Multi-Conductor Transmission Line Networks in Analysis of Side-CoupledMetal-Insulator-Metal Plasmonic Structures

Ali Eshaghian, Meisam Bahadori, Mohsen Rezaei, Amin Khavasi, Hossein Hodaei, and Khashayar Mehrany?

Department of Electrical Engineering, Sharif University of Technology, Tehran 11155–4363, Iran

?Corresponding author: mehrany@sharif.edu

Abstract

An approximate and accurate enough multi-conductor transmission line model is developed for analysis of side-coupled metal-insulator-metal (MIM) waveguides. MIM waveguides have been already modeled by single conductor transmission lines. Here,the side coupling effects that exist between neighboring plasmonic structures are taken into account by finding appropriate valuesfor distributed mutual inductance and mutual capacitance between every two neighboring conductors in the conventional single-conductor transmission line models. In this manner, multi-conductor transmission line models are introduced. Closed-form expres-sions are given for the transmission and reflection of miscellaneous MIM plasmonic structures, e.g. dual-stop-band and band-passfilters. In all examples, the results of the proposed model are compared against the fully numerical finite-difference time-domain(FDTD) method. The results of the analytical model are in good agreement with the numerical results.

Keywords: Distributed circuits, Optical filters, Side-coupling, Surface waves, Transmission lines

1. Introduction

The characteristic feature of plasmonics is to confine theelectromagnetic energy within a sub-wavelength scale far be-low the diffraction limit [1, 2, 3, 4]. As far as analysis anddesign of plasmonic structures are concerned, such a strongconfinement has two opposite aspects. The negative side is thenecessity of going through time-consuming brute-force numer-ical analysis, which does not impart much physical insight. Thepositive side is the availability of accurate enough approximatemethods based on easy-to-solve distributed circuit networks,which can facilitate the design procedure considerably [5, 6, 7].

Since the planar Metal-Insulator-Metal (MIM) waveguidehas been successfully modeled by a simple distributed circuit;viz. a single conductor transmission line [5, 8], much atten-tion is directed toward plasmonic structures that are made ofMIM waveguides. Namely, MIM bends [5], splitters [5, 7], de-multiplexers [9, 10, 11, 12], stub filters [13, 14, 6], resonators[15, 16], and junctions [13, 17, 18] are all successfully mod-eled by replacing each waveguide section in the structure withits corresponding single conductor transmission line. Recently,the idea of using single conductor transmission line in model-ing of plasmonic structures made of MIM waveguides has beenextended to model side coupling between two adjacent singlemode MIM waveguides in the structure, e.g. directional cou-pler, and side coupled waveguide resonator [19, 20]. The ex-tended model is made of two coupled single conductor trans-mission lines and thus cannot account for the coupling effectsthat exist among more than two neighboring MIM waveguides.Although the coupling between non-adjacent MIM waveguidesis practically negligible, a multi-conductor transmission line

model is still needed to analyze structures such as dual-stop-band filters [21], cavity based band-pass filters, and coupledstub configurations.

Here, for the first time to the best of our knowledge, a multi-conductor transmission line with distributed capacitance and in-ductance matrices is introduced to model the coupling effectsthat exist among more than two neighboring MIM waveguides.The electrical characteristics of the multi-conductor transmis-sion line in the proposed model are given by closed-form ex-pressions. Thanks to the negligibility of the coupling betweennon-adjacent MIM waveguides, the per-unit-length capacitanceand inductance matrices of the proposed multi-conductor trans-mission line can be approximated by symmetric tridiagonal ma-trices [22]. Every diagonal element of the matrix representsthe per-unit-length self-capacitance or self-inductance of oneof the conductors in the proposed multi-conductor transmis-sion line model. Since each conductor in the model representsone of the coupled MIM waveguides in the structure and sinceself-coupling effect is reasonably weaker than mutual-couplingeffect, the diagonal elements associated with that conductorcan be approximately obtained in terms of the per-unit-lengthself-capacitance and self-inductance of the single conductortransmission line that would represent the same MIM waveg-uide in the absence of coupling effect. Furthermore, the up-per and lower diagonal elements are approximated by the per-unit-length mutual-capacitance and mutual-inductance of thealready introduced two coupled single-conductor transmissionlines representing the coupling between adjacent MIM waveg-uides in the structure. The success of the proposed model isindebted to the fact that accurate analysis of side-coupling ef-fects between neighboring MIM structures does not necessitate

Preprint submitted to Elsevier October 16, 2013

consideration of the below cut-off higher order modes. Rather,consideration of the coupling between fundamental modes ofthe structures, which are represented by the conductor lines ofthe proposed multi-conductor transmission lines, is usually suf-ficient. This fact is numerically demonstrated.

This paper is organized as follows: first, the formulation ofthe proposed multi-conductor transmission line model, its pa-rameters, and its current and voltage distributions, are givenby closed-form expressions. The proposed distributed circuitmodel is then applied to analyze miscellaneous MIM structures,viz. a dual-stop-band filter based on rectangular cavity res-onators, a band-pass filter, and MIM waveguide with coupledstubs. The accuracy of the proposed model in all cases is jus-tified by the well-known finite-difference time-domain (FDTD)method. Finally, the conclusions are drawn in section 4.

2. Formulation of the proposed model

The schematic view of the considered structure made of nparallel MIM plasmonic waveguides is shown in Fig. 1. Thewidth of the ith waveguide is denoted by wi, and the distancebetween ith and (i + 1)th waveguides is denoted by di. The per-mittivity of the dielectric and metallic regions are representedby εd, and εm, respectively. It is the aim of this section to pro-pose an n-conductor transmission line model that accounts forthe coupling effects between the n parallel MIM waveguides inthe structure. This model is to be applied in analysis of miscel-laneous devices in the following sections.

The telegrapher’s equations for the n-conductor transmissionline model read as [23]:

∂

∂zV = − jωL I (1a)

∂

∂zI = − jωC V (1b)

where ω stands for the angular frequency of the harmonic field,V and I are voltage and current vectors whose ith elements rep-resent transverse electric and magnetic fields of the ith MIMwaveguide, respectively:

V =[V1 V2 . . . Vn

]T(2a)

I =[I1 I2 . . . In

]T(2b)

In these expressions, the per-unit-length inductance; L =

[Li j]n×n, and capacitance; C = [Ci j]n×n, matrices of the n-conductor transmission line model are to be found in terms ofwi, di, εd, εm, and ω. Inasmuch as L and C matrices do notvary along the propagation direction; z, the voltage and currentdistributions are given as:(

VI

)(z) = exp(−Gz)

(VI

)(0) (3)

where

G = jω(0n×n LC 0n×n

)(4)

Figure 1: A typical structure made of n side-coupled MIM plasmonic waveg-uides.

As already mentioned, the coupling effects between nonadja-cent waveguides are usually negligible and thus the n-conductortransmission line model can be approximated by neglectingLi j and Ci j for |i − j| > 1. This is schematically shown inFig. 2, where the self-inductance and the self-capacitance ofthe ith conductor, and the mutual-inductance and the mutual-capacitance between the ith and (i+1)th conductors are denotedby Lsi , Csi , Lmi , and Cmi , respectively. Applying Kirchhoff’svoltage and current laws proves that the L and C matrices inthe telegrapher’s equations can be approximated by the follow-ing tridiagonal matrices:

L ≈[Lsiδi j + Lmi

(δi( j+1) + δ(i+1) j

)]n×n

(5a)

C ≈[(

Csi + Cm(i−1) + Cmi

)δi j −Cmi

(δi( j+1) + δ(i+1) j

)]n×n

(5b)

in which δi j is the kronecker delta:

δi j =

1 if i = j0 if i , j

(6)

It is worth noting that Lsi and Csi stand for the per-unit-lengthself-inductance and the self-capacitance of the ith conductor inthe presence of the coupling between neighboring conductors,and therefore, differ from the per-unit-length self-inductanceand the self-capacitance of the ith conductor in isolation. Still,the coupling-induced perturbation of Lsi and Csi is weak enoughto justify resorting to the closed-form expressions that havebeen already proposed for an isolated MIM waveguide [19]:

Csi = ε0εd/wi (7a)

Lsi = wi β2i /(ω

2ε0εd) (7b)

where βi is the complex propagation constant of the ith MIMwaveguide in isolation, and wi is its width normalized to theunit length in the y direction [7].

Similarly, the coupling-induced perturbation of Lmi , and Cmi

caused by the presence of all other conductors (save the ithand (i + 1)th conductors themselves) can be, to some good ex-tent, neglected and thus the mutual-inductance and the mutual-capacitance between the ith and (i + 1)th conductors can beapproximated by resorting to the closed-form expressions that

2

Figure 2: The proposed n-conductor transmission line model for the structurein Fig. 1. It is assumed that Li j = 0 and Ci j = 0 for |i − j| > 1.

have been already proposed for the mutual-inductance and themutual-capacitance between two adjacent waveguides [19]:

Lmi = g0i −

√g2

0i+ g1i (8a)

Cmi =β2

ei+ β2

oi− β2

i − β2i+1

ω2(Lsi + Ls(i+1) − 2Lmi

) (8b)

in which

g0i =1

4CsiCs(i+1)

[(Csi + Cs(i+1) )(β

2ei

+ β2oi

)/ω2 − LsiC2si

− Ls(i+1)C2s(i+1)

]−

(Lsi + Ls(i+1)

)/4 (9a)

g1i =(βi βi+1)2 − (βei βoi )

2

ω4 CsiCs(i+1)

+Lsi Ls(i+1) (Csi + Cs(i+1) )CsiCs(i+1) (Lsi + Ls(i+1) )

×(β2

ei+ β2

oi− β2

i − β2i+1

)/ω2 (9b)

where βei and βoi denote the propagation constant of the evenand odd supermodes in a structure made of the ith and (i + 1)thMIM waveguides when the rest of the waveguides are absent.

3. Applications of the proposed model

The accuracy of the proposed model in analysis of miscel-laneous MIM structures, viz. a dual-stop-band filter based onrectangular cavity resonators [21], a band-pass filter based onslot cavity, and MIM waveguide with coupled stubs is numeri-cally shown in this section.

It should be pointed out that all MIM structures includingthose considered in this manuscript are three-dimensional in

practice. Therefore, they are inevitably made of the three-dimensional counterpart of the MIM waveguide, viz. the plas-monic slot waveguide [24, 25, 26]. Fortunately, the plasmonicslot waveguide can be fabricated by using the standard nano-fabrication techniques including electron beam lithography, fo-cused ion milling, and etching processes [24, 27]. It shouldbe also noted that field effect modulation of charge carriersin metal-oxide-silicon (MOS) geometry can successfully imi-tate MIM waveguides and structures [28, 29]. Therefore, thestandard CMOS process can also be employed to realize MIMstructures.

The major deviation of the two-dimensional theoretical anal-ysis in this paper from the experimental three-dimensional real-ization would be due to the out-of-plane leakage from the struc-ture, which is certain to occur at junctions and interfaces, e.g.at bends, stubs, and splitters [24]. The out-of-plane leakageis neglected in this work because the considered structures aretwo-dimensional and therefore uniform along the out-of-planedirection. Fortunately, the out-of-plane leakage becomes lesspronounced at longer wavelengths.

Through out this section, the well-known Drude model [3] isemployed for the permittivity of the metallic region:

εm(ω) = ε∞ −ω2

p

ω2 − jωγ(10)

The dielectric permittivity is set to εd = 1 (the relative per-mittivity of vacuum), the relative permittivity of silver at veryhigh frequencies is ε∞ = 3.7, the bulk plasma frequency andthe collision frequency is respectively set to ωp = 1.38 × 1016

Hz, and γ = 2.73 × 1013 Hz [30]. The obtained results of theproposed model are justified by two-dimensional FDTD solu-tions. The FDTD simulations are carried out by using perfectlymatched layers (PMLs) boundary conditions and the grid sizesof 2 nm.

3.1. Dual-stop-band plasmonic filter

As the first example, a typical dual-stop-band plasmonic fil-ter based on two side-coupled nano cavities (SCNCs) [21] isanalyzed by using the proposed model. The schematic of thestructure is shown in Fig. 3(a). The widths of the upper SCNC,the waveguide, and the lower SCNC are denoted by w1, w2, andw3 respectively. The coupling length and the distances betweenthe waveguide and the upper and lower SCNCs are denoted byl, d1, and d2 respectively.

The corresponding model of this structure is shown in Fig.3(b). The upper SCNC, the waveguide, and the lower SCNCare each represented by a transmission line whose per-unit-length self-inductance and self-capacitance are denoted by Lsi ,and Csi (i=1, 2, 3), respectively. In accordance with Eq. (7)of the previous section, these per-unit-length circuit elementscan be written in terms of the complex propagation constantsof the individual MIM waveguides that form the two SCNCs(β1 and β3) and the middle waveguide of the structure (β2).The side coupling between the upper SCNC, the waveguide,and the lower SCNC is modeled by considering the per-unit-length mutual-inductance and mutual-capacitance between the

3

Figure 3: (a) A typical dual-stop-band plasmonic filter based on side-couplednano cavities (SCNCs). (b) Its corresponding multi-conductor transmission linemodel.

upper (lower) SCNC, and the waveguide. In accordance withEqs. (8) and (9) of the previous section, these per-unit-lengthcircuit elements can be written in terms of the complex prop-agation constants of the even and odd supermodes supportedby the coupled MIM waveguides that represent the couplingbetween the upper (lower) SCNC, and the waveguide. In thisfashion, the side-coupling effects are taken into account by athree-conductor transmission line whose per-unit-length induc-tance, L, and capacitance, C, matrices are as follows:

L ≈

Ls1 Lm1 0Lm1 Ls2 Lm2

0 Lm2 Ls3

(11a)

C ≈

Cs1 + Cm1 −Cm1 0−Cm1 Cm1 + Cs2 + Cm2 −Cm2

0 −Cm2 Cs3 + Cm2

(11b)

Here, the per-unit-length mutual-inductance and mutual-capacitance between the upper (lower) SCNC, and the waveg-uide are denoted by Lm1 (Lm2 ), and Cm1 (Cm2 ), respectively.

As it is shown in Fig. 3(b), the transmission lines that cor-respond to the upper and lower SCNCs should be of length land be terminated by appropriate load impedance, ZLi (i=1, 3).Since the complex value of the appropriate load impedance atthe terminal ports of the transmission line should comply withthe reflection coefficient of the fundamental mode of the MIMwaveguide that forms the SCNC, and since that reflection canbe accurately approximated by the Fresnel’s reflection coeffi-cient [6, 13], ZLi (i=1, 3) can be written as:

ZLi = Zwi

√εd

εm(12)

Figure 4: The transmitted power spectrum of the dual-stop-band filter in Fig.3. The results of the proposed model and of the FDTD are plotted by solid anddotted lines, respectively. Inset shows transverse magnetic field profiles at threedifferent wavelengths.

where Zwi is the characteristic impedance of the ith MIMwaveguide [5] and can be written as :

Zwi = wiβi

ωε0εd(13)

Now that the parameters of the multi-conductor transmissionline model are all given, the reflection; R, and transmission; T ,coefficients of the dual-stop-band plasmonic filter shown in Fig.3(a) can be easily obtained:

T = det(AT)/det(A) (14a)R = det(AR)/det(A) (14b)

where

A =[A1 A2 A3 A4 A5 A6

](15a)

AT =[A1 b A3 A4 A5 A6

](15b)

AR =[A1 A2 A3 A4 b A6

](15c)

and

A1 =

(ZL1 e1

e1

), A2 =

(e2

e2/Zw2

), A3 =

(ZL3 e3

e3

)(16a)

A4 = E(ZL1 e1−e1

), A5 = E

(−e2

e2/Zw2

), A6 = E

(ZL3 e3−e3

)(16b)

b = E(

e2e2/Zw2

), E = exp(−Gl) (16c)

e1 =

100 , e2 =

010 , e3 =

001 (16d)

The accuracy of the above-mentioned expressions for a casewhere w1 = 50 nm, w2 = 60 nm, w3 = 80 nm, d1 = 20 nm, d2= 20 nm, and l = 225 nm is shown in Fig. 4, where the trans-mission coefficient is plotted versus the free space wavelength(solid) and is compared against the FDTD result (dotted).

4

3.2. Plasmonic band-pass filter based on slot cavity

As another example, a typical band-pass plasmonic filterbased on a single SCNC is considered. The schematic of thestructure is shown in Fig. 5(a). In accordance with the figure,the widths of the upper MIM waveguide, the SCNC, and thelower MIM waveguide are denoted by w1, w2, and w3, respec-tively. The coupling length between the SCNC and the MIMwaveguides is l1 and the length of the SCNC is l1 + l2. The dis-tance between the SCNC and the upper and lower MIM waveg-uides are d1, and d2, respectively.

The corresponding model of this structure is shown in Fig.5(b). The upper and lower MIM waveguides, and the SCNC intheir between are each represented by a transmission line whoseper-unit-length self-inductance and self-capacitance are de-noted by Lsi , and Csi (i=1, 2, 3), respectively. Once again, theseper-unit-length circuit elements can be written in terms of thecomplex propagation constants of the individual MIM waveg-uides that correspond to the upper and lower MIM waveguidesand the SCNC of the structure. The side-coupling between theSCNC and the upper and lower MIM waveguides is then mod-eled by considering the per-unit-length mutual-inductance andmutual-capacitance between the upper (lower) MIM waveg-uide, and the SCNC. They are respectively denoted by Lm1

(Lm2 ), and Cm1 (Cm2 ), and can be written in terms of the com-plex propagation constants of the even and odd supermodessupported by the coupled MIM waveguides that represent thecoupling between the upper (lower) MIM waveguide and theSCNC. Much like the previous subsection, the side-couplingeffects are taken into account by a three-conductor transmis-sion line whose per-unit-length inductance, L, and capacitance,C, matrices are governed by Eqs. (11a) and (11b).

It is worth noting that the length of the three-conductor trans-mission line is equal to the coupling length; l1, and differs fromthe length of the SCNC; l1+l2. Therefore, the distributed circuitthat represents the SCNC is made of two sections. One sectionis side coupled to the upper and lower MIM waveguides, andis therefore part of the three-conductor transmission line whoselength is l1. The other section represents the uncoupled partof the SCNC, and is therefore modeled by a single-conductortransmission line whose length and per-unit-length inductanceand capacitance are l2, Ls2 , and Cs2 , respectively.

The transmission lines that represent the upper and lowerMIM waveguides are terminated at their right-hand-side portby load impedances ZL1 , and ZL3 , respectively. The transmis-sion line that represents the SCNC is terminated at its both portsby load impedance ZL2 . The complex values of the appropriateload impedances can be found by using Eq. (12).

Now that the parameters of the multi-conductor transmissionline model are all given, the reflection; R, and transmission; T ,coefficients of the band-pass plasmonic filter shown in Fig. 5(a)can be easily obtained:

T = det(AT)/det(A) (17a)R = det(AR)/det(A) (17b)

Figure 5: (a) A typical band-pass plasmonic filter based on a side-coupled slotcavity. (b) Its corresponding multi-conductor transmission line model.

where

A =[A1 A2 A3 A4 A5 A6

](18a)

AT =[A1 A2 A3 b A5 A6

](18b)

AR =[A1 A2 A3 A4 b A6

](18c)

and

A1 =

(ZL1 e1

e1

), A2 =

(Zine2

e2

), A3 =

(ZL3 e3

e3

)(19a)

A4 = E(−e1

e1/Zw1

), A5 = E

(−e3

e3/Zw3

), A6 = E

(ZL2 e2−e2

)(19b)

b = E(

e3e3/Zw2

), Zin = Zw2

ZL2 + jZw2 tan(β2l2)Zw2 + jZL2 tan(β2l2)

(19c)

e1 =

100 , e2 =

010 , e3 =

001 , E = exp(−Gl1) (19d)

The accuracy of the above-mentioned expressions for a band-pass filter tuned at 0.7µm with parameters w1 = 50 nm, w2 = 50nm, w3 = 50 nm, d1 = 30 nm, d2 = 30 nm, l1 = 205 nm, andl2 = 0 nm is shown in Fig. 6, where the transmission coeffi-cient is plotted versus the free space wavelength (solid) and iscompared against the FDTD result (dotted).

3.3. Plasmonic stub filterWithout losing generality, a typical plasmonic stub filter

made of an MIM waveguide of width w3, and two stubs ofwidths w1, and w2 is considered. Fig. 7(a) shows the schematicview of this structure. The lengths of the first and second stubsare h + ∆h, and h, respectively. The distance between the twostubs is denoted by d0.

5

Figure 6: The transmitted power spectrum of the band-pass filter in Fig. 5.The results of the proposed model and of the FDTD are plotted by solid anddotted lines, respectively. Inset shows transverse magnetic field profiles at twodifferent wavelengths.

The corresponding model of this structure is shown in Fig.7(b). The MIM waveguide is modeled by a single conductortransmission line whose per-unit-length circuit elements are de-noted by Ls3 and Cs3 . The terminations of the two stubs aremodeled by complex load impedances; ZL1 and ZL2 . It is worthnoting that ZL1 , ZL2 , Ls3 and Cs3 are all already given [see Eqs.(7) and (12)]. In the conventional transmission line model [13],the first and second stubs are modeled by two uncoupled single-conductor transmission lines of length h + ∆h, and h, respec-tively. The per-unit-length self-inductance and self-capacitanceof these two transmission lines; Lsi and Csi (i = 1, 2), are deter-mined in terms of the propagation constants and widths of theMIM waveguides that form the stub sections. What makes theproposed model different from the conventional transmissionline model is that the coupling between the neighboring stubsis no longer neglected. This could be a matter of consequencewhen d0 is small enough to cause a strong coupling.

To include the coupling effects in the proposed model, theper-unit-length mutual inductance, Lm1 , and mutual capaci-tance, Cm1 , between the hitherto uncoupled single-conductortransmission lines that represent the stub sections are takeninto account. Much like the previous subsections, these per-unit-length mutual circuit elements can be found in terms ofthe complex propagation constants of the even and odd super-modes. These supermodes are supported by the coupled MIMwaveguides that represent the coupling between the neighbor-ing stubs. In this manner, the first and second stubs are mod-eled together; even though the extra length of the first stub isleft to be modeled by a single conductor transmission line oflength ∆h. Evidently, the per-unit-length self-inductance andself-capacitance of this single-conductor transmission line areLs1 and Cs1 .

Two numerical examples are given. First, it is assumed thatw1 = 50 nm, w2 = 50 nm, w3 = 50 nm, h = 300 nm, ∆h = 0nm, and d0 = 100 nm. The transmission coefficient of the struc-

Figure 7: (a) A typical MIM plasmonic structure with two neighboring stubs.(b) Its corresponding multi-conductor transmission line model.

ture is obtained by using the proposed circuit model and theFDTD method. The obtained results are plotted versus the freespace wavelength in Fig. 8(a). It is worth noting that the resultsof the proposed model and the conventional model [13] wherethe coupling effects are neglected virtually overlap. Second,the distance between the neighboring stubs in the same struc-ture is decreased to d0 = 20 nm. The transmission coefficientof the structure is once again obtained by using the proposedcircuit model, the conventional transmission line model [13],and the FDTD method. The obtained results are plotted versusthe free space wavelength in Fig. 8(b). This time, the resultsof the conventional approach differ from the results of the pro-posed model. Since the latter is more close to the FDTD results,

(a) (b)

Figure 8: The transmitted power spectrum of the structure in Fig. 7 when (a)d0 = 100 nm and the coupling is weak, and (b) d0 = 20 nm and the coupling isstrong. The results of the proposed multi-conductor transmission line model, ofthe conventional transmission line model, and of the FDTD method are plottedby solid, dashed, and dotted lines respectively.

6

Figure 9: (a) The structure of a plasmonic loop feedback. (b) The proposedtransmission line model for analyzing the structure.

the coupling effects should be included, particularly at shorterwavelengths.

3.4. Plasmonic feedback configurationFinally, a recently proposed structure having a plasmonic

feedback loop [31] is considered . The schematic view of thisstructure is shown in Fig. 9(a). The corresponding multi-conductor transmission line model of this structure is shownin Fig. 9(b). It is not dissimilar to the multi-conductor trans-mission line model of the structure in Fig. 7. Only this timethe two conductors of the transmission line model represent-ing the coupled stub sections are not terminated by complexload impedances. Rather, they are connected to each othervia a single conductor transmission line whose per-unit-lengthself-inductance and self-capacitance; Ls4 and Cs4 , depend onthe propagation constant and width of the MIM waveguide thatforms the upper side of the square feedback loop.

As an example, it is assumed that h = 200 nm, d0 = 20 nm,w1 = 50 nm, w2 = 50 nm, w3 = 50 nm, and w4 = 50 nm. Thetransmission coefficient of the structure is obtained by using theproposed circuit model and the FDTD method. The obtainedresults are plotted versus the free space wavelength in Fig. 10.They are in very good agreement with each other.

4. Conclusion

In this paper, a multi-conductor transmission line modelwas proposed for miscellaneous side-coupled MIM plasmonicstructures. Closed-form expressions were given for the pa-rameters of the proposed model. The accuracy of the pro-posed model was assessed by using the FDTD method. It was

Figure 10: The transmitted power spectrum of the structure with square loopfeedback in Fig. 9. The results of the proposed model and of the FDTD are plot-ted by solid and dashed lines, respectively. Inset shows the transverse magneticfield profile at λ = 0.8 and 1.2 µm.

shown that the proposed model is capable of providing accu-rate enough results. The results of the proposed model can beobtained at almost no computational cost.

It is worth noting that the proposed model is also applicableto non-plasmonic structures whenever the electromagnetic en-ergy is highly confined, i.e. whenever the higher order modesare below cut-off and thus do not play a critical role. One no-ticeable example is the possibility of applying our method tophotonic crystal based devices and structures. Interestingly,every single example in this manuscript has a photonic crys-tal counterpart, which can be successfully modeled by similarmulti-conductor transmission lines. Although there have beensome efforts to model photonic crystal structures via single con-ductor transmission line models [32, 33, 34, 35, 36], the ideaof using multi-conductor transmission lines has never been ap-plied to photonic crystal structures.

References

[1] S. Maier, Plasmonics: The promise of highly integrated optical devices,Selected Topics in Quantum Electronics, IEEE Journal of 12 (6) (2006)1671–1677.

[2] D. K. Gramotnev, S. I. Bozhevolnyi, Plasmonics beyond the diffractionlimit, Nature Photonics 4 (2) (2010) 83–91.

[3] S. Maier, Plasmonics: fundamentals and applications, Springer, 2007.[4] S. I. Bozhevolnyi, Plasmonic nano-guides and circuits, in: Plasmonics

and Metamaterials, Optical Society of America, 2008.[5] G. Veronis, S. Fan, Bends and splitters in metal-dielectric-metal subwave-

length plasmonic waveguides, Applied Physics Letters 87 (13) (2005)131102–131102.

[6] A. Pannipitiya, I. D. Rukhlenko, M. Premaratne, H. T. Hattori, G. P.Agrawal, Improved transmission model for metal-dielectric-metal plas-monic waveguides with stub structure, Optics Express 18 (6) (2010)6191–6204.

[7] H. Nejati, A. Beirami, Theoretical analysis of the characteristicimpedance in metal–insulator–metal plasmonic transmission lines, OpticsLetters 37 (6) (2012) 1050–1052.

[8] M. Staffaroni, J. Conway, S. Vedantam, J. Tang, E. Yablonovitch, Circuitanalysis in metal-optics, Photonics and Nanostructures-Fundamentals andApplications 10 (1) (2012) 166–176.

[9] M. Bahadori, A. Eshaghian, H. Hodaei, M. Rezaei, K. Mehrany, Analy-sis and design of optical demultiplexer based on arrayed plasmonic slotcavities: Transmission line model, Photonics Technology Letters, IEEE25 (8) (2013) 784–786.

7

[10] N. Nozhat, N. Granpayeh, Analysis of the plasmonic power splitter andmux/demux suitable for photonic integrated circuits, Optics Communica-tions 284 (13) (2011) 3449–3455.

[11] F. Hu, H. Yi, Z. Zhou, Wavelength demultiplexing structure based onarrayed plasmonic slot cavities, Optics letters 36 (8) (2011) 1500–1502.

[12] Y. Guo, L. Yan, W. Pan, B. Luo, K. Wen, Z. Guo, H. Li, X. Luo, Aplasmonic splitter based on slot cavity, Optics Express 19 (15) (2011)13831–13838.

[13] A. Pannipitiya, I. Rukhlenko, M. Premaratne, Analytical modeling of res-onant cavities for plasmonic-slot-waveguide junctions, Photonics Journal,IEEE 3 (2) (2011) 220–233.

[14] Y. Matsuzaki, T. Okamoto, M. Haraguchi, M. Fukui, M. Nakagaki, Char-acteristics of gap plasmon waveguide with stub structures, Optics Express16 (21) (2008) 16314–16325.

[15] Q. Zhang, X. Huang, X. Lin, J. Tao, X. Jin, A subwavelength coupler-typemim optical filter, Optics express 17 (9) (2009) 7549–7555.

[16] M. Rezaei, M. Miri, A. Khavasi, K. Mehrany, B. Rashidian, An efficientcircuit model for the analysis and design of rectangular plasmonic res-onators, Plasmonics 7 (2) (2012) 245–252.

[17] G. Veronis, S. E. Kocabas, D. A. Miller, S. Fan, Modeling of plasmonicwaveguide components and networks, Journal of Computational and The-oretical Nanoscience 6 (8) (2009) 1808–1826.

[18] H. Hodaei, M. Rezaei, M. Miri, M. Bahadori, A. Eshaghian, K. Mehrany,Easy-to-design nano-coupler between metal–insulator–metal plasmonicand dielectric slab waveguides, Plasmonics 8 (2013) 1123–1128.

[19] M. Rezaei, S. Jalaly, M. Miri, A. Khavasi, A. Fard, K. Mehrany,B. Rashidian, A distributed circuit model for side-coupled nanoplasmonicstructures with metal–insulator–metal arrangement, Selected Topics inQuantum Electronics, IEEE Journal of 18 (6) (2012) 1692–1699.

[20] M. Bahadori, A. Eshaghian, M. Rezaei, H. Hodaei, K. Mehrany, Cou-pled transmission line model for planar metal-dielectric-metal plasmonicstructures: inclusion of the first non-principal mode, Quantum Electron-ics, IEEE Journal of 49 (9) (2013) 777–784.

[21] H. Lu, X. Liu, L. Wang, D. Mao, Y. Gong, Nanoplasmonic triple-wavelength demultiplexers in two-dimensional metallic waveguides, Ap-plied Physics B 103 (4) (2011) 877–881.

[22] W. Barth, R. Martin, J. Wilkinson, Calculation of the eigenvalues of asymmetric tridiagonal matrix by the method of bisection, NumerischeMathematik 9 (5) (1967) 386–393.

[23] D. Pozar, Microwave engineering, John Wiley & Sons, 2009.[24] W. Cai, W. Shin, S. Fan, M. L. Brongersma, Elements for plasmonic

nanocircuits with three-dimensional slot waveguides, Advanced materi-als 22 (45) (2010) 5120–5124.

[25] G. Veronis, S. Fan, Modes of subwavelength plasmonic slot waveguides,Journal of Lightwave Technology 25 (9) (2007) 2511–2521.

[26] Z. Han, A. Elezzabi, V. Van, Experimental realization of subwavelengthplasmonic slot waveguides on a silicon platform, Optics letters 35 (4)(2010) 502–504.

[27] J. Dionne, H. Lezec, H. A. Atwater, Highly confined photon transport insubwavelength metallic slot waveguides, Nano Letters 6 (9) (2006) 1928–1932.

[28] J. A. Dionne, K. Diest, L. A. Sweatlock, H. A. Atwater, Plasmostor:a metal- oxide- si field effect plasmonic modulator, Nano Letters 9 (2)(2009) 897–902.

[29] J. Dionne, L. Sweatlock, M. Sheldon, A. Alivisatos, H. Atwater, Silicon-based plasmonics for on-chip photonics, Selected Topics in QuantumElectronics, IEEE Journal of 16 (1) (2010) 295–306.

[30] J. Tao, X. Huang, X. Lin, Q. Zhang, X. Jin, A narrow-band sub-wavelength plasmonic waveguide filter with asymmetrical multiple-teeth-shaped structure, Optics express 17 (16) (2009) 13989–13994.

[31] M. Swillam, A. Helmy, Feedback effects in plasmonic slot waveguidesexamined using a closed form model, Photonics Technology Letters,IEEE 24 (6) (2012) 497–499.

[32] M. Miri, A. Khavasi, M. Miri, K. Mehrany, A transmission line resonatormodel for fast extraction of electromagnetic properties of cavities in two-dimensional photonic crystals, Photonics Journal, IEEE 2 (4) (2010) 677–685.

[33] A. Khavasi, M. Rezaei, M. Miri, K. Mehrany, Circuit model for efficientanalysis and design of photonic crystal devices, Journal of Optics 14 (12)(2012) 125502.

[34] N. Habibi, A. Khavasi, M. Miri, K. Mehrany, Circuit model for mode

extraction in lossy/lossless photonic crystal waveguides, JOSA B 29 (1)(2012) 170–177.

[35] A. Khavasi, M. Miri, M. Rezaei, K. Mehrany, B. Rashidian, Transmis-sion line model for extraction of transmission characteristics in photoniccrystal waveguides with stubs: optical filter design, Optics Letters 37 (8)(2012) 1322–1324.

[36] N. Nozhat, R. McPhedran, C. de Sterke, N. Granpayeh, The plasmonicfolded directional coupler, Photonics and Nanostructures-Fundamentalsand Applications 9 (4) (2011) 308–314.

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