Abstract

We show that, under certain conditions, a negative Goos–Hänchen shift—a longitudinal displacement of a totally internally reflected wave packet—occurs in periodic media such as waveguide arrays.

© 2011 Optical Society of America

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References

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  1. F. Goos and H. Lindberg-Hänchen, Ann. Phys. 440, 251 (1949).
    [CrossRef]
  2. M. Merano, A. Aiello, G. ’t Hooft, M. V. Exter, E. Eliel, and J. Woerdman, Opt. Express 15, 15928 (2007).
    [CrossRef] [PubMed]
  3. I. Shadrivov, A. Zharov, and Y. Kivshar, Appl. Phys. Lett. 83, 2713 (2003).
    [CrossRef]
  4. D. Qing and G. Chen, Opt. Lett. 29, 872 (2004).
    [CrossRef] [PubMed]
  5. C. Luo, S. Johnson, J. Joannopoulos, and J. Pendry, Phys. Rev. B 65, 201104 (2002).
    [CrossRef]
  6. Y. Urzhumov and D. Smith, Phys. Rev. Lett. 105, 163901(2010).
    [CrossRef]
  7. T. Ergin, N. Stenger, P. Brenner, J. Pendry, and M. Wegener, Science 328, 337 (2010).
    [CrossRef] [PubMed]
  8. J. He, J. Yi, and S. He, Opt. Express 14, 3024 (2006).
    [CrossRef] [PubMed]
  9. D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
    [CrossRef] [PubMed]
  10. A. Szameit and S. Nolte, J. Phys. B 43, 163001 (2010).
    [CrossRef]
  11. O. Emile, T. Galstyan, A. L. Floch, and F. Bretenaker, Phys. Rev. Lett. 75, 1511 (1995).
    [CrossRef] [PubMed]
  12. A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, and A. Tünnermann, New J. Phys. 10, 103020 (2008).
    [CrossRef]
  13. A. Snyder and J. Love, Optical Waveguide Theory (Kluwer, 1983).
  14. A. Snyder and J. Love, Appl. Opt. 15, 236 (1976).
    [CrossRef] [PubMed]

2010 (3)

Y. Urzhumov and D. Smith, Phys. Rev. Lett. 105, 163901(2010).
[CrossRef]

T. Ergin, N. Stenger, P. Brenner, J. Pendry, and M. Wegener, Science 328, 337 (2010).
[CrossRef] [PubMed]

A. Szameit and S. Nolte, J. Phys. B 43, 163001 (2010).
[CrossRef]

2008 (1)

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, and A. Tünnermann, New J. Phys. 10, 103020 (2008).
[CrossRef]

2007 (1)

2006 (1)

2004 (1)

2003 (2)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

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

2002 (1)

C. Luo, S. Johnson, J. Joannopoulos, and J. Pendry, Phys. Rev. B 65, 201104 (2002).
[CrossRef]

1995 (1)

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

1976 (1)

1949 (1)

F. Goos and H. Lindberg-Hänchen, Ann. Phys. 440, 251 (1949).
[CrossRef]

’t Hooft, G.

Aiello, A.

Brenner, P.

T. Ergin, N. Stenger, P. Brenner, J. Pendry, and M. Wegener, Science 328, 337 (2010).
[CrossRef] [PubMed]

Bretenaker, F.

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

Chen, G.

Christodoulides, D. N.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

Dreisow, F.

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, and A. Tünnermann, New J. Phys. 10, 103020 (2008).
[CrossRef]

Eliel, E.

Emile, O.

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

Ergin, T.

T. Ergin, N. Stenger, P. Brenner, J. Pendry, and M. Wegener, Science 328, 337 (2010).
[CrossRef] [PubMed]

Exter, M. V.

Floch, A. L.

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

Galstyan, T.

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

Goos, F.

F. Goos and H. Lindberg-Hänchen, Ann. Phys. 440, 251 (1949).
[CrossRef]

He, J.

He, S.

Heinrich, M.

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, and A. Tünnermann, New J. Phys. 10, 103020 (2008).
[CrossRef]

Joannopoulos, J.

C. Luo, S. Johnson, J. Joannopoulos, and J. Pendry, Phys. Rev. B 65, 201104 (2002).
[CrossRef]

Johnson, S.

C. Luo, S. Johnson, J. Joannopoulos, and J. Pendry, Phys. Rev. B 65, 201104 (2002).
[CrossRef]

Kivshar, Y.

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

Lederer, F.

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, and A. Tünnermann, New J. Phys. 10, 103020 (2008).
[CrossRef]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

Lindberg-Hänchen, H.

F. Goos and H. Lindberg-Hänchen, Ann. Phys. 440, 251 (1949).
[CrossRef]

Love, J.

A. Snyder and J. Love, Appl. Opt. 15, 236 (1976).
[CrossRef] [PubMed]

A. Snyder and J. Love, Optical Waveguide Theory (Kluwer, 1983).

Luo, C.

C. Luo, S. Johnson, J. Joannopoulos, and J. Pendry, Phys. Rev. B 65, 201104 (2002).
[CrossRef]

Merano, M.

Nolte, S.

A. Szameit and S. Nolte, J. Phys. B 43, 163001 (2010).
[CrossRef]

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, and A. Tünnermann, New J. Phys. 10, 103020 (2008).
[CrossRef]

Pendry, J.

T. Ergin, N. Stenger, P. Brenner, J. Pendry, and M. Wegener, Science 328, 337 (2010).
[CrossRef] [PubMed]

C. Luo, S. Johnson, J. Joannopoulos, and J. Pendry, Phys. Rev. B 65, 201104 (2002).
[CrossRef]

Pertsch, T.

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, and A. Tünnermann, New J. Phys. 10, 103020 (2008).
[CrossRef]

Peschel, U.

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, and A. Tünnermann, New J. Phys. 10, 103020 (2008).
[CrossRef]

Qing, D.

Shadrivov, I.

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

Silberberg, Y.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

Smith, D.

Y. Urzhumov and D. Smith, Phys. Rev. Lett. 105, 163901(2010).
[CrossRef]

Snyder, A.

A. Snyder and J. Love, Appl. Opt. 15, 236 (1976).
[CrossRef] [PubMed]

A. Snyder and J. Love, Optical Waveguide Theory (Kluwer, 1983).

Stenger, N.

T. Ergin, N. Stenger, P. Brenner, J. Pendry, and M. Wegener, Science 328, 337 (2010).
[CrossRef] [PubMed]

Szameit, A.

A. Szameit and S. Nolte, J. Phys. B 43, 163001 (2010).
[CrossRef]

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, and A. Tünnermann, New J. Phys. 10, 103020 (2008).
[CrossRef]

Trompeter, H.

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, and A. Tünnermann, New J. Phys. 10, 103020 (2008).
[CrossRef]

Tünnermann, A.

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, and A. Tünnermann, New J. Phys. 10, 103020 (2008).
[CrossRef]

Urzhumov, Y.

Y. Urzhumov and D. Smith, Phys. Rev. Lett. 105, 163901(2010).
[CrossRef]

Wegener, M.

T. Ergin, N. Stenger, P. Brenner, J. Pendry, and M. Wegener, Science 328, 337 (2010).
[CrossRef] [PubMed]

Woerdman, J.

Yi, J.

Zharov, A.

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

Ann. Phys. (1)

F. Goos and H. Lindberg-Hänchen, Ann. Phys. 440, 251 (1949).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

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

J. Phys. B (1)

A. Szameit and S. Nolte, J. Phys. B 43, 163001 (2010).
[CrossRef]

Nature (1)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

New J. Phys. (1)

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, and A. Tünnermann, New J. Phys. 10, 103020 (2008).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. B (1)

C. Luo, S. Johnson, J. Joannopoulos, and J. Pendry, Phys. Rev. B 65, 201104 (2002).
[CrossRef]

Phys. Rev. Lett. (2)

Y. Urzhumov and D. Smith, Phys. Rev. Lett. 105, 163901(2010).
[CrossRef]

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

Science (1)

T. Ergin, N. Stenger, P. Brenner, J. Pendry, and M. Wegener, Science 328, 337 (2010).
[CrossRef] [PubMed]

Other (1)

A. Snyder and J. Love, Optical Waveguide Theory (Kluwer, 1983).

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Figures (4)

Fig. 1
Fig. 1

(a) Schematic of a waveguide array with two regions separated by an interface. If a broad initial wave packet reflects within the left array, in general it experiences a GHS. (b) Band structure diagram for the left and right array (β versus κ L , R ). Where the bands do not overlap, waves totally internally reflect, and a GHS occurs.

Fig. 2
Fig. 2

(a) GHS Δ plotted as a function of β / c L for δ β = 2 c L + 2 c R and c L = c R . The blue dashed line denotes analytical results, and the solid red line is from simulations. The inset is the band diagram denoting regions of negative and positive effective mass in the bands of the left and right regions. (b) Simulation of the propagation of a wave packet in our system. The center of gravity is marked with a dashed line and the respective “rays” in the geometric-optics approximation with a solid blue line. In the magnification is shown that the rays intersect before the interface.

Fig. 3
Fig. 3

GHS, Δ, plotted as a function of β / c L with δ β = 2 ( c L + c R ) and c R = c L . Plotted are both the exact results (dashed blue line) and the expansion given in Eq. (4) (solid green line). The expansion is taken around the continuum limit at β / c L = 2 .

Fig. 4
Fig. 4

GHS Δ plotted (a) as a function of β / c L and γ / c L and (b) as a function of β / c L and c R / c L . As γ / c L and c R / c L grow, the crossover from negative to positive GHS occurs closer to β = 0 .

Equations (4)

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[ i z + β L , R ] Ψ n ( z ) + c L , R ( Ψ n 1 ( z ) + Ψ n + 1 ( z ) ) = 0 ,
[ i z + β R ] Ψ 0 ( z ) + c L Ψ 1 ( z ) + c R Ψ 1 ( z ) = 0 ,
β = 2 c L , R cos ( κ L , R ) + β L , R ,
Δ = c L ( c L + c R ) 1 2 c L β + 2 c L c R ( c L + c R ) 2 + O ( 2 c L β ) .

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