An ultra-short contra-directional coupler utilizing surface plasmon-polaritons at optical frequencies

An ultra-short contra-directionalcoupler utilizing surface

plasmon-polaritons at opticalfrequencies

Yan Wang, Rubaiyat Islam and George V. EleftheriadesThe Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of

TorontoToronto, ON, Canada, M5S 3G4, 2006 5

[email protected]

http://www.waves.utoronto.ca/prof/gelefth/main.html

Abstract: A nano-scaled coupled-line coupler based on the guidanceof surface plasmon-polaritons (SPPs) is proposed, designed and simulatedat optical frequencies. The coupler comprises layered dielectric materialsand silver, which serve as two stacked nano-transmission lines to achievebroadside coupling. The key property of this coupler is thatit operatesbased on the principle of contra-directional coupling between a forwardand a backward wave giving rise to supermodes that are characterizedby complex-conjugate eigenvalues (even when the materialsare assumedlossless). The resulting exponential attenuation along the coupler leads todramatically reduced coupling lengths compared to previously reportedco-directional SPP couplers (e.g. from millimeters to submicrons). Theeffect of material losses and finite coupler width are also analyzed.

© 2006 Optical Society of America

OCIS codes: (240.6680) Surface Plasmons; (240.5420) Polaritons; (240.0310) Thin films;(230.3990) Microstructure devices

References and links1. J. Takahara and T. Kobayashi, “From subwavelength opticsto nano-optics,” Opt. Photonics News15, 54-59

(2004).2. G. Shvets, “Photonic approach to making a material with a negative index of refraction,” Phys. Rev. B67, 035109

(2003).3. E. N. Economou, “Surface plasmons in thin films,” Phys. Rev.182, 539-554 (1969).4. A. Alu and N. Engheta, “Optical nanotransmission lines: synthesis of planar left-handed metamaterials in the

infrared and visible regimes,” J. Opt. Soc. Am. B23, 571-583 (2006).5. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric

structures,” Phy. Rev. B61, 10484-10503 (2000).6. G.V. Eleftheriades, A.K. Iyer, and P.C. Kremer, “Planar negative refractive index media using periodically L-C

loaded transmission lines,” IEEE Trans. Microwave Theory Technol.50, 2702-2712 (2002).7. R. Islam, F. Elek, and G.V. Eleftheriades, “A coupled-linemetamaterial coupler having co-directional phase but

contra-directional power flow,” IEE Electron. Lett.40, 315–317 (2004).8. A. Boltasseva, T. Nikolajsen, K. Leosson, K. Kjaer, M.S. Larsen, and S.I. Bozhevolnyi, “Integrated optical com-

ponents utilizing long-range surface plasmon polaritons,”J. Lightwave Technol.23, 413–422 (2005).9. R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, “Demonstration of integrated optics elements based on

long-range surface plasmon polaritons,” Opt. Express13, 977–984 (2005).

#71016 - $15.00 USD Received 17 May 2006; revised 26 July 2006; accepted 26 July 2006

(C) 2006 OSA 7 August 2006 / Vol. 14, No. 16 / OPTICS EXPRESS 7279

10. R. Islam and G.V. Eleftheriades, “Printed high-directivity metamaterial MS/NRI coupled-line coupler for signalmonitoring applications,” IEEE Microwave Wirel. Compon. Lett. 16, 164–166 (2006).

11. E.T. Arakawa, M.W. Williams, R.N. Hamm, and R.H. Ritchie. “Effect of damping on surface plasmon disper-sion,” Phy. Rev. Lett.31, 1127–1129 (1973).

12. P.B. Johnson and R.W. Christy, “Optical constants of thenoble metals,” Phy. Rev. B6, 4370–4379 (1972).

1. Introduction

The concept of utilizing low-dimensional waves to overcomethe transverse diffraction limit ofoptical guided-wave structures has received considerableattention in recent years. In particular,surface plasmon-polaritons (SPPs) guided at the interfaceof media with positive and negativepermittivities are used to illustrate such a concept [1]. Some interesting structures that supportSPPs include the negative-dielectric-film (ND-film) and thenegative-dielectric-gap (ND-gap),where a thin layer of ND material is sandwiched between positive dielectric media or viceversa (see Fig. 1). Noble metals, such as gold and silver, arewell known candidates for theND material because they exhibit small negative electric permittivities at optical frequencies.Some polaritonic materials, such as SiC, can also be employed as the ND medium for mid/longinfrared applications [2]. The theoretical analysis of SPPs has continued for more than threedecades, during which the characteristics of SPPs guided atthe interface of a semi-infinite di-electric medium and metal, as well as those guided by thin metal films with various dimensionshave been explored ([1], [3], [4] and [5]). One particular interesting concept in [4] is to recog-nize the ND-film and the ND-gap structures as backward-wave (BW-wave) and forward-wave(FW-wave) transmission lines at optical frequencies, respectively. The reader is reminded thatbackward-wave propagation is equivalent to left-handed propagation (̄E, H̄ andK̄ form a left-handed triplet), for which the Poynting vector and the phasevelocity have opposite directions.Therefore, this concept could potentially open up the possibility of extending many applica-tions derived from left-handed transmission-line metamaterials at microwave frequencies ([6])to the optical domain. Indeed, the concept of the short coupler at optical frequencies presentedin this paper is inspired by the microwave transmission-line couplers reported in [7] and [10]. In[7] and [10], the couplers consist of a forward transmission-line edge-coupled to a left-handed(backward) transmission-line. This leads to contra-directional coupling between the two linesand the resulting two coupled eigenmodes have propagation constants that form a complex-conjugate pair. The corresponding exponential attenuation along the coupler (which is valideven for lossless lines) indicates a rapid coupling rate, therefore leading to very short couplinglengths. In this paper, we explain how to extend these short couplers to the optical domain usingplasmonic films (e.g. silver).

Fig. 1. Geometry and possible materials for the contra-directional SPP coupler featuringthe stacked ND-film and ND-gap topology. (The arrows represent the power flow.)

The physical realization of discrete plasmonic componentsat the nanometer scale is made



possible with recent developments in fabrication technologies. Some previous works, such as[8] and [9], have been carried out on co-directional plasmonic couplers. The operation of thosecouplers depends on the small difference between the propagation constants of the two coupledmodes, which leads to long coupling lengths on the millimeter scale. In contrast, couplers thatoperate under the contra-directional coupling condition exhibit much shorter coupling lengths.At microwave frequencies, the two transmission line branches in [7] are implemented by em-ploying microstrip technology in conjunction with loadinglumped L-C elements. At opticalfrequencies, this paper proposes to replace the microstriplines with their nano-transmissionline counterparts. In addition, the FW-wave and the BW-wavebranches are stacked to achievebroadside coupling (instead of edge coupling) as shown in Fig. 1. As predicted by contra-directional coupled-mode theory, this coupler supports complex coupled modes, whose attenu-ation constant increases drastically at a certain frequency range to form a leaky stopband ([7]and [10]). The very short coupling length exhibited by the coupler is a direct result of the largeattenuation constant which holds true even when the materials are completely lossless. Thisattenuation constant physically signifies that the energy carried by the FW-wave line is effi-ciently redirected (coupled) into the BW-wave line or vise versa at a high exponential rate. Theplasmonic coupler proposed in this paper features almost 0dB coupling within 1µm for ideal(lossless) metal, and -3.5dB coupling when losses are accounted for.

In Section 2, theω −β dispersion relations for SPPs guided by the metal-gap and the metal-film, as well as for the complex coupled modes are derived. Allderivations assume the metalis lossless and the structure is infinite in one of the transverse directions. In Section 3, thefull-wave simulations of the dispersion diagrams, wave propagation and coupling, and the S-parameter analysis of the coupler are conducted using Comsol Multiphysics simulation pack-age. Finally, the effect of the metal loss (described by the Drude model) and the finite transverseconfinement are presented in Section 4 and 5 respectively.

2. Theory

It is well known that noble metals, such as gold and silver, have negative electric permittivitiesbelow their plasma frequencies. Such materials can be treated as the ND medium at optical fre-quencies. The frequency dispersion characteristic of the relative electric permittivity of metalsis described by Drude-Sommerfeld’s theorem in Eq. (1).

εm(ω) = 1−ω2

p

ω2 (1)

where the plasma frequencyωp = 1.29×1016rad/s, which corresponds to a free-space wave-lengthλp of 146nm for silver.

2.1. Dispersion Characteristics of SPPs Guided by the Metal-film and the Metal-gap

Both the metal-film and the metal-gap in Fig. 2 support twoTMz SPP modes with even andodd H-field distributions in the transverse direction. Theω −β dispersion relations of the SPPmodes are derived from Maxwell’s equations and the consistency conditions.

We first start with the metal-film structure. For theTMz SPP modes, the transverse H-fieldsin the three layers are expressed in Eq. (2).

Hx1 = Ae−ky,dye− jβz

Hx2 =(

Be−ky,my +Ceky,my)

e− jβz

Hx3 = Deky,dye− jβz (2)



whereky,d andky,m are the transverse wave-numbers in the dielectric medium and the metalrespectively, andβ is the phase constant. The four unknownsA, B, C andD represent the fieldamplitudes. From Maxwell’s equations, the electric field that is tangential to the interface canbe expressed through Eq. (3) for each layer,

Ez = −1

jωε∂Hx

∂y(3)

The boundary conditions imply thatHx andEz for each layer must be continuous at the inter-faces, from which we obtain a 4×4 homogeneous linear system of equations. The eigenvaluesolution reveals that the dispersion relations of the even and the odd SPPs for the metal-film aredescribed in Eqs. (4) and (5) respectively,

ky,dh

εd= −

ky,mh

εmtanh

(

ky,mh

2

)

(4)

ky,dh

εd= −

ky,mh

εmcoth

(

ky,mh

2

)

(5)

whereh is the film thickness. Finally, the consistency conditions in Eq. (6) relate the transversewave numbersky,d andky,m to the phase constantβ .

β 2−k2

y,d = εoεdk20

β 2−k2

y,m = εoεmk20 (6)

The closed form expressions of theω − β dispersion relations can be found through directsubstitution. Finally, the dispersion characteristic forthe metal-gap structure can be representedin exactly the same manner when interchangingky,d with ky,m andεd with εm.

Fig. 2. Dispersion diagrams (The relative electric permittivity of the dielectric media are:εair = 1,εsilica−glass = 2.09,εglass = 4.2,εsilicon = 12.1. The ND-film guides BW evenmodes and the ND-gap guides FW even modes.)

The correspondingω−β dispersion diagrams with various dielectric materials areillustratedin Fig. 2. The results match with the SPP dispersion behaviors described in [3]. In summary,we can see that the metal-film and the metal-gap can indeed actas surface-wave mode guid-ing structures at optical frequencies. More specifically, the metal-film guides backward even



modes while the metal-gap guides forward even modes. However, the two even modes residein different frequency bands for the same dielectric material. It should be noted that the evenmode is of interest here because this mode profile facilitates its excitation with a Laser viathe end-fire coupling technique [9]. Therefore, from now on,we will focus on the even modeonly. The horizontal asymptotes that separate the backwardand forward modes represent thesurface resonance frequencies, at whichεd = −εm. Note that the waves are tightly bounded atthe metal-dielectric interfaces near the surface resonances. In this regime, the phase constantβbecomes very large, which indicates that the effective wavelength becomes very small. This isparticularly desirable for designing electrically small devices. However, this advantage is com-promised when material losses are taken into considerationsince the attenuation increases withβ .

2.2. Dispersion Characteristics of the Coupled Modes

Since the surface resonance frequency decreases whenεd increases as indicated in Fig. 2, theapproach of stacking metal-film and metal-gap structures consisting of different dielectrics (seeFig. 1) renders the possibility of producing both FW and BW even modes at the same frequency.One interesting application of having both modes co-existing at the same frequency has beenpresented in [7] and [10], where an extended contra-directional coupled-mode theory was de-rived to demonstrate that short couplers comprising a BW-wave and a FW-wave transmissionline can produceco-directional phasebut contra-directional power flowin the two branches.Such anomalous coupling is due to the fact that this structure supports two supermodes hav-ing complex-conjugate propagation constants(γ = α ± jβ ), of which the real partα increasesdrastically around the intersection of theω −β dispersion curves of the isolated BW and FWmodes.

The dispersion relations for the coupled-mode system (see Fig. 3) can be derived in a similarfashion as we did for the individual metal-film or metal-gap structure. Assuming the coupledmodes are alsoTMz modes, the transverse H-fields in the six layers are expressed in Eq. (7).

Hx1 = Ae−ky1ye− jβz

Hx2 =(

Be−ky2y +Ceky2y)

e− jβz

Hx3 =(

De−ky3y +Feky3y)

e− jβz

Hx4 =(

Geky4y +He−ky4y)

e− jβz

Hx5 =(

Keky5y +Le−ky5y)

e− jβz

Hx6 = Meky6ye− jβz (7)

where there are ten unknowns (A,B,C, etc.) that are associated with the field amplitude, andthe transverse wave numbers (ky’s) are related to the phase constantβ through the consistencyconditions. Together with the tangentialEz’s derived from Eq. (3), the boundary conditions atfive interfaces render a 10×10 homogeneous system of equations. Since it is awkward to obtainthe closed-form expressions for theω −β dispersion equations, the corresponding eigenvaluesolutions are now obtained by finding the roots of the 10×10 determinant numerically.

The comparison between the dispersion diagrams of the isolated FW (metal-gap) and theBW (metal-film) waves and of the coupled system is illustrated in Fig. 3. As shown, a stop-band forms for the two coupled modes close to the intersection of the isolatedω −β curves.By adopting the conventions used in [7], the two coupled modes are termed as thec-mode(e−(α− jβ )z) and theπ-mode (e−(α+ jβ )z). Figure 3 shows that the large attenuation in the stop-band is purely due to the coupling effect since no material losses are included in the analysis.



1 1.5 2 2.5

x 108

0.45

0.46

0.47

0.48

0.49

0.5

0.51

0.52

β (rad/m)

ω/ω

p

βπ (Comsol simulation)

βc (Comsol simulation)

βπ (Modal theory)

βc (Modal theory)

isolated FW−wave(metal−gap ε

d=2.09)

isolated BW−wave(metal−film ε

d=4.2)

(a) phase constantβ (βc andβπ have equal magnitude inthe stopband)

0 50 100 150 2000.45

0.46

0.47

0.48

0.49

0.5

0.51

0.52

α (dB/µm)

ω/ω

p

απ (Comsol simulation)

αc (Comsol simulation)

απ (Modal theory)

αc (Modal theory)


d=2.09)


d=4.2)

(b) attenuation constantα (αc and απ are zero in thepassband, but increase drastically in the stopband)

Fig. 3. Dispersion diagram using the lossless Drude model of silver

The corresponding coupled eigenmodes are characterized bycomplex-conjugate propagationconstants.

The formation of the stopband at the intersection of the isolatedω −β curves can be under-stood more intuitively from the phase-matching condition,which requires that the fields acrosstwo stacked branches must incur the same phase when propagating along the z-direction. Thiscondition is easiest to be satisfied when the two isolated modes exhibit the same phase con-stants. Furthermore, since one line supports a BW wave and the other supports a FW wave, thePoynting vectors along the two branches must point in opposite directions. In other words, thiscoupled system provides animmediatereturn path for the incident power (see the arrows inFig. 1 that represent the power flow). Moreover, the large attenuation constant indicates that theenergy leaks from the input to the coupled port at an exponential rate, hence manifesting a veryshort coupling length with a high coupling level. In addition, since the power decays exponen-tially along the direction of propagation, the coupling level increases as the length increases.Theoretically, 0dB coupling is achieved when the line is semi-infinite. This is different from theco-directional couplers, where the maximum power transferoccurs periodically along the line.Additionally, any power splitting ratio between the coupled and through ports can be achievedby adjusting the length and separation distance according to different design specifications. Thedimensions and materials for the coupler presented in this paper are shown in Fig. 1 and showthe feasibility of realizing this kind of a coupler. A side note is that the small dimension of thecoupler may present a challenge in directly coupling light into the structure. However, thesecouplers can be used as interconnects in futuristic denselypacked integrated optical circuits,where the neighbor components are on the same order of scale.In terms of characterizing thecoupler through experimental measurements directly, one likely needs to design transitions thatcouple light in the structure utilizing near-field microscope tip devices at the nanometer scale.

3. Simulation Results

In order to verify the modal theory presented above, fullwave simulations have been carriedout using Comsol Multiphysics software package (formerly known as FEMLAB). The disper-sion diagrams are extracted through the application mode2D Perpendicular Wave EigenmodeAnalysisin Comsol’sElectromagnetics Module. Figure 3 shows that the results match with thetheoretical dispersion diagrams very well.



The characteristics of theπ-mode and thec-mode shown in Fig. 4(a) and 4(b) can be sum-marized as follows. Theπ-mode carries power forward in the metal-film layer and backwardin the metal-gap layer, but the opposite is true for thec-mode. Finally, the two modes alwaysexist simultaneously in the coupler. However, if the metal-gap is excited, then thec-mode dom-inates in order to satisfy the condition that the input poweris flowing forward into the structure,similarly theπ-mode dominates if the metal-film is excited.

(a) π-mode profile along the y transverse direction (refer to Fig. 1for the dimension andmaterials of each layer)

(b) c-mode profile along the y transverse direction (refer to Fig. 1for the dimension andmaterials of each layer)

Fig. 4. Coupled eigenmodes at the center frequency of the stopband (ω/ωp = 0.487)

To visualize the wave propagation and the coupling effect, simulations using applicationmode2D TM In-Plane Harmonic Propagationin theElectromagnetics Moduleare conductedfor both the passband and the stopband, and the results are presented in Fig. 5. We can see thatthe power flows from the input port to the through port directly in the passband, but rapidlyleaks to the coupled port in the stopband.

The performance of the coupler is analyzed in terms of the coupling level and the couplinglength. The coupling level can be interpreted as the coefficientS21 in the S-parameter analysisof a four-port network. Figure 6 shows that almost 0dB coupling level with very high isolationis achieved in the stopband for the lossless metal. The coupling length can be determined qual-



Fig. 5. Power flow along the 1µm coupler assuming the metal is lossless (Note: the powerscale shown in the figure is normalized with respect to the incident power)

0.46 0.48 0.5 0.52−80

−70

−60

−50

−40

−30

−20

−10

0

10

ω/ωP

dB

S21

S31

S41

Fig. 6. S-parameter analysis for 1µmcoupler assuming the metal is lossless



itatively from Fig. 5 and more quantitatively from Fig. 3(b), where the power attenuation (dBper micron) is plotted. As shown, at the center of the stopband, the power attenuation is morethan 70dB/µm. This suggests that the coupler acts almost as being semi-infinite even over asubmicron scale.

4. Effect of Loss

A more accurate Drude-Sommerfeld’s model that also takes loss into consideration is given inEq. (8).

εm(ω) = 1−ω2

p

ω2− j ωτ

(8)

where the relaxation timeτ = 1.25× 10−14s for silver. The effect of loss on the dispersion

1 1.5 2 2.5

x 108

0.45

0.46

0.47

0.48

0.49

0.5

0.51

0.52

β (rad/m)

ω/ω

p

βπ (Comsol simulation)

βc (Comsol simulation)

βπ (Modal theory)

βc (Modal theory)


d=4.2)


d=2.09)

(a) phase constantβ (βc andβπ are not exactly equal inmagnitude in the stopband)

0 50 100 150 2000.45

0.46

0.47

0.48

0.49

0.5

0.51

0.52

α (dB/µm)

ω/ω

p

απ (Comsol simulation)

αc (Comsol simulation)

απ (Modal theory)

αc (Modal theory)


d=2.09)


d=4.2)

(b) attenuation constantα (αc andαπ are dominated bythe material losses in the passband and the couplingeffect in the stopband)

Fig. 7. Dispersion diagram using the lossy Drude model of silver

property of the isolated FW and BW SPP’s, as well as the coupled modes can be observedin Fig. 7. This figure shows that the coupled modes are no longer lossless in the passband;Instead, they follow the lossy pattern of the isolated FW andBW waves. In the stopband, thetwo coupled modes do not have the same phase constantβ except at the center of the stopband,and theπ-mode attenuates slightly faster than thec-mode. However, the coupling effect forboth supermodes still dominates over the material loss in the stopband since the attenuationconstants are very close to the ones extracted from the lossless model. The power attenuationdue to material losses when the ND-gap is excited is illustrated in Fig. 8. In the passband,the power attenuates along the lower branch. In the stopband, we can also observe the contra-directional coupling phenomenon, but the intensity of the coupled power is diminished.

A more quantitative analysis is conducted by comparing Fig.6 and 9, we can see that thematerial losses reduce the coupling level from 0dB to -3.5dBwhen the coupler operates at thecenter frequency of the stopband. The loss is stronger away from the stopband center due tothe fact that the wave experiences more attenuation since ittravels longer before coupling intothe other branch. Figure 7(b) shows the loss-per-micron dueto both the material loss and thecoupling effect at each frequency, from which the power attenuation along the line and thecoupling length can be estimated.



Fig. 8. Power flow along 1µmcoupler assuming the metal loss in Eq. (8) (Note: the powerscale shown in the figure is normalized with respect to the incident power)

0.46 0.48 0.5 0.52−80

−70

−60

−50

−40

−30

−20

−10

0

10

ω/ωP

dB

S21

S31

S41

Fig. 9. S-parameter analysis for 1µmcoupler assuming the metal loss in Eq. (8)



5. Effect of Finite Width

In practice, the metal film/gap has finite width as well as finite thickness. Therefore, the corre-sponding guided modes contain coupled components of the surface waves at four edges. Theeffect of the 2D confinement in the transverse direction on SPPs is discussed extensively in [5].This discussion suggests that the fundamental even mode hasa near Gaussian profile, thereforecan be excited using a simple end-fire technique. It also suggests that the dispersion behaviorof the fundamental even mode approximates the one of the infinite width when the aspect ra-tio of width/thicknessis large. This section explores the effect of finite width on the coupledmodes by comparing the dispersion diagram with the one for infinite width, and determinesthe minimum width necessary to match their dispersion properties for the chosen thickness andseparation distance. Figure 10(a) shows that a decrease in the width results in smaller phaseconstants for both coupled modes, but it does not alter the frequency range where the stop-band occurs. Figure 10(b) also shows that the attenuation constant for theπ-mode increasessignificantly as the width decreases below 100nm. For designpurposes, Fig. 10 suggests thatany width larger than 200nm is a good approximation of the onedesigned assuming an infinitewidth.

0.8 1 1.2 1.4 1.6 1.8

x 108

0.46

0.47

0.48

0.49

0.5

0.51

β (rad/m)

ω/ω

p

infinite widthw = 200nmw = 100nmw = 60nm

(a) phase constantβ

20 40 60 80 100 120 1400.46

0.47

0.48

0.49

0.5

0.51

α (dB/µm)

ω/ω

p

(b) attenuation constantα

Fig. 10. Dispersion diagram of the coupled modes near the stopband with finite width

6. Conclusion

In conclusion, this paper presents the theory and simulation of a nano-scaled coupled-line cou-pler based on the guidance of surface plasmon-polaritons atoptical frequencies. The couplercomprises layered dielectric materials and silver, which serve as two stacked nano-transmissionlines to achieve broadside coupling. Comparing to conventional optical dielectric couplers, theguided modes are surface waves that can overcome the diffraction limit in the transverse di-rection of the guided-wave structures. Therefore it has thepotential to be used in nano-scaledcircuits at optical frequencies.

More importantly, this plasmonic coupler is based on the principle of the contra-directionalcoupling phenomenon that involves complex modes previously observed in a microstrip linecoupler designed to operate at microwave frequencies. Thisleads to dramatically reduced cou-pling lengths. The characteristics of the coupled modes areanalyzed from the theoretical deriva-tions as well as simulations using Comsol Multiphysics finite-element solver. Theω −β dis-persion diagrams generated by both approaches agree very well to each other. We observe



that the power can efficiently couple between the FW-wave andthe BW-wave branches givingrise to supermodes that are characterized by complex-conjugate eigenvalues, whose real part(attenuation constant) increases drastically for certainfrequencies. This results in exponentialattenuation along the coupler that leads to dramatically reduced coupling lengths compared topreviously reported co-directional SPP couplers (e.g. from millimeters to submicrons).

The coupler shows 0dB coupling within submicron coupling length for the stopband if con-sidering the lossless Drude model for silver. The coupling level reduces to -3.5dB when theeffect of loss is considered. In addition, the effect of a finite coupler width is also investigated,and it is observed that any width that is greater than 200nm will approximate the coupled modesof the infinite width very well.

Finally, we would like to thank the reviewers who brought to our attention that the exper-imentally measuredεr values of silver depart from the Drude model significantly ata certainfrequency range. This has been investigated in several papers, including [11] and [12]. We haveconducted some preliminary dispersion analysis on the ND-film and ND-gap structures basedon the data in [12]. The results suggest that the operating frequency shifts to a lower value,but the condition that facilitates the concept of contra-directional coupling, that is the couplingbetween a forward and a backward wave as implied by Fig. 3(a),remains intact. Moreover, atthis new operating frequency, the imaginary part of the permittivity of silver is well predictedby the Drude model. The ultimate characterization of this coupler can only be assessed in detailthrough an actual experiment, but this is beyond the scope ofthis present paper.



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An ultra-short contra-directional coupler utilizing surface plasmon-polaritons at optical frequencies