Abstract

We derive the polarization-dependent displacements parallel and perpendicular to the plane of incidence for a Gaussian light beam reflected from a planar interface, taking into account the propagation of the beam. Using a classical-optics formalism we show that beam propagation may greatly affect both Goos–Hänchen and Imbert–Fedorov shifts when the incident beam is focused.

© 2008 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2003).
  2. F. Goos and H. Hanchen, Ann. Phys. 1, 333 (1947).
    [CrossRef]
  3. O. Costa de Beauregard and C. Imbert, Phys. Rev. Lett. 28, 1211 (1972).
    [CrossRef]
  4. C. Imbert, Phys. Rev. D 5, 787 (1972).
    [CrossRef]
  5. F. Pillon, H. Gilles, and S. Girard, Appl. Opt. 43, 1863 (2004).
    [CrossRef] [PubMed]
  6. W. Nasalski, Phys. Rev. E 74, 056613 (2006).
    [CrossRef]
  7. K. Yu. Bliokh and Yu. P. Bliokh, Phys. Rev. Lett. 96, 073903 (2006).
    [CrossRef] [PubMed]
  8. H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, Phys. Rev. E 62, 7330 (2000).
    [CrossRef]
  9. A. Aiello and J. P. Woerdman, arXiv:0710.1643v2 [physics.optics] (2007).
  10. The literature on these topics is so vast that it is impossible to cite the complete bibliography. As an example, see and references therein.
  11. O. Hosten and P. Kwiat, Science 319, 787 (2008).
    [CrossRef] [PubMed]
  12. It should be noted, however, that in an additional signal enhancement technique based upon polarization control is employed. In this Letter we do not cover such issues since for Gaussian beams the propagation factor is independent of polarization.
  13. Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988).
    [CrossRef] [PubMed]
  14. M. Merano, A. Aiello, G. W. 't Hooft, M. P. van Exter, E. R. Eliel, and J. P. Woerdman, Opt. Express 15, 15928 (2007).
    [CrossRef] [PubMed]
  15. H. A. Haus and J. L. Pan, Am. J. Phys. 61, 818 (1993).
    [CrossRef]
  16. C.-F. Li, Phys. Rev. A 76, 013811 (2007).
    [CrossRef]
  17. K. Artmann, Ann. Phys. 437, 87 (1948).
    [CrossRef]
  18. K. Yu. Bliokh and Yu. P. Bliokh, Phys. Rev. E 75, 066609 (2007).
    [CrossRef]
  19. It should be noted for these “experimental” data the angular aperture ϑ=λ/(2πw0) of the beam is roughly ϑ≈1/130 so that second-order effects (O[ϑ]2) are negligible as compared with the first-order effects given by Eqs. . Second-order effects may become relevant, for example, in the proximity of the Brewster angle (typically within 10−4rad) in air-to-glass reflection processes. We do have full second-order versions of Eqs. , but they are so cumbersome that there is no room for them in the format of a Letter. They will appear in a full research paper that is in preparation.
  20. Note that in the authors use a lens L2 with focal length zeff to collimate the beam so that ⟨yr⟩ remains constant after passing L2.

2008

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

2007

M. Merano, A. Aiello, G. W. 't Hooft, M. P. van Exter, E. R. Eliel, and J. P. Woerdman, Opt. Express 15, 15928 (2007).
[CrossRef] [PubMed]

C.-F. Li, Phys. Rev. A 76, 013811 (2007).
[CrossRef]

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

2006

W. Nasalski, Phys. Rev. E 74, 056613 (2006).
[CrossRef]

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

2004

2000

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, Phys. Rev. E 62, 7330 (2000).
[CrossRef]

1993

H. A. Haus and J. L. Pan, Am. J. Phys. 61, 818 (1993).
[CrossRef]

1988

Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988).
[CrossRef] [PubMed]

1972

O. Costa de Beauregard and C. Imbert, Phys. Rev. Lett. 28, 1211 (1972).
[CrossRef]

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

1948

K. Artmann, Ann. Phys. 437, 87 (1948).
[CrossRef]

1947

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

Aharonov, Y.

Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988).
[CrossRef] [PubMed]

Aiello, A.

Albert, D. Z.

Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988).
[CrossRef] [PubMed]

Artmann, K.

K. Artmann, Ann. Phys. 437, 87 (1948).
[CrossRef]

Bliokh, K. Yu.

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

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

Bliokh, Yu. P.

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

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

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2003).

Costa de Beauregard, O.

O. Costa de Beauregard and C. Imbert, Phys. Rev. Lett. 28, 1211 (1972).
[CrossRef]

Eliel, E. R.

Gilles, H.

Girard, S.

Goos, F.

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

Hanchen, H.

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

Haus, H. A.

H. A. Haus and J. L. Pan, Am. J. Phys. 61, 818 (1993).
[CrossRef]

Hosten, O.

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

Imbert, C.

O. Costa de Beauregard and C. Imbert, Phys. Rev. Lett. 28, 1211 (1972).
[CrossRef]

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

Kwiat, P.

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

Kwok, C. W.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, Phys. Rev. E 62, 7330 (2000).
[CrossRef]

Lai, H. M.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, Phys. Rev. E 62, 7330 (2000).
[CrossRef]

Li, C.-F.

C.-F. Li, Phys. Rev. A 76, 013811 (2007).
[CrossRef]

Loo, Y. W.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, Phys. Rev. E 62, 7330 (2000).
[CrossRef]

Merano, M.

Nasalski, W.

W. Nasalski, Phys. Rev. E 74, 056613 (2006).
[CrossRef]

Pan, J. L.

H. A. Haus and J. L. Pan, Am. J. Phys. 61, 818 (1993).
[CrossRef]

Pillon, F.

't Hooft, G. W.

Vaidman, L.

Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988).
[CrossRef] [PubMed]

van Exter, M. P.

Woerdman, J. P.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2003).

Xu, B. Y.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, Phys. Rev. E 62, 7330 (2000).
[CrossRef]

Am. J. Phys.

H. A. Haus and J. L. Pan, Am. J. Phys. 61, 818 (1993).
[CrossRef]

Ann. Phys.

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

K. Artmann, Ann. Phys. 437, 87 (1948).
[CrossRef]

Appl. Opt.

Opt. Express

Phys. Rev. A

C.-F. Li, Phys. Rev. A 76, 013811 (2007).
[CrossRef]

Phys. Rev. D

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

Phys. Rev. E

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

W. Nasalski, Phys. Rev. E 74, 056613 (2006).
[CrossRef]

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, Phys. Rev. E 62, 7330 (2000).
[CrossRef]

Phys. Rev. Lett.

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

Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988).
[CrossRef] [PubMed]

O. Costa de Beauregard and C. Imbert, Phys. Rev. Lett. 28, 1211 (1972).
[CrossRef]

Science

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

Other

It should be noted, however, that in an additional signal enhancement technique based upon polarization control is employed. In this Letter we do not cover such issues since for Gaussian beams the propagation factor is independent of polarization.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2003).

A. Aiello and J. P. Woerdman, arXiv:0710.1643v2 [physics.optics] (2007).

The literature on these topics is so vast that it is impossible to cite the complete bibliography. As an example, see and references therein.

It should be noted for these “experimental” data the angular aperture ϑ=λ/(2πw0) of the beam is roughly ϑ≈1/130 so that second-order effects (O[ϑ]2) are negligible as compared with the first-order effects given by Eqs. . Second-order effects may become relevant, for example, in the proximity of the Brewster angle (typically within 10−4rad) in air-to-glass reflection processes. We do have full second-order versions of Eqs. , but they are so cumbersome that there is no room for them in the format of a Letter. They will appear in a full research paper that is in preparation.

Note that in the authors use a lens L2 with focal length zeff to collimate the beam so that ⟨yr⟩ remains constant after passing L2.

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Figures (1)

Fig. 1
Fig. 1

Geometry of beam reflection at the air–medium interface.

Equations (10)

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E inc exp [ i Z i X i 2 + Y i 2 2 ( Λ + i Z i ) ] ( x ̂ i f P + y ̂ i f S i z ̂ i f P X i + f S Y i Λ + i Z i ) ,
E ref exp [ i Z r X r 2 + Y r 2 2 ( Λ + i Z r ) ] { x ̂ r [ f P r P ( 1 i X r Λ + i Z r ln r P θ i ) + i f S Y r Λ + i Z r ( r P + r S ) cot θ i ] + y ̂ r [ f S r S ( 1 i X r Λ + i Z r ln r S θ i ) i f P Y r Λ + i Z r ( r P + r S ) cot θ i ] i z ̂ r ( r P f P X r + r S f S Y r Λ + i Z r ) } ,
M = X I ( X r , Y r , Z r ) d X r d Y r I ( X r , Y r , Z r ) d X r d Y r ,
X r = φ P R P 2 a P 2 + φ S R S 2 a S 2 R P 2 a P 2 + R S 2 a S 2 Z r Λ ρ P R P 2 a P 2 + ρ S R S 2 a S 2 R P 2 a P 2 + R S 2 a S 2 ,
Y r = a P a S cot θ i R P 2 a P 2 + R S 2 a S 2 { [ ( R P 2 + R S 2 ) sin η + 2 R P R S sin ( η ϕ P + ϕ S ) ] Z r Λ ( R P 2 R S 2 ) cos η } ,
x ̂ r E ref exp [ X r 2 2 ( Λ + i Z r ) ] ( 1 i X r Λ + i Z r ln r P θ i ) exp [ ( X r φ P + i ρ P ) 2 2 ( Λ + i Z r ) ] .
X r = φ P ( Z r Λ ) ρ P ,
v ̂ E ref X r = 0 exp [ Y r 2 2 ( Λ + i Z r ) ] ( 1 + i Y r D cot Δ Λ + i Z r ) exp [ ( Y r i D cot Δ ) 2 2 ( Λ + i Z r ) ] ,
Y r = ( Z r Λ ) D cot Δ , Y r 2 = ( Λ 2 + Z r 2 ) ( 2 Λ ) ,
y r = 2 k y r 2 R ( z ) δ cot Δ 4 π y r 2 z λ δ cot Δ ,

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