Abstract

We report a unifying approach to the Goos-Hänchen (GH) shifts on external optical reflection for metals and dielectrics in particular for the case of high losses, that is for a large imaginary part of the dielectric constant. In this regime metals and dielectrics have a similar GH shift which is in contrast to the low-loss regime where the metallic and dielectric forms of the GH shift are very different. When going from the low-loss to the high-loss regime we find that metals show a much more prominent transition; we present a condition on the dielectric constant which characterizes this transition. We illustrate our theoretical analysis with a realistic example of seven lossy materials.

© 2008 Optical Society of America

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References

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  1. F. Goos and H. Hanchen, "Ein neuer und fundamentaler Versuch zur Totalreflexion," Annalen der Physik 436, 333-346 (1947).
    [CrossRef]
  2. W. J. Wild and C. L. Giles, "Goos-H¨anchen shifts from absorbing media," Phys. Rev. A 25, 2099-2101 (1982)http://link.aps.org/abstract/PRA/v25/p2099.
    [CrossRef]
  3. H. M. Lai and S. W. Chan, "Large and negative Goos-Hanchen shift near the Brewster dip on reflection from weakly absorbing media," Opt. Lett. 27, 680-682 (2002)http://www.opticsinfobase.org/abstract.cfm?URI=ol-27-9-680.
    [CrossRef]
  4. H. M. Lai, S. W. Chan, and W. H. Wong, "Nonspecular effects on reflection from absorbing media at and around Brewster’s dip," J. Opt. Soc. Am. A 23, 3208-3216 (2006)http://www.opticsinfobase.org/abstract.cfm?URI=josaa-23-12-3208.
    [CrossRef]
  5. D. Felbacq, A. Moreau, and R. Smaali, "Goos- Hanchen effect in the gaps of photonic crystals," Opt. Lett. 28, 1633-1635 (2003)http://www.opticsinfobase.org/abstract.cfm?URI=ol-28-18-1633.
    [CrossRef] [PubMed]
  6. N.-H. Shen, J. Chen, Q.-Y. Wu, T. Lan, Y.-X. Fan, and H.-T. Wang, "Large lateral shift near pseudo-Brewster angle on reflection from a weakly absorbing double negative medium," Opt. Express 14, 10,574-10,579 (2006). URL http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-22-10574.
    [CrossRef]
  7. T. Tamir and H. L. Bertoni, "Lateral Displacement of Optical Beams at Multilayered and Periodic Structures," J. Opt. Soc. Am. 61, 1397-1413 (1971)http://www.opticsinfobase.org/abstract.cfm?URI=josa-61-10-1397.
    [CrossRef]
  8. Y. Xiang, X. Dai, and S. Wen, "Negative and positive Goos-Hanchen shifts of a light beam transmitted from an indefinite medium slab," Appl. Phys. A 87, 285-290 (2007).
    [CrossRef]
  9. P. T. Leung, C. W. Chen, and H.-P. Chiang, "Large negative Goos-Hanchen shift at metal surfaces," Opt. Commum. 276, 206-208 (2007)http://dx.doi.org/10.1016/j.optcom.2007.04.019.
    [CrossRef]
  10. M. Merano, A. Aiello, G. W. ’t Hooft, M. P. van Exter, E. R. Eliel, and J. P. Woerdman, "Observation of Goos-H¨anchen shifts in metallic reflection," Opt. Express 15, 15,928-15,934 (2007)http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-24-15928.
    [CrossRef]
  11. P. T. Leung, C. W. Chen, and H.-P. Chiang, "Addendum to ‘Large negative Goos-H¨anchen shift at metal surfaces’," Opt. Commun. 281, 1312-1313 (2008)http://dx.doi.org/10.1016/j.optcom.2007.11.061.
    [CrossRef]
  12. M. Dressel and G. Gruner, Electrodynamics of Solids (Cambridge University Press, Cambridge, 2002).
    [CrossRef]
  13. H. Wolter, "Untersuchungen zur Strahlversetzung bei Totalreflexion des Lichtes mit der Methode der Minimumstrahlkennzeichnung," Zeitschrift fur Naturforschung 5a, 143-153 (1950).
  14. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic Press, San Diego, 1985).
  15. K. Artmann, "Berechnung der Seitenversetzung des totalreflektierten Strahles," Annalen der Physik 437, 87-102 (1948).
    [CrossRef]
  16. M. McGuirk and C. K. Carniglia, "An angular spectrum representation approach to the Goos-Hanchen shift," J. Opt. Soc. Am. 67, 103-107 (1977)http://www.opticsinfobase.org/abstract.cfm?URI=josa-67-1-103.
    [CrossRef]
  17. A. Aiello and J. P. Woerdman, "The reflection of a Maxwell-Gaussian beam by a planar surface," (2007). Posted to arXiv physics/0710.1643,http://arxiv.org/abs/0710.1643.
  18. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, Cambridge, 1999). Reprinted 2003.
  19. P. B. Johnson and R. W. Christy, "Optical constants of transition metals: Ti, V, Cr, Mn, Fe, Co, Ni and Pd," Phys. Rev. B 9, 5056-5070 (1974)http://link.aps.org/abstract/PRB/v9/p5056.
    [CrossRef]
  20. D. Han, X. Li, F. Wu, X. Liu, and J. Zi, "Enhanced transmission of optically thick metallic films at infrared wavelengths," Appl. Phys. Lett. 88, 161110 (2006)http://link.aip.org/link/?APPLAB/88/161110/1.
    [CrossRef]

2007 (1)

Y. Xiang, X. Dai, and S. Wen, "Negative and positive Goos-Hanchen shifts of a light beam transmitted from an indefinite medium slab," Appl. Phys. A 87, 285-290 (2007).
[CrossRef]

1950 (1)

H. Wolter, "Untersuchungen zur Strahlversetzung bei Totalreflexion des Lichtes mit der Methode der Minimumstrahlkennzeichnung," Zeitschrift fur Naturforschung 5a, 143-153 (1950).

1948 (1)

K. Artmann, "Berechnung der Seitenversetzung des totalreflektierten Strahles," Annalen der Physik 437, 87-102 (1948).
[CrossRef]

1947 (1)

F. Goos and H. Hanchen, "Ein neuer und fundamentaler Versuch zur Totalreflexion," Annalen der Physik 436, 333-346 (1947).
[CrossRef]

Artmann, K.

K. Artmann, "Berechnung der Seitenversetzung des totalreflektierten Strahles," Annalen der Physik 437, 87-102 (1948).
[CrossRef]

Dai, X.

Y. Xiang, X. Dai, and S. Wen, "Negative and positive Goos-Hanchen shifts of a light beam transmitted from an indefinite medium slab," Appl. Phys. A 87, 285-290 (2007).
[CrossRef]

Goos, F.

F. Goos and H. Hanchen, "Ein neuer und fundamentaler Versuch zur Totalreflexion," Annalen der Physik 436, 333-346 (1947).
[CrossRef]

Hanchen, H.

F. Goos and H. Hanchen, "Ein neuer und fundamentaler Versuch zur Totalreflexion," Annalen der Physik 436, 333-346 (1947).
[CrossRef]

Wen, S.

Y. Xiang, X. Dai, and S. Wen, "Negative and positive Goos-Hanchen shifts of a light beam transmitted from an indefinite medium slab," Appl. Phys. A 87, 285-290 (2007).
[CrossRef]

Wolter, H.

H. Wolter, "Untersuchungen zur Strahlversetzung bei Totalreflexion des Lichtes mit der Methode der Minimumstrahlkennzeichnung," Zeitschrift fur Naturforschung 5a, 143-153 (1950).

Xiang, Y.

Y. Xiang, X. Dai, and S. Wen, "Negative and positive Goos-Hanchen shifts of a light beam transmitted from an indefinite medium slab," Appl. Phys. A 87, 285-290 (2007).
[CrossRef]

Annalen der Physik (2)

F. Goos and H. Hanchen, "Ein neuer und fundamentaler Versuch zur Totalreflexion," Annalen der Physik 436, 333-346 (1947).
[CrossRef]

K. Artmann, "Berechnung der Seitenversetzung des totalreflektierten Strahles," Annalen der Physik 437, 87-102 (1948).
[CrossRef]

Appl. Phys. A (1)

Y. Xiang, X. Dai, and S. Wen, "Negative and positive Goos-Hanchen shifts of a light beam transmitted from an indefinite medium slab," Appl. Phys. A 87, 285-290 (2007).
[CrossRef]

Zeitschrift fur Naturforschung (1)

H. Wolter, "Untersuchungen zur Strahlversetzung bei Totalreflexion des Lichtes mit der Methode der Minimumstrahlkennzeichnung," Zeitschrift fur Naturforschung 5a, 143-153 (1950).

Other (16)

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic Press, San Diego, 1985).

M. McGuirk and C. K. Carniglia, "An angular spectrum representation approach to the Goos-Hanchen shift," J. Opt. Soc. Am. 67, 103-107 (1977)http://www.opticsinfobase.org/abstract.cfm?URI=josa-67-1-103.
[CrossRef]

A. Aiello and J. P. Woerdman, "The reflection of a Maxwell-Gaussian beam by a planar surface," (2007). Posted to arXiv physics/0710.1643,http://arxiv.org/abs/0710.1643.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, Cambridge, 1999). Reprinted 2003.

P. B. Johnson and R. W. Christy, "Optical constants of transition metals: Ti, V, Cr, Mn, Fe, Co, Ni and Pd," Phys. Rev. B 9, 5056-5070 (1974)http://link.aps.org/abstract/PRB/v9/p5056.
[CrossRef]

D. Han, X. Li, F. Wu, X. Liu, and J. Zi, "Enhanced transmission of optically thick metallic films at infrared wavelengths," Appl. Phys. Lett. 88, 161110 (2006)http://link.aip.org/link/?APPLAB/88/161110/1.
[CrossRef]

P. T. Leung, C. W. Chen, and H.-P. Chiang, "Large negative Goos-Hanchen shift at metal surfaces," Opt. Commum. 276, 206-208 (2007)http://dx.doi.org/10.1016/j.optcom.2007.04.019.
[CrossRef]

M. Merano, A. Aiello, G. W. ’t Hooft, M. P. van Exter, E. R. Eliel, and J. P. Woerdman, "Observation of Goos-H¨anchen shifts in metallic reflection," Opt. Express 15, 15,928-15,934 (2007)http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-24-15928.
[CrossRef]

P. T. Leung, C. W. Chen, and H.-P. Chiang, "Addendum to ‘Large negative Goos-H¨anchen shift at metal surfaces’," Opt. Commun. 281, 1312-1313 (2008)http://dx.doi.org/10.1016/j.optcom.2007.11.061.
[CrossRef]

M. Dressel and G. Gruner, Electrodynamics of Solids (Cambridge University Press, Cambridge, 2002).
[CrossRef]

W. J. Wild and C. L. Giles, "Goos-H¨anchen shifts from absorbing media," Phys. Rev. A 25, 2099-2101 (1982)http://link.aps.org/abstract/PRA/v25/p2099.
[CrossRef]

H. M. Lai and S. W. Chan, "Large and negative Goos-Hanchen shift near the Brewster dip on reflection from weakly absorbing media," Opt. Lett. 27, 680-682 (2002)http://www.opticsinfobase.org/abstract.cfm?URI=ol-27-9-680.
[CrossRef]

H. M. Lai, S. W. Chan, and W. H. Wong, "Nonspecular effects on reflection from absorbing media at and around Brewster’s dip," J. Opt. Soc. Am. A 23, 3208-3216 (2006)http://www.opticsinfobase.org/abstract.cfm?URI=josaa-23-12-3208.
[CrossRef]

D. Felbacq, A. Moreau, and R. Smaali, "Goos- Hanchen effect in the gaps of photonic crystals," Opt. Lett. 28, 1633-1635 (2003)http://www.opticsinfobase.org/abstract.cfm?URI=ol-28-18-1633.
[CrossRef] [PubMed]

N.-H. Shen, J. Chen, Q.-Y. Wu, T. Lan, Y.-X. Fan, and H.-T. Wang, "Large lateral shift near pseudo-Brewster angle on reflection from a weakly absorbing double negative medium," Opt. Express 14, 10,574-10,579 (2006). URL http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-22-10574.
[CrossRef]

T. Tamir and H. L. Bertoni, "Lateral Displacement of Optical Beams at Multilayered and Periodic Structures," J. Opt. Soc. Am. 61, 1397-1413 (1971)http://www.opticsinfobase.org/abstract.cfm?URI=josa-61-10-1397.
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

Comparison of typical GH shifts Dp,s for a low-loss metal (Au) and a low-loss dielectric (Ge) as a function of the angle of incidence θ. The dielectric constants for the chosen materials are: ε = -29.0+2.03i (Au) and ε = 21.6+2.77i (Ge) at a wavelength of λ = 826 nm [14]. a): p-polarization: The curve for Au is 5-fold magnified. b) s-polarization: The curve for Ge is 10-fold magnified. The curves are calculated from Eqs. (6) and (7) below (for Au see also the experimental result in [10]).

Fig. 2.
Fig. 2.

Series of graphs comparing the GH shift Dp as a function of the angle of incidence θ for the metallic and dielectric case on transition from the low-loss to the high-loss regime. We consider the artifical case of a metal and dielectric with a paired dielectric constant such that εr is of equal magnitude but opposite sign and εi is equal for the pair. The curves are calculated from Eq. (6) below.

Fig. 3.
Fig. 3.

Contour plot of the curvature D p at θ = 90° for metals (εr < 0) and dielectrics (εr >0). The curvature is calculated from Eq. (6) and the contour lines are numerically evaluated. The small numbers give the value of D p at θ = 90° along the contour. The sign of this number and hence the sign of the curvature determines whether the metal has the maximum GH shift at θ = 90° or if the maximum GH shift occurs at a second extremum. The red curve shows the approximated separation between the two regimes according to Eq. (10); it is almost a straight line. We have excluded a small, gray area which corresponds to internal reflection. The change from external to internal reflection causes a concentration of contour curves at the point εr = 1, εi = 0.

Fig. 4.
Fig. 4.

Plot of the GH shift for p-polarized light for the 7 materials specified in Tab. 1. The figure shows that for an increased loss the GH curves for metals and dielectrics are becoming more similar. The arrows indicate the direction of an increasing ratio εi /|εr |.

Fig. 5.
Fig. 5.

Plot of the reflectivity for p-polarized light for the materials listed in Tab. 1. As in Fig. 4 the arrows indicate the increase in the ratio εi /|εr |. One can see that an increased loss effects metals and dielectrics differently; whereas the reflectivity of a metal is reduced for increased loss, a dielectric shows an enhanced reflectivity.

Tables (1)

Tables Icon

Table 1. Table listing the dielectric constants ε = εr + iεi at λ = 496 nm and calculated values based upon these dielectric constants for the materials shown in Figs. 4 and 5. The angle θ max denotes the angle of the maximum GH shift, while θB is Brewster angle as defined by the minimum in the reflectivity. Both quantities have been determined numerically from the graphs in Figs. 4 and 5. An estimate of the Brewster angle according to [20] is given by θ |ε|. The residual reflectivity �� p,s and the differential reflectivity Δ�� are all evaluated at the Brewster angle θB . The entries for ’Source’ refer to the dielectric constants.

Equations (10)

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D μ = λ 2 π ϕ μ θ ,
r p = ε cos ( θ ) ε sin 2 ( θ ) ε cos ( θ ) + ε sin 2 ( θ ) , r s = cos ( θ ) ε sin 2 ( θ ) cos ( θ ) + ε sin 2 ( θ ) .
D μ = λ 2 π Im r μ r μ .
D p = λ 2 π Im ( 2 ε sin ( θ ) ε sin 2 ( θ ) ( ε cos 2 ( θ ) sin 2 ( θ ) ) ) ,
D S = λ 2 π Im ( 2 sin ( θ ) ε sin 2 ( θ ) ) .
D P = λ 2 π sin ( θ ) ε i ε 2 cos 2 ( θ ) + sin 2 ( θ ) ( ε sin 2 ( θ ) sin 2 ( θ ) ) sin 2 ( θ ) ε cos 2 ( θ ) 2 ε sin 2 ( θ ) ε sin 2 ( θ ) + ε r sin 2 ( θ ) ,
D s = λ 2 π sin ( θ ) ε i ε sin 2 ( θ ) ε sin 2 ( θ ) + ε r sin 2 ( θ ) .
D P D S = ε 2 cos 2 ( θ ) + sin 2 ( θ ) ( ε sin 2 ( θ ) sin 2 ( θ ) ) sin 2 ( θ ) ε cos 2 ( θ ) 2 .
1 sin 2 ( θ ) ε cos 2 ( θ ) 2 = 1 ( sin 2 ( θ ) ε r cos 2 ( θ ) ) 2 + ( ε i cos ( θ ) ) 2 .
( ε r + 0.67 ) 2 ( 0.12 ) 2 ε i 2 ( 0.21 ) 2 = 1 ,

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