## The characterization of GH shifts of surface plasmon resonance in a waveguide using the FDTD method

Optics Express, Vol. 17, Issue 23, pp. 20714-20720 (2009)

http://dx.doi.org/10.1364/OE.17.020714

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### Abstract

We have explicated the Goos-Hänchen (GH) shift in a μm-order Kretchmann-Raether configuration embedded in an optical waveguide structure by using the finite-difference time-domain method. For optical waveguide-type surface plasmon resonance (SPR) devices, the precise derivation of the GH shift has become critical. Artmann’s equation, which is accurate enough for bulk optics, is difficult to apply to waveguide-type SPR devices. This is because Artmann's equation, based on the differentiation of the phase shift, is inaccurate at the critical and resonance angles where drastic phase changes occur. In this study, we accurately identified both the positive and the negative GH shifts around the incidence angle of resonance. In a waveguide-type Kretchmann-Raether configuration with an Au thin film of 50 nm, positive and negative lateral shifts of −0.75 and + 1.0 μm are obtained on the SPR with the incident angles of 44.4 ° and 47.5 °, respectively, at a wavelength of 632.8 nm.

© 2009 OSA

## 1. Introduction

3. B. Liedberg, I. Lundstrom, and E. Stenberg, “Principles of biosensing with an extended coupling matrix and surface plasmon resonance,” Sens. Actuators B Chem. **11**(1-3), 63–72 (
1993). [CrossRef]

4. F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalreflexion,” Ann. Phys. **436**(7-8), 333–346 (
1947). [CrossRef]

## 2. Theoretical analysis

### 1) The Goos-Hänchen shift

*D*, occurs in the positive lateral direction at the interface of the dielectric materials. The GH shift, as proposed by Artmann, is seen in the phase transition of the reflected light, and is given by [6

6. K. Artmann, “Berechnung der Seitenversetzung des totalreflektierten Strahles,” Ann. Phys. **437**(1-2), 87–102 (
1948). [CrossRef]

*δ*by p-polarized light is:The complex admittance of incident medium

*η*is:and the admittance of free space

### 2) The Drude model used for the FDTD analysis

7. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Propag. **14**(3), 302–307 (
1966). [CrossRef]

8. P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B **6**(12), 4370–4379 (
1972). [CrossRef]

9. S. K. Gray and T. Kupka, “Propagation of light in metallic nanowire arrays: Finite-difference time-domain studies of silver cylinders,” Phys. Rev. B **68**(4), 045415 (
2003). [CrossRef]

**J**is initially at zero in the regions of free space and having some dielectric material with a real dielectric constant. However,

## 3. The simulation results and a discussion

10. H. M. Lai, F. C. Cheng, and W. K. Tang, “Goos-Hänchen effect around and off the critical angle,” J. Opt. Soc. Am. A **3**(4), 550–557 (
1986). [CrossRef]

11. P. R. Berman, “Goos-Hänchen shift in negatively refractive media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **66**(6), 067603 (
2002). [CrossRef]

12. H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the goos-Hanchen effect,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **62**(55 Pt B), 7330–7339 (
2000). [CrossRef] [PubMed]

13. M. Merano, A. Aiello, G. W. ‘t Hooft, M. P. van Exter, E. R. Eliel, and J. P. Woerdman, “Observation of Goos-Hänchen shifts in metallic reflection,” Opt. Express **15**(24), 15928–15934 (
2007). [CrossRef] [PubMed]

## 4. Conclusions

## Acknowledgments

## References and links

1. | E. Kretschmann and H. Raether, “Radiative decay of non-radiative surface plasmons excited by light,” Z. Naturforsch. A |

2. | A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Zeitschrift fur Physik A Hadrons and Nuclei |

3. | B. Liedberg, I. Lundstrom, and E. Stenberg, “Principles of biosensing with an extended coupling matrix and surface plasmon resonance,” Sens. Actuators B Chem. |

4. | F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalreflexion,” Ann. Phys. |

5. | P. Drude, “Zur Elektronentheorie I/II,” Ann. Phys. |

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

7. | K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Propag. |

8. | P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B |

9. | S. K. Gray and T. Kupka, “Propagation of light in metallic nanowire arrays: Finite-difference time-domain studies of silver cylinders,” Phys. Rev. B |

10. | H. M. Lai, F. C. Cheng, and W. K. Tang, “Goos-Hänchen effect around and off the critical angle,” J. Opt. Soc. Am. A |

11. | P. R. Berman, “Goos-Hänchen shift in negatively refractive media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

12. | H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the goos-Hanchen effect,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

13. | M. Merano, A. Aiello, G. W. ‘t Hooft, M. P. van Exter, E. R. Eliel, and J. P. Woerdman, “Observation of Goos-Hänchen shifts in metallic reflection,” Opt. Express |

**OCIS Codes**

(130.3120) Integrated optics : Integrated optics devices

(240.4350) Optics at surfaces : Nonlinear optics at surfaces

(240.6680) Optics at surfaces : Surface plasmons

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: August 31, 2009

Revised Manuscript: October 1, 2009

Manuscript Accepted: October 1, 2009

Published: October 27, 2009

**Citation**

Geum-Yoon Oh, Doo Gun Kim, and Young-Wan Choi, "The characterization of GH shifts of surface plasmon resonance in a waveguide using the FDTD method," Opt. Express **17**, 20714-20720 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-23-20714

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### References

- E. Kretschmann and H. Raether, “Radiative decay of non-radiative surface plasmons excited by light,” Z. Naturforsch. A 23, 2135–2136 (1968).
- A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Zeitschrift fur Physik A Hadrons and Nuclei 216, 398–410 (1968).
- B. Liedberg, I. Lundstrom, and E. Stenberg, “Principles of biosensing with an extended coupling matrix and surface plasmon resonance,” Sens. Actuators B Chem. 11(1-3), 63–72 (1993). [CrossRef]
- F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalreflexion,” Ann. Phys. 436(7-8), 333–346 (1947). [CrossRef]
- P. Drude, “Zur Elektronentheorie I/II,” Ann. Phys. 3, 4 (1900).
- K. Artmann, “Berechnung der Seitenversetzung des totalreflektierten Strahles,” Ann. Phys. 437(1-2), 87–102 (1948). [CrossRef]
- K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Propag. 14(3), 302–307 (1966). [CrossRef]
- P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]
- S. K. Gray and T. Kupka, “Propagation of light in metallic nanowire arrays: Finite-difference time-domain studies of silver cylinders,” Phys. Rev. B 68(4), 045415 (2003). [CrossRef]
- H. M. Lai, F. C. Cheng, and W. K. Tang, “Goos-Hänchen effect around and off the critical angle,” J. Opt. Soc. Am. A 3(4), 550–557 (1986). [CrossRef]
- P. R. Berman, “Goos-Hänchen shift in negatively refractive media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(6), 067603 (2002). [CrossRef]
- H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the goos-Hanchen effect,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(55 Pt B), 7330–7339 (2000). [CrossRef] [PubMed]
- M. Merano, A. Aiello, G. W. ‘t Hooft, M. P. van Exter, E. R. Eliel, and J. P. Woerdman, “Observation of Goos-Hänchen shifts in metallic reflection,” Opt. Express 15(24), 15928–15934 (2007). [CrossRef] [PubMed]

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