## Backpropagating modes of surface polaritons on a cross-negative interface

Optics Express, Vol. 13, Issue 2, pp. 417-427 (2005)

http://dx.doi.org/10.1364/OPEX.13.000417

Acrobat PDF (530 KB)

### 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

## 1. Introduction

1. 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]

2. 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]

*ε*<0 and

*µ*<0) simultaneously in the THz or potentially optical regimes. Negative-index metamaterials, whose permittivity and permeability are simultaneously negative and are known not to exist in a natural form in materials, were made by composing an array of metallic wires and split-ring resonators (SRRs) [3

3. 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]

4. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science **292**, 77–79 (2001). [CrossRef] [PubMed]

5. V. G. Veselago, “The electromagnetics of substances with simultaneously negative values of ε and µ,” Sov. Phys. Usp. **10**, 509–514 (1968). [CrossRef]

*µ*(

*ω*)=1-(

*Fω*

^{2})/(

*ω*

^{2}-

*i*Γ

*ω*), where the strength (

*F*), resonance frequency (

*ω*

_{0}), and lifetime (Γ) of the magnetic dipole resonance are defined mainly by the structure parameters of SRRs, and the wire grids placed between the SRRs also have the relative permittivity of

*ε*(

*ω*)=1-

*ω*

^{2}with the plasma frequency (

*ω*

_{p}) given by the width and radius of the wires [6

6. 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]

7. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech.47, 2075–2084 (1999). [CrossRef]

*ε*and

*µ*are negative simultaneously, the planar structure must satisfy the necessary condition of

*ω*

_{p}/

*ω*

_{0}>1. After further consideration under the assumption that the period of the SRRs is 50

*µm*for a typical THz response, we can conclude that the gap size between the inner and the outer rings in the SRR must be of the order of nanometers [8

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

*µ*<0 and wire grids for

*ε*<0. It has also been reported that periodic assembly of two alternating layers, one with negative permittivity and the other with negative permeability, can support backpropagating modes with an effectively negative index of refraction [9

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

*µ*<0 and

*ε*<0, respectively. The boundary surface or cross-negative interface can be formed in a layered structure composed of artificial magnetic (SRRs) and electrical (plasmonic wires) layers since the surface polaritons that propagate on the cross-negative interface can have magnetic fields that oscillate normal to the surface of the SRR layer. We first find relative conditions for the material parameters of the two semi-infinite layers that support the backpropagating surface polaritons by considering dispersion relation and energy flow density transported by the surface polaritons. We then confirm our findings numerically by simulating backpropagation of the surface polaritons that are resonantly excited by attenuated total reflection (ATR) of a Gaussian beam incident on the cross-negative interface.

11. 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]

*p*-polarized surface electric polaritons (SEPs), which are surface localized longitudinal oscillations of the electric dipoles, whereas the negative permeability makes possible the s-polarized surface magnetic polaritons (SMPs), which are surface localized longitudinal oscillations of the magnetic dipoles. The surface polaritons propagate along the boundary surface with their electric and magnetic fields localized and evanesce into both adjoined materials. Excitation of a single normal mode of the surface polaritons is influenced simultaneously by the material parameters of both media adjoined at the interface. Therefore, one can expect backpropagating modes of surface polaritons that have negative group velocity not only in double-negative media, but also at the cross-negative interface of two media in which only

*ε*<0 is in one medium and only

*µ*<0 is in the other. It is also known that, in thin slab geometry such as plasma or an ionic crystal [12

12. A. A. Oliner and T. Tamir, “Backward waves on isotropic plasma slabs,” J. Appl. Phys. **33**, 231–233 (1962). [CrossRef]

13. 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]

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

1. 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]

5. V. G. Veselago, “The electromagnetics of substances with simultaneously negative values of ε and µ,” Sov. Phys. Usp. **10**, 509–514 (1968). [CrossRef]

## 2. Generalized conditions of material parameters for surface polariton excitation

*φ*(

*x*,

*z*)=

*A*

*exp*(

*iβx*-

*γ*

_{s}|

*z*|), localized near the interface (

*z*=0 plane) of two media for which relative material parameters are given by (

*ε*

_{s},

*µ*

_{s}), where medium index

*s*has a value of 1 for

*z*<0 or 2 for

*z*>0.

*β*is the propagation constant in the direction of the

*x*axis and

*γ*

_{s}is the decay constant in each medium. Wave amplitude

*A*represents the

*y*component of the magnetic field or the electric field when the surface-localized electromagnetic wave is coupled to a SEP or a SMP, respectively. By requiring that all the field components satisfy Maxwell equations for all the positions including the interface, we can obtain a relationship between the propagation and the decay constants of the SMP-coupled waves (SMP modes) and the SEP-coupled waves (SEP modes) in terms of material parameters (

*ε*

_{1},

*µ*

_{1}) and (

*ε*

_{2},

*µ*

_{2}). For SMP modes, the relationship is

*k*

_{0}=

*ω*/

*c*[15]. The relative conditions of the four material parameters that support excitation of the SMP modes can be derived from Eqs. (1) to (3). For an additional procedure, assume that the all-material parameters are complex valued but dominated by their real parts such that

*ε*

_{s}=

*ε′*

_{s}(1+

*ie*

_{s}), |

*e*

_{s}|□ 1 and

*µ*

_{s}=

*µ′*

_{s}(1+

*im*

_{s}), |

*m*

_{s}|□ 1, where all the symbols on the right-hand sides are real. When the propagation and decay constants are expanded by powers of

*e*

_{s}and

*m*

_{s}and terms up to their first orders are taken, they can be rewritten as

*b*

_{SMP}and

*g*

_{SMP}have values of the same orders of magnitude as

*e*

_{s}and

*m*

_{s}, because the terms that include the real parts of the material parameters in Eqs. (6) and (7) have numerators with multiplication factors of

*e*

_{s}and

*m*

_{s}, which are the same combinations of

*ε′*

_{s}and

*µ′*

_{s}as their denominators, respectively, and this results in overall factors of

*e*

_{s}and

*m*

_{s}to be of the order of 1. Therefore, the terms that include

*b*

_{SMP}and

*g*

_{SMP}in Eqs. (4) and (5) are negligible for the propagation and decay constants according to the previous assumption for the imaginary parts of the material parameters. If

*β′*

_{SMP}is a real value, the propagation constant βSMP has a dominant real part and the corresponding wave function expresses the propagation state with a relatively slow decaying profile along the interface. Otherwise, if

*β′*

_{SMP}is an imaginary value, the wave hardly propagates and decays rapidly. Thus it is reasonable to regard real-valued

*β′*

_{SMP}as a necessary condition, namely, in-plane propagation condition, for SMPs. To apply a similar manner to decay constant

*γ*

_{SMP,s}, a surface localization condition for SMPs could be obtained as

*γ′*

_{SMP,s}must be a real value. To complete the surface localization condition, it is necessary for

*γ′*

_{SMP,s}to be positive for both media. This condition requires

*µ′*

_{1}

*µ′*

_{2}<0, when we approximate Eq. (3) to be

*γ′*

_{SMP},

_{1}/

*µ′*

_{1}=-

*γ′*

_{SMP,}

_{2}/

*µ′*

_{2}. In summary, the in-plane propagation condition reduces to

*µ′*

_{1}/

*ε′*

_{1}<

*µ′*

_{2}/

*ε′*

_{2}when -1<

*µ′*

_{2}/

*µ′*

_{1}<0, or

*µ′*

_{1}/

*ε′*

_{1}>

*µ′*

_{2}/

*ε′*

_{2}when

*µ′*

_{2}/

*µ′*

_{1}<-1. And the surface localization condition for a SMP reduces to

*ε′*

_{1}

*µ′*

_{1}<

*ε′*

_{2}

*µ′*

_{2}when -1<

*µ′*

_{2}/

*µ′*

_{1}<0, or

*ε′*

_{1}

*µ′*

_{1}>

*ε′*

_{2}

*µ′*

_{2}when

*µ′*

_{2}/

*µ′*

_{1}<-1. For SEP modes the relative conditions for in-plane propagation and surface localization can be obtained simply by interchanging the relative permittivity and permeability positions with each other. Figure 1 shows diagrams for visualization of these relative conditions for SMP modes in the blue region and SEP modes in the red region. To represent four possible sign combinations of (a) (

*ε′*

_{1}>0,

*µ′*

_{1}>0), (b) (

*ε′*

_{1}/<0,

*µ′*

_{1}>0), (c) (

*ε′*

_{1}>0,

*µ′*

_{1}<0), and (d) (

*ε′*

_{1}<0,

*µ′*

_{1}<0). the normalized values of (

*ε′*

_{1}/|

*ε′*

_{1}|,

*µ′*

_{1}/|

*µ′*

_{1}|) are marked by black dots and their complementary values by white dots. The dashed curves in Figs. 1(a) and 1(d) represent the critical boundaries of

*ε′*

_{2}

*µ′*

_{2}=

*ε′*

_{1}

*µ′*

_{1}for the surface localization conditions; the dashed lines in Figs. 1(b) and 1(c) reveal

*ε′*

_{2}/

*µ′*

_{2}=

*ε′*

_{1}/

*µ′*

_{1}for the in-plane propagation conditions. The color densities in the figure indicate the normalized total energy flow densities of the SMP modes (

*η*

_{SMP}) and the SEP modes (

*η*

_{SEP}) that are defined in Eq. (3).

*ε′*

_{1}/|

*ε′*

_{1}|,-

*µ′*

_{1}/|

*µ′*

_{1}|). Physical implication of the two distinguished parts becomes obvious after evaluation of normalized total energy flow densities transported by the SEP and SMP modes:

*a*represents SEP or SMP,

*ξ′*

_{SEP,s}≡

*ε′*

_{s}, and

*ξ′*

_{SMP,s}≡

*µ′*

_{s}.

*p⃑*

_{a,s}represent the energy flow density that is derived by integrating time-averaged Poynting vector

*S⃑*

_{a}(

*z*) over the respective surface normal distance through medium

*s*:

_{a,1}ξ′

_{a,2}<0 the propagation directions of

*p⃑*

_{a,1}and

*p⃑*

_{a,2}are always opposite each other [16

16. D. L. Mills and E. Burstein, “Polaritons: the electromagnetic modes of media,” Rep. Prog. Phys. **37**, 817–926 (1974). [CrossRef]

_{a,1}<0 and |

*p⃑*

_{a,1}|>|

*p⃑*

_{a,2}|, for example, the normalized total energy flow density

*η*

_{a}is negative. This can be thought of as the mode with negative group velocity with respect to phase velocity because the Poynting vector direction is always equal to the group velocity for linear waves that propagate in a homogeneous medium with arbitrary spatial and temporal dispersion [17

17. A. Bers, “Note on group velocity and energy propagation,” Amer. J. Phys. **68**, 482–484 (2000). [CrossRef]

18. B. E. A. Saleh and M. C. Teich, “Polarization and crystal optics,” in *Fundamentals of Photonics* (Wiley, New York, 1991), Chap. 6. [CrossRef]

*η*

_{SEP}≥-1 in red and the 1≥

*η*

_{SMP}≥-1 in blue. The surface polaritons are excited at the regions in which 0>

*η*

_{a}≥-1 (1≥

*η*

_{a}>0) have a negative (positive) group velocity. These density plots obviously show that the regions that allow negative group velocity do not consist of only the double-negative areas in which either of the two media has ε

*′*

_{s}<0 and

*µ′*

_{s}<0 simultaneously, but also the cross-negative areas in which

*ε′*

_{2}>0 and

*µ′*

_{2}<0 while

*ε′*

_{1}<0 and

*µ′*

_{1}>0 or vice versa, such as is indicated by the arrows in Figs. 1(b) and 1(c).

*ε′*

_{1},

*µ′*

_{1},

*ε′*

_{2},

*µ′*

_{2}) to (-

*ε′*

_{1},-

*µ′*

_{1},-

*ε′*

_{2},-

*µ′*

_{2}). The excitation conditions of surface polaritons were derived from kinetic consideration of the propagation and decay constants, therefore it can be concluded that, if a SEP (SMP) can propagate on a boundary with material parameters of (

*ε′*

_{1},

*µ′*

_{1},

*ε′*

_{2},

*µ′*

_{2}), the inverted material parameters of (-

*ε′*

_{1},-

*µ′*

_{1},-

*ε′*

_{2},-

*µ′*

_{2}) also support a SEP (SMP) with the same propagation and decay constants as the original ones. For this reason, Figs. 1(a) and 1(d) and 1(b) and 1(c) show centrosymmetry with each other. However, there are differences in the field distributions between the two SEP (SMP) modes with (

*ε′*

_{1},

*µ′*

_{1},

*ε′*

_{2},

*µ′*

_{2}) and (-

*ε′*

_{1},-

*µ′*

_{1},-

*ε′*

_{2},-

*µ′*

_{2}). For example, the magnetic field of a SEP mode can be expressed by

**H**=

*H*

_{0}

*exp*(

*iβx*-

*γ*|

*z*|)

**e**

_{y}. According to the Maxwell equations the corresponding electric field is given by

**E**=(

*β*

**e**

_{x}±

*iγ*

**e**

_{z})×

**H**/

*ωε*

_{i}where the +(-) sign is taken for z>0 (z<0). Inverting the signs of material parameters does not alter the magnetic field, whereas the electric field is changed by a

*π*shift in its phase because of

*ε*

_{i}. As a consequence, only the direction of the Poynting vector is reversed. This is also confirmed by comparison of the upper-left region with the lower-right region in Figs. 1(a) and 1(d) or the upper-right region with the lower-left region in Figs. 1(b) and 1(c), where distribution of the normalized total energy flow density is centrosymmetric with different propagation directions. This dependence of surface polariton modes on simultaneous sign inversion of the real parts of the material parameters can be explained more generally by conjugation symmetry of the frequency domain Maxwell equations [19

19. A. Lakhtakia, “Conjugation symmetry in linear electromagnetism in extension of materials with negative real permittivity and permeability scalars,” Microwave Opt. Technol. Lett. **40**, 160–161 (2004). [CrossRef]

**E**(

**r**,

*ω*) and

**H**(

**r**,

*ω*) are solutions of the Maxwell equations in system

**Σ**with the spatial distributions of material parameters

*ε*(

**r**,

*ω*) and

*µ*(

**r**,

*ω*), then

**Ē**(

**r**,

*ω*)=

**E***(

**r**,

*ω*) and

**H̄**(

**r**,

*ω*)=

**H***(

**r**,

*ω*) are also exact solutions for conjugate system

**, whose relative permittivity and permeability are -**Σ ¯

*ε**(

**r**,

*ω*) and -

*µ**(

**r**,

*ω*), respectively. Note that

**is given by the sign inversion of the real parts of the material parameters for**Σ ¯

**Σ**. It is also noted that the time-averaged Poynting vector given by Re[

**E**(

**r**,

*ω*)×

**H***(

**r**,

*ω*)] is invariant under such transformation, and. as a consequence, the group velocity direction of the conjugate modes between

**Σ**and

**is equal for each mode. But complex conjugate operation on the spatial field amplitudes changes the signs of the wave vectors of plane-wave components of which**Σ ¯

**E**(

**r**,

*ω*) and

**H**(

**r**,

*ω*) are composed, resulting in inversion of the phase velocities.

## 3. Backpropagating modes of surface magnetic and surface electric polaritons on crossnegative interfaces

*ε′*

_{1}<0,

*µ*

_{1}=1) and a metamaterial with (

*ε*

_{2}=1,

*µ′*

_{2}<0) over a frequency range lower neighbor of Ω

_{0}. The material parameters of

*ε*

_{1}and

*µ*

_{2}can be expressed in the form of

*ω*/

*ω*

_{p}where

*ω*

_{p}is the plasma frequency of the metal.

*ε*

_{1}(Ω) is a plasmonic form of a Drude model and

*µ*

_{2}(Ω) is from an array of planar SRRs with a resonance frequency of Ω

_{0}[7

7. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech.47, 2075–2084 (1999). [CrossRef]

*F*is the area fraction of the internal opening in the SRR and set to be 0.5, hereafter. To show schematically the frequency dependence of excitation of the SMP and SEP modes, we assume that Γ

_{1}=0 and Γ

_{2}=0 under the approximation of small damping losses in the media. Figure 2(a) shows a parametric plot of (

*ε′*

_{2}/|

*ε′*

_{1}|,

*µ′*

_{2}/|

*µ′*

_{1}|) as a function of Ω for three different Ω

_{0}of 0.8(>Ω

_{c}), Ω

_{c}, and 0.4(<Ω

_{c}), where

*ε′*

_{1}<0, and the positions marked by A’s and B’s represent the boundary 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, andΩ(B4)=0.8944. At the critical resonance frequency of Ω

_{0}=Ω

_{c}(dash-dot curves), there is no surface polariton mode in the cross-negative areas (

*ε′*

_{2}/|

*ε′*

_{1}|>0 and

*µ′*

_{2}/|

*µ′*

_{1}|<0). When Ω

_{0}=0.8 (curves with open circles) as an example of Ω

_{0}>Ω

_{c}, a SEP(-) band that supports the SEP modes with negative group velocity is located in the frequency range of Ω(B2)<Ω<Ω(B3), whereas a SEP(+) band with positive group velocity is in 0<Ω<Ω(B1) and a SMP(+) band is in Ω(B3)<Ω<Ω(B4). In contrast, when Ω

_{0}=0.4 (curves with solid circles), a single SMP(-) band appears in Ω(A2)<Ω<Ω(A3), and two SEP(+) bands appear in 0<Ω<Ω(A1) and (A3) Ω <Ω<Ω(A4), respectively. After further analysis, we can finally conclude that, for the cross-negative areas shown in Fig. 1(b), the SEP(-) modes appear only for the case of

_{0}<1 ; the SMP(-) modes only for

_{0}=0.4 and Ω

_{0}=0.8, respectively, and

*B*≡

*cβ′*/

*ω*

_{p}. For comparison, that of the SEP modes is also presented by solid circles and that of bulk modes that propagate in the metamaterial is represented by dashed curves. The negative slopes between Ω(A2) and Ω(A3) in Fig. 2(b) and between Ω(

*B*2) and Ω(

*B*3) in Fig. 2(c) clearly show the negative group velocities of the SMP(-) and SEP(-) modes, respectively.

## 4. Numerical demonstration of backpropagating modes on cross-negative interfaces

*ε′*<0,

*µ′*>0) and the metamaterial (

*ε′*>0,

*µ′*<0) by using the plane-wave expansion method. A Gaussian beam that is incident from the dielectric with a finite waist is assumed to be TE and TM polarized for SMP and SEP modes, respectively. The calculation results are shown in Fig. 3. In the calculation it is assumed that small damping constants of Γ

_{1}=10

^{-3}and Γ

_{2}=10

^{-3}Ω

_{0}and that the dielectric (

*ε*=2.25) and air (

*ε*=1.0) are semi-infinite. The thickness of the metamaterial is 10×

*ƛ*

_{p}for all cases, where

*ƛ*

_{p}(≡

*c*/

*ω*

_{p}) is the plasma wavelength. The thicknesses of the metal layers are given differently by 2.7×

*ƛ*

_{p}, 4×

*ƛ*

_{p}, 3×

*ƛ*

_{p}and 3.25×

*ƛ*

_{p}for Figs. 3(a), 3(b), 3(c), and 3(d), respectively, to guarantee high coupling efficiency from the incident beam to the surface modes. It is apparent that the excited modes are not coupled modes but single-interface modes at the cross-negative (metal–metamaterial) interface, because no field enhancement is seen at the dielectric–metal or the metamaterial–air interfaces for all the cases. Figure 3(a) clearly shows the leftward propagation of the electric fields (

*E*

_{y}) of the SMP(-) mode, as depicted by a dotted arrow near the cross-negative interface. The Gaussian beam incident from the dielectric is not only reflected from the dielectric–metal boundary, but it is coupled resonantly to the SMP(-) mode near the cross-negative interface. The electric field of the SMP(-) mode is more concentrated on the metamaterial layer with a group velocity antiparallel to its phase velocity. Evidence of the negative group velocity can also be found intuitively by observing the reemitted fields that radiate back into the dielectric medium. The reemitted fields that are shown just under the SMP(-) propagation region reveal wave fronts parallel to the reflected beam. As a consequence we can confirm that the phase velocity of the SMP(-) mode is positive in the

*x*direction, but the group velocity is negative. The SMP(+) mode at point M2, on the other hand, is stretched toward the metal layer as shown in Fig. 3(b). It has a group velocity parallel to its phase velocity, which can also be checked by the wave fronts of the reemitted fields parallel to the reflected wave fronts. For the SEP modes depicted in Figs. 3(c) and 3(d), the SEP(+) mode has its magnetic field concentrated more in the metamaterial layer than in the metal layer, similar to the SMP(-) mode in Fig. 3(a), but it has rightward or forward propagation. The reverse is shown for the SEP(-) mode in Fig. 3(d) with the SMP(+) mode in Fig. 3(b). If we recall that the energy flow density is directly proportional to |

*E*

_{y}|

^{2}(SMP modes) or |

*H*

_{y}|

^{2}(SEP modes) as described in Eq. (10), these differences in field concentration can easily be understood by the fact that backpropagating SMP or SEP modes should have more energy in a diamagnetic or a metal layer, respectively.

*b*

_{SMP}in Eq. (6). We introduced

*Q*

_{SMP}≡1/

*b*

_{SMP}as the effective number of spatial oscillations in the same manner as the definition of the resonance quality factor for temporal oscillations. For the SMP(-) mode in Fig. 3 (a), the relative damping constants of Γ

_{1}=10

^{-3}and Γ

_{2}=10

^{-3}Ω

_{0}reveal

*Q*

_{SMP}=45.71. If we take into consideration ten times larger damping constants of Γ

_{1}=10

^{-2}and Γ

_{2}=10

^{-2}Ω

_{0},

*Q*

_{SMP}=4.56, which is a ten times smaller value as expected from the dependence of

*b*

_{SMP}on the imaginary parts of the material parameters. For a more practical case, we consider an SMP(-) mode excited at the interface between a gold and a two-dimensional magnetic metamaterial as reported in Ref. 20. When

*ω*

_{p}=1.367×10

^{16}Hz,

*γ*

_{1}=4.08×10

^{13}Hz for gold [21

21. M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander Jr., and C. A. Ward, “Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt. **22**, 1099–1120 (1983). [CrossRef] [PubMed]

*F*=0.2,

*ω*

_{0}=1.489×10

^{14}Hz,

*γ*

_{2}=2.4×10

^{12}Hz for metamaterial, excitation of the SMP(-) modes can occur in the frequency range from 1.56×10

^{14}Hz to 1.63×10

^{14}Hz. At 1.5836×10

^{14}Hz, for example, (

*ε′*

_{2}/|

*ε′*

_{1}|,

*µ′*

_{2}/|

*µ′*

_{1}|)=(10

^{-4},-0.5) and the corresponding SMP(-) mode has

*Q*

_{SMP}=3.06, which means that the mode undergoes a significant amount of propagation loss.

## 5. Conclusion

*ε*<0 in one medium and only

*µ*<0 in the other. Backpropagating modes of surface magnetic polaritons at cross-negative interfaces have been confirmed in detail by evaluation of their dispersion relations and ATR coupling behavior based on the plane-wave expansion method. Third, the two propagation directions, parallel and antiparallel to the phase velocity, are inherently determined by the values of the material parameters, regardless of their frequency dispersive characteristics. In particular, antiparallel propagation is also possible even when no double-negative medium is involved, such as the cross-negative media composed of two nontransparent media: one with negative permittivity only and the other with negative permeability only. Fourth, if a set of material parameters supports a parallel (antiparallel) propagating SEP (SMP), the sign inverted set supports antiparallel (parallel) propagating SEP (SMP) without changing the propagation and decay constants.

*B*

_{x}) of the SMP(-) mode always have a varying flux normal to the metamaterial surface, which proves that efficient production of a negative magnetic response (

*µ*<0) can be achieved in an artificial planar structure. Therefore, it is possible that such a cross-negative interface with a single-negative requirement imposed on the material parameters of two adjoined media can be implemented by stacking two different types of planar structure: one with a negative permeability, such as a SRR; the other with a negative permittivity, such as a metallic grid. The scalability of these separated planar structures could enable us to realize surface polaritonic devices with lefthanded behavior in THz and potentially optical frequencies.

## Acknowledgments

## References and links

1. | 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 |

2. | S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science |

3. | 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. |

4. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science |

5. | V. G. Veselago, “The electromagnetics of substances with simultaneously negative values of ε and µ,” Sov. Phys. Usp. |

6. | J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low frequency plasmons in thin-wire structures,” J. Phys. Condens. Matter |

7. | J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech.47, 2075–2084 (1999). [CrossRef] |

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

9. | D. R. Fredkin and A. Ron, “Effectively left-handed (negative index) composite material,” Appl. Phys. Lett. |

10. | A. D. Boardman, |

11. | 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. |

12. | A. A. Oliner and T. Tamir, “Backward waves on isotropic plasma slabs,” J. Appl. Phys. |

13. | K. L. Kliewer and R. Fuchs, “Optical modes of vibration in an ionic crystal slab including retardation. I. Nonradiative region,” Phys. Rev. |

14. | P. Tournois and V. Laude, “Negative group velocities in metal-film optical waveguides,” Opt. Commun. |

15. | H. Raether, |

16. | D. L. Mills and E. Burstein, “Polaritons: the electromagnetic modes of media,” Rep. Prog. Phys. |

17. | A. Bers, “Note on group velocity and energy propagation,” Amer. J. Phys. |

18. | B. E. A. Saleh and M. C. Teich, “Polarization and crystal optics,” in |

19. | A. Lakhtakia, “Conjugation symmetry in linear electromagnetism in extension of materials with negative real permittivity and permeability scalars,” Microwave Opt. Technol. Lett. |

20. | 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 |

21. | M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander Jr., and C. A. Ward, “Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt. |

**OCIS Codes**

(240.5420) Optics at surfaces : Polaritons

(240.6690) Optics at surfaces : Surface waves

**ToC Category:**

Research Papers

**History**

Original Manuscript: September 1, 2004

Revised Manuscript: December 14, 2004

Published: January 24, 2005

**Citation**

Jaewoong Yoon, Seok Song, Cha-Hwan Oh, and Pill-Soo Kim, "Backpropagating modes of surface polaritons on a cross-negative interface," Opt. Express **13**, 417-427 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-2-417

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### References

- 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]
- 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]
- R. A. Shelby, D. R. Smith, and S. Schultz, �??Experimental Verification of a Negative Index of Refraction�??, Science 292, 77-79 (2001). [CrossRef] [PubMed]
- V. G. Veselago, �??The electromagnetics of substances with simultaneously negative values of E and µ�??, Sov. Phys. Usp. 10, 509-514 (1968). [CrossRef]
- 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]
- 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]
- Details are available at http://microoptics.hanyang.ac.kr/home/DNMMbySurfacePatterning.pdf.
- D. R. Fredkin and A. Ron, �??Effectively left-handed (negative index) composite material�??, Appl. Phys. Lett. 81, 1753-1755 (2002). [CrossRef]
- A. D. Boardman, Electromagnetic Surface Modes (John Wiley & Sons, New York, 1982).
- 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]
- 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]
- P. Tournois and V. Laude, �??Negative group velocities in metal-film optical waveguides�??, Opt. Commun. 137, 41-45 (1997). [CrossRef]
- Heinz Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag Berlin Heidelberg, 1988).
- D. L. Mills and E. Burstein, �??Polaritons: the electromagnetic modes of media�??, Rep. Prog. Phys. 37, 817-926 (1974). [CrossRef]
- A. Bers, �??Note on group velocity and energy propagation�??, Ame. J. Phys. 68, 482-484 (2000) [CrossRef]
- B. A. Saleh and M. C. Teich, �??Polarization and crystal optics�?? in Fundamentals of Photonics (John Wiley & Sons, Inc., 1991), pp. 214-210 [CrossRef]
- 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]
- 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]
- M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander, Jr., and C. A. Ward, �??Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared�??, Appl. Opt. 22, 1099-1120 (1983). [CrossRef] [PubMed]

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