Abstract

The Goos–Haenchen shift of a totally reflected beam at the planar interface of two dielectric media, as if the incident beam is reflected from beneath the interface between the incident and transmitted media, has been geometrically associated with the penetration of the incident photons in the less-dense forbidden transmission region. This geometrical approach is here generalized to analytically calculate the Goos–Haenchen shift in one- and two-dimensional periodic structures. Several numerical examples are presented, and the obtained results are successfully tested against the well-known Artman’s formula. The proposed approach is shown to be a fast, simple, and efficient method that can provide good physical insight to the nature of the phenomenon.

© 2008 Optical Society of America

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  1. F. Goos and H. Hänchen, Ann. Phys. (Paris) 1, 333 (1947).
  2. K. Artmann, Ann. Phys. (Paris) 2, 87 (1948).
  3. R. H. Renard, J. Opt. Soc. Am. 54, 1190 (1964).
    [CrossRef]
  4. S. R. Seshadri, J. Opt. Soc. Am. A 5, 583 (1988).
    [CrossRef]
  5. J. P. Hugonin and R. Petit, J. Opt. (Paris) 8, 73 (1977).
  6. J. J. Cowan and B. Anicin, J. Opt. Soc. Am. 67, 1307 (1977).
    [CrossRef]
  7. E. Pfleghaar, A. Marseille, and A. Weis, Phys. Rev. Lett. 70, 2281 (1993).
    [CrossRef] [PubMed]
  8. B. M. Jost, A.-A. R. Al-Rashed, and B. E. A. Saleh, Phys. Rev. Lett. 81, 2233 (1998).
    [CrossRef]
  9. X. Hu, Y. Huang, W. Zhang, D. K. Qing, and J. Peng, Opt. Lett. 30, 899 (2005).
    [CrossRef] [PubMed]
  10. A. Lakhtakia, Electromagnetics 23, 71 (2003).
    [CrossRef]
  11. I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, Appl. Phys. Lett. 83, 2713 (2003).
    [CrossRef]
  12. H. M. Lai and S. W. Chan, Opt. Lett. 27, 680 (2002).
    [CrossRef]
  13. P. T. Leung, C. W. Chen, and H. P. Chiang, Opt. Commun. 276, 206 (2007).
    [CrossRef]
  14. O. Emile, T. Galstyan, A. Le Floch, and F. Bretenaker, Phys. Rev. Lett. 75, 1511 (1995).
    [CrossRef] [PubMed]
  15. D. Felbacq, A. Moreau, and R. Smaâli, Opt. Lett. 28, 1633 (2003).
    [CrossRef] [PubMed]
  16. D. Felbacq and R. Smaâli, Phys. Rev. Lett. 92, 193902 (2004).
    [CrossRef] [PubMed]
  17. J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 1999).
  18. A. Rung and C. G. Ribbing, Phys. Rev. Lett. 92, 123901 (2004).
    [CrossRef] [PubMed]
  19. A. Khavasi, A. K. Jahromi, and K. Mehrany, J. Opt. Soc. Am. A 25, 1564 (2008).
    [CrossRef]

2008

2007

P. T. Leung, C. W. Chen, and H. P. Chiang, Opt. Commun. 276, 206 (2007).
[CrossRef]

2005

2004

D. Felbacq and R. Smaâli, Phys. Rev. Lett. 92, 193902 (2004).
[CrossRef] [PubMed]

A. Rung and C. G. Ribbing, Phys. Rev. Lett. 92, 123901 (2004).
[CrossRef] [PubMed]

2003

D. Felbacq, A. Moreau, and R. Smaâli, Opt. Lett. 28, 1633 (2003).
[CrossRef] [PubMed]

A. Lakhtakia, Electromagnetics 23, 71 (2003).
[CrossRef]

I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, Appl. Phys. Lett. 83, 2713 (2003).
[CrossRef]

2002

1998

B. M. Jost, A.-A. R. Al-Rashed, and B. E. A. Saleh, Phys. Rev. Lett. 81, 2233 (1998).
[CrossRef]

1995

O. Emile, T. Galstyan, A. Le Floch, and F. Bretenaker, Phys. Rev. Lett. 75, 1511 (1995).
[CrossRef] [PubMed]

1993

E. Pfleghaar, A. Marseille, and A. Weis, Phys. Rev. Lett. 70, 2281 (1993).
[CrossRef] [PubMed]

1988

1977

J. P. Hugonin and R. Petit, J. Opt. (Paris) 8, 73 (1977).

J. J. Cowan and B. Anicin, J. Opt. Soc. Am. 67, 1307 (1977).
[CrossRef]

1964

1948

K. Artmann, Ann. Phys. (Paris) 2, 87 (1948).

1947

F. Goos and H. Hänchen, Ann. Phys. (Paris) 1, 333 (1947).

Al-Rashed, A.-A. R.

B. M. Jost, A.-A. R. Al-Rashed, and B. E. A. Saleh, Phys. Rev. Lett. 81, 2233 (1998).
[CrossRef]

Anicin, B.

Artmann, K.

K. Artmann, Ann. Phys. (Paris) 2, 87 (1948).

Bretenaker, F.

O. Emile, T. Galstyan, A. Le Floch, and F. Bretenaker, Phys. Rev. Lett. 75, 1511 (1995).
[CrossRef] [PubMed]

Chan, S. W.

Chen, C. W.

P. T. Leung, C. W. Chen, and H. P. Chiang, Opt. Commun. 276, 206 (2007).
[CrossRef]

Chiang, H. P.

P. T. Leung, C. W. Chen, and H. P. Chiang, Opt. Commun. 276, 206 (2007).
[CrossRef]

Cowan, J. J.

Emile, O.

O. Emile, T. Galstyan, A. Le Floch, and F. Bretenaker, Phys. Rev. Lett. 75, 1511 (1995).
[CrossRef] [PubMed]

Felbacq, D.

Galstyan, T.

O. Emile, T. Galstyan, A. Le Floch, and F. Bretenaker, Phys. Rev. Lett. 75, 1511 (1995).
[CrossRef] [PubMed]

Goos, F.

F. Goos and H. Hänchen, Ann. Phys. (Paris) 1, 333 (1947).

Hänchen, H.

F. Goos and H. Hänchen, Ann. Phys. (Paris) 1, 333 (1947).

Hu, X.

Huang, Y.

Hugonin, J. P.

J. P. Hugonin and R. Petit, J. Opt. (Paris) 8, 73 (1977).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 1999).

Jahromi, A. K.

Jost, B. M.

B. M. Jost, A.-A. R. Al-Rashed, and B. E. A. Saleh, Phys. Rev. Lett. 81, 2233 (1998).
[CrossRef]

Khavasi, A.

Kivshar, Y. S.

I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, Appl. Phys. Lett. 83, 2713 (2003).
[CrossRef]

Lai, H. M.

Lakhtakia, A.

A. Lakhtakia, Electromagnetics 23, 71 (2003).
[CrossRef]

Le Floch, A.

O. Emile, T. Galstyan, A. Le Floch, and F. Bretenaker, Phys. Rev. Lett. 75, 1511 (1995).
[CrossRef] [PubMed]

Leung, P. T.

P. T. Leung, C. W. Chen, and H. P. Chiang, Opt. Commun. 276, 206 (2007).
[CrossRef]

Marseille, A.

E. Pfleghaar, A. Marseille, and A. Weis, Phys. Rev. Lett. 70, 2281 (1993).
[CrossRef] [PubMed]

Mehrany, K.

Moreau, A.

Peng, J.

Petit, R.

J. P. Hugonin and R. Petit, J. Opt. (Paris) 8, 73 (1977).

Pfleghaar, E.

E. Pfleghaar, A. Marseille, and A. Weis, Phys. Rev. Lett. 70, 2281 (1993).
[CrossRef] [PubMed]

Qing, D. K.

Renard, R. H.

Ribbing, C. G.

A. Rung and C. G. Ribbing, Phys. Rev. Lett. 92, 123901 (2004).
[CrossRef] [PubMed]

Rung, A.

A. Rung and C. G. Ribbing, Phys. Rev. Lett. 92, 123901 (2004).
[CrossRef] [PubMed]

Saleh, B. E. A.

B. M. Jost, A.-A. R. Al-Rashed, and B. E. A. Saleh, Phys. Rev. Lett. 81, 2233 (1998).
[CrossRef]

Seshadri, S. R.

Shadrivov, I. V.

I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, Appl. Phys. Lett. 83, 2713 (2003).
[CrossRef]

Smaâli, R.

Weis, A.

E. Pfleghaar, A. Marseille, and A. Weis, Phys. Rev. Lett. 70, 2281 (1993).
[CrossRef] [PubMed]

Zhang, W.

Zharov, A. A.

I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, Appl. Phys. Lett. 83, 2713 (2003).
[CrossRef]

Ann. Phys. (Paris)

F. Goos and H. Hänchen, Ann. Phys. (Paris) 1, 333 (1947).

K. Artmann, Ann. Phys. (Paris) 2, 87 (1948).

Appl. Phys. Lett.

I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, Appl. Phys. Lett. 83, 2713 (2003).
[CrossRef]

Electromagnetics

A. Lakhtakia, Electromagnetics 23, 71 (2003).
[CrossRef]

J. Opt. (Paris)

J. P. Hugonin and R. Petit, J. Opt. (Paris) 8, 73 (1977).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

P. T. Leung, C. W. Chen, and H. P. Chiang, Opt. Commun. 276, 206 (2007).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

D. Felbacq and R. Smaâli, Phys. Rev. Lett. 92, 193902 (2004).
[CrossRef] [PubMed]

O. Emile, T. Galstyan, A. Le Floch, and F. Bretenaker, Phys. Rev. Lett. 75, 1511 (1995).
[CrossRef] [PubMed]

E. Pfleghaar, A. Marseille, and A. Weis, Phys. Rev. Lett. 70, 2281 (1993).
[CrossRef] [PubMed]

B. M. Jost, A.-A. R. Al-Rashed, and B. E. A. Saleh, Phys. Rev. Lett. 81, 2233 (1998).
[CrossRef]

A. Rung and C. G. Ribbing, Phys. Rev. Lett. 92, 123901 (2004).
[CrossRef] [PubMed]

Other

J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 1999).

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Figures (4)

Fig. 1
Fig. 1

Geometrical interpretation of the GHS.

Fig. 2
Fig. 2

The GHS at the interface of the considered 1DPC: the Artman’s formula (solid curve), the proposed geometrical approximation by using the EPSD (dashed curve), the proposed geometrical approximation by using the PSD (dotted curve).

Fig. 3
Fig. 3

The GHS at the interface of a 2DPC whose basic cell is shown in the inset: the Artman’s formula (solid curve), the proposed geometrical approximation by using the one-dimensional EPSD based on the refractive index averaging (dashed curve), and the proposed geometrical approximation by using the two-dimensional EPSD based on the field averaging (dotted curve).

Fig. 4
Fig. 4

FDTD simulation of a Gaussian plane wave incident upon the interface of a typical sqaure 2DPC.

Equations (6)

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Δ = x x Ψ r ( x , 0 ) 2 d x x Ψ r ( x , 0 ) 2 d x .
Δ = 2 d tan ( θ ) ,
d exp = y y ψ t ( x , y ) 2 d y y ψ t ( x , y ) 2 d y ,
Δ = 2 d exp tan ( θ ) .
n ( x , y ) = n ( y ) = { 1.0 p Λ < y ( 4 p + 1 ) Λ 4 2.5 ( 4 p + 1 ) Λ 4 < y ( p + 1 ) Λ } ; p Z + .
d eff = y y x Ψ t d x 2 d y y x Ψ t d x 2 d y .

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