## The Goos-Hänchen effect for surface plasmon polaritons |

Optics Express, Vol. 19, Issue 16, pp. 15483-15489 (2011)

http://dx.doi.org/10.1364/OE.19.015483

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### 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.

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*e.g.*negative refraction [3

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]

*ɛ*

_{1}(

*ω*) is incident on a planar interface with an optically less dense metal whose dielectric function is

*ɛ*

_{2}(

*ω*) (|

*ɛ*

_{2}(

*ω*)| ≫ |

*ɛ*

_{1}(

*ω*)|). We consider the two cases in which the second metal is either infinitely long (single interface) or of finite length

*L*(double interface). The electromagnetic field of the SPP is determined by use of an impedance boundary condition [13

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

*x*

_{3}= 0. Scattering amplitudes

*A*(

_{i}**p**

_{||}) for

*p*- and

*s*-polarized fields (

*i*=

*p,s*) can be obtained from the solution of a pair of coupled integral equations

*M*(

_{i, j}**p**

_{||}|

**q**

_{||}) are given by

*p*- and

*s*-polarization, where

*β*

_{0}(

*q*

_{||}) > 0 and Im

*β*

_{0}(

*q*

_{||})< 0.

*S̃*(

**Q**

_{||}=

**p**

_{||}–

**q**

_{||}) is the Fourier transformed of the surface profile function where for the single and double interface, respectively. Due to the translational invariance of the system in the

*x*

_{2}-direction

*A*(

_{p,s}**q**

_{||}) have the general form Substituting Eq. (6) into Eq. (2) leads to a pair of effective one-dimensional integral equations

**p̄**

_{||}= (

*p*

_{1},

*k*

_{2}) and

**q̄**

_{||}= (

*q*

_{1},

*k*

_{2}). Equations (7) are solved numerically using the Nystrom method [14

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

*q*

_{∞},

*q*

_{∞}]. The resulting integrals over

*q*

_{1}were converted to sums using a

*N*-point extended midpoint method.

*p*

_{1}was given the values of the abscissas used in the evaluation of the integrals and a square 2

*N*× 2

*N*supermatrix equation with

*N*= 18001 for

*a*(

_{p,s}*p*

_{1}) is solved by a standard linear equation solver. The convergence of the solution was monitored by increasing

*q*

_{∞}and

*N*systematically until the solution did not change upon further increases of these parameters. A lateral displacement of the incident SPP beam is identifiable in the far field region by the intensity distribution of the scattered electromagnetic field of the propagating

*p*-polarized SPP mode with wave number

*k*

_{||}. For an incident plane wave, the scattered field is given by where

*p*-polarized SPP. The contribution to this field in the region

*x*

_{1}< 0 from the reflected surface plasmon polariton is given by the residue of the integrand at the simple pole it has at

*q*

_{1}= –

*k*

_{1}(

*ω*) = –cos(

*θ*)

*k*

_{||}(

*ω*). With the assumption that

*ɛ*

_{1}(

*ω*) has an infinitesimal positive imaginary part, this pole lies in the lower half of the complex

*q*

_{1}plane. It can be shown that

*a*(

_{p}*q*

_{1}) has no pole in this region. On evaluating the residue at this pole we obtain for the electric field of the reflected SPP in the region

*x*

_{1}< 0,

*x*

_{3}> 0 where

*R*as a function of the angle of incidence when a SPP in the form of a plane wave whose wavelength is

*λ*= 632.8 nm, propagating on a gold surface with

*ɛ*

_{1}(

*ω*) = −11.8 at the corresponding frequency, is incident on its planar interface with aluminum (

*ɛ*

_{2}(

*ω*) = −64.07). Since the mean free paths of SPP on these two surfaces are

*L*

_{1}= 7

*μ*m and

*L*

_{2}= 30

*μ*m, we expect the effect of ohmic losses on the our results obtained for real-valued

*ɛ*(

_{i}*ω*) to be small. The angle of incidence is

*θ*= 78°, and the 1/

*e*half width of the beam is

*w*= 20

*c*/

*ω*. The critical angle for total internal reflection, given by

*θ*= 75.4° in this case. Note that preliminary, non-converged results were shown unanalyzed as work in progress in Ref. [15

_{c}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]

*R*is small (∼10

^{−4}) for all angles smaller than

*θ*, and equal to unity for angles greater than

_{c}*θ*

_{c}.

*R*is not a monotonically increasing function of

*θ*, but has a pronounced minimum at the angle of incidence

*θ*≈ 45°. The occurrence of this dip has been explained for a somewhat different SPP scattering problem as the Brewster effect for the incident SPP [16]. The shift of the position of the minimum in Fig. 2 from

*θ*= 45° is due to a small imaginary part added to

*ɛ*

_{1}(

*ω*). The phase shift

*φ*(

*θ*) at the interface is close to

*π*for angles smaller than

*θ*= 45°, and jumps to nearly 2

*π*at this angle. In the rather narrow interval [

*θ*

_{c}, 90°] the phase decreases continuously from 2

*π*to

*π*. Because of the derivative of

*φ*(

*θ*) in Artmann’s result one can therefore expect a significant lateral displacement of the reflected beam.

*θ*with 1/

*e*half width 2/[

*k*

_{||}(

*ω*)

*w*], centered at

*θ*=

*θ*

_{0}, and normalized to unity, which yields a Gaussian beam of 1/

*e*half width

*w*whose angle of incidence is

*θ*

_{0}.

*x*

_{1}= 0) are marked with a dashed line, showing a displacement of

*D*= 60.3

*c*/

*ω*= 9.6

*λ*.

*D*on the angle of incidence: the smaller the width, the fewer the structures in

*D*(

*θ*

_{0}),

*e.g.*as in Fig. 4 for beams with

*w*= 30

*c*/

*ω*(dashed line) or

*w*= 10

*c*/

*ω*(dotted line). For instance, the negative displacement at

*θ*

_{0}= 45° becomes less pronounced and the feature in the curve smears out. Similar observations can be made for the structures close to the critical angle for total internal reflection. In particular,

*D*remains finite when

*θ*

_{0}→ 90°.

*L*= 20

*c*/

*ω*the phase of a reflected SPP plane wave (see inset of Fig. 5) is identical to the one at the single interface in the interval [

*θ*

_{c}, 90°] since the wave is totally reflected at the first interface. The major difference is the oscillating behavior at smaller angles, which is due to multiple scattering at the two interfaces leading to destructive or constructive interference at different angles as in a Fabry-Pérot interferometer. These oscillations also appear in

*D*(

*θ*

_{0}) shown in Fig. 5 and decay with larger

*L*.

*λ*are about one order of magnitude larger than the corresponding results for volume waves [1

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

*θ*

_{c}, 90°] in the case of SPP. The values are also sensitive to changes in the dielectric functions. Figure 6 shows the change in

*D*upon variation of either

*ɛ*

_{1}(

*ω*) or

*ɛ*

_{2}(

*ω*) while the other one is fixed. This results were obtaind by the use of Artmann’s formula (1). In particular, the strong dependence of the calculated lateral displacement on the value of

*ɛ*

_{1}(

*ω*), i.e. the dielectric function of the gold surface, indicates that small modifications of the latter may be resolved. Adsorption of molecules changes the dielectric environment of surfaces. Experimental measurements of the Goos-Hänchen effect for SPP for

*θ*

_{0}>

*θ*

_{c}, i.e., at grazing incidence, depending on molecular coverage may prove useful in sensing this change and thereby allowing drawing conclusions on adsorption or desorption processes, complementing techniques such as surface plasmon resonance spectroscopy.

*θ*

_{c}≃ 75° for total internal reflection of the SPP, lateral displacements of several times the wavelength of the incident beam occur. The sensitivity of the displacement to changes of the surface optical properties may be exploited to measure, for instance, the modification of the dielectric environment of the metal upon molecular adsorption.

## Acknowledgments

## References and links

1. | F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. |

2. | K. Artmann, “Berechnung der Seitenversetzung des totalreflektierten Strahles,” Ann. Phys. |

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. |

4. | H. Lezec, J. Dionne, and H. Atwater, “Negative refraction at visible frequencies,” Science |

5. | M. Dennis, N. Zheludev, and F. Garcia de Abajo, “The plasmon Talbot effect,” Opt. Express |

6. | A. Maradudin and T. Leskova, “The Talbot effect for a surface plasmon polariton,” New J. Phys. |

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. |

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. |

9. | Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. |

10. | P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Transformation optics for plasmonics,” Nano Lett. |

11. | J. Renger, M. Kadic, G. Dupont, S. Acimovic, S. Guenneau, and R. Quidant, “Hidden progress: broadband plasmonic invisibility,” Opt. Express |

12. | R. Zia and M. L. Brongersma, “Surface plasmon polariton analogue to young’s double-slit experiment,” Nat. Nanotechnol. |

13. | A. A. Maradudin, “The impedance boundary condition at a two-dimensional rough metal surface,” Optics Commun. |

14. | K. Atkinson, “The numerical solution of Fredholm integral equations of the second kind with singular kernels,” Numerische Mathematik |

15. | F. Huerkamp, T. A. Leskova, and A. A. Maradudin, “Surface plasmon polariton analogues of volume electromagnetic wave effects,” Proc. SPIE |

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 |

**OCIS Codes**

(240.5420) Optics at surfaces : Polaritons

(240.6680) Optics at surfaces : Surface plasmons

(290.5825) Scattering : Scattering theory

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: May 24, 2011

Revised Manuscript: June 21, 2011

Manuscript Accepted: June 28, 2011

Published: July 28, 2011

**Citation**

Felix Huerkamp, Tamara A. Leskova, Alexei A. Maradudin, and Björn Baumeier, "The Goos-Hänchen effect for surface plasmon polaritons," Opt. Express **19**, 15483-15489 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-16-15483

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### References

- 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]
- 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]
- H. Lezec, J. Dionne, and H. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007). [CrossRef] [PubMed]
- M. Dennis, N. Zheludev, and F. Garcia de Abajo, “The plasmon Talbot effect,” Opt. Express 15, 9692–9700 (2007). [CrossRef] [PubMed]
- A. Maradudin and T. Leskova, “The Talbot effect for a surface plasmon polariton,” New J. Phys. 11, 033004 (2009). [CrossRef]
- 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]
- 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]
- 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]
- R. Zia and M. L. Brongersma, “Surface plasmon polariton analogue to young’s double-slit experiment,” Nat. Nanotechnol. 2, 426–429 (2007). [CrossRef]
- A. A. Maradudin, “The impedance boundary condition at a two-dimensional rough metal surface,” Optics Commun. 116, 452 – 467 (1995). [CrossRef]
- K. Atkinson, “The numerical solution of Fredholm integral equations of the second kind with singular kernels,” Numerische Mathematik 19, 248–259 (1972). [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]
- 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).

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