Abstract

We study, analytically and numerically, reflection and transmission of an arbitrarily polarized vortex beam on an interface separating two dielectric media and derive general expressions for linear and angular Goos–Hänchen (GH) and Imbert–Fedorov shifts. We predict a novel vortex-induced GH shift and also reveal a direct connection between the spin-induced angular shifts and the vortex-induced linear shifts.

© 2009 Optical Society of America

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References

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  1. F. Goos and H. Hänchen, Ann. Phys. 1, 333 (1947).
    [CrossRef]
  2. K. Artmann, Ann. Phys. 2, 87 (1948).
    [CrossRef]
  3. F. I. Fedorov, Dokl. Akad. Nauk SSSR 105, 465 (1955).
  4. C. Imbert, Phys. Rev. D 5, 787 (1972).
    [CrossRef]
  5. H. Schilling, Ann. Phys. 16, 122 (1965).
    [CrossRef]
  6. J. Ricard, Nouv. Rev. Opt. 5, 7 (1974).
    [CrossRef]
  7. J. P. Hugonin and R. Petit, J. Opt. 8, 73 (1977).
    [CrossRef]
  8. V. G. Fedoseev, Opt. Spectrosc. 58, 296 (1985).
  9. V. S. Liberman and B. Y. Zel'dovich, Phys. Rev. A 46, 5199 (1992).
    [CrossRef] [PubMed]
  10. M. A. Player, J. Phys. A 20, 3667 (1987).
    [CrossRef]
  11. V. G. Fedoseyev, J. Phys. A 21, 2045 (1988).
    [CrossRef]
  12. M. Onoda, S. Murakami, and N. Nagaosa, Phys. Rev. Lett. 93, 083901 (2004).
    [CrossRef] [PubMed]
  13. K. Y. Bliokh and Y. P. Bliokh, Phys. Rev. Lett. 96, 073903 (2006).
    [CrossRef] [PubMed]
  14. K. Y. Bliokh and Y. P. Bliokh, Phys. Rev. E 75, 066609 (2007).
    [CrossRef]
  15. L. Dutriaux, A. Le Floch, and F. Bretenaker, Europhys. Lett. 24, 345 (1993).
    [CrossRef]
  16. F. Pillon, H. Gilles, and S. Girard, Appl. Opt. 43, 1863 (2004).
    [CrossRef] [PubMed]
  17. O. Hosten and P. Kwiat, Science 319, 787 (2008).
    [CrossRef] [PubMed]
  18. A. Aiello and J. P. Woerdman, Opt. Lett. 33, 1437 (2008).
    [CrossRef] [PubMed]
  19. V. G. Fedoseyev, Opt. Commun. 193, 9 (2001).
    [CrossRef]
  20. R. Dasgupta and P. K. Gupta, Opt. Commun. 257, 91 (2006).
    [CrossRef]
  21. L. Allen, M. J. Padgett, and M. Babiker, Prog. Opt. 39, 291 (1999).
    [CrossRef]
  22. V. G. Fedoseyev, J. Phys. A 41, 505202 (2008).
    [CrossRef]
  23. V. G. Fedoseyev, arXiv:0810.4407.

2008 (3)

O. Hosten and P. Kwiat, Science 319, 787 (2008).
[CrossRef] [PubMed]

A. Aiello and J. P. Woerdman, Opt. Lett. 33, 1437 (2008).
[CrossRef] [PubMed]

V. G. Fedoseyev, J. Phys. A 41, 505202 (2008).
[CrossRef]

2007 (1)

K. Y. Bliokh and Y. P. Bliokh, Phys. Rev. E 75, 066609 (2007).
[CrossRef]

2006 (2)

K. Y. Bliokh and Y. P. Bliokh, Phys. Rev. Lett. 96, 073903 (2006).
[CrossRef] [PubMed]

R. Dasgupta and P. K. Gupta, Opt. Commun. 257, 91 (2006).
[CrossRef]

2004 (2)

F. Pillon, H. Gilles, and S. Girard, Appl. Opt. 43, 1863 (2004).
[CrossRef] [PubMed]

M. Onoda, S. Murakami, and N. Nagaosa, Phys. Rev. Lett. 93, 083901 (2004).
[CrossRef] [PubMed]

2001 (1)

V. G. Fedoseyev, Opt. Commun. 193, 9 (2001).
[CrossRef]

1999 (1)

L. Allen, M. J. Padgett, and M. Babiker, Prog. Opt. 39, 291 (1999).
[CrossRef]

1993 (1)

L. Dutriaux, A. Le Floch, and F. Bretenaker, Europhys. Lett. 24, 345 (1993).
[CrossRef]

1992 (1)

V. S. Liberman and B. Y. Zel'dovich, Phys. Rev. A 46, 5199 (1992).
[CrossRef] [PubMed]

1988 (1)

V. G. Fedoseyev, J. Phys. A 21, 2045 (1988).
[CrossRef]

1987 (1)

M. A. Player, J. Phys. A 20, 3667 (1987).
[CrossRef]

1985 (1)

V. G. Fedoseev, Opt. Spectrosc. 58, 296 (1985).

1977 (1)

J. P. Hugonin and R. Petit, J. Opt. 8, 73 (1977).
[CrossRef]

1974 (1)

J. Ricard, Nouv. Rev. Opt. 5, 7 (1974).
[CrossRef]

1972 (1)

C. Imbert, Phys. Rev. D 5, 787 (1972).
[CrossRef]

1965 (1)

H. Schilling, Ann. Phys. 16, 122 (1965).
[CrossRef]

1955 (1)

F. I. Fedorov, Dokl. Akad. Nauk SSSR 105, 465 (1955).

1948 (1)

K. Artmann, Ann. Phys. 2, 87 (1948).
[CrossRef]

1947 (1)

F. Goos and H. Hänchen, Ann. Phys. 1, 333 (1947).
[CrossRef]

Aiello, A.

Allen, L.

L. Allen, M. J. Padgett, and M. Babiker, Prog. Opt. 39, 291 (1999).
[CrossRef]

Artmann, K.

K. Artmann, Ann. Phys. 2, 87 (1948).
[CrossRef]

Babiker, M.

L. Allen, M. J. Padgett, and M. Babiker, Prog. Opt. 39, 291 (1999).
[CrossRef]

Bliokh, K. Y.

K. Y. Bliokh and Y. P. Bliokh, Phys. Rev. E 75, 066609 (2007).
[CrossRef]

K. Y. Bliokh and Y. P. Bliokh, Phys. Rev. Lett. 96, 073903 (2006).
[CrossRef] [PubMed]

Bliokh, Y. P.

K. Y. Bliokh and Y. P. Bliokh, Phys. Rev. E 75, 066609 (2007).
[CrossRef]

K. Y. Bliokh and Y. P. Bliokh, Phys. Rev. Lett. 96, 073903 (2006).
[CrossRef] [PubMed]

Bretenaker, F.

L. Dutriaux, A. Le Floch, and F. Bretenaker, Europhys. Lett. 24, 345 (1993).
[CrossRef]

Dasgupta, R.

R. Dasgupta and P. K. Gupta, Opt. Commun. 257, 91 (2006).
[CrossRef]

Dutriaux, L.

L. Dutriaux, A. Le Floch, and F. Bretenaker, Europhys. Lett. 24, 345 (1993).
[CrossRef]

Fedorov, F. I.

F. I. Fedorov, Dokl. Akad. Nauk SSSR 105, 465 (1955).

Fedoseev, V. G.

V. G. Fedoseev, Opt. Spectrosc. 58, 296 (1985).

Fedoseyev, V. G.

V. G. Fedoseyev, J. Phys. A 41, 505202 (2008).
[CrossRef]

V. G. Fedoseyev, Opt. Commun. 193, 9 (2001).
[CrossRef]

V. G. Fedoseyev, J. Phys. A 21, 2045 (1988).
[CrossRef]

V. G. Fedoseyev, arXiv:0810.4407.

Gilles, H.

Girard, S.

Goos, F.

F. Goos and H. Hänchen, Ann. Phys. 1, 333 (1947).
[CrossRef]

Gupta, P. K.

R. Dasgupta and P. K. Gupta, Opt. Commun. 257, 91 (2006).
[CrossRef]

Hänchen, H.

F. Goos and H. Hänchen, Ann. Phys. 1, 333 (1947).
[CrossRef]

Hosten, O.

O. Hosten and P. Kwiat, Science 319, 787 (2008).
[CrossRef] [PubMed]

Hugonin, J. P.

J. P. Hugonin and R. Petit, J. Opt. 8, 73 (1977).
[CrossRef]

Imbert, C.

C. Imbert, Phys. Rev. D 5, 787 (1972).
[CrossRef]

Kwiat, P.

O. Hosten and P. Kwiat, Science 319, 787 (2008).
[CrossRef] [PubMed]

Le Floch, A.

L. Dutriaux, A. Le Floch, and F. Bretenaker, Europhys. Lett. 24, 345 (1993).
[CrossRef]

Liberman, V. S.

V. S. Liberman and B. Y. Zel'dovich, Phys. Rev. A 46, 5199 (1992).
[CrossRef] [PubMed]

Murakami, S.

M. Onoda, S. Murakami, and N. Nagaosa, Phys. Rev. Lett. 93, 083901 (2004).
[CrossRef] [PubMed]

Nagaosa, N.

M. Onoda, S. Murakami, and N. Nagaosa, Phys. Rev. Lett. 93, 083901 (2004).
[CrossRef] [PubMed]

Onoda, M.

M. Onoda, S. Murakami, and N. Nagaosa, Phys. Rev. Lett. 93, 083901 (2004).
[CrossRef] [PubMed]

Padgett, M. J.

L. Allen, M. J. Padgett, and M. Babiker, Prog. Opt. 39, 291 (1999).
[CrossRef]

Petit, R.

J. P. Hugonin and R. Petit, J. Opt. 8, 73 (1977).
[CrossRef]

Pillon, F.

Player, M. A.

M. A. Player, J. Phys. A 20, 3667 (1987).
[CrossRef]

Ricard, J.

J. Ricard, Nouv. Rev. Opt. 5, 7 (1974).
[CrossRef]

Schilling, H.

H. Schilling, Ann. Phys. 16, 122 (1965).
[CrossRef]

Woerdman, J. P.

Zel'dovich, B. Y.

V. S. Liberman and B. Y. Zel'dovich, Phys. Rev. A 46, 5199 (1992).
[CrossRef] [PubMed]

Ann. Phys. (3)

F. Goos and H. Hänchen, Ann. Phys. 1, 333 (1947).
[CrossRef]

K. Artmann, Ann. Phys. 2, 87 (1948).
[CrossRef]

H. Schilling, Ann. Phys. 16, 122 (1965).
[CrossRef]

Appl. Opt. (1)

Dokl. Akad. Nauk SSSR (1)

F. I. Fedorov, Dokl. Akad. Nauk SSSR 105, 465 (1955).

Europhys. Lett. (1)

L. Dutriaux, A. Le Floch, and F. Bretenaker, Europhys. Lett. 24, 345 (1993).
[CrossRef]

J. Opt. (1)

J. P. Hugonin and R. Petit, J. Opt. 8, 73 (1977).
[CrossRef]

J. Phys. A (3)

M. A. Player, J. Phys. A 20, 3667 (1987).
[CrossRef]

V. G. Fedoseyev, J. Phys. A 21, 2045 (1988).
[CrossRef]

V. G. Fedoseyev, J. Phys. A 41, 505202 (2008).
[CrossRef]

Nouv. Rev. Opt. (1)

J. Ricard, Nouv. Rev. Opt. 5, 7 (1974).
[CrossRef]

Opt. Commun. (2)

V. G. Fedoseyev, Opt. Commun. 193, 9 (2001).
[CrossRef]

R. Dasgupta and P. K. Gupta, Opt. Commun. 257, 91 (2006).
[CrossRef]

Opt. Lett. (1)

Opt. Spectrosc. (1)

V. G. Fedoseev, Opt. Spectrosc. 58, 296 (1985).

Phys. Rev. A (1)

V. S. Liberman and B. Y. Zel'dovich, Phys. Rev. A 46, 5199 (1992).
[CrossRef] [PubMed]

Phys. Rev. D (1)

C. Imbert, Phys. Rev. D 5, 787 (1972).
[CrossRef]

Phys. Rev. E (1)

K. Y. Bliokh and Y. P. Bliokh, Phys. Rev. E 75, 066609 (2007).
[CrossRef]

Phys. Rev. Lett. (2)

M. Onoda, S. Murakami, and N. Nagaosa, Phys. Rev. Lett. 93, 083901 (2004).
[CrossRef] [PubMed]

K. Y. Bliokh and Y. P. Bliokh, Phys. Rev. Lett. 96, 073903 (2006).
[CrossRef] [PubMed]

Prog. Opt. (1)

L. Allen, M. J. Padgett, and M. Babiker, Prog. Opt. 39, 291 (1999).
[CrossRef]

Science (1)

O. Hosten and P. Kwiat, Science 319, 787 (2008).
[CrossRef] [PubMed]

Other (1)

V. G. Fedoseyev, arXiv:0810.4407.

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Figures (2)

Fig. 1
Fig. 1

Geometry of the beam reflection and transmission at an interface. Projections of the GH shifts, X r cos θ , X t cos θ , and the IF shifts, Y r , t , are shown by white arrows.

Fig. 2
Fig. 2

Numerical (curves) and theoretical (symbols) results for the linear and angular IF and GH shifts of the reflected ( r ) and transmitted ( t ) vortex beam with complex polarization as a function of the angle of incidence θ. Parameters are n = 1.5 , μ = 1 , e = 1 3 , e = ( 1 + i ) 3 , and m = 1 . For the wavelength of 632 nm , X r = 44 nm for θ = 40 ° and Y r = 240 nm for θ 90 ° . These values are proportionally enhanced for higher m.

Equations (21)

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E trans i ( e X ̂ i + e Y ̂ i ) L n , m ( X i , Y i , Z i ) .
R = R e 2 + R e 2 , T = T e 2 + T e 2 ,
Q r = R 2 , Q t = n cos θ μ cos θ T 2 , Q r + Q t = 1 ,
( e r e r ) = 1 R ( R e R e ) , ( e t e t ) = 1 T ( T e T e ) .
X r tot = 1 k ( e 2 ϕ θ + e 2 ϕ θ ) ,
Y r tot = cot θ k [ σ ( 1 + cos δ ) + χ sin δ ] ,
Y r 0 = σ cot θ 2 k ( R + R ) 2 R 2 ,
Y t 0 = σ cot θ 2 k T 2 + T 2 2 T T cos θ cos θ T 2 .
K x r 0 = 1 2 D d ln Q r d θ , K x t 0 = cos θ 2 D cos θ d ln Q t d θ ,
K y r 0 = χ cot θ 2 D R 2 R 2 R 2 , K y t 0 = χ cot θ 2 D T 2 T 2 T 2 .
L r = m Z ̂ r , L t = 1 2 ( cos θ cos θ + cos θ cos θ ) m Z ̂ t .
Y r 1 = 0 , Y t 1 = m 2 k tan θ ( 1 n 2 ) .
[ γ r , t X r , t + i sgn ( m ) ( Y r , t γ r , t D x r , t k K x r , t 0 ) ] m .
Y r 2 = m D k K x r 0 , Y t 2 = m cos θ cos θ D k K x t 0 .
Y r , t = Y r , t 0 + Y r , t 1 + Y r , t 2 .
X r = m D k K y r 0 , X t = m cos θ cos θ D k K y t 0 .
K x , y r , t = ( 1 + m ) K x , y r , t 0 .
Q r K x r cos θ + Q t K x t cos θ = 0 ,
Q r K y r + Q t K y t = 0 ,
Q r ( J z r Y r k sin θ ) + Q t ( J z t Y t k sin θ ) = J z i .
J a = L a + S a , S a = σ a Z ̂ a ,

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