Rapidly convergent representations for 2D and 3D Green s functions for a linear periodic array of dipole sources

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1 Rapily convergent repreentation for D an reen function for a linear perioic array of ipole ource Derek Van Oren an Vitaliy Lomakin Department of Electrical an Computer Engineering, Univerity of California, San Diego, 95 ilman Dr., La Jolla, CA 993 Accepte for publication in IEEE Tranaction on Antenna an Propagation Abtract Hybri pectral-patial repreentation are introuce to rapily calculate perioic calar an yaic reen function of the Helmholtz euation for D an configuration with a 1D (linear) perioicity. The preente cheme work eamlely for any obervation location near the array an for any practical array perioicitie, incluing electrically mall an large perioicitie. The repreentation are bae on the epanion of the perioic reen function in term of the continuou pectral integral over the tranvere (to the array) pectral parameter. To achieve high convergence an numerical efficiency, the introuce integral repreentation are cat in a hybri form in term of (i) a mall number of contribution ue to ource locate aroun the unit cell of interet, (ii) a mall number of ymmetric combination of the Flouet moe, an (iii) an integral evaluate along the teepet ecent path (). The integral i regularize by etracting the ingular behavior near the ale point of the integran an integrating the etracte component in cloe form. Efficient uarature rule are etablihe to evaluate thi integral uing a mall number of uarature noe with arbitrary mall error for a wie range of tructure parameter. Strength of the introuce approach are emontrate via etenive numerical eample. 1

2 1. Introuction Structure compriing infinite perioic array of element fin many application in phyic an engineering incluing microwave an optical filter, printe an leaky-wave antenna, laer, an waveguie. An efficient metho to compute electromagnetic fiel in uch tructure i to ue the integral approach [1], in which fiel are foun in term of a uperpoition integral (patial convolution) of a ource itribution, given within a unit cell, an a perioic reen function (PF), which incorporate the perioic bounary conition. Efficient metho for computing PF are important to enable the ue of integral-euation metho, to compute fiel raiate an cattere from perioic configuration, an to tuy their iperion propertie. It i well known that repreentation of PF via irect patial ummation are etremely lowly convergent, an iverge when the perioicity i efine with a comple phae hift between element [1]. The convergence can be improve by uing pectral ummation in term of Flouet erie. However Flouet erie repreentation uffer from low convergence in the important cae where the obervation point reie near a perioic tructure. For problem with 1D (linear) perioicity the Flouet erie are ivergent along the perioicity ai. Several approache have been propoe to accelerate the computation of PF. Some are bae on pectral an patial formulation with Poion, Kummer, an Shank tranformation [1-5], which accelerate the original patial an pectral erie repreentation but o not lea to eponentially rapi convergence an may be low in variou ituation [1,, 6]. The Ewal approach, in contrat, uing Ewal an Poion tranformation, lea to eponentially convergent PF erie repreentation. However, it reuire ouble ummation, can uffer high-freuency breakown, an involve chooing a proper plitting parameter, which may be not traightforwar to implement [1,, 6-16]. Moreover, the Ewal approach ha never been etene to yaic reen function. Veyoglu tranformation [1,, 6, 17] an perfectly matche layer (PML) approache [18, 19] alo can be ue to lea to eponential convergence. However, a

3 eplaine below, thee approache are not applicable to comple phae hifte conition an may uffer from low-freuency breakown. In thi paper we introuce a novel metho for computing calar an yaic PF for D an problem compriing a 1D perioicity with phae hifte conition. The metho i bae on the alternative reen function pectral repreentation formalim [] an involve the tranvere (to the perioicity ai) pectral epanion with the longituinal (along the ai) pectral 1D PF available in cloe form. Thee repreentation lea to eponential computational convergence for any location near the array ai an for any practical perioicitie, incluing electrically mall an large perioicitie. Moreover, the introuce erie have a clear phyical interpretation, being epree in term of a mall number of irect fiel contribution, a mall number of Flouet moe, an a rapily convergent integral repreenting the combine contribution of the remaining ource. The paper i organize a follow. Section preent the problem formulation. Section 3 i a mathematical preentation of the reolvent formalim, which i then ue in ection 4 to erive alternative pectral repreentation for PF in term of the longituinal repreentation (conventional Flouet moe epanion) an tranvere repreentation (novel pectral approach). The latter repreentation i ue in Sec. 5 with relate Appenice A an B to evelop highly efficient (eponentially accurate an rapi) cheme for computing calar D an PF. Section 6 outline how the proceure in Sec. 5 can be etene to rapily compute the yaic D an PF. Section 7 preent an etenive numerical tuy upporting the theoretical reult in Sec Section 8 raw concluion an ummarize fining of the paper.. Problem formulation Conier a 1D perioic array of ientical electric ipole ource in free pace. The ipole are arrange along the ai with pacing L an irecte along a general unit vector ˆp (Fig. 1). Thee are linear ource for D configuration an point ource for problem. The ource have a linear phae hift jk e nl etermine by the (generally comple) parameter k, where n i an integer counting the ource uch that the ource 3

4 with n = lie at the origin of the coorinate ytem (Fig. 1). The time epenence with ω being the angular freuency, i aume an uppree everywhere in the paper. j t e ω, Electromagnetic fiel raiate by thi array can be foun uing yaic PF. Specifically, the electric fiel raiate by the array in Fig. 1 can be foun from D, D, ˆ E = Il p. (1) where Il i the current moment of each ipole. (Here an in the ret of the paper the upercript D, enote a function for D or cae.) The D an yaic PF D, are given by D, jη D, () r = ( k I + ). () k Here, η i the free-pace characteritic impeance, k i the free-pace wavenumber, an D an are repectively D an calar PF that are efine via the uperpoition principle D,3 D jknl D,3 r = D n= () e (, k R ), jkrn D 1 () e ( k, Rn) = H ( krn), ( k, Rn) =. 4j 4π R n n (3) D where an, the calar D an free-pace reen function ue to a ingle iolate ource, are umme over all the ource in the array. Here, R = r r i the itance from the oberver locate at r to the n -th ource at r n. The calar PF n n D i a function of the two patial variable an z, while epen only on an y z 1 = ( + ). Clearly, the calar (an thu yaic) PF ehibit the phae hifte perioicity property with repect to the perioicity L 4

5 D, jk nl D, ( r+ nlˆ ) = e ( r), ( ˆ) ( ). D, jk nl D, r+ nl = e r (4) The conition in E. (4) can be regare not only a a property of the PF but alo a bounary conition, which relate the fiel on the two bounarie of each unit cell. For lole meia the erie in E. (3) converge etremely lowly. Therefore, more efficient repreentation for contructing the PF are to be evelope. 3. Spectral reolvent formalim The calar PF in E. (3) can be foun by eparation of variable into the tranvere ( z or ) an longituinal ( ) variable. To thi en, the PF are epane over a erie of bai function contituting an orthogonal bai for the problem either in the longituinal or tranvere irection. The epanion over the longituinal pectrum lea to the well known Flouet moe repreentation. The epanion over the tranvere pectrum ha never been eploite for PF an i tuie below. More generally, the PF can be epree a a contour integration of a prouct of pectral 1D or D reen function a efine in the framework of the reolvent approach by Felen an Marcuvitz []. The PF are repreente a 1 1 ( ) λ (, λ ) (, λ ( λ )) λ (, λ ) (, λ ( λ )), π j π j D r = g gz z z = zgz z z g z 1 1 () (, ) (, ( )) (, ) (, ( )). π j π j r = λg λ g λ λ = λg λ g λ λ (5) Here, λ an z, λ are pectral variable aociate with the longituinal an tranvere eparable variable an z or, repectively. Thee pectral parameter are relate via λ =. λz, k In E. (5), the function g ( z, λ ) an (, λ ) are 1D an D PF for the pectral problem in the tranvere irection. They are given by z z g 5

6 j λz z e gz(, z λz) =, j λ z ( λ ) 1 () g(, λ) = H. 4 j (6) Thee function have branch point at λ z, =, an the aociate branch cut are choen along the poitive real ai (ee Fig. (a)). The top Riemann heet, in which integration 1 are carrie out, i choen uch that { λ z, } Im ( ) <. In both euation of (5), the function g (, λ ) i the 1D PF for the pectral problem in the irection. It i given by 1 e g (, λ ) = j λ j λ j( λ k) L j( λ + k) L j λ 1 e 1 e e. (7) for < L. It can be foun by umming up contribution from all point ource for the 1D pectral problem in the irection 1 jk ( λ ) 1 nl j λ nl (, λ) ( ) n= g = j e e. (8) The pectral PF g (, λ ) atifie the 1D pectral Helmholtz euation ( λ) g(, λ) δ( ) = in the n = unit cell with the bounary conition g + L = e g. (9) (, ) jk L λ (, λ) The PF g (, λ ) ha an infinite number of pole in the comple λ plane but it oe not have any branch point an branch cut. In E. (5), the integration contour in the comple λ plane encloe the ingularitie of g (, λ ) but not thoe of g ( z, λ ( λ )) an (, λ ( λ )). Similarly, the integration z z contour over λ z an λ encloe the ingularitie of gz( z, λ z) an g (, λ ) but not thoe of g(, λ( λ z, )) []. g 6

7 The repreentation of E. (5) in term of the longituinal an tranvere pectral parameter yiel ientical reult. The choice between them i motly one of computational convenience an convenience of repreentation. The two repreentation choice are analyze net. 4. Alternative pectral repreentation The formalim in Sec. 3 can be ue to evelop two alternative repreentation for the D an PF by uing the integration over the longituinal an tranvere pectral variable in E. (5). Section 4.1 preent the longituinal (Flouet moe) pectrum repreentation. Section 4. introuce the novel tranvere pectrum repreentation. 4.1 Longituinal (Flouet moe) repreentation In thi repreentation, the integration over the longituinal pectral parameter λ in E. (5) i carrie out. Becaue g (, λ ) ha no branch cut in thi comple plane, the only contribution to the integral come from the reiue of it infinite number of pole at ( k πm L) λm = + / (Fig. (b)). Each reiue contribution i ientifie a the contribution of the correponing Flouet (iffraction) moe. (Here an in the ret of the paper the integer m will be ue to count the pole of g an correponing Flouet moe of the array.) The reulting um of thee reiue lea to the conventional Flouet moe epanion D jkm jkzm z 1 e e () r =, L jk m= 1 1 () e H ( k ), zm 3 D jkm () r = L m= 4 j m (1) where k = k + π m L an m k{, } = ( k k ) are referre to a the longituinal an 1 z m m tranvere Flouet moe wavenumber, repectively. The choice of the uare root for k epen on the application. In many cae, e.g. when the interet i to obtain the { z, } m cattering coefficient of a perioic tructure, the uare root i choen o that k{ z, } m i 7

8 on the upper Riemann heet with Im{ k{ z, } m} for all m. In other cae, uch a the analyi of comple iperion relation of leaky-wave antenna, one may chooe Im{ k } > for a finite number of Flouet moe m. { z, } m The um (1) converge eponentially fat when z, are large with the number of term reuire to achieve a high accuracy being greater than L ( π { z, } ). On the other han, a, z, the ummation in (1) converge etremely lowly. However, it i thi latter cae, where the fiel are oberve along the ai of the ource, that i often of interet. Therefore, alternative rapily convergent repreentation for an are to be evelope. It i note that the Flouet moe repreentation (1) can be obtaine in other way. In thi paper, the erivation of (1) eemplifie the ue of the reolvent formalim an emphaize an intimate relation between the Flouet moe pectrum an the 1D pectral PF g (, λ ). Thi approach alo emontrate the relation between the longituinal an tranvere repreentation. 4. Tranvere repreentation D Integration over the tranvere pectral parameter λ z an λ in E. (5) yiel the tranvere repreentation. The only contribution to the integral come from integrating aroun the branch cut along the poitive part of the real ai (Fig. (a)). Changing variable from the pectral variable λ z, to g i a ymmetric function of z, 1 kz, ( λz, ) PF in term of the tranvere pectrum epanion = an eploiting the fact that k lea to an alternative repreentation for the D an D 1 r = kz kzz g k kz π () co( ) (, ( )), 1 r = kkj k g k k π () ( ) (, ( )). (11) 8

9 Here, J ( k ) i the zeroth-orer Beel function, kz, i ientifie a the wavenumber in the tranvere irection, k = ( λ ) = ( k k ) i ientifie a the wavenumber 1 1 z, z, in the irection, an g (, k ) i the longituinal 1D PF that i reefine in term of the longituinal wavenumber k 1 e g (, k ) = jk jk j( k k) L j( k+ k) L jk 1 e 1 e e. (1) A clear from the icuion after E. (1), the pectral PF g (, k ) ha pole in the comple z, k plane at k ( k ( k π m L) ) 1 = +, which correpon to the Flouet { z, } m moe wavenumber km = k + π m L. For lole configuration an purely real k, thee pole form pair with oppoite ign on either the imaginary or real ai of k z. For obervation point within the n = unit cell, i.e. for < L, g ( ; k ) ecay eponentially fat for large z,. Thi ecay lea to an eponentially fat convergence k of the repreentation (11) even for the cae z = =, provie that. For thi reaon the longituinal repreentation i the better choice for fining the fiel at or near the ai of the ource. The reult of the tranvere repreentation in (11) i ientical to that in (1) if all uare root in (1) are efine on the upper Riemann heet, i.e. if Im{ k{ z, } m}. To efine the PF on the lower Riemann heet aociate with a wavenumber k { z, } M efine in (1) via { z, } M Im{ k } >, the epreion in (11) i moifie a jkm co( k z) e () r () r, L jk D D zm j () () ( ). L 4 jkm r r J km e zm (13) Here, D, () r in the left an right-han ie are PF efine on the lower an upper Riemann heet of k { z, } M, repectively. Thi epreion can be obtaine by taking the 9

10 ifference between D, () r on the lower an upper Riemann heet. The ret of the analyi in the paper aume that the PF are efine on the upper Riemann heet of k for all m. The PF on the lower Riemann heet can be eaily obtaine via (13). { z, } m It i alo note that in all practical ituation only a mall number of lower Riemann heet (typically 1 or ) i reuire to be coniere. Compare to the Flouet moe epanion (1), the repreentation in (11) are better uite for computing the PF near the array. However, they till may uffer from problem in their numerical implementation. For the cae r (i.e. for z,, ) the repreentation in (11) are lowly convergent with convergence rate imilar to that of (1). In aition, the integran in (11) may have pole on the real (integration) ai. Furthermore, for large kl, i.e. in the high-freuency regime, the integran are highly ocillatory, which can complicate their irect evaluation. Thee potential problem are reolve in Sec Fat computation of the calar perioic reen function Thi ection preent a proceure that ue the tranvere pectral epanion in Sec. 4. to erive rapily convergent repreentation for the D an PF. The repreentation are vali for any ource-oberver location near the array an can be eaily implemente in a computer coe. The proceure involve manipulating the integran in (11) to make them lowly varying an thu eaily integrable. Section 5.1 preent uch a rapily convergent repreentation. Section 5. preent an efficient numerical implementation of the repreentation in Sec Regularize tranvere repreentation The convergence of the integral (11) can be improve by eplicitly etracting the contribution of a certain number of ource in an aroun the unit cell of interet an evaluating them irectly uing the patial PF in E. (3). The remaining infinite ource may be evaluate uing the tranvere repreentation (11). To formulate thi, the longituinal pectral PF g (, k ) in (8) can be rewritten a 1

11 jknl jk nl jkl( + 1) jkl( + 1) jk jkl( + 1) jk e e e e e e e g(, k) = + + j( k k) L j( k+ k) L. (14) n= jk 1 1 jk e e Here, the firt term in the right-han ie i a ummation of free-pace longituinal pectral 1D PF, taken over + 1 ource reiing at location = nl aroun the origin. The econ term in the right-han of (14), which comprie the contribution from the remaining infinite number of ource, ecay eponentially fat for large k z, regarle of the obervation location within the n = (zeroth) unit cell (an even for = ). To avoi the poible ocillatory behavior of the integran in (11) for large L, the integration path can be eforme from the original path along the real ai to the to the teepet ecent path () [, 1] (ee Fig. 3). In general, the path houl pa through a ale point whoe poition in the comple kz, plane epen on L,, an z. Here, to obtain a robut cheme for any r withunit cell, it i aume that L> r an the ale point i given by k, =. Reiue at pole that are croe while eforming the original integration path to the mut then be taken into account. To thi en, the calar PF can be written a z p jkm D jk 1 () co( ) nl kzmz e D r = n + + n= 4j L 1 m= jkzm () e H ( kr ) (), r jkr n p jk nl e j jkm r = + m + n= 4π R 1 4 n L m= () e J ( k ) e (). r (15) Here, the firt ummation are obtaine by uing the firt term in the right-han ie of (14) in (11). Thee ummation repreent contribution ue to + 1 ource locate in an aroun the origin (Fig. 1), an can alo be foun by taking the um in (3) over thee ource. The econ ummation in (15) repreent contribution of p pole croe while the eforming the original integration path to the. Comparing with the Flouet moe repreentation (1), thee pole contribution are ientifie a combination of the Flouet moe with ymmetric behavior with repect to the irection tranvere to the 11

12 array ai ( z an irection). The function integral given by D () r an () r in (15) are D jkl( + 1) jk jkl( + 1) jk jkl( + 1) 1 e e e e e co( kz z) () r = k, j( k k) L j( k k) L z π e 1 e jk (16) jkl( + 1) 1 e e e e e J( k ) kk. jk jkl( + 1) jk jkl( + 1) jk () r = j( k k) L j( k k) L π e 1 e Thee function repreent the contribution of the remaining ource. Unlike the integran in E. (11), the integran in E. (16) have branch point aociate with the 1 uare root k = ( k kz, ), choen uch that Im{ k } <. Thee branch point, however, can be eliminate by a change of variable. The integran in (16) ecay very rapily with an increae of k z,. To how thi eplicitly the integral can be formulate by making a change in variable from kz, to the imenionle variable, efine via right-han ie of E. (16) can be rewritten a k k j = (1 ). The integral on the D D α r = () f () e, + j α r = () f () e, (17) where α = kl( + 1) an ( 1) ( ) ( 1) ( ) 1 jk L + k j + jk L + k j+ D e e e e f () = + co( kz + j), jkl kl( j+ ) jkl kl( j+ ) π 1 e e 1 e e f 1 e e e e () = + π 1 e e 1 e e jkl( + 1) k ( j + ) jkl( + 1) k( j+ ) jkl kl ( j J + ) jkl kl( j ) ( k + j + ). (18) The factor of e α i more ignificant for large in each integral enure a rapi ecay of the integran an thi ecay. Typically, a mall i aeuate to rener ufficient 1

13 convergence, an for mot practical problem only a mall number of pole p nee to be taken into account. An important ifference between the integran in (17) i that the integran of ha a branch cut ( + j) 1, which appear ue to the variable D tranformation from kz, to, wherea the integran of oe not. A eplaine in Sec. 5., the preence of the branch cut may lea to a low-freuency breakown for the D cae an reuire moification in haning the low-freuency regime ( L abence of the branch cut in the integran of λ ). The in (16) i ue to preence of the k factor in the integran for thi cae. Robut numerical implementation for the integration in (17) are given in Sec Robut numerical implementation of the tranvere repreentation When the pole in the integran in E. (18) reie near the ale point =, the integran vary rapily along the near thee pole. Such ituation are encountere when the pole are within the contributing zone of the integran, i.e. when <Ω m p α 1 with Ω being a parameter of orer O (1). Thi occur in the low-freuency regime p ( L λ ) where α 1 an/or when the wavenumber of one of the Flouet moe i cloe to the free-pace wavenumber, i.e. when k + π m L ± k. In the latter cae the pole m can be very cloe to = an 1 m <Ω p α even for a large α. In uch cae, the rate of the integran variation ue to the preence of the pole may be much greater than that ue to other term. To mooth thee variation the reiue of the integran at thee pole can be eplicitly etracte an integrate in a cloe form. It i note that the ymmetry of g (, k ) with repect to kz, an lea to the eitence of pair of pole that are ymmetric about the origin in the comple plane. Denote a ± the m th pair m of pole that reie near the ale point = an aume that there are a total (typically mall) number of p pole pair to account for. One can then rewrite the integral in (17) a 13

14 p D D α D D r = + m m m + j n= 1 () f () e Re{ f ( )} I, p α r = + m m m n= 1 () f () e Re{ f ( )} I. (19) Here, p D D, D, mre{ f ( m) () = () m= 1 m f f () are non-reonant function that have no ingularitie near the integration path. The integral D I m an I m in (19) are given by I erfc( ign( )) 1 1, + j + j ign( j ) + j + j α α m α D e πe jmα jm e m = = + m m m m m α e 1 α m m = = α m m I e Ei( ), (1) where erfc i the complementary error function, an Ei i the eponential integral function []. The integral for I D m can be rapily evaluate numerically uing variou aaptive integration metho. The integran of the firt integral for D I m ha a pole an a branch-cut ingularity, wherea the integran of the econ integral ha only the branchcut ingularity, which allow for a impler numerical evaluation. It i important to mention that the integral an pecial function in (1) o not epen on the obervation location an nee to be evaluate only once for a given problem. Therefore, they o not contribute ignificantly to the overall computational cot. The firt integration term in the right-han ie of (19) can be evaluate uing variou uarature rule, e.g. thoe evelope in Appeni A an B. The reulting integral are given by 14

15 D, ( ) p D, D, D, D, D, D, () r = f ( ) e α w + Re{ f ( )} I, () m m m = 1 n= 1 where are uarature noe for D an problem given in Appeni A an B, D, w are correponing uarature weight, an i the number of uarature noe D, (an weight). For the D cae, the uarature rule are efine ifferently for highfreuency ( α 1, i.e. L ~ λ or L λ ) an low-freuency ( α 1, i.e. λ L ) regime. For the high freuency regime ( α 1), a imple eually-pace noe uarature rule i very efficient (Appeni A.1). For the low-freuency regime, the uare root ( ) 1 + j varie much fater than the eponential term ep( α ), an reuire a pecial treatment. A hown in Appeni A., the uare-root term can be epane over a erie of coine function an thi erie can be truncate to reult in a uarature rule with a mall number of uarature noe for any α. For the cae, an efficient uarature rule i obtaine in Appeni B bae on the integran epanion over Beel function erie. Thi rule applie to any perioicity (electrically mall an large) without moification becaue the integran in (19) oe not have a branch-cut ingularity. Repreentation in E. (15) with the contribution in () are highly efficient for computing the D an PF for any obervation location near the tructure (whether for z = =, or for r =, or for z, of everal λ ) an for any practical perioicitie L (whether electrically mall with L λ, intermeiate with L ~ λ, an electrically large L λ ). The number of uarature noe, pole contribution term p, an irect component to ouble preciion. p, etracte pole combine can be mall to achieve any accuracy up 6. Fat computation of the yaic perioic reen function Having etablihe a proceure for calculating the calar PF D,, one may apply the yaic operator in E. () to fin the full yaic PF D,3 D. The yaic operator, involving erivative with repect to patial coorinate, i brought inie the integral in E. (11) 15

16 D jη = kz k kzz g k kz π k + () r ( I )co( ) (, ( )), jη = kk k J k g k k π k + () r ( I ) ( ) (, ( )). (3) Following the proceure leaing to E. (15), a number + 1 of irect fiel contribution near the unit cell of interet are etracte an evaluate in cloe form. The integration path of the integral i then eforme to the with pole reiue taken into account. Finally, the reulting integral are regularize a in Sec. 5 an are evaluate uing the ame uarature rule a for the calar PF in E. (). ote that, epite the application of the ifferential operator, the integran of o not have branch cut an therefore o not reuire the uare-root truncation for the low-freuency regime. The implementation of the yaic PF allow one, via E. (1), to fin all the electricfiel component reulting from an infinite array of ientical ipole ource oriente in any irection. To fin the magnetic-fiel component a imilar approach may be ue, involving a ifferent yaic operator but thi icuion i omitte for brevity. 7. umerical reult Thi ection preent numerical analyi that how how the formulation in Sec. 5 an 6 can be ue to calculate calar an yaic D an PF. It alo tuie the convergence rate of the obtaine repreentation an how that thee repreentation are very efficient for a wie range of tructure parameter. The numerical analyi tart with calculating D an fiel near a 1D array for ifferent tructure parameter an procee with the tuy of the error behavior. Figure 4(a) how the magnitue of the calar PF an the yaic component, an, plotte very cloe to the ai along the horizontal line < <.5L yy (half of the unit cell i ufficient ue to ymmetry). The calculation are one uing E. (15) with L = λ, y =, z 1 3 = L, an k (.85.1 ) = j k. The contribution from the ource at the zeroth unit cell (i.e., D, D, ) i ubtracte to avoi ivergence at 16

17 the unit cell center. Cloe to the ai the off-iagonal component of vanih, while yy an zz are nearly eual. Phyically, thi inicate that a the fiel from a perioic array of ipole oriente parallel to an tranvere to the ai have TE an TM polarization, repectively. The fiel how ymmetry in the y an z irection. Figure 4(b) how D plotte along the ame line, but for z = an z =.1L. The correponing yaic component are not hown ince for mot D problem the fiel may be eparate into TE z an TM z component, each of which may be olve uing the calar PF D, o the yaic operator D i of limite interet. In Fig. 5(a) an 5(b) the magnitue of the D an calar PF are plotte along the ame line with L =.4λ an z 1 3 = L but for ifferent value of k. Real value of k with k plane wave, wherea k < k may correpon to the ituation of ecitation of the array by an incient > k may correpon to a ituation where an array i ecite by a low wave propagating along the array for which the fiel ecay away from the array. Comple value of k, though unphyical, are ueful in plane-wave epanion of the fiel, an are neceary for fining cattering from uch an array. All reult in Fig. 4-5 were obtaine uing = 3, = 1, an Ω = 3, an Ω =.5 D ( Ω efine in Appeni A etermine the integral truncation reuire for contructing a uarature rule). Thee reult were compare with reult obtaine via the conventional pectral repreentation in E. (1) with 8 ummation term, aume to be computationally eact. The reult were foun to be fully convergent (the curve for the repreentation E. (1) an (15) cannot be itinguihe viually). To further emontrate the rapi convergence of the tranvere repreentation (15) over a wie range of value of the perio L, Fig. 6(a) an 6(b) how the relative error between the reult of the longituinal an tranvere repreentation obtaine via E. (1) an 17

18 (15) a a function of the number of uarature noe ue in the integration. The error of the tranvere repreentation i efine a Error L / D, D, eact ( ) numeric ( ) r r = D, D, eact ( ) ( ) r r D, L / 1, (4) where are obtaine via (1) with a very large to achieve ouble-preciion D, eact accuracy an are compute uing the rapi formulation of (15) with the D, numeric integral evaluate via () with uarature rule in Appeni A an B. ote that in the enominator of (4) the zeroth irect term (numerically eact ) PF D, eact () D, () r i etracte from the reference r to avoi iviion by a large number for r. Without etracting thi zeroth term, the reulting error woul artificially become ignificantly lower for mall r. Calculation were one uing z 1 3 = = L, k = (.85.1 j) k, = 5 for three value of perioicity ranging from etremely 6 mall ( L = 1 λ ) to moerate ( L =.6λ ), an to very large ( L = 1λ ). Very mall perio correpon to ene array, uch a mehe. Moerate perio (on the orer of a wavelength) are characteritic of filter a well a phae array, leaky-wave, an traveling/urface wave antenna. Large perio can alo moel certain antenna array. The parameter Ω wa choen in the range from.4 to 6.. For the PF, the uarature rule in Appeni B wa ue. For the D PF, the uarature rule in (7) an (3) were ue for L 1 6 = λ an L.6 = λ, L = 1λ, repectively. It i evient that the reult converge very rapily with an increae of the number of integration noe for any value of the perio that woul be of practical interet. The component of the yaic PF how imilar convergence behavior an, therefore, are not hown. To emontrate the effect of pole reiing near the integration contour, figure 7(a) how the PF calculate with an without pole ingularitie etracte for L =.4λ, k = 1.5k, 3 = 1 L, 5 =, an Ω = 5.5. Thi value of k wa choen to be cloe to k uch that the preence of the pole affect the rate of the variation of the integran in (17). Such a value of k may correpon to an array ecite by a low wave, e.g. in the 18

19 cae of low wave traveling on a Yagi-Ua array of ipole. It i evient that without pole etraction, the reult o not converge rapily. For k even cloer to k the reult become abolutely inaccurate. On the other han, after the pole etraction, a very rapi convergence i obtaine. et, to how the importance of the uare-root truncation of the D PF integran at low freuencie, figure 7(b) how the D PF calculate uing the non-truncate uarature rule in (7) an the truncate uarature rule in (3) with k = (.85.1 j) k, an ifferent value of z 1 For thi low-freuency regime ( L 3 = L, 5 L 3 = 1 λ, =, an Ω ranging from 1.8 to 4.6 for. In both cae, the pole were etracte a ecribe in (19)-(). λ ) the preence of the uare root for D in (19) affect ignificantly the rate of the integran variation. A a reult, the non-truncate uarature rule in (7) oe not lea to a rapi convergence. For even maller L the obtaine non-truncate reult become abolutely inaccurate. However, with the truncate uarature rule in (3) a rapi convergence i achieve. Figure 8(a) an 8(b) tuy the convergence of the tranvere repreentation for a range of obervation itance z an from the array for which the Flouet moe repreentation (1) may till be impractical. The tructure parameter are choen a L =.6λ, k = (.85.1 j) k, an Ω, fie for each curve, range from 3.8 to 5.5. The error increae away from the array, which i aociate with an increae rate of the integran variation for larger z an. Clearly the larget error are obtaine for the mallet an ( = 3, = 6 ). The error can be reuce by only a light increae of an/or. It i alo important to mention that very low error with a mall an are obtaine even for relatively larger iplacement from the array (, z ~5λ ). Thi i an important property of the preente formulation a it can allow uing the ame formulation without witching it with other formulation (e.g. with the Flouet ummation in (1)) for a wie range of tructure parameter. 19

20 8. Summary an concluion Highly rapi an accurate numerical implementation for calar an yaic PF for D an configuration with 1D (linear) perioicitie are introuce. The implementation are bae on the epanion of the PF in term of the continuou pectrum integral over the tranvere pectral parameter (E. (11)). To achieve a high convergence, the introuce integral repreentation are cat in a hybri form in term of (i) a mall number of contribution ue to ource locate in an aroun the origin of a perioic unit cell of interet, (ii) a mall number of the integran pole reiue contribution correponing to ymmetric combination of Flouet moe generate by the array, an (iii) the remaining integral evaluate along the. The integran of the integral are regularize by etracting the ingular pole behavior near the ale point of the integran an integrating it in cloe form. The reulting regularize integran ehibit a lowly varying epenence of the integration variable, which allow implementing highly efficient uarature rule for the integral evaluation. The integral for the D cae are further regularize by truncating the uare-root behavior at low freuencie. umerical imulation were eecute to emontrate trength of the introuce approach. It wa hown that the relative accuracy of.1% can be achieve with only about total number of 8 ummation component. Double-preciion accuracy can be achieve for a total number of about 5 ummation component. It houl be note that the preente cheme for the PF reuce to Veyoglu tranformation approach [1,, 6, 17] for =, real k, an without pole etraction or uare-root truncation. However, Veyoglu approach cannot be ue for general comple k an cannot be efine on lower Riemann heet becaue it oe not eplicitly take into account the pole reiue a one in E. (13) an (15). In aition, ue to the ingular behavior of the integran, the Veyoglu approach become lowly convergent in the low-freuency regime an for the important cae of the blin angle regime, where one of the Flouet wavenumber i near the free-pace wavenumber. The cheme preente in thi paper efficiently reolve all the potential problem of the Veyoglu an other alternative approache a eplaine further net.

21 1. Low-an high-freuency regime: The preente cheme work eamlely in the low-freuency regime with L λ, intermeiate freuency regime with L ~ λ, an high-freuency regime with L L 6 = 1 λ an L 1 λ. For eample, in Fig. 6 the convergence rate for = λ were nearly ientical. Thi i achieve ue to the pole ingularity etraction in (19), which make the integran vary lowly for any L, an uare-root truncation in (3), which allow contructing an efficient uarature rule for D PF in the low-freuency regime.. Robutne an convenience: The preente cheme reuire chooing only a few convergence parameter. Thee inclue the parameter Ω etermining the integration range limit ma, the number of integration noe, the number of irectly etracte term, an the parameter Ω p etermining the raiu of the integral contributing zone, in which the integran pole are etracte. All thee parameter can be choen only once to aure a ufficient accuracy, inepenently of particular problem parameter. For eample, Ω = 6, = 1, = 5, Ω = woul lea to the ouble-preciion accuracy for any practical array parameter an obervation location near the array. Furthermore, the preente cheme lea to a very imilar performance for D an configuration for calar an yaic PF. The cheme work eamlely for any location near the tructure, whether along the ai (incluing the origin) or at a few wavelength from the ai. 3. Computational compleity: The preente cheme are highly computationally efficient. The total number of ummation term i eplicitly given by total = ( + 1) + + p + p with total = O(1) for any practical tructure parameter an obervation location. For eample, for the convergence parameter choice in item reulting in ouble-preciion accuracy total = 5 (auming p p = = reuire for mot practical value of L ). If lower accuracy i p acceptable, total can be reuce further conierably. A icue in Sec. 5, the preente cheme o not reuire etenive computational time for evaluating variou pecial-function integral at every obervation location. 1

22 4. Singularity at r : The preente cheme eplicitly etract the fiel contribution from the ource locate at the origin of the n = unit cell. A a reult the ingular behavior of the calar an yaic PF for r i ientical to that of the calar an yaic free-pace PF. Thi fact may be of importance for integral euation metho to evaluate the elf-term where the teting an bai function coincie [1]. The preente cheme can be etene to more complicate environment like layere meia. For D problem thee etenion are traightforwar: One only a in E. (11) a ummation over the icrete pectrum of guie moe. For problem thee etenion are omewhat more involve but poible. Etenion can be evelope for problem involving D perioicitie. The preente iea alo can be ue to calculate more complicate reen function, e.g. reen function for an aperioic ource near an infinitely perioic tructure. Such etenion will be reporte elewhere. The calar an yaic PF can be ue in many practical problem incluing the evaluation of antenna raiation pattern, integral euation for perioic tructure, an the tuy of iperion propertie of perioic array. Acknowlegment Thi reearch wa upporte by anocale Intericiplinary Reearch Team (IRT) program, ational Science Founation (SF) an by DARPA ACHOS program. Appeni A: Quarature rule for the D cae For evaluating the D PF in E. (17) an (19), one nee to contruct a uarature rule for integral of the following form D D = 1 α f () e f( ) w, (5) + j where D are the uarature noe, D w are the correponing uarature weight, i the number of the uarature noe, an f ( ) i an even function of. ote that ince the uare-root 1 ( + j) i a comple value function, the uarature rule efinition

23 in (5) reult in comple-value weight. Thi efinition i employe for convenience of the numerical repreentation for the integration in (). Due to the eponential ecay of the integran, the upper integration limit of the integral (5) can be truncate to a value ma, at which the integran ha a ufficiently low magnitue. From (33), ma can be choen a ma 1 =Ω α, where Ω i a parameter of O (1) choen a Ω = log(1 ε ) with ε being a precribe error of the integration truncation. For a given function f ( ), the accuracy of the evelope uarature rule epen on the parameter Ω. Thi parameter, through ma, etermine the integral truncation error ε an the number of uarature noe, which in turn etermine how many epanion function in a repreentation of f ( ) are integrate eactly (E. (6) an (31)). Large value of ma reuire larger to achieve a certain precribe error, while mall value of ma reult in a large integral truncation error. For a given there i an optimal Likewie, for a given practice one may chooe Ω (i.e. optimal ma ) that lea to the minimal achievable error. Ω there i a minimal an ufficient convergence for any et of tructure parameter. that will lea to a minimal error. In Ω to be lightly greater than neceary to enure The uarature rule in (5) i contructe ifferently in the high-freuency regime ( α 1) an low-freuency regime ( α 1). A. 1 High-freuency regime ( α 1, i.e. L ~ λ or L λ ) For α 1, the uare root in (5) varie more lowly than the ret of the integran. It i therefore aume to be a lowly varying function that can be epane over a mall number P of epanion function in the truncate interval account the ymmetry of the integran, it can be epane a [, ]. Taking into ma α P 1 f () e pπ = a co p. (6) + j p= ma 3

24 Bae on thi epanion a uarature rule with eually pace noe ( + 1 ) = ; = 1,...,, ( 1) D ma w D ( 1) 1 = ma ( + 1 ) + j D ( ) (7) integrate P= epanion (coine) function eactly. Thi imple uarature rule, which lea to very high accuracy an convergence, i ue in E. (17) an (19) to evaluate the integral an lea to () for the D cae for the high-freuency regime. A. Low-freuency regime ( α 1, i.e. L λ ) In the low-freuency regime the uare-root term 1 ( + j) in (5) (an in (19)) for D varie much fater than the eponential term ep( α ) for near the origin. To epan thi uare-root function in the truncate integration range ( [, ] ), a large number of coine function i reuire ma 1 pπ = b co p. (8) + j p= ma Therefore, uarature rule that aume lowly varying integran (a in Appeni A.1) woul lea to a low-freuency breakown where no accurate olution can be obtaine. However, it i note that the function ( )ep( α ) f epane over a mall number P of coine function till varie lowly an can be P 1 α pπ f() e = ap co p=. (9) ma Therefore, the reult of the integration in (5) will not change if the epanion of 1 ( + j) i truncate to the ame number of P term P 1 pπ h() = bp co p=, (3) ma 4

25 where h() i the truncate repreentation of ( + j) 1. The integran with the truncate function h () can be epane over P coine function P 1 α pπ f() h() e = cp co p=. (31) ma For uch an integran the following uarature rule can be ue ( + 1 ) = ; = 1,..., ( 1) D ma w = ( 1) h( ) D D ma ( + 1 ) (3) to integrate eactly P epanion (coine) function. Thi uarature rule i ue in () for the low-freuency cae ( α 1, L λ ). It houl be note that the uarature rule in (3) i efficient not only for α 1 but alo for α 1. However, for α 1 the imple uarature in (7) lea to a maller number of uarature noe an, therefore, i ue to obtain the reult in the paper in thi cae. Appeni B: Quarature rule for the cae For evaluating the PF in E. (17) an (19), one nee to eign a uarature rule for integral of the following form = 1 α f () e f( ) w, (33) where are the uarature noe for the cae an w are the correponing uarature weight. A in Appeni A, the upper integration limit of the integral (33) can be truncate to a value ma. The integran in (33) i aume to be a lowly varying function that can be epane over a mall number P of epanion function in the truncate interval [, ]. Taking into account the weighting function, the emi-infinite integration ma 5

26 range, an the behavior of the integran in (33), thi integran ecluing the weighting function can be epane over an orthogonal et of Beel function f() e P p α χ apj p= 1 ma =, (34) where χ p i the are orthogonal over the range th p zero of the Beel function ma J. The bai function J( χ p ma) [, ] with repect to the weighting function []. Following the conventional approach for eriving a uarature rule for function with a known epanion an bae on (33) with (34), the noe by olving the following ytem of euation an weight w are foun χ χ, (35) ma p p J = J w ma = 1 ma where, the integral in the left han ie i given in cloe form by ma J( χ p ma) = ma J1( χp) χp. Euation (35) repreent a ytem of non-linear euation that can be olve for noe an weight w (total unknown) uing variou moification of the ewton metho. Such olution reult in a auian uarature rule that integrate eactly P= epanion (Beel) function J ( p ) χ of (34). The uarature rule (33) with noe an weight obtaine from ma (35) i ue in () to evaluate the integral for the cae in (19). 6

27 Reference [1] A. F. Peteron, S. L. Ray, an R. Mittra, Computational Metho for Electromagnetic. Picataway,.J.: IEEE Pre, [] A. W. Mathi an A. F. Peteron, "A comparion of acceleration proceure for the two-imenional perioic reen' function," IEEE Tran. Antenna Propagat., vol. 44, pp , [3] S. Singh an R. Singh, "Efficient computation of the free-pace perioic reen' function," IEEE Tran. Microwave Theory Tech., vol. 39, pp , [4] P. Wynn, "On a evice for computing the e_m(s_n) tranformation," Math Table Ai Comput., vol. 1, pp. 9-96, [5] A. L. Fructo, R. R. Boi, F. Mea, an F. Meina, "Application of the Kummer' tranformation to the efficient computation of 3-D reen' function with oneimenional perioicity in homogeneou meia," preente at URSI at. Raio Sci. Meeting, San Diego (CA), 8. [6]. Valerio, P. Baccarelli, P. Burghignoli, an A. alli, "Comparative Analyi of Acceleration Techniue for -D an 3-D reen' Function in Perioic Structure Along One an Two Direction," IEEE Tran. Antenna an Propagat., vol. 55, pp [7] P. P. Ewal, "Die berechnung opticher un elektrotatichen gitterpotentiale," Ann. er Phy., vol. 64, pp , 191. [8] K. E. Joran,. R. Richter, an P. Sheng, "An efficient numerical evaluation of the reen' function for the Helmholtz operator on perioic tructure," J. Comp. Phy., vol. 63, pp. -35, [9] V.. Papanicolaou, "Ewal' metho reviite: rapily convergent erie repreentation of certain reen' function," J. Comp. Anal. Appl., vol. 1, pp , [1] A. Kutepely an A. Q. Martin, "On the plitting parameter in the Ewal metho," IEEE Microwave uie Wave Lett., vol. 1, pp ,. [11] S. Orokar, D. R. Jackon, an D. R. Wilton, "Efficient Computation of the D perioic reen' function uing the Ewal metho," J. Comp. Phy., vol. 19, pp , 6. [1] F. Capolino, D. R. Wilton, an W. A. Johnon, "Efficient computation of the D reen' function for 1D perioic tructure uing the Ewal metho," IEEE Tran. Antenna Propagat., vol. 53, pp , 5. [13] F. Capolino, D. R. Wilton, an W. A. Johnon, "Efficient computation of the reen function for the Helmholtz operator for a linear array of point ource uing the Ewal metho," J. Comp. Phy., vol. 3, pp. 5 61, 7. [14]. Kobize, B. Shanker, an D. P. yuit, "Efficient integral-euation-bae metho for accurate analyi of cattering from perioically arrange nanotructure," Phy. Rev. E, vol. 7, pp. 567, 5. 7

28 [15] I. Stevanovic an J. R. Moig, "Perioic reen' function for kewe 3-D lattice uing the Ewal tranformation," Microwave Optical Tech. Lett., vol. 49, pp , 7. [16] F. T. Celepcikay, D. R. Wilton, D. R. Jackon, an F. Capolino, "Chooing plitting parameter an ummation limit in the numerical evaluation of 1-D an -D perioic reen' function uing the Ewal metho," Raio Sci., vol. in pre, 8. [17] M. E. Veyoglu, H. A. Yueh, R. T. Shin, an J. A. Kong, "Polarimetric paive remote ening of perioic urface," J. Electromag. Wave Appl., vol. 5, pp. 67-8, [18] H. Rogier, "ew erie epanion for the 3-D reen function of multilayere meia with 1-D perioicity bae on perfectly matche layer," IEEE Tran. Microwave Theory Tech., vol. 55, pp , 7. [19] H. Rogier an D. D. Zutter, "A fat converging erie epanion for the -D perioic reen function bae on perfectly matche Layer," IEEE Tran. Microwave Theory Tech., vol. 5, pp , 4. [] L. B. Felen an. Marcuvitz, Raiation an Scattering of Wave. Englewoo Cliff,.J.: Prentice-Hall, [1] W. C. Chew, Wave an fiel in inhomogeneou meia. ew York: IEEE Pre, [] I. S. rahteyn an I. M. Ryzhik, Table of Integral, Serie, an Prouct. San Diego, CA: Acaemic Pre,. 8

29 Figure z, L unit cell Fig. 1: Structure configuration repreenting a perioic array of arbitrarily irecte ipole ource. Im { λ } Im { λ } z Re { λ } z Re { λ } (a) Fig. : The 1D pectral PF g in the λ an λ z, comple plane. (a) gz( z, λ z) ha a (b) branch cut in the comple λ z plane. The integration aroun it from E. (5) give the tranvere repreentation in E. (11); (b) g (, λ ) ha an infinite number of pole in the comple λ plane. The integration contour in (5) encloe all of them, thu giving the Flouet erie in E. (1). Im{ } kz, Re{ } kz, 9

30 Fig. 3: The comple k z, plane with pole an. k,, yy k yy D z = z =.1L /L /L (a) (b) Fig. 4: (a) Magnitue of the calar PF an the yaic component an yy at z 1 3 = L, y = an (b) the magnitue of the calar PF D for two value of z, all plotte along half the unit cell uing L = λ, k = (.85.1 j) k, = 3, = 1, Ω = 3, an Ω =.5. D k =.8k k =1.k k =(.8-.1j)k D k =.8k k =1.k k =(.8-.1j)k /L /L (a) (b) 3

31 Fig. 5: Magnitue of the calar PF (a) an (b) D plotte along half the unit cell for ifferent value of the linear phae hift parameter k. Both have L =.4λ, 3 = z = 1 L, 3 =, = 1, Ω = 3, an Ω =.5. D L=1-6 λ L=.6λ L=1λ L=1-6 λ L=.6λ L=1λ Error 1-1 Error D (# of integration noe) (# of integration noe) (a) (b) Fig. 6: Error of the PF (a) an (b) noe for three value of the pacing D a a function of the number of integration L, with parameter k = (.85.1 j) k, an = 5. The value of Ω vary from.4 to = z = 1 L, Error 1-1 Error D without pole etraction with pole etraction (# of integration noe) 1-15 without truncation with truncation (# of integration noe) (a) (b) Fig. 7: Error of PF (a) k = 1.5k, with an without pole etraction in (19), uing L.4 3 = 1 L, = λ, Ω = with pole etraction, Ω = without pole 31

32 etraction; (b) k = (.85.1 j) k, without truncation. D with an without uare-root truncation in (3), uing z 1 3 = L, L 3 = 1 λ, Ω = with truncation, an Ω =.6. 1 =3, =6 1 =3, =6 1-5 =6, =6 =6, =1 1-5 =6, =6 =6, =1 Error 1-1 Error D /L z/l (a) (b) Fig. 8: Error of PF (a) an (b) D a a function of itance from the array for ifferent choice of convergence parameter, uing L =.6λ, k = (.85.1 j) k, an Ω range from 3.8 to