## Goos-Hänchen shifts in harmonic generation from metals |

Optics Express, Vol. 21, Issue 9, pp. 10878-10885 (2013)

http://dx.doi.org/10.1364/OE.21.010878

Acrobat PDF (1343 KB)

### Abstract

We present the first calculation of the Goos-Hänchen shifts in the context of the nonlinear generation of fields. We specifically concentrate on shifts of second harmonic generated at metallic surfaces. At metallic surfaces the second harmonic primarily arises from discontinuities of the field at surfaces which not only result in large harmonic generation but also in significant Goos-Hänchen shifts of the generated second harmonic. Our results can be extended to other shifts like angular shifts and Fedorov-Imbert shifts.

© 2013 OSA

## 1. Introduction

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

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

3. B. R. Horowitz and T. Tamir, “Lateral displacement of a light beam at a dielectric interface,” J. Opt. Soc. Am. **61**(5), 586 (1971). [CrossRef]

4. W. J. Wild and C. L. Giles, “Goos-Hänchen shifts from absorbing media,” Phys. Rev. A **25**(4), 2099–2101 (1982). [CrossRef]

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

5. H. G. L. Schwefel, W. Köhler, Z. H. Lu, J. Fan, and L. J. Wang, “Direct experimental observation of the single reflection optical Goos-Hänchen shift,” Opt. Lett. **33**(8), 794–796 (2008). [CrossRef] [PubMed]

8. M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics **3**(6), 337–340 (2009). [CrossRef]

5. H. G. L. Schwefel, W. Köhler, Z. H. Lu, J. Fan, and L. J. Wang, “Direct experimental observation of the single reflection optical Goos-Hänchen shift,” Opt. Lett. **33**(8), 794–796 (2008). [CrossRef] [PubMed]

4. W. J. Wild and C. L. Giles, “Goos-Hänchen shifts from absorbing media,” Phys. Rev. A **25**(4), 2099–2101 (1982). [CrossRef]

*p*polarized incident light [6

6. 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]

7. C. Bonnet, D. Chauvat, O. Emile, F. Bretenaker, A. Le Floch, and L. Dutriaux, “Measurement of positive and negative Goos-Hänchen effects for metallic gratings near Wood anomalies,” Opt. Lett. **26**(10), 666–668 (2001). [CrossRef] [PubMed]

9. J. He, J. Yi, and S. He, “Giant negative Goos-Hänchen shifts for a photonic crystal with a negative effective index,” Opt. Express **14**(7), 3024–3029 (2006). [CrossRef] [PubMed]

8. M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics **3**(6), 337–340 (2009). [CrossRef]

12. O. de Beauregard and C. Imbert, “Quantized longitudinal and transverse shifts associated with total internal reflection,” Phys. Rev. D Part. Fields **7**(12), 3555–3563 (1973). [CrossRef]

13. Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. **101**(4), 043903 (2008). [CrossRef] [PubMed]

16. D. Haefner, S. Sukhov, and A. Dogariu, “Spin Hall effect of light in spherical geometry,” Phys. Rev. Lett. **102**(12), 123903 (2009). [CrossRef] [PubMed]

16. D. Haefner, S. Sukhov, and A. Dogariu, “Spin Hall effect of light in spherical geometry,” Phys. Rev. Lett. **102**(12), 123903 (2009). [CrossRef] [PubMed]

18. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. **3**(2), 161 (2011). [CrossRef]

19. A. Dogariu and C. Schwartz, “Conservation of angular momentum of light in single scattering,” Opt. Express **14**(18), 8425–8433 (2006). [CrossRef] [PubMed]

20. N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. **128**(2), 606–622 (1962). [CrossRef]

21. S. S. Jha, “Theory of optical harmonic generation at a metal surface,” Phys. Rev. **140**(6A), A2020–A2030 (1965). [CrossRef]

22. N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. **174**(3), 813–822 (1968). [CrossRef]

22. N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. **174**(3), 813–822 (1968). [CrossRef]

23. P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B Condens. Matter **33**(12), 8254–8263 (1986). [CrossRef] [PubMed]

24. C. F. Li, “Unified theory for Goos-Haenchen and Imbert-Fedorov effects,” Phys. Rev. A **76**(1), 013811 (2007). [CrossRef]

## 2. GH shift for fundamental beam

*s*and

*p*component associated with it. Such a treatment is beneficial, as it correctly describes beams encountered experimentally. For example, in the geometry we have considered, say a polarizer is placed in the path of the incident beam that produces beam with its electric field vectors in the

*xz*plane throughout the extent of the beam. However, owing to the spread in wave vector space, this does not imply that the entire beam is

**or**

*s***polarized. The**

*p**s*and

*p*polarization directions depend on the in-plane component of the wave vector

**z**being the unit vector in the positive

*z*direction. In this basis, the electric field can be represented as,

*s*and

*p*field components using the Fresnel formulae for the air–metal interface. Our next step is to calculate the shift in the reflected beam. The reflected field at (

*z*= 0) is of the form,where

**ρ**= (

*x,y,*0). The transmitted electric field at (

*z*= 0) can also be expressed in the same way.

*s*and

*p*components along the lines of Eq. (4). To calculate the GH shift, we calculate the position of the x-coordinate of centroid of the reflected field. This can be expressed as the following integral.

*s*and

*p*components of the reflected electric field are calculated using the Fresnel formulae for the air-metal interface, and in the current notation, can be expressed as

*k*and

_{z0}*k*are the z-components of the propagation vector in air and the metal, respectively, calculated using Eq. (2). It may be noted that, k

_{zm}_{zm}can be purely imaginary for metals (ε is negative and large). As in the case of the incident field, the total reflected field is given by

_{pr}is the reflected u

_{p}component. For an incident field whose centroid lies at

**ρ**= (0,0,0), the GH shift is therefore,where we have not shown explicit dependence of quantities on

*k*

_{x}and

*k*

_{y}. To make a correspondence with the lateral beam shift calculated by Artmann, we use the quantity D = <x>cos(θ). It can be shown that the value of D calculated from Eq. (8), in the limit of slowly varying phase, reduces to the formula for the GH shift derived by Artmann. Figure 2 compares the GH-shift calculated using Eq. (8) with the Artmann’s formula for the two polarizations.

## 3. GH shift for second harmonic generated beam

20. N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. **128**(2), 606–622 (1962). [CrossRef]

21. S. S. Jha, “Theory of optical harmonic generation at a metal surface,” Phys. Rev. **140**(6A), A2020–A2030 (1965). [CrossRef]

25. G. S. Agarwal and S. S. Jha, “Surface-enhanced second-harmonic generation at a metallic grating,” Phys. Rev. B **26**(2), 482–496 (1982). [CrossRef]

26. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B **6**(12), 4370–4379 (1972). [CrossRef]

*x*– direction similar to the GH shift of the fundamental beam. For the purpose of this paper we will use the form of the second harmonic response at the metal surface derived by Agarwal and Jha [25

25. G. S. Agarwal and S. S. Jha, “Surface-enhanced second-harmonic generation at a metallic grating,” Phys. Rev. B **26**(2), 482–496 (1982). [CrossRef]

*ε*(

*2ω*) being the dielectric function of the metal at the second harmonic frequency

*2ω*. Since we need the shift of the beam in reflection, we shall compute the centroid of the reflected second harmonic field distribution. The reflected second harmonic field amplitudes in

*k*-space are given by,andThe vectors

**A**and

**B**, which determine the second harmonic response, are given by,

_{z0}and k

_{zm}are in air and the medium, respectively. The reflected field

*2k*. Equations (8) and (16) are the main results of our work, which allow us to treat the GH shift of the reflected fundamental beam, and the second harmonic beam in reflection within the same theoretical framework.

_{x}6. 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]

*s*polarized input is negligible [22

22. N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. **174**(3), 813–822 (1968). [CrossRef]

## 4. Discussion

26. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B **6**(12), 4370–4379 (1972). [CrossRef]

27. J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B Condens. Matter **35**(3), 1129–1141 (1987). [CrossRef] [PubMed]

29. B. S. Mendoza and W. L. Mochán, “Exactly solvable model of surface second-harmonic generation,” Phys. Rev. B Condens. Matter **53**(8), 4999–5006 (1996). [CrossRef] [PubMed]

30. X. Yin and L. Hesselink, “Goos-Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. **89**(26), 261108 (2006). [CrossRef]

31. M. Kumari and S. Dutta Gupta, “Positive and negative Giant Goos--Hänchen shift in a Near-symmetric layered medium for illumination from opposite ends,” Opt. Commun. **285**(5), 617–620 (2012). [CrossRef]

5. H. G. L. Schwefel, W. Köhler, Z. H. Lu, J. Fan, and L. J. Wang, “Direct experimental observation of the single reflection optical Goos-Hänchen shift,” Opt. Lett. **33**(8), 794–796 (2008). [CrossRef] [PubMed]

8. M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics **3**(6), 337–340 (2009). [CrossRef]

13. Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. **101**(4), 043903 (2008). [CrossRef] [PubMed]

16. D. Haefner, S. Sukhov, and A. Dogariu, “Spin Hall effect of light in spherical geometry,” Phys. Rev. Lett. **102**(12), 123903 (2009). [CrossRef] [PubMed]

## 5. Summary

## Acknowledgments

## References and links

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

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

3. | B. R. Horowitz and T. Tamir, “Lateral displacement of a light beam at a dielectric interface,” J. Opt. Soc. Am. |

4. | W. J. Wild and C. L. Giles, “Goos-Hänchen shifts from absorbing media,” Phys. Rev. A |

5. | H. G. L. Schwefel, W. Köhler, Z. H. Lu, J. Fan, and L. J. Wang, “Direct experimental observation of the single reflection optical Goos-Hänchen shift,” Opt. Lett. |

6. | 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 |

7. | C. Bonnet, D. Chauvat, O. Emile, F. Bretenaker, A. Le Floch, and L. Dutriaux, “Measurement of positive and negative Goos-Hänchen effects for metallic gratings near Wood anomalies,” Opt. Lett. |

8. | M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics |

9. | J. He, J. Yi, and S. He, “Giant negative Goos-Hänchen shifts for a photonic crystal with a negative effective index,” Opt. Express |

10. | F. I. Fedorov, “K teorii polnovo otrazenija,” Dokl. Akad. Nauk. SSR |

11. | C. Imbert, “Calculation and experimental proof of the transverse shift Induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D Part. Fields |

12. | O. de Beauregard and C. Imbert, “Quantized longitudinal and transverse shifts associated with total internal reflection,” Phys. Rev. D Part. Fields |

13. | Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. |

14. | O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science |

15. | N. Hermosa, A. M. Nugrowati, A. Aiello, and J. P. Woerdman, “Spin Hall effect of light in metallic reflection,” Opt. Lett. |

16. | D. Haefner, S. Sukhov, and A. Dogariu, “Spin Hall effect of light in spherical geometry,” Phys. Rev. Lett. |

17. | L. Allen, S. M. Barnett, and M. J. Padgett, eds., |

18. | A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. |

19. | A. Dogariu and C. Schwartz, “Conservation of angular momentum of light in single scattering,” Opt. Express |

20. | N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. |

21. | S. S. Jha, “Theory of optical harmonic generation at a metal surface,” Phys. Rev. |

22. | N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. |

23. | P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B Condens. Matter |

24. | C. F. Li, “Unified theory for Goos-Haenchen and Imbert-Fedorov effects,” Phys. Rev. A |

25. | G. S. Agarwal and S. S. Jha, “Surface-enhanced second-harmonic generation at a metallic grating,” Phys. Rev. B |

26. | P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

27. | J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B Condens. Matter |

28. | W. L. Schaich and B. S. Mendoza, “simple model of second-harmonic generation,” Phys. Rev. B Condens. Matter |

29. | B. S. Mendoza and W. L. Mochán, “Exactly solvable model of surface second-harmonic generation,” Phys. Rev. B Condens. Matter |

30. | X. Yin and L. Hesselink, “Goos-Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. |

31. | M. Kumari and S. Dutta Gupta, “Positive and negative Giant Goos--Hänchen shift in a Near-symmetric layered medium for illumination from opposite ends,” Opt. Commun. |

**OCIS Codes**

(240.4350) Optics at surfaces : Nonlinear optics at surfaces

(260.3910) Physical optics : Metal optics

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: March 20, 2013

Revised Manuscript: April 17, 2013

Manuscript Accepted: April 19, 2013

Published: April 26, 2013

**Citation**

V. J. Yallapragada, Achanta Venu Gopal, and G. S. Agarwal, "Goos-Hänchen shifts in harmonic generation from metals," Opt. Express **21**, 10878-10885 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-9-10878

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### References

- F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. (Berlin)436(7-8), 333–346 (1947). [CrossRef]
- K. Artmann, “Berechnung der Seitenversetzung des totalreflektierten Strahles,” Ann. Phys. (Berlin)437(1-2), 87–102 (1948). [CrossRef]
- B. R. Horowitz and T. Tamir, “Lateral displacement of a light beam at a dielectric interface,” J. Opt. Soc. Am.61(5), 586 (1971). [CrossRef]
- W. J. Wild and C. L. Giles, “Goos-Hänchen shifts from absorbing media,” Phys. Rev. A25(4), 2099–2101 (1982). [CrossRef]
- H. G. L. Schwefel, W. Köhler, Z. H. Lu, J. Fan, and L. J. Wang, “Direct experimental observation of the single reflection optical Goos-Hänchen shift,” Opt. Lett.33(8), 794–796 (2008). [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. Express15(24), 15928–15934 (2007). [CrossRef] [PubMed]
- C. Bonnet, D. Chauvat, O. Emile, F. Bretenaker, A. Le Floch, and L. Dutriaux, “Measurement of positive and negative Goos-Hänchen effects for metallic gratings near Wood anomalies,” Opt. Lett.26(10), 666–668 (2001). [CrossRef] [PubMed]
- M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics3(6), 337–340 (2009). [CrossRef]
- J. He, J. Yi, and S. He, “Giant negative Goos-Hänchen shifts for a photonic crystal with a negative effective index,” Opt. Express14(7), 3024–3029 (2006). [CrossRef] [PubMed]
- F. I. Fedorov, “K teorii polnovo otrazenija,” Dokl. Akad. Nauk. SSR105, 465 (1955).
- C. Imbert, “Calculation and experimental proof of the transverse shift Induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D Part. Fields5(4), 787–796 (1972). [CrossRef]
- O. de Beauregard and C. Imbert, “Quantized longitudinal and transverse shifts associated with total internal reflection,” Phys. Rev. D Part. Fields7(12), 3555–3563 (1973). [CrossRef]
- Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett.101(4), 043903 (2008). [CrossRef] [PubMed]
- O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science319(5864), 787–790 (2008). [CrossRef] [PubMed]
- N. Hermosa, A. M. Nugrowati, A. Aiello, and J. P. Woerdman, “Spin Hall effect of light in metallic reflection,” Opt. Lett.36(16), 3200–3202 (2011). [CrossRef] [PubMed]
- D. Haefner, S. Sukhov, and A. Dogariu, “Spin Hall effect of light in spherical geometry,” Phys. Rev. Lett.102(12), 123903 (2009). [CrossRef] [PubMed]
- L. Allen, S. M. Barnett, and M. J. Padgett, eds., Optical Angular Momentum (Taylor & Francis, 2003).
- A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon.3(2), 161 (2011). [CrossRef]
- A. Dogariu and C. Schwartz, “Conservation of angular momentum of light in single scattering,” Opt. Express14(18), 8425–8433 (2006). [CrossRef] [PubMed]
- N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev.128(2), 606–622 (1962). [CrossRef]
- S. S. Jha, “Theory of optical harmonic generation at a metal surface,” Phys. Rev.140(6A), A2020–A2030 (1965). [CrossRef]
- N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev.174(3), 813–822 (1968). [CrossRef]
- P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B Condens. Matter33(12), 8254–8263 (1986). [CrossRef] [PubMed]
- C. F. Li, “Unified theory for Goos-Haenchen and Imbert-Fedorov effects,” Phys. Rev. A76(1), 013811 (2007). [CrossRef]
- G. S. Agarwal and S. S. Jha, “Surface-enhanced second-harmonic generation at a metallic grating,” Phys. Rev. B26(2), 482–496 (1982). [CrossRef]
- P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972). [CrossRef]
- J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B Condens. Matter35(3), 1129–1141 (1987). [CrossRef] [PubMed]
- W. L. Schaich and B. S. Mendoza, “simple model of second-harmonic generation,” Phys. Rev. B Condens. Matter45(24), 14279–14292 (1992). [CrossRef] [PubMed]
- B. S. Mendoza and W. L. Mochán, “Exactly solvable model of surface second-harmonic generation,” Phys. Rev. B Condens. Matter53(8), 4999–5006 (1996). [CrossRef] [PubMed]
- X. Yin and L. Hesselink, “Goos-Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett.89(26), 261108 (2006). [CrossRef]
- M. Kumari and S. Dutta Gupta, “Positive and negative Giant Goos--Hänchen shift in a Near-symmetric layered medium for illumination from opposite ends,” Opt. Commun.285(5), 617–620 (2012). [CrossRef]

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