New Perspective on the Reciprocity Theorem of Classical Electrodynamics

Transcription

1 New Perpective n the Reciprcity Therem f Claical Electrdynamic Maud Manuripur 1,2) and Din Ping Tai 2) 1) Cllege f Optical Science, The Univerity f Arizna, Tucn, Arizna ) Department f Phyic, Natinal Taiwan Univerity, Taipei 106, Taiwan (Publihed in Optic Cmmunicatin 284, , 2011, di: /j.ptcm ) Abtract. We prvide a imple phyical prf f the reciprcity therem f claical electrdynamic in the general cae f material media that cntain linearly plarizable a well a linearly magnetizable ubtance. The excitatin urce i taken t be a pint-diple, either electric r magnetic, and the mnitred field at the bervatin pint can be electric r magnetic, regardle f the nature f the urce diple. The electric and magnetic uceptibility tenr f the material ytem may vary frm pint t pint in pace, but they cannt be functin f time. In the cae f patially nn-diperive media, the nly ther cntraint n the lcal uceptibility tenr i that they be ymmetric at each and every pint. The prf i readily extended t media that exhibit patial diperin: Fr reciprcity t hld, the electric uceptibility tenr χ E_ mn that relate the cmplex-valued magnitude f the electric diple at lcatin r m t the trength f the electric field at r n mut be the tranpe f χ E_ nm. Similarly, the neceary and ufficient cnditin fr the magnetic uceptibility tenr i χ M_ mn = χ T M_ nm. 1. Intrductin. The principle f reciprcity in acutic a well a electrmagnetic (EM) ytem wa firt enunciated by Lrd Rayleigh [1]. Sn afterward, H. A. Lrentz and J. R. Carn extended the cncept and prvided und phyical and mathematical argument that underlie the rigru prf f the reciprcity therem [2,3]. Over the year, the therem ha been embellihed and extended t cver a brader range f pibilitie, and t apply with fewer cntraint [4-11]. The baic cncept and it prf baed n Maxwell macrcpic equatin are dicued in tandard textbk n electrmagnetim [12,13]. Fr a recent review f reciprcity in ptic, the reader i referred t the cmprehenive article by Pttn [14]. Rughly peaking, the idea f reciprcity can be tated a fllw: In a linear, time-invariant electrmagnetic ytem ubject t certain retrictin, if the urce f radiatin i placed in regin A and the reulting EM field i mnitred in regin B, then witching the lcatin f the urce and the berver will reult in the field mnitred at A t be intimately related t that previuly berved at B. Example include: i) The radiatin pattern f an antenna i clely related t it receptin pattern. ii) Thin-film multilayer tack f metal and dielectric may have different reflectivitie when illuminated frm ppite directin (i.e., frnt illuminatin veru rear illuminatin), but they alway exhibit preciely the ame tranmiivity at any given angle f incidence, irrepective f which ide f the tack i illuminated [14,15]. iii) When a diffractin grating i illuminated at an arbitrary angle f incidence, everal diffractin rder uually emerge, each prpagating in a different directin and each having a pecific diffractin efficiency. If nw the directin f incidence i made t cincide with the path taken by ne f the emergent rder, ay, the ne cming ff at an angle θ m and with a diffractin efficiency η m, then ne f the newly emerging diffractin rder will fllw the previu path f incidence (in the revere directin, f cure), and will have the ame efficiency η m [16].

2 Thi paper preent a imple yet general prf f the reciprcity therem, which bring ut the eential phyic f the phenmenn and clarifie the need fr the retrictin under which the therem applie. The nly mathematical fact needed in ur prf i that the prduct f any number f matrice, ay, M 1, M 2,, M n, when tranped, will be equal t the prduct f the tranped matrice in revere rder, that i, ( M MM... M M) = M M... M M M. (1) T T T T T T n 1 n n n Generally peaking, reciprcity in electrdynamic ytem applie when the material media that urrund the urce f radiatin are linearly plarizable and/r magnetizable. The electric and magnetic uceptibility tenr f the material ytem can vary frm pint t pint in pace, but they mut be time-independent. When the media are patially nn-diperive, the nly ther cntraint n the uceptibility tenr i the requirement f ymmetry at each and every pint. In ther wrd, if the plarizatin denity P(r,ω ) and magnetizatin denity M(r,ω ) at a given pint r in pace and at a fixed frequency ω are related t the lcal electric and magnetic field, E(r,ω ) and H(r,ω ), thrugh the fllwing linear relatin: P(r,ω )=ε χ E (r,ω )E(r,ω ), (2a) M(r,ω )=μ χ M (r,ω )H(r,ω ), then we mut have χ (r,ω )=χ T (r,ω ) fr electric plarizatin (ubcript E) a well a magnetizatin (ubcript M). In the abve equatin, ε and μ are the permittivity and permeability f free pace, the ytem f unit emplyed i MKSA, and bth χ E and χ M are dimeninle entitie. We hall impe n ther retrictin n χ E and χ M, allwing their cmpnent t be arbitrary cmplex-valued functin f r and ω. The excitatin urce will be taken t be a tatinary, mnchrmatic pint-diple, either electric r magnetic, lcated at r. The mnitred field at the bervatin pint r will be either electric r magnetic, regardle f the nature f the urce. When the urce and the berved field are f the ame type, i.e., bth electric r bth magnetic, ne mut ue in the revere path the ame type f urce and meaure the ame type f field a in the frward path. In cntrat, when the urce and the berved field are f different type, then, upn revering the path, ne mut witch bth the urce type and the mnitred field. Fr example, if the urce at r i an electric pint-diple, p exp( iω t), while the berved field at r i the H-field, then, in the revere path, the urce placed at r mut be a magnetic pint-diple, m exp( iω t), and the field mnitred at r mut be the E-field. (Nte that the ubcript ued in cnjunctin with the bervatin pint r i italicized. Thi huld nt be cnfued with the ubcript ued with the frequency ω f the cillatin, with the amplitude p and m f the diple, and with the EM field amplitude E and H.) The paper i rganized a fllw. In Sectin 2 we decribe the radiatin field f an electric pint-diple in the urrunding free pace, expreing the radiated EM field in Carteian crdinate fr an arbitrary electric pint-diple p exp( iω t) with cmpnent alng the x-, y-, and z-axe. The crrepnding frmula fr the EM field radiated by a magnetic pint-diple m exp( iω t) are given in Sectin 3. In Sectin 4 we prve the reciprcity therem in the imple cae where the media urrunding the urce are electrically plarizable, having a lcal, ymmetric, time-independent electric uceptibility χ E (r,ω ). The retrictin t electrically plarizable media will be lifted in Sectin 5, where the media urrunding the urce are allwed (2b) 2

3 t have an electric a well a a magnetic uceptibility, χ M (r,ω ). Sectin 6 generalize the reult t patially-diperive media, where bth uceptibilitie will be delcalized. In Sectin 7 and 8 we remark n the difference between ur apprach t reciprcity and the cnventinal methd f prving the therem. Sectin 9 ummarize the reult f the paper and cnclude with an bervatin regarding the Feld-Tai reciprcity lemma [8,9]. 2. Electrmagnetic field radiated by an cillating electric diple. With reference t Fig. 1, cnider the electric pint-diple pr (,) t = p ˆ zδ ( r r)exp( i ωt), z lcated at the fixed pint r and cillating alng the z-axi with cntant amplitude p z and at fixed frequency ω. At anther pint, ay r, the radiated field given by an exact lutin f Maxwell equatin are [17,18]: Er (, t) = E( r r )exp( i ωt) = ( pz /4 πεr) (3a) { 2c θ [(1/ r ) i( ω / cr)] ˆ ρ + in θ [(1/ r ) i( ω / cr) ( ω / c) ] ˆ θ } exp[ i ω ( t r/ c)], Hr (, t) = Hr ( r)exp( i ω t) = ( cp /4 πr) { in θ [( ω / c) + i( ω / cr)] ˆ φ } exp[ i ω ( t r/ c)]. (3b) 2 z In the abve equatin, c =1/ μ ε i the peed f light in vacuum (with μ and ε being the permeability and permittivity f free-pace), r = r r i the eparatin vectr between the urce diple and the bervatin pint, r = r i the length f r, and (θ, φ) are the plar and azimuthal crdinate f the bervatin pint a een frm the lcatin r f the diple. z θ r r (pz,ω ) y x φ Fig. 1. An electric pint-diple lcated at r and riented alng the z-axi cillate with frequency ω and (cmplex) amplitude p z. The electric and magnetic field radiated by the diple are berved at r, whe pitin relative t r i given by r =r r. The pherical crdinate f the eparatin r between the urce and the berver are (r,θ,φ). Defining Δ x = x x, (4a) Δ y = y y, (4b) Δ z = z z, (4c) c θ = ( Δ z)/ r, (4d) in [( x) ( y) ] 2 / r, θ = Δ + Δ (4e) 3

4 2 2 1 c ( x) /[( x) ( y) ] 2, φ = Δ Δ + Δ (4f) in ( y)/[( x) ( y) ] 2, φ = Δ Δ + Δ (4g) we prceed t expre the unit vectr ( ˆ ρ, ˆ θφ, ˆ) f the pherical crdinate ytem in term f ( xˆ, yˆ, z ˆ) f Carteian crdinate, namely, ˆ ρ = [( Δ x) xˆ + ( Δ y) yˆ + ( Δz) zˆ ]/ r, (5a) ˆ θ = cθ cφ xˆ + cθinφ yˆ in θ zˆ, (5b) ˆ φ = inφ xˆ + c φ yˆ. (5c) T implify the ntatin further belw, we define the fllwing functin f r and ω : f ( r, ω ) = (1/4 π) [(1/ r ) i( ω / cr ) ( ω / c r) ] exp(i ω r/ c), (6a) gr (, ω ) = (1/4 π) [(3/ r) i(3 ω / cr) ( ω / cr) ] exp(i ω rc / ), (6b) hr (, ω ) = (1/4 π)( [ ω / cr) + i( ω / cr) ] exp(i ω rc /). (6c) The EM field f Eq.(3), prduced at r by the electric pint-diple lcated at r, may nw be written in Carteian crdinate a Er (, t) = ε p { f( r, ω ) zˆ+ g( r, ω )[( ΔxΔ z) xˆ+ ( ΔyΔ z) yˆ+ ( Δz) z ˆ] } exp( i ω t), (7a) 1 2 z Hr (, t) = cp h( r, ω )[( Δy) xˆ ( Δx) y ˆ]exp( i ω t). (7b) z Frm the abve exprein f the cmplex field amplitude crrepnding t a z-riented diple, ne can readily determine the crrepnding amplitude when the pint-diple cillate either alng the x- r the y-axi. Cnequently, the EM field at r, prduced by the electric pint-diple p= p ˆ ˆ ˆ xx+ py y+ pz z at r, are given by the fllwing matrix equatin: 2 ε E (, ) (, )( ) (, ) (, ) x frω + grω Δx grω ΔxΔy grω ΔxΔz p x ε E = p 2 gr (, ω ) ΔΔ x y fr (, ω ) + gr (, ω )( Δy) gr (, ω ) ΔΔ y z, y y 2 ε E gr (, ω ) ΔΔ x z gr (, ω ) ΔΔ y z fr (, ω ) + gr (, ω )( Δz) p z z (8a) μ H 0 (, ) (, ) x hrω Δz hrω Δy px μ H = Z hr (, ω ) Δz 0 hr (, ω ) Δx p. y y μ H hr (, ω ) Δy hr (, ω ) Δx 0 p z z (8b) In Eq.(8b), Z = μ /ε 377Ω i the impedance f the free pace. Denting the 3 3 cefficient matrice in Eq.(8a) and (8b) by U (r,ω ) and V (r,ω ), repectively, the cmpact verin f Eq.(8) will be ε E ( r ) = U ( r, ω ) p, (9a) μ H ( r ) = Z V ( r, ω ) p. (9b) 4

5 Thee U and V matrice have an imprtant prperty that i f crucial ignificance fr the reciprcity therem. If the lcatin f the urce and the berver are witched, that i, if the diple i mved t r and the berver placed at r, then Δx, Δy, and Δz will change ign. A can be een in Eq.(8a), ince the cefficient matrix U (r,ω ) cntain prduct f pair f Δx, Δy, and Δz, it will nt change at all when the prpagatin directin i revered. Furthermre, ince U (r,ω ) i ymmetric, it tranpe will be the ame matrix. In cntrat, the cefficient matrix V (r,ω ) f Eq.(8b) cntain element that are prprtinal t Δx, Δy, r Δz; a uch, the matrix will change ign upn revering the prpagatin directin. Hwever, V (r,ω ) i antiymmetric, which mean that it ign will change again when tranped. Bth U and V thu have the eential prperty that when the prpagatin directin i revered and the matrix i tranped, each matrix will remain intact. Thi feature f the cefficient matrice in Eq.(9a) and (9b) i the fundamental phyic behind the reciprcal prpertie f a wide cla f linear, time-invariant electrmagnetic ytem. 3. Electrmagnetic field radiated by an cillating magnetic diple. A magnetic pint-diple i the dual f an electric pint-diple, radiating an EM field imilar t that given by Eq.(8), albeit with the rle f E and H witched. T retain ymmetry between the tw type f diple radiatr, we define the magnetic mment m f a mall lp f area A and current I uch that it magnitude will be m =μ IA. [Thi definitin f the magnetic diple i cnitent with the B-field f Maxwell equatin being written a B =μ H +M, a pped t B =μ (H+M), which crrepnd t the definitin f the magnetic diple mment a m =IA.] Fr a magnetic pint-diple m= m ˆ ˆ ˆ xx+ my y + mzz lcated at r, the EM field at r = r +r, btained frm an exact lutin f Maxwell equatin are μ H ( r ) = U ( r, ω ) m, (10a) ε E ( r ) = Z V ( r, ω ) m. (10b) 1 Aide frm the rle-reveral f ε E and μ H, the main difference between the field in the abve equatin and the in Eq.(9) i the change f the cntant cefficient Z f Eq.(9b) t Z 1 in Eq.(10b). 4. Reciprcity in a ytem cntaining electrically-plarizable media. Let the electric pintdiple p(r,t)=p exp( iω t) be lcated at r in the ytem depicted in Fig.2. Thi will be the nly urce diple in the ytem, whe radiatin will excite the electric diple f the urrunding media. The E-field arriving at the bervatin pint r i the uperpitin f the E-field frm all the diple in the ytem. The firt diple, p, being the urce, ha a fixed amplitude, f cure, but the diple lcated at r 1, r 2,, r N will have their amplitude determined by the lcal E-field. The diple at r 1, r 2,, r N repreent the entirety f the linear media that urrund the urce and the bervatin pint. Taking a traight path frm r t r, the E-field that reache directly frm the urce t the berver will be given by ε E =U p, where U i the prpagatin matrix frm r t r. Next, take a different path frm r t r, thi time viiting ne r mre diple f the urrunding media. Figure 2 hw a particular path that ge thrugh r 1, r 2, r 3, and r 4, befre arriving at r. Alng thi path, p excite the diple at r 1, which excite that at r 2, which excite that at r 3, which excite that at r 4, whe field then reache the berver at r. The E-field that reache r frm thi particular path will be given by 5

6 ε E ( r ) = U χ U χ U χ U χ U p. (11) 4 E 4 43 E 3 32 E 2 21 E 1 1 In the abve equatin, χ E_ n i the electric uceptibility tenr aciated with the diple at r n. We hall aume that all uch tenr are ymmetric, but impe n ther retrictin n them. Fr intance, the cmpnent f χ E_ n may be real- r cmplex-valued, crrepnding, repectively, t tranparent r abrbing media. Prceeding in imilar fahin, we can cmpute the E-field at r prduced by each and every diple in the ytem, excited via all pible path thrugh the urrunding media. The ttal field at r will then be btained by adding up all the field prduced via all pible path frm r t r. Nte that a given path, after leaving the urce at r and befre arriving at the bervatin pint, may viit a given diple (aciated with the material media) any number f time, r it may nt viit that particular diple at all. The nly diple that mut appear nce and nly nce in the beginning f each and every path i the urce diple, becaue it i the urce that initiate all ther excitatin, while it wn excitatin i nt influenced in any way by radiatin frm the urrunding diple. z (p,ω ) r 1 r r 7 r + r 2 + r 5 y x r 6 r 3 r4 Fig. 2. An electric pint-diple lcated at r cillate with frequency ω and cmplex amplitude p = p x x^ +p y y^ +p z z^. The ytem cntain N ther fixed diple, lcated at r 1, r 2,, r N, each f which repnd linearly t the lcal EM field. The bervatin pint r i where the electric field E(r, t) i mnitred. If the diple attached t r n i an electric diple, it mut repnd t the lcal E-field with an electric uceptibility ε χ E_ n (ω) that i a ymmetric tenr. Similarly, if the diple attached t r m i magnetic, it mut repnd t the lcal H-field with a magnetic uceptibility tenr μ χ M_ m (ω) that i al ymmetric. There i n pecific relatin between the uceptibility tenr aciated with different diple f the ytem. The implet path between the urce and the berver i the direct path frm r t r. Hwever, any path that tart at r and viit ne r mre f the diple befre arriving at r i al legitimate. There i n limit a t hw many time a particular diple can be viited, nr are there any retrictin n the equence f media diple that are included in a given path. The partial E-field prduced at r by the lat diple in each path mut all be added tgether t prduce the ttal E-field at the bervatin pint. A an example, cnider a imple ytem cniting f a urce diple p lcated at r, tw electric diple with uceptibility tenr χ 1 and χ 2, lcated at r 1 and r 2, and an bervatin pint r, where the E-field i mnitred. The firt few term f the infinite erie that cnverge t the berved E-field are 6

7 ε E (r )=U p +U 1 χ 1 U 1 p +U 2 χ 2 U 2 p +U 2 χ 2 U 21 χ 1 U 1 p +U 1 χ 1 U 12 χ 2 U 2 p +U 1 χ 1 U 12 χ 2 U 21 χ 1 U 1 p +U 2 χ 2 U 21 χ 1 U 12 χ 2 U 2 p +. (12) Cntinuing nw with the example depicted in Fig. 2, if we place the ame, r perhap a different urce diple, ay, p, at the bervatin pint r, and mve the berver t r, the cntributin f each path thrugh the ytem will be btained by a imilar prcedure a befre, except, f cure, fr the reveral f the directin in which each path i travered. The cntributin t the E-field by the revere f the path taken in Eq.(11), fr example, will be ε E' ( r ) = U χ U χ U χ U χ U p'. (13) 1 E 1 12 E 2 23 E 3 34 E 4 4 Nte that the cefficient matrix f Eq.(13) i btained frm that in Eq.(11) by a imple tranpitin. The ame prcedure will apply t all the term in the um ver the variu path. Therefre, the cefficient matrix relating the E-field berved at r when the urce diple i at r, i the tranpe f the cefficient matrix relating the E-field berved at r when the urce diple i at r. Thi cmplete the prf f reciprcity in the cae f electrically plarizable media excited by an electric diple. Given a urce diple p(t) aligned with the x-, y-, and z-axe in three eparate experiment, ne can make nine meaurement f the variu E-field cmpnent t btain the cmplete 3 3 tranfer matrix fr the E-field prduced by an electric diple aciated with given urce and berver lcatin. (In a typical experiment, ne wuld firt rient p alng the x-axi and meaure E x, E y, E z at the bervatin pint, then repeat the experiment with p aligned with the y- axi, and then again with p aligned with the z-axi.) Accrding t the reciprcity therem, the tranpe f the abve 3 3 matrix will be the tranfer matrix when the urce diple (electric) i placed at the riginal bervatin pint, and the riginal urce lcatin i chen a the place t mnitr the E-field. Next, let the urce diple be a magnetic pint-diple, m, while the urrunding material media are the ame a befre (i.e., cntaining electric diple with ymmetric, time-independent uceptibility tenr). Thi time the berved field at r i the magnetic field μ H. The field arriving directly frm the urce i given by μ H =U m, which remain intact when the urce and berver pitin are exchanged. Fr all the ther path that g thrugh the urrunding media, the prcedure i the ame a befre, except that Eq.(11) and (13) will nw becme: μ H ( r ) = V χ U χ U χ U χ V m, (14a) 4 E 4 43 E 3 32 E 2 21 E 1 1 μ H' ( r ) = V χ U χ U χ U χ V m'. (14b) 1 E 1 12 E 2 23 E 3 34 E 4 4 Nte that the cntant cefficient Z and Z 1 which, accrding t Eq.(9b) and (10b), mut multiply V 1 and V 4, end up cancelling each ther ut, leaving nly an incnequential minu ign in frnt f bth Eq.(14a) and (14b). A befre, the cefficient matrix f Eq.(14b) i the tranpe f that in Eq.(14a). The berved H-field being the uperpitin f an infinite number f term imilar t the in Eq.(14), we cnclude that reciprcity cntinue t hld when the urce diple i magnetic and the berved field i the H-field. By a cmpletely analgu argument ne can readily prve the reciprcity therem in the cae where the urce diple i electric while the berved field i magnetic, in which cae the revere path mut have a magnetic diple a the urce and an E-field mnitr fr an berver. 7

8 5. Reciprcity in ytem cntaining bth electric and magnetic media. Let u firt cnider a tatinary ytem f magnetic diple nly, each diple having it wn ymmetric, timeindependent uceptibility tenr χ M_ n. Suppe thee diple cntitute the magnetic media that urrund the urce diple at r and the bervatin pint r. The prf f reciprcity fr thi ytem fllw exactly the ame line f reaning a wa purued in the preceding ectin in cnnectin with media cmpried lely f electric diple. Fr example, if the urce i a magnetic diple m lcated at r, and we che t mnitr the H-field at r, all the matrice ued alng the variu path thrugh the ytem will be U matrice. If, hwever, the urce i an electric pint-diple p, and we mnitr the E-field at r, then the firt and the lat matrice aciated with any path thrugh the urrunding media will be V matrice, while the matrice aciated with interactin between pair f magnetic diple will cntinue t be U matrice. In general, the urrunding media will cntain bth electric and magnetic diple, with ymmetric, time-independent uceptibility tenr χ E_ n and χ M_ m, repectively. The tring f matrice aciated with an arbitrary path frm the urce t the berver will then cntain a U matrix whenever a diple i fllwed by the ame type f diple (i.e., electric diple fllwed by electric diple, r magnetic diple fllwed by magnetic diple), and a V matrix whenever a magnetic diple fllw an electric diple, r vice-vera. When the urce and the berved field are f the ame type, the V matrice alway ccur in pair, irrepective f the path taken frm r t r. T ee thi, cnider a ituatin where the urce i an electric diple and the berver mnitr the E-field. In thi ytem, whenever a tranitin ccur frm an electric diple t a magnetic diple alng a given path, either the ppite tranitin take place befre the path terminate, r the lat diple t tranmit it E-field t the berver will be a magnetic diple. Either way, fr every V matrix that prduce an H-field frm an electric diple, there will be a cmpenating V matrix that generate an E-field frm a magnetic diple. Thu the cefficient Z and Z 1 d nt fail t alternate alng the path and alway end up cancelling ut. It may happen that bth an electric diple and a magnetic diple reide at a given pint r n within the urrunding media. Thee diple, therefre, cannt influence each ther directly, althugh they definitely affect each ther thrugh their interactin with ther diple. Any path that include r n mut therefre che between the electric diple and the magnetic diple reiding at that lcatin. If, befre arriving at the bervatin pint, the ame path happen t viit r n again, the new chice between the tw reident diple will be independent f the previu chice() made at that ame lcatin. In thi way, there will be many path thrugh the ytem which are gemetrically identical (becaue they cntain the ame equence f pint), yet they are phyically ditinct becaue f the chice() made with regard t electing the electric r the magnetic diple at the lcatin where bth diple reide. In general, if the path cntain k lcatin where electric and magnetic diple verlap, there will be 2 k ditinct phyical equence f diple crrepnding t the ame gemetrical path. Thee are minr detail that ne mut take int cnideratin when cntructing a equence f diple fr a given path, but they d nt alter the prf f reciprcity given in the preceding ectin. Perhap the eaiet way t analyze a path thrugh a ytem that cntain bth electric and magnetic diple i t keep track f bth the E-field and the H-field alng the entire path. The prcedure i bet decribed by a imple example. Suppe that, aide frm r and r, there are nly tw pint alng a path, r 1 and r 2, each cntaining an electric a well a a magnetic diple. Frming 6 1 matrice uch a [p n, m n ] and [ε E n, μ H n ] t keep track f the field and the diple, ne can travere the path r r 1 r 2 r uing the fllwing equence f peratin: 8

9 1 1 1 ε E U Z V 2 χ 2 E 0 2 U Z V χ E 1 0 U1 Z V p 1 = μ H 0 0 O ZV U 2 χ 2 M 2 ZV U χ M m 1 ZV 1 U 1 (15) Carrying ut the multiplicatin in the abve equatin, we find (fr the path under cnideratin) all allwed equence f the U and V matrice fr different cmbinatin f urce diple and berved field, namely, ε = ( χ χ χ χ χ χ χ χ ) (16a) E U, 2 E 2U21 E 1U1 U2 E 2V21 M 1V1 V2 M 2V21 E 1U1 V2 M 2U21 M 1V1 p μ = ( χ χ χ χ χ χ χ χ ) (16b) H U, 2 M 2U21 M 1U1 V2 E 2U21 E 1V1 V2 E 2V21 M 1U1 U2 M 2V21 E 1V1 m μ = ( χ χ + χ χ + χ χ χ χ ) (16c) H Z V, 2 E 2U21 E 1U1 U2 M 2V21 E 1U1 U2 M 2U21 M 1V1 V2 E 2V21 M 1V1 p ε = ( χ χ + χ χ + χ χ χ χ ) (16d) 1 E Z U. 2 E 2U21 E 1V1 U2 E 2V21 M 1U1 V2 M 2U21 M 1U1 V2 M 2V21 E 1V1 m In the abve equatin, each term crrepnd t a different equence f electric and magnetic diple alng the path. The methd autmatically find all pible cmbinatin. Al nte in Eq.(16c) and (16d) that reciprcity hld even when the urce i an electric diple while the berved field at r i magnetic, and vice-vera. The trick in uch cae, when exchanging the pitin f the urce and the berver, i t witch the type f the urce diple ( p m ) and al the type f field that i being mnitred ( E H ). S 6. Reciprcity in the preence f patial diperin. In a medium exhibiting patial diperin, the diple repnd nt jut t the lcal field but al t field at ther lcatin within the material media. Fr example, the electric diple reiding at r n will repnd t the lcal E-field with a uceptibility χ E _ nn, but al t the E-field at r m with a uceptibility χ E _ nm. In what fllw, we will hw that reciprcity will apply t uch ytem prvided that χ nm =χ T mn. The prf f reciprcity in the preence f patial diperin fllw the ame line f reaning a wa ued in the preceding ectin. The main difference here i that, a we prceed alng a given path frm the urce at r t the bervatin pint r, the cntributin f each pint uch a r n t it immediate neighbr r n +1 will be via the field at that pint, ay, E(r n ), a well a radiatin by the induced diple at that pint, p(r n ). It i thu neceary at each tep alng the path t keep track f bth E(r n ) and the radiated field prduced by p(r n ) that reache r n +1. A imple example will clarify the prcedure. Suppe the chen path i r r 1 r 2 r 3 r, and that the urce diple a well a the media diple are all electric. The urce diple p excite the firt diple p 1 directly, which repnd with a uceptibility χ 11 t thi excitatin field. The effect f p 1 n p 2, hwever, will be via E(r 1 ) a well a the radiated E-field f p(r 1 ), that i, p(r 2 )=ε χ 21 E(r 1 )+χ 22 U 12 p(r 1 ). Similarly, the effect f p 2 n p 3 i via E(r 2 ) a well a the radiated E-field f p(r 2 ), that i, p(r 3 )=ε χ 32 E(r 2 )+χ 33 U 23 p(r 2 ). The diple at r 3 then cntribute ε E(r )=U 3 p(r 3 ) t the field at the bervatin pint. Uing the 6 1 matrix [ε E(r n ), p(r n )] a a vehicle t keep track f bth the lcal field and the lcal diple, the berved field at the end f the r r 1 r 2 r 3 r path may be written a fllw: ε E( r ) 0 U3 0 U 0 U 0 U 0 = χ32 χ33u32 χ21 χ22u 21 0 χ11u1 p( r ) (17) 9

10 Straightfrward multiplicatin f the abve matrice yield ε E ( r ) = U ( χ U χ + χ U χ + χ U χ U χ ) U p ( r ). (18) The firt term n the right-hand-ide f Eq.(18) repreent the cntributin f the diple at r 3 induced by the E-field at r 2, which i the reult f radiatin frm the diple at r 1, which diple i directly excited by the urce p. The ecnd term repreent the cntributin t the berved E- field by the diple at r 3 induced by the lcal field, which i prduced by radiatin frm the diple at r 2, which i in turn excited by the E-field that reache r 1 directly frm the urce p. The lat term i the cntributin t the berved E-field by the diple at r 3 induced by the lcal field, which ha arrived there frm the diple at r 2, al induced by the lcal field, which i the reult f prpagatin frm the diple at r 1, al induced by the lcal field, which field ha prpagated t r 1 frm the urce p. It i eay t ee frm Eq.(18) that reciprcity will hld if, upn revering the prpagatin path (i.e., ging frm r t r ), χ T 32 wuld becme χ 23 and χ T 21 wuld becme χ 12. Needle t ay, the lcal uceptibility tenr, χ 11, χ 22, and χ 33 mut al be ymmetric, a befre. In general, therefre, the cnditin fr reciprcity in the preence f patial diperin i χ E_ mn =χ T E_ nm. The abve reult can readily be generalized t the cae f material media cntaining bth electric and magnetic diple. With the bviu exceptin f the urce lcatin r and the bervatin pint r, every pint alng any path thrugh the media mut nw be ht t bth an electric diple and a magnetic diple. A dicued at the end f Sectin 5, at each pint r n alng the gemetric path, fr any particular realizatin f a tring f diple, we mut che the reident electric diple r the reident magnetic diple (but nt bth). If the path cntain k uch pint, there will be 2 k phyically ditinct path crrepnding t the ame gemetrical path. Subequently, in paing frm r n t r n +1 alng any uch tring f diple, we mut allw the E(r n ) t excite the next diple nly if the next diple i an electric diple. Similarly, H(r n ) can excite the diple at r n +1 nly if that diple i magnetic. Hwever, the radiatin frm either type f diple at r n cntain bth an E-field and an H-field, ne f which will excite the diple at r n +1, depending n whether the diple reiding at r n +1 i electric r magnetic. A gd way t keep track f all the interactin i via the 12 1 matrice [ε E(r n ), μ H(r n ), p(r n ), m(r n )] and the crrepnding matrice cmped f U mn, V mn, and the relevant uceptibility tenr that prpagate the field and the diple trength frm r n t r m. In thi way, any patially diperive ytem cntaining bth electric and magnetic diple can be prperly mdeled with the aid f ur dicrete diple. Fllwing the ame line f reaning a in the earlier dicuin, ne can nw hw that reciprcity will hld prvided that χ E_ mn =χ T E_ nm and χ M_ mn =χt M_ nm. 7. Cmparin with the tandard prf f reciprcity. The riginal prf f reciprcity fr linear media in the abence f patial diperin i baed n Maxwell macrcpic equatin thrugh the fllwing line f argument. Suppe the nly urce f radiatin in the ytem i an cillating electric pint-diple f magnitude p and frequency ω, lcated at r. The plarizatin denity ditributin, P 1 (r,t), may then be plit int a urce term, p δ(r r )exp( iω t), and a media term, ε χ E (r)e 1 (r)exp( iω t), where the uceptibility tenr χ E (r) i ymmetric at all pint r within the urrunding media. In thi firt arrangement invlving urce at r and bervatin pint at r, the E-field magnitude i dented by E 1 (r). Al aumed i a magnetizatin ditributin thrughut the urrunding media given by μ χ M (r)h 1 (r)exp( iω t). Maxwell curl equatin may nw be written a fllw: 10

11 H () r = i ω p δ( r r ) i ω ε [1 + χ ()] r E(), r (19a) 1 E 1 E () r = i ω μ [1 + χ ()] r H(). r (19b) 1 M 1 In the revere cnfiguratin, let the urce be p δ(r r )exp( iω t), which i a pint-diple f magnitude p lcated at r. The E- and H-field in thi cae will be identified by the ubcript 2, and the curl equatin will be H () r = i ω p' δ( r r ) i ω ε [1 + χ ()] r E (), r (20a) 2 E 2 E () r = i ω μ [1 + χ ()] r H (). r (20b) 2 M 2 We nw multiply E 2 (r) int Eq.(19a) and H 1 (r) int Eq.(20b), then ubtract the latter equatin frm the frmer, t btain E ( r) [ H ( r)] H ( r) [ E ( r)] = [ H ( r) E ( r )] = i ω E ( r) p δ( r r ) i ω ε E ( r)[1 + χ ( r)] E( r) i ω μ H ( r)[1 + χ ( r)] H ( r ). (21a) 2 2 E 1 1 M 2 In like manner, we multiply H 2 (r) int Eq.(19b) and E 1 (r) int Eq.(20a), then ubtract the frmer equatin frm the latter, t btain E ( r) [ H ( r)] H ( r) [ E ( r)] = [ H ( r) E ( r )] = i ω E () r p' δ( r r ) i ω ε E ()[1 r + χ ()] r E () r i ω μ H ()[1 r + χ ()] r H(). r (21b) 1 1 E 2 2 M 1 If Eq.(21b) i nw ubtracted frm Eq.(21a), the ymmetry f χ E (r) and χ M (r) caue the term cntaining thee tenr t cancel ut. The reulting equatin will be [ E () r H () r E () r H ()] r = i ω [ E () r p' δ( r r ) E () r p δ( r r )]. (22) Upn integrating Eq.(22) ver the entire pace, the integrated divergence n the left-hand ide, in accrdance with Gau therem, reduce t the urface integral f E1 H2 E2 H1 at infinity, which can be argued t apprach zer when the integratin urface i ufficiently far frm all urce f radiatin (thi invlve a nn-trivial argument). Setting the integral f the right-hand ide f Eq.(22) equal t zer then yield E 1 ( r) p' = E 2 ( r) p, which i the tatement f reciprcity fr the cae under cnideratin. In a cmpletely analgu way, we let the urce at r remain the electric pint-diple p, but, in the revere path, we ubtitute a magnetic pint-diple m fr the urce at r. Equatin (19) thu remain the ame, but, in Eq.(20), the urce term mve t the ecnd curl equatin and appear a +i ωm' δ ( r r ). The ret f the prf remain unchanged, and the final reult will be H1( r) m' = E2( r) p. Similarly, when bth urce are magnetic diple, the abve derivatin yield H1( r) m' = H2( r) m. Thee claical reult, f cure, are identical with the btained uing the prped cheme f the preent paper. One advantage f the prped methd i that it can be readily generalized t the cae f patially diperive media, a wa hwn in Sectin 6. Anther advantage i that it de nt require the urface integral f E1 H2 E2 H1 t vanih in the limit when the integratin urface mve t infinity. 11

12 8. Cmparin with the prf f reciprcity baed n Green functin. A pwerful methd f prving reciprcity invlve the ue f Green functin. In the fllwing paragraph we derive the dicrete verin f Green functin fr a imple urrunding medium that cntain nly tw pair f pint-diple, ( p 1,m 1 ) c-lcated at r 1, and ( p 2,m 2 ) c-lcated at r 2. Depite it implicity, thi example embdie the eence f the Green functin apprach t lving electrmagnetic prblem. Let the pair f pint-diple (p,m ) be c-lcated at the urce pint r, cillating with the cntant frequency ω. The electrmagnetic radiatin frm thee urce diple will excite the diple at r 1 and r 2, which have lcal uceptibilitie (χ E_11,χ M_11 ) and (χ E_22,χ M_22 ), repectively. Additinally, t intrduce patial diperin int thi dicrete ytem, we define the uceptibilitie (χ E_12,χ M_12 ) fr the diple at r 1 repnding t the E- and H-field at r 2, and al (χ E_21,χ M_21 ) fr the diple at r 2 repnding t the E- and H-field at r 1. T keep track f the field a well a the diple trength at variu lcatin, we ue the 1 12 vectr [ε E(r), μ H(r), p(r), m(r)], writing the field and the induced diple at r 1 a fllw: ε E 0 0 U Z V ε E U Z V μ H ZV U μ H ZV U1 = + p p 1 χ 0 χ U Z χ V p χ U E 12 E E E 11 1 Z χ V E 11 1 m1 0 χ Z χ V χ U m Z χ V M 12 M M M 11 1 χ U M 11 1 (23) Intrducing an bviu ntatin t expre the abve equatin in a mre cmpact frm, Eq.(23) may be rewritten C = W C + X p + Y m (24a) In like manner, the crrepnding equatin fr the field and diple at r 2, excited by the at r 1 and r, will be C = W C + X p + Y m (24b) The lutin t thee equatin i readily btained by lving the matrix equatin fr [C 1,C 2 ], namely, X1 12 Y1 = p X2 21 Y2 C I W I W m. C W I W I The field prduced at the bervatin pint r by the diple at r 1 and r 2 may nw be btained by multiplying int [C 1,C 2 ] the fllwing 6 24 prpagatin matrix: 0 0 U Z V 0 0 U Z V 0 0 ZV 1 U ZV 2 U In thi way we find the field at the bervatin pint due t the excitatin f the media by the urce diple. We mut al remember t add t thee field the direct cntributin frm the urce diple (p,m ) lcated at r.. m (25) (26) 12

13 The cntinuum analg f the inverted quare matrix appearing in Eq.(25) i a Green functin. Given the ymmetrie inherent in thi matrix a well a in the prpagatin matrice between the urce and the media, and al the between the media and the bervatin pint, it i pible t prve reciprcity in a ytem with ymmetric uceptibility tenr, namely, χ E_ mn = χ T E_ nm and χ M_ mn = χ T M_ nm [10,11]. The prf i cniderably mre cmplicated, hwever, than the ne preented here in Sectin 6. The advantage f ur methd f prf i that it nly require partial field (and the crrepndingly induced partial diple) at each pint within the urrunding media. In cntrat, Eq.(25) cntain the cmplete lutin [C 1,C 2 ] f Maxwell equatin fr all the diple f the urrunding media. In ther wrd, there i mre cmplexity in the Green functin apprach t prving reciprcity than i actually needed fr the therem. 9. Summary and cncluding remark. In thi paper we have intrduced an elementary yet pwerful apprach t prving the reciprcity therem f claical electrdynamic, a well a gaining a better undertanding f the phyical bai f the therem. Our reult pertain t the cae when an electric r a magnetic pint-diple, p exp( iω t) r m exp( iω t), lcated at a urce pint r and urrunded by linear, time-invariant media pecified by their electric and magnetic uceptibilitie, ε χ E (r, r ) and μ χ M (r, r ), prduce the EM field E (r )exp( iω t) and H (r )exp( iω t) at an bervatin pint r. If the urce and the berver pitin are exchanged, and if a different diple, either p exp( iω t) r m exp( iω t), i placed at r, the berved field at r will be E (r )exp( iω t) and H (r )exp( iω t). Accrding t the reciprcity therem, when the uceptibility tenr are ymmetric, that i, when χ E (r,r )=χ T E (r,r) and χ M (r,r )=χ T M (r,r), the fllwing relatin hld amng the variu urce and berved field. i) p E (r )=p E (r ) when the urce diple in bth frward and revere path are electric. ii) m H (r )=m H (r ) when the urce diple in bth frward and revere path are magnetic. iii) m H (r )=p E (r ) when the urce diple in the frward path i electric, while that in the revere path i magnetic. iv) p E (r )=m H (r ) when the urce diple in the frward path i magnetic, while that in the revere path i electric. It huld be bviu that the abve reult can be readily generalized t the cae f extended urce, a an extended urce i nthing but a cllectin f diple at different patial lcatin, whe berved field at any given pint r i the linear uperpitin f the field prduced by each and every diple aciated with the urce. In the literature, the Rayleigh-Carn-Lrentz reciprcity therem i ften tated in term f the urce current denity J (r)exp( iω t) and the berved E-field E (r)exp( iω t), a fllw: J () r E' ()d r r = J' () r E ()d. r r (27) V1 V2 In the abve equatin, the urce in the frward path ccupie a vlume V 1, while that in the revere path ccupie a vlume V 2. Equatin (27) i equivalent t ur verin f the therem, p E (r )=p E (r ), fr an extended urce, the rean being that, far a Maxwell equatin are cncerned, J (r)exp( iω t) and Pr (,)/ t t= i ωp()exp(i r ωt) are 13

14 inditinguihable, prvided, f cure, that ω 0. Thi i al the rean why we have left the term J free (r,t) ut f the Maxwell-Ampere equatin Hr (,) t = Jfree(,) rt + Dr (,)/ t t; ee Eq.(19a). Any part f J free (r,t) that i aciated with the urce may be replaced with Pr (,)/ t t, and any part f it that i aciated with the urrunding (linear) media thrugh the cnductivity tenr σ(r,r ) may be replaced with electric diple having uceptibility χ E (r, r )=iσ (r,r )/(ε ω ). The tatement f the therem in term f electric pint-diple i, therefre, cmpletely equivalent t that in term f current denity ditributin, a in Eq.(27). A verin f the reciprcity therem, knwn a the Feld-Tai lemma [8,9], i uually tated a fllw: J () r H' ()d r r = J' () r H ()d. r r (28) V1 V2 The Feld-Tai lemma i nt a general a the Rayleigh-Carn-Lrentz therem tated in Eq.(27). Fr example, in the prf prvided by C. T. Tai [9], me f the urrunding media in the frward path mut be replaced by cmplementary media in the revere path. The electric and magnetic uceptibilitie f the urrunding media mut be calar entitie (i.e., itrpic media), the dielectric and magnetic media mut be piece-wie hmgeneu (e.g., tratified media), and, in ging frm the frward t the revere path, the uceptibilitie f thee itrpic and tratified media mut be mdified in accrdance with a certain algrithm. Mrever, any perfect electrical cnductr in the frward path becme a perfect magnetic cnductr in the revere path. We have nt been able t prve the Feld-Tai lemma f Eq.(28) uing ur prped methd. Hwever, we can prve Eq.(28) under the far le tringent cnditin that, in ging frm the frward t the revere path, all the urrunding media be replaced with their cmplement, in the ene that the electric and magnetic uceptibility tenr χ E (r, r ) and χ M (r, r ) be exchanged. Thi verin f the reciprcity therem can be prven in very much the ame way a the Rayleigh-Carn-Lrentz verin wa prven in the preceding ectin. Once again, the nly cntraint n uceptibility tenr i the requirement f ymmetry, namely, χ E (r,r )=χ T E (r,r) and χ M (r,r )=χ T M (r,r). Finally, it mut be pinted ut that, in recent year, the angular pectrum f the electrmagnetic field emitted by diple ha been ued t acertain fr all pint in pace (including the near field) the reciprcity and unitarity f the cattering (r S) matrix [19-21]. Thi imprtant extenin f the reciprcity therem cnfirm the cnervatin f infrmatin in the near field, where evanecent and inhmgeneu field predminate. The extenin ha al hed light n the cnnectin amng reciprcity, unitarity, and time-reveral invariance in claical ptic. Our prped frmulatin f reciprcity in the preent paper i in cmplete accrd with the afrementined extenin f the claical therem t ituatin invlving the near-field. Thi huld be evident frm Eq.(9) and (10), which are the exact lutin f Maxwell equatin fr radiating pint-diple; lutin that are applicable t all pint in the urrunding pace, frm the immediate vicinity f the diple all the way acr t the far field. Acknwledgement. The authr are grateful t Pui-Tak Leung fr many helpful dicuin. We al thank the annymu referee wh drew ur attentin t reference One f the authr (M.M.) al wuld like t acknwledge the upprt frm the Natinal Science Cuncil f Taiwan while he wa n abbatical leave at the Natinal Taiwan Univerity in Taipei. 14

15 Reference 1. J. W. S. Rayleigh, Treatie n Sund, Vl. II, McMillan, Lndn (1878). 2. H. A. Lrentz, The therem f Pynting cncerning the energy in the electrmagnetic field and tw general prpitin cncerning the prpagatin f light, Verl. Gewne Vergad. Afd. Natuurkd. K. Ned. Akad. Wet. Amterdam 4, 176 (1896); H. A. Lrentz, Cllected Paper, Vl. III, pp1-11, Martinu Nijhff, Hague (1936). 3. J. R. Carn, "A generalizatin f reciprcal therem," Bell Sy. Tech. J. 3, (1924); al, "The reciprcal energy therem," Bell Sy. Tech. J. 9, (1930). 4. A. T. de Hp, Reciprcity f the electrmagnetic field, Appl. Sci. Re. B 8, (1959). 5. A. T. de Hp, Time-dmain reciprcity therem fr electrmagnetic field in diperive media, Radi Sci. 22, (1987). 6. V. L. Ginburg, Sme remark n the electrdynamic reciprcity therem, Radiphyic and Quantum Electrnic 28, , Springer, New Yrk, C. Altman and K. Such, Reciprcity, Spatial Mapping, and Time Reveral in Electrmagnetic, Kluwer, Drdrecht, Ya. N. Feld, On the quadratic lemma in electrdynamic, Sv. Phy. Dkl. 37, (1992). 9. C. T. Tai, Cmplementary reciprcity therem in electrmagnetic thery, IEEE Tran. Antenna Prp. 40, (1992). 10. H. Y. Xie, P. T. Leung, and D. P. Tai, General prf f ptical reciprcity fr nnlcal electrdynamic, J. Phy. A: Math. Ther. 42, (2009). 11. H. Y. Xie, P. T. Leung, and D. P. Tai, Clarificatin and extenin f the ptical reciprcity therem, J. Math. Phy. 50, (2009). 12. L. D. Landau and E. M. Lifhitz, Electrdynamic f Cntinuu Media, Pergamn, Oxfrd, W. K. H. Panfky and M. Phillip, Claical Electricity and Magnetim, Addin-Weley, Reading, MA, R. J. Pttn, Reciprcity in Optic, Rep. Prg. Phy. 67, (2004). 15. M. Manuripur, Claical Optic and It Applicatin, 2 nd editin, Cambridge Univerity Pre, Cambridge, U.K. (2009). 16. R. Petit, ed., Electrmagnetic Thery f Grating, Springer, Berlin, M. Brn and E. Wlf, Principle f Optic, 7 th editin, Cambridge Univ. Pre, Cambridge, U.K. (2003). 18. J. A. Kng, Electrmagnetic Wave Thery, EMW Publihing, Btn, MA, R. Carminati, M. Niet-Veperina, and J.-J. Greffet, Reciprcity f evanecent electrmagnetic wave, JOSA A 15, (1998). 20. R. Carminati, J. J. Sáenz, J.-J. Greffet, and M. Niet-Veperina, Reciprcity, unitarity, and time-reveral ymmetry f the S matrix f field cntaining evanecent cmpnent, Phy. Rev. A 62, (2000). 21. M. Niet-Veperina, Scattering and Diffractin in Phyical Optic, 2 nd editin, Wrld Scientific, Singapre (2006). 15