Abstract

We present a theory for Goos-Hänchen (GH) and Imbert–Fedorov (IF) shifts for beams of light with arbitrary spatial coherence. By applying the well-known theory of partial spatial coherence, we can calculate explicitly spatial and angular GH and IF shifts for completely polarized beams of any shape and spatial coherence. For the specific case of a Gauss-Schell source, we find that only the angular part of GH and IF shifts is affected by the spatial coherence of the beam. A physical explanation of our results is given.

© 2011 Optical Society of America

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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. K. Yu. Bliokh and Y. P. Bliokh, Phys. Rev. E 96, 243903(2006).
  6. A. Aiello and J. P. Woerdman, Opt. Lett. 33 (13), 1437(2008).
    [CrossRef] [PubMed]
  7. M. V. Berry, Proc. R. Soc. A, published online before print March 23, 2011 doi: 10.1098/rspa.2011.0081 (2011).
  8. M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, Nat. Photonics 3, 337 (2009).
    [CrossRef]
  9. M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, Phys. Rev. A 82, 023817 (2010).
    [CrossRef]
  10. H. Schomerus and M. Hentschel, Phys. Rev. Lett. 75, 066609 (2007).
  11. C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydło, Phys. Rev. Lett. 102, 146804 (2009).
    [CrossRef] [PubMed]
  12. M. Onoda, S. Murakami, and N. Nagaosa, Phys. Rev. Lett. 93, 083901 (2004).
    [CrossRef] [PubMed]
  13. K. Yu. Bliokh and Y. P. Bliokh, Phys. Rev. Lett. 96, 073903(2006).
    [CrossRef] [PubMed]
  14. O. Hosten and P. Kwiat, Science 319, 787 (2008).
    [CrossRef] [PubMed]
  15. R. Simon and T. Tamir, J. Opt. Soc. Am. A 6, 18 (1989).
    [CrossRef]
  16. L.-Q. Wang, L.-G. Wang, S.-Y. Zhu, and M. S. Zubairy, J. Phys. B 41, 055401 (2008).
    [CrossRef]
  17. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
  18. We do not consider here unpolarized beams. This case is not interesting because both spatial and angular GH shifts naturally split in two independent components for linear polarization parallel (p) and perpendicular (s) to the plane of incidence. For the IF shifts the situation is analogous with the two independent polarization components being now the left-circular and the right-circular ones. From this fact it follows that, for example, a completely unpolarized beam will simply undergo a spatial GH shift equal to the sum of the 50% of the shift of a p-polarized beam and the 50% of the shift of a s-polarized beam.
  19. F. Gori, Opt. Commun. 34, 301 (1980).
    [CrossRef]
  20. A. Starikov and E. Wolf, J. Opt. Soc. Am. 72, 923 (1982).
    [CrossRef]
  21. J. W. Goodman, Statistical Optics (John Wiley & Sons, Inc., NY, 2000).
  22. A. Aiello, M. Merano, and J. P. Woerdman, Phys. Rev. A 80, 061801(R) (2009).
    [CrossRef]

2011 (1)

M. V. Berry, Proc. R. Soc. A, published online before print March 23, 2011 doi: 10.1098/rspa.2011.0081 (2011).

2010 (1)

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, Phys. Rev. A 82, 023817 (2010).
[CrossRef]

2009 (3)

M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, Nat. Photonics 3, 337 (2009).
[CrossRef]

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydło, Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef] [PubMed]

A. Aiello, M. Merano, and J. P. Woerdman, Phys. Rev. A 80, 061801(R) (2009).
[CrossRef]

2008 (3)

A. Aiello and J. P. Woerdman, Opt. Lett. 33 (13), 1437(2008).
[CrossRef] [PubMed]

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

L.-Q. Wang, L.-G. Wang, S.-Y. Zhu, and M. S. Zubairy, J. Phys. B 41, 055401 (2008).
[CrossRef]

2007 (1)

H. Schomerus and M. Hentschel, Phys. Rev. Lett. 75, 066609 (2007).

2006 (2)

K. Yu. Bliokh and Y. P. Bliokh, Phys. Rev. E 96, 243903(2006).

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

2004 (1)

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

2000 (1)

J. W. Goodman, Statistical Optics (John Wiley & Sons, Inc., NY, 2000).

1995 (1)

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

1989 (1)

1982 (1)

1980 (1)

F. Gori, Opt. Commun. 34, 301 (1980).
[CrossRef]

1972 (1)

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

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, Phys. Rev. A 82, 023817 (2010).
[CrossRef]

A. Aiello, M. Merano, and J. P. Woerdman, Phys. Rev. A 80, 061801(R) (2009).
[CrossRef]

M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, Nat. Photonics 3, 337 (2009).
[CrossRef]

A. Aiello and J. P. Woerdman, Opt. Lett. 33 (13), 1437(2008).
[CrossRef] [PubMed]

Akhmerov, A. R.

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydło, Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef] [PubMed]

Artmann, K.

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

Beenakker, C. W. J.

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydło, Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef] [PubMed]

Berry, M. V.

M. V. Berry, Proc. R. Soc. A, published online before print March 23, 2011 doi: 10.1098/rspa.2011.0081 (2011).

Bliokh, K. Yu.

K. Yu. Bliokh and Y. P. Bliokh, Phys. Rev. E 96, 243903(2006).

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

Bliokh, Y. P.

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

K. Yu. Bliokh and Y. P. Bliokh, Phys. Rev. E 96, 243903(2006).

Fedorov, F. I.

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

Goodman, J. W.

J. W. Goodman, Statistical Optics (John Wiley & Sons, Inc., NY, 2000).

Goos, F.

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

Gori, F.

F. Gori, Opt. Commun. 34, 301 (1980).
[CrossRef]

Hänchen, H.

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

Hentschel, M.

H. Schomerus and M. Hentschel, Phys. Rev. Lett. 75, 066609 (2007).

Hermosa, N.

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, Phys. Rev. A 82, 023817 (2010).
[CrossRef]

Hosten, O.

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

Imbert, C.

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

Kwiat, P.

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

Mandel, L.

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

Merano, M.

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, Phys. Rev. A 82, 023817 (2010).
[CrossRef]

A. Aiello, M. Merano, and J. P. Woerdman, Phys. Rev. A 80, 061801(R) (2009).
[CrossRef]

M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, Nat. Photonics 3, 337 (2009).
[CrossRef]

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]

Schomerus, H.

H. Schomerus and M. Hentschel, Phys. Rev. Lett. 75, 066609 (2007).

Sepkhanov, R. A.

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydło, Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef] [PubMed]

Simon, R.

Starikov, A.

Tamir, T.

Tworzydlo, J.

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydło, Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef] [PubMed]

van Exter, M. P.

M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, Nat. Photonics 3, 337 (2009).
[CrossRef]

Wang, L.-G.

L.-Q. Wang, L.-G. Wang, S.-Y. Zhu, and M. S. Zubairy, J. Phys. B 41, 055401 (2008).
[CrossRef]

Wang, L.-Q.

L.-Q. Wang, L.-G. Wang, S.-Y. Zhu, and M. S. Zubairy, J. Phys. B 41, 055401 (2008).
[CrossRef]

Woerdman, J. P.

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, Phys. Rev. A 82, 023817 (2010).
[CrossRef]

M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, Nat. Photonics 3, 337 (2009).
[CrossRef]

A. Aiello, M. Merano, and J. P. Woerdman, Phys. Rev. A 80, 061801(R) (2009).
[CrossRef]

A. Aiello and J. P. Woerdman, Opt. Lett. 33 (13), 1437(2008).
[CrossRef] [PubMed]

Wolf, E.

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

A. Starikov and E. Wolf, J. Opt. Soc. Am. 72, 923 (1982).
[CrossRef]

Zhu, S.-Y.

L.-Q. Wang, L.-G. Wang, S.-Y. Zhu, and M. S. Zubairy, J. Phys. B 41, 055401 (2008).
[CrossRef]

Zubairy, M. S.

L.-Q. Wang, L.-G. Wang, S.-Y. Zhu, and M. S. Zubairy, J. Phys. B 41, 055401 (2008).
[CrossRef]

Ann. Phys. (2)

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

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

Dokl. Akad. Nauk SSSR (1)

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

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Phys. B (1)

L.-Q. Wang, L.-G. Wang, S.-Y. Zhu, and M. S. Zubairy, J. Phys. B 41, 055401 (2008).
[CrossRef]

Nat. Photonics (1)

M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, Nat. Photonics 3, 337 (2009).
[CrossRef]

Opt. Commun. (1)

F. Gori, Opt. Commun. 34, 301 (1980).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (2)

A. Aiello, M. Merano, and J. P. Woerdman, Phys. Rev. A 80, 061801(R) (2009).
[CrossRef]

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, Phys. Rev. A 82, 023817 (2010).
[CrossRef]

Phys. Rev. D (1)

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

Phys. Rev. E (1)

K. Yu. Bliokh and Y. P. Bliokh, Phys. Rev. E 96, 243903(2006).

Phys. Rev. Lett. (4)

H. Schomerus and M. Hentschel, Phys. Rev. Lett. 75, 066609 (2007).

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydło, Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef] [PubMed]

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

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

Science (1)

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

Other (4)

M. V. Berry, Proc. R. Soc. A, published online before print March 23, 2011 doi: 10.1098/rspa.2011.0081 (2011).

J. W. Goodman, Statistical Optics (John Wiley & Sons, Inc., NY, 2000).

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

We do not consider here unpolarized beams. This case is not interesting because both spatial and angular GH shifts naturally split in two independent components for linear polarization parallel (p) and perpendicular (s) to the plane of incidence. For the IF shifts the situation is analogous with the two independent polarization components being now the left-circular and the right-circular ones. From this fact it follows that, for example, a completely unpolarized beam will simply undergo a spatial GH shift equal to the sum of the 50% of the shift of a p-polarized beam and the 50% of the shift of a s-polarized beam.

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Equations (23)

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W i j ( r I 1 , r I 2 , ν ) = a i * a j U * ( r I 1 , ν ) U ( r I 2 , ν ) ν ,
U ( r ) ( r ; ν ) = a 1 r 1 ( θ ) U ( x + X 1 , y Y 1 , z ; ν ) x ^ + a 2 r 2 ( θ ) U ( x + X 2 , y Y 2 , z ; ν ) y ^ ,
X 1 = i ln r 1 θ , Y 1 = i a 2 a 1 ( 1 + r 2 r 1 ) cot θ ,
X 2 = i ln r 2 θ , Y 2 = i a 1 a 2 ( 1 + r 1 r 2 ) cot θ .
R 0 ( z ) = i = 1 2 w i ( Δ i + Z θ 0 2 2 Θ i ) ,
w i ( θ ) = | a i r i ( θ ) | 2 | a 1 r 1 ( θ ) | 2 + | a 2 r 2 ( θ ) | 2 ,
Δ i = Re X i Y i , Θ i = Im X i Y i ,
I ν ( r ) = i = 1 2 | a i r i | 2 | U ( x + X i , y Y i , z ; ν ) | 2 ν ,
R i ( z ) = ρ | U ( x + X i , y Y i , z ; ν ) | 2 ν d x d y | U ( x + X i , y Y i , z ; ν ) | 2 ν d x d y ,
| U ( x + X i , y Y i , z ; ν ) | 2 ν = n α n ( ν ) | ψ n ( x + X i , y Y i , z ; ν ) | 2 ,
R i , n ( z ) = ρ ψ n ( x + X i , y Y i , z ; ν ) | 2 d x d y | ψ n ( x + X i , y Y i , z ; ν ) | 2 d x d y .
[ M ( n ; z ) ] i j = 2 Im ψ n * ( x i x j ) ψ n d x d y | ψ n | 2 d x d y ,
R ν ( z ) = i = 1 2 w i ( Δ i + M ¯ ( z ) Θ i ) ,
W ( 0 ) ( ρ 1 , ρ 2 , ν ) = S ( ρ 1 , ν ) S ( ρ 2 , ν ) g ( ρ 2 ρ 1 , ν ) ,
S ( ρ , ν ) = A 2 ( ν ) e ρ 2 2 σ S 2 ( ν ) , g ( ρ , ν ) = e ρ 2 2 σ g 2 ( ν ) .
ψ n m ( ρ ) = C n m v 0 H n ( x 2 v 0 ) H m ( y 2 v 0 ) e x 2 + y 2 v 0 2 ,
α n m ( ν ) = α 0 q n + m , ( n , m = 0 , 1 , 2 , ) .
v 0 2 = 4 σ S 2 / 1 + ( 2 σ S / σ g ) 2 ,
α 0 = A 2 ( ν ) ( 2 σ S / σ g ) 1 + 1 + ( 2 σ S / σ g ) 2 ,
q = [ α 0 / A 2 ( ν ) ] 2 .
M ¯ ( z ) = Z θ ¯ S 2 1 0 0 1 ,
θ ¯ S 2 2 v 0 2 1 + q 1 q = 1 2 k 2 σ S 2 [ 1 + ( 2 σ S σ g ) 2 ] .
R ν ( z ) = i = 1 2 w i ( Δ i + Z θ ¯ S 2 Θ i ) .

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