Abstract

We present an analytical study of the dynamic interplay among surface plasmon polarization charges, electromagnetic fields, and energy flow in the metal/dielectric interface and metal nanoslit structure. Particular attention is given to the regime where the energy flow in the metal side is significant compared to that in the dielectric side. The study reveals that a vortex-like circulation of energy is an intrinsic feature of surface plasmon propagation supported by a metal/dielectric interface, and, in general, a vortex can form when the permittivity and permeability values of the materials involved satisfy the following condition: {(εm/εd) <-1 and (µm/µd) >-1} or {(εm/εd)>-1 and (µm/µd)<-1}.

© 2006 Optical Society of America

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References

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  1. H. Raether, Surface plasmons on smooth and rough surfaces and on gratings, (Springer-Verlag, Berlin 1988).
  2. E. N. Economou, "Surface plasmons in thin films," Phys. Rev. 182, 539-554 (1969).
    [CrossRef]
  3. J. Nkoma, R. Loudon, and D. R. Tilley, "Elementary properties of surface polaritons," J. Phys. C: Solid State Phys. 7, 3547-3559 (1974).
    [CrossRef]
  4. P. Tournois and V. Laude, "Negative group velocities in metal-film optical waveguides," Opt. Commun. 137, 41-45 (1997).
    [CrossRef]
  5. J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
    [CrossRef]
  6. S. Astilean, Ph. Lalanne, and M. Palamaru, "Light transmission through metallic channels much smaller than the wavelength," Opt. Commun. 175, 265-273 (2000).
    [CrossRef]
  7. P. N. Stavrinou and L. Solymar, "The propagation of electromagnetic power through subwavelength slits in a metallic grating," Opt. Commun. 206, 217-223 (2002).
    [CrossRef]
  8. I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, "Guided modes in negative-refractive-index waveguides," Phys. Rev. E 67 057602(1)-(4) (2003).
    [CrossRef]
  9. H. Shin, and S. Fan, "All-angle negative refraction for surface plasmons using a metal-dielectric-metal structure," Phys. Rev. Lett. 96, 073907(1)-(4) (2006).
  10. K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, "Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability," Phys. Rev. B 73, 085104(1)-(11) (2006).
    [CrossRef]
  11. P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
    [CrossRef]
  12. E. D. Palik, Optical constants of solids, (Academic Press, New York 1998).
  13. J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, "Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model," Phys. Rev. B 72, 075405(1)-(11) (2005).
    [CrossRef]
  14. J. D. Jackson, Classical electrodynamics 3rd Edition, (John Wiley and Sons, New York 1999).
  15. Z. Sun, Y. S. Jung, and H. K. Kim, "Dynamic evolution of surface plasmon resonances in metallic nanoslit arrays," Appl. Phys. Lett. 86, 023111(1)-(3) (2005).
    [CrossRef]
  16. C. Liu, C. Yan, H. Chen, Y. Liu, and S. Gao, "Evanescent field on the surface of a negative-index planar lens," Appl. Phys. Lett. 88, 231102(1)-(3) (2006).
    [CrossRef]

2002

P. N. Stavrinou and L. Solymar, "The propagation of electromagnetic power through subwavelength slits in a metallic grating," Opt. Commun. 206, 217-223 (2002).
[CrossRef]

2000

S. Astilean, Ph. Lalanne, and M. Palamaru, "Light transmission through metallic channels much smaller than the wavelength," Opt. Commun. 175, 265-273 (2000).
[CrossRef]

1999

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

1997

P. Tournois and V. Laude, "Negative group velocities in metal-film optical waveguides," Opt. Commun. 137, 41-45 (1997).
[CrossRef]

1974

J. Nkoma, R. Loudon, and D. R. Tilley, "Elementary properties of surface polaritons," J. Phys. C: Solid State Phys. 7, 3547-3559 (1974).
[CrossRef]

1972

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

1969

E. N. Economou, "Surface plasmons in thin films," Phys. Rev. 182, 539-554 (1969).
[CrossRef]

Astilean, S.

S. Astilean, Ph. Lalanne, and M. Palamaru, "Light transmission through metallic channels much smaller than the wavelength," Opt. Commun. 175, 265-273 (2000).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Economou, E. N.

E. N. Economou, "Surface plasmons in thin films," Phys. Rev. 182, 539-554 (1969).
[CrossRef]

Garcia-Vidal, F. J.

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

Johnson, P. B.

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Lalanne, Ph.

S. Astilean, Ph. Lalanne, and M. Palamaru, "Light transmission through metallic channels much smaller than the wavelength," Opt. Commun. 175, 265-273 (2000).
[CrossRef]

Laude, V.

P. Tournois and V. Laude, "Negative group velocities in metal-film optical waveguides," Opt. Commun. 137, 41-45 (1997).
[CrossRef]

Loudon, R.

J. Nkoma, R. Loudon, and D. R. Tilley, "Elementary properties of surface polaritons," J. Phys. C: Solid State Phys. 7, 3547-3559 (1974).
[CrossRef]

Nkoma, J.

J. Nkoma, R. Loudon, and D. R. Tilley, "Elementary properties of surface polaritons," J. Phys. C: Solid State Phys. 7, 3547-3559 (1974).
[CrossRef]

Palamaru, M.

S. Astilean, Ph. Lalanne, and M. Palamaru, "Light transmission through metallic channels much smaller than the wavelength," Opt. Commun. 175, 265-273 (2000).
[CrossRef]

Pendry, J. B.

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

Porto, J. A.

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

Solymar, L.

P. N. Stavrinou and L. Solymar, "The propagation of electromagnetic power through subwavelength slits in a metallic grating," Opt. Commun. 206, 217-223 (2002).
[CrossRef]

Stavrinou, P. N.

P. N. Stavrinou and L. Solymar, "The propagation of electromagnetic power through subwavelength slits in a metallic grating," Opt. Commun. 206, 217-223 (2002).
[CrossRef]

Tilley, D. R.

J. Nkoma, R. Loudon, and D. R. Tilley, "Elementary properties of surface polaritons," J. Phys. C: Solid State Phys. 7, 3547-3559 (1974).
[CrossRef]

Tournois, P.

P. Tournois and V. Laude, "Negative group velocities in metal-film optical waveguides," Opt. Commun. 137, 41-45 (1997).
[CrossRef]

J. Phys. C: Solid State Phys.

J. Nkoma, R. Loudon, and D. R. Tilley, "Elementary properties of surface polaritons," J. Phys. C: Solid State Phys. 7, 3547-3559 (1974).
[CrossRef]

Opt. Commun.

P. Tournois and V. Laude, "Negative group velocities in metal-film optical waveguides," Opt. Commun. 137, 41-45 (1997).
[CrossRef]

S. Astilean, Ph. Lalanne, and M. Palamaru, "Light transmission through metallic channels much smaller than the wavelength," Opt. Commun. 175, 265-273 (2000).
[CrossRef]

P. N. Stavrinou and L. Solymar, "The propagation of electromagnetic power through subwavelength slits in a metallic grating," Opt. Commun. 206, 217-223 (2002).
[CrossRef]

Phys. Rev.

E. N. Economou, "Surface plasmons in thin films," Phys. Rev. 182, 539-554 (1969).
[CrossRef]

Phys. Rev. B

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Phys. Rev. Lett.

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

Other

H. Raether, Surface plasmons on smooth and rough surfaces and on gratings, (Springer-Verlag, Berlin 1988).

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, "Guided modes in negative-refractive-index waveguides," Phys. Rev. E 67 057602(1)-(4) (2003).
[CrossRef]

H. Shin, and S. Fan, "All-angle negative refraction for surface plasmons using a metal-dielectric-metal structure," Phys. Rev. Lett. 96, 073907(1)-(4) (2006).

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, "Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability," Phys. Rev. B 73, 085104(1)-(11) (2006).
[CrossRef]

E. D. Palik, Optical constants of solids, (Academic Press, New York 1998).

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, "Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model," Phys. Rev. B 72, 075405(1)-(11) (2005).
[CrossRef]

J. D. Jackson, Classical electrodynamics 3rd Edition, (John Wiley and Sons, New York 1999).

Z. Sun, Y. S. Jung, and H. K. Kim, "Dynamic evolution of surface plasmon resonances in metallic nanoslit arrays," Appl. Phys. Lett. 86, 023111(1)-(3) (2005).
[CrossRef]

C. Liu, C. Yan, H. Chen, Y. Liu, and S. Gao, "Evanescent field on the surface of a negative-index planar lens," Appl. Phys. Lett. 88, 231102(1)-(3) (2006).
[CrossRef]

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Figures (8)

Fig. 1.
Fig. 1.

Slit of width a, filled with dielectric material with dielectric constant εD and bounded by semi-infinite regions of metal with dielectric constant εM .

Fig. 2.
Fig. 2.

The time-dependent Poynting vectors (top panel) and time-averaged Poyning vectors (bottom panel) calculated for a surface plasmon wave propagating along the interface of silver (x<0) and air (x>0) at (a,d) 650 nm, (b,e) 400 nm, and (c,f) 350 nm free-space wavelength.

Fig. 3.
Fig. 3.

Materials that support surface bound waves (TE or TM polarized surface plasmons) at a generalized metal (ε 2, µ 2)/dielectric (ε 1, µ 1) interface. Here it is assumed that all permittivity and permeability values are real, and ε 1>0 and µ 1>0. The 3rd quadrant (ε 2<0 and µ 2<0) represents left-handed materials. The case of a silver/air interface that supports surface plasmons locates in the TM mode zone of the 2nd quadrant.

Fig. 4.
Fig. 4.

Snapshot images of the electromagnetic and Poynting vector fields and the polarization charge distribution calculated at a single air (x>0)/silver (x<0) interface at 400 nm free-space wavelength.

Fig. 5.
Fig. 5.

Snapshot images of the electromagnetic and Poynting vector fields and the polarization charge distribution calculated for a 100-nm-wide silver slit at 400 nm free-space wavelength.

Fig. 6.
Fig. 6.

Snapshot images of the electromagnetic and Poynting vector fields and the polarization charge distribution calculated for a 25-nm-wide silver slit at 400 nm free-space wavelength.

Fig. 7.
Fig. 7.

Ratio of the total power propagating in the metal (silver) region (backward propagating) to the total power propagating in the air filled slit region (forward propagating). A ratio greater than -1 indicates that the overall result is forward propagating.

Fig. 8.
Fig. 8.

The orientation of the time-averaged Poynting vector in the metal (silver) and slit (air) regions, calculated at 650 nm wavelength [(a) and (b)] and 400 nm wavelength [(c) and (d)], and for slit width of 200 nm (blue), 100 nm (green), 50 nm (red), and 25 nm (violet). The right panels [(b) and (d)] show a magnified view of the slit region. Here the orientation is expressed as the tilt angle with respect to the +y direction in the slit region and to the -y direction in the metal region. (See the right-most panel.)

Equations (42)

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k sp = k 0 ε M ε D ε M + ε D
γ D = k sp 2 ε D k 0 2
γ M = k sp 2 ε M k 0 2
H z ( 1 ) ( x , y ) = H 0 [ 1 ± exp ( γ D a ) ] exp ( γ M x + i k sp y )
H z ( 2 ) ( x , y ) = H 0 { exp ( γ D x ) ± exp [ γ D ( a x ) ] } exp ( i k sp y )
H z ( 3 ) ( x , y ) = H 0 [ exp ( γ D a ) ± 1 ] exp [ γ M ( x a ) + i k sp y ]
[ 1 exp ( γ D a ) 1 + exp ( γ D a ) ] ± 1 γ D ε D = γ M ε M
· P = · ε 0 E
σ P ( y ) = lim x 0 ε 0 [ E x ( 1 ) ( x , y ) E x ( 2 ) ( x , y ) ]
σ P ( y ) = H 0 c ( 1 ε M 1 ε D ) ( k sp k 0 ) [ 1 + exp ( γ D a ) ] exp ( i k sp y )
σ P + ( y ) = H 0 c ( 1 ε M 1 ε D ) ( k sp k 0 ) [ 1 + exp ( γ D a ) ] exp ( i k sp y )
S ( x , y , t ) = E ( x , y , t ) × H ( x , y , t )
< S > ( x , y ) = 1 2 Re [ E ( x , y ) × H * ( x , y ) ]
< S y > ( 1 ) ( x , y ) = H 0 2 2 η 0 1 + exp ( γ D a ) 2 Re ( N eff ε M )
× exp [ 2 Re ( γ M ) x 2 Im ( k sp ) y ]
< S y > ( 2 ) ( x , y ) = H 0 2 η 0 Re ( N eff ε D ) exp [ Re ( γ D ) a 2 Im ( k sp ) y ]
× { cosh [ Re ( γ D ) ( a 2 x ) ] + cos [ Im ( γ D ) ( a 2 x ) ] }
< S y > ( 3 ) ( x , y ) = H 0 2 2 η 0 1 + exp ( γ D a ) 2 Re ( N eff ε M )
× exp [ 2 Re ( γ M ) ( a x ) 2 Im ( k sp ) y ]
< S x > ( 1 ) ( x , y ) = H 0 2 2 η 0 1 + exp ( γ D a ) 2 Im ( γ M k 0 ε M )
× exp [ 2 Re ( γ M ) x 2 Im ( k sp ) y ]
< S x > ( 2 ) ( x , y ) = H 0 2 η 0 exp [ Re ( γ D ) a 2 Im ( k sp ) y ]
× { Re ( γ D ε D k 0 ) s in [ Im ( γ D ) ( 2 x a ) ] Im ( γ D ε D k 0 ) s inh [ Re ( γ D ) ( a 2 x ) ] }
< S x > ( 3 ) ( x , y ) = H 0 2 2 η 0 1 + exp ( γ D a ) 2 Im ( γ M k 0 ε M )
× exp [ 2 Re ( γ M ) ( a x ) 2 Im ( k sp ) y ]
ϕ M = tan 1 [ Im ( γ M ε M ) Re ( k sp ε M ) ] in the metal side
ϕ D = tan 1 [ Im ( γ D ε D ) Re ( k sp ε D ) ] in the dielectric side
( P δz ) ( 1 ) = ( P δz ) ( 3 ) = H 0 2 4 η 0 1 + exp ( γ D a ) 2 Re ( γ M ) Re ( N eff ε M ) exp [ 2 Im ( k sp ) y ]
( P δz ) ( 2 ) = H 0 2 η 0 Re ( N eff ε D ) exp [ 2 Im ( k sp ) y Re ( γ D ) a ]
× { sinh [ Re ( γ D ) a ] Re ( γ D ) + sin [ Im ( γ D ) a ] Im ( γ D ) }
k sp 2 γ 1 2 = ε 1 μ 1 k 0 2
k sp 2 γ 2 2 = ε 2 μ 2 k 0 2
γ 2 μ 2 = γ 1 μ 1 ( for TE ) ; γ 2 ε 2 = γ 1 ε 1 ( for TM )
{ μ 1 > 0 μ 2 < 0 for TE } and { ε 1 > 0 ε 2 < 0 for TM }
N eff 2 = ε 1 μ 2 ( μ 2 μ 1 ) ( ε 2 ε 1 ) ( μ 2 μ 1 ) 2 1
( γ 1 k 0 ) 2 = ε 1 μ 1 1 ( ε 2 ε 1 ) ( μ 2 μ 1 ) ( μ 2 μ 1 ) 2 1
( γ 2 k 0 ) 2 = ( μ 2 μ 1 ) 2 ( γ 1 k 0 ) 2
{ [ ( μ 2 μ 1 ) < 1 ] [ ( μ 2 μ 1 ) < ( ε 2 ε 1 ) ] [ ( μ 2 μ 1 ) ( ε 2 ε 1 ) < 1 ] }
{ [ 1 < ( μ 2 μ 1 ) < 0 ] [ ( μ 2 μ 1 ) > ( ε 2 ε 1 ) ] [ ( μ 2 μ 1 ) ( ε 2 ε 1 ) > 1 ] }
{ [ ( ε 2 ε 1 ) < 1 ] [ ( ε 2 ε 1 ) < ( μ 2 μ 1 ) ] [ ( ε 2 ε 1 ) ( μ 2 μ 1 ) < 1 ] }
{ [ 1 < ( ε 2 ε 1 ) < 0 ] [ ( ε 2 ε 1 ) > ( μ 2 μ 1 ) ] [ ( ε 2 ε 1 ) ( μ 2 μ 1 ) > 1 ] }
× S = z ̂ ρ s ε 0 H z

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