Abstract

We study the Goos–Hänchen shift (GHS) on a curved surface through numerical simulation by the boundary element method. A negative GHS is first discovered on a concave dielectric interface below the critical angle, accompanied by a large positive GHS on the convexity. The simulation shows that the GHS on a planar interface is the composition of the GHS from a concave and the corresponding convex interface. This work will enrich the study of the GHS for different curved surfaces, which will have potential applications in micro-optics and near-field optics.

© 2011 Optical Society of America

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  1. F. Goos and H. Hänchen, Ann. Physics 436, 333 (1947).
    [CrossRef]
  2. H. G. L. Schwefel, W. Köhler, Z. H. Lu, J. Fan, and L. J. Wang, Opt. Lett. 33, 794 (2008).
    [CrossRef] [PubMed]
  3. Ziauddin, S. Qamar, and M. S. Zubairy, Phys. Rev. A 81, 023821 (2010).
    [CrossRef]
  4. X. Yin and L. Hesselink, Appl. Phys. Lett. 89, 261108 (2006).
    [CrossRef]
  5. J. Huang, Z. Duan, H. Y. Ling, and W. Zhang, Phys. Rev. A 77, 063608 (2008).
    [CrossRef]
  6. V.-O. de Haan, J. Plomp, T. M. Rekveldt, W. H. Kraan, and A. A. van Well, Phys. Rev. Lett. 104, 010401 (2010).
    [CrossRef] [PubMed]
  7. C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydlo, Phys. Rev. Lett. 102, 146804 (2009).
    [CrossRef] [PubMed]
  8. P. R. Berman, Phys. Rev. E 66, 067603 (2002);
    [CrossRef]
  9. P. R. Berman, Phys. Rev. E 71, 039903(2005).
    [CrossRef]
  10. Y. Wang, Z. Yue, Y. Liu, and J. Xu, Optik 121, 307 (2010).
    [CrossRef]
  11. C.-F. Li, Phys. Rev. Lett. 91, 133903 (2003).
    [CrossRef] [PubMed]
  12. X. Chen, C.-F. Li, R.-R. Wei, and Y. Zhang, Phys. Rev. A 80, 015803 (2009).
    [CrossRef]
  13. H. Schomerus and M. Hentschel, Phys. Rev. Lett. 96, 243903 (2006).
    [CrossRef] [PubMed]
  14. J. Unterhinninghofen, J. Wiersig, and M. Hentschel, Phys. Rev. E 78, 016201 (2008).
    [CrossRef]
  15. M. Hentschel and H. Schomerus, Phys. Rev. E 65, 045603 (2002).
    [CrossRef]
  16. K. Artmann, Ann. Phys. 437, 87 (1948).
    [CrossRef]
  17. H. M. Lai, F. C. Cheng, and W. K. Tang, J. Opt. Soc. Am. A 3, 550 (1986).
    [CrossRef]
  18. J.-L. Shi, C.-F. Li, and Q. Wang, Int. J. Mod. Phys. B 21, 2777 (2007).
    [CrossRef]
  19. R. H. Renard, J. Opt. Soc. Am. 54, 1190 (1964).
    [CrossRef]
  20. Exactly, it is a GH-like shift. The profile of the reflection beam always has an approximate Gaussian shape. So we still call it GHS.
  21. W. T. Ang, A Beginner’s Course in Boundary Element Methods (Universal Publishers, 2007).
  22. J. D. Jackson, Classical Electrondynamics, 3rd ed.(Wiley, 1999).
  23. G.-Y. Oh, D. G. Kim, and Y.-W. Choi, Opt. Express 17, 20714 (2009).
    [CrossRef] [PubMed]
  24. C.-L. Zou, Y. Yang, Y.-F. Xiao, C.-H. Dong, Z.-F. Han, and G.-C. Guo, J. Opt. Soc. Am. B 26, 2050 (2009).
    [CrossRef]

2010 (3)

Ziauddin, S. Qamar, and M. S. Zubairy, Phys. Rev. A 81, 023821 (2010).
[CrossRef]

V.-O. de Haan, J. Plomp, T. M. Rekveldt, W. H. Kraan, and A. A. van Well, Phys. Rev. Lett. 104, 010401 (2010).
[CrossRef] [PubMed]

Y. Wang, Z. Yue, Y. Liu, and J. Xu, Optik 121, 307 (2010).
[CrossRef]

2009 (4)

X. Chen, C.-F. Li, R.-R. Wei, and Y. Zhang, Phys. Rev. A 80, 015803 (2009).
[CrossRef]

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydlo, Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef] [PubMed]

G.-Y. Oh, D. G. Kim, and Y.-W. Choi, Opt. Express 17, 20714 (2009).
[CrossRef] [PubMed]

C.-L. Zou, Y. Yang, Y.-F. Xiao, C.-H. Dong, Z.-F. Han, and G.-C. Guo, J. Opt. Soc. Am. B 26, 2050 (2009).
[CrossRef]

2008 (3)

J. Huang, Z. Duan, H. Y. Ling, and W. Zhang, Phys. Rev. A 77, 063608 (2008).
[CrossRef]

H. G. L. Schwefel, W. Köhler, Z. H. Lu, J. Fan, and L. J. Wang, Opt. Lett. 33, 794 (2008).
[CrossRef] [PubMed]

J. Unterhinninghofen, J. Wiersig, and M. Hentschel, Phys. Rev. E 78, 016201 (2008).
[CrossRef]

2007 (1)

J.-L. Shi, C.-F. Li, and Q. Wang, Int. J. Mod. Phys. B 21, 2777 (2007).
[CrossRef]

2006 (2)

H. Schomerus and M. Hentschel, Phys. Rev. Lett. 96, 243903 (2006).
[CrossRef] [PubMed]

X. Yin and L. Hesselink, Appl. Phys. Lett. 89, 261108 (2006).
[CrossRef]

2005 (1)

P. R. Berman, Phys. Rev. E 71, 039903(2005).
[CrossRef]

2003 (1)

C.-F. Li, Phys. Rev. Lett. 91, 133903 (2003).
[CrossRef] [PubMed]

2002 (2)

M. Hentschel and H. Schomerus, Phys. Rev. E 65, 045603 (2002).
[CrossRef]

P. R. Berman, Phys. Rev. E 66, 067603 (2002);
[CrossRef]

1986 (1)

1964 (1)

1948 (1)

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

1947 (1)

F. Goos and H. Hänchen, Ann. Physics 436, 333 (1947).
[CrossRef]

Akhmerov, A. R.

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydlo, Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef] [PubMed]

Ang, W. T.

W. T. Ang, A Beginner’s Course in Boundary Element Methods (Universal Publishers, 2007).

Artmann, K.

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

Beenakker, C. W. J.

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydlo, Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef] [PubMed]

Berman, P. R.

P. R. Berman, Phys. Rev. E 71, 039903(2005).
[CrossRef]

P. R. Berman, Phys. Rev. E 66, 067603 (2002);
[CrossRef]

Chen, X.

X. Chen, C.-F. Li, R.-R. Wei, and Y. Zhang, Phys. Rev. A 80, 015803 (2009).
[CrossRef]

Cheng, F. C.

Choi, Y.-W.

de Haan, V.-O.

V.-O. de Haan, J. Plomp, T. M. Rekveldt, W. H. Kraan, and A. A. van Well, Phys. Rev. Lett. 104, 010401 (2010).
[CrossRef] [PubMed]

Dong, C.-H.

Duan, Z.

J. Huang, Z. Duan, H. Y. Ling, and W. Zhang, Phys. Rev. A 77, 063608 (2008).
[CrossRef]

Fan, J.

Goos, F.

F. Goos and H. Hänchen, Ann. Physics 436, 333 (1947).
[CrossRef]

Guo, G.-C.

Han, Z.-F.

Hänchen, H.

F. Goos and H. Hänchen, Ann. Physics 436, 333 (1947).
[CrossRef]

Hentschel, M.

J. Unterhinninghofen, J. Wiersig, and M. Hentschel, Phys. Rev. E 78, 016201 (2008).
[CrossRef]

H. Schomerus and M. Hentschel, Phys. Rev. Lett. 96, 243903 (2006).
[CrossRef] [PubMed]

M. Hentschel and H. Schomerus, Phys. Rev. E 65, 045603 (2002).
[CrossRef]

Hesselink, L.

X. Yin and L. Hesselink, Appl. Phys. Lett. 89, 261108 (2006).
[CrossRef]

Huang, J.

J. Huang, Z. Duan, H. Y. Ling, and W. Zhang, Phys. Rev. A 77, 063608 (2008).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrondynamics, 3rd ed.(Wiley, 1999).

Kim, D. G.

Köhler, W.

Kraan, W. H.

V.-O. de Haan, J. Plomp, T. M. Rekveldt, W. H. Kraan, and A. A. van Well, Phys. Rev. Lett. 104, 010401 (2010).
[CrossRef] [PubMed]

Lai, H. M.

Li, C.-F.

X. Chen, C.-F. Li, R.-R. Wei, and Y. Zhang, Phys. Rev. A 80, 015803 (2009).
[CrossRef]

J.-L. Shi, C.-F. Li, and Q. Wang, Int. J. Mod. Phys. B 21, 2777 (2007).
[CrossRef]

C.-F. Li, Phys. Rev. Lett. 91, 133903 (2003).
[CrossRef] [PubMed]

Ling, H. Y.

J. Huang, Z. Duan, H. Y. Ling, and W. Zhang, Phys. Rev. A 77, 063608 (2008).
[CrossRef]

Liu, Y.

Y. Wang, Z. Yue, Y. Liu, and J. Xu, Optik 121, 307 (2010).
[CrossRef]

Lu, Z. H.

Oh, G.-Y.

Plomp, J.

V.-O. de Haan, J. Plomp, T. M. Rekveldt, W. H. Kraan, and A. A. van Well, Phys. Rev. Lett. 104, 010401 (2010).
[CrossRef] [PubMed]

Qamar, S.

Ziauddin, S. Qamar, and M. S. Zubairy, Phys. Rev. A 81, 023821 (2010).
[CrossRef]

Rekveldt, T. M.

V.-O. de Haan, J. Plomp, T. M. Rekveldt, W. H. Kraan, and A. A. van Well, Phys. Rev. Lett. 104, 010401 (2010).
[CrossRef] [PubMed]

Renard, R. H.

Schomerus, H.

H. Schomerus and M. Hentschel, Phys. Rev. Lett. 96, 243903 (2006).
[CrossRef] [PubMed]

M. Hentschel and H. Schomerus, Phys. Rev. E 65, 045603 (2002).
[CrossRef]

Schwefel, H. G. L.

Sepkhanov, R. A.

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydlo, Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef] [PubMed]

Shi, J.-L.

J.-L. Shi, C.-F. Li, and Q. Wang, Int. J. Mod. Phys. B 21, 2777 (2007).
[CrossRef]

Tang, W. K.

Tworzydlo, J.

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydlo, Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef] [PubMed]

Unterhinninghofen, J.

J. Unterhinninghofen, J. Wiersig, and M. Hentschel, Phys. Rev. E 78, 016201 (2008).
[CrossRef]

van Well, A. A.

V.-O. de Haan, J. Plomp, T. M. Rekveldt, W. H. Kraan, and A. A. van Well, Phys. Rev. Lett. 104, 010401 (2010).
[CrossRef] [PubMed]

Wang, L. J.

Wang, Q.

J.-L. Shi, C.-F. Li, and Q. Wang, Int. J. Mod. Phys. B 21, 2777 (2007).
[CrossRef]

Wang, Y.

Y. Wang, Z. Yue, Y. Liu, and J. Xu, Optik 121, 307 (2010).
[CrossRef]

Wei, R.-R.

X. Chen, C.-F. Li, R.-R. Wei, and Y. Zhang, Phys. Rev. A 80, 015803 (2009).
[CrossRef]

Wiersig, J.

J. Unterhinninghofen, J. Wiersig, and M. Hentschel, Phys. Rev. E 78, 016201 (2008).
[CrossRef]

Xiao, Y.-F.

Xu, J.

Y. Wang, Z. Yue, Y. Liu, and J. Xu, Optik 121, 307 (2010).
[CrossRef]

Yang, Y.

Yin, X.

X. Yin and L. Hesselink, Appl. Phys. Lett. 89, 261108 (2006).
[CrossRef]

Yue, Z.

Y. Wang, Z. Yue, Y. Liu, and J. Xu, Optik 121, 307 (2010).
[CrossRef]

Zhang, W.

J. Huang, Z. Duan, H. Y. Ling, and W. Zhang, Phys. Rev. A 77, 063608 (2008).
[CrossRef]

Zhang, Y.

X. Chen, C.-F. Li, R.-R. Wei, and Y. Zhang, Phys. Rev. A 80, 015803 (2009).
[CrossRef]

Zou, C.-L.

Zubairy, M. S.

Ziauddin, S. Qamar, and M. S. Zubairy, Phys. Rev. A 81, 023821 (2010).
[CrossRef]

Ann. Phys. (1)

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

Ann. Physics (1)

F. Goos and H. Hänchen, Ann. Physics 436, 333 (1947).
[CrossRef]

Appl. Phys. Lett. (1)

X. Yin and L. Hesselink, Appl. Phys. Lett. 89, 261108 (2006).
[CrossRef]

Int. J. Mod. Phys. B (1)

J.-L. Shi, C.-F. Li, and Q. Wang, Int. J. Mod. Phys. B 21, 2777 (2007).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Opt. Express (1)

Opt. Lett. (1)

Optik (1)

Y. Wang, Z. Yue, Y. Liu, and J. Xu, Optik 121, 307 (2010).
[CrossRef]

Phys. Rev. A (3)

X. Chen, C.-F. Li, R.-R. Wei, and Y. Zhang, Phys. Rev. A 80, 015803 (2009).
[CrossRef]

Ziauddin, S. Qamar, and M. S. Zubairy, Phys. Rev. A 81, 023821 (2010).
[CrossRef]

J. Huang, Z. Duan, H. Y. Ling, and W. Zhang, Phys. Rev. A 77, 063608 (2008).
[CrossRef]

Phys. Rev. E (4)

P. R. Berman, Phys. Rev. E 66, 067603 (2002);
[CrossRef]

P. R. Berman, Phys. Rev. E 71, 039903(2005).
[CrossRef]

J. Unterhinninghofen, J. Wiersig, and M. Hentschel, Phys. Rev. E 78, 016201 (2008).
[CrossRef]

M. Hentschel and H. Schomerus, Phys. Rev. E 65, 045603 (2002).
[CrossRef]

Phys. Rev. Lett. (4)

H. Schomerus and M. Hentschel, Phys. Rev. Lett. 96, 243903 (2006).
[CrossRef] [PubMed]

C.-F. Li, Phys. Rev. Lett. 91, 133903 (2003).
[CrossRef] [PubMed]

V.-O. de Haan, J. Plomp, T. M. Rekveldt, W. H. Kraan, and A. A. van Well, Phys. Rev. Lett. 104, 010401 (2010).
[CrossRef] [PubMed]

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydlo, Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef] [PubMed]

Other (3)

Exactly, it is a GH-like shift. The profile of the reflection beam always has an approximate Gaussian shape. So we still call it GHS.

W. T. Ang, A Beginner’s Course in Boundary Element Methods (Universal Publishers, 2007).

J. D. Jackson, Classical Electrondynamics, 3rd ed.(Wiley, 1999).

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of reflection. Dielectric with refractive index n 1 ( n 2 ) in domain A 1 ( A 2 ) is bounded by the curve a 1 ( a 2 ) . D denotes the GHS between the realistic (blue solid) and geometrical (blue dashed) ray. The concave and convex interfaces are illustrated in the insets.

Fig. 2
Fig. 2

Field distribution of the reflection of an s-polarized Gaussian beam on (a) planar and (b) concave interfaces. At the top, the curves show the normalized field distributions of incident (red dashed) and reflection (black solid) beams on the boundary. At the bottom, the GHS is the displacement between the realistic (blue solid) and geometrical ray (blue dashed). On the planar interface D Planar = 0.22 λ 0 with incident angle α = 14.8 ° , while on the concave with k R p = 800 , the negative GHS is 0.15 λ 0 with α = 12.0 ° . The parameters are: n 1 = 4.0 , n 2 = 1.0 , and beam waist w 0 = 3 λ 0 . The center of the incident beam is at x = 0 . λ 0 is the vacuum wavelength. The error is estimated to be less than 0.02 λ 0 . All the parameters are the same in the following simulations.

Fig. 3
Fig. 3

GHS on planar interface versus incident angle α and number of boundary elements N.

Fig. 4
Fig. 4

GHS on curved interface versus incident angle α and curvature. N = 1600 .

Fig. 5
Fig. 5

(a)  D k R + and (b)  D k R versus incident angle α and curvature.

Equations (3)

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U I = d k x A ( k x ) e i k I · r ,
U R = d k x A ( k x ) R ( k x , x ) e i k R · r ,
U R d k x A ( k x ) [ R ( k x , 0 ) + R ( k x , x ) x = 0 Δ x ] × e i ( k R Planar + Δ k R ) · r .

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