Abstract

In this paper we extend the fast-all-modes method and the numerical modified steepest-descent-path method to the optical frequency range by finding all modes and solving the total electric field in three dimensions that is due to a point source above a lossy thin metal film with a negative permittivity situated between two dissimilar dielectric materials. We show that up to four proper surface wave modes may propagate on the film surface, including both backward and forward waves. We also solve for the electric field below the lossy thin metal film and verify the existence of superlensing of the electric field, comparing that case to the case of a dielectric film where no superlensing occurs. The CPU time using the fast-all-modes method and the numerical modified steepest-descent-path method is considerably less than that using the conventional method of integration along the Sommerfeld integration path.

© 2009 Optical Society of America

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References

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  1. J. Q. Bagley, B. Wu, and L. Tsang, “Electromagnetic fields of Hertzian dipoles in layered negative refractive index materials,” IEEE Antennas Wireless Propag. Lett. 7, 749-752 (2008).
    [CrossRef]
  2. L. Tsang and B. Wu, “Electromagnetic fields of Hertzian dipoles in layered media of moderate thickness including the effects of all modes,” IEEE Antennas Wireless Propag. Lett. 6, 316-319 (2007).
    [CrossRef]
  3. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  4. N. Fang, Z. Liu, T.-J. Yen, and X. Zhang, “Regenerating evanescent waves from a silver superlens,” Opt. Express 11, 682-287 (2003).
    [CrossRef] [PubMed]
  5. N. Fang and X. Zhang, “Imaging properties of a metamaterials superlens,” Appl. Phys. Lett. 82, 161-163 (2003).
    [CrossRef]
  6. D. O. S. Melville and R. J. Blaikie, “Super-resolution imaging through a planar silver layer,” Opt. Express 13, 2127-2134 (2005).
    [CrossRef] [PubMed]
  7. S. Durant, Z. Liu, J. M. Steele, and X. Zhang, “Theory of the transmission properties of an optical far-field superlens for imaging beyond the diffraction limit,” J. Opt. Soc. Am. B 23, 2383-2392 (2006).
    [CrossRef]
  8. T. Tamir and A. A. Oliner, “The spectrum of electromagnetic waves guided by a plasma layer,” Proc. IEEE 51, 317-332 (1963).
    [CrossRef]
  9. A.D.Boardman (editor), Electromagnetic Surface Modes (Wiley, 1982).
  10. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

2008

J. Q. Bagley, B. Wu, and L. Tsang, “Electromagnetic fields of Hertzian dipoles in layered negative refractive index materials,” IEEE Antennas Wireless Propag. Lett. 7, 749-752 (2008).
[CrossRef]

2007

L. Tsang and B. Wu, “Electromagnetic fields of Hertzian dipoles in layered media of moderate thickness including the effects of all modes,” IEEE Antennas Wireless Propag. Lett. 6, 316-319 (2007).
[CrossRef]

2006

2005

2003

N. Fang, Z. Liu, T.-J. Yen, and X. Zhang, “Regenerating evanescent waves from a silver superlens,” Opt. Express 11, 682-287 (2003).
[CrossRef] [PubMed]

N. Fang and X. Zhang, “Imaging properties of a metamaterials superlens,” Appl. Phys. Lett. 82, 161-163 (2003).
[CrossRef]

2000

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

1963

T. Tamir and A. A. Oliner, “The spectrum of electromagnetic waves guided by a plasma layer,” Proc. IEEE 51, 317-332 (1963).
[CrossRef]

Bagley, J. Q.

J. Q. Bagley, B. Wu, and L. Tsang, “Electromagnetic fields of Hertzian dipoles in layered negative refractive index materials,” IEEE Antennas Wireless Propag. Lett. 7, 749-752 (2008).
[CrossRef]

Blaikie, R. J.

Durant, S.

Fang, N.

N. Fang and X. Zhang, “Imaging properties of a metamaterials superlens,” Appl. Phys. Lett. 82, 161-163 (2003).
[CrossRef]

N. Fang, Z. Liu, T.-J. Yen, and X. Zhang, “Regenerating evanescent waves from a silver superlens,” Opt. Express 11, 682-287 (2003).
[CrossRef] [PubMed]

Liu, Z.

Melville, D. O. S.

Oliner, A. A.

T. Tamir and A. A. Oliner, “The spectrum of electromagnetic waves guided by a plasma layer,” Proc. IEEE 51, 317-332 (1963).
[CrossRef]

Pendry, J. B.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Raether, H.

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

Steele, J. M.

Tamir, T.

T. Tamir and A. A. Oliner, “The spectrum of electromagnetic waves guided by a plasma layer,” Proc. IEEE 51, 317-332 (1963).
[CrossRef]

Tsang, L.

J. Q. Bagley, B. Wu, and L. Tsang, “Electromagnetic fields of Hertzian dipoles in layered negative refractive index materials,” IEEE Antennas Wireless Propag. Lett. 7, 749-752 (2008).
[CrossRef]

L. Tsang and B. Wu, “Electromagnetic fields of Hertzian dipoles in layered media of moderate thickness including the effects of all modes,” IEEE Antennas Wireless Propag. Lett. 6, 316-319 (2007).
[CrossRef]

Wu, B.

J. Q. Bagley, B. Wu, and L. Tsang, “Electromagnetic fields of Hertzian dipoles in layered negative refractive index materials,” IEEE Antennas Wireless Propag. Lett. 7, 749-752 (2008).
[CrossRef]

L. Tsang and B. Wu, “Electromagnetic fields of Hertzian dipoles in layered media of moderate thickness including the effects of all modes,” IEEE Antennas Wireless Propag. Lett. 6, 316-319 (2007).
[CrossRef]

Yen, T.-J.

Zhang, X.

Appl. Phys. Lett.

N. Fang and X. Zhang, “Imaging properties of a metamaterials superlens,” Appl. Phys. Lett. 82, 161-163 (2003).
[CrossRef]

IEEE Antennas Wireless Propag. Lett.

J. Q. Bagley, B. Wu, and L. Tsang, “Electromagnetic fields of Hertzian dipoles in layered negative refractive index materials,” IEEE Antennas Wireless Propag. Lett. 7, 749-752 (2008).
[CrossRef]

L. Tsang and B. Wu, “Electromagnetic fields of Hertzian dipoles in layered media of moderate thickness including the effects of all modes,” IEEE Antennas Wireless Propag. Lett. 6, 316-319 (2007).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Express

Phys. Rev. Lett.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Proc. IEEE

T. Tamir and A. A. Oliner, “The spectrum of electromagnetic waves guided by a plasma layer,” Proc. IEEE 51, 317-332 (1963).
[CrossRef]

Other

A.D.Boardman (editor), Electromagnetic Surface Modes (Wiley, 1982).

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

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Figures (9)

Fig. 1
Fig. 1

Side view of the geometry.

Fig. 2
Fig. 2

Graphical method of finding surface wave modes for lossless, symmetric epsilon-negative film. Equations (7, 8, 9) are represented by the red dashed-dotted line, the blue dashed line, and the black solid line, respectively.

Fig. 3
Fig. 3

Initial and refined mode locations in complex u plane.

Fig. 4
Fig. 4

Captured and noncaptured modes, Sommerfeld integration path, and steepest-descent path in the complex θ plane.

Fig. 5
Fig. 5

Surface electric field strength | E 1 z | verses distance, h = 0.095 λ 0 .

Fig. 6
Fig. 6

Surface electric field strength | E 1 z | versus thickness of the film, ρ = 0.1 λ 0 , ε r 1 = ± 2.4 j 0.4 .

Fig. 7
Fig. 7

Comparison of integrands along the steepest-descent path of silver versus dielectric in region 2, ρ = 0.01 λ 0 . Integration points are linearly spaced.

Fig. 8
Fig. 8

Electric field strength | E 2 z | in the near field below the silver film.

Fig. 9
Fig. 9

Electric field strength | E 2 z | in the far field in region 2, 5 λ 0 below the silver film.

Equations (21)

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E 1 z = ( I l 8 π ω ε 1 ) d k ρ ( k ρ 3 k z ) exp ( j k z z ) H 0 ( 2 ) ( k ρ ρ ) [ 1 + R TM ] ,
R TM = R 01 TM + R 12 TM exp ( 2 j k 1 z h ) 1 + R 01 TM R 12 TM exp ( 2 j k 1 z h ) ,
( ε 1 k z ε 1 k 2 z + ε k 1 z ε 2 k 1 z ) sin ( k 1 z h ) j ( ε 1 k z ε 2 k 1 z + ε k 1 z ε 1 k 2 z ) cos ( k 1 z h ) = 0 .
j ( ε 1 2 k z 2 + ε 2 k 1 z 2 ) sin ( k 1 z h ) + 2 ε k z ε 1 k 1 z cos ( k 1 z h ) = 0 .
v = u ( ε ε 1 ) tan ( u ) Even Mode,
v = u ( ε ε 1 ) cot ( u ) Odd Mode.
v = ( ε | ε 1 | ) | u | tanh | u | Even Mode,
v = ( ε | ε 1 | ) | u | coth | u | Odd Mode,
u 2 + v 2 = ( | k 1 2 | + k 2 ) ( h 2 ) 2 .
( ε 1 2 v w ε ε 2 u 2 ) sin ( u ) u + ε 1 ( ε 2 v + ε w ) cos ( u ) = 0 .
v 2 + u 2 = ( k 1 2 k 2 ) h 2 = r 2 ,
w 2 + u 2 = ( k 1 2 k 2 2 ) h 2 = t 2 ,
v 2 w 2 = ( k 2 2 k 2 ) h 2 = s 2 ,
[ ε 1 ε 2 k = 0 N max n = k N max c n ( 1 ) k ( n k ) r 2 ( n k ) v 2 k + 1 + ε ε 2 k = 1 N max n = k 1 N max s n ( 1 ) k ( n + 1 k ) r 2 ( n + 1 k ) v 2 k + ε ε 2 n = 0 N max s n r 2 ( n + 1 ) ] 2
= ( v 2 s 2 ) [ ε 1 ε 1 k = 0 N max n = k N max s n ( 1 ) k ( n k ) r 2 ( n k ) v 2 k + 1 + ε ε 1 k = 0 N max n = k N max c n ( 1 ) k ( n k ) r 2 ( n k ) v 2 k ] 2 ,
p 0 = tanh 1 ( ε 1 ε + ε 2 ε 1 2 + ε ε 2 ) j m π ,
p 0 = coth 1 ( ε 1 ε + ε 2 ε 1 2 + ε ε 2 ) j π 2 j m π .
p = tanh 1 ( ε 1 ( 1 + r 2 ( 2 p 0 2 ) ) ε 2 + ( 1 + t 2 ( 2 p 0 2 ) ) ε ( 1 + r 2 ( 2 p 0 2 ) ) ( 1 + t 2 ( 2 p 0 2 ) ) ε 1 2 + ε ε 2 ) j m π ,
p = coth 1 ( ε 1 ( 1 + r 2 ( 2 p 0 2 ) ) ε 2 + ( 1 + t 2 ( 2 p 0 2 ) ) ε ( 1 + r 2 ( 2 p 0 2 ) ) ( 1 + t 2 ( 2 p 0 2 ) ) ε 1 2 + ε ε 2 ) j π 2 j m π .
tanh 1 ( ε 1 ε + ε 2 ε 1 2 + ε ε 2 ) = tanh 1 ( 1 ) = .
E 2 z = ( I l 8 π ω ε 2 ) d k ρ ( k ρ 3 k z ) exp ( j k z z ) H 0 ( 2 ) ( k ρ ρ ) T TM exp ( + j k 2 z z ) ,

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