LETTER A Mathematical Proof of Physical Optics Equivalent Edge Currents Based upon the Path of Most Rapid Phase Variation

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1 659 A Mathematcal Proof of Physcal Optcs Equvalent Ege Currents Base upon the Path of Most Rap Phase Varaton Suomn CUI a), Nonmember, Ken-ch SAKINA, an Makoto ANDO, Members SUMMARY Mathematcal proof for the equvalent ege currents for physcal optcs (EECs) s gven for plane wave ncence an the observer n far zone; the perfect accuracy of EECs for plane wave ncence as well as the egraaton for the pole source closer to the scatterer s clearly explane for the frst tme. EECs for perfectly conuctng plates are extene to those for mpeance plates. key wors: physcal optcs, equvalent ege currents, planar scatterer, ntegral reucton 1. Introucton Surface raaton ntegrals can be asymptotcally reuce to lne ntegraton of fcttous currents, name equvalent ege currents (EECs), along the ege of the scatterer [1. The ervaton of EECs for physcal optcs (EECs) contrbutes not only to effcent computaton of fel but also to constructon of the accurate frnge wave n the physcal theory of ffracton (PTD). The ervaton of EECs for every pont on the ege an for an arbtrary recton of llumnaton an observaton traces back to Mchael s work [1. All the EECs [1 [3, such as those for, the geometrcal theory of ffracton (GTD) an PTD, are erve asymptotcally for hgh frequency. EECs are erve by conuctng the ntegraton along the nner coornate complementary to the ege coornate frst an retanng only the en pont contrbuton; fels are then calculate by ntegratng EECs along the ege coornate. The choce of the recton of the nner coornate for the frst ntegraton, affects the expressons of EECs an the resultant fels serously. Among varous proposals [1 [3, the authors have propose a unque recton of nner ntegraton for, name the path of most rap phase varaton [3, an succeee n elmnatng all the false sngulartes of EECs. However no mathematcal proof has been gven to these EECs except some numercal comparsons [3, as s often the case wth asymptotc methos. In ths letter, we show for the frst tme that Manuscrpt receve November 2, The authors are wth the Department of Electrcal an Electronc Engneerng, Facultyof Engneerng Tokyo Insttute of Technology, Tokyo, Japan. a) E-mal: cusm@antenna.pe.ttech.ac.jp EECs base on the path of rap phase varaton [3 are rgorous for plane wave ncence an for far fel observaton. We frst ecompose Goron s exact expresson for surface ntegraton over a polyheron plate [4nto components from respectve ses. Each component s rewrtten n terms of the ege fxe coornate system at the pont of nterest an s compare wth the EECs [3. It s foun that the EECs resultng from the path of most rap phase varaton are rgorous an more than asymptotc for the plane wave ncence. Therefore, for general llumnaton, accuracy of the EECs for surface-to-ege ntegral reucton epens upon the qualty of plane wave approxmaton of the ncence. For pole wave scatterng from a sk, for example, not only the frequency but also the angle subtene by the sk affect the accuracy. Fnally, EECs for perfectly conuctng ege are extene to those for mpeance ege. 2. Dervaton of EECs for Perfectly Conuctng Ege We assume the plane wave ncence; E ( r)= E 0 exp( jkˆk r), H ( r)= H 0 exp( jkˆk r) (1) where E 0 an H 0 are constant ampltue vectors, k s the free space propagaton constant, an ˆk s n the recton nto whch the ncence propagates. The far fel scattere by the polyheron shown n Fg. 1 can be expresse as E exp( jkr 0 ) = jkη 0 ˆks 4πr ˆk s 0 2ˆn H 0 exp(jk w r )S (2) S where η 0 enotes the ntrnsc mpeance of free space, ˆk S enotes the recton of the observaton pont, ˆn s the unt normal vector on the llumnate face of the plate, an w = ˆk S ˆk. Goron has erve exact close form expresson for polyheron [4

2 660 ˆk = ˆx sn θ cos ϕ ŷ sn θ sn ϕ ẑ cos θ (5) ˆk S =ˆx sn θ S cos ϕ S +ŷ sn θ S sn ϕ s +ẑ cos θ S (6) By usng the followng enttes ˆn H 0 = H 0 ˆtˆx + E η 0 ˆt sn ϕ + H 0 sn θ 0 ˆt cos θ ) cos ϕ ẑ sn θ (7) Fg. 1 The geometry of a polyheron plate an local system for one of the ses. S exp(k w r )S = w ˆn 2 k snc 2 k a m w ( w ˆn) a m ) exp(k b m w) (3) where a m = r m+1 r m, b m = 1 2 ( r m+1 + r m ) wth r m ncatng the tp of the polyheron as s shown n Fg. 1, snc(x) = sn(x)/x, N s the total number of the eges of the plate an r N+1 = r 1. Note Eq. (3) s true for ˆn w 0. When ˆn w =0, the left han se of Eq. (3) s equal to the area of the polyheron plate whle the rght one verges. Ths fact correspons to the nvalaton of EECs scusse n ths paper on the reflecton an shaow bounares. By usng Eqs. (2) an (3), we can express fel as E exp( jkr 0 ) 2j(ˆn H ê θ,ϕ = jkη 0 ) ê θ,ϕ 0 4πr 0 ˆn w 2 k (ˆn w) a m snc 2 k a m w exp(jk b m w) (4) where ê θ,ϕ represents the polarzaton of the recever an ê θ,ϕ ˆk S 0, an lmt N wll be taken for smooth scatterer. Equaton (4) suggests that the fel scattere from the polyheral can be gven as the summaton of the contrbuton from each ege. Ths observaton s entcal wth that n EECs an each contrbuton s reay for comparson wth EECs for that ege. Now, we focus on the scattere fel from the ege #m. The ege-fxe local coornate system s consere at the mpont P of the ege, the rght-han systems wth ẑ n the recton of the ege ˆt s so efne that the y-axs conces wth ˆn on the llumnate face of the plate as shown n the Fg. 1. The ncent an observaton vectors ˆk an ˆk S are efne n ths system as follows: ) (ˆn H 0) ê ϕ = H 0 ˆt sn ϕ S (8) (ˆn H 0) ê θ = H 0 ˆt(ctgθ S cos ϕ S ctgθ cos ϕ ) sn θ S 1 E η 0 ˆt sn ϕ sn θ S (9) 0 sn θ an nsertng Eqs. (8) an (9) nto Eq. (4), we can get the fel from the gven ege #m n the local coornate system as followng E θ Eϕ [ exp( jkr 0 ) 2j 1 = jkη 0 4πR 0 kc 2 l snc 2 kl w ˆt H 0 ˆt(ctgθ S cos ϕ S ctgθ cos ϕ ) sn θ S 1 η 0 E 0 ˆt sn ϕ sn θ S sn θ H 0 ˆt sn ϕ S where l s the length of the ege of nterest, an = sn θ cos ϕ + sn θ S cos ϕ S, (10) c= (sn θ cos ϕ +sn θ S cos ϕ) 2 +(cos θ +cos θ S ) 2 (11) Now, ffracte fel from ths ege s wrtten n terms of equvalent electrc (I e ) an magnetc (I m ) ege currents as, E = jk exp( jkr 0) 4πR 0 l/2 l/2 {η 0 I e ( r )ˆk S (ˆk S ˆt)+I m ( r )ˆk S ˆt} exp(jkˆk S r )t (12) The coeffcents I0 e an I0 m are efne as I e ( r )= I0 e exp( jkˆk r ) an I m ( r )=I0 m exp( jkˆk r ) for plane wave, respectvely. Prove that I0 e an I0 m are constants on the ege, we get

3 661 [ E θ E ϕ = jk exp( jkr 0) 4πR 0 [ 1 l snc 2 kl w ˆt 0 sn θ S I0 m [ η0 sn θ S 0 [ I e 0 (13) Comparng Eqs. (13) wth (10), we get I e 0 = I m 0 = 2j E 0 ˆt kη 0 sn 2 D e + 2j H 0 ˆt D em, θ k sn θ 2j H 0 ˆt ky 0 sn θ sn θ S D m (14) Fg. 2 Geometry for ffracton from a flat sk. D e = sn θ sn ϕ D em = sn θ (ctgθ S cos ϕ S ctgθ cos ϕ ) D m = sn θ sn ϕ S c 2. (15) Usng the relatonshp E ˆt = sn θ Eθ, H ˆt = sn θ Eϕη 1, we can get the EECs n Eq. (12) as follows I e = I = 2 jkη 0 c [ 2 ( cos θ cos ϕ sn θ ) cos θ S cos ϕ S Eϕ sn θ S + sn ϕ Eθ (16) I m = M = 2 sn θ sn ϕ S jk sn θ S c 2 E ϕ (17) whch are perfectly entcal wth the soluton n reference [3. The EECs have only real sngulartes at shaow an reflecton bounares where c =0(θ S = π θ, ϕ S = π±ϕ ). In these cases, the change of phase ue to exp(jk w ˆr ) n the ntegran of Eq. (2) vanshes, an the scatterng phenomena s no longer local but s global; the behavor of hgh frequency ffracton vanshes an asymptotc evaluaton breaks own naturally. So the remanng sngulartes n EECs are not false an all reasonable [3. Snce EECs are expresse completely n the local coornate system, we can apply them to wer class of ffracton problems for general llumnaton. Ths s an asymptotc approxmaton an reasonable accuracy s expecte prove the plane wave approxmaton s val for llumnaton, such as a pole n far zone [3. Numercal results for a crcular sk llumnate by a pole parallel to the sk (Fg. 2) are presente to llustrate ths stuaton. Fgures 3(a), (b) an (c) show the ffracton fels precte by the lne ntegraton of EECs an surface ntegraton as the reference for the fxe value of /a = 5 an fferent frequences. As the frequency becomes lower an becomes much smaller than the wavelength (0.2λ), accuracy s egrae. In ths case, the hgher orer terms of the pole raaton can not be neglecte an the plane wave approxmaton s volate. Ths shows EECs have asymptotc features an are hgh frequency approxmaton. As another check to hghlght the global conton for the plane wave approxmaton, the rato of /a s vare. The stance s kept constant ( = 5λ) whle the sk raus a s vare from 5λ to 0.2λ n Fgs. 4(a), 3(a) an 4(b). By comparng Fgs. 4(a), 3(a) an 4(b), we can see hgher accuracy s attane for smaller value of a. Ths seems contractory n the sense of hgh frequency asymptotc approxmaton. For larger value of a, the angle subtene by the sk becomes large an the angle of ncence upon the ege evates from parallel. So, the conton of plane wave approxmaton s globally volate whle the pole s far way from the ege an the ncence at every ege pont satsfes the local plane wave approxmaton. From above results, even for very small sk, the EECs can prect very accurate results prove the plane wave approxmaton s satsfe. It s worth notng that even for the plane wave ncence, the EECs n [1, [2 prect worse results [5. 3. EECs for Impeance Structures Now we exten the EECs for mpeance plates. After we use the technque propose by Goron, the fel for mpeance plates s expresse as E = jk exp( jkr 0) ˆkS (ˆk S η 0 j 0 + m 0 ) 4πr 0 ˆn w 2 j k

4 662 (a) (a) (b) (b) Fg. 4 Dffracton fels precte by the EECs an from a flat sk ( =5λ, ϕ =45 ). (a) a =5λan (b) =0.25λ. γ sn ϕ (α E 0 ˆt + βη 0 H 0 ˆt) γ sn ϕ (δe 0 ˆt ση 0H 0 ˆt) (19) (c) Fg. 3 Dffracton fels precte by the EECs an from a flat sk (/a =5,ϕ =45 ). (a) =5λ, (b) =1λ an (c) =0.2λ. ( ) 1 (ˆn w) a m snc 2 k a m w exp(jk b m w) (18) where j 0 an m 0 are the equvalent electrcal an magnetc currents evaluate at the orgn of the coornate system [6. In the ege-fxe coornate system, the far fel for a gven ege s expresse as [ E θ E ϕ = jk exp( jkr [ 0) 2j 1 4πR 0 kc 2 l snc 2 kl w ˆt where γ =(1+η sn θ sn ϕ ) 1 (1 + η 1 sn θ sn ϕ ) 1, α = A A S + C B S, β = B A S A B S, δ = C A S A C S, σ = A A S B C S, A = cos θ cos ϕ, B = sn ϕ + η 1 sn θ, C = sn ϕ + η sn θ, A S = cos θ S cos ϕ S, B S = sn ϕ S η 1 sn θ S, C S = sn ϕ S η sn θ S, an η s the normalze surface mpeance of the plate. Comparng wth Eq. (12), we can wrte the currents as followng I e = I m = 2j E ˆt kη 0 sn 2 D e + 2j H ˆt D em, θ k sn θ 2jH ˆt D m + 2j E ˆt D me (20) ky 0 sn θ sn θ S k sn θ S D e = γ sn ϕ sn 2 θ [(A A S + C B S ) sn θ S D em = γ sn ϕ sn θ (B A S A B S ) sn θ S D m = γ sn ϕ sn θ (A A S B C S )

5 663 D me = γ sn ϕ sn θ (C A S A C S ) c 2. (21) whch are the equvalent ege currents for the mpeance plate wthout false sngulartes. If η = 0, EECs for conuctng surface (16) an (17) s obtane. When θ S = θ, the EECs can reuce to the non-unform EECs shown n [7. 4. Concluson Mathematcal proof for EECs base upon the path of most rap phase varaton s gven for plane wave ncence. It s clearly explane that EECs [3as apple for general problem have not only local but also global conton n terms of plane wave approxmaton. Analogous EECs are propose for mpeance plates as well. Mathematcal an comparatve stuy of EECs of other types nclung Mofe ege representaton [8wth hgher accuracy for pole s stll left for future work. Acknowlegement Ths work was supporte n part by the Japan Socety for Promoton of Scence (JSPS) uner the JSPS Fellowshp for Foregn Researchers n Japan. References [1 A. Mchael, Equvalent ege currents for arbtrary aspects of observaton, IEEE Trans. Antennas & Propag., vol.ap- 32, pp , [2 A. Mchael, Elmnaton of nfntes n equvalent ege currents. Part 1 Frnge current components, IEEE Trans. Antennas & Propag., vol.ap-34, pp , [3 M. Ano, T. Murasak, an T. Knoshta, Elmnaton of false sngulartes n GTD equvalent ege currents, IEE Proc. H, vol.138, no.4, pp , [4 W.B. Goron, Far fel approxmatons to Krchhoff- Helmholtz representatons of the scattere fels, IEEE Trans. Antennas & Propag., vol.ap-23, pp , [5 S. Cu, Z. Wu, an M. Wang, Generalze expressons for the frst an hgher orer equvalent ege currents an applcaton n bstatc scatterng, ACTA electronca Snca, no.3, [6 T.B.A. Senor an J.L. Volaks, Approxmate bounary contons n electromagnetcs, The Insttute of Electrcal Engneers, Lonon, Unte Kngom, [7 M. Ooo an M. Ano, Unform physcal optcs ffracton coeffcents for mpeance surfaces an apertures, IEICE Trans. Electron., vol.e80-c, no.7, pp , July [8 T. Murasak an M. Ano, Equvalent ege currents by mofe ege representaton: Physcal optcs components, IEICE Trans. Electron., vol.e75-c, no.5, pp , May 1992.