Efficient mode-matching analysis of 2-D scattering by periodic array of circular cylinders
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Efficient mode-matching analysis of 2-D scattering by periodic array of circular cylinders Document Transcript

  • 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. 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