IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 61, NO. 4, APRIL 2013 2019

Circuit Model for Extraordinary TransmissionThrough Periodic Array of Subwavelength

Stepped SlitsAmin Khavasi, Masoud Edalatipour, and Khashayar Mehrany

Abstract—In this paper, two circuit models are proposed for an-alytical investigation of extraordinary transmission through peri-odic array of subwavelength stepped slits in a thick metallic plate.A cascade model is first developed and then a stub model is sug-gested for cases that the cascade model is not accurate. The param-eters of the proposed circuits are given by closed-form expressions.Both symmetric and asymmetric stepped slits are considered.

Index Terms—Circuit model, extraordinary transmission,stepped slit.

I. INTRODUCTION

E XTRAORDINARY TRANSMISSION (EOT) throughsubwavelength periodic array of holes or slits, reported

by Ebbesen et al. [1], has been a hot topic since its discovery[2]–[5]. Although this phenomenon has been qualitativelydescribed by surface waves [7]–[9], the resonance of transverseelectromagnetic (TEM) modes [10], and artificial surfaceconductivity [11], some circuit models have been recentlyproposed to provide both qualitative and quantitative insightto the subject [12]–[15]. Circuit models have been developedfor 2-D array of holes [12] and for 1-D array of metallic slits[13]–[15].In the aforementioned works on 1-D array of metallic slits,

the slits are assumed to have no cut. Introducing a cut leads to astepped slit which has been recently investigated in the contextof EOT [16]–[19], and it has been shown that the introductionof the step changes the resonant wavelength substantially [17].It has also been demonstrated that adjustable phase resonancescan be realized in compound metallic gratings [18].The aim of this paper is to generalize the circuit model pre-

sented in [13]–[15], to the metallic gratings with stepped slits.First, a cascade model is developed. Although, this model is ac-curate in many cases, it will fail if the junctions are too close toeach other. For these cases, a stub model is proposed.Similar to our previous work [15], arbitrary angle of inci-

dence and arbitrary surrounding media are studied. Parameters

Manuscript received April 04, 2012; revised October 23, 2012; accepted De-cember 26, 2012. Date of publication January 23, 2013; date of current versionApril 03, 2013. This work was supported in part by the Iran’s National ElitesFoundation under Shahid Dr. Chamran Award.The authors are with the Electrical Engineering Department, Sharif Univer-

sity of Technology, Tehran 11155-4363, Iran (e-mail: mehrany@sharif.ir).Digital Object Identifier 10.1109/TAP.2012.2237152

of the appropriate circuit model are given by closed-form ex-pressions to avoid resorting to additional numerical efforts fortheir extraction.In this paper, we also show that by adjusting the length and

place of a step, a flat-top bandpass filter can be designed. An an-alytical circuit model can be very useful for tailoring the spectralresponse of these structures.This paper is organized as follows. Sections II and III are de-

voted to the detailed description of the proposed circuit models:the cascade model and the stub model, respectively. Numer-ical results are then given in Section IV, where the accuracyof the proposed models is compared against numerical resultsobtained by using rigorous approaches. Finally, the conclusionsare made in Section V.

II. THE CASCADE MODEL

The geometry of the structure to be studied is shown inFig. 1. The periodic arrangement of stepped slits forms ametallic grating with period . The slits are filled with a di-electric with refractive index of . The grating is assumedto have three layers in the -direction. Each layer is a simplelamellar grating enumerated by . The width of slitsand their height in the th layer are denoted by and ,respectively. The grating region is sandwiched between twohomogenous regions with refractive indices of and . Themagnetic permeability is everywhere equal to that of the freespace. The structure is illuminated by a transverse-magnetic(TM) polarized (magnetic field along the -axis) uniform planewave whose wave vector is incident at angle with respect tothe -axis. The free-space wavelength of the wave is .The proposed circuit model is made of a transmission line and

a reactive lumped circuit element. The former represents prop-agating waves within layers of the grating (between the slits),and the latter represents the evanescent waves storing electro-magnetic energy at either the junctions formed between adjacentlayers of the grating, or the two flat interfaces formed betweenthe grating region and the surrounding homogenous media.If is large enough, the two junctions inside the grating can

be assumed to be decoupled. With this assumption, a cascademodel, as shown in Fig. 2, is proposed for the structure. Thelength of the transmission line corresponding to the grating layeris naturally set to be . Similarly, its propagation constant

and characteristic impedance are set to be equal tothe propagation constant and characteristic impedance of the

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2020 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 61, NO. 4, APRIL 2013

Fig. 1. The geometry of the structure under study: periodic array of steppedslits under the incidence of a uniform plane wave.

Fig. 2. The proposed circuit to model the structure shown in Fig. 1 for TMpolarized incident waves.

dominant TEM mode supported by a parallel plate waveguidewhose plates are separated by a distance [15]

(1)

(2)

where and are the free-space wavenumber and impedance, respectively. Equation (1) has beengiven by assuming perfect conductivity; however, it can bestraightforwardly modified to take into account the effect oflosses [13].The circuit is terminated by , and , at its

input and output terminals, respectively. The andimpedances represent electromagnetic wave impedances in theincident and transmission homogenous regions having refrac-tive indices of and , respectively. It is, therefore, easy toshow that [15]

(3)

where and are the zeroth diffracted order angles in regions1 and 3, respectively. It is worth noting that .The wavelength-dependent capacitors and account

for the evanescent fields at the upper and lowers interfaces

Fig. 3. (a) Symmetric and (b) asymmetric junction of two guides.

between the grating region and the surrounding homogenousmedia. Their approximate values are presented in [15]

(4)

where and

(5)

Here, denotes the speed of light in free space.As suggested in [15], a good empirical formula for is

, where stands for the fill factor ofthe th layer of the grating.It should be mentioned that (5) is valid in the range .The only parameters that remain to be determined are

and , representing the effect of junctions between the ad-jacent grating layers. Fortunately, a very accurate closed-formexpression has been provided in [20] to take the effects of higherorder modes excited at the junction of two waveguides.Fig. 3(a) and (b) shows symmetric and asymmetric junctions

of two waveguides with arbitrary widths and , respec-tively. The susceptance of the capacitor modeling the symmetricjunction [Fig. 3(a)] is [20]

(6)

where and

(7)

In accordance with Fig. 3, the larger and smaller widths aredenoted by and , respectively.To obtain the susceptance of the capacitor modeling the

asymmetric junction, in (6) and (7) should be replaced with[20]. It should be noted that the aforementioned equations

are invalid in the range and forsymmetric and asymmetric cases, respectively.

III. THE STUB MODEL

For a cut with small height the structure will be similar toFig. 4. In this case, the evanescent waves excited at the twojunctions inside the grating can couple to each other through thesecond layer. This makes the cascade model inaccurate, which

KHAVASI et al.: CIRCUIT MODEL FOR EXTRAORDINARY TRANSMISSION THROUGH PERIODIC ARRAY OF SUBWAVELENGTH STEPPED SLITS 2021

Fig. 4. The (a) symmetric and (b) asymmetric structure of Fig. 1 with small , for which the stub model should be used.

Fig. 5. (a) The proposed stub model for the structure shown in Fig. 4(b). (b)The circuit whose input impedance is .

will be demonstrated in Section IV. Here, we develop anothercircuit model for such cases.Consider a unit cell of the structure shown in Fig. 4(b); it re-

sembles a waveguide that a stub is in series with it. From thispoint of view, the circuit model shown in Fig. 5(a) can be at-tributed to the asymmetric structure. In this circuit, the stub ismodeled with the series impedance , which is the inputimpedance of the circuit shown in Fig. 5(b). In this circuit,and are the characteristic impedance and the propagationconstant of a parallel plate waveguide whose plates are sepa-rated by a distance . These parameters for a parallelplate waveguide were already given in (1) and (2). It is also ob-vious that the length of the stub and its corresponding transmis-sion line is . It is here assumed that .The impedance is the impedance of the metal that the stub

is terminated to it. The value of can be calculated by [21]:

(8)

where is the permittivity of the metal.

The reactive elements whose values are denoted by , ,, and are for taking into account the effect of T-junction.

Approximate closed-form expressions for these elements havebeen given in [20]

(9)

(10)

(11)

(12)

where , , and is the Nepernumber.For the symmetric case [see Fig. 4(a)], the junction’s model

has not been given in [20], however we extend the proposedmodel to the symmetric case by using an intuitive reasoning. Inthis case, instead of one stub of length , we have two stubsof length . Therefore, in Fig. 5(b) must be replaced with

. Furthermore, in this case, two symmetric stubs are on twosides of the waveguide. Thanks to the symmetry of the structure,each stub is separated from the other by a perfect electric con-ductor wall dividing the main waveguide into two equal halves.Hence, in (11) and (12), should be changed to . Finally, inFig. 5(a), we put instead of the series impedance .

IV. NUMERICAL EXAMPLES

Let us examine the accuracy of the proposed model in a nu-merical example. Consider a structure with symmetric junc-tions whose parameters are in accordance with Fig. 1: 0 ,

, , ,2 cm, and 0.1 cm. The aluminum is used as theconductor whose direct current (dc) conductivity is set to3.65 10 S/m. The transmission spectra of this structure for

2022 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 61, NO. 4, APRIL 2013

Fig. 6. Transmitted power through an array of stepped slits versus cal-culated by using the proposed cascade model (solid line), HFSS simulations(circles), and the rigorous approach of [22] (points). The structure parametersare: 0 , , , ,2 cm, and 0.1 cm. The results are illustrated for three differentwidths of : (a) 0.2 cm, (b) 0.4 cm, and (c) 0.6 cm.The junctions are assumed to be symmetric.

0.2 cm, 0.4 cm, and 0.6 cm are plottedin Fig. 6(a)–(c), respectively. The results have been obtained byusing the proposed cascade model (solid line), high-frequencystructural simulator (HFSS) simulations (circles), and the rig-orous approach of [22] (points), in which the regularization pre-sented in [23] has been applied. Excellent agreement is observedbetween all methods.It can be seen from these figures that the presence of stepped

slits results in the displacement of resonance frequencies. It isfortunate that such a displacement is geometrically adjustable.For example, two resonance frequencies are merged together inFig. 6(c) and a relatively flat-top spectral response has been cre-ated around . A notable benefit of the analyticalcircuit model, proposed in the letter, is that the spectral responseof EOT can be efficiently tailored.As another example, consider a structure with the same pa-

rameters as in the previous example, but its junctions are asym-metric. The accuracy of the proposed model for this case isdemonstrated in Fig. 7(a)–(c), where the transmission is de-picted in terms of for 0.2 cm, 0.4 cm, and

0.6 cm, respectively. The results are obtained by using

Fig. 7. Transmitted power through an array of stepped slits versus cal-culated by using the proposed cascade model (solid line), HFSS simulations(circles), and the rigorous approach of [22] (points). The structure parametersare: 0 , , , ,2 cm, and 0.1 cm. The results are illustrated for three differentwidths of : (a) 0.2 cm, (b) 0.4 cm, and (c) 0.6 cm.The junctions are assumed to be asymmetric.

the proposed cascade model (solid line), HFSS simulations (cir-cles), and the rigorous approach of [22] (points). Once again, anexcellent agreement is observed between the proposed modeland the rigorous approaches.Now, let us compare the accuracy of the cascade model

against the stub model. The structure has many parameters, andeach of them can contribute to the accuracy of results. However,from Fig. 5 and (9)–(12), it is obvious that ,

, and are the three parameters thatdirectly affect the accuracy of the stub model. For the sake ofsimplicity, we only examine the effect of these three crucialparameters on both models. Although it is not a completecomparison, it can give us a rule of thumb for selecting theappropriate model. Consider a structure under normal incidenceof a plane wave with . The structure’s parameters are:

and 2 cm. Therelative error in calculating power transmission is computedas a function of and for both models. Regions atwhich the relative error of the cascade and stub models is morethan 10% are illustrated by black and gray colors in Fig. 8,

KHAVASI et al.: CIRCUIT MODEL FOR EXTRAORDINARY TRANSMISSION THROUGH PERIODIC ARRAY OF SUBWAVELENGTH STEPPED SLITS 2023

Fig. 8. Regions with more than 10% relative error in calculation of transmittedpower: the black and gray regions for the cascade and stub models, respectively.Four cases are depicted: (a) , (b) , (c) , and (d)

. The junctions are assumed to be asymmetric.

Fig. 9. Transmitted power through an array of stepped slits versus cal-culated by using the proposed stub model (solid line), HFSS simulations (cir-cles), and the rigorous approach of [22] (points). The structure parameters are:

0 , , , , 2 cm,0.1 cm, and 0.6 cm. The results are illustrated

for both (a) asymmetric and (b) symmetric junctions.

respectively. The error maps are plotted in Fig. 8(a)–(d), forfour cases, , , , and , respectively.As a rule of thumb, it can be concluded that the stub model

should be used for . Although it cannot be taken as ageneral and accurate rule, it is useful in many cases.To further demonstrate the accuracy of the proposed stub

model, the transmission of this structure is plotted as a func-tion of in Fig. 9(a) and (b) for asymmetric and symmetricjunctions, respectively. The height of the second layer is chosensmall enough ( 0.1 cm) so that the stub model remains ac-curate.

V. CONCLUSION

In conclusion, extraordinary transmission through subwave-length array of stepped slits has been studied by using analyticalcircuit models. First, a cascade model was presented, which iscomposed of transmission lines corresponding to different partsof slits, and capacitances for modeling junctions. Another modelhas been also proposed in which the second layer of the gratinghas been regarded as a stub. It is demonstrated that the cascademodel is suitable when the two junctions inside the grating arefar enough apart, otherwise the stub model should be used.The circuit parameters have been given in closed-form ex-

pression, thus the model can be employed for efficient investi-gation of extraordinary transmission in such structures.The results of the proposed circuit models have been com-

pared with rigorous methods in numerical examples and excel-lent agreements have been observed.

REFERENCES

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[15] A. Khavasi and K.Mehrany, “Circuit model for Lamellar metallic grat-ings in the sub-wavelength regime,” IEEE J. Quantum Electron., vol.47, no. 10, pp. 1330–1335, Oct. 2011.

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2024 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 61, NO. 4, APRIL 2013

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Amin Khavasi was born in Zanjan, Iran, on January22, 1984. He received the B.Sc., M.Sc., and Ph.D.degrees from Sharif University of Technology,Tehran, Iran, in 2006, 2008, and 2012, respectively,all in electrical engineering.Since then, he has been with the Department of

Electrical Engineering, Sharif University of Tech-nology, where he is currently an Assistant Professor.His research interests include photovoltaics, plas-monics, and circuit modeling of photonic structures.

Masoud Edalatipour was born in Ghaenat, Iran, onMarch 6, 1985. He received the B.Sc. degree fromFerdowsi University of Mashhad, Mashhad, Iran, in2007 and the M.Sc. degree from Sharif Universityof Technology, Tehran, Iran, in 2009, both in elec-trical engineering. He is currently working toward thePh.D. degree at Sharif University of Technology.His research interests include terahertz, plas-

monics, and photonics.

Khashayar Mehrany was born in Tehran, Iran,on September 16, 1977. He received the B.Sc.,M.Sc., and Ph.D. (magna cum laude) degrees fromSharif University of Technology, Tehran, Iran, in1999, 2001, and 2005, respectively, all in electricalengineering.Since then, he has been with the Department of

Electrical Engineering, Sharif University of Tech-nology, where he is currently an Associate Professor.His current research interests include photonics,plasmonics, and integral imaging.