Abstract

We theoretically investigate the tight focusing properties of hybridly polarized vector beams. Some numerical results are obtained to illustrate the intensity, phase, and polarization of tightly focused hybridly polarized vector beams. It is shown that the shape of the focal pattern may change from an elliptical beam to a ring focus with increasing radial index. The phase distribution around the tightly focused ring is shown to be the helical phase profile, indicating that the radial-variant spin angular momentum of hybridly polarized vector beams can be converted into the radial-variant orbital angular momentum.

© 2012 Optical Society of America

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  23. X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Reply to the comment on “optical orbital angular momentum from the curl of polarization”,” Phys. Rev. Lett. 106, 189302 (2011).
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    [CrossRef]

2011 (4)

K. Huang, P. Shi, G. W. Cao, K. Li, X. B. Zhang, and Y. P. Li, “Vector-vortex Bessel-Gauss beam and their tightly focusing properties,” Opt. Lett. 36, 888–890 (2011).
[CrossRef]

J. Shu, J. Pu, and Y. Liu, “Angular momentum conversion of elliptically polarized beams focused by high numerical-aperture phase Fresnel zone plates,” Appl. Phys. B 104, 639–646(2011).
[CrossRef]

S. H. Yan, B. L. Yao, and M. Lei, “Comment on “optical orbital angular momentum from the curl of polarization”,” Phys. Rev. Lett. 106, 189301 (2011).
[CrossRef]

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Reply to the comment on “optical orbital angular momentum from the curl of polarization”,” Phys. Rev. Lett. 106, 189302 (2011).
[CrossRef]

2010 (3)

2009 (2)

2008 (3)

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photon. 2, 501–505 (2008).
[CrossRef]

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[CrossRef]

Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel-Gaussian beams through a high numerical-aperture objective,” Opt. Lett. 33, 49–51 (2008).
[CrossRef]

2007 (5)

2006 (2)

Z. Bomzon, M. Gu, and J. Shamir, “Angular momentum and geometrical phase in tightly focused circularly polarized plane waves,” Appl. Phys. Lett. 89, 241104 (2006).
[CrossRef]

A. F. Abouraddy and K. C. Toussainnt, “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96, 153901 (2006).
[CrossRef]

2003 (2)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef]

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[CrossRef]

2002 (1)

L. E. Helseth, “Focusing of atoms with strongly confined light potentials,” Opt. Commun. 212, 343–352 (2002).
[CrossRef]

2001 (1)

E. P. Walker and T. D. Milster, “Beam shaping for optical data storage,” Proc. SPIE 4443, 73–92 (2001).
[CrossRef]

2000 (1)

1996 (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. A 253, 358–379 (1959).
[CrossRef]

Abouraddy, A. F.

A. F. Abouraddy and K. C. Toussainnt, “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96, 153901 (2006).
[CrossRef]

Amato-Grill, J.

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[CrossRef]

Bomzon, Z.

Z. Bomzon, M. Gu, and J. Shamir, “Angular momentum and geometrical phase in tightly focused circularly polarized plane waves,” Appl. Phys. Lett. 89, 241104 (2006).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Brown, T. G.

Cao, G. W.

Chen, B. S.

Chen, J.

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Reply to the comment on “optical orbital angular momentum from the curl of polarization”,” Phys. Rev. Lett. 106, 189302 (2011).
[CrossRef]

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[CrossRef]

X. L. Wang, Y. N. Li, J. Chen, C. S. Guo, J. P. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2010).
[CrossRef]

Chiu, D. T.

Y. Q. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef]

Chong, C. T.

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photon. 2, 501–505 (2008).
[CrossRef]

Ding, J. P.

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Reply to the comment on “optical orbital angular momentum from the curl of polarization”,” Phys. Rev. Lett. 106, 189302 (2011).
[CrossRef]

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[CrossRef]

X. L. Wang, Y. N. Li, J. Chen, C. S. Guo, J. P. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2010).
[CrossRef]

X. L. Wang, J. P. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32, 3549–3551 (2007).
[CrossRef]

DiTrapani, P.

N. K. Efremidis, K. Hizanidis, B. A. Malomed, and P. DiTrapani, “Three-dimensional vortex solitons in self-defocusing media,” Phys. Rev. Lett. 98, 113901 (2007).
[CrossRef]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef]

Edgar, J. S.

Y. Q. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef]

Efremidis, N. K.

N. K. Efremidis, K. Hizanidis, B. A. Malomed, and P. DiTrapani, “Three-dimensional vortex solitons in self-defocusing media,” Phys. Rev. Lett. 98, 113901 (2007).
[CrossRef]

Grier, D. G.

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[CrossRef]

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[CrossRef]

Gu, M.

Z. Bomzon, M. Gu, and J. Shamir, “Angular momentum and geometrical phase in tightly focused circularly polarized plane waves,” Appl. Phys. Lett. 89, 241104 (2006).
[CrossRef]

M. Gu, Advanced Optical Imaging Theory (Springer, 2000).

Guo, C. S.

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Reply to the comment on “optical orbital angular momentum from the curl of polarization”,” Phys. Rev. Lett. 106, 189302 (2011).
[CrossRef]

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[CrossRef]

X. L. Wang, Y. N. Li, J. Chen, C. S. Guo, J. P. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2010).
[CrossRef]

X. L. Wang, J. P. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32, 3549–3551 (2007).
[CrossRef]

Helseth, L. E.

L. E. Helseth, “Focusing of atoms with strongly confined light potentials,” Opt. Commun. 212, 343–352 (2002).
[CrossRef]

Hizanidis, K.

N. K. Efremidis, K. Hizanidis, B. A. Malomed, and P. DiTrapani, “Three-dimensional vortex solitons in self-defocusing media,” Phys. Rev. Lett. 98, 113901 (2007).
[CrossRef]

Huang, K.

Jeffries, G. D. M.

Y. Q. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef]

Lei, M.

S. H. Yan, B. L. Yao, and M. Lei, “Comment on “optical orbital angular momentum from the curl of polarization”,” Phys. Rev. Lett. 106, 189301 (2011).
[CrossRef]

Lerman, G. M.

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef]

Levy, U.

Li, C. F.

Li, K.

Li, Y. N.

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Reply to the comment on “optical orbital angular momentum from the curl of polarization”,” Phys. Rev. Lett. 106, 189302 (2011).
[CrossRef]

X. L. Wang, Y. N. Li, J. Chen, C. S. Guo, J. P. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2010).
[CrossRef]

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[CrossRef]

Li, Y. P.

Liu, Y.

J. Shu, J. Pu, and Y. Liu, “Angular momentum conversion of elliptically polarized beams focused by high numerical-aperture phase Fresnel zone plates,” Appl. Phys. B 104, 639–646(2011).
[CrossRef]

Lukyanchuk, B.

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photon. 2, 501–505 (2008).
[CrossRef]

Malomed, B. A.

N. K. Efremidis, K. Hizanidis, B. A. Malomed, and P. DiTrapani, “Three-dimensional vortex solitons in self-defocusing media,” Phys. Rev. Lett. 98, 113901 (2007).
[CrossRef]

McGloin, D.

Y. Q. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef]

Milster, T. D.

E. P. Walker and T. D. Milster, “Beam shaping for optical data storage,” Proc. SPIE 4443, 73–92 (2001).
[CrossRef]

Ni, W. J.

Pu, J.

J. Shu, J. Pu, and Y. Liu, “Angular momentum conversion of elliptically polarized beams focused by high numerical-aperture phase Fresnel zone plates,” Appl. Phys. B 104, 639–646(2011).
[CrossRef]

Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel-Gaussian beams through a high numerical-aperture objective,” Opt. Lett. 33, 49–51 (2008).
[CrossRef]

Pu, J. X.

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[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. A 253, 358–379 (1959).
[CrossRef]

Roichman, Y.

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[CrossRef]

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[CrossRef]

Shamir, J.

Z. Bomzon, M. Gu, and J. Shamir, “Angular momentum and geometrical phase in tightly focused circularly polarized plane waves,” Appl. Phys. Lett. 89, 241104 (2006).
[CrossRef]

Sheppard, C.

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photon. 2, 501–505 (2008).
[CrossRef]

Shi, L. P.

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photon. 2, 501–505 (2008).
[CrossRef]

Shi, P.

Shu, J.

J. Shu, J. Pu, and Y. Liu, “Angular momentum conversion of elliptically polarized beams focused by high numerical-aperture phase Fresnel zone plates,” Appl. Phys. B 104, 639–646(2011).
[CrossRef]

Stalder, M.

Stern, L.

Sun, B.

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[CrossRef]

Toussainnt, K. C.

A. F. Abouraddy and K. C. Toussainnt, “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96, 153901 (2006).
[CrossRef]

Walker, E. P.

E. P. Walker and T. D. Milster, “Beam shaping for optical data storage,” Proc. SPIE 4443, 73–92 (2001).
[CrossRef]

Wang, H. F.

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photon. 2, 501–505 (2008).
[CrossRef]

Wang, H. T.

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Reply to the comment on “optical orbital angular momentum from the curl of polarization”,” Phys. Rev. Lett. 106, 189302 (2011).
[CrossRef]

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[CrossRef]

X. L. Wang, Y. N. Li, J. Chen, C. S. Guo, J. P. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2010).
[CrossRef]

X. L. Wang, J. P. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32, 3549–3551 (2007).
[CrossRef]

Wang, X.

Wang, X. L.

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Reply to the comment on “optical orbital angular momentum from the curl of polarization”,” Phys. Rev. Lett. 106, 189302 (2011).
[CrossRef]

X. L. Wang, Y. N. Li, J. Chen, C. S. Guo, J. P. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2010).
[CrossRef]

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[CrossRef]

X. L. Wang, J. P. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32, 3549–3551 (2007).
[CrossRef]

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. A 253, 358–379 (1959).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Yan, S. H.

S. H. Yan, B. L. Yao, and M. Lei, “Comment on “optical orbital angular momentum from the curl of polarization”,” Phys. Rev. Lett. 106, 189301 (2011).
[CrossRef]

Yao, B. L.

S. H. Yan, B. L. Yao, and M. Lei, “Comment on “optical orbital angular momentum from the curl of polarization”,” Phys. Rev. Lett. 106, 189301 (2011).
[CrossRef]

Youngworth, K. S.

Zhan, Q.

Zhang, X. B.

Zhang, Z.

Zhao, Y. Q.

Y. Q. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef]

Adv. Opt. Photon. (1)

Appl. Opt. (1)

Appl. Phys. B (1)

J. Shu, J. Pu, and Y. Liu, “Angular momentum conversion of elliptically polarized beams focused by high numerical-aperture phase Fresnel zone plates,” Appl. Phys. B 104, 639–646(2011).
[CrossRef]

Appl. Phys. Lett. (1)

Z. Bomzon, M. Gu, and J. Shamir, “Angular momentum and geometrical phase in tightly focused circularly polarized plane waves,” Appl. Phys. Lett. 89, 241104 (2006).
[CrossRef]

Nat. Photon. (1)

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photon. 2, 501–505 (2008).
[CrossRef]

Nature (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[CrossRef]

Opt. Commun. (1)

L. E. Helseth, “Focusing of atoms with strongly confined light potentials,” Opt. Commun. 212, 343–352 (2002).
[CrossRef]

Opt. Express (3)

Opt. Lett. (6)

Phys. Rev. Lett. (8)

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[CrossRef]

A. F. Abouraddy and K. C. Toussainnt, “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96, 153901 (2006).
[CrossRef]

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[CrossRef]

N. K. Efremidis, K. Hizanidis, B. A. Malomed, and P. DiTrapani, “Three-dimensional vortex solitons in self-defocusing media,” Phys. Rev. Lett. 98, 113901 (2007).
[CrossRef]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef]

Y. Q. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef]

S. H. Yan, B. L. Yao, and M. Lei, “Comment on “optical orbital angular momentum from the curl of polarization”,” Phys. Rev. Lett. 106, 189301 (2011).
[CrossRef]

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Reply to the comment on “optical orbital angular momentum from the curl of polarization”,” Phys. Rev. Lett. 106, 189302 (2011).
[CrossRef]

Proc. R. Soc. A (1)

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

Proc. SPIE (1)

E. P. Walker and T. D. Milster, “Beam shaping for optical data storage,” Proc. SPIE 4443, 73–92 (2001).
[CrossRef]

Other (2)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

M. Gu, Advanced Optical Imaging Theory (Springer, 2000).

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Figures (8)

Fig. 1.
Fig. 1.

Scheme of tight focusing system.

Fig. 2.
Fig. 2.

(a), (b), (c) radial-variant linearly polarized vector fields. (a) n=0.5, α=0, (b) n=1.0, α=0, (c) n=1.0, α=π/2; (d), (e), (f) radial-variant vector fields with hybrid SoPs. (d) n=0.5 and α=0, (e) n=0.5, α=π/4, (f) n=1.0, α=π/4.

Fig. 3.
Fig. 3.

The total intensity distributions in the focal plane for different radial index, where (a) n=0.1, (b) n=0.5, (c) n=1, (d) n=2, (e) n=3, (f) n=4, respectively. (g), (h), (i) The intensity distributions in the focal plane for x, y, and z components with n=3, respectively. The other parameters are chosen as NA=0.9, A0=1, α=0.

Fig. 4.
Fig. 4.

Phase and SoP distributions of the electric field in the focused ring for different r with n=3. (a) The phase difference between the x and y components in the ring focus of the different radius. (b) The phase difference between the x and y components in the cross section of the ring focus. (c) The SoP and radial intensity distributions of the ring focus. (d) The phase distributions in the z component of the electric field. The other parameters are the same as in Fig. 3.

Fig. 5.
Fig. 5.

Influence of the initial phase α on the phase distributions of x, y, and z components of the electric field in the maximum light intensity of the ring focus with n=3. (a) α=0, (b) α=π/4, (c) α=π/2. The other parameters are chosen as NA=0.8, A0=1.

Fig. 6.
Fig. 6.

Influence of the radial index on the phase distributions of x, y, and z components of the electric field in the maximum light intensity of the ring focus. (a) n=2, (b) n=2.5, (c) n=3. The other parameters are chosen as NA=0.8, A0=1, α=0.

Fig. 7.
Fig. 7.

Influence of the positive and negative of the radial index on the phase distributions of x, y, and z components of the electric field in the maximum light intensity of the ring focus. (a), (b) n=2, n=2, NA=0.8; n=3, n=3, NA=0.6, respectively. The other parameters are the same as in Fig. 3.

Fig. 8.
Fig. 8.

Phase distribution of the x, y, and z components of the vector fields with localized linear polarization and with hybrid SoPs. (a) n=3, the phase distributions of the x and y components in the maximum light intensity of the ring focus, (b) n=3, the phase distribution of the z component in the maximum light intensity of the ring focus, NA=0.8. The other parameters are the same as in Fig. 3.

Equations (13)

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E(r,φ,z)=[ExEyEz]=ikf2π0α02πA(θ,ϕ)exp[ik(zcosθ+rsinθcos(ϕφ))]sinθcos(θ)×[cos2ϕcosθ+sin2ϕcosϕsinϕ(cosθ1)sinθcos(ϕ)]dϕdθ,
{e^x=12(e^r+e^l)e^y=j12(e^re^l),
{e^r=12(e^x+je^y)e^l=12(e^xje^y),
E(r,φ,z)=[Ex+ExEy+EyEz+Ez]=ikf2π0α02πsinθcosθexp[ik(zcosθ+rsinθcos(ϕφ))]×[Ax(θ,ϕ)(cos2ϕcosθ+sin2ϕ)+Ay(θ,ϕ)cosϕsinϕ(cosθ1)Ax(θ,ϕ)cosϕsinϕ(cosθ1)+Ay(θ,ϕ)(cos2ϕ+sin2ϕcosθ)Ax(θ,ϕ)sinθcos(ϕ)+Ay(θ,ϕ)sinθsin(ϕ)]dϕdθ.
E(r)=A0circ(r/r0)[cosδe^x+sinδe^y],
E(r)=A0circ(r/r0)[eiδe^x+eiδe^y],
Ax(θ)=A0cos(2nπfsinθ+α),
Ay(θ)=A0sin(2nπfsinθ+α).
Ax(θ)=A0exp[i(2nπfsinθ/r0+α)],
Ay(θ)=A0exp[i(2nπfsinθ/r0+α)].
Ex(r,φ,z)=C·i0αsinθcosθexp(ikzcosθ){Ax(θ)J0(krsinθ)(cosθ+1)+J2(krsinθ)(1cosθ)[Ax(θ)cos(2φ)+Ay(θ)sin(2φ)]}dθ,
Ey(r,φ,z)=C·i0αsinθcosθexp(ikzcosθ){Ay(θ)J0(krsinθ)(cosθ+1)+J2(krsinθ)(1cosθ)[Ax(θ)sin(2φ)Ay(θ)cos(2φ)]}dθ,
Ez(r,φ,z)=2C·i0αsin2θcosθexp(ikzcosθ){iJ1(krsinθ)[Ax(θ)cosφ+Ay(θ)sinφ]}dθ,

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