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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 23 — Nov. 9, 2009
  • pp: 21313–21319
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Propagation-dependent beam profile distortion associated with the Goos-Hanchen shift

Yuhang Wan, Zheng Zheng, and Jinsong Zhu  »View Author Affiliations


Optics Express, Vol. 17, Issue 23, pp. 21313-21319 (2009)
http://dx.doi.org/10.1364/OE.17.021313


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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

The nonspecular reflection phenomena of a bounded light beam from a flat surface, where the actual reflected beam deviates from the path predicted by the ray optics with lateral/focal/angular shifts and/or beam waist modifications, have long been studied [1

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

4

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

]. Among them, the lateral shift, named Goos-Hanchen (GH) shift, is the most widely studied one. Artmann first theoretically identified the cause of the GH effect as the different phases experienced by different spatial frequency components of the beam using the stationary-phase approach [5

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

]. While the GH shift is negligible under common conditions, various configurations using absorbing media [6

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]

], left-handed metamaterial [7

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

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]

] and surface plasmon resonance (SPR) devices [9

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]

] could hugely enhance the effect. Many experimental demonstrations [9

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

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]

] and applications [11

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

13

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]

] of the enhanced GH effect have ensued.

Traditionally, the beam profile affected by the angular-dependent reflectivity is analyzed at the 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]

] or the centroid [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]

] of the profile. In contrast, in the experiments, this displacement is measured by position sensitive detectors placed in the far field, whose output is proportional to the weighted average position of the beam. More recently, the role of beam propagation in the GH phenomenon has attracted more attention. Theoretical studies reveal that the GH centroid shift could vary significantly at different distances from the reflection plane, and the propagation-dependent GH shift variation is derived based on the second-order truncated functional expansion of the reflectivity [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]

].

With the emerging of novel micro-photonic devices leveraging the GH effect [16

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]

], understanding the impact of the GH effect on the propagating wavefront, instead of just its averaged position, becomes increasingly important. Also, in the applications with potentially large GH effect, such as SPR sensing systems [17

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]

], accurately measuring and interpreting the beam shape reflected from a surface becomes essential.

2. Theoretical model

The schematic diagram of a focused beam incident upon a reflecting interface whose reflectivity can give rise to significant GH shift is shown in Fig. 1
Fig. 1 Schematic diagram of the configuration under study
. Three different coordinate systems are employed in our model: the incident coordinate (xi,zi) for the input Gaussian beam focused at the zi=0 plane (Bi plane), the interface coordinate (xo,zo) in which the zo=0 plane (Bo plane) is the physical reflecting surface, and the reflected coordinate (xr,zr) for the reflected beam [18

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]

]. Two-dimensional coordinates are assumed for simplicity.

The optical field distribution along the Bi plane with the beam-waist radius w0 is given by:
EBi(xi,0)=exp[(xi/w0)2]/πw0
(1)
From Eq. (1), following an approach similar to that in [19

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]

], the field distribution along the Br plane (zr=0) can be found via coordinate transformations and Fourier transforms as:
EBr(xr,0)=12πcosθ+r(kx)exp[(kxksinθ2cosθw0)2]exp(ikxxrcosθikzxrsinθ)dkx
(2)
where θ is the incident angle, k is the wavenumber in the incident medium, r(kx) is the angular-dependent, complex reflectivity, kx and kz are the wavenumbers in the xo and zo directions respectively, and kx2+kz2=k2.

As Eq. (2) cannot be integrated directly, further simplification is needed for later analytical analysis. Under the paraxial approximation, kz can be expanded at kx=ksinθ as: kz=kcosθ -(kx-ksinθ)tanθ. 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:
EBr(xr,0)=12πcosθ+r(kx)exp[(kxksinθ2cosθw0)2]exp[ixr(kxksinθcosθ)]dkx
(3)
As the field distribution on the 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:
ECr(x,Z)=Ziλ+EBr(xr,0)exp(ikrxxr)rxxr2dxr
(4)
where rxxris the distance from a diffraction point xr in the Br plane to an observation point x in the Cr plane. For simplicity, we use E(x) in place of ECr(x,Z) in the following derivations.

Near-field approximation: Under the Fresnel approximation, the observed field distribution E(x), which we use in place of ECr(x,Z) in the following derivations, is written as:
E(x)=1iλZeikZeik2Zx2+EBr(xr,0)exp(ik2Zxr2)exp(ikZxxr)dxr
(5)
By denoting the normalized deviation of the incident kx vector σ=(kx-ksinθ)/(kcosθ) and substituting Eq. (3) into Eq. (5), E(x) can be written as:
E(x)=1iλZeikZeik2Zx2k2π+r(σ)exp[(kσw02)2]dσ+exp(ikxrσ)exp(ik2Zxr2)exp(ikZxxr)dxr
(6)
As the second integral on the right-hand side can be integrated over xr and leaves only kx term 2πZ/keiπ/4exp[ik(Zσx)2/(2Z)], by denotingC0=k2/2πkZeiπ/4eikZ, we get:
E(x)=C02π+r(σ)exp[(kσw02)2]exp(ikZ2σ2)exp(ikxσ)dσ
(7)
Far-field approximation: Under the Fraunhofer approximation, the exp[ikxr2/(2Z)] term in Eq. (5) is negligible, and the integral term is similar to the Fourier transform of EBr(xr,0). From Eq. (3), we get:
EBr(u,0)=kFT1{r(σ)exp[(kσw0/2)2]}
(8)
where u=kxr. That is the inversed Fourier transform from the spatial angular variable σ to u. Substituting Eq. (8) into Eq. (5), E(x) can be written as the Fourier transform of EBr(u,0) from the variable u to the spatial angular variable x/Z:
E(x)=1ikλZeikZeik2Zx2+EBr(u,0)exp(ixZu)du
(9)
E(x/Z)FT[EBr(u,0)]FT(FT1{r(σ)exp[(kσw0/2)2]})r(σ)exp[(kσw0/2)2]
(10)
As σ=x/Z is approximated in the far field, the field distribution along Cr plane is written as:
E(x)=C1r(x/Z)exp[(x/wz)2
(11)
where C1=i/λZeikZeikx2/(2Z), and the waist radius at Z is wz=2Z/kw 0 in the far-field area.

From the above approximations, we can see from Eq. (11) that, in the far field, the intensity profile of the reflected beam is directly modulated by the angular-dependent reflectance as people normally observe in the experiments. While in near field, however, the profile of the reflected beam is determined by both the intensity and the phase term of the reflectivity as shown in Eq. (7), where the phase term of the integral varies fast at different Z, and can cause strong and fast-varying distortions.

3. Results and discussions

To further investigate the phenomenon, we apply the above theoretical model to an SPR configuration, where SPR with a sharp spectral response results in strongly enhanced GH effect [9

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]

]. The medium that the beam propagates in is assumed to be the BK7 glass (ε1 = 2.295), and a 35nm thick layer of silver (ε2 = −18+0.5i) is considered to be at the glass and air (ε3 = 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

20. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

]. Figure 2(a)
Fig. 2 (a) Reflectivity under different incident angles; (b)The field distribution in the Br plane with different focused beams at the incident angle of 42.99° (soild line: calculated by Eq. (3), dashed line: calculated by Eq. (12))
shows the calculated reflectivity (amplitude and phase) around the SPR angle. For a focused beam that covers a range of different kx vectors, the r(σ) (i.e. r(kx)) is calculated through the relationship θ=sin−1(kx/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.

First, we consider the beam profile observed at the 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 ϕ01σ+ϕn(σ), with ϕ1 the first derivative of ϕ and ϕn(σ) the higher order term, the ϕn(σ) term can be omitted. Thus, Eq. (3) becomes:
EBr(xr,0)=eiϕ0k2π+R(σ)exp[(kσw02)2]exp[ikσ(xr+ϕ1k)]dσ
(12)
It is the Fourier transform of the Gaussian function modulated by R(σ)with a shift of ϕ1/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]

].

However, for a relatively tightly focused beam (such as w0 = 20 μm, beam divergence angle = 1.15°), the higher order terms of 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
Fig. 3 Goos-Hanchen centroid shift in the Br plane under different incident angles for different input beam sizes. Solid line: calculated with the Artmann’s formula
shows that the traditional Artmann’s formula [5

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

] accurately predicts the GH shift under different incident angles when the beam size is relatively large but fails to match the results under a tightly focused beam, similar to what had been reported in [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]

].

The difference in the propagation-dependent beam profile variations is even more profound. With the loosely focused beam (w 0 = 200 μm), Fig. 5(a)
Fig. 5 Propagation-dependent profile distortion of the focused reflected beam
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 > ZF), 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 Z-dependent phase term exp(-ikZσ2/2) and the higher-order reflection phase terms of ϕ(σ).

We also note that the phenomenon we discussed here is not limited to the SPR effect and could happen in other configurations of GH effect, where either the beam covers a large angular spectrum or the phase change is steep and, thus, highly nonlinear. For applications like high-resolution SPR array sensing, where the measurement of the reflected beam profile is used to re-construct the angular-dependent reflectivity, the distortions and their spatial variations might cause uncertainties in the measurement results if their presence is ignored, especially in a compact or integrated system.

4. Conclusions

By deriving the theoretical model governing the optical field distribution of a propagating reflected beam from a flat surface, we show that the beam profile can strongly deviate from the previously widely-studied shape at the reflecting plane in the near-field, before it converges to a profile decided by the intensity reflectance in the far-field. Our study shows that this fast-varying, propagation-dependent profile distortion cannot be accurately predicted by theories using only the lower-order terms of the reflectivity. Our results offer useful guidelines for future studies on devices and experiments with strong GH effects.

Acknowledgments

The work at Beihang University was supported by NSFC (60877054/60921001), 973 Program (2009CB930701), and the Innovation Foundation of BUAA for PhD Graduates.

References and links

1.

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

2.

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]

3.

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]

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]

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]

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. 46(22), 5347–5351 ( 2007). [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]

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]

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]

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]

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]

20.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

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

  1. F. Goos and H. Hanchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436(7-8), 333–346 (1947). [CrossRef]
  2. 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]
  3. 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]
  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]
  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]
  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. 46(22), 5347–5351 (2007). [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]
  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]
  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]
  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]
  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]
  20. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

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