Abstract

By means of an impedance boundary condition and numerical solution of integral equations for the scattering amplitudes to which its use gives rise, we study as a function of its angle of incidence the reflection of a surface plasmon polariton beam propagating on a metal surface whose dielectric function is ɛ 1(ω) when it is incident on a planar interface with a coplanar metal surface whose dielectric function is ɛ 2(ω). When the surface of incidence is optically more dense than the surface of scattering, i.e. when |ɛ 2(ω)| ≫ |ɛ 1(ω)|, the reflected beam undergoes a lateral displacement whose magnitude is several times the wavelength of the incident beam. This displacement is the surface plasmon polariton analogue of the Goos-Hänchen effect. Since this displacement is sensitive to the dielectric properties of the surface, this effect can be exploited to sense modifications of the dielectric environment of a metal surface, e.g. due to adsorption of atomic or molecular layers on it.

© 2011 OSA

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Corrections

Felix Huerkamp, Tamara A. Leskova, Alexei A. Maradudin, and Björn Baumeier, "The Goos-Hänchen effect for surface plasmon polaritons: erratum," Opt. Express 19, 18807-18807 (2011)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-19-20-18807

References

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  1. F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436, 333–346 (1947).
    [CrossRef]
  2. K. Artmann, “Berechnung der Seitenversetzung des totalreflektierten Strahles,” Ann. Phys. 437, 87–102 (1948).
    [CrossRef]
  3. H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure (vol 96, pg 073907, 2006),” Phys. Rev. Lett. 96, 073907 (2006).
    [CrossRef] [PubMed]
  4. H. Lezec, J. Dionne, and H. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
    [CrossRef] [PubMed]
  5. M. Dennis, N. Zheludev, and F. Garcia de Abajo, “The plasmon Talbot effect,” Opt. Express 15, 9692–9700 (2007).
    [CrossRef] [PubMed]
  6. A. Maradudin and T. Leskova, “The Talbot effect for a surface plasmon polariton,” New J. Phys. 11, 033004 (2009).
    [CrossRef]
  7. A. Tredicucci, C. Gmachl, F. Capasso, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “Single-mode surface-plasmon laser,” Appl. Phys. Lett. 76, 2164–2166 (2000).
    [CrossRef]
  8. B. Baumeier, T. A. Leskova, and A. A. Maradudin, “Cloaking from surface plasmon polaritons by a circular array of point scatterers,” Phys. Rev. Lett. 103, 246803 (2009).
    [CrossRef]
  9. Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10, 1991–1997 (2010).
    [CrossRef] [PubMed]
  10. P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Transformation optics for plasmonics,” Nano Lett. 10, 1985–1990 (2010).
    [CrossRef] [PubMed]
  11. J. Renger, M. Kadic, G. Dupont, S. Acimovic, S. Guenneau, and R. Quidant, “Hidden progress: broadband plasmonic invisibility,” Opt. Express 18, 15757–15768 (2010).
    [CrossRef] [PubMed]
  12. R. Zia and M. L. Brongersma, “Surface plasmon polariton analogue to young’s double-slit experiment,” Nat. Nanotechnol. 2, 426–429 (2007).
    [CrossRef]
  13. A. A. Maradudin, “The impedance boundary condition at a two-dimensional rough metal surface,” Optics Commun. 116, 452 – 467 (1995).
    [CrossRef]
  14. K. Atkinson, “The numerical solution of Fredholm integral equations of the second kind with singular kernels,” Numerische Mathematik 19, 248–259 (1972).
    [CrossRef]
  15. F. Huerkamp, T. A. Leskova, and A. A. Maradudin, “Surface plasmon polariton analogues of volume electromagnetic wave effects,” Proc. SPIE 7467, 74670H (2009).
    [CrossRef]
  16. Y. A. Nikitin, G. Brucoli, F. J. García-Vidal, and L. Martín-Moreno, “Scattering of surface plasmon polaritons by impedance barriers: Dependence on angle of incidence,” Phys. Rev. B 77, 195441 (2008).

2010 (3)

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10, 1991–1997 (2010).
[CrossRef] [PubMed]

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Transformation optics for plasmonics,” Nano Lett. 10, 1985–1990 (2010).
[CrossRef] [PubMed]

J. Renger, M. Kadic, G. Dupont, S. Acimovic, S. Guenneau, and R. Quidant, “Hidden progress: broadband plasmonic invisibility,” Opt. Express 18, 15757–15768 (2010).
[CrossRef] [PubMed]

2009 (3)

B. Baumeier, T. A. Leskova, and A. A. Maradudin, “Cloaking from surface plasmon polaritons by a circular array of point scatterers,” Phys. Rev. Lett. 103, 246803 (2009).
[CrossRef]

A. Maradudin and T. Leskova, “The Talbot effect for a surface plasmon polariton,” New J. Phys. 11, 033004 (2009).
[CrossRef]

F. Huerkamp, T. A. Leskova, and A. A. Maradudin, “Surface plasmon polariton analogues of volume electromagnetic wave effects,” Proc. SPIE 7467, 74670H (2009).
[CrossRef]

2007 (3)

M. Dennis, N. Zheludev, and F. Garcia de Abajo, “The plasmon Talbot effect,” Opt. Express 15, 9692–9700 (2007).
[CrossRef] [PubMed]

R. Zia and M. L. Brongersma, “Surface plasmon polariton analogue to young’s double-slit experiment,” Nat. Nanotechnol. 2, 426–429 (2007).
[CrossRef]

H. Lezec, J. Dionne, and H. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
[CrossRef] [PubMed]

2006 (1)

H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure (vol 96, pg 073907, 2006),” Phys. Rev. Lett. 96, 073907 (2006).
[CrossRef] [PubMed]

2000 (1)

A. Tredicucci, C. Gmachl, F. Capasso, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “Single-mode surface-plasmon laser,” Appl. Phys. Lett. 76, 2164–2166 (2000).
[CrossRef]

1995 (1)

A. A. Maradudin, “The impedance boundary condition at a two-dimensional rough metal surface,” Optics Commun. 116, 452 – 467 (1995).
[CrossRef]

1972 (1)

K. Atkinson, “The numerical solution of Fredholm integral equations of the second kind with singular kernels,” Numerische Mathematik 19, 248–259 (1972).
[CrossRef]

1954 (1)

Y. A. Nikitin, G. Brucoli, F. J. García-Vidal, and L. Martín-Moreno, “Scattering of surface plasmon polaritons by impedance barriers: Dependence on angle of incidence,” Phys. Rev. B 77, 195441 (2008).

1948 (1)

K. Artmann, “Berechnung der Seitenversetzung des totalreflektierten Strahles,” Ann. Phys. 437, 87–102 (1948).
[CrossRef]

1947 (1)

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436, 333–346 (1947).
[CrossRef]

Acimovic, S.

Artmann, K.

K. Artmann, “Berechnung der Seitenversetzung des totalreflektierten Strahles,” Ann. Phys. 437, 87–102 (1948).
[CrossRef]

Atkinson, K.

K. Atkinson, “The numerical solution of Fredholm integral equations of the second kind with singular kernels,” Numerische Mathematik 19, 248–259 (1972).
[CrossRef]

Atwater, H.

H. Lezec, J. Dionne, and H. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
[CrossRef] [PubMed]

Bartal, G.

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10, 1991–1997 (2010).
[CrossRef] [PubMed]

Baumeier, B.

B. Baumeier, T. A. Leskova, and A. A. Maradudin, “Cloaking from surface plasmon polaritons by a circular array of point scatterers,” Phys. Rev. Lett. 103, 246803 (2009).
[CrossRef]

Brongersma, M. L.

R. Zia and M. L. Brongersma, “Surface plasmon polariton analogue to young’s double-slit experiment,” Nat. Nanotechnol. 2, 426–429 (2007).
[CrossRef]

Brucoli, G.

Y. A. Nikitin, G. Brucoli, F. J. García-Vidal, and L. Martín-Moreno, “Scattering of surface plasmon polaritons by impedance barriers: Dependence on angle of incidence,” Phys. Rev. B 77, 195441 (2008).

Capasso, F.

A. Tredicucci, C. Gmachl, F. Capasso, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “Single-mode surface-plasmon laser,” Appl. Phys. Lett. 76, 2164–2166 (2000).
[CrossRef]

Cho, A. Y.

A. Tredicucci, C. Gmachl, F. Capasso, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “Single-mode surface-plasmon laser,” Appl. Phys. Lett. 76, 2164–2166 (2000).
[CrossRef]

Dennis, M.

Dionne, J.

H. Lezec, J. Dionne, and H. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
[CrossRef] [PubMed]

Dupont, G.

Fan, S.

H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure (vol 96, pg 073907, 2006),” Phys. Rev. Lett. 96, 073907 (2006).
[CrossRef] [PubMed]

Garcia de Abajo, F.

García-Vidal, F. J.

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Transformation optics for plasmonics,” Nano Lett. 10, 1985–1990 (2010).
[CrossRef] [PubMed]

Y. A. Nikitin, G. Brucoli, F. J. García-Vidal, and L. Martín-Moreno, “Scattering of surface plasmon polaritons by impedance barriers: Dependence on angle of incidence,” Phys. Rev. B 77, 195441 (2008).

Gmachl, C.

A. Tredicucci, C. Gmachl, F. Capasso, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “Single-mode surface-plasmon laser,” Appl. Phys. Lett. 76, 2164–2166 (2000).
[CrossRef]

Goos, F.

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436, 333–346 (1947).
[CrossRef]

Guenneau, S.

Hänchen, H.

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436, 333–346 (1947).
[CrossRef]

Huerkamp, F.

F. Huerkamp, T. A. Leskova, and A. A. Maradudin, “Surface plasmon polariton analogues of volume electromagnetic wave effects,” Proc. SPIE 7467, 74670H (2009).
[CrossRef]

Huidobro, P. A.

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Transformation optics for plasmonics,” Nano Lett. 10, 1985–1990 (2010).
[CrossRef] [PubMed]

Hutchinson, A. L.

A. Tredicucci, C. Gmachl, F. Capasso, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “Single-mode surface-plasmon laser,” Appl. Phys. Lett. 76, 2164–2166 (2000).
[CrossRef]

Kadic, M.

Leskova, T.

A. Maradudin and T. Leskova, “The Talbot effect for a surface plasmon polariton,” New J. Phys. 11, 033004 (2009).
[CrossRef]

Leskova, T. A.

F. Huerkamp, T. A. Leskova, and A. A. Maradudin, “Surface plasmon polariton analogues of volume electromagnetic wave effects,” Proc. SPIE 7467, 74670H (2009).
[CrossRef]

B. Baumeier, T. A. Leskova, and A. A. Maradudin, “Cloaking from surface plasmon polaritons by a circular array of point scatterers,” Phys. Rev. Lett. 103, 246803 (2009).
[CrossRef]

Lezec, H.

H. Lezec, J. Dionne, and H. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
[CrossRef] [PubMed]

Liu, Y.

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10, 1991–1997 (2010).
[CrossRef] [PubMed]

Maradudin, A.

A. Maradudin and T. Leskova, “The Talbot effect for a surface plasmon polariton,” New J. Phys. 11, 033004 (2009).
[CrossRef]

Maradudin, A. A.

B. Baumeier, T. A. Leskova, and A. A. Maradudin, “Cloaking from surface plasmon polaritons by a circular array of point scatterers,” Phys. Rev. Lett. 103, 246803 (2009).
[CrossRef]

F. Huerkamp, T. A. Leskova, and A. A. Maradudin, “Surface plasmon polariton analogues of volume electromagnetic wave effects,” Proc. SPIE 7467, 74670H (2009).
[CrossRef]

A. A. Maradudin, “The impedance boundary condition at a two-dimensional rough metal surface,” Optics Commun. 116, 452 – 467 (1995).
[CrossRef]

Martín-Moreno, L.

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Transformation optics for plasmonics,” Nano Lett. 10, 1985–1990 (2010).
[CrossRef] [PubMed]

Y. A. Nikitin, G. Brucoli, F. J. García-Vidal, and L. Martín-Moreno, “Scattering of surface plasmon polaritons by impedance barriers: Dependence on angle of incidence,” Phys. Rev. B 77, 195441 (2008).

Nesterov, M. L.

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Transformation optics for plasmonics,” Nano Lett. 10, 1985–1990 (2010).
[CrossRef] [PubMed]

Nikitin, Y. A.

Y. A. Nikitin, G. Brucoli, F. J. García-Vidal, and L. Martín-Moreno, “Scattering of surface plasmon polaritons by impedance barriers: Dependence on angle of incidence,” Phys. Rev. B 77, 195441 (2008).

Quidant, R.

Renger, J.

Shin, H.

H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure (vol 96, pg 073907, 2006),” Phys. Rev. Lett. 96, 073907 (2006).
[CrossRef] [PubMed]

Sivco, D. L.

A. Tredicucci, C. Gmachl, F. Capasso, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “Single-mode surface-plasmon laser,” Appl. Phys. Lett. 76, 2164–2166 (2000).
[CrossRef]

Tredicucci, A.

A. Tredicucci, C. Gmachl, F. Capasso, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “Single-mode surface-plasmon laser,” Appl. Phys. Lett. 76, 2164–2166 (2000).
[CrossRef]

Zentgraf, T.

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10, 1991–1997 (2010).
[CrossRef] [PubMed]

Zhang, X.

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10, 1991–1997 (2010).
[CrossRef] [PubMed]

Zheludev, N.

Zia, R.

R. Zia and M. L. Brongersma, “Surface plasmon polariton analogue to young’s double-slit experiment,” Nat. Nanotechnol. 2, 426–429 (2007).
[CrossRef]

Ann. Phys. (2)

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436, 333–346 (1947).
[CrossRef]

K. Artmann, “Berechnung der Seitenversetzung des totalreflektierten Strahles,” Ann. Phys. 437, 87–102 (1948).
[CrossRef]

Appl. Phys. Lett. (1)

A. Tredicucci, C. Gmachl, F. Capasso, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “Single-mode surface-plasmon laser,” Appl. Phys. Lett. 76, 2164–2166 (2000).
[CrossRef]

Nano Lett. (2)

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10, 1991–1997 (2010).
[CrossRef] [PubMed]

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Transformation optics for plasmonics,” Nano Lett. 10, 1985–1990 (2010).
[CrossRef] [PubMed]

Nat. Nanotechnol. (1)

R. Zia and M. L. Brongersma, “Surface plasmon polariton analogue to young’s double-slit experiment,” Nat. Nanotechnol. 2, 426–429 (2007).
[CrossRef]

New J. Phys. (1)

A. Maradudin and T. Leskova, “The Talbot effect for a surface plasmon polariton,” New J. Phys. 11, 033004 (2009).
[CrossRef]

Numerische Mathematik (1)

K. Atkinson, “The numerical solution of Fredholm integral equations of the second kind with singular kernels,” Numerische Mathematik 19, 248–259 (1972).
[CrossRef]

Opt. Express (2)

Optics Commun. (1)

A. A. Maradudin, “The impedance boundary condition at a two-dimensional rough metal surface,” Optics Commun. 116, 452 – 467 (1995).
[CrossRef]

Phys. Rev. B (1)

Y. A. Nikitin, G. Brucoli, F. J. García-Vidal, and L. Martín-Moreno, “Scattering of surface plasmon polaritons by impedance barriers: Dependence on angle of incidence,” Phys. Rev. B 77, 195441 (2008).

Phys. Rev. Lett. (2)

B. Baumeier, T. A. Leskova, and A. A. Maradudin, “Cloaking from surface plasmon polaritons by a circular array of point scatterers,” Phys. Rev. Lett. 103, 246803 (2009).
[CrossRef]

H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure (vol 96, pg 073907, 2006),” Phys. Rev. Lett. 96, 073907 (2006).
[CrossRef] [PubMed]

Proc. SPIE (1)

F. Huerkamp, T. A. Leskova, and A. A. Maradudin, “Surface plasmon polariton analogues of volume electromagnetic wave effects,” Proc. SPIE 7467, 74670H (2009).
[CrossRef]

Science (1)

H. Lezec, J. Dionne, and H. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
[CrossRef] [PubMed]

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Figures (6)

Fig. 1
Fig. 1

Scattering geometry of the double interface system. The blue (gray) vectors are the beam reflected from the first interface, the red (black) vectors are the beams reflected from the second interface, and the green (light gray) vectors are the actual reflected beams.

Fig. 2
Fig. 2

The modulus of the reflection amplitude R and phase shift φ as functions of the angle of incidence θ for an incident SPP plane wave at the single interface.

Fig. 3
Fig. 3

(a) Color-level plot of the intensity of the incident beam (left), and the reflected beam (right). The angle of incidence is θ 0 = 78°, the beam width is w = 20c/ω and x 3 = 0.1c/ω. The maxima of the incident and reflected beam are marked with dashed lines, the displacement is D = 60.3c/ω = 9.6λ. (b) Plot of the respective intensities along the x 2-direction at the interface (x 1 = 0).

Fig. 4
Fig. 4

Calculated lateral displacement D for different beam widths (200, 30, and 10 c/ω) as a function of the angle of incidence of the beam θ 0.

Fig. 5
Fig. 5

Lateral displacement as a function of angle of incidence θ 0 at a double interface with L = 20c/ω (solid line) compared to the result for a single interface (dashed line). The width of the SPP beam is w = 30c/ω. The inset shows the phase of a reflected SPP plane wave as a function of θ.

Fig. 6
Fig. 6

Change of D upon variation of the dielectric functions of the two metals at an angle of incidence of θ 0 = 80°.

Equations (9)

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

D ( θ 0 ) = λ 2 π 1 k | | cos θ 0 d φ ( θ ) d θ | θ = θ 0 .
A i ( p | | ) + ζ ( ω ) j = p , s S ˜ ( p | | q | | ) M i , j ( p | | | q | | ) d j ( q | | ) A j ( q | | ) d 2 q | | ( 2 π ) 2 = ζ ( ω ) S ˜ ( p | | k | | ) M i , p ( p | | | k | | ) .
M p , p ( p | | | q | | ) = M s , s ( p | | | q | | ) = i p ^ | | q ^ | | M p , s ( p | | | q | | ) = M s , p ( p | | | q | | ) = ( p ^ | | × q ^ | | ) 3 .
S ˜ ( Q | | ) = R 2 S ( x | | ) e Q | | x | | d 2 x | | = 2 π δ ( Q 2 ) f ( Q 1 ) ,
f I ( Q 1 ) = 1 i ( Q 1 i η ) , f II ( Q 1 ) = L sinc ( Q 1 ) 2 exp ( i Q 1 L 2 )
A p , s ( q | | ) = 2 π δ ( q 2 k 2 ) a p , s ( q 1 ) .
a i ( p 1 ) + ζ ( ω ) j = p , s M i , j ( p ¯ | | | q ¯ | | ) f ( p 1 q 1 ) a j ( q 1 ) d j ( q ¯ | | ) d q 1 2 π = ζ ( ω ) M i , p ( p ¯ | | | k | | ) f ( p 1 k 1 ) ,
E sc ( x | ω ) = e ^ p ( q | | ) A p ( q | | ) β 0 ( q | | ) + i ω c κ 1 ( ω ) e i q | | x | | β 0 ( q | | ) x 3 d 2 q | | ( 2 π ) 2
E ref ( x | ω ) = r ( k 1 ) c ω e ^ p ( k 1 , k 2 ) e i k 1 x 1 + i k 2 x 2 β 0 ( k | | ) x 3 ,

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