Abstract

Under the approximation of small birefringence, the properties of radially and azimuthally polarized vortex beams tightly focused through a uniaxial birefringent crystal are investigated. The contour plots of intensity distribution near the focus and in the real focal plane are illustrated by performing numerical calculations. The dependence of the focal shift on numerical aperture and birefringence are analyzed. Moreover, the Strehl ratio in the real focal plane as a function of birefringence is also analyzed. It is revealed that the variation of birefringence has no influence on the focal shift and the Strehl ratio of azimuthally polarized vortex beams.

© 2008 Optical Society of America

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References

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  1. A. Ciattoni, G. Ciattoni, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A 19, 792-796 (2002).
    [CrossRef]
  2. M. Avendaño-Alejo and M. Rosete-Aguilar, “Optical path difference in a plane-parallel uniaxial plate,” J. Opt. Soc. Am. A 23, 926-932 (2006).
    [CrossRef]
  3. G. Ciattoni, A. Ciattoni, and C. Sapia, “Radially and azimuthally polarized vortices in uniaxial crystals,” Opt. Commun. 220, 33-40 (2003).
    [CrossRef]
  4. D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281, 202-209 (2008).
    [CrossRef]
  5. B. E. Bernacki and M. Mansuripur, “Investigation of substrate birefringence effects on optical-disk performance,” Appl. Opt. 32, 6547-6555 (1993).
    [CrossRef] [PubMed]
  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. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77-87(2000).
    [CrossRef] [PubMed]
  8. G. M. Lerman and U. Levy, “Tight focusing of spatially variant vector optical fields with elliptical symmentry of liner polarization,” Opt. Lett. 32, 2194-2196 (2007).
    [CrossRef] [PubMed]
  9. E. P. Walker and T. D. Milster, “Beam shaping for optical data storage,” Proc. SPIE 4443, 73-92 (2001).
    [CrossRef]
  10. L. E. Helseth, “Focusing of atoms with strongly confined light potentials,” Opt. Commun. 212, 343-352 (2002).
    [CrossRef]
  11. S. Stallinga, “Axial birefringence in high-numerical-aperture optical systems and the light distribution close to focus,” J. Opt. Soc. Am. A 18, 2846-2859 (2001).
    [CrossRef]
  12. J. J. Stamnes, G. S. Sithambaranathan, M. Jain, J. K. Lotsberg, and V. Dhayalan, “Focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 226, 107-123(2003).
    [CrossRef]
  13. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713-4716 (1997).
    [CrossRef]
  14. P. Török, P. Varga, and G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. I,” J. Opt. Soc. Am. A 12, 2136-2144(1995).
    [CrossRef]
  15. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
  16. M. Gu, Advanced Optical Imaging Theory (Springer, 2000).

2008 (2)

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281, 202-209 (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 (1)

2006 (1)

2003 (2)

G. Ciattoni, A. Ciattoni, and C. Sapia, “Radially and azimuthally polarized vortices in uniaxial crystals,” Opt. Commun. 220, 33-40 (2003).
[CrossRef]

J. J. Stamnes, G. S. Sithambaranathan, M. Jain, J. K. Lotsberg, and V. Dhayalan, “Focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 226, 107-123(2003).
[CrossRef]

2002 (2)

2001 (2)

2000 (1)

1997 (1)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

1995 (1)

1993 (1)

Appl. Opt. (1)

J. Opt. Soc. Am. A (4)

Opt. Commun. (4)

J. J. Stamnes, G. S. Sithambaranathan, M. Jain, J. K. Lotsberg, and V. Dhayalan, “Focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 226, 107-123(2003).
[CrossRef]

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

G. Ciattoni, A. Ciattoni, and C. Sapia, “Radially and azimuthally polarized vortices in uniaxial crystals,” Opt. Commun. 220, 33-40 (2003).
[CrossRef]

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281, 202-209 (2008).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713-4716 (1997).
[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 U. Press, 1999).

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

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Figures (6)

Fig. 1
Fig. 1

Scheme of the optical system.

Fig. 2
Fig. 2

Intensity distribution of BG beams near the focus in the x z plane. (a) Radial polarization with topological n = 1 , (b) radial polarization with topological n = 2 , (c) azimuthal polarization with topological n = 1 , (d) azimuthal polarization with topological n = 2 . The parameters for calculation are Δ n = 2 × 10 3 , NA = 0.9 , λ = 633 nm , d = 1 λ , β = 0.25 mm 1 , f = 1 cm , w 0 = 2 cm .

Fig. 3
Fig. 3

Intensity distribution in the real focal plane with topological charge n = 1 . (a), (c), (e), (g) Radial polarization; (b), (d), (f), (h) azimuthal polarization. (a), (b) Total intensity pattern; (c), (d) intensity pattern of the component in the x direction; (d), (e) intensity pattern of the component in the y direction; (g), (h) intensity pattern of the component in the z direction. The parameters for calculation are the same as in Fig. 2.

Fig. 4
Fig. 4

Intensity distribution in the real focal plane with topological charge n = 2 . (a), (c), (e), (g) Radial polarization; (b), (d), (f), (h) azimuthal polarization. (a), (b) Total intensity pattern; (c), (d) intensity pattern of the component in the x direction; (d), (e) intensity pattern of the component in the y direction; (g), (h) intensity pattern of the component in the z direction. The parameters for calculation are the same as in Fig. 2.

Fig. 5
Fig. 5

Dependence of the focal shift on (a) NA and (b) birefringence. The parameters for calculation are the same as in Fig. 2.

Fig. 6
Fig. 6

The Strehl ratio as a function of birefringence. The parameters for calculation are the same as in Fig. 2.

Equations (13)

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E ( r 2 ) = - i E 0 π 0 α 0 2 π sin θ 1 cos θ 1 P ( θ 1 , ϕ ) [ t p exp ( i W p ) ( cos ϕ e x + sin ϕ e y ) p ^ 2 + t s exp ( i W s ) ( - sin ϕ e x + cos ϕ e y ) s ^ 2 ] exp ( i k ^ 2 r 2 ) d ϕ d θ 1 ,
t s = 2 sin θ 2 cos θ 1 sin ( θ 1 + θ 2 ) ,
t p = 2 sin θ 2 cos θ 1 sin ( θ 1 + θ 2 ) cos ( θ 1 θ 2 ) .
W = k d ( n 1 cos θ 1 n 2 cos θ 2 ) ,
Δ W = k ( d + z ) Δ n sin 2 θ 2 / cos θ 2 ,
E n ( r , ϕ , z ) = E 0 J n ( β r ) exp ( r 2 / w 0 2 ) exp ( i n ϕ ) exp ( i k z ) ,
P n ( θ , ϕ ) = A ( θ ) exp ( i n ϕ ) ,
A ( θ ) = E 0 J n ( β f sin θ ) exp ( f 2 sin 2 θ / w 0 2 ) .
E ( r , ψ , z ) = [ E x ( r , ψ , z ) E y ( r , ψ , z ) E z ( r , ψ , z ) ] = i n + 1 E 0 [ i ( I n + 1 e i ψ I n 1 e i ψ ) I n + 1 e i ψ + I n 1 e i ψ 2 I n ] e i n ψ ,
I n ( r , z ) = 0 α exp [ i k ( W + Δ W ) ] A ( θ 1 ) cos θ 1 sin θ 1 t p sin θ 2 J n ( k 1 r sin θ 1 ) exp ( i k 2 z cos θ 2 ) d θ 1 ,
I n ± 1 ( r , z ) = 0 α exp [ i k ( W + Δ W ) ] A ( θ 1 ) cos θ 1 sin θ 1 t p cos θ 2 J n ± 1 ( k 1 r sin θ 1 ) exp ( i k 2 z cos θ 2 ) d θ 1 .
E ( r , ψ , z ) = [ E x ( r , ψ , z ) E y ( r , ψ , z ) E z ( r , ψ , z ) ] = i n + 1 E 0 [ I n + 1 e i ψ + I n 1 e i ψ i ( I n + 1 e i ψ I n 1 e i ψ ) 0 ] e i n ψ ,
I n ± 1 ( r , z ) = 0 α exp ( i k W ) A ( θ 1 ) cos θ 1 sin θ 1 t s J n ± 1 ( k 1 r sin θ 1 ) exp ( i k 2 z cos θ 2 ) d θ 1 .

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