Abstract

We resolve the existing controversy concerning the selection of the sign of the normal-to-the-interface component of the wave vector kz of an electromagnetic wave in an active (gain) medium. Our method exploits the fact that no ambiguity exists in the case of a film of the active medium, since its coefficient of reflectance is invariant under the inversion of the sign of kz. Then we show that the limit of the infinite film thickness determines a unique and physically consistent choice of the wave vector and the refractive index. Important practical implications of the theory are identified and discussed.

© 2007 Optical Society of America

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Corrections

V. U. Nazarov and Y.-C. Chang, "Resolving the wave-vector and the refractive index from the coefficient of reflectance: erratum," Opt. Lett. 32, 3345-3345 (2007)
https://www.osapublishing.org/ol/abstract.cfm?uri=ol-32-22-3345

References

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  1. S. A. Ramakrishna and O. J. Martin, Opt. Lett. 30, 2626 (2005).
    [CrossRef] [PubMed]
  2. Y.-F. Chen, P. Fischer, and F. W. Wise, Phys. Rev. Lett. 95, 067402 (2005).
    [CrossRef] [PubMed]
  3. T. G. Mackay and A. Lakhtakia, Phys. Rev. Lett. 96, 159701 (2006).
    [CrossRef] [PubMed]
  4. Y.-F. Chen, P. Fischer, and F. W. Wise, Phys. Rev. Lett. 96, 159702 (2006).
    [CrossRef]
  5. S. A. Ramakrishna, Phys. Rev. Lett. 98, 059701 (2007).
    [CrossRef] [PubMed]
  6. Y.-F. Chen, P. Fischer, and F. W. Wise, Phys. Rev. Lett. 98, 059702 (2007).
    [CrossRef]
  7. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, Vol. 111 of Springer Tracts in Modern Physics (Springer-Verlag, 1988).
  8. J. Skaar, Opt. Lett. 31, 3372 (2006).
    [CrossRef] [PubMed]
  9. Equation for the reflectance of a four-component system can be derived in the same way as Eq. for a three-component system by applying Maxwell's equations and the standard boundary conditions at interfaces.
  10. J. Wei and M. Xiao, Opt. Commun. 270, 455 (2007).
    [CrossRef]
  11. J. Seidel, S. Grafstrom, and L. Eng, Phys. Rev. Lett. 94, 177401 (2005).
    [CrossRef] [PubMed]
  12. J. B. Geddes III, T. G. Mackay, and A. Lakhtakia, "On the refractive index for a nonmagnetic two-component medium: resolution of a controversy," Opt. Commun. (2007), doi:10.1016/j.optcom.2007.08.025.

2007 (3)

S. A. Ramakrishna, Phys. Rev. Lett. 98, 059701 (2007).
[CrossRef] [PubMed]

Y.-F. Chen, P. Fischer, and F. W. Wise, Phys. Rev. Lett. 98, 059702 (2007).
[CrossRef]

J. Wei and M. Xiao, Opt. Commun. 270, 455 (2007).
[CrossRef]

2006 (3)

J. Skaar, Opt. Lett. 31, 3372 (2006).
[CrossRef] [PubMed]

T. G. Mackay and A. Lakhtakia, Phys. Rev. Lett. 96, 159701 (2006).
[CrossRef] [PubMed]

Y.-F. Chen, P. Fischer, and F. W. Wise, Phys. Rev. Lett. 96, 159702 (2006).
[CrossRef]

2005 (3)

S. A. Ramakrishna and O. J. Martin, Opt. Lett. 30, 2626 (2005).
[CrossRef] [PubMed]

Y.-F. Chen, P. Fischer, and F. W. Wise, Phys. Rev. Lett. 95, 067402 (2005).
[CrossRef] [PubMed]

J. Seidel, S. Grafstrom, and L. Eng, Phys. Rev. Lett. 94, 177401 (2005).
[CrossRef] [PubMed]

Opt. Commun. (1)

J. Wei and M. Xiao, Opt. Commun. 270, 455 (2007).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (6)

Y.-F. Chen, P. Fischer, and F. W. Wise, Phys. Rev. Lett. 95, 067402 (2005).
[CrossRef] [PubMed]

T. G. Mackay and A. Lakhtakia, Phys. Rev. Lett. 96, 159701 (2006).
[CrossRef] [PubMed]

Y.-F. Chen, P. Fischer, and F. W. Wise, Phys. Rev. Lett. 96, 159702 (2006).
[CrossRef]

S. A. Ramakrishna, Phys. Rev. Lett. 98, 059701 (2007).
[CrossRef] [PubMed]

Y.-F. Chen, P. Fischer, and F. W. Wise, Phys. Rev. Lett. 98, 059702 (2007).
[CrossRef]

J. Seidel, S. Grafstrom, and L. Eng, Phys. Rev. Lett. 94, 177401 (2005).
[CrossRef] [PubMed]

Other (3)

J. B. Geddes III, T. G. Mackay, and A. Lakhtakia, "On the refractive index for a nonmagnetic two-component medium: resolution of a controversy," Opt. Commun. (2007), doi:10.1016/j.optcom.2007.08.025.

Equation for the reflectance of a four-component system can be derived in the same way as Eq. for a three-component system by applying Maxwell's equations and the standard boundary conditions at interfaces.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, Vol. 111 of Springer Tracts in Modern Physics (Springer-Verlag, 1988).

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Figures (4)

Fig. 1
Fig. 1

Systems under study: (a) Semi-infinite dielectric ( ϵ 0 ) , metal film ( ϵ 1 , d 1 ) , and a semi-infinite active medium ( ϵ 2 ) . (b) As in (a) but the active medium constitutes a film of thickness d 2 , and there is a semi-infinite dielectric on the top with the dielectric function ϵ 3 .

Fig. 2
Fig. 2

Coefficient of reflectance of the system of a glass prism, silver film, and cresyl violet dye with ( ϵ 2 = 1.85 9 × 10 6 i , solid curve) and without ( ϵ 2 = 1.85 + 0 × i , dashed curve) gain. Other parameters are those from [11] (see text).

Fig. 3
Fig. 3

Convergence of the coefficient of reflectance of the system with a finite film of the active dielectric to that with the semi-infinite one. Parameters are those from [11] (see text).

Fig. 4
Fig. 4

Implications of the selection of the three different branch cuts in the k z 2 complex plane. Dashed curve (red), branch cut along positive real axis ( 0 ϕ < 2 π ) ; dotted curve (green), branch cut along negative imaginary axis [ π 2 ϕ < ( 3 2 ) π ] [1]; dashed–dotted curve (blue), the branch cut along negative real axis ( π ϕ < π ) [3, 10]; solid curve (black), a film of the active dielectric ( d 2 = 9 cm ) . Other parameters are those from [11] (see text).

Equations (4)

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R = r 01 + r 12 e 2 i k 1 z d 1 1 + r 01 r 12 e 2 i k 1 z d 1 2 ,
r i j = ( k i z ϵ j k j z ϵ i ) ( k i z ϵ j + k j z ϵ i )
k i z = ± 2 π λ ϵ i ϵ 0 sin 2 θ ,
R = e 2 i k 1 z d 1 ( r 12 + e 2 i k 2 z d 2 r 23 ) + r 01 ( 1 + e 2 i k 2 z d 2 r 12 r 23 ) 1 + e 2 i k 2 z d 2 r 12 r 23 + e 2 i k 1 z d 1 r 01 ( r 12 + e 2 i k 2 z d 2 r 23 ) 2 .

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