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

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    [CrossRef]
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    [CrossRef]
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2008

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]

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]

2007

2006

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

2005

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]

2004

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]

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]

2003

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]

2002

2000

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]

1986

1977

1972

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]

1971

1948

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

1947

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

Aiello, A.

Artmann, K.

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

Bertoni, H. L.

Boardman, A. D.

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

Brownstein, K. R.

Cao, Z.

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]

Cao, Z. Q.

Carniglia, C. K.

Chan, S. W.

Chen, C. W.

Chen, L.

Chiang, H. P.

de Beauregard, O. C.

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]

Fang, N.

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]

Goos, F.

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

Hanchen, H.

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

He, Y.

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]

Hess, O.

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

Hesselink, L.

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

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]

Horowitz, B. R.

Imbert, C.

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]

Johansen, K.

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]

Kivshar, Y. S.

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]

Lai, H. M.

Leung, P. T.

Li, C. F.

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]

Li, H.

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]

Li, H. G.

Liao, L. S.

Liedberg, B.

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]

Lin, W. C.

Lin, Z. H.

Liu, Z. W.

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]

Lundstrom, I.

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]

Ou, F.

Qiao, H. C.

Shadrivov, I. V.

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]

Shen, Q.

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]

Shen, Q. S.

Sijercic, E.

Stalberg, R.

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]

Tamir, T.

Tsakmakidis, K. L.

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

Tse, W. S.

Wang, L.-G.

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]

Wang, Q.

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]

Wang, Y.

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]

Woerdman, J. P.

Yin, X. B.

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

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]

Yu, T.

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]

Zhang, X.

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]

Zharov, A. A.

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]

Zhu, S.-Y.

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]

Ann. Phys.

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

Ann. Phys.

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

Appl. Opt.

Appl. Phys. Lett.

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]

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]

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

J. Appl. Phys.

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]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Meas. Sci. Technol.

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]

Nature

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

Opt. Lett.

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

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]

Phys. Rev. Lett.

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]

Other

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

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of the configuration under study

Fig. 2
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))

Fig. 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

Fig. 4
Fig. 4

The centroid of the reflected beam at different distances under different incident angles. In (a), solid line: results based on our Eq. (7); dashed line: results based on Eq. (7) in [15].

Fig. 5
Fig. 5

Propagation-dependent profile distortion of the focused reflected beam

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

E B i ( x i , 0 ) = exp [ ( x i / w 0 ) 2 ] / π w 0
E B r ( x r , 0 ) = 1 2 π cos θ + r ( k x ) exp [ ( k x k sin θ 2 cos θ w 0 ) 2 ] exp ( i k x x r cos θ i k z x r sin θ ) d k x
E B r ( x r , 0 ) = 1 2 π cos θ + r ( k x ) exp [ ( k x k sin θ 2 cos θ w 0 ) 2 ] exp [ i x r ( k x k sin θ cos θ ) ] d k x
E C r ( x , Z ) = Z i λ + E B r ( x r , 0 ) exp ( i k r x x r ) r x x r 2 d x r
E ( x ) = 1 i λ Z e i k Z e i k 2 Z x 2 + E B r ( x r , 0 ) exp ( i k 2 Z x r 2 ) exp ( i k Z x x r ) d x r
E ( x ) = 1 i λ Z e i k Z e i k 2 Z x 2 k 2 π + r ( σ ) exp [ ( k σ w 0 2 ) 2 ] d σ + exp ( i k x r σ ) exp ( i k 2 Z x r 2 ) exp ( i k Z x x r ) d x r
E ( x ) = C 0 2 π + r ( σ ) exp [ ( k σ w 0 2 ) 2 ] exp ( i k Z 2 σ 2 ) exp ( i k x σ ) d σ
E B r ( u , 0 ) = k F T 1 { r ( σ ) exp [ ( k σ w 0 / 2 ) 2 ] }
E ( x ) = 1 i k λ Z e i k Z e i k 2 Z x 2 + E B r ( u , 0 ) exp ( i x Z u ) d u
E ( x / Z ) F T [ E B r ( u , 0 ) ] F T ( F T 1 { r ( σ ) exp [ ( k σ w 0 / 2 ) 2 ] } ) r ( σ ) exp [ ( k σ w 0 / 2 ) 2 ]
E ( x ) = C 1 r ( x / Z ) exp [ ( x / w z ) 2
E B r ( x r , 0 ) = e i ϕ 0 k 2 π + R ( σ ) exp [ ( k σ w 0 2 ) 2 ] exp [ i k σ ( x r + ϕ 1 k ) ] d σ

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