ e x ( B +1 cos Θ 1+ B 1 cos Θ 1 ) + (1) n ˜ +k e y ( B +1 sin Θ 1+ + B 1 sin Θ 1 )].

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