Abstract

The magnitudes of beam shifts (Goos–Hänchen and Imbert–Fedorov, spatial and angular) are greatly enhanced when a reflected light beam is postselected by an analyzer, by analogy with superweak measurements in quantum theory. Particularly strong enhancements can be expected close to angles at which no light is transmitted for fixed initial and final polarizations. We derive a formula for the angular and spatial shifts at such angles (which includes the Brewster angle), and we show that their maximum size is limited by higher-order terms from the reflection coefficients occurring in the Artmann shift formula.

© 2013 Optical Society of America

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  1. M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, Nat. Photonics 3, 337 (2009).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  4. B. R. Horowitz and T. Tamir, J. Opt. Soc. Am. 61, 586 (1971).
    [CrossRef]
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    [CrossRef]
  6. I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, Phys. Rev. D 402112 (1989).
    [CrossRef]
  7. O. Hosten and P. Kwiat, Science 319, 787 (2008).
    [CrossRef]
  8. J. B. Götte and M. R. Dennis, New J. Phys. 14, 073016 (2012).
    [CrossRef]
  9. M. R. Dennis and J. B. Götte, New J. Phys. 14, 073013 (2012).
    [CrossRef]
  10. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
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    [CrossRef]
  12. R. M. A. Azzam and T. F. Thonn, Appl. Opt. 22, 4155 (1983).
    [CrossRef]
  13. M. V. Berry and P. Shukla, J. Phys. A 43, 354024 (2010).
    [CrossRef]
  14. K. Artmann, Ann. Phys. 437, 87 (1948).
    [CrossRef]
  15. A. G. Kofman, S. Ashbab, and F. Nori, Phys. Rep. 520, 43 (2012).
    [CrossRef]
  16. Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, Phys. Rev. Lett. 109, 013901 (2012).
    [CrossRef]
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    [CrossRef]
  18. A. Aiello and J. P. Woerdman, “Theory of angular Goos-Hänchen shift near Brewster incidence,” arxiv:0903.3730v2 (2009).
  19. F. I. Fedorov, Doklady Akademii Nauk SSSR 105, 465 (1955).
  20. G. Jayaswal, G. Mistura, and M. Merano, Opt. Lett. 38, 1232 (2013).
    [CrossRef]
  21. R. Jozsa, Phys. Rev. A 76, 044103 (2007).
    [CrossRef]
  22. H. Kobayashi, G. Puentes, and Y. Shikano, Phys. Rev. A 86, 053805 (2012).
    [CrossRef]
  23. M. R. Dennis and J. B. Götte, Phys. Rev. Lett. 109, 183903 (2012).
    [CrossRef]

2013 (2)

2012 (6)

H. Kobayashi, G. Puentes, and Y. Shikano, Phys. Rev. A 86, 053805 (2012).
[CrossRef]

M. R. Dennis and J. B. Götte, Phys. Rev. Lett. 109, 183903 (2012).
[CrossRef]

J. B. Götte and M. R. Dennis, New J. Phys. 14, 073016 (2012).
[CrossRef]

M. R. Dennis and J. B. Götte, New J. Phys. 14, 073013 (2012).
[CrossRef]

A. G. Kofman, S. Ashbab, and F. Nori, Phys. Rep. 520, 43 (2012).
[CrossRef]

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, Phys. Rev. Lett. 109, 013901 (2012).
[CrossRef]

2010 (1)

M. V. Berry and P. Shukla, J. Phys. A 43, 354024 (2010).
[CrossRef]

2009 (2)

A. Aiello, M. Merano, and J. P. Woerdman, Opt. Lett. 34, 1207 (2009).
[CrossRef]

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

2008 (1)

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

2007 (1)

R. Jozsa, Phys. Rev. A 76, 044103 (2007).
[CrossRef]

1989 (2)

I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, Phys. Rev. D 402112 (1989).
[CrossRef]

A. Lakthakia, Optics News 15(6), 14 (1989).
[CrossRef]

1988 (1)

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

1985 (1)

1983 (1)

1971 (1)

1955 (1)

F. I. Fedorov, Doklady Akademii Nauk SSSR 105, 465 (1955).

1948 (1)

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

Aharonov, Y.

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

Aiello, A.

K. Y. Bliokh and A. Aiello, J. Opt. 15, 014001 (2013).
[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, Opt. Lett. 34, 1207 (2009).
[CrossRef]

A. Aiello and J. P. Woerdman, “Theory of angular Goos-Hänchen shift near Brewster incidence,” arxiv:0903.3730v2 (2009).

Albert, D. Z.

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

Artmann, K.

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

Ashbab, S.

A. G. Kofman, S. Ashbab, and F. Nori, Phys. Rep. 520, 43 (2012).
[CrossRef]

Azzam, R. M. A.

Berry, M. V.

M. V. Berry and P. Shukla, J. Phys. A 43, 354024 (2010).
[CrossRef]

Bliokh, K. Y.

K. Y. Bliokh and A. Aiello, J. Opt. 15, 014001 (2013).
[CrossRef]

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, Phys. Rev. Lett. 109, 013901 (2012).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Chan, C. C.

Dennis, M. R.

J. B. Götte and M. R. Dennis, New J. Phys. 14, 073016 (2012).
[CrossRef]

M. R. Dennis and J. B. Götte, New J. Phys. 14, 073013 (2012).
[CrossRef]

M. R. Dennis and J. B. Götte, Phys. Rev. Lett. 109, 183903 (2012).
[CrossRef]

Duck, I. M.

I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, Phys. Rev. D 402112 (1989).
[CrossRef]

Ebbesen, T. W.

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, Phys. Rev. Lett. 109, 013901 (2012).
[CrossRef]

Fedorov, F. I.

F. I. Fedorov, Doklady Akademii Nauk SSSR 105, 465 (1955).

Genet, C.

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, Phys. Rev. Lett. 109, 013901 (2012).
[CrossRef]

Gorodetski, Y.

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, Phys. Rev. Lett. 109, 013901 (2012).
[CrossRef]

Götte, J. B.

M. R. Dennis and J. B. Götte, Phys. Rev. Lett. 109, 183903 (2012).
[CrossRef]

J. B. Götte and M. R. Dennis, New J. Phys. 14, 073016 (2012).
[CrossRef]

M. R. Dennis and J. B. Götte, New J. Phys. 14, 073013 (2012).
[CrossRef]

Hasman, E.

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, Phys. Rev. Lett. 109, 013901 (2012).
[CrossRef]

Horowitz, B. R.

Hosten, O.

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

Jayaswal, G.

Jozsa, R.

R. Jozsa, Phys. Rev. A 76, 044103 (2007).
[CrossRef]

Kleiner, V.

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, Phys. Rev. Lett. 109, 013901 (2012).
[CrossRef]

Kobayashi, H.

H. Kobayashi, G. Puentes, and Y. Shikano, Phys. Rev. A 86, 053805 (2012).
[CrossRef]

Kofman, A. G.

A. G. Kofman, S. Ashbab, and F. Nori, Phys. Rep. 520, 43 (2012).
[CrossRef]

Kwiat, P.

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

Lakthakia, A.

Merano, M.

Mistura, G.

Nori, F.

A. G. Kofman, S. Ashbab, and F. Nori, Phys. Rep. 520, 43 (2012).
[CrossRef]

Puentes, G.

H. Kobayashi, G. Puentes, and Y. Shikano, Phys. Rev. A 86, 053805 (2012).
[CrossRef]

Shikano, Y.

H. Kobayashi, G. Puentes, and Y. Shikano, Phys. Rev. A 86, 053805 (2012).
[CrossRef]

Shitrit, N.

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, Phys. Rev. Lett. 109, 013901 (2012).
[CrossRef]

Shukla, P.

M. V. Berry and P. Shukla, J. Phys. A 43, 354024 (2010).
[CrossRef]

Stein, B.

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, Phys. Rev. Lett. 109, 013901 (2012).
[CrossRef]

Stevenson, P. M.

I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, Phys. Rev. D 402112 (1989).
[CrossRef]

Sudarshan, E. C. G.

I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, Phys. Rev. D 402112 (1989).
[CrossRef]

Tamir, T.

Thonn, T. F.

Vaidman, L.

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

van Exter, M. P.

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

Woerdman, J. P.

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, Opt. Lett. 34, 1207 (2009).
[CrossRef]

A. Aiello and J. P. Woerdman, “Theory of angular Goos-Hänchen shift near Brewster incidence,” arxiv:0903.3730v2 (2009).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Ann. Phys. (1)

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

Appl. Opt. (1)

Doklady Akademii Nauk SSSR (1)

F. I. Fedorov, Doklady Akademii Nauk SSSR 105, 465 (1955).

J. Opt. (1)

K. Y. Bliokh and A. Aiello, J. Opt. 15, 014001 (2013).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. A (1)

M. V. Berry and P. Shukla, J. Phys. A 43, 354024 (2010).
[CrossRef]

Nat. Photonics (1)

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

New J. Phys. (2)

J. B. Götte and M. R. Dennis, New J. Phys. 14, 073016 (2012).
[CrossRef]

M. R. Dennis and J. B. Götte, New J. Phys. 14, 073013 (2012).
[CrossRef]

Opt. Lett. (3)

Optics News (1)

Phys. Rep. (1)

A. G. Kofman, S. Ashbab, and F. Nori, Phys. Rep. 520, 43 (2012).
[CrossRef]

Phys. Rev. A (2)

R. Jozsa, Phys. Rev. A 76, 044103 (2007).
[CrossRef]

H. Kobayashi, G. Puentes, and Y. Shikano, Phys. Rev. A 86, 053805 (2012).
[CrossRef]

Phys. Rev. D (1)

I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, Phys. Rev. D 402112 (1989).
[CrossRef]

Phys. Rev. Lett. (3)

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

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, Phys. Rev. Lett. 109, 013901 (2012).
[CrossRef]

M. R. Dennis and J. B. Götte, Phys. Rev. Lett. 109, 183903 (2012).
[CrossRef]

Science (1)

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

Other (2)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

A. Aiello and J. P. Woerdman, “Theory of angular Goos-Hänchen shift near Brewster incidence,” arxiv:0903.3730v2 (2009).

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Figures (2)

Fig. 1.
Fig. 1.

Angular shifts close to a conventional Brewster angle. The θ-dependent angular shift Δx (thick red curve), with F=(1,0), is compared with the shift of the total intensity [1] (light pink curve) for a reduced refractive index of n=1.5 (e.g., glass–air interface). Both shifts follow a resonance-like curve close to the Brewster angle at θB=56.31°. Black circles indicate Δx obtained by numerical integration of Eq. (3). The moments of the spectrum δ2 and δ4 are obtained for a Gaussian beam with beam waist of 50 μm and a wavelength of 0.526 μm. On the same abscissa, we plot the ratio of reflected to incident intensity (thin gray curve), showing a minimum close to the maximum shift. The figure also shows three contour plots (insets) of the angular spectrum of the light beam at angles (a) θ=56.02°, (b) 56.32°, and (c) 57.87°. Orange crosses indicate the origin from which the shifts are measured. The strength of the contour is shown in the colorbar in the top right corner.

Fig. 2.
Fig. 2.

Large, weak spatial shift close to a null-reflection angle θnr. The shift Dy (thick red curve) is a function of θ for a refractive index of n=3/10, showing the resonance-like curve around θnr41.76°. The light pink curve is the shift for Imbert–Fedorov eigenpolarization, which is smaller than 0.15 μm across the shown range. On the same axis, the ratio of reflected to incident intensity is plotted (thin gray curve). The black circles are the shift for a numerical experiment for a Gaussian beam with a waist of 50 μm and a wavelength of 0.526 μm. The insets show contour plots of the beam’s amplitude at angles (a) θ=41.60°, (b) 41.77°, and (c) 42.86°. Orange crosses indicate the origin for the contour plots.

Equations (7)

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ρ(K)=F*·R(K)·E,
ρ(K)ρ0+δρ1·(c(α)s(α))+12δ2(c(α)s(α))·P2·(c(α)s(α)),
0r|φ(r)|2dr=0K|φ˜(K)|2dK=1.
Δ=1k0dKK02πdαK|ψ˜(K)|20dKK02πdα|ψ˜(K)|20dδ02πdαδ2(cosα,sinα)|ψ˜(K)|20dδ02πdα|ψ˜(K)|2,
δ2Re{ρ0*ρ1+δ48δ2(2P2·ρ1*+ρ1*trP2}|ρ0|2+δ2[ρ12+Re(ρ0*trP2)].
D=0drr02πdϕr|ψ(r)|20drr02πdϕ|ψ(r)|2
1kIm{ρ0*ρ1+δ24(2P2·ρ1*+ρ1*trP2)}|ρ0|2+δ2[ρ12+Re(ρ0*trP2)],

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