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- 1. 1Eﬃcient Mode-Matching Analysis ofTwo-Dimensional Scattering by Periodic Array ofCircular CylindersYong Heui Cho and Do-Hoon KwonAbstract—Based on a common-area concept and insertionof an inﬁnitesimal PMC (Perfect Magnetic Conductor) wire,a new mode-matching method for mixed coordinate systemsis proposed for the analysis of periodic magnetodielectric,PEC (Perfect Electric Conductor), and PMC circularcylinder arrays. Our scattering solutions were computedand they showed favorable agreements with other knownresults in terms of power scattering coeﬃcients, resonantfrequency of nano-structures, and sum rule of extinctionwidth. The low-frequency solutions of periodic arrays werealso formulated and compared with the full-wave counterparts.Index Terms—Electromagnetic diﬀraction, periodic struc-tures, mode-matching methods, electromagnetic resonance,nanowires, sum rule.I. IntroductionAPeriodic array of dielectric circular cylinders is acanonical diﬀraction structure and has been exten-sively studied in [1]-[5]. Since most of dielectric gratingscan be approximately regarded as circular cylinders, itis of practical interest to obtain the simple yet analyticscattering solutions of a periodic two-dimensional (2D)dielectric cylinder array. In [1] and [2], diﬀraction byperiodic circular cylinders was analyzed with inﬁnitesummation of circular cylindrical wave functions thatrepresent the electromagnetic ﬁelds of a single circularcylinder. This fundamental approach has been widelyutilized in the analyses of periodic gratings [3]-[5]. Avariety of numerical methods for periodic dielectric andmetallic structures were discussed in [3] as well. Recently,periodic gratings with very small radius have been usedfor nano-structures to invoke the resonance characteristics.This type of periodic gratings is called a nanorod [6]or nanowire [7]. In optics, a nanorod structure is afundamental geometry to design various optical devicessuch as biosensors, nanoantennas [8], polarization selec-tive surface, terahertz transmission lines, and near-ﬁeldscanning optical microscopes.In this work, a novel mode-matching technique formixed coordinate systems is proposed to present theThis work was supported in part by the US Army Research OﬃceGrant No. W911NF-12-1-0289.Y. H. Cho is with the School of Information and CommunicationEngineering, Mokwon University, Doanbuk-ro 88, Seo-gu, Daejeon,302-729, Korea (e-mail: yongheuicho@gmail.com).D.-H. Kwon is with the Department of Electrical and ComputerEngineering, University of Massachusetts Amherst, Amherst, MA,01003, USA (e-mail: dhkwon@ecs.umass.edu).simple yet exact scattering equations for a periodic 2Dcircular cylinder array. A standard mode-matching tech-nique in [3], [9] is still used to formulate the analyticdispersion and scattering equations for rectangular [10]and circular cylindrical [11] structures. Using staircaseapproximation and generalized scattering matrix, openperiodic waveguides with arbitrary unit-cell can be ap-proximately analyzed in the Cartesian coordinate system[12]. A complex mode-matching method combined withthe perfectly matched layer (PML) [13] opened a newway to get the reﬂection and transmission spectra ofoptical waveguide interconnects. In addition, the fast scat-tering analyses of electrically large substrate integratedwaveguide (SIW) devices were performed with a mode-matching technique using a cylindrical mode expansion[14]. Although a standard mode-matching technique iswell-known, it is diﬃcult to apply this method to ageometry with both rectangular and circular cylindricalfeatures [15]-[18]. To analyze such a geometry, we needto introduce a common-area concept proposed in [15].The common area concept has been widely used toanalyze waveguide problems [16]-[18]. In this approach, theﬁelds expanded in the rectangular and circular cylindricalcoordinate systems should have a so-called common areawhere all boundary conditions are consistently satisﬁed.Since our periodic problem requires the periodic boundarycondition on each side, our geometry is surely diﬀerentfrom those in [16]-[18] where the PEC (Perfect ElectricConductor) region is essential to apply a mode-matchingtechnique based on the common area. In our periodic case,there is no PEC region to match the boundary conditionsand get a set of simultaneous equations. Therefore, we willuse an inﬁnitesimal PMC (Perfect Magnetic Conductor)wire to overcome this diﬃculty and match the boundaryconditions of the TE-mode (Ez = 0, Hz = 0). It shouldbe noted that the inﬁnitesimal PMC wire is transparentand does not interact with the TE-mode incident ﬁeld.A common-area concept and the insertion of inﬁnitesi-mal PMC wires allows us to obtain the analytic scatteringrelations of a periodic 2D circular cylindrical array inboth rectangular and circular cylindrical features. Usingthe ﬁnal set of simultaneous equations for a periodic2D circular cylinder array, we can obtain the resonancecharacteristics of nano-structures in the optical band andthe sum rule for the extinction width in the microwaveband.
- 2. 2TRegion (I)Region (II)Region (IV)xyz2aiθIncidence11,µε22,µεRegion (III)PeriodicboundaryconditionPeriodicboundaryconditionFig. 1. Unit-cell of a periodic two-dimensional circular cylinderarray in a free spaceII. Mode-Matching in Mixed Coordinate SystemsConsider a periodic 2D circular cylinder array in a freespace shown in Fig. 1. In order to compute the scatteringcharacteristics of an inﬁnite number of circular cylinders,we introduce periodic boundary conditions at x = ±T/2.Fig. 1 illustrates a unit-cell of the inﬁnite number ofcircular cylinders. In the following development, an e−iωttime convention is assumed and omitted throughout. Anincident electric ﬁeld in region (I) (ρ > T/2, |x| ≤ T/2, y >0) is given byEiz(x, y) = eik1(sin θix−cos θiy), (1)where k1 = ω√µ1 1, θi is the angle of incidence of theTE-mode (Ez = 0, Hz = 0) plane wave with respect tothe φ-axis, and 0 ≤ θi ≤ π/2. For convenience, we deﬁnea transmitted ﬁeld in region (IV) (ρ > T/2, |x| ≤ T/2, y <0)Etz(x, y) = eik1(sin θix−cos θiy)(2)as the incident ﬁeld evaluated behind the array. Utilizinga standard mode-matching method and the Floquet anal-ysis, we represent the electric ﬁelds for regions (I) to (IV)asEIz (x, y) =∞∑m=−∞Amei(Tmx+ξmy)(3)EIIz (ρ, φ) =∞∑m=−∞[BmJm(k1ρ) + CmNm(k1ρ)]× eimφ(4)EIIIz (ρ, φ) =∞∑m=−∞DmJm(k2ρ)eimφ(5)EIVz (x, y) =∞∑m=−∞Emei(Tmx−ξmy), (6)where Am to Em are unknown modal coeﬃcients, Tm =k1 sin θi + 2mπ/T, ξm =√k21 − T2m, k2 = ω√µ2 2,ρ =√x2 + y2, φ = tan−1(y/x), and Jm(·), Nm(·) are themth-order Bessel functions of the ﬁrst and second kinds,respectively. It should be noted that the electric ﬁelds inregions (I) to (IV) are formulated in the mixed coordinatesystems based on a common-area concept [15].By enforcing the tangential ﬁeld continuities at ρ = aand ρ = T/2, we can constitute the simultaneous equa-tions for the TE-mode. Multiplying the equations forthe Ez- and Hφ-ﬁeld continuities at ρ = a by e−ilφ(l = 0, ±1, ±2, · · · ) and integrating from φ = 0 to φ = 2πgivesBmJm(k1a) + CmNm(k1a) = DmJm(k2a) , (7)1η1[BmJm(k1a) + CmNm(k1a)] =1η2DmJm(k2a) , (8)where η1,2 =√µ1,2/ 1,2 and (·) denotes the diﬀerentia-tion with respect to the argument k1,2ρ. Then,Bm = B(0)m Dm (9)Cm = C(0)m Dm , (10)whereB(0)m =πk1a2[Jm(k2a)Nm(k1a)−η1η2Jm(k2a)Nm(k1a)](11)C(0)m =πk1a2[η1η2Jm(k2a)Jm(k1a)− Jm(k2a)Jm(k1a)]. (12)Similarly, multiplying the equations for the Hφ-ﬁeld con-tinuity at ρ = T/2 by e−ilφ(l = 0, ±1, ±2, · · · ) andintegrating from φ = 0 to φ = 2π yieldsDl =1H(0)l[ ∞∑m=−∞(AmKml + EmLml) + Sφ,l], (13)whereKml = ik1e−ilφmJcel(k1T2, −φm)(14)Lml = ik1eilφmJcel(k1T2, π + φm)(15)Jcem (x, φ) =∂Jem(x, φ)i∂x(16)Jem(x, φ) = im[πJm(x)− 2∞∑n=−∞ei(2n+1)φ (−1)n2n + 1Jm+2n+1(x)](17)Sφ,l = 2πk1e−ilθiJ−l(k1T/2) (18)H(0)l = 2πk1[B(0)l Jl (k1T/2) + C(0)l Nl (k1T/2)], (19)and φm = tan−1(ξm/Tm). In the next step, we multiplythe Ez-ﬁeld continuity equation at ρ = T/2 by e−ilφ(l =
- 3. 30, ±1, ±2, · · · ) and integrate on the two intervals, φ =(0, π) and φ = (π, 2π). Combining (13) and the Ez-ﬁeldcontinuity equation, we obtain the ﬁnal set of simultaneousequations for Am and Em as∞∑m=−∞[Am(Iml −2πE(0)lH(0)lKml)+ Em(Jml −2πE(0)lH(0)lLml) ]=2πE(0)lH(0)lSφ,l − Sz,l , (20)∞∑m=−∞[Am∞∑k=−∞(−1)k−lKmkD(0)kl+ Em( ∞∑k=−∞(−1)k−lLmkD(0)kl − Jml) ]= SIVz,l −∞∑k=−∞(−1)k−lSφ,kD(0)kl , (21)whereIml = e−ilφmJel(k1T2, −φm)(22)Jml = eilφmJel(k1T2, π + φm)(23)E(0)k = B(0)k Jk(k1T/2) + C(0)k Nk(k1T/2) (24)D(0)kl =E(0)kH(0)kGe(k − l) (25)Ge(m) =(−1)m− 1im(26)SIVz,l = ile−ilθiJel(k1T2,3π2− θi)(27)Sz,l = 2πe−ilθiJ−l(k1T/2) . (28)After obtaining Am and Em, we can get Dm with (13).Similarly, Bm and Cm are obtained with Dm based on(9)–(10).The Ez-ﬁeld continuity at ρ = T/2 should be enforcedon two diﬀerent intervals, φ = (0, π) and φ = (π, 2π) toobtain the additional boundary condition. Since we haveﬁve sets of unknown modal coeﬃcients, Am to Em in (3)to (6), we should get ﬁve tangential electric and magneticboundary conditions at ρ = a and T/2 to solve the ﬁnalsimultaneous equations. Applying the Ez-ﬁeld matchingcondition on two intervals means that the PMC wires existat (x, y) = (±T/2, 0) in Fig. 1. Although the PMC wiresshould be at (x, y) = (±T/2, 0) to match the boundaryconditions, no magnetic current can ﬂow through the PMCwires due to the TE-mode excitation, thus indicating thatthe lines can be ignored.Since the geometry in Fig. 1 is composed of magnetodi-electric materials, we can easily obtain the simultaneousscattering equations of the TM-mode (Ez = 0, Hz = 0)based on the duality of the Maxwell’s equations. In viewof the duality theorem, we need to replace → µ, µ → ,¯Ee → ¯Hm, and ¯He → − ¯Em. Then, (20)–(21) can be usedfor the TM-mode with substituting → µ, µ → . Thesimultaneous equations for the PEC and PMC materialsare given in Appendix A based on (20)–(21).When 1 = 2 and µ1 = µ2, we get the result of Am =Em = 0 which means that there is no scattered ﬁeld inregions (I) and (IV). This behavior partly conﬁrms thevalidity of our simultaneous equations, (20)–(21).Using the scattered ﬁelds, (3) and (6), we can deﬁne thereﬂectance, transmittance, and absorbance, respectively,asRtot =PrPi=M−∑m=−M+ρm (29)Ttot =PtPi=M−∑m=−M+τm (30)Atot = 1 − Rtot − Ttot , (31)where Pi, Pr, Pt are incident, reﬂected, and transmittedpowers per unit-cell in Fig. 1, respectively, M± = [k1T(1±sin θi)/(2π)], [x] is the maximum integer less than x,ρm = |Am|2[ξ∗mk1 cos θi](32)τm = |Em + δm0|2[ξ∗mk1 cos θi], (33)δml is the Kronecker delta, (·)∗is the complex conjugateof (·), and [·] is the real part of (·).III. Low-Frequency SolutionsEven though the exact simultaneous equations, (20)–(21), can be used to analyze the low-frequency behaviorswhen frequency approaches zero, it is convenient and nu-merically eﬃcient to deduce new low-frequency solutionsbased on (20)–(21). Taking the low-frequency limit of(51)–(52) in Appnedix A yields the following simpliﬁed
- 4. 4low-frequency equations of the PEC cylinders:∞∑m=−∞[(Am + δm0)( ∞∑k=−∞KmkD(E)kl − Iml)+ (Em + δm0)∞∑k=−∞Km,−kD(E)kl]= k1{Ti1+l[e−ilθiJcel (0, π/2 − θi)]− 2∞∑k=−∞UkD(E)kl}, (34)∞∑m=−∞[(Am + δm0)∞∑k=−∞(−1)k−lKmkD(E)kl+ (Em + δm0)( ∞∑k=−∞(−1)k−lKm,−kD(E)kl − Im,−l) ]= −2k1∞∑k=−∞(−1)k−lUkD(E)kl . (35)When k1 → 0, the low-frequency forms of the parametersin (34)–(35) are given byKml ∼4π|m|T×{(−i) · si(1 − l, |m|π) for m > 0i · si∗(1 + l, |m|π) for m < 0(36)K0l ∼ k1i1−leilθiJcel (0, θi − π/2) (37)Iml ∼ −2 ×{si(−l, |m|π) for m ≥ 0si∗(l, |m|π) for m < 0(38)Um = πe−imθiJ−m(0)− i ·[eim(θi−π/2)Jcem (0, θi − π/2)](39)D(E)ml ∼T4πGe(m − l)×{1|m|1−(2a/T )2|m|1+(2a/T )2|m| for m = 0− log(2aT)for m = 0(40)Jcem (0, φ) =e−imφi(m2 − 1)[1 + (−1)m]× (m cos φ + i sin φ) (41)si(m, a) = −12∫ π0ei[a exp(iφ)+mφ])dφ . (42)Eqn. (42) is a generalized sine integral such that si(0, a) =si(a) and its recurrence relation is given bya · si(m + 1, a) − im · si(m, a)=(−1)me−ia− eia2. (43)The sine integral si(a) is deﬁned in [19, Eqn. (5.2.5)].The simultaneous equations, (34)–(35), can be repre-sented in a matrix form asM(E)[Am + δm0Em + δm0]= k1S(E), (44)TABLE ITM-mode power reﬂection coeﬃcient of the zeroth-order Floquetmode ρ0 with the same parameters in Fig. 2T/λ0 M = 3 M = 7 M = 11 M = 21 Lattice sum [4]0.4 0.02825 0.02703 0.02703 0.02703 0.02710.5 0.02416 0.02273 0.02273 0.02273 0.02220.6 0.01356 0.01259 0.01258 0.01258 0.01330.7 0.002776 0.002588 0.002586 0.002586 0.0020.8 0.001326 0.001279 0.001282 0.001282 00.9 0.3029 0.4820 0.4824 0.4824 0.5341.0 0.008885 0.004154 0.004193 0.004193 0.0043where M(E)and S(E)are corresponding scattering andsource matrices of (34)–(35), respectively. Since M(E)andS(E)are independent of k1, Am + δm0 and Em + δm0 arelinearly proportional to k1.Similar to the PEC case, the low-frequency solutions forthe PMC and magnetodielectric materials are given by∞∑m=−∞[Am( ∞∑k=−∞KmkD(p)kl − Iml)+ Em∞∑k=−∞Km,−kD(p)kl]= Sz,l − SIVz,l −∞∑k=−∞Sφ,kD(p)kl , (45)∞∑m=−∞[Am∞∑k=−∞(−1)k−lKmkD(p)kl+ Em( ∞∑k=−∞(−1)k−lKm,−kD(p)kl − Im,−l) ]= SIVz,l −∞∑k=−∞(−1)k−lSφ,kD(p)kl , (46)where p = M for PMC, p = 0 for magnetodielectricmaterial,D(M)ml ∼T4πGe(m − l)×{1|m|1+(2a/T )2|m|1−(2a/T )2|m| for m = 08(k1T )2[(2a/T )2−1] for m = 0(47)D(0)ml ∼T4πGe(m − l)×1|m|1+µ1µ2+“1−µ1µ2”(2a/T )2|m|1+µ1µ2−“1−µ1µ2”(2a/T )2|m|for m = 08(k1T )2h“1− 21”(2a/T )2−1i for m = 0. (48)IV. DiscussionsTo verify our mode-matching formulations in mixedcoordinate systems, the scattering equations, (20)–(21),were computed to get the resonance characteristics of aperiodic 2D dielectric cylinder array.Fig. 2 and Table I show the power reﬂection coeﬃcientof the zeroth-order Floquet mode ρ0 in (32) with respect
- 5. 50.4 0.5 0.6 0.7 0.8 0.9 100.20.40.60.81Normalized period, T/λ0Powerreflectioncoefficient,ρ0M = 3M = 7M = 11M = 21Lattice sum [4]Fig. 2. TE-mode power reﬂection coeﬃcient of the zeroth-orderFloquet mode ρ0 versus normalized period T/λ0 with θi = 0, a =0.3T, 1 = 0, 2 = 2 0, and µ1 = µ2 = µ0400 600 800 1000 1200 1400 1600 1800 200000.20.40.60.81Wavelength [nm]TE−modereflectance,RtotM = 11M = 3M = 1Fig. 3. TE-mode reﬂectance of TiO2 nanorods versus wavelengthwith θi = 0, a = 40 [nm], T = 783.6 [nm], 1 = 0, µ1 = µ2 = µ0,and 2 obtained from [20]to the normalized period. In addition, the convergencecharacteristics of modal coeﬃcients are shown in Fig. 2and Table I, where M denotes the truncated numberof modes in regions (I) or (IV). As M increases, thepower reﬂection coeﬃcient approximately converges to theresults computed by the lattice sum technique [4]. TableI indicates that the cases of M = 3 [m = 0, ±1 in (20)–(21)] and M = 7 [m = 0, ±1, ±2, ±3] converge to that ofM = 21 with errors less than 7.2% and 0.2%, respectively,when T/λ0 ≤ 0.8. For the TM-mode case in Table I, weused the duality form of (20)–(21) with which 2 and µ2are replaced with µ2 and 2, respectively.Figs. 3 and 4 illustrate the characteristics of scatteredpowers for an array of nanorods composed of TiO2 (di-electric) and silver (metal). We computed the reﬂectance(29) and transmittance (30) based on (20)–(21) using the300 350 400 450 50000.20.40.60.81Wavelength [nm]NormalizedTM−modepowerReflectance, RtotTransmittance,TtotAbsorbance, AtotFig. 4. Normalized TM-mode powers of silver nanowires versuswavelength with M = 11, θi = 0, a = 70 [nm], T = 375 [nm],1 = 0, µ1 = µ2 = µ0, and 2 taken from [21]refractive index (n = n + iκ, 2 = n 2) data [20], [21].The geometrical parameters of the nanorod array for Fig.3 were taken from [6]. The refractive index of TiO2 foran ordinary ray was obtained as a form of the Sellmeierequation from [20]. The resonant wavelength from ourmode-matching analysis is around 814.6 [nm], which isvery close to 800 [nm] predicted in [6]. The convergenceresults show that a dominant-mode solution with M = 3(m = 0, ±1) yields a very good approximation for λ >700 [nm]. This means that three modes (M = 3) areenough to predict resonance behaviors of TiO2 properly.The dominant-mode M = 1 (m = 0) only gives favorableresults when λ > 1400 [nm]. In Fig. 4, the TM-modereﬂectance, transmittance, and absorbance of a silvernanowire array are shown. Since the measured refractiveindex of silver is complex (κ = 0) [21], the resonance peakin Fig. 4 is not as sharp as that in Fig. 3, where the lossof TiO2 is assumed to be zero (κ = 0) [20]. The resonancepeak of reﬂectance in Fig. 4 is at 374.1 [nm] which is veryclose to the approximate value of 377 [nm] in [7]. Thisconﬁrms that our approach is valid for nano-structures.As a second example, TM-mode scattering by an arrayof PEC cylinders in the microwave frequency band isconsidered. For an array of closely spaced cylinders withT = 50 [mm], a = 20 [mm], Fig. 5 plots the zeroth-orderFloquet mode reﬂectance ρ0 and transmittance τ0 withrespect to frequency and compares them with the nu-merical solutions obtained using the commercial analysispackage HFSS from Ansys. The mode-matching solutionwas obtained from the dual conﬁguration of the TE-mode. For both incident angles θi = 0, 30◦considered inFig. 5, mode-matching and HFSS results show an excellentagreement, validating the proposed solution methodology.It is noted that grating lobes begin to appear at 6 [GHz]and 4 [GHz] for the θi = 0 and θi = 30◦cases, respectively,and the mode-matching solution recovers transmission and
- 6. 60 2 4 6 8 10 1200.20.40.60.81Frequency [GHz]Reflectanceandtransmittancespectra ρ0: θi=0τ0: θi=0ρ0: θi=30°τ0: θi=30°HFSSFig. 5. The zeroth-order Floquet mode reﬂectance and transmit-tance spectra for a PEC cylinder array with M = 11, T = 50 [mm],and a = 20 [mm] for two diﬀerent incident angles θi = 0 and 30◦.Circled data points (◦) represent simulation results obtained usingAnsys HFSS.reﬂection coeﬃcients accurately and eﬃciently.A sum rule [22], [23] relates the dynamic scatteringcharacteristics integrated over all frequency to the staticand low-frequency scattering responses. The sum rule is auseful tool to estimate the fundamental limit of practicalantennas [23]. Let S0 denote the complex scatteringcoeﬃcient of the zeroth-order Floquet mode. From theﬁeld deﬁnitions, (2) and (6), S0 = E0 for the PMC arrayin the TE-mode case (and the PEC array in the TM-mode via duality). Appropriately for 2D conﬁgurations,let the extinction width σext be deﬁned as the sum of thescattering width σs and the absorption width σa (which isequal to zero for lossless scatterers). The optical theoremfor doubly-periodic scatterers [22] can be extended toobtain the optical theorem for singly-periodic 2D arraysasσext(k1) = σs + σa = −2T · [S0] cos θi . (49)If we deﬁne a function ρext(k1) = S0/k21, the sum rule forσext for doubly-periodic scatterers [22] is modiﬁed to read∫ ∞0σext(k1)k21dk1 = −T cos θi∫ ∞−∞ρext(k1) dk1= πT · [Res(ρext, k1 = 0)] cos θi . (50)where ρext(k1) is analytic in the complex upper half-plane ( [k1] > 0), [·] is the imaginary part of (·),and Res[f(z), z = 0] is the residue of f(z) at z = 0.For θi = 0, Fig. 6(a) plots S0 at low frequencies usingdiﬀerent number of terms M = 3 and M = 11. Atboth values of M, the full-wave solutions (51)–(52) andthe low-frequency solutions (45)–(46) show an excellentagreement. In addition, Fig. 6(a) also shows that onlya small number of terms with M = 3 are enough toobtain accurate low-frequency solutions. It is observedthat S0 = O(k1) and purely imaginary as k1 → 0. Hence,the residue in (50) is related to the slope of [S0] inFig. 6(a). Fig. 6(b) shows the extinction width σext for0 20 40 60 80 100 120 140 160 180 200−0.1−0.0500.050.10.150.2Frequency [MHz]Complexscatteringcoefficient,S0imagreal(51) & (52) with M=3(51) & (52) with M=11(45) & (46) with M=3(45) & (46) with M=11(a)0 2 4 6 8 10 12020406080100Frequency [GHz]Extinctionwidth,σext[mm]θi=0θi=30°(b)Fig. 6. Low-frequency scattering behavior of a PEC cylinderarray considered in Fig. 5 and the extinction scattering width σext.(a) The zeroth-order Floquet mode scattering coeﬃcient S0 at lowfrequencies for the θi = 0 case. (b) The extinction width σext withrespect to frequency.the two incident angles as a function of frequency from(49). The sum rule (50) can be tested, where the integral in(50) is obtained numerically from Fig. 6(b) and the residueis obtained from the low-frequency solutions [Fig. 6(a) forθi = 0]. With the numerical integration performed from0 to 20 [GHz], (50) was found to be accurate with errorsless than 4.2% for the two cases considered. The accuracywill improve as numerical integrations are performed overa wider frequency range. This example provides anothervalidation of the full-wave and eﬃcient low-frequencysolutions of the proposed analysis method.V. ConclusionsA novel mode-matching method in mixed coordinatesystems was proposed to analyze plane-wave scatteringcharacteristics of periodic circular cylinders composed ofmagnetodielectric, PEC, and PMC materials. Numericalcomputations were performed to check the accuracy of ouranalytic formulations. Numerical experiments show thata three-term approximation (M = 3) is enough to obtainaccurate scattering results in most practical cases. Forinstance, the result of M = 3 for a = 0.3T, 2 = 2 0,
- 7. 7and T/λ0 ≤ 0.8 is within the maximum error of 7.2%compared to that of M = 21. A three-term approximationis also good for eﬃcient computation of the low-frequencyﬁeld quantities. The proposed method has been testedand validated for the resonance characteristics of nano-structures in the optical regime and for the sum rule forthe extinction width in the microwave regime.Appendix A: Equations for PEC and PMCFor the cases of PEC ( 2 → ∞ and µ2 = µ0) andPMC ( 2 = 0 and µ2 → ∞) cylinders, the simultaneousequations for magnetodielectric, (20) and (21), are refor-mulated as∞∑m=−∞[Am(Iml −2πE(p)lH(p)lKml)+ Em(Jml −2πE(p)lH(p)lLml) ]=2πE(p)lH(p)lSφ,l − Sz,l , (51)∞∑m=−∞[Am∞∑k=−∞(−1)k−lKmkD(p)kl+ Em( ∞∑k=−∞(−1)k−lLmkD(p)kl − Jml) ]= SIVz,l −∞∑k=−∞(−1)k−lSφ,kD(p)kl , (52)where p = E for PEC, p = M for PMC,H(E)k = 2πk1[Jk(k1T/2) −Jk(k1a)Nk(k1a)Nk(k1T/2)](53)E(E)k = Jk(k1T/2) −Jk(k1a)Nk(k1a)Nk(k1T/2) (54)H(M)k = 2πk1[Jk(k1T/2) −Jk(k1a)Nk(k1a)Nk(k1T/2)](55)E(M)k = Jk(k1T/2) −Jk(k1a)Nk(k1a)Nk(k1T/2) (56)D(p)kl =E(p)kH(p)kGe(k − l)p=E,M. (57)References[1] V. Twersky, “On scattering of waves by the inﬁnite grating ofcircular cylinders,” IRE Trans. Antennas Propag., vol. 10, no. 6,pp. 737-765, Nov. 1962.[2] O. Kavakhoglu and B. Schneider, “On multiple scattering ofradiation by an inﬁnite grating of dielectric circular cylindersat oblique incidence,” Int. J. Infrared Milli. Waves, vol. 29, no.4, pp. 329-352, Apr. 2008.[3] K. Yasumoto (Ed.), Electromagnetic Theory and Applicationsfor Photonic Crystals, Taylor & Francis CRC Press, 2005.[4] T. Kushta and K. Yasumoto, “Electromagnetic scattering fromperiodic arrays of two circular cylinders per unit cell,” Prog.Electromagn. Res., vol. 29, pp. 69-85, 2000.[5] R. Gomez-Medina, M. Laroche, and J. J. Saenz, “Extraordinaryoptical reﬂection from sub-wavelength cylinder arrays,” Opt.Express, vol. 14, no. 9, pp. 3730-3737, May 2006.[6] M. Laroche, S. Albaladejo, R. Carminati, and J. J. Saenz,“Optical resonances in one-dimensional dielectric nanorod arrays:ﬁeld-induced ﬂuorescence enhancement,” Opt. Lett. vol. 32, no.18, pp. 2762-2764, Sep. 2007.[7] D. M. Natarov, V. O. Byelobrov, R. Sauleau, T. M. Benson, andA. I. Nosich, “Periodicity-induced eﬀects in the scattering andabsorption of light by inﬁnite and ﬁnite gratings of circular silvernanowires,” Opt. Express, vol. 19, no. 22, pp. 22176-22190, Oct.2011.[8] S. V. Boriskina and L. D. Negro, “Multiple-wavelength plasmonicnanoantennas,” Opt. Lett., vol. 35, no. 4, pp. 538-540, Feb. 2010.[9] R. Mittra and S. W. Lee, Analytical Techniques in the Theoryof Guided Waves, New York: Macmillan, 1971.[10] A. K. Rashid and Z. Shen, “Scattering by a two-dimensionalperiodic array of vertically placed microstrip lines,” IEEE Trans.Antennas Propag., vol. 59, no. 7, pp. 2599-2606, Jul. 2011.[11] A. C. Polycarpou and M. A. Christou, “Full-wave scatteringfrom a grooved cylinder-tipped conducting wedge,” IEEE Trans.Antennas Propag., vol. 59, no. 7, pp. 2732-2735, Jul. 2011.[12] J. E. Varela and J. Esteban, “Analysis of laterally open periodicwaveguides by means of a generalized transverse resonanceapproach,” IEEE Trans. Microw. Theory Tech., vol. 59, no. 4,pp. 816-826, Apr. 2011.[13] R. Wang, L. Han, J. Mu and W. Huang, “Simulation ofwaveguide crossings and corners with complex mode-matchingmethod,” J. Lightwave Technol., vol. 30, no. 12, pp. 1795-1801,Jun. 2012.[14] M. Casaletti, R. Sauleau, S. Maci, and M. Ettorre, “Modematching method for the analysis of substrate integrated waveg-uides,” in Proc. Eur. Conf. Antennas Propag., Mar. 2012, pp.691-694.[15] L. Lewin, “On the inadequacy of discrete mode-matching tech-niques in some waveguide discontinuity problems,” IEEE Trans.Microw. Theory Tech., vol. 18, no. 7, pp. 364-369, Jul. 1970.[16] H.-S. Yang, J. Ma, and Z.-Z. Lu, “Circular groove guide forshort millimeter and submillimeter waves,” IEEE Trans. Microw.Theory Tech., vol. 43, no. 2, pp. 324-330, Feb. 1995.[17] H.-W. Yao and K. A. Zaki, “Modeling of generalized coaxialprobes in rectangular waveguides,” IEEE Trans. Microw. TheoryTech., vol. 43, no. 12, pp. 2805-2811, Dec. 1995.[18] J. Zheng and M. Yu, “Rigorous mode-matching method of cir-cular to oﬀ-center rectangular side-coupled waveguide junctionsfor ﬁlter applications,” IEEE Trans. Microw. Theory Tech., vol.55, no. 11, pp. 2365-2373, Nov. 2007.[19] M. Abramowitz and I. A. Stegun, Handbook of MathematicalFunctions with Formulas, Graphs, and Mathematical Tables,National Bureau of Standards: Applied Mathematics Series-55,1964.[20] M. Bassis (Ed.), Handbook of Optics, Volume IV: OpticalProperties of Materials, Nonlinear Optics, Quantum Optics,McGraw-Hill Professional, 2009, 3rd ed.[21] P. B. Johnson and R. W. Christy, “Optical constants of thenoble metals,” Phys. Rev. B, vol. 6, no. 12, pp. 4370-4379, Dec.1972.[22] C. Sohl and C. Larsson and M. Gustafsson and G. Kristensson,“A scattering and absorption identity for metamaterials: experi-mental results and comparison with theory,” J. Appl. Phys., vol.103, pp. 054906/1–5, 2008.[23] M. Gustafsson, C. Sohl, and G. Kristensson, “Illustrations ofnew physical bounds on linearly polarized antennas,” IEEETrans. Antennas Propag., vol. 57, no. 5, pp.1319-1327, May 2009.

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