Abstract

We show that backpropagating modes of surface polaritons can exist at the interface between two semi-infinite cross-negative media, one with negative permittivity (ε<0) and the other with negative permeability (µ<0). These single-interface modes that propagate along the surface of a cross-negative interface are physically of interest, since the single-negative requirements imposed on the material parameters can easily be achieved at terahertz and potentially optical frequencies by scaling the dimension of artificially structured planar materials. Conditions for material parameters that support a backpropagating mode of the surface polaritons are obtained by considering dispersion relation and energy flow density transported by surface polaritons and confirmed numerically by simulation of surface polariton propagation resonantly excited at a cross-negative interface by attenuated total reflection.

© 2005 Optical Society of America

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References

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Ame. J. Phys.

A. Bers, �??Note on group velocity and energy propagation�??, Ame. J. Phys. 68, 482-484 (2000)
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

D. R. Fredkin and A. Ron, �??Effectively left-handed (negative index) composite material�??, Appl. Phys. Lett. 81, 1753-1755 (2002).
[CrossRef]

IEEE Trans. Microw. Theory Tech.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, �??Magnetism from conductors and enhanced nonlinear phenomena�??, IEEE Trans. Microw. Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

J. Appl. Phys.

J. Yoon, G. Lee, S. H. Song, C. �??H. Oh, and P. �??S. Kim, �??Surface-plasmon photonic band gaps in dielectric gratings on a flat metal surface�??, J. Appl. Phys. 94, 123-129 (2002).
[CrossRef]

A. A. Oliner and T. Tamir, �??Backward waves on isotropic plasma slabs�??, J. Appl. Phys. 33, 231-233 (1962).
[CrossRef]

J. Phys.: Condens. Matter

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, �??Low frequency plasmons in thin-wire structures�??, J. Phys.: Condens. Matter 10, 4785-4809 (1998).
[CrossRef]

Opt. Commun.

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

Opt. Technol. Lett.

A. Lakhtakia, �??Conjugation symmetry in linear electromagnetism in extension of materials with negative real permittivity and permeability scalars�??, Microw. Opt. Technol. Lett. 40, 160-161 (2004).
[CrossRef]

Phys. Rev.

K. L. Kliewer and R. Fuchs, �??Optical modes of vibration in an ionic crystal slab including retardation. I. Nonradiative region�??, Phys. Rev. 144, 495-503 (1966).
[CrossRef]

Phys. Rev. E

N. �??C. Panoiu and R. M. Osgood, Jr., �??Influence of the dispersive properties of metals on the transmission characteristics of left-handed materials�??, Phys. Rev. E 68, 016611 (2003).
[CrossRef]

Phys. Rev. Lett.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, �??Composite medium with simultaneously negative permeability and permittivity�??, Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Rep. Prog. Phys.

D. L. Mills and E. Burstein, �??Polaritons: the electromagnetic modes of media�??, Rep. Prog. Phys. 37, 817-926 (1974).
[CrossRef]

Science

R. A. Shelby, D. R. Smith, and S. Schultz, �??Experimental Verification of a Negative Index of Refraction�??, Science 292, 77-79 (2001).
[CrossRef] [PubMed]

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, �??Terahertz Magnetic Response from Artificial Materials�??, Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, �??Magnetic response of metamaterials at 100 terahertz�??, Science 306, 1351-1353 (2004).
[CrossRef] [PubMed]

Sov. Phys. Usp.

V. G. Veselago, �??The electromagnetics of substances with simultaneously negative values of E and µ�??, Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Other

Details are available at http://microoptics.hanyang.ac.kr/home/DNMMbySurfacePatterning.pdf.

A. D. Boardman, Electromagnetic Surface Modes (John Wiley & Sons, New York, 1982).

B. A. Saleh and M. C. Teich, �??Polarization and crystal optics�?? in Fundamentals of Photonics (John Wiley & Sons, Inc., 1991), pp. 214-210
[CrossRef]

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

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Figures (3)

Fig. 1.
Fig. 1.

Density plots of the normalized energy flow densities of the SEP (in red) and SMP (in blue) modes for four possible combinations of ε′ 1 and µ′ 1. The filled and open circles in (a)–(d) indicate the positions of (ε′ 1/|ε′ 1|,µ′ 1/µ′ 1) and (-ε′ 1/|ε′ 1|,-µ′ 1/|µ′ 1|), respectively; the white dashed boundaries represent the relations of ε′ 1 µ′ 1=ε′ 2 µ′ 2 in (a) and (d) and ε′ 1/µ′ 1=ε′ 2/µ′ 2 in (b) and (c). The arrows in (b) and (c) indicate the cross-negative areas where the surface polaritons have a negative group velocity.

Fig. 2.
Fig. 2.

(a) Parametric plot of (ε′ 2/|ε′ 1|,µ′ 2/|µ′ 1|) as a function of normalized frequency Ω. The positions marked by A’s and B’s represent the cutoff frequencies of Ω(A1)=0.4, Ω(A2)=0.4472, Ω(A3)=0.4714, Ω(A4)=1/√2, Ω(B1)=1/√2, Ω(B2)=0.8, Ω(B3)=0.8528, and Ω(B4)=0.8944. (b), (c), Dispersion curves of the SEP and SMP modes when (b) Ω0=0.4 and (c) Ω0=0.8, where the dashed curves represent dispersion relations of the bulk propagating mode in the metamaterial. The points marked by M1, M2, E1 and E2 are used in Fig. 3.

Fig. 3.
Fig. 3.

Simulation results of the ATR coupling from an incident Gaussian beam to (a) the SMP(-) mode of point M1 (Ω=0.46, B=0.41), (b) the SMP(+) mode at point M2 (Ω=0.87, B=0.53), (c) the SEP(+) mode of point E1 (Ω=0.50, B=0.35), (d) the SEP(-) mode of point E2 (Ω=0.84, B=0.49). Color density represents the y component of the electric field for (a) and (b) and the magnetic field for (c) and (d). Note that the y axis is normal to the page surface.

Equations (12)

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β SMP = [ μ 1 μ 2 ( μ 2 ε 1 μ 1 ε 2 ) μ 2 2 μ 1 2 ] 1 2 k 0 ,
γ SMP , s = [ μ s 2 ( μ 1 ε 1 μ 2 ε 2 ) μ 2 2 μ 1 2 ] 1 2 k 0 ,
γ SMP , 1 μ 1 = γ SMP , 2 μ 2 ,
β SMP [ μ 1 μ 2 ( μ 2 ε 1 μ 1 ε 2 ) μ 2 2 μ 1 2 ] 1 2 k 0 ( 1 + i b SMP ) = β SMP ( 1 + i b SMP ) ,
γ SMP , s [ μ s 2 ( μ 1 ε 1 μ 2 ε 2 ) μ 2 2 μ 1 2 ] 1 2 k 0 ( 1 + i g SMP ) = γ SMP , s ( 1 + i g SMP ) ,
b SMP = m 1 + m 2 2 + μ 2 ε 1 ( m 2 + e 1 ) μ 1 ε 2 ( m 1 + e 2 ) 2 ( μ 2 ε 1 μ 1 ε 2 ) μ 2 2 m 2 μ 1 2 m 1 μ 2 2 μ 1 2 ,
g SMP = m s + μ 1 ε 1 ( m 1 + e 1 ) μ 2 ε 2 ( m 2 + e 2 ) 2 ( μ 1 ε 1 μ 2 ε 2 ) μ 2 2 m 2 μ 1 2 m 1 μ 2 2 μ 1 2 ,
η a p a · x ̂ p a , 1 + p a , 2 = ξ a , 1 γ a , 1 + ξ a , 2 γ a , 2 ξ a , 1 γ a , 1 + ξ a , 2 γ a , 2 ,
p a = p a , 1 + p a , 2 ,
p a , 1 = 0 S a ( z ) dz = A 2 4 ω ( β ξ a , 1 γ a , 1 ) x ̂ , p a , 2 = 0 S a ( z ) dz = A 2 4 ω ( β ξ a , 2 γ a , 2 ) x ̂ .
{ E ( ω ) E * ( ω ) , H ( ω ) H * ( ω ) ε ( ω ) ε * ( ω ) , μ ( ω ) μ * ( ω ) } .
ε 1 ( Ω ) = 1 1 Ω 2 + i Γ 1 Ω , μ 2 ( Ω ) = 1 F Ω 0 2 Ω 2 Ω 0 2 + i Γ 2 Ω .

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