Abstract

We study the focusing properties of elliptically polarized vortex beams. Based on vectorial Debye theory, some numerical calculations are given to illustrate the intensity and phase distribution properties of tightly focused vortex beams. It is found that the spin angular momentum of the elliptically polarized vortex beam will convert to orbital angular momentum by the focusing. The influence of corresponding parameters on focusing properties is also investigated in great detail. It is shown that elliptical light spots can be obtained in the focal plane. Moreover the elliptical spot may rotate and the spot shape may change with the change of certain parameters. These properties are quite important for application of this kind of elliptically polarized vortex beam.

© 2009 Optical Society of America

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  1. E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349-357 (1959).
    [CrossRef]
  2. A. Boivin and E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138, B1561-B1565 (1965).
    [CrossRef]
  3. J. W. M. Chon, X. Gan, and M. Gu, “Splitting of the focal spot of a high numerical-aperture objective in free space,” Appl. Phys. Lett. 81, 1576-1578 (2002).
    [CrossRef]
  4. N. Bokor and N. Davidson, “A three dimensional dark focal spot uniformly surrounded by light,” Opt. Commun. 279, 229-234 (2007).
    [CrossRef]
  5. T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272, 314-319 (2007).
    [CrossRef]
  6. 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]
  7. G. M. Lerman and U. Levy, “Tight focusing of spatially variant vector optical fields with elliptical symmetry of liner polarization,” Opt. Lett. 32, 2194-2196 (2007).
    [CrossRef] [PubMed]
  8. E. P. Walker and T. D. Milster, “Beam shaping for optical data storage,” Proc. SPIE 4443, 73-92 (2001).
    [CrossRef]
  9. L. E. Helseth, “Focusing of atoms with strongly confined light potentials,” Opt. Commun. 212, 343-352 (2002).
    [CrossRef]
  10. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77-87 (2000).
    [CrossRef] [PubMed]
  11. S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7 (2000).
    [CrossRef]
  12. Z. Zhang, J. Pu, and X. Wang, “Tight focusing of radially and azimuthally polarized vortex beams through a uniaxial birefringent crystal,” Appl. Opt. 47, 1963-1967 (2008).
    [CrossRef] [PubMed]
  13. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
    [CrossRef] [PubMed]
  14. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219-276 (2001).
    [CrossRef]
  15. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185-8189 (1992).
    [CrossRef] [PubMed]
  16. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Ween, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123-132(1993).
    [CrossRef]
  17. Z. Bomzon, M. Gu, and J. Shamir, “Angular momentum and geometrical phases in tightly focused circularly polarized plane waves,” Appl. Phys. Lett. 89, 241104 (2006).
    [CrossRef]
  18. F. S. Roux, “Distribution of angular momentum and vortex morphology in optical beams,” Opt. Commun. 242, 45-55(2004).
    [CrossRef]
  19. J. Courtial and M. J. Padgett, “Limit to the orbital angular momentum per unit energy in a light beam that can be focused onto a small particle,” Opt. Commun. 173, 269-274 (2000).
    [CrossRef]
  20. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064-4075 (1997).
    [CrossRef]
  21. M. Gu, Advanced Optical Imaging Theory (Springer, 2000).

2008 (2)

2007 (3)

N. Bokor and N. Davidson, “A three dimensional dark focal spot uniformly surrounded by light,” Opt. Commun. 279, 229-234 (2007).
[CrossRef]

T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272, 314-319 (2007).
[CrossRef]

G. M. Lerman and U. Levy, “Tight focusing of spatially variant vector optical fields with elliptical symmetry of liner polarization,” Opt. Lett. 32, 2194-2196 (2007).
[CrossRef] [PubMed]

2006 (1)

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

2004 (1)

F. S. Roux, “Distribution of angular momentum and vortex morphology in optical beams,” Opt. Commun. 242, 45-55(2004).
[CrossRef]

2003 (1)

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

2002 (2)

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

J. W. M. Chon, X. Gan, and M. Gu, “Splitting of the focal spot of a high numerical-aperture objective in free space,” Appl. Phys. Lett. 81, 1576-1578 (2002).
[CrossRef]

2001 (2)

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219-276 (2001).
[CrossRef]

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

2000 (3)

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7 (2000).
[CrossRef]

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

J. Courtial and M. J. Padgett, “Limit to the orbital angular momentum per unit energy in a light beam that can be focused onto a small particle,” Opt. Commun. 173, 269-274 (2000).
[CrossRef]

1997 (1)

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064-4075 (1997).
[CrossRef]

1993 (1)

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Ween, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123-132(1993).
[CrossRef]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

1965 (1)

A. Boivin and E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138, B1561-B1565 (1965).
[CrossRef]

1959 (1)

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349-357 (1959).
[CrossRef]

Allen, L.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Ween, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123-132(1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Ween, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123-132(1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Boivin, A.

A. Boivin and E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138, B1561-B1565 (1965).
[CrossRef]

Bokor, N.

N. Bokor and N. Davidson, “A three dimensional dark focal spot uniformly surrounded by light,” Opt. Commun. 279, 229-234 (2007).
[CrossRef]

Bomzon, Z.

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

Brown, T. G.

Chon, J. W. M.

J. W. M. Chon, X. Gan, and M. Gu, “Splitting of the focal spot of a high numerical-aperture objective in free space,” Appl. Phys. Lett. 81, 1576-1578 (2002).
[CrossRef]

Courjon, D.

T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272, 314-319 (2007).
[CrossRef]

Courtial, J.

J. Courtial and M. J. Padgett, “Limit to the orbital angular momentum per unit energy in a light beam that can be focused onto a small particle,” Opt. Commun. 173, 269-274 (2000).
[CrossRef]

Davidson, N.

N. Bokor and N. Davidson, “A three dimensional dark focal spot uniformly surrounded by light,” Opt. Commun. 279, 229-234 (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] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Gan, X.

J. W. M. Chon, X. Gan, and M. Gu, “Splitting of the focal spot of a high numerical-aperture objective in free space,” Appl. Phys. Lett. 81, 1576-1578 (2002).
[CrossRef]

Glockl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Gorshkov, V. N.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064-4075 (1997).
[CrossRef]

Grosjean, T.

T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272, 314-319 (2007).
[CrossRef]

Gu, M.

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

J. W. M. Chon, X. Gan, and M. Gu, “Splitting of the focal spot of a high numerical-aperture objective in free space,” Appl. Phys. Lett. 81, 1576-1578 (2002).
[CrossRef]

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

Heckenberg, N. R.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064-4075 (1997).
[CrossRef]

Helseth, L. E.

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

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Levy, U.

Malos, J. T.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064-4075 (1997).
[CrossRef]

Milster, T. D.

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

Padgett, M. J.

J. Courtial and M. J. Padgett, “Limit to the orbital angular momentum per unit energy in a light beam that can be focused onto a small particle,” Opt. Commun. 173, 269-274 (2000).
[CrossRef]

Pu, J.

Quabis, S.

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

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Roux, F. S.

F. S. Roux, “Distribution of angular momentum and vortex morphology in optical beams,” Opt. Commun. 242, 45-55(2004).
[CrossRef]

Shamir, J.

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

Soskin, M. S.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219-276 (2001).
[CrossRef]

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064-4075 (1997).
[CrossRef]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

van der Ween, H. E. L. O.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Ween, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123-132(1993).
[CrossRef]

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219-276 (2001).
[CrossRef]

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064-4075 (1997).
[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, X.

Woerdman, J. P.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Ween, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123-132(1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Wolf, E.

A. Boivin and E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138, B1561-B1565 (1965).
[CrossRef]

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349-357 (1959).
[CrossRef]

Youngworth, K. S.

Zhang, Z.

Appl. Opt. (1)

Appl. Phys. Lett. (2)

J. W. M. Chon, X. Gan, and M. Gu, “Splitting of the focal spot of a high numerical-aperture objective in free space,” Appl. Phys. Lett. 81, 1576-1578 (2002).
[CrossRef]

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

Opt. Commun. (7)

F. S. Roux, “Distribution of angular momentum and vortex morphology in optical beams,” Opt. Commun. 242, 45-55(2004).
[CrossRef]

J. Courtial and M. J. Padgett, “Limit to the orbital angular momentum per unit energy in a light beam that can be focused onto a small particle,” Opt. Commun. 173, 269-274 (2000).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Ween, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123-132(1993).
[CrossRef]

N. Bokor and N. Davidson, “A three dimensional dark focal spot uniformly surrounded by light,” Opt. Commun. 279, 229-234 (2007).
[CrossRef]

T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272, 314-319 (2007).
[CrossRef]

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

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. (1)

A. Boivin and E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138, B1561-B1565 (1965).
[CrossRef]

Phys. Rev. A (2)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064-4075 (1997).
[CrossRef]

Phys. Rev. Lett. (1)

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

Proc. R. Soc. London Ser. A (1)

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349-357 (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]

Prog. Opt. (1)

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219-276 (2001).
[CrossRef]

Other (1)

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

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Figures (11)

Fig. 1
Fig. 1

Scheme of tight focusing.

Fig. 2
Fig. 2

Intensity distributions in the focal plane for different topological charges. (a), (d), (g), (j) m = 0 (RHE-polarized beam); (b), (e), (h), (k) m = 1 (RHE-polarized beam); (c), (f), (i), (l) m = 1 (LHE-polarized beam). (a)–(c) total intensity I t , (d)–(f) x component I x , (g)–(i) y component I y , (j)–(l) z component I z . The other parameters are chosen as NA = 0.9 , β = π / 4 , E 0 x = 1 , E 0 y = 1.5 .

Fig. 3
Fig. 3

Phase contours of E z ( r , ϕ , z ) in the focal plane. (a), (b) m = 0 ; (c), (d) m = 1 ; (e), (f) m = 2 ; (a), (c), (e) RHE-polarized beam; (b), (d), (f) LHE-polarized beam. The other parameters are the same as in Fig. 2.

Fig. 4
Fig. 4

Phase contours of the x, y, and z components in the focal plane with m = 1 . (a), (b), (c) RHE-polarized beam; (d), (e), (f) LHE-polarized beam; (a), (d) E x ( r , ϕ , z ) ; (b), (e) E y ( r , ϕ , z ) ; (c), (f) E z ( r , ϕ , z ) . The other parameters are the same as in Fig. 2.

Fig. 5
Fig. 5

Influence of varying E 0 y on I t in the focal plane with m = 1 (RHE-polarized beam). (a)  E 0 y = 1 , (b)  E 0 y = 2 , (c)  E 0 y = 4 , (d)  E 0 y = 40 . The other parameters are the same as in Fig. 2.

Fig. 6
Fig. 6

Influence of varying E 0 y on I x in the focal plane with m = 1 (RHE-polarized beam). (a)  E 0 y = 1 , (b)  E 0 y = 4 , (c)  E 0 y = 14 , (d)  E 0 y = 40 . The other parameters are the same as in Fig. 2.

Fig. 7
Fig. 7

Influence of varying E 0 y on I y in the focal plane with m = 1 (RHE-polarized beam). (a)  E 0 y = 1 , (b)  E 0 y = 4 , (c)  E 0 y = 14 , (d)  E 0 y = 40 . The other parameters are the same as in Fig. 2.

Fig. 8
Fig. 8

Influence of varying E 0 y on I z in the focal plane with m = 1 (RHE-polarized beam). (a)  E 0 y = 1 , (b)  E 0 y = 2 , (c)  E 0 y = 4 , (d)  E 0 y = 40 . The other parameters are the same as in Fig. 2.

Fig. 9
Fig. 9

Influence of varying phase retardation β on I t in the focal plane with m = 1 (RHE-polarized beam). (a)  β = 0 , (b)  β = π / 6 , (c)  β = π / 3 , (d)  β = π / 2 , (e)  β = 2 π / 3 , (f)  β = π . E 0 y = 1 . The other parameters are the same as in Fig. 2.

Fig. 10
Fig. 10

Influence of varying NA on the total intensity in the focal plane with m = 1 (RHE-polarized beam). (a)  NA = 0.8 , (b)  NA = 0.85 , (c)  NA = 0.9 , (d)  NA = 0.95 . E 0 y = 1 . The other parameters are the same as in Fig. 2.

Fig. 11
Fig. 11

Influence of varying phase retardation β on phase distributions of E z ( r , ϕ , z ) with m = 1 (RHE-polarized beam) in the focal plane. (a)  β = 0 , (b)  β = π / 6 , (c)  β = π / 3 , (d)  β = π / 2 , (e)  β = 2 π / 3 , (f)  β = π . E 0 y = 1 . The other parameters are the same as in Fig. 2.

Equations (8)

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E ( r , φ , z ) = [ E x E y E z ] = i k f 2 π 0 α 0 2 π A ( θ , ϕ ) exp [ i k ( z cos θ + r sin θ cos ( ϕ φ ) ) ] sin θ cos θ [ cos 2 ϕ cos θ + sin 2 ϕ cos ϕ sin ϕ ( cos θ 1 ) sin θ cos ( ϕ ) ] d ϕ d θ .
E ( r , φ , z ) = [ E x ± E x E y ± E y E z ± E z ] = i k f 2 π 0 α 0 2 π sin θ cos θ exp [ i k ( z cos θ + r sin θ cos ( ϕ φ ) ) ] × [ A x ( θ , ϕ ) ( cos 2 ϕ cos θ + sin 2 ϕ ) ± A y ( θ , ϕ ) e i β cos ϕ sin ϕ ( cos θ 1 ) A x ( θ , ϕ ) cos ϕ sin ϕ ( cos θ 1 ) ± A y ( θ , ϕ ) e i β ( cos 2 ϕ + sin 2 ϕ cos θ ) A x ( θ , ϕ ) sin θ cos ( ϕ ) ± A y ( θ , ϕ ) e i β sin θ sin ( ϕ ) ] d ϕ d θ .
E ± ( r ) = E x e x ± E y e i β e y .
E m j ( r , ϕ ) = E 0 j ( 2 r w 0 ) | m | exp ( r 2 w 0 2 ) exp ( i m ϕ ) , ( j = x , y ) ,
A m j ( θ , ϕ ) = A m j ( θ ) exp ( i m ϕ ) = E 0 j ( 2 f sin θ w 0 ) | m | exp ( f 2 sin 2 θ w 0 2 ) exp ( i m ϕ ) , ( j = x , y ) .
E ± , x ( r , φ , z ) = i k f 2 0 α sin θ cos θ exp ( i k z cos θ ) { A m x ( θ ) ( 1 + cos θ ) i m J m ( k r sin θ ) exp ( i m φ ) + 1 2 ( A m x ( θ ) ± ( i ) A m y ( θ ) e i β ) ( cos θ 1 ) i m + 2 J m + 2 ( k r sin θ ) exp [ i ( m + 2 ) φ ] + 1 2 ( A m x ( θ ) ± i A m y ( θ ) e i β ) ( cos θ 1 ) i m 2 J m 2 ( k r sin θ ) exp [ i ( m 2 ) φ ] } d θ ,
E ± , y ( r , φ , z ) = i k f 2 0 α sin θ cos θ exp ( i k z cos θ ) { ± A m y ( θ ) e i β ( 1 + cos θ ) i m J m ( k r sin θ ) exp ( i m φ ) + 1 2 [ ( i ) A m x ( θ ) A m y ( θ ) e i β ] ( cos θ 1 ) i m + 2 J m + 2 ( k r sin θ ) exp [ i ( m + 2 ) φ ] + 1 2 ( i A m x ( θ ) A m y ( θ ) e i β ) ( cos θ 1 ) i m 2 J m 2 ( k r sin θ ) exp [ i ( m 2 ) φ ] } d θ ,
E ± , z ( r , φ , z ) = i k f 2 0 α sin 2 θ cos θ exp ( i k z cos θ ) { [ A m x ( θ ) ± ( i ) A m y ( θ ) e i β ] i m + 1 J m + 1 ( k r sin θ ) exp [ i ( m + 1 ) φ ] + ( A m x ( θ ) ± i A m y ( θ ) e i β ) i m 1 J m 1 ( k r sin θ ) exp [ i ( m 1 ) φ ] } d θ .

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