Abstract

In this Letter, we describe the behavior of partially coherent, partially polarized focused vector beams after passing a linear polarizer placed at the focal plane of a high numerical aperture microscope lens. In particular, we develop a mathematical framework for such beams that helps the understanding of the performance of polarizers when interact with non-paraxial beams. The features of the focused field after the polarizer are numerically evaluated for some illustrative examples.

© 2018 Optical Society of America

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References

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  9. P. Yeh, Optical Waves in Layered Media (Wiley, 1988), Vol. 95.
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    [Crossref]
  11. H. L. Ong, Jpn. J. Appl. Phys. 30, L1028 (1991).
    [Crossref]
  12. R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, Sci. Rep. 7, 42122 (2017).
    [Crossref]
  13. S. Zhang, H. Partanen, C. Hellmann, and F. Wyrowski, Opt. Express 26, 9840 (2018).
    [Crossref]
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    [Crossref]
  17. B. Richards and E. Wolf, Proc. R. Soc. London A 253, 358 (1959).
    [Crossref]
  18. R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, Opt. Lasers Eng. 98, 176 (2017).
    [Crossref]
  19. J. Tervo, T. Setälä, and A. T. Friberg, J. Opt. Soc. Am. A 21, 2205 (2004).
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  20. R. Martínez-Herrero and P. M. Mejías, Opt. Lett. 34, 2303 (2009).
    [Crossref]
  21. T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, Phys. Rev. E 66, 016615 (2002).
    [Crossref]
  22. R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009), Vol. 147.
  23. M. Santarsiero, J. Opt. Soc. Am. A 24, 3493 (2007).
    [Crossref]
  24. R. Martínez-Herrero and P. M. Mejías, Opt. Lett. 32, 1471 (2007).
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  25. R. Martínez-Herrero and P. M. Mejías, Opt. Lett. 32, 1504 (2007).
    [Crossref]
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    [Crossref]
  27. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
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    [Crossref]

2018 (1)

2017 (2)

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, Sci. Rep. 7, 42122 (2017).
[Crossref]

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, Opt. Lasers Eng. 98, 176 (2017).
[Crossref]

2016 (1)

2015 (1)

2013 (1)

2009 (3)

2007 (3)

2004 (1)

2002 (1)

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, Phys. Rev. E 66, 016615 (2002).
[Crossref]

1993 (1)

1991 (2)

H. L. Ong, Appl. Phys. Lett. 59, 155 (1991).
[Crossref]

H. L. Ong, Jpn. J. Appl. Phys. 30, L1028 (1991).
[Crossref]

1984 (1)

1982 (1)

1980 (1)

P. Yeh, Surf. Sci. 96, 41 (1980).
[Crossref]

1979 (1)

1972 (1)

1959 (1)

B. Richards and E. Wolf, Proc. R. Soc. London A 253, 358 (1959).
[Crossref]

Aiello, A.

Banzer, P.

Berreman, D. W.

Cai, Y.

X. Liu, F. Wang, L. Liu, C. Zhao, and Y. Cai, J. Opt. Soc. Am. A 32, 2058 (2015).
[Crossref]

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, Progress in Optics (Elsevier, 2017), Vol. 62, pp. 157–223.

Carnicer, A.

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, Sci. Rep. 7, 42122 (2017).
[Crossref]

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, Opt. Lasers Eng. 98, 176 (2017).
[Crossref]

Chen, Y.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, Progress in Optics (Elsevier, 2017), Vol. 62, pp. 157–223.

Fainman, Y.

Friberg, A. T.

Gbur, G.

G. Gbur and T. Visser, Progress in Optics (Elsevier, 2010), Vol. 55, pp. 285–341.

Gu, C.

Hellmann, C.

Juvells, I.

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, Sci. Rep. 7, 42122 (2017).
[Crossref]

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, Opt. Lasers Eng. 98, 176 (2017).
[Crossref]

Kaivola, M.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, Phys. Rev. E 66, 016615 (2002).
[Crossref]

Kolb, T.

Korger, J.

Leuchs, G.

Lindfors, K.

Liu, L.

X. Liu, F. Wang, L. Liu, C. Zhao, and Y. Cai, J. Opt. Soc. Am. A 32, 2058 (2015).
[Crossref]

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, Progress in Optics (Elsevier, 2017), Vol. 62, pp. 157–223.

Liu, X.

X. Liu, F. Wang, L. Liu, C. Zhao, and Y. Cai, J. Opt. Soc. Am. A 32, 2058 (2015).
[Crossref]

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, Progress in Optics (Elsevier, 2017), Vol. 62, pp. 157–223.

Maluenda, D.

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, Opt. Lasers Eng. 98, 176 (2017).
[Crossref]

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, Sci. Rep. 7, 42122 (2017).
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Marquardt, C.

Martínez-Herrero, R.

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, Sci. Rep. 7, 42122 (2017).
[Crossref]

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, Opt. Lasers Eng. 98, 176 (2017).
[Crossref]

R. Martínez-Herrero and P. M. Mejías, Opt. Lett. 34, 2303 (2009).
[Crossref]

R. Martínez-Herrero and P. M. Mejías, Opt. Lett. 32, 1471 (2007).
[Crossref]

R. Martínez-Herrero and P. M. Mejías, Opt. Lett. 32, 1504 (2007).
[Crossref]

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009), Vol. 147.

Mejías, P. M.

Ong, H. L.

H. L. Ong, Appl. Phys. Lett. 59, 155 (1991).
[Crossref]

H. L. Ong, Jpn. J. Appl. Phys. 30, L1028 (1991).
[Crossref]

Partanen, H.

Piquero, G.

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009), Vol. 147.

Richards, B.

B. Richards and E. Wolf, Proc. R. Soc. London A 253, 358 (1959).
[Crossref]

Santarsiero, M.

Setälä, T.

Shamir, J.

Shevchenko, A.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, Phys. Rev. E 66, 016615 (2002).
[Crossref]

Tervo, J.

Visser, T.

G. Gbur and T. Visser, Progress in Optics (Elsevier, 2010), Vol. 55, pp. 285–341.

Wang, F.

Wittmann, C.

Wolf, E.

B. Richards and E. Wolf, Proc. R. Soc. London A 253, 358 (1959).
[Crossref]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Wyrowski, F.

Yeh, P.

Yu, J.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, Progress in Optics (Elsevier, 2017), Vol. 62, pp. 157–223.

Zhang, S.

Zhao, C.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

H. L. Ong, Appl. Phys. Lett. 59, 155 (1991).
[Crossref]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (5)

Jpn. J. Appl. Phys. (1)

H. L. Ong, Jpn. J. Appl. Phys. 30, L1028 (1991).
[Crossref]

Opt. Express (2)

Opt. Lasers Eng. (1)

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, Opt. Lasers Eng. 98, 176 (2017).
[Crossref]

Opt. Lett. (5)

Phys. Rev. E (1)

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, Phys. Rev. E 66, 016615 (2002).
[Crossref]

Proc. R. Soc. London A (1)

B. Richards and E. Wolf, Proc. R. Soc. London A 253, 358 (1959).
[Crossref]

Sci. Rep. (1)

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, Sci. Rep. 7, 42122 (2017).
[Crossref]

Surf. Sci. (1)

P. Yeh, Surf. Sci. 96, 41 (1980).
[Crossref]

Other (5)

P. Yeh, Optical Waves in Layered Media (Wiley, 1988), Vol. 95.

G. Gbur and T. Visser, Progress in Optics (Elsevier, 2010), Vol. 55, pp. 285–341.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, Progress in Optics (Elsevier, 2017), Vol. 62, pp. 157–223.

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009), Vol. 147.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

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Figures (4)

Fig. 1.
Fig. 1. Coordinate system.
Fig. 2.
Fig. 2. Non-polarized coherent beam a = b = 1 , NA = 0.9 , f 0 = 1 , and β = 0 . The minimum value of P 3 D is 0.52. These distributions are calculated at the plane x y (see Fig. 1). Axis values in λ units.
Fig. 3.
Fig. 3. Almost incoherent beam ( L c / f = 0.3 ), NA = 0.9 , f 0 = 1 , β = 0 : (a) radially polarized ( a = 0 , b = 1 ) , min ( P 3 D ) = 0.51 ; (b) azimuthally polarized ( a = 1 , b = 0 ) , min ( P 3 D ) = 0.64 . These distributions are calculated at the plane x y (see Fig. 1). Axis values in λ units.
Fig. 4.
Fig. 4. Partially coherent beam ( L c / f = 3 ), NA = 0.9 , f 0 = 1 , β = 0 : (a) radially polarized ( a = 0 , b = 1 ) , min ( P 3 D ) = 0.5 ; (b) azimuthally polarized ( a = 1 , b = 0 ) , min ( P 3 D ) = 0.5 . These distributions are calculated at the plane x y (see Fig. 1). Axis values in λ units.

Equations (22)

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E ( r ) = A 0 θ M 0 2 π cos θ E 0 ( θ , ϕ ) exp ( i k r · s ) sin θ d θ d ϕ ,
E ( r ) = A 0 θ M 0 2 π cos θ ( E 0 ( θ , ϕ ) · q ( θ , ϕ , β ) ) p ( θ , ϕ , β ) × exp ( i k r · s ) sin θ d θ d ϕ ,
q ( θ , ϕ , β ) = t s ( θ ) cos θ 0 cos ( ϕ β ) 1 sin 2 θ 0 cos 2 ( ϕ β ) e 1 t p ( θ ) sin ( ϕ β ) 1 sin 2 θ 0 cos 2 ( ϕ β ) e 2 ,
p ( θ , ϕ , β ) = t s ( θ ) cos θ 0 cos ( ϕ β ) 1 sin 2 θ 0 cos 2 ( ϕ β ) e 1 + t p ( θ ) sin ( ϕ β ) 1 sin 2 θ 0 cos 2 ( ϕ β ) e 2 ,
t s ( θ ) = 2 cos θ cos θ + n o cos θ 0 t p ( θ ) = 2 cos θ cos θ 0 + n o cos θ ,
t s ( θ ) = 2 n o cos θ 0 cos θ + n o cos θ 0 t p ( θ ) = 2 n o cos θ 0 cos θ 0 + n o cos θ .
W ^ p ( r 1 , r 2 ) = | A | 2 0 θ M 0 θ M 0 2 π 0 2 π W ^ o ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) × exp ( i k ( r 1 s 1 r 2 s 2 ) ) sin θ 1 sin θ 2 d θ 1 d θ 2 d ϕ 1 d ϕ 2 ,
W ^ o ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) = cos θ 1 cos θ 2 Γ ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) × p ( θ 1 , ϕ 1 , β ) p ( θ 2 , ϕ 2 , β ) ,
Γ ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) = q ( θ 1 , ϕ 1 ) W ^ i ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) q ( θ 2 , ϕ 2 ) .
W ^ i ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) = n λ n 2 F n ( θ 1 , ϕ 1 ) F n ( θ 2 , ϕ 2 ) .
W ^ P ( r 1 , r 2 ) = n λ n 2 H n ( r 1 ) H n ( r 2 ) , with
H n ( r ) = 0 θ M 0 2 π cos θ ( F n ( θ , ϕ ) · q ( θ , ϕ ) ) p ( θ , ϕ ) exp ( i k rs ) sin θ d θ d ϕ .
P 3 D 2 = 3 2 ( Tr [ W ^ p ( r , r ) W ^ p ( r , r ) ] [ Tr [ W ^ p ( r , r ) ] ] 2 1 3 ) ,
P 3 D 2 = 1 3 n m λ n 2 λ m 2 | H n ( r ) | 2 | H m ( r ) | 2 sin 2 α n m ( r ) 2 ( n λ n 2 | H n ( r ) | 2 ) 2 ,
sin 2 α n m ( r ) = 1 | H n ( r ) H m ( r ) | 2 | H n ( r ) | 2 | H m ( r ) | 2 .
W ^ i ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) = Γ i ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) M ^ ,
| μ W | max = | Γ i ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) | Γ i ( θ 1 , ϕ 1 , θ 1 , ϕ 1 ) Γ i ( θ 2 , ϕ 2 , θ 2 , ϕ 2 ) .
P 3 D 2 = 1 3 a b | H 1 ( r ) | 2 | H 2 ( r ) | 2 sin 2 α 12 ( r ) ( a | H 1 ( r ) | 2 + b | H 2 ( r ) | 2 ) 2 ,
H j ( r ) = A 0 θ M 0 2 π cos θ ( q U ) j * g ( θ , ϕ ) p ( θ , ϕ ) × exp ( i k rs ) sin θ d θ d ϕ ,
P 3 D j 2 = 1 3 n m λ n 2 λ m 2 | H n j ( r ) | 2 | H m j ( r ) | 2 sin 2 α n m j ( r ) 2 ( n λ n 2 | H n j ( r ) | 2 ) 2 ,
H n j ( r ) = A 0 θ M 0 2 π cos θ ( q U ) j * g n ( θ , ϕ ) p ( θ , ϕ ) × exp ( i k rs ) sin θ d θ d ϕ ,
Γ i ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) sin θ 1 sin θ 2 f o 2    sin 2 θ M exp ( sin 2 θ 1 + sin 2 θ 2 f o 2 sin 2 θ M ) × exp ( sin 2 θ 1 + sin 2 θ 2 2 sin θ 1 sin θ 2 cos ( ϕ 1 ϕ 2 ) L c 2 / f 2 ) × exp ( i ( ϕ 2 ϕ 1 ) ) .

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