Abstract

We propose and demonstrate an interferometric method to measure the Goos–Hänchen (GH) shift, which is based on observing the interference between p- and s-polarized beams. In our method both p- and s-polarized beams are observed simultaneously and across the entire beam profile. To demonstrate our method, we measured the GH shift of aluminum (Al) and glass at different values of the incidence angle ranging from 20° to 70°, with a helium–neon laser as source. We compared the experimental result with theoretical calculations and found a good agreement between them. Our method also enables us to measure the GH shift at any point across the entire beam profile, for arbitrary beam profiles. This is not possible with the methods currently in use. We presented the observed values for the Gaussian mode used, which enables us to find the relative shifts between the p and s components at various points on the incident profile after reflection.

© 2013 Optical Society of America

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  1. F. Goos and H. Hanchen, “Ein neuer and fundamentaler Versuch zur total Reflection,” Ann. Phys. 436, 333–346 (1947).
  2. K. Artmann, “Calculation of the lateral shift of totally reflected beams,” Ann. Phys. 437, 87–102 (1948).
  3. R. H. Renard, “Total reflection: a new evaluation of the Goos–Hänchen shift,” J. Opt. Soc. Am. 54, 1190–1197 (1964).
    [CrossRef]
  4. C. Bonnet, D. Chauvat, O. Emile, F. Bretenaker, A. L. Floch, and L. Dutriaux, “Measurement of positive and negative Goos–Hänchen effects for metallic gratings near wood anomalies,” Opt. Lett. 26, 666–668 (2001).
    [CrossRef]
  5. C. F. Li, “Negative lateral shift of a light beam transmitted through a dielectric slab and interaction of boundary effects,” Phys. Rev. Lett. 91, 133903 (2003).
    [CrossRef]
  6. H. M. Lai and S. W. Chan, “Large and negative Goos–Hänchen shift near the Brewster dip on reflection from weakly absorbing media,” Opt. Lett. 27, 680–682 (2002).
    [CrossRef]
  7. W. J. Wild and C. L. Giles, “Goos–Hänchen shifts for absorbing media,” Phys. Rev. A 25, 2099–2101 (1982).
    [CrossRef]
  8. P. T. Leung, C. W. Chen, and H. P. Chiang, “Large negative Goos–Hanchen shift at metal surfaces,” Opt. Commun. 276, 206–208 (2007).
    [CrossRef]
  9. Y. Wang, Y. Liu, J. Xu, H. Zhang, L. Bai, Y. Xiao, J. Yan, and X. Zhang, “Numerical study of lateral displacements of Gaussian beams reflected from weakly absorbing media near Brewster dip and reflected from strongly absorbing media,” J. Opt. A 11, 105701 (2009).
    [CrossRef]
  10. M. Merano, A. Aiello, G. W. ’t Hooft, M. P. van Exter, E. R. Eliel, and J. P. Woerdman, “Observation of Goos–Hanchen shifts in metallic reflection,” Opt. Express 15, 15928–15934 (2007).
    [CrossRef]
  11. J. B. Gotte, A. Aiello, and J. P. Woerdman, “Loss-induced transition of the Goos–Hänchen effect for metals and dielectrics,” Opt. Express 16, 3961–3969 (2008).
    [CrossRef]
  12. H. M. Lai, S. W. Chan, and W. H. Wong, “Nonspecular effects on reflection from absorbing media at and around the Brewster dip,” J. Opt. Soc. Am. A 23, 3208–3216 (2006).
    [CrossRef]
  13. J. He, J. Yi, and S. He, “Giant negative Goos–Hänchen shifts for a photonic crystal with a negative effective index,” Opt. Express 14, 3024–3029 (2006).
    [CrossRef]
  14. L. G. Wang and S. Y. Zhu, “Large negative lateral shifts from the Kretschmann–Raether configuration with left-handed materials,” Appl. Phys. Lett. 87, 221102 (2005).
    [CrossRef]
  15. M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, “Observing angular deviations in the specular reflection of light beam,” Nat. Photonics 3, 337–340 (2009).
    [CrossRef]
  16. H. G. L. Schwefel, W. Köhler, Z. H. Lu, J. Fan, and L. J. Wang, “Direct experimental observation of the single reflection optical Goos–Hänchen shift,” Opt. Lett. 33, 794–796 (2008).
    [CrossRef]
  17. D. J. Rhodes and C. K. Carniglia, “Measurement of Goos–Hänchen shift at grazing incidence using Lloyd’s mirror,” J. Opt. Soc. Am. 67, 679–683 (1977).
    [CrossRef]
  18. Chun-Fang-Li, “Unified theory for Goos–Hänchen and Imbert–Fedorov effects,” Phys. Rev. A 76, 013811 (2007).
    [CrossRef]
  19. C. Prajapati and D. Ranganathan, “Goos–Hänchen and Imbert–Fedorov shifts for Hermite–Gauss beams,” J. Opt. Soc. Am. A 29, 1377–1382 (2012).
    [CrossRef]
  20. N. Hermosa, M. Merano, A. Aiello, and J. P. Woerdman, “Orbital angular momentum induced beam shifts,” Proc. SPIE 7950, 79500F (2011).
    [CrossRef]
  21. M. Merano, N. Hermosa, and J. P. Woerdman, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82, 023817 (2010).
    [CrossRef]
  22. A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos–Hanchen and Imbert–Fedorov shifts,” Opt. Lett. 33, 1437–1439 (2008).
    [CrossRef]
  23. The refractive index of aluminium is taken from the website http://refractiveindex.info at 633 nm wavelength .

2012 (1)

2011 (1)

N. Hermosa, M. Merano, A. Aiello, and J. P. Woerdman, “Orbital angular momentum induced beam shifts,” Proc. SPIE 7950, 79500F (2011).
[CrossRef]

2010 (1)

M. Merano, N. Hermosa, and J. P. Woerdman, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82, 023817 (2010).
[CrossRef]

2009 (2)

Y. Wang, Y. Liu, J. Xu, H. Zhang, L. Bai, Y. Xiao, J. Yan, and X. Zhang, “Numerical study of lateral displacements of Gaussian beams reflected from weakly absorbing media near Brewster dip and reflected from strongly absorbing media,” J. Opt. A 11, 105701 (2009).
[CrossRef]

M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, “Observing angular deviations in the specular reflection of light beam,” Nat. Photonics 3, 337–340 (2009).
[CrossRef]

2008 (3)

2007 (3)

M. Merano, A. Aiello, G. W. ’t Hooft, M. P. van Exter, E. R. Eliel, and J. P. Woerdman, “Observation of Goos–Hanchen shifts in metallic reflection,” Opt. Express 15, 15928–15934 (2007).
[CrossRef]

Chun-Fang-Li, “Unified theory for Goos–Hänchen and Imbert–Fedorov effects,” Phys. Rev. A 76, 013811 (2007).
[CrossRef]

P. T. Leung, C. W. Chen, and H. P. Chiang, “Large negative Goos–Hanchen shift at metal surfaces,” Opt. Commun. 276, 206–208 (2007).
[CrossRef]

2006 (2)

2005 (1)

L. G. Wang and S. Y. Zhu, “Large negative lateral shifts from the Kretschmann–Raether configuration with left-handed materials,” Appl. Phys. Lett. 87, 221102 (2005).
[CrossRef]

2003 (1)

C. F. Li, “Negative lateral shift of a light beam transmitted through a dielectric slab and interaction of boundary effects,” Phys. Rev. Lett. 91, 133903 (2003).
[CrossRef]

2002 (1)

2001 (1)

1982 (1)

W. J. Wild and C. L. Giles, “Goos–Hänchen shifts for absorbing media,” Phys. Rev. A 25, 2099–2101 (1982).
[CrossRef]

1977 (1)

1964 (1)

1948 (1)

K. Artmann, “Calculation of the lateral shift of totally reflected beams,” Ann. Phys. 437, 87–102 (1948).

1947 (1)

F. Goos and H. Hanchen, “Ein neuer and fundamentaler Versuch zur total Reflection,” Ann. Phys. 436, 333–346 (1947).

’t Hooft, G. W.

Aiello, A.

Artmann, K.

K. Artmann, “Calculation of the lateral shift of totally reflected beams,” Ann. Phys. 437, 87–102 (1948).

Bai, L.

Y. Wang, Y. Liu, J. Xu, H. Zhang, L. Bai, Y. Xiao, J. Yan, and X. Zhang, “Numerical study of lateral displacements of Gaussian beams reflected from weakly absorbing media near Brewster dip and reflected from strongly absorbing media,” J. Opt. A 11, 105701 (2009).
[CrossRef]

Bonnet, C.

Bretenaker, F.

Carniglia, C. K.

Chan, S. W.

Chauvat, D.

Chen, C. W.

P. T. Leung, C. W. Chen, and H. P. Chiang, “Large negative Goos–Hanchen shift at metal surfaces,” Opt. Commun. 276, 206–208 (2007).
[CrossRef]

Chiang, H. P.

P. T. Leung, C. W. Chen, and H. P. Chiang, “Large negative Goos–Hanchen shift at metal surfaces,” Opt. Commun. 276, 206–208 (2007).
[CrossRef]

Chun-Fang-Li,

Chun-Fang-Li, “Unified theory for Goos–Hänchen and Imbert–Fedorov effects,” Phys. Rev. A 76, 013811 (2007).
[CrossRef]

Dutriaux, L.

Eliel, E. R.

Emile, O.

Fan, J.

Floch, A. L.

Giles, C. L.

W. J. Wild and C. L. Giles, “Goos–Hänchen shifts for absorbing media,” Phys. Rev. A 25, 2099–2101 (1982).
[CrossRef]

Goos, F.

F. Goos and H. Hanchen, “Ein neuer and fundamentaler Versuch zur total Reflection,” Ann. Phys. 436, 333–346 (1947).

Gotte, J. B.

Hanchen, H.

F. Goos and H. Hanchen, “Ein neuer and fundamentaler Versuch zur total Reflection,” Ann. Phys. 436, 333–346 (1947).

He, J.

He, S.

Hermosa, N.

N. Hermosa, M. Merano, A. Aiello, and J. P. Woerdman, “Orbital angular momentum induced beam shifts,” Proc. SPIE 7950, 79500F (2011).
[CrossRef]

M. Merano, N. Hermosa, and J. P. Woerdman, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82, 023817 (2010).
[CrossRef]

Köhler, W.

Lai, H. M.

Leung, P. T.

P. T. Leung, C. W. Chen, and H. P. Chiang, “Large negative Goos–Hanchen shift at metal surfaces,” Opt. Commun. 276, 206–208 (2007).
[CrossRef]

Li, C. F.

C. F. Li, “Negative lateral shift of a light beam transmitted through a dielectric slab and interaction of boundary effects,” Phys. Rev. Lett. 91, 133903 (2003).
[CrossRef]

Liu, Y.

Y. Wang, Y. Liu, J. Xu, H. Zhang, L. Bai, Y. Xiao, J. Yan, and X. Zhang, “Numerical study of lateral displacements of Gaussian beams reflected from weakly absorbing media near Brewster dip and reflected from strongly absorbing media,” J. Opt. A 11, 105701 (2009).
[CrossRef]

Lu, Z. H.

Merano, M.

N. Hermosa, M. Merano, A. Aiello, and J. P. Woerdman, “Orbital angular momentum induced beam shifts,” Proc. SPIE 7950, 79500F (2011).
[CrossRef]

M. Merano, N. Hermosa, and J. P. Woerdman, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82, 023817 (2010).
[CrossRef]

M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, “Observing angular deviations in the specular reflection of light beam,” Nat. Photonics 3, 337–340 (2009).
[CrossRef]

M. Merano, A. Aiello, G. W. ’t Hooft, M. P. van Exter, E. R. Eliel, and J. P. Woerdman, “Observation of Goos–Hanchen shifts in metallic reflection,” Opt. Express 15, 15928–15934 (2007).
[CrossRef]

Prajapati, C.

Ranganathan, D.

Renard, R. H.

Rhodes, D. J.

Schwefel, H. G. L.

van Exter, M. P.

M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, “Observing angular deviations in the specular reflection of light beam,” Nat. Photonics 3, 337–340 (2009).
[CrossRef]

M. Merano, A. Aiello, G. W. ’t Hooft, M. P. van Exter, E. R. Eliel, and J. P. Woerdman, “Observation of Goos–Hanchen shifts in metallic reflection,” Opt. Express 15, 15928–15934 (2007).
[CrossRef]

Wang, L. G.

L. G. Wang and S. Y. Zhu, “Large negative lateral shifts from the Kretschmann–Raether configuration with left-handed materials,” Appl. Phys. Lett. 87, 221102 (2005).
[CrossRef]

Wang, L. J.

Wang, Y.

Y. Wang, Y. Liu, J. Xu, H. Zhang, L. Bai, Y. Xiao, J. Yan, and X. Zhang, “Numerical study of lateral displacements of Gaussian beams reflected from weakly absorbing media near Brewster dip and reflected from strongly absorbing media,” J. Opt. A 11, 105701 (2009).
[CrossRef]

Wild, W. J.

W. J. Wild and C. L. Giles, “Goos–Hänchen shifts for absorbing media,” Phys. Rev. A 25, 2099–2101 (1982).
[CrossRef]

Woerdman, J. P.

N. Hermosa, M. Merano, A. Aiello, and J. P. Woerdman, “Orbital angular momentum induced beam shifts,” Proc. SPIE 7950, 79500F (2011).
[CrossRef]

M. Merano, N. Hermosa, and J. P. Woerdman, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82, 023817 (2010).
[CrossRef]

M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, “Observing angular deviations in the specular reflection of light beam,” Nat. Photonics 3, 337–340 (2009).
[CrossRef]

J. B. Gotte, A. Aiello, and J. P. Woerdman, “Loss-induced transition of the Goos–Hänchen effect for metals and dielectrics,” Opt. Express 16, 3961–3969 (2008).
[CrossRef]

A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos–Hanchen and Imbert–Fedorov shifts,” Opt. Lett. 33, 1437–1439 (2008).
[CrossRef]

M. Merano, A. Aiello, G. W. ’t Hooft, M. P. van Exter, E. R. Eliel, and J. P. Woerdman, “Observation of Goos–Hanchen shifts in metallic reflection,” Opt. Express 15, 15928–15934 (2007).
[CrossRef]

Wong, W. H.

Xiao, Y.

Y. Wang, Y. Liu, J. Xu, H. Zhang, L. Bai, Y. Xiao, J. Yan, and X. Zhang, “Numerical study of lateral displacements of Gaussian beams reflected from weakly absorbing media near Brewster dip and reflected from strongly absorbing media,” J. Opt. A 11, 105701 (2009).
[CrossRef]

Xu, J.

Y. Wang, Y. Liu, J. Xu, H. Zhang, L. Bai, Y. Xiao, J. Yan, and X. Zhang, “Numerical study of lateral displacements of Gaussian beams reflected from weakly absorbing media near Brewster dip and reflected from strongly absorbing media,” J. Opt. A 11, 105701 (2009).
[CrossRef]

Yan, J.

Y. Wang, Y. Liu, J. Xu, H. Zhang, L. Bai, Y. Xiao, J. Yan, and X. Zhang, “Numerical study of lateral displacements of Gaussian beams reflected from weakly absorbing media near Brewster dip and reflected from strongly absorbing media,” J. Opt. A 11, 105701 (2009).
[CrossRef]

Yi, J.

Zhang, H.

Y. Wang, Y. Liu, J. Xu, H. Zhang, L. Bai, Y. Xiao, J. Yan, and X. Zhang, “Numerical study of lateral displacements of Gaussian beams reflected from weakly absorbing media near Brewster dip and reflected from strongly absorbing media,” J. Opt. A 11, 105701 (2009).
[CrossRef]

Zhang, X.

Y. Wang, Y. Liu, J. Xu, H. Zhang, L. Bai, Y. Xiao, J. Yan, and X. Zhang, “Numerical study of lateral displacements of Gaussian beams reflected from weakly absorbing media near Brewster dip and reflected from strongly absorbing media,” J. Opt. A 11, 105701 (2009).
[CrossRef]

Zhu, S. Y.

L. G. Wang and S. Y. Zhu, “Large negative lateral shifts from the Kretschmann–Raether configuration with left-handed materials,” Appl. Phys. Lett. 87, 221102 (2005).
[CrossRef]

Ann. Phys. (2)

F. Goos and H. Hanchen, “Ein neuer and fundamentaler Versuch zur total Reflection,” Ann. Phys. 436, 333–346 (1947).

K. Artmann, “Calculation of the lateral shift of totally reflected beams,” Ann. Phys. 437, 87–102 (1948).

Appl. Phys. Lett. (1)

L. G. Wang and S. Y. Zhu, “Large negative lateral shifts from the Kretschmann–Raether configuration with left-handed materials,” Appl. Phys. Lett. 87, 221102 (2005).
[CrossRef]

J. Opt. A (1)

Y. Wang, Y. Liu, J. Xu, H. Zhang, L. Bai, Y. Xiao, J. Yan, and X. Zhang, “Numerical study of lateral displacements of Gaussian beams reflected from weakly absorbing media near Brewster dip and reflected from strongly absorbing media,” J. Opt. A 11, 105701 (2009).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Nat. Photonics (1)

M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, “Observing angular deviations in the specular reflection of light beam,” Nat. Photonics 3, 337–340 (2009).
[CrossRef]

Opt. Commun. (1)

P. T. Leung, C. W. Chen, and H. P. Chiang, “Large negative Goos–Hanchen shift at metal surfaces,” Opt. Commun. 276, 206–208 (2007).
[CrossRef]

Opt. Express (3)

Opt. Lett. (4)

Phys. Rev. A (3)

M. Merano, N. Hermosa, and J. P. Woerdman, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82, 023817 (2010).
[CrossRef]

Chun-Fang-Li, “Unified theory for Goos–Hänchen and Imbert–Fedorov effects,” Phys. Rev. A 76, 013811 (2007).
[CrossRef]

W. J. Wild and C. L. Giles, “Goos–Hänchen shifts for absorbing media,” Phys. Rev. A 25, 2099–2101 (1982).
[CrossRef]

Phys. Rev. Lett. (1)

C. F. Li, “Negative lateral shift of a light beam transmitted through a dielectric slab and interaction of boundary effects,” Phys. Rev. Lett. 91, 133903 (2003).
[CrossRef]

Proc. SPIE (1)

N. Hermosa, M. Merano, A. Aiello, and J. P. Woerdman, “Orbital angular momentum induced beam shifts,” Proc. SPIE 7950, 79500F (2011).
[CrossRef]

Other (1)

The refractive index of aluminium is taken from the website http://refractiveindex.info at 633 nm wavelength .

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Figures (11)

Fig. 1.
Fig. 1.

Schematic diagram of experimental setup.

Fig. 2.
Fig. 2.

Geometry of reflection of p- and s-polarized beams from the absorbing media, which gives the relative GH shift experimentally. For reflection from absorbing media, the GH shift of one polarized component is negative and the other polarized component is positive, as can be seen from the theoretical graph Fig. 7.

Fig. 3.
Fig. 3.

(a) Fringe pattern observed with both p- and s-polarized beams. (b) Shows the p-polarized beam blocked and (c) shows the s-polarized beam blocked.

Fig. 4.
Fig. 4.

Fringe pattern observed when both p-and s-polarized beams interfere at various angles of incidence approaching the Brewster angle, (a) 22°, (b) 36°, and (c) 48°.

Fig. 5.
Fig. 5.

Actual experimental phase shift between the p-and s-polarized beams with angle of incidence ranging from 20° to 70°.

Fig. 6.
Fig. 6.

GH shift as obtained from the phase shifts of Fig. 5 together with theoretical curve.

Fig. 7.
Fig. 7.

Theoretical result of variation of GH shift as a function of angle of incidence varying from 20° to 90°. Dashed line is the GH shift of s-polarized beam, dotted line is the GH shift of the p-polarized beam, and solid line is difference between p- and s-polarized beams and is the theoretical curve plotted in Fig. 6.

Fig. 8.
Fig. 8.

Actual experimental phase shift between the p- and s-polarized beams with angle of incidence ranging from 20° to 70°.

Fig. 9.
Fig. 9.

GH shift as obtained from the phase shifts of Fig. 8 together with theoretical curve.

Fig. 10.
Fig. 10.

Theoretical result of variation of GH shift as a function of angle of incidence varying from 20° to 90°. Dashed line is the GH shift of the s-polarized beam, dotted line is the GH shift of the p-polarized beam, and the solid line is difference between p- and s-polarized beams and is the theoretical curve plotted in Fig. 9.

Fig. 11.
Fig. 11.

Variation of difference between GH shift of p- and s-polarized beam along a transverse section of the Gaussian beam at 30° incidence angle, the horizontal axis is labeled by the observed fringe number.

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

E⃗(x,y,z)=12πA⃗(ky,kz)ei(kxx+kyy+kzzωt)dkydkz,
Dp=1|Rs|2+|Rp|2(|Rp|2ϕpky)A2dkydkz.
Ds=1|Rs|2+|Rp|2(|Rs|2ϕsky)A2dkydkz.
A(ky,kz)=ωyωzπexp(ωy22(kyky0)2)exp(ωz2kz22).
Rp=n^22cosθn^22sin2θn^22cosθ+n^22sin2θ,Rs=cosθn^22sin2θcosθ+n^22sin2θ,
ϕp=arctan[2κ2cosθκ2+sin2θκ4cos2θκ2sin2θ],ϕs=arctan[2cosθκ2+sin2θκ2+sin2θcos2θ].
ϕp=arctan[2ncosθn2sin2θ1cos2θn2(n2sin2θ1)],ϕs=arctan[2ncosθn2sin2θ1n2cos2θ(n2sin2θ1)].
DpDs=λ2π(ϕpϕs)sinθ,

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