## Propagation-dependent beam profile distortion associated with the Goos-Hanchen shift

Optics Express, Vol. 17, Issue 23, pp. 21313-21319 (2009)

http://dx.doi.org/10.1364/OE.17.021313

Acrobat PDF (190 KB)

### Abstract

The propagation-dependent profile distortion of the reflected beam is studied via deriving the theoretical model of the optical field distribution in both the near and far field. It is shown that strong and fast-varying beam distortions can occur along the propagation path, compared to the profile on the reflecting surface. Numerical simulations for the case of a typical SPR configuration with a sharp angular response curve reveal that, when the phase distribution in the angular range covered by the input beam becomes nonlinear, previous theories based on the linear phase approximation fail to predict the Goos-Hanchen shift and its propagation-dependent variations precisely. Our study could shed light on more accurate modeling of the Goos-Hanchen effect’s impact on the relevant photonic devices and measurement applications.

© 2009 OSA

## 1. Introduction

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

4. T. Tamir, “Nonspecular phenomena in beam fields reflected by multilayered media,” J. Opt. Soc. Am. A **3**(4), 558–565 (
1986). [CrossRef]

5. K. Artmann, “Berechnung der Seitenversetzung des totalreflektieren Strahles,” Ann. Phys. **6**, 87–102 (
1948). [CrossRef]

6. H. M. Lai and S. W. Chan, “Large and negative Goos-Hänchen shift near the Brewster dip on reflection from weakly absorbing media,” Opt. Lett. **27**(9), 680–682 (
2002). [CrossRef] [PubMed]

7. I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, “Giant Goos-Hanchen effect at the reflection from left-handed metamaterials,” Appl. Phys. Lett. **83**(13), 2713–2715 (
2003). [CrossRef]

8. L.-G. Wang and S.-Y. Zhu, “Large positive and negative Goos-Hanchen shifts from a weakly absorbing left-handed slab,” J. Appl. Phys. **98**(4), 043522–043524 (
2005). [CrossRef]

9. X. B. Yin, L. Hesselink, Z. W. Liu, N. Fang, and X. Zhang, “Large positive and negative lateral optical beam displacements due to surface plasmon resonance,” Appl. Phys. Lett. **85**(3), 372–374 (
2004). [CrossRef]

9. X. B. Yin, L. Hesselink, Z. W. Liu, N. Fang, and X. Zhang, “Large positive and negative lateral optical beam displacements due to surface plasmon resonance,” Appl. Phys. Lett. **85**(3), 372–374 (
2004). [CrossRef]

10. L. Chen, Z. Q. Cao, F. Ou, H. G. Li, Q. S. Shen, and H. C. Qiao, “Observation of large positive and negative lateral shifts of a reflected beam from symmetrical metal-cladding waveguides,” Opt. Lett. **32**(11), 1432–1434 (
2007). [CrossRef] [PubMed]

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

13. Y. Wang, H. Li, Z. Cao, T. Yu, Q. Shen, and Y. He, “Oscillating wave sensor based on the Goos-Hanchen effect,” Appl. Phys. Lett. **92**(6), 061117 (
2008). [CrossRef]

*reflection plane*in the theoretical studies, and the GH shift is quantitatively evaluated as the change in the peak position [14

14. C. F. Li and Q. Wang, “Prediction of simultaneously large and opposite generalized Goos-Hänchen shifts for TE and TM light beams in an asymmetric double-prism configuration,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **69**(5), 055601 (
2004). [CrossRef] [PubMed]

15. A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hänchen and Imbert-Fedorov shifts,” Opt. Lett. **33**(13), 1437–1439 (
2008). [CrossRef] [PubMed]

15. A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hänchen and Imbert-Fedorov shifts,” Opt. Lett. **33**(13), 1437–1439 (
2008). [CrossRef] [PubMed]

16. K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature **450**(7168), 397–401 (
2007). [CrossRef] [PubMed]

17. K. Johansen, R. Stalberg, I. Lundstrom, and B. Liedberg, “Surface plasmon resonance: instrumental resolution using photo diode arrays,” Meas. Sci. Technol. **11**(11), 1630–1638 (
2000). [CrossRef]

*profile*of the reflected focused Gaussian beam associated with the GH effect using the diffraction theory. Taking an SPR configuration as an example, our results show that strong and fast-varying distortions not yet fully elucidated in previous studies can occur in the near-field region mostly caused by the nonlinear phase shift introduced by the reflection, while in far field, the beam distribution is mainly modulated by the intensity reflectance as people would commonly observe.

## 2. Theoretical model

*x*,

_{i}*z*) for the input Gaussian beam focused at the

_{i}*z*=0 plane (

_{i}*Bi*plane), the interface coordinate (

*x*,

_{o}*z*) in which the

_{o}*z*=0 plane (

_{o}*Bo*plane) is the physical reflecting surface, and the reflected coordinate (

*x*,

_{r}*z*) for the reflected beam [18

_{r}18. T. Tamir and H. L. Bertoni, “Lateral Displacement of Optical Beams at Multilayered and Periodic Structures,” J. Opt. Soc. Am. **61**(10), 1397–1413 (
1971). [CrossRef]

*Bi*plane with the beam-waist radius

*w*is given by:From Eq. (1), following an approach similar to that in [19

_{0}19. B. R. Horowitz and T. Tamir, “Lateral Displacement of a Light Beam at a Dielectric Interface,” J. Opt. Soc. Am. **61**(5), 586–594 (
1971). [CrossRef]

*Br*plane (

*z*=0) can be found via coordinate transformations and Fourier transforms as:where

_{r}*θ*is the incident angle,

*k*is the wavenumber in the incident medium,

*r*(

*k*) is the angular-dependent, complex reflectivity,

_{x}*k*and

_{x}*k*are the wavenumbers in the

_{z}*x*and

_{o}*z*directions respectively, and

_{o}*k*=

_{x}^{2}+k_{z}^{2}*k*.

^{2}*k*can be expanded at

_{z}*k*as:

_{x}=ksinθ*k*. Our studies show that for the cases examined in this paper the above 1st-order Taylor series expansion is accurate enough. Then Eq. (2) can be rewritten as:As the field distribution on the

_{z}=kcosθ -(k_{x}-ksinθ)tanθ*Br*plane is obtained, the distribution at the observation plane

*Cr*after propagating a distance of

*Z*can be calculated by applying the diffraction theory:where

*x*in the

_{r}*Br*plane to an observation point

*x*in the

*Cr*plane. For simplicity, we use

*E(x)*in place of

*E*in the following derivations.

_{Cr}(x,Z)*Near-field approximation*: Under the Fresnel approximation, the observed field distribution

*E(x)*, which we use in place of

*E*in the following derivations, is written as:By denoting the normalized deviation of the incident

_{Cr}(x,Z)*k*vector

_{x}*σ=(k*and substituting Eq. (3) into Eq. (5),

_{x}-ksinθ)/(kcosθ)*E(x)*can be written as:As the second integral on the right-hand side can be integrated over

*x*and leaves only

_{r}*k*term

_{x}*Far-field approximation*: Under the Fraunhofer approximation, the

*exp[ikx*term in Eq. (5) is negligible, and the integral term is similar to the Fourier transform of

_{r}^{2}/(2Z)]*E*0

_{Br}(x_{r},*)*. From Eq. (3), we get:where

*u=kx*. That is the inversed Fourier transform from the spatial angular variable

_{r}*σ*to

*u*. Substituting Eq. (8) into Eq. (5),

*E(x)*can be written as the Fourier transform of

*E*from the variable

_{Br}(u,0)*u*to the spatial angular variable

*x/Z*: As

*σ=x/Z*is approximated in the far field, the field distribution along

*Cr*plane is written as:where

*Z*is

*w*

_{z}=2Z/kw_{0}in the far-field area.

*Z*, and can cause strong and fast-varying distortions.

## 3. Results and discussions

9. X. B. Yin, L. Hesselink, Z. W. Liu, N. Fang, and X. Zhang, “Large positive and negative lateral optical beam displacements due to surface plasmon resonance,” Appl. Phys. Lett. **85**(3), 372–374 (
2004). [CrossRef]

_{1}= 2.295), and a 35nm thick layer of silver (ε

_{2}= −18+0.5i) is considered to be at the glass and air (ε

_{3}= 1) interface, which generates the SPR effect for the TM-polarized input beam. The optical wavelength is set at 633 nm and the incident angle is set around 42.99

^{◦}(the SPR angle). The reflectivity from the glass/silver/air interface is calculated using the Fresnel equations [20]. Figure 2(a) shows the calculated reflectivity (amplitude and phase) around the SPR angle. For a focused beam that covers a range of different

*k*vectors, the

_{x}*r(σ)*(i.e.

*r(k*) is calculated through the relationship

_{x})*θ=sin*.

^{−1}(k_{x}/k)*r(σ)*can also be written as

*R(σ)exp(iϕ(σ))*, where

*R(σ)*is the amplitude response and

*ϕ(σ)*is the phase response.

*Br*plane, as most theoretical work studying the GH shift assumed. For a quasi-collimated or loosely focused beam (

*w*

_{0}= 200 μm, beam divergence angle = 0.12°, or a larger beam), as shown in Fig. 2(a), the corresponding range of

*σ*is small. When

*ϕ(σ)*’s expanded into

*ϕ*, with

_{0}+ϕ_{1}σ+ϕ_{n}(σ)*ϕ*the first derivative of

_{1}*ϕ*and

*ϕ*the higher order term, the

_{n}(σ)*ϕ*term can be omitted. Thus, Eq. (3) becomes:It is the Fourier transform of the Gaussian function modulated by

_{n}(σ)*R(σ)*with a shift of

*ϕ*/

_{1}*k*, just as the classical theoretical model of the Goos-Hanchen shift predicts [5

5. K. Artmann, “Berechnung der Seitenversetzung des totalreflektieren Strahles,” Ann. Phys. **6**, 87–102 (
1948). [CrossRef]

*w*= 20 μm, beam divergence angle = 1.15°), the higher order terms of

_{0}*ϕ(σ)*over a much larger angular spectrum range become non-negligible. We found that their effect on the spatial distribution of the beam is significant. Figure 2(b) shows the field amplitude distribution of the reflected beam under different input beam sizes. For a quasi-collimated or loosely focused beam, the beam profile calculated using Eq. (3) agrees very well with the result calculated using Eq. (12) by omitting the higher order phase terms. While when the focus is relatively tight, the beam profile not only significantly deviates from the input beam shape, but also shows strong difference with the result calculated using Eq. (12), illustrating the effect of the nonlinear phase on the beam shape. Correspondingly, the centroid position of the reflected beam in the

*Br*plane, as well as the GH shift, diverges from the traditional theoretical prediction. Figure 3 shows that the traditional Artmann’s formula [5

5. K. Artmann, “Berechnung der Seitenversetzung des totalreflektieren Strahles,” Ann. Phys. **6**, 87–102 (
1948). [CrossRef]

14. C. F. Li and Q. Wang, “Prediction of simultaneously large and opposite generalized Goos-Hänchen shifts for TE and TM light beams in an asymmetric double-prism configuration,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **69**(5), 055601 (
2004). [CrossRef] [PubMed]

*Br*plane, the beam position and shape also changes. Figure 4 shows the calculated beam centroid position at different distances from the

*Br*plane under different incident angles. The propagation distance is normalized by the Fraunhofer distance

*Z*, defined by

_{F}*Z*= 180*

_{F}*w*

_{0}

^{2}/

*λ*. For a loosely focused beam, our simulated results based on Eq. (7) match well with the results obtained using the equations given in [15

15. A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hänchen and Imbert-Fedorov shifts,” Opt. Lett. **33**(13), 1437–1439 (
2008). [CrossRef] [PubMed]

*w*

_{0}=20μm, the magnitude and the trend of the GH shift change significantly as shown in Fig. 4(b). In contrast, the results based on previous studies are invariant to the beam size [15

**33**(13), 1437–1439 (
2008). [CrossRef] [PubMed]

*w*

_{0}= 200 μm), Fig. 5(a) shows that the shape of the beam hardly changes when it moves from the near field region into the far field region. To better illustrate the possible changes in the shape of the beam, the scale of the x axis is normalized by the beam size of a similar Gaussian beam at that distance in order to remove the effect of size expansion due to beam divergence. The more tightly focused beam experiences a much more significant change through the propagation, as shown in Fig. 5(b). Only after a certain distance (when approaching the far field area,

*Z*>

*Z*), the beam shape becomes stable and maintains that shape afterwards, and the steady beam profile matches what is described in Eq. (11) and is proportional to the angular-dependent reflectance. The strong distance-dependent distortion is caused by the interaction of the

_{F}*Z*-dependent phase term

*exp*(-

*ikZσ*/2) and the higher-order reflection phase terms of

^{2}*ϕ(σ)*.

**33**(13), 1437–1439 (
2008). [CrossRef] [PubMed]

*R*, e.g. the SPR angle in this case (42.99°). The result in Fig. 4 (a) is in line with that. Under that incident angle, when the beam is large, the reflected beam under the GH effect is just moving in parallel to that predicted by the geometric optics shifted by a fixed GH distance with little variation in its shape, as shown in Fig. 5 (a), just as many previous studies on GH effects envisioned. However, for the tightly focused beam, not only this angular position is shifted, to 43.06° in our case here, but the change in the beam shape is very large

*even*when the centroid position remains unchanged (see Fig. 5(c)).

## 4. Conclusions

## Acknowledgments

## References and links

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

2. | O. C. de Beauregard and C. Imbert, “Quantized Longitudinal and Transverse Shifts Associated with Total Internal Reflection,” Phys. Rev. Lett. |

3. | C. K. Carniglia and K. R. Brownstein, “Focal shift and ray model for total internal reflection,” J. Opt. Soc. Am. |

4. | T. Tamir, “Nonspecular phenomena in beam fields reflected by multilayered media,” J. Opt. Soc. Am. A |

5. | K. Artmann, “Berechnung der Seitenversetzung des totalreflektieren Strahles,” Ann. Phys. |

6. | H. M. Lai and S. W. Chan, “Large and negative Goos-Hänchen shift near the Brewster dip on reflection from weakly absorbing media,” Opt. Lett. |

7. | I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, “Giant Goos-Hanchen effect at the reflection from left-handed metamaterials,” Appl. Phys. Lett. |

8. | L.-G. Wang and S.-Y. Zhu, “Large positive and negative Goos-Hanchen shifts from a weakly absorbing left-handed slab,” J. Appl. Phys. |

9. | X. B. Yin, L. Hesselink, Z. W. Liu, N. Fang, and X. Zhang, “Large positive and negative lateral optical beam displacements due to surface plasmon resonance,” Appl. Phys. Lett. |

10. | L. Chen, Z. Q. Cao, F. Ou, H. G. Li, Q. S. Shen, and H. C. Qiao, “Observation of large positive and negative lateral shifts of a reflected beam from symmetrical metal-cladding waveguides,” Opt. Lett. |

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

12. | C. W. Chen, W. C. Lin, L. S. Liao, Z. H. Lin, H. P. Chiang, P. T. Leung, E. Sijercic, and W. S. Tse, “Optical temperature sensing based on the Goos-Hänchen effect,” Appl. Opt. |

13. | Y. Wang, H. Li, Z. Cao, T. Yu, Q. Shen, and Y. He, “Oscillating wave sensor based on the Goos-Hanchen effect,” Appl. Phys. Lett. |

14. | C. F. Li and Q. Wang, “Prediction of simultaneously large and opposite generalized Goos-Hänchen shifts for TE and TM light beams in an asymmetric double-prism configuration,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

15. | A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hänchen and Imbert-Fedorov shifts,” Opt. Lett. |

16. | K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature |

17. | K. Johansen, R. Stalberg, I. Lundstrom, and B. Liedberg, “Surface plasmon resonance: instrumental resolution using photo diode arrays,” Meas. Sci. Technol. |

18. | T. Tamir and H. L. Bertoni, “Lateral Displacement of Optical Beams at Multilayered and Periodic Structures,” J. Opt. Soc. Am. |

19. | B. R. Horowitz and T. Tamir, “Lateral Displacement of a Light Beam at a Dielectric Interface,” J. Opt. Soc. Am. |

20. | H. Raether, |

**OCIS Codes**

(240.0240) Optics at surfaces : Optics at surfaces

(240.6680) Optics at surfaces : Surface plasmons

(260.0260) Physical optics : Physical optics

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: August 19, 2009

Revised Manuscript: October 11, 2009

Manuscript Accepted: October 12, 2009

Published: November 6, 2009

**Citation**

Yuhang Wan, Zheng Zheng, and Jinsong Zhu, "Propagation-dependent beam profile distortion associated with the Goos-Hanchen shift," Opt. Express **17**, 21313-21319 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-23-21313

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### References

- F. Goos and H. Hanchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436(7-8), 333–346 (1947). [CrossRef]
- O. C. de Beauregard and C. Imbert, “Quantized Longitudinal and Transverse Shifts Associated with Total Internal Reflection,” Phys. Rev. Lett. 28(18), 1211–1213 (1972). [CrossRef]
- C. K. Carniglia and K. R. Brownstein, “Focal shift and ray model for total internal reflection,” J. Opt. Soc. Am. 67, 121–122 (1977). [CrossRef]
- T. Tamir, “Nonspecular phenomena in beam fields reflected by multilayered media,” J. Opt. Soc. Am. A 3(4), 558–565 (1986). [CrossRef]
- K. Artmann, “Berechnung der Seitenversetzung des totalreflektieren Strahles,” Ann. Phys. 6, 87–102 (1948). [CrossRef]
- H. M. Lai and S. W. Chan, “Large and negative Goos-Hänchen shift near the Brewster dip on reflection from weakly absorbing media,” Opt. Lett. 27(9), 680–682 (2002). [CrossRef] [PubMed]
- I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, “Giant Goos-Hanchen effect at the reflection from left-handed metamaterials,” Appl. Phys. Lett. 83(13), 2713–2715 (2003). [CrossRef]
- L.-G. Wang and S.-Y. Zhu, “Large positive and negative Goos-Hanchen shifts from a weakly absorbing left-handed slab,” J. Appl. Phys. 98(4), 043522–043524 (2005). [CrossRef]
- X. B. Yin, L. Hesselink, Z. W. Liu, N. Fang, and X. Zhang, “Large positive and negative lateral optical beam displacements due to surface plasmon resonance,” Appl. Phys. Lett. 85(3), 372–374 (2004). [CrossRef]
- L. Chen, Z. Q. Cao, F. Ou, H. G. Li, Q. S. Shen, and H. C. Qiao, “Observation of large positive and negative lateral shifts of a reflected beam from symmetrical metal-cladding waveguides,” Opt. Lett. 32(11), 1432–1434 (2007). [CrossRef] [PubMed]
- X. B. Yin and L. Hesselink, “Goos-Hanchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. 89(26), 261108 (2006). [CrossRef] [PubMed]
- C. W. Chen, W. C. Lin, L. S. Liao, Z. H. Lin, H. P. Chiang, P. T. Leung, E. Sijercic, and W. S. Tse, “Optical temperature sensing based on the Goos-Hänchen effect,” Appl. Opt. 46(22), 5347–5351 (2007). [CrossRef] [PubMed]
- Y. Wang, H. Li, Z. Cao, T. Yu, Q. Shen, and Y. He, “Oscillating wave sensor based on the Goos-Hanchen effect,” Appl. Phys. Lett. 92(6), 061117 (2008). [CrossRef]
- C. F. Li and Q. Wang, “Prediction of simultaneously large and opposite generalized Goos-Hänchen shifts for TE and TM light beams in an asymmetric double-prism configuration,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(5), 055601 (2004). [CrossRef] [PubMed]
- A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hänchen and Imbert-Fedorov shifts,” Opt. Lett. 33(13), 1437–1439 (2008). [CrossRef] [PubMed]
- K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450(7168), 397–401 (2007). [CrossRef] [PubMed]
- K. Johansen, R. Stalberg, I. Lundstrom, and B. Liedberg, “Surface plasmon resonance: instrumental resolution using photo diode arrays,” Meas. Sci. Technol. 11(11), 1630–1638 (2000). [CrossRef]
- T. Tamir and H. L. Bertoni, “Lateral Displacement of Optical Beams at Multilayered and Periodic Structures,” J. Opt. Soc. Am. 61(10), 1397–1413 (1971). [CrossRef]
- B. R. Horowitz and T. Tamir, “Lateral Displacement of a Light Beam at a Dielectric Interface,” J. Opt. Soc. Am. 61(5), 586–594 (1971). [CrossRef]
- H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

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