JP_Nanocircuit Loading of Plasmonic Waveguides

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 9, SEPTEMBER 2012 4381

Nanocircuit Loading of Plasmonic WaveguidesAlessia Polemi, Member, IEEE, Andrea Al, Member, IEEE, and Nader Engheta, Fellow, IEEE

AbstractWe apply the optical nanocircuit concepts to modeland design optical nanofilters in realistic plasmonic waveguidesof different nature, including strips and groove waveguides. Thenanocircuit elements are designed to fit the waveguide geometry,and its equivalent impedance is analytically calculated by substi-tuting the role of the conduction current with displacement cur-rent. The effect of plasmonic waveguide walls is rigorously mod-eled in terms of an extra nanocircuit loading that is included inour model. We show via numerical results that the nanocircuit ap-proach may be effectively applied to the design of nanofilters, anal-ogous to familiar concepts at radio-frequencies.

Index TermsNanocircuit, nanoparticles, plasmonics.

I. INTRODUCTION

I N the past few years, the interest in scaling radio frequencyand microwave circuits, such as transmission lines, filters,printed circuit boards (PCBs), to higher frequency regimeshas rapidly grown [1][13]. In particular, nanophotonics andplasmonics represent rapidly emerging research areas that mayenable the realization of novel devices at the nanoscale, inparticular for chemical and biomedical sensing [7], [8], infor-mation and communications technologies [9], [10], enhancedenergy harvesting [11], [12], environmental applications [13],and several other applications [14]. Plasmonics has the poten-tial of scaling the circuit concepts to the visible, as recentlyenvisioned in [15]. The importance of extending the familiarelectronic concepts to the visible range may allow drasticallyreducing the size of circuits and simultaneously increasing theoperating speed by several orders of magnitude. Simply scalingclassic circuit elements to the infrared and visible frequen-cies is not straightforward, since metals lose their conductiveproperties at high frequencies; thus, for the introduction of themetatronic paradigm for optical nanocircuits has been a keyrecent development that may lead to the realization of novelnanoelectronic devices [15][33]. Within this paradigm, fun-damental concepts inspired in plasmonics and metamaterialshave been introduced in order to develop optical nanocircuits,

Manuscript received October 31, 2011; revised February 20, 2012; acceptedApril 09, 2012. Date of publication July 03, 2012; date of current version August30, 2012.A. Polemi is with the Department of Chemistry, Drexel Univer-

sity, Philadelphia, PA 19104 USA (e-mail: [email protected];[email protected]).A. Al is with Department of Electrical and Computer Engineering, The Uni-

versity of Texas at Austin, Austin, TX 78712 USA (e-mail: [email protected]).N. Engheta is with Department of Electrical and Systems Engineering, School

of Engineering and Applied Science, University of Pennsylvania, Philadelphia,PA 19104 USA (e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TAP.2012.2207065

showing how plasmonic and non-plasmonic particles may ef-fectively be employed to realize complex nanocircuit elementsat infrared and optical frequencies. In [16] the nanocircuitdescription of an isolated element has been developed bysubstituting the role of conduction current with displacementcurrent. This way, the ratio between average applied voltageand the total displacement current flowing through the elementhas been used to define an equivalent optical impedance forthe individual element. Nanoparticles with positive real part ofpermittivity (non-plasmonic or dielectric) operate as nanoca-pacitors, while particles with negative real part of permittivity(plasmonic) operate as nanoinductors. Inherent losses of opticalmaterials behave within this paradigm as nanoresistors. Theconnections and coupling among nanocircuit elements has beeninvestigated more closely in [17][21], leading to the conceptof a full optical nanocircuit board [22]. Application of theseideas have been considered in recent years to realize opticalsub-diffractive waveguides [23][25] and nanowires [26] or toload optical nanoantennas [27][30]. One of the most direct ap-plications of this paradigm has been pursued in [31], where theauthors have applied familiar circuit concepts of filter theoryto derive and tailor the transfer function of optical nanofilters(low-pass, high-pass, stop- and band-pass). For this purpose,they embedded suitably designed nanocircuit elements in ametalinsulatormetal (MIM) plasmonic waveguide, whichmay guide the input optical excitation. In this configuration,all the nanoelements were considered infinite in the transversedirection (nanorods). In [31], for simplicity the host waveguidehas been assumed as made by two perfectly electric conducting(PEC) walls, thus a transverse electromagnetic (TEM) modewas assumed as exciting the nanofilters, and no modal dis-persion was taken into account. In a more realistic scenario,at optical frequencies the metallic walls of a parallel platewaveguide are also dispersive [3], [32], and thus we expect anadditional effect on the transfer function of the filter.In this paper, we address in detail this issue, presenting for the

first time the modeling of nanofilters embedded in realistic op-tical waveguides. We assume that the optical excitation is con-fined within plasmonic optical waveguides. We focus on silver,although other metals may be considered with no conceptualdifference. We numerically investigate the transfer function ofthe nanofilter, by means of full wave simulations based on fi-nite integration technique (FIT) [34]. For the sake of simplicity,we start with the case of a MIM with silver walls, in orderto understand and model the effect of plasmonic walls on thenanocircuit model. We then show how the penetration of theelectric field in the silver, due to its finite skin depth, may betreated within the optical nanocircuit theory as an additional se-ries element. The transfer function is modified to take these ef-fects into account, and numerical results are shown to validateour theory. Then, we move to more realistic three dimensional

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4382 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 9, SEPTEMBER 2012

plasmonic waveguides, such as nanostrip and nanogroove plas-monic waveguides. In particular, we focus our attention on therectangular and triangular groove waveguides. For three-dimen-sional nanostrip waveguides, the equivalent impedance of theloading nanoelements is analytically calculated and the effectof field penetration into silver walls is similarly taken into ac-count. Also in this case, the transfer function is obtained withfull-wave simulations and compared with analogous circuit de-signs at lower frequencies. We obtain very good agreement overa wide range of frequencies, before higher order effects appear,not captured by our model. Further complications arise in thecase of triangular waveguides, which is also extensively ana-lyzed in the following.The paper is organized as follows. We first analyze the

concept of optical nanofilters in a 2-D MIM loaded withnanorods, as proposed in [31], but considering here the plas-monic nature of the waveguide metallic walls (Section II). Inorder to be consistent with the terminology adopted in [31]and used throughout the rest of this paper, we refer to MIMas a parallel plate waveguide (PPW) with plasmonic walls.Then, we move to 3-D optical waveguides, and we investigatenanofiltering in nanostrip waveguides (Section III) and rectan-gular nanogrooves (Section IV). Finally, we use nanoprisms toload triangular nanogrooves and we derive the nano-elementequivalent impedance in this scenario (Section V).

II. NANOFILTERING IN PLASMONIC PARALLEL-PLATEWAVEGUIDES

In order to understand how nanoparticles and plasmonicwaveguides interact in terms of optical nanocircuits, in thissection we start analyzing the case of a PPW with silverwalls, loaded by -negative and -positive nanorods, andcombinations thereof. In [31], the transfer functions of suchcombinations of nanoparticles have been investigated to realizeoptical nanofilters, but in that case the waveguide walls havebeen considered as perfectly conducting. Here, we expandthe optical nanocircuit paradigm to also take into account thevariation of the transfer function due to the dispersion featuresof the plasmonic waveguide and the finite skin depth of thewaveguide.

A. PPW Nanofilter With an -Positive NanorodThe waveguide geometry analyzed in this section is shown in

Fig. 1(a), and consists of a PPW with a gap width mmfilled with air. In this first example, the waveguide is loadedwith a simple dielectric nanorod, which, following the theorydeveloped in [31], constitutes an RC nanofilter, where the resis-tance (R) is provided by the characteristic impedance of the lineand the nanocapacitor is given by the optical impedance ofa dielectric nanorod with permittivity , basemm and height mm. In this scenario, we have

chosen the height to be smaller than the waveguide width ,in order to increase the value of , consistent with the designin [31]. For this reason, we use a silver protrusion underneaththe nanorod. The capacitance per unit length of the nanopar-ticle may be evaluated [31] as . The resistanceof the nanofilter is provided by the characteristic impedanceof the plasmonic waveguide. There is no general expression

Fig. 1. (a) Geometry of the PPW loaded by a -positive nanorod. The penetra-tion of the field into the PPW silver wall is qualitatively shown. (b) Equivalentcircuit including the effect of the silver walls as an additional impedance whosevalue is given in (3).

Fig. 2. Amplitude (a) and phase (b) of the transfer function for an opticalnanofilter made of a dielectric nanorod inside a plasmonic PPW, as depictedin Fig. 1(a) with mm. The equivalent circuit is depicted in the inset ofFig. 1(b).

for the line impedance of an inhomogeneous dielectric wave-guide, but within the focus of this paper we can safely assumethat the guided mode is mostly confined in the insulator. Underthis condition, the main effect of the silver walls is to slowdown the modal propagation, producing a longitudinal com-ponent of the electric field that lowers the effective character-istic impedance in the insulator. Its value is given by

, with being the free space impedance, andthe effective refractive index of the plasmonic PPW [25], [31],[35]. This impedance model for plasmonic waveguides has beensuccessfully employed to predict anomalous matched transmis-sion through narrow gratings [36]. By applying this model, wecan calculate the transfer function for this optical nanofilter, asshown in Fig. 2, in amplitude (a) and phase (b), where the dashedline refers to full-wave simulations, and the dotted line refers tocircuit theory applied to the circuit in Fig. 1(b), where the wallsof the waveguide are considered impenetrable [ in themodel of Fig. 1(b)].It is seen that the circuit theory successfully applied in

[31] for a PEC waveguide is not adequate to model realisticsilver walls. Since these walls are plasmonic, the field pen-etrates into the metal, as qualitatively shown in Fig. 1(a),consistent with the silver skin depth , where

is the wavenumber in silver, and is the speedof light. We assume here a Drude model for the silver relativepermittivity , where ,

and .Due to these simple considerations, we can take into accountthese effects considering an equivalent silver nanorod in serieswith the dielectric one, with cross sectional dimensions and

POLEMI et al.: NANOCIRCUIT LOADING OF PLASMONIC WAVEGUIDES 4383

. Since at the frequencies under investigation is negativeand complex, this load has an inductive nature [16], [31]. Wedefine

(1)

and

(2)

such that the additional silver impedance is approximately givenby

(3)

This modifies the basic RC circuit modeling the dielectricnanorod into the one shown in Fig. 1(b), converting the RClow-pass response obtained in a PEC waveguide [31] into astop-band filter. This effect is clearly visible in the amplitudeand phase of the calculated transfer function, as shown in Fig. 2(solid line), which matches with very good approximationthe full-wave simulations. The introduction of an additionalseries impedance modeling the plasmonic effects of the wallseffectively restores the accuracy of the circuit model in thisconfiguration. Notice that if , its contribution inthe shunt connection may be neglected. This happens when

. In the following plots and results,we will similarly associate a solid line to our circuit theoryincluding the plasmonic effect, a dotted line to the circuit theorywithout this effect, and a dashed line to full-wave simulations.

B. PPW Nanofilter Loaded by an -Negative Nanorod

In this section, we analyze a nanofilter formed by an -neg-ative nanorod, as shown in Fig. 3(a). This geometry realizes anRL nanofilter, where in this case the nanoinductor is a plas-monic nanorod with cross sectional dimensions mmand . This nanorod has a permittivity

where we set here rad/s andrad/s. The value of the inductance can be cal-

culated as in [31] as , and a shuntresistance accounts for the losses . Thetransfer function for this optical nanofilter is shown in Fig. 4, inamplitude (a) and phase (b). A full wave simulation is carriedout (dashed line) and compared with circuit theory, with andwithout the extra load correction represented by the samecalculated in the previous subsection. The effect of the silverwalls also in this case is taken into account by adding the sametwo additional series impedances to the nanorod impedance [seeFig. 3(b)]. It is clear from Fig. 1 and Fig. 3 that the effect of theadditional inductance on the overall circuit response is differentfor different configurations, consistent with the different circuitconnection. The filter response is well predicted by the circuitmodel in both scenarios. It is not surprising that when the induc-tance introduced by the skin depth in the silver is in series withanother inductance (Fig. 3) the effect is less drastic than whenit is in series with a capacitance (Fig. 1).

Fig. 3. (a) Geometry of the PPW filled with an -negative nanorod. The pene-tration of the field into the PPW silver wall is qualitatively depicted. (b) Equiv-alent circuit including the effect of the silver walls as an additional impedancewhose value is given in (3).

Fig. 4. Amplitude (a) and phase (b) of the transfer function for the opticalnanofilter made of an -negative nanorod in a plasmonic PPW, as depicted inFig. 3(a). The equivalent circuit is depicted in Fig. 3(b).

Fig. 5. (a) Geometry of a silver PPW loaded by a shunt combination ofand nanorods. The penetration of the field in the silver walls is qualita-tively depicted. (b) Equivalent circuit including the effect of the silver walls asan additional impedance, whose value is given in (3).

C. PPW Nanofilter Formed by Parallel and SeriesCombinations of -Negative and -Positive NanorodsBy combining inductors and capacitors, we can realize a

second-order filter response with more interesting features. Theparallel combination of -negative and -positive nanorods isfirst considered in this subsection, as shown in Fig. 5(a). Aparallel LC filter realizes a passband transfer function, whichwe verify in our design.The two nanoelements considered here are characterized by

the same values of permittivity used in Sections II-A and II-B.They are positioned on top of a silver protrusion forming a gap

mm with the top wall of the PPW. Dimensions are thesame used before, i.e., mm. Results for this con-figuration are shown in Fig. 6 for amplitude (a) and phase (b) ofthe transfer function. The full-wave simulation and the theoret-ical results using the circuit theory considering the correctionassociated with the skin depth of silver agree reasonably well, inboth amplitude and phase, despite a small shift in the passband


Fig. 6. Amplitude (a) and phase (b) of the transfer function for an opticalnanofilter made of a shunt combination of and nanorodsin a silver PPW, as depicted in Fig. 5(a). The equivalent circuit is depicted inFig. 6(b).

Fig. 7. (a) Geometry of the PPW loaded by a series combination ofand nanorods. The penetration of the field into the PPW silver wallis also qualitatively depicted. (b) Equivalent circuit including the effect of thesilver walls as an additional impedance whose value is given in (3).

frequency, associated with local fringing effects not captured bythe nanocircuit model. Also in this case, the correction due to thesilver walls is particularly important to restore the accuracy ofthe nanocircuit model.Then, we investigate the series connection of -negative and-positive nanorods, whose arrangement is shown in Fig. 7(a)and forms a series LC nanofilter, which supports a stop-bandtransfer function. The two nanoelements fill symmetrically thePPW, and they are characterized by the same values of permit-tivity used in Sections II-A and II-B. Results for this configu-ration are shown in Fig. 8 for the amplitude (a) and phase (b)of the transfer function. We emphasize again here that the ef-fect on the overall circuit response of the field penetration insidethe silver walls, which is described by the extra inductance inthe equivalent circuit, is different for different configurations.Nevertheless, the different response is nicely predicted by thedifferent connection between the inductance and the rest of thecircuit. It is expected, for example, that when the extra induc-tance is in series with an LC parallel circuit (Fig. 6) the effect ismore drastic than when it is in series with an LC series (Fig. 7).In both cases, the pass-band and stop-band are captured by ourmodel.

III. PLASMONIC NANOSTRIP WAVEGUIDE

After having confirmed that nanofilters embedded in a plas-monic parallel-plate waveguide may be described using the op-tical circuit theory, properly modified to include the finite con-ductivity of plasmonic metals, in this section, we analyze a morerealistic 3-Dwaveguide configuration. In this case, the plates are

Fig. 8. Amplitude (a) and phase (b) of the transfer function for an opticalnanofilter made of a series combination of and nanorods in aplasmonic PPW, as depicted in Fig. 7(a). The equivalent nanocircuit is depictedin Fig. 7(b).

Fig. 9. (a) Geometry of the PSW, loaded by a parallelepiped dielectric nano-particle. The equivalent filter circuit is shown in (b).

Fig. 10. Amplitude (a) and phase (b) of the transfer function for the low-passoptical nanofilter made of a dielectric nanorod in a PSW.

not infinite in the lateral direction, but they have a finite width. The dispersion properties of this plasmonic strip waveguide(PSW), whose geometry is shown in Fig. 9(a), is similar to thePPW, and its effective refractive index can be efficientlycalculated by applying the effective index method [35][40].As done in Section II, we load the PSW with different

nanoparticles, in order to realize optical nanofilters. In this casethe load is obtained by inserting a parallelepiped nanoparticle,which fills the waveguide cross section, as shown in Fig. 9(a).We apply the circuit theory illustrated for the PPW, wherenow the characteristic impedance of the transmission line is

.

A. RC Nanofilter With an -Positive Nanoparticle Load

The geometry is shown in the inset of Fig. 10, and it refersto a PSW where mm and mm. The dielectricnanoparticle has dimensions mm andmm, and a value of permittivity . The nanofilter

circuit is depicted in Fig. 9(b), where is the value of the ca-pacitive load provided by . The effect of the


Fig. 11. Amplitude (a) and phase (b) of the transfer function for an opticalnanofilter made of an -negative nanoparticle in a PSW.

silver walls can be taken into account with the extra load , asin (3), which is now modified into

(4)

and

(5)

being the silver skin depth. Also in this case,if , then ; thus, in the shuntconnection, the resistance contribution is very small.The transfer function, in amplitude and phase, is shown in

Fig. 10(a) and (b), respectively. Also in this example, the effectof the silver walls introduces a stop-band at higher-frequency,which is well modeled by our improved nanocircuit model, con-sidering the inductance of the walls. Despite the slight devia-tion from the full-wave simulations, the stop-band feature ofthe nanofilter is correctly predicted. The main reason behindthe discrepancy lies in the inadequacy of the EIM to determinethe propagation and attenuation constant of the waveguide, es-pecially when open boundaries are involved, as pointed out in[35].

B. PSW Nanofilter With an -Negative Nanoparticle Load

In this case, we realize an RL nanofilter design by in-serting in the PSW an -negative nanoelement with depth

mm and permittivity following a Drude modelwith rad/s

and rad/s. The geometry is shown in the insetof Fig. 11. The results from the equivalent circuit modelis shown in Fig. 9(b), where represents the nanopar-ticle load, provided by the shunt connection of the in-ductance and the resistance

. The values of and are the sameas in (4) and (5).The amplitude and phase of the transfer function are shown

in Fig. 11(a) and (b), respectively. The high-pass feature of thisnanofilter are also preserved in this 3-D optical waveguide andthe nanocircuit theory predicts the response of this filter withgood accuracy.

Fig. 12. Amplitude (a) and phase (b) of the transfer function for a nanofiltermade of a parallel combination of and nanoparticles in aplasmonic PSW (see inset).

Fig. 13. Amplitude (a) and phase (b) of the transfer function for an opticalnanofilter made of a parallel combination of and nanoparticlein a plasmonic PSW (see inset).

C. PSW Nanofilter Loaded by a Combination of -Negativeand -Positive Nanoparticles

In this subsection, we design shunt and series combinations of-negative and -positive nanoparticles to realize second-ordernanofilters in a PSW. The shunt connection is depicted in theinset of Fig. 12. The two nanoelements have dimensions

mm and mm. The permittivities ofboth the C and L elements are the same as for the cases investi-gate earlier, in Section III-A and III-B, respectively. The equiv-alent circuit model is the one in Fig. 9(b), where is the shuntconnection of the two nanoparticles, and accounts for thesilver effect. Results for this configuration are shown in Fig. 12,showing the amplitude (a) and phase (b) of the transfer function.The series connection of -negative and -positive nanopar-

ticles is shown in the inset of Fig. 13. The two nanoelementshave dimensions mm and mm.in Sections III-A and III-B, respectively. The equivalent circuittheory is shown in Fig. 9(b), where is the series connectionof the two nanoparticles, and accounts for the silver effect.Results for this configuration are shown in Fig. 13. Consistentwith the results for the PPW, also in this 3-D configuration it isindeed possible to tailor the design of combinations of nanopar-ticles to realize low-pass, high-pass, stop-band and pass-bandfilters. The effect of the finite conductivity of silver walls is ac-counted for with additional series inductances, which are wellmodeled by equivalent inductive loads associated with the skindepth field penetration in silver.


Fig. 14. Real and imaginary part of the effective refractive index for the RGWshown on the right, where mm and mm.

Fig. 15. (a) 2-D section of a RGW loaded with a dielectric nanoparticle: thefield penetration into the silver walls is qualitatively sketched. (b) The effect ofthe silver walls is included as loads in the equivalent filter circuit.

Fig. 16. Amplitude (a) and phase (b) of the transfer function for the opticalnanofilter loaded with a -positive nanoparticle . Dimensions:mm, mm, mm.

IV. PLASMONIC RECTANGULAR GROOVE WAVEGUIDEIn this section, in an effort to model even more realistic plas-

monic waveguide designs, we investigate a plasmonic rectan-gular groove waveguide (RGW), whose geometry is shown inFig. 14. In [35], we have shown how this configuration can pro-vide a good approximation of more complicated groove wave-guides, such as the triangular groove waveguide, analyzed inthe next section. The dispersion properties of the RGW may becalculated applying the effective index method (EIM) [3], [35].The effective refractive index (real and imaginary part) inFig. 14 for a RGW where mm and mm. Thefundamental mode here has a cut-off around 300 THz. As donein the previous sections, we investigate some basic filter de-signs, obtained by loading the plasmonic waveguide with -neg-ative and -positive nanoparticles. In this case, the nanopar-ticle used to load the channel is a dielectric nanoparallelepiped,

Fig. 17. Amplitude (a) and phase (b) of the transfer function for an opticalnanofilter loaded with a series combination of and nanoparti-cles (see inset). Dimensions: mm, mm, mm.

with crossectional dimensions , and depth . Hereinafter:mm, mm. Although the field inside the

channel is not necessarily uniform along the dimension asin the previous configurations, we apply the same circuit modelused for the strip waveguide to describe the loads. In the presentcase, as shown in the 2-D sketch in Fig. 15(a), when the grooveis loaded, additional field penetration is expected through thebottom wall, giving rise to an extra in shunt with the particleload , as shown in Fig. 15(b). As explained in Section II, be-cause of the silver properties at these frequencies, the load hasan inductive nature. In particular, we define

(6)

and

(7)

such that

(8)

In Fig. 15(b), the characteristic impedance of the RGW iscalculated as the impedance of the TE mode, normalized to thewaveguide dimensions, i.e., .For brevity, we include here only two results, relevant to an-positive loading (see Fig. 16) and to a series combination of-positive and -negative nanoparticles (see Fig. 17). Detailsof dimensions and permittivities are given in the pertinent cap-tions. The amplitude and phase of the transfer functions for RCand for series LC filters are shown, where the circuit takes intoaccount the effect of all the lateral walls through the loads(3) and (8). These results are then compared with a CST fullwave simulation. The geometry of each filter is shown in theinset.Also for this more complicated 3-D geometry, the agreement

with full-wave simulations is good overall, and both the pre-sented results show how the modeling of extra inductive loadsassociated with the plasmonic nature of the channel is essentialto capture the filter behavior. In fact, the resonance of the filterwould not be modeled otherwise, as it is clear from both Fig. 16and Fig. 17. Results for the RL and shunt RLC circuits, whichare not included here, show a similar accuracy.


Fig. 18. Real and imaginary part of the effective refractive index for the TGWwhose geometry is shown on the right, where mm and mm.

V. PLASMONIC TRIANGULAR GROOVE WAVEGUIDEAs the last example of plasmonic waveguides loaded by

nanofilters, we consider here the triangular groove waveguide(TGW) [37][39], whose geometry is depicted in Fig. 18(b).This can be regarded as a modification of the RGW described inthe previous section, where the lateral silver walls converge to apoint. This is one of the most common plasmonic waveguides,and its analysis in terms of nanocircuit theory may be carriedout building on the previous results with some modifications.The dispersion characteristic for a TGW with dimensions

mm and mm is shown in Fig. 18(a) in termsof real and imaginary part of the effective index .The fundamental mode has a cut off around 250 THz, and it

crosses the light line around 320 THz. Higher order modes aresupported by the waveguide beyond 550 THz, which are notcaptured by the nanocircuit theory. In order to approximate thecharacteristic impedance of this waveguide, we assume that anincremental characteristic admittance in the transverse directioncan be approximately defined as

(9)

where

(10)

By integrating along the groove depth , we obtain

(11)

where has a small, but finite value. In particular,throughout the following sections, we will use mm.In the Section V-D we will explicitly discuss the effect ofthe choice of this value on the overall result. From (11), thecharacteristic impedance may be then derived as

(12)

In order to load the TGW, we will use prism-shaped nanopar-ticles. The -positive and -negative nanoparticles, and a com-bination of them, will be placed in such a way that they fill thetip of the TGW, where the field is more intense [see Fig. 19(a)].This will lead to an equivalent nanofilter circuit as shown inFig. 19(a), where represents the nanoparticle load, andaccounts for silver wall effect.

Fig. 19. (a) Sketch of the cross-section of a TGW loaded by a prism nano-particle: the field penetration in the silver walls is qualitatively sketched.(b) Equivalent nanofilter circuit.

Fig. 20. Amplitude (a) and phase (b)of the transfer function for the nanofiltermade by a prism nanoparticle in a plasmonic TGW. Dimensions:

mm, mm, mm.

A. Loading by an -Positive Nanoparticle

In this section, a small -positive prism nanoparticle (see insetof Fig. 20). The sizes of the nanoparticle are: mm,

mm, mm, while the permit-tivity is . In order to calculate the capacitance associ-ated with the -positive nanoparticle, we define an incrementalcapacitance, in agreement with the formulation used above todefine the line impedance of the TGW:

(13)

where is the same as in (10). By integrating along thenanoparticle depth , we obtain

(14)

Then, the value of in the equivalent circuit model is easilycalculated as . The finite conductivity is also in-cluded here through the extra load , whose inductance andresistance are calculated as in (4) and (5). The effect of is sub-stituted here by [Fig. 19(a)], since the field is concentrated al-most exclusively around the nanoparticle, due to the waveguideshape. The results for this nanofilter are depicted in Fig. 20, con-sistent with the previous figures. It is recognized that the effectof silver walls here is dominant, and the nanofilter has a typicalstop-band response, associated with the series resonance of the(capacitive) dielectric particle and the (inductive) silver walls.Failing to include these effects in the nanocircuit model cannotcapture correctly the nanofilter response.


Fig. 21. Transfer function in amplitude (a) and phase (b) for the nanofilter madeby a prism nanoparticle in a plasmonic TGW( rad/s and rad/s). Dimensions: mm,

mm, mm.

B. Loading With an -Negative NanoparticleIn this case, the prism nanoparticle is realized with an-negative material. In particular, we use a Drude model

where rad/sand rad/s. The sizes of the nanoparticle are:

mm, mm, mm (seeinset of Fig. 21). The inductance associated with the -nega-tive nanoparticle can be obtained, as done above, through anincremental . After some algebraic steps, we can write

(15)

and

(16)

The value of in the nanocircuit model is easily calculated asThe results for this nanofilter are depicted in

Fig. 20, where the transfer function of the filter is shown in am-plitude (a) and phase (b). Although some discrepancies in theupper frequency range, due to higher order modes, the levelof the amplitude of the transmission function is captured onlywith the appropriate inclusion of the silver effect. The improve-ment is less visible in the phase, since both the nanoparticle andthe silver act as equivalent inductors, thus showing a similarbehavior.

C. Combination of -Positive and -Negative NanoparticlesWe combine now -positive and -negative nanoparticles, in

shunt and series connections, as done in the previous sections,to realize second-order nanofilter responses.For the shunt connection, we use a pair of prism nanoparti-

cles, as in the inset of Fig. 22, with same values of permittivi-ties and sizes equal to those used in the sections above. For thecase of a series connection, we use a configuration of -positiveand -negative as depicted in the inset of Fig. 23. The resultsfor these two nanofilters are depicted in Fig. 22 and in Fig. 23,where the transfer functions of the filters are shown in amplitude(a) and phase (b). Also these two results show how the penetra-tion of the field in the silver walls represents a crucial effect tobe taken into account. We have modeled this effect on the basisof a skin depth rule which, except for higher order effects due

Fig. 22. Amplitude (a) and phase (b) of the transfer function for a nanofiltermade by a shunt combination of andprism nanoparticles in a plasmonic TGW ( rad/s and

rad/s). We compare full-wave (CST) simulations (dashed line) to circuittheory with (solid line) and without the inclusions of the silver effects (dottedline). Dimensions: mm, mm, mm.

Fig. 23. Amplitude (a) and phase (b) of the transfer function for a nanofiltermade by a series combination of andprism nanoparticles in a plasmonic TGW ( rad/s and

rad/s). Dimensions: mm, mm,mm.

to the non uniformity of the field in the triangular groove, ap-pears to be reliable, and it allows us to capture the filter responsein all the cases presented so far.

D. Influence of the Choice of on the Transfer Function

We briefly discuss in this section how the choice of theparameter affects the transfer function of the filter in thisnanogroove example. We refer to the two basic cases of -pos-itive and -negative loading, as described in Section V-A andV-B, respectively. Fig. 24 shows the amplitude of the transferfunction for the above mentioned cases, as a function of theparameter (small values), for different frequencies. Even ifthe impedances in [(12), (14), (15)] individually tend to zero forsmall , only their relative values are relevant to calculate thetransfer function of the nanofilter. Except for values of veryclose to zero, the transfer function has constant values with ,ensuring that we can choose arbitrary values, as long as theyare not too small (above few nm) to calculate the impedancesinvolved in the nanocircuit model. The non-singular behaviorof the transfer function for , may also be demonstratedanalytically. In fact, the transfer function may be written ingeneral as

(17)


Fig. 24. Amplitude of the transfer function in terms of the parameter fordifferent frequencies. (a) -positive loaded filter; (b) -negative loaded filter.

where is defined in (12), andis the total load of the filter (see Fig. 9 as a refer-

ence). For the -positive loaded filter, , whereis defined in (14). Thus, after some algebraic manipulations,

the transfer function can be rewritten as

(18)

where and . It is easy tosee that . Similarly for -positive loaded filter,where with and defined as in(15) and (16), respectively, the transfer function can be rewrittenas

(19)

where.

Also in this case, . Evidently, too smallvalues of provide a meaningless transfer function, since thesingularity at the tip dominates the integrals. For this reason, inthe previous simulations we have used mm, whichprovides excellent numerical agreement between ourfull-wave simulations and the nanocircuit model. Variationsof around this value would not perturb the resultsshown in the previous figures.

VI. CONCLUSIONIn this paper, we have shown how optical nanocircuit theory

may be used to tailor and design optical nanofilters loading re-alistic plasmonic waveguides. We have extended the recentlyproposed concept of optical nanofilters to consider 2-D and 3-Drealistic plasmonic waveguides, such as strips and groove wave-guides. We have shown that the circuit concepts may be con-sistently applied to different designs and waveguide configu-rations. The obtained transfer function is affected by the dis-persion of the waveguide walls and finite skin depth, but theseeffects may be accurately taken into account in the frameworkof optical nanocircuit theory, by considering additional induc-tive loads, simply associated with the field penetration in themetal. Full-wave numerical simulations have been carried onto test the effectiveness of our method, which opens interestingscenarios in applying simple circuit formulas to the design ofoptical nanofilters.

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Alessia Polemi (S00M04) was born inCasteldelpiano, Italy, on July 10, 1973. Shereceived the Dr. Ing. degree (cum laude) in telecom-munications engineering and the Ph.D. degree ininformation engineering-electromagnetic fields fromthe University of Siena, Siena, Italy, in July 1999and March 2003, respectively.From January 2003 to October 2006, she was a

Postdoctoral Researcher at the University of Siena.In November 2006, she was an Assistant Professorof electromagnetic fields in the Department of Infor-

mation Engineering, University of Modena and Reggio Emilia, and was alsothe Italian Student Adviser for the Institution of Engineering and Technology(IET). In 2008, she was a Visiting Professor at the University of Pennsylvania,Philadelphia, working on plasmonic structures. She is now a Research Scientistat Drexel University, Philadelphia, working on the electromagnetic interactionbetween molecules and nanoparticles, on the optimization of SERS nanosur-faces, and on nanoantennas. Her scientific background includes high-frequencyscattering theories, asymptotics electromagnetic methods, numerical electro-magnetic methods, periodic structures, bandgap structures, antenna design, andRFID systems. Her current research is oriented to plasmonics, optical nanoan-tennas, thin film photovoltaics, molecule-metal energy transfer.

Andrea Al (S03M07) received the laurea, M.S.,and Ph.D. degrees from the University of Roma Tre,Rome, Italy, in 2001, 2003 and 2007, respectively.He is an Assistant Professor at The University of

Texas at Austin. From 2001 to 2008, he has been pe-riodically working at the University of Pennsylvania,Philadelphia, where he has also developed signif-icant parts of his Ph.D. and postgraduate research.He is the coauthor of over 160 journal papers and 16book chapters, with over 3000 citations. His currentresearch interests span over a broad range of areas,

including metamaterials and plasmonics, electromagnetics, optics and pho-tonics, acoustics, cloaking and transparency, nanocircuits and nanostructuresmodeling, miniaturized antennas and nanoantennas, RF antennas and circuits.Dr. Al is currently an Associate Editor of the IEEE ANTENNAS AND

WIRELESS PROPAGATION LETTERS and of Optics Express, Guest Editor ofseveral, special issues on metamaterials and an OSA Traveling Lecturer. Hehas been the recipient of several international awards and recognitions forhis research studies, among which the 2012 SPIE Early Career Achievement

Award, the 2011 Issac Koga Gold Medal from URSI, an NSF Faculty Early Ca-reer Development (CAREER) Award, an AFOSR Young Investigator Award,a DTRA Young Investigator Award, the L. B. Felsen Award for Excellence inElectrodynamics, four URSI Young Scientist Awards and the Raj Mittra TravelGrant Young Researcher Award. His students have also received several bestpaper awards, including the 1st prize student paper award at the IEEE Antennasand Propagation Symposium in 2011.

Nader Engheta (S80M82SM89F96) re-ceived the BS degree in electrical engineering fromthe University of Tehran, the MS and Ph.D. degreesin electrical engineering both from the CaliforniaInstitute of Technology (Caltech) in Pasadena,California.He is the H. Nedwill Ramsey Professor of Elec-

trical and Systems Engineering, and Professor of Bio-engineering at the University of Pennsylvania. Afterspending one year as a Postdoctoral Research Fellowat Caltech and four years as a Senior Research Sci-

entist as Kaman Sciences Corporations Dikewood Division in Santa Monica,he joined the faculty of the University of Pennsylvania, where he is currentlythe Ramsey Professor. He is also a member of the Mahoney Institute of Neu-rological Sciences. He was the graduate group chair of electrical engineeringfrom July 1993 to June 1997. His current research interests and activities spanover a broad range of areas including metamaterials and plasmonics, nanoop-tics and nanophotonics, nanocircuits and nanostructures modeling, bio-inspired/biomimetic polarization imaging and reverse engineering of polarization vision,miniaturized antennas and nanoantennas, hyperspectral sensing, biologicallybased visualization and physics of sensing and display of polarization imagery,through-wall microwave imaging, fields and waves phenomena, fractional op-erators and fractional paradigm in electrodynamics. He has published numerousjournal papers, book chapters, and conference articles in his fields of research.Prof. Engheta was selected as one of the Scientific American Magazine 50

Leaders in Science and Technology in 2006 for developing the concept of opticallumped nanocircuits, and he is a Guggenheim Fellow, an IEEE Third Millen-nium Medalist, a Fellow of the American Physical Society (APS), Optical So-ciety of America (OSA), American Association for the Advancement of Science(AAAS), and of the SPIE-International Society for Optical Engineering, and therecipient of the 2008 George H. Heilmeier Award for Excellence in Researchfrom UPenn, the Fulbright Naples Chair Award, NSF Presidential Young In-vestigator award, theUPS Foundation Distinguished Educator term Chair, andseveral teaching awards including theChristian F. and Mary R. Lindback Foun-dation Award, theW. M. Keck Foundations 1995 Engineering Teaching Excel-lence Award, and two times recipient of S. ReidWarren, Jr. Award. He is selectedto receive the 2012 IEEE Electromagnetics Award. He was an Associate Editorof The IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS (20022007),of the IEEE TRANSACTIONS ON ANTENNA AND PROPAGATION (19962001), andof Radio Science (19911996). He was on the Editorial Board of the Journal ofElectromagnetic Waves and Applications. He is currently on the Editorial boardof journal Metamaterials, on the board of Journal Waves in Random and Com-plex Media, and on the Editorial Board of the Istituto Superiore Mario BoellaBook Series in Radio Science. He served as an IEEE Antennas and PropagationSociety Distinguished Lecturer for the period 19971999. He is a member ofSigma Xi, Commissions B, D, and K of the U.S. National Committee (USNC) ofthe International Union of Radio Science (URSI), and a member of the Electro-magnetics Academy. He was the Chair of the Commission B of USNC-URSI for20092011. He is the Chair of the Gordon Research Conference on Plasmonicsin 2012. He has organized and chaired various special sessions in internationalsymposia and conferences, and has guest edited/coedited several special issues,namely, the special issue of Journal of Electromagnetic Waves and Applicationson the topic of Wave Interaction with Chiral and Complex Media in 1992,part special issue of the Journal of the Franklin Institute on the topic of An-tennas and Microwaves (from the 13th Annual Benjamin Franklin Symposium)in 1995, special issue of the Wave Motion on the topic of Electrodynamics inComplex Environments (with L. B. Felsen) in 2001, special issue of the IEEETRANSACTIONS ON ANTENNAS AND PROPAGATION on the topic of Metamate-rials (with R. W. Ziolkowski) in 2003, special issue of Solid State Commu-nications on the topic of Negative Refraction and Metamaterials for OpticalScience and Engineering (with G. Shvets) in 2008, and the special issue ofthe IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS on thetopic of Metamaterials (with V. Shalaev, N. Litchinitser, R. McPhedran, E.Shamonina, and T. Klar) in 2010. He coedited (with R. W. Ziolkowski) thebook Metamaterials: Physics and Engineering Explorations byWileyIEEEPress, 2006

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