A Theory of Antenna Electromagnetic Near Field Part I

Transcription

1 1 A Theoy of Antenna Electomagnetic Nea Field Pat I Said M. Mikki and Yahia M. Anta Abstact We pesent in this wok a compehensive theoy of antenna nea fields in two pats, highlighting in paticula the engineeing pespective. Pat I stats by poviding a geneal conceptual famewok fo the moe detailed spectal theoy to be developed in Pat II. The pesent pape poceeds by poposing a geneal spatial desciption fo the electomagnetic field in the antenna exteio egion based on an asymptotic intepetation of the Wilcox expansion. This desciption is then extended by constucting the fields in the entie exteio domain by a diect computation stating fom the fa-field adiation patten. This we achieve by deiving the Wilcox expansion fom the multipole expansion, which allows us to analyze the enegy exchange pocesses between vaious egions in the antenna suounding domain, spelling out the effect and contibution of each mode in an analytical fashion. The esults ae used subsequently to evaluate the eactive enegy of abitay antennas in a complete fom witten in tems of the TE and TM modes. Finally, the concept of eactive enegy is eexamined in depth to illustate the inheent ambiguity of the cicuit total electic and magnetic eactive enegies. We conclude that the eactive field concept is inadequate to the chaacteization of the antenna nea field in geneal. I. INTRODUCTION A. Motivations fo the Seach fo a Theoy of Antenna Nea Fields Antenna pactice has been dominated since its inception in the eseaches of Hetz by pagmatic consideations, such as how to geneate and eceive electomagnetic waves with the best possible efficiency, how to design and build lage and complex systems, including aays, cicuits to feed these aays, and the natual extension towad a moe sophisticated signal pocessing done on site. Howeve, we believe that the othe aspects of the field, such as the puely theoetical, nonpagmatic study of antennas fo the sake of knowledge-foitself, is in a state altogethe diffeent. We believe that to date the available liteatue on antennas still appeas to equie a sustained, compehensive, and igoous teatment fo the topic of nea fields, a teatment that takes into account the peculia natue of the electomagnetic behavio at this zone. Nea fields ae impotant because they ae opeationally complex and stuctually ich. Away fom the antenna, in the fa zone, things become pedictable; the fields take simple fom, and appoach plane waves. Thee is not much to know about the behavio of the antenna aside fom the adiation patten. Howeve, in the nea zone, the field fom cannot be anticipated in advance like the coesponding case in the fa zone. Instead, we have to live with a geneally vey complicated field patten that may vay consideably in qualitative fom fom one point to anothe. In such situation, it is meaningless to seach fo an answe to the question: What is the nea field eveywhee? since one has at least to specify what kinds of stuctues he is looking fo. In light of being totally ignoant about the paticula souce excitation of the antenna, the best one can do is to ely on geneal theoems deived fom Maxwell s equations, most pominently the dyadic Geens function theoems. But even this is not enough. It is equied, in ode to develop a significant, nontivial theoy of nea fields, to look fo futhe stuctues sepaated off fom this Geens functions of the antenna. We popose in this wok (Pat II) the idea of popagating and nonpopagating fields as the emakable featues in the electomagnetic fields of elevance to undestanding how antennas wok. The common liteatue on antenna theoy does not seem to offe a systematic teatment of the nea field in a geneal way, i.e., when the type and excitation of the antenna ae not known a pioi. In this case, one has to esot to the highest possible abstact level of theoy in ode to fomulate popositions geneal enough to include all antennas of inteest. The only level in theoy whee this can be done, of couse, is the mathematical one. Since this epesents the innemost coe of the stuctue of antennas, one can postulate valid conclusions that may descibe the majoity of applications, being cuent o potential. In this context, engineeing pactice is viewed methodologically as being commensuate with physical theoy as such, with the diffeence that the main object of study in the fome, antennas, is an atificially ceated system, not a natual object pe se. Antenna theoy has focused fo a long time on the poblems of analysis and design of adiating elements suitable fo a wide vaiety of scientific and engineeing applications. The demand fo a eliable tool helping to guide the design pocess led to the invention and devolvement of seveal numeical tools, like method of moments, finite element method, finite diffeence time-domain method, etc, which can efficiently solve Maxwell s equations fo almost any geomety, and coesponding to a wide ange of impotant mateials. While this development is impotant fo antenna engineeing pactice, the numeical appoach, obviously, does not shed light on the deep stuctue of the antenna system in geneal. The eason fo this is that numeical tools accept a given geomety and geneate a set of numeical data coesponding to cetain electomagnetic popeties of inteest elated to that paticula poblem at hand. The esults, being fistly numeical, and secondly elated only to a paticula poblem, cannot lead to significant insights on geneal questions, such as the natue of electomagnetic adiation o the inne stuctue of the antenna nea field. Such insight, howeve, can be gained by eveting to some taditional methods in the liteatue, most conveniently expansion theoems fo quantities that poved to

2 2 be of inteest in electomagnetic theoy, and then applying such tools ceatively to the antenna poblem in ode to gain a knowledge as geneal as possible. The engineeing community ae geneally inteested in this kind of eseach fo seveal easons. Fist, the antenna system is an engineeing system pa excellence; it is not a natual object, but an atificial entity ceated by man to satisfy cetain pagmatic needs. As such, the theoetical task of studying the geneal behavio of antennas, especially the stuctual aspects of the system, falls, in ou opinion, into the lot of engineeing science, not physics pope. Second, the woking enginee can make use of seveal geneal esults obtained within the theoetical pogam of the study of antenna systems as poposed in this pape, and pioneeed peviously by many [1], [2], [3], [4], [5], [6]. Such geneal esults can give useful infomation about the fundamental limitation on cetain measues, such as quality facto, bandwidth, coss-polaization, gain, etc. It is exactly the geneality of such theoetical deivations what makes them extemely useful in pactice. Thid, moe knowledge about fields and antennas is always a positive contibution even if it does not lead to pactical esults at the immediate level. Indeed, futue eseaches, with fetile imagination, may manage to convet some of the mathematical esults obtained though a theoetical pogam of eseach into a valuable design and devolvement citeion. B. Oveview of the Pesent Pape At the most geneal level, this pape, Pat I, will study the antenna nea field stuctue in the spatial domain, while the main emphasis of Pat II will be the analysis this time conducted in the spectal domain. The spatial domain analysis will be pefomed via the Wilcox expansion while the spectal appoach will be pusued using the Weyl expansion. The elation between the two appoaches will be addessed in the final stages of Pat II [9]. In Section II, we clealy fomulate the antenna system poblem at the geneal level elated to the nea field theoy to be developed in the following sections. We don t conside at this stage additional specifications like dispesion, losses, anisotopicity, etc, since these ae not essential factos in the nea field desciption to be developed in Pat I using the Wilcox expansions and in Pat II using the Weyl expansion. Ou goal will be to set the antenna poblem in tems of powe and enegy flow in ode to satisfy the demands of the subsequent sections, paticulay ou teatment of eactive enegy in Section VI. In Section III we stat ou conceptualization of the nea field by poviding a physical intepetation of the Wilcox expansion of the adiation field in the antenna exteio egion. Hee, the spatial stuctue is defined as a layeing of this egion into spheical egions undestood in the asymptotic sense such that each egion coesponds to a tem in the Wilcox expansion. In Section IV, we suppot this desciption by showing how to constuct the electomagnetic field in all these egions stating fom the fa-field adiation patten and in a diect, nonecusive fashion. This will povide a complete and exact mathematical desciption fo the nea field of a class of antennas that ae compatible with a given adiation patten and also can be fit inside the innemost egion defined in the spatial configuation intoduced in Section III. We then use these esults to study the phenomenon of electomagnetic inteaction between all the spheical egions compising the antenna field in the exteio egion. Section V povides a complete set of expessions fo the self and mutual inteactions, quantifying then the details of the enegy exchange pocesses occuing between vaious spatial egions in the antenna suounding domain. Of paticula inteest, we pove that the mutual inteaction between half of these egions is exactly zeo. In Section VI, we eexamine the taditional concept of eactive enegy. The main contibution hee esides in utilizing the Wilcox expansion of the exteio electomagnetic fields in ode to compute the eactive enegy in a complete analytical fom. As it tuns out, no infinite numeical integal is needed in pinciple fo computing the antenna eactive enegy and hence the quality facto. We also show that the eason why the eactive enegy is finite has its oots in the geneal theoem poved in Section V, which states that the enegy exchange between some egions in the exteio domain is exactly zeo. The application of this theoem will show that a tem in the enegy density seies cancels out which would othewise give ise to logaithmic divegence in the total eactive enegy. We then povide a demonstation of the inheent ambiguity in the definition of the eactive enegy when the field distibution in the nea zone is examined moe caefully. The existence of such ambiguity endes the concept of eactive enegy, designed oiginally fo the study of the RLC cicuit model of the antenna input impedance, of limited value in descibing the antenna as a field oscillato, athe than a cicuit. Finally, to pepae fo the tansition to Pat II, we compute the total enegy in a spheical shell aound the antenna and expess it as powe seies in 1/. This analysis of the nea-field shell eveals the maximum infomation that can be discened about the nea field stuctue in the spatial domain fom the fa-field pespective. II. GENERAL CONSIDERATION FOR ENERGETICS AND POWER FLOW IN ANTENNA SYSTEMS The pupose of this section is to caefully eview the geneal knowledge we can infe fom Maxwell s equations egading the enegy and powe dynamics suounding abitay antenna systems. The adiation poblem is vey complicated. At this peliminay stage, what is needed to be examined is how much infomation can be deduced fom the mathematical fomalism of electomagnetic theoy about adiation poblems in a way that does not fall unde estictions of paticula antenna geometies and/od excitations. Given the complexity of the poblem thus descibed, we need to citically eflect on what has been aleady achieved so fa in antenna theoy, paticulay as developed by the electical engineeing community. Conside the geneal adiation poblem in Figue 1. We assume that an abitay electic cuent J() exists inside a volume V 0 enclosed by the suface S 0. Let the antenna be suounded by an infinite, isotopic, and homogenous space with electic pemittivity ε and magnetic pemeability µ. The

3 3 Fig. 1. Geneal desciption of antenna system. antenna cuent will adiate electomagnetic fields eveywhee and we ae concened with the egion outside the souce volume V 0. We conside two chaacteistic egions. The fist is the egion V enclosed by the spheical suface S and this will be the setting fo the nea fields. The second egion V is the one enclosed by the spheical suface S taken at infinity and it coesponds to the fa fields. The complex Poynting theoem states that [15] S = 1 2 J E + 2iω (w h w e ), (1) whee the complex Poynting vecto is defined as S = (1/2) E H and the magnetic and electic enegy densities ae given, espectively, by w e = 1 4 εe E, w h = 1 4 µh H. (2) Let us integate (1) thoughout the volume V egion.) We find ds 1 2 (E H ) = dv ( 1 2 J E ) V 0 S +2iω dv (w m w e ). V (nea field The divegence theoem was employed in witing the LHS while the integal of the fist tem in the RHS was esticted to the volume V 0 because the souce cuent is vanishing outside this egion. The imaginay pat of this equation yields Im S ds 1 2 (E H ) = Im V 0 dv ( 1 2 J E ) +2ω V dv (w (4) h w e ). The eal pat leads to Re ds 1 2 (E H ) = Re dv V 0 S (3) ( 1 ) 2 J E. (5) This equation stipulates that the eal time-aveaged powe, which is conventionally defined as the eal pat of the complex Poynting vecto, is given in tems of the wok done by the souce on the field ight at the antenna cuent. Moeove, since this wok is evaluated only ove the volume V 0, while the suface S is chosen at abitay distance, we can see then that the net time-aveaged enegy flux geneated by the antenna is the same thoughout any closed suface as long as it does enclose the souce egion V 0. 1 We need to eliminate the souce-field inteaction (wok) tem appeaing in equation (3) in ode to focus entiely on the fields. To do this, conside the spheical suface S at infinity. Applying the complex Poynting theoem thee and noticing that the fa-field expessions give eal powe flow, we conclude fom (4) that Im dv V 0 ( 1 2 J E ) = 2ω dv (w h w e ). V (6) Substituting (6) into the nea-field enegy balance (4), we find Im ds 1 2 (E H ) = 2ω dv (w h w e ). (7) S V V This equation suggests that the imaginay pat of the complex Poynting vecto, when evaluated in the nea field egion, is dependent on the diffeence between the electic and magnetic enegy in the egion enclosed between the obsevation suface S and the suface at infinity S, i.e., the total enegy diffeence outside the obsevation volume V. In othe wods, we now know that the enegy diffeence W h W e is a convegent quantity because the LHS of (7) is finite. 2 Since this condition is going to play impotant ole late, we stess it again as dv (w h w e ) <. (8) V V Combining equations (5) and (7), we each ds 1 2 (E H ) = P ad 2iω dv (w h w e ), (9) S V V whee the adiated enegy is defined as P ad = Re ds 1 2 (E H ). (10) S We need to be caeful about the intepetation of equation (9). Stictly speaking, what this esult tells us is only the following. Fom an obsevation sphee S at an abitay distance in the nea-field zone. As long as this sphee encloses the souce egion V 0, then the eal pat of the powe flux, the suface integal of the complex Poynting vecto, will give the net eal powe flow though S, while the imaginay pat is the total diffeence between the electic and magnetic enegies in the infinite egion outside the obsevation volume V. We epeat: the condition (8) is satisfied and this enegy diffeence is finite. Relation (9) is the theoetical basis fo the taditional expession of the antenna input impedance in tems of fields suounding the adiating stuctue [15], [6]. III. THE STRUCTURE OF THE ANTENNA NEAR FIELD IN THE SPATIAL DOMAIN We now tun to a close examination of the natue of the antenna nea fields in the spatial domain, while the spectal 1 That is, the suface need not be spheical. Howeve, in ode to facilitate actual calculations in late pats of this pape, we estict ouselves to spheical sufaces. 2 We emind the eade that all souce singulaises ae assumed to be inside the volume V 0.

4 4 appoach is defeed to Pat II of this pape [9]. Hee, we conside the fields geneated by the antenna that lying in the intemediate zone, i.e., the inteesting case between the fa zone k and the static zone k 0. The objective is not to obtain a list of numbes descibing the numeical spatial vaiation of the fields away fom the antenna, a task well-achieved with pesent day compute packages. Instead, we aim to attain a conceptual insight on the natue of the nea field by mapping out its inne stuctue in details. We suggest that the natual way to achieve this is the use of the Wilcox expansion [12]. Indeed, since ou fields in the volume outside the souce egion satisfy the homogenous Helmholtz equation, we can expand the electic and magnetic fields as [12] E () = eik A n (θ, ϕ) n, H () = eik B n (θ, ϕ) n, (11) whee A n and B n ae vecto angula functions dependent on the fa-field adiation patten of the antenna and k = ω εµ is the wavenumbe. The fa fields ae the asymptotic limits of the expansion. That is, e E () ik A e 0 (θ, ϕ), H () ik B 0 (θ, ϕ). (12) The eason why this appoach is the convenient one can be given in the following manne. We ae inteested in undestanding the stuctue of the nea field of the antenna. In the fa zone, this stuctue is extemely simple; it is nothing but the zeoth-ode tem of the Wilcox expansion as singled out in (12). Now, as we leave the fa zone and descend towad the antenna cuent distibution, the fields stat to get moe complicated. Mathematically speaking, this coesponds to the addition of moe tems into the Wilcox seies. The implication is that moe tems (and hence the emeging complexity in the spatial stuctue) ae needed in ode to convege to accuate solution of the field as we get close and close to the cuent distibution. Let us then divide the entie exteio egion suounding the antenna into an infinite numbe of spheical layes as shown in Figue 2. The outemost laye R 0 is identified with the fa zone while the innemost laye R is defined as the minimum sphee totally enclosing the antenna cuent distibution. 3 In between these two egions, an infinite numbe of layes exists, each coesponding to a tem in the Wilcox expansion as we now explain. The boundaies between the vaious egions ae not shaply defined, but taken only as indicatos in the asymptotic sense to be descibed momentaily. 4 The outemost egion R 0 coesponds to the fa zone. The value of, say, the electic field thee is A 0 exp (ik)/. As we stat to descend towad the antenna, we ente into the next egion R 1, whee the mathematical expession of the fa field given in (12) is no longe valid and has to be augmented by the next tem in the Wilcox expansion. Indeed, we find that fo R 1, the electic field takes (appoximately, 3 Stictly speaking, thee is no eason why R should be the minimum sphee. Any sphee with lage size satisfying the mentioned condition will do in theoy. 4 To be pecise, by definition only egion R possesses a clea-cut bounday (the minimum sphee enclosing the souce distibution.) Fig. 2. Geneal desciption of antenna nea-field spatial stuctue. asymptotically) the fom A 0 exp (ik) / + A 1 exp (ik) / 2. Subtacting the two fields fom each othe, we obtain the diffeence A 1 exp (ik) / 2. Theefoe, it appeas to us vey natual to intepet the egion R 1 as the seat of a field in the fom A 1 exp (ik) / 2. Similaly, the nth egion R n is associated (in the asymptotic sense just sketched) with the field fom A n exp (ik) / n+1. We immediately mention that this individual fom of the field does not satisfy Maxwell s equations. The nth field fom given above is a mathematical depiction of the effect of getting close to the antenna on the total (Maxwellian) field stuctue; it epesents the contibution added by the laye unde consideation when passed though by the obseve while descending fom the fa zone to the antenna cuent distibution. By dividing the exteio egion in this way, we become able to mentally visualize pogessively the vaious contibutions to the total nea field expession as they ae mapped out spatially. 5 It is impotant hee to mention that, as will be poved in Pat II [9], localized and nonlocalized enegies exist in each laye in tun; that is, each egion R n contains both popagating and nonpopagating enegies, which amounts to the obsevation that in each egion pat of the field emains thee, while the emaining pat of the field moves to the next lage laye. 6 What concens us hee (Pat I) is not this moe sophisticated spectal analysis of the field associated with each laye, but the simple mapping out of the antenna nea fields into such ough spatial distibution of concentic layes undestood in the asymptotic sense. To be sue, this spatial pictue, illuminating as it is, will emain a mee definition unless it is cooboated by some inteesting consequences. This actually tuns out to be the case. As pointed out in the pevious paagaph, it is possible to show that cetain theoems about the physical behavio of each laye can be poved. Bette still, it is possible to investigate the issue of the mutual electomagnetic inteaction between diffeent egions appeaing in Figue 2. It tuns out that a 5 It is fo this eason that we efain fom igouously defining the nea field as all the tems in the Wilcox expansion with n 1 as is the habit with some wites. The eason is that such field is not Maxwellain. 6 The pocess is still even moe complicated because of the inteaction (enegy exchange) between the popagating and nonpopagating pats. See [9] fo analysis and conclusions.

5 5 geneal theoem (to be poved in Section V) can be established, which shows that exactly half of these layes don t electomagnetically inteact with each othe. In ode to undestand the meaning of this emak, we need fist to define pecisely what is expessed in the tem inteaction. Let us use the Wilcox expansion (11) to evaluate the electic and magnetic enegies appeaing in (2). Since the seies expansion unde consideation is absolutely convegent, and the conjugate of an absolutely convegent seies is still absolutely convegent, the two expansions of E and E can be feely multiplied and the esulting tems can be aanged as we please. The esult is w e = ε 4 E E = ε A n A n 4 n+n +2, (13) w h = µ 4 H H = µ 4 n =0 n =0 B n B n n+n +2. (14) We eaange the tems of these two seies to poduce the following illuminating fom w e () = ε A n A n 4 2n+2 + ε Re {A n A n }, (15) 2 n+n +2 w h () = µ 4 B n B n 2n+2 + µ 2 n,n =0 n>n n,n =0 n>n Re {B n B n } n+n +2. (16) In witing equations (15) and (16), we made use of the ecipocity in which the enegy tansfe fom laye n to laye n is equal to the coesponding one fom laye n to laye n. The fist sums in the RHS of (15) and (16) epesent the self inteaction of the nth laye with itself. Those ae the self inteaction of the fa field, the so-called adiation density, and the self inteactions of all the emanning (inne) egions R n with n 1. The second sum in both equations epesents the inteaction between diffeent layes. Notice that those inteactions can be gouped into two categoies, the inteaction of the fa field (0th laye in the Wilcox expansion) with all othe layes, and the emaining mutual inteactions between diffeent layes befoe the fa-field zone (again R n with n 1.) Now because we ae inteested in the spatial stuctue of nea field, that is, the vaiation of the field as we move close to o fathe fom the antenna physical body whee the cuent distibution esides, it is natual to aveage ove all the angula infomation contained in the enegy expessions (15) and (16). That is, we intoduce the adial enegy density function of the electomagnetic fields by integating (15) and (16) ove the entie solid angle Ω in ode to obtain w e () = ε 4 w h () = µ 4 A n, A n 2n+2 + ε 2 B n, B n 2n+2 + µ 2 n,n =0 n>n n,n =0 n>n A n, A n n+n +2, (17) B n, B n n+n +2, (18) whee the mutual inteaction between two angula vecto fields F and G is defined as 7 F (θ, ϕ), G (θ, ϕ) dω Re {F (θ, ϕ) G (θ, ϕ)}. 4π (19) In deiving (17) and (18), we made use of the fact that the enegy seies is unifomly convegent in θ and ϕ in ode to intechange the ode of integation and summation. 8 Equations (17) and (18) clealy demonstate the consideable advantage gained by expessing the enegy of the antenna fields in tems of Wilcox expansion. The angula functional dependence of the enegy density is completely emoved by integation ove all the solid angles, and we ae left aftewads with a powe expansion in 1/, a esult that povides diect intuitive undestanding of the stuctue of the nea field since in such type of seies moe highe-ode tems ae needed fo accuate evaluation only when we get close to the antenna body, i.e., fo lage 1/. Moeove, the total enegy is then obtained by integating ove the emaining adial vaiable, which is possible in closed fom as we will see late in Section VI-B. A paticulay inteesting obsevation, howeve, is that almost half of the mutual inteaction tems appeaing in in (17) and (18) ae exactly zeo. Indeed, we will pove late that if the intege n + n is odd, then the inteactions ae identically zeo, i.e., A n, A n = B n, B n = 0 fo n + n = 2k + 1 and k is intege. This epesents, in ou opinion, a significant insight on the natue of antenna nea fields in geneal. In ode to pove this theoem and deduce othe esults, we need to expess the angula vecto fields A n (θ, ϕ) and B n (θ, ϕ) in tems of the antenna spheical TE and TM modes. This we accomplish next by deiving the Wilcox expansion fom the multipole expansion. IV. DIRECT CONSTRUCTION OF THE ANTENNA NEAR-FIELD STARTING FROM A GIVEN FAR-FIELD A. Intoduction RADIATION PATTERN We have seen how the Wilcox expansion can be physically intepeted as the mathematical embodiment of a spheical layeing of the antenna exteio egion undestood in a convenient asymptotic sense. The localization of the electomagnetic field within each of the egions appeaing in Figue 2 suggests that the outemost egion R 0, the fa zone, coesponds to the simplest field stuctue possible, while the fields associated with the egions close to the antenna exclusion sphee, R, ae consideably moe complex. Howeve, as was pointed long ago, the entie field in the exteio egion can be completely detemined ecusively fom the adiation patten [12]. In this section we futhe develop this idea by showing that the entie egion field can be detemined fom the fa field diectly, i.e., nonecusively, by a simple constuction based on the analysis of the fa field into its spheical wavefunctions. In othe wods, 7 Fo example, in tems of this notation, the pinciple of ecipocity used in deiving (15) and (16) can now be expessed economically in the fom A n, A n = A n, A n. 8 See Appendix A.

6 6 we show that a modal analysis of the adiation patten, a pocess that is computationally obust and staightfowad, can lead to complete knowledge of the exteio domain nea field, in an analytical fom, as it is inceasing in complexity while pogessing fom the fa zone to the nea zone. This desciption is meaningful because it has been expessed in tems of physical adiation modes. The deivation will help to appeciate the geneal natue of the nea field spatial stuctue that was given in Section III by gaining some insight into the mechanism of electomagnetic coupling between the vaious spatial egions defined in Figue 2, a task we addess in details in Section V. B. Mathematical Desciption of the Fa-Field Radiation Patten and the Concomitant Nea-Field Ou point of depatue is the fa-field expessions (12), whee we obseve that because A 0 (θ, ϕ) and B 0 (θ, ϕ) ae well-behaved angula vecto fields tangential to the sphee, it is possible to expand thei functional vaiations in tems of infinite sum of vecto spheical hamonics [14], [15]. That is, we wite E () ( 1) l+1 [a E (l, m) X lm η eik k l=0 m= l a M (l, m) ˆ X lm ], (20) H () e ik k ( 1) l+1 [a M (l, m) X lm l=0 m= l +a E (l, m) ˆ X lm ], (21) the seies being absolutely-unifomly convegent [13], [17]. Hee, η = µ/ε is the wave impedance. a E (l, m) and a M (l, m) stand fo the coefficients of the expansion TE lm and TM lm modes, espectively. 9 The definition of these modes will be given in a moment. ( / The vecto spheical hamonics l ) ae defined as X lm = 1 (l + 1) LY lm (θ, ϕ), whee L = i is the angula momentum opeato; Y lm is the spheical hamonics of degee l and ode m defined as (2l + 1) (l m)! Y lm (θ, ϕ) = Pl m (cos θ) e imϕ, (22) 4π (l + m)! whee Pl m stands fo the associated Legende function. Since the asymptotic expansion of the spheical vecto wavefunctions is exact, 10 the electomagnetic fields thoughout the entie exteio egion of the antenna poblem can be expanded as a seies of complete set of of vecto multipoles [15] E () = η l=0 m= l [ a E (l, m) h (1) l (k) X lm ] + i k a M (l, m) h (1) l (k) X lm, (23) 9 These coefficients can also be detemined fom the antenna cuent distibution, i.e., the souce point of view. Fo deivations and discussion, see [15]. 10 That is, exact because of the expansion of the spheical Hankel function given in (28.) H () = [ a M (l, m) h (1) l (k) X lm l=0 m= l ] i k a E (l, m) h (1) l (k) X lm, (24) which is absolutely and unifomly convegent. The spheical Hankel function of the fist kind h (1) l (k) is used to model the adial dependence of the outgoing wave in antenna systems. In this fomulation, we define the TE and TM modes as follows H TE lm TE lm mode = a E (l, m) l(l+1) k h (1) l (k) Y lm (θ, ϕ), E TE lm = 0, (25) E TE lm TM lm mode = a M (l, m) l(l+1) k h (1) l (k) Y lm (θ, ϕ), H TE lm = 0. (26) Stictly speaking, the adjective tansvese in the labels TE and TM is meaningless fo the fa field because thee both the electic and magnetic fields have zeo adial components. Howeve, the teminology is still mathematically petinent because the two linealy independent angula vecto fields X lm and ˆ X lm fom complete set of basis functions fo the space of tangential vecto fields on the sphee. Fo this eason, and only fo this, we still may fequently use phases like fa field TE and TM modes. In conclusion we find that the fa-field adiation patten (20) and (21) detemines exactly the electomagnetic fields eveywhee in the antenna exteio egion. This obsevation was cooboated by deiving a ecusive set of elations constucting the entie Wilcox expansion stating only fom the fa field [12]. In the emaining pat of this section, we povide an altenative nonecusive deivation of the same esult in tems of the fa-field spheical TE and TM modes. The upshot of ou agument is the unique deteminability of the antenna nea field in the vaious spheical egions appeaing in Figue 2 by a specified fa field taken as the stating point of the engineeing analysis of geneal adiating stuctues. C. Deivation of the Exteio Domain Nea-Field fom the Fa-Field Radiation Patten The second tems in the RHS of (23) and (24) can be simplified with the help of the following elation 11 h (1) l l(l+1) h (1) (k) X lm = ˆi [ l (k) ] Y lm (θ, ϕ) + 1 h (1) l (k) ˆ X lm (θ, ϕ). (27) We expand the outgoing spheical Hankel function h (1) l (k) in a powe seies of 1/ using the following well-known seies [14],[18] h (1) l (k) = eik b l n n, (28) 11 Equation (27) can be eadily deived fom the definition of the opeato L = i above and the expansion = ˆ (ˆ ) ˆ ˆ, and by making use of the elation L 2 Y lm = l (l + 1) Y lm.

7 7 whee b l n = ( i) l+1 i n n!2 n k n+1 (l + n)! (l n)!. (29) That is, in contast to the situation with cylindical wavefunctions, the spheical Hankel function can be expanded only in finite numbe of powes of 1/, the highest powe coinciding with the ode of the Hankel function l. Substituting (28) into (27), we obtain afte some manipulations h (1) l X lm = i l (l + 1) eik eik nb l n ˆ X n+1 lm + eik b l n n+1 ˆY lm ikb l n n ˆ X lm. (30) By elabeling the indices in the summations appeaing in the RHS of (30) involving powes 1/ n+2, the following is obtained h (1) l (k) X lm = i l (l + 1) eik eik l+1 n=1 (n 1)b l n 1 n ˆ X lm + eik l+1 n=1 b l n 1 ˆY n lm ik bl n n ˆ X lm. (31) Now it will be convenient to wite this expession in the following succinct fom whee and h (1) l X lm = eik c l n = l+1 c l n ˆY lm + d l n ˆ X lm n, (32) { 0, n = 0, i l (l + 1)b l n 1, 1 n l + 1. (33) ikb l 0, n = 0, d l n = ikb l n (n 1) b l n 1, 1 n l, (34) lb l l, n = l + 1. Using (32), the expansions (23) and (24) can be ewitten as E () = η H () = l=0 m= l + i k a M (l, m) eik l=0 m= l i k a E (l, m) eik [ a E (l, m) eik l+1 g l n X lm n l+1 c l n ˆY lm+d l n ˆ X lm ] n [ a M (l, m) eik l+1 g l n X lm n l+1 c l n ˆY lm+d l n ˆ X lm ] n,, (35) (36) Assuming that the electomagnetic field in the antenna exteio egion is well-behaved, it can be shown that the infinite double seies in (35) and (36) involving the l- and n- sums ae absolutely convegent, and subsequently invaiant to any pemutation (eaangement) of tems [16]. Now let us conside the fist seies in the RHS of (36). We can easily see that each powe n will aise fom contibutions coming fom all the multipoles of degee l n. That is, we eaange as a M (l, m) eik b l n X n lm l=0 m= l = eik 1 n l=n m= l a M (l, m) b l nx lm. (37) The situation is diffeent with the second seies in the RHS of (36). In this case, contibutions to the 0th and 1st powes of 1/ oiginate fom the same multipole, that of degee l = 0. Aftewads, all highe powe of 1/, i.e., tems with n 2, will eceive contibutions fom multipoles of the (n 1)th degee, but yet with diffeent weighting coefficients. We unpack this obsevation by witing l=0 m= l = eik ik + n=1 i k a E (l, m) eik [ 1 n l=0 m= l l=n 1 m= l l+1 (c l n ˆY lm+d l n ˆ X lm) n a E (l, m) ( c l 0 ˆY lm + d l 0 ˆ X lm ) a E (l, m) ( c l n ˆY lm + d l n ˆ X lm ) ], (38) That is, fom (37) and (38) equation (36) takes the fom H () = eik B n (θ, ϕ) n, (39) whee B 0 (θ, ϕ) = B n (θ, ϕ) = l=n 1 m= l l=0 m= l l=n m= l ia E (l,m) k ( i) l+1 k [a M (l, m) X lm +a E (l, m) ˆ X lm ], a M (l, m) b l nx lm ( c l n ˆY lm + d l n ˆ X lm ), n 1. (40) (41) By exactly the same pocedue, we deive fom equation (35) the following esult whee A 0 (θ, ϕ) = η A n (θ, ϕ) = η +η l=n 1 m= l E () = eik l=0 m= l l=n m= l ia M (l,m) k ( i) l+1 k A n (θ, ϕ) n, (42) [a E (l, m) X lm a M (l, m) ˆ X lm ], a E (l, m) b l nx lm (43) ( c l n ˆY lm + d l n ˆ X lm ), n 1. (44) Theefoe, the Wilcox seies is deived fom the multipole expansion and the exact vaiation of the angula vecto fields A n and B n ae diectly detemined in tems of the spheical fa-field modes of the antenna. We notice that these two nth vecto fields take the fom of infinite seies of spheical

8 8 hamonics of degees l n, i.e., the fom of the tail of the infinite seies appeaing in the fa field expession (20) and (21). The coefficients, howeve, of the same modes appeaing in the latte seies ae now modified by the simple n- dependence of c l n and d l n as given in (33) and (34). Convesely, the contibution of each l-multipole to the espective tems in the Wilcox expansion is detemined by the weights c l n and d l n, which ae vaying with l. Thee is no dependence on m in this deivation of the Wilcox tems in tems of the electomagnetic field multipoles. D. Geneal Remaks As can be seen fom the diect elations (43), (44), (40), and (41), the antenna nea field in the vaious egions R n defined in Figue 2 is developable in a seies of highe-ode TE and TM modes, those modes being uniquely detemined by the content of the fa-field adiation patten. Some obsevations on this deivation ae wothy mention. We stat by noticing that the expessions of the fa field (43) and (40), the initial stage of the analysis, ae not homogenous with the expessions of the inne egions (44) and (41). This can be attibuted to mixing between two adjacent egions. Indeed, in the scala poblem only modes of ode l n contibute to the content of the egion R n. Howeve, due to the effect of adial diffeentiation in the second tem of the RHS of (27), the afoementioned mixing between two adjacent egions emeges to the scene, manifesting itself in the appeaance of contibutions fom modes with ode n 1 in the egion R n. This, howeve, always comes fom the dual polaization. Fo example, in the magnetic field, the TM lm modes with l n contibute to the field localized in egion R n, while the contibution of the TE lm modes comes fom ode l n 1. The dual statement holds fo the electic field. As will be seen in Section V, this will lead to simila conclusion fo electomagnetic inteactions between the vaious egions. We also bing to the eade s attention the fact that the deivation pesented in this section does not imply that the adiation patten detemines the antenna itself, if by the antenna we undestand the cuent distibution inside the innemost egion R. Thee is an infinite numbe of cuent distibutions that can poduce the same fa-field patten. Ou esults indicate, howeve, that the entie field in the exteio egion, i.e., outside the egion R, is detemined exactly and nonecusively by the fa field. We believe that the advantage of this obsevation is consideable fo the engineeing study of electomagnetic adiation. Antenna designes usually specify the goals of thei devices in tems of adiation patten chaacteistics like sidelobe level, diectivity, coss polaization, null location, etc. It appeas fom ou analysis that an exact analytical elation between the nea field and these design goals do exist in the fom deived above. Since the enginee can still choose any type of antenna that fits within the enclosing egion R, the esults of this pape should be viewed as a kind of canonical machiney fo geneating fundamental elations between the fa-field pefomance and the lowe bound fomed by the field behavio in the entie exteio egion compatible with any antenna cuent distibution that can be enclosed inside R. Fo example, elations (69) and (70) povide the exact analytical fom fo the eactive enegy in the exteio egion. This then foms a lowe bound on the actual eactive enegy fo a specific antenna, because the field inside R will only add to the eactive enegy calculated fo the exteio egion. To summaize this impotant point, ou esults in this pape apply only to a class 12 of antennas compatible with a given adiation patten, not to a paticula antenna cuent distibution. 13 This, we epeat, is a natual theoetical famewok fo the engineeing analysis of antenna fundamental pefomance measues. 14 V. A CLOSER LOOK AT THE SPATIAL DISTRIBUTION OF ELECTROMAGNETIC ENERGY IN THE ANTENNA EXTERIOR A. Intoduction REGION In this section, we utilize the esults obtained in Section IV in ode to evaluate and analyze the enegy content of the antenna nea field in the spatial domain. We continue to wok within the oveall pictue sketched in Section III in which the antenna exteio domain was divided into spheical egions undestood in the asymptotic sense (Figue 2), and the total enegy viewed as the sum of self and mutual inteactions of among these egions. Indeed, we will teat now in details the vaious types of inteactions giving ise to the adial enegy density function in the fom intoduced in (17) and (18). The calculation will make use of the following standad othogonality elations 4π dω X lm X l m = δ ll δ mm, 4π dω X lm (ˆ X l m ) = 0, (45) 4π dω (ˆ X lm) (ˆ X l m ) = δ ll δ mm, ˆ (ˆ X lm ) = ˆ X lm = 0, whee δ lm stands fo the Konecke delta function. B. Self Inteaction of the Outemost Region (Fa Zone, Radiation Density) The fist type of tems is the self inteaction of the fields in egion R 0, i.e., the fa zone. These ae due to the tems involving A 0, A 0 and B 0, B 0 fo the electic and magnetic fields, espectively. Fom (19), (43), (40), and (45), we eadily obtain the familia expessions A 0, A 0 = η2 k 2 l=0 m= l [ a E (l, m) 2 + a M (l, m) 2], (46) 12 Potentially infinite in size. 13 This pogam will be studied thooughly in [11]. 14 The extensively-eseached aea of fundamental limitations of electically small antennas is a special case in this geneal study. We don t pesuppose any estiction on the size of the innemost egion R, which is equied only to enclose the entie antenna in ode fo the vaious seies expansions used in this pape to convege nicely. Stictly speaking, electically small antennas ae moe challenging fo the impedance matching poblem than the field point of view. The field stuctue of an electically small antenna appoaches the field of an infinitesimal dipole and hence does not motivate the moe sophisticated teatment developed in this pape, paticulay the spectal appoach of Pat II.

9 9 B 0, B 0 = 1 k 2 l=0 m= l [ a M (l, m) 2 + a E (l, m) 2]. (47) That is, all TE lm and TM lm modes contibute to the self inteaction of the fa field. As we will see immediately, the pictue is diffeent fo the self inteactions of the inne egions. C. Self Inteactions of the Inne Regions Fom (19), (44), and (45), we obtain A n, A n = η 2 a E (l, m) b l n + η2 k 2 l=n 1 m= l l=n m= l ( a M (l, m) 2 c l 2 n + d l 2) n, n 1, Similaly, fom (19), (41), and (45) we find B n, B n = a M (l, m) b l 2 n + 1 k 2 l=n m= l l=n 1 m= l ( a E (l, m) 2 c l 2 n + d l 2) n, n 1. 2 (48) (49) Theefoe, in contast to the case with the adiation density, the 0th egion, the self inteaction of the nth inne egion (n > 0) consists of two types: the contibution of TE lm modes to the electic enegy density, which involves only modes with l n; and the contibution of the TM lm modes to the same enegy density, which comes this time fom modes with ode l n 1. The dual situation holds fo the magnetic enegy density. This qualitative splitting of the modal contibution to the enegy density into two distinct types is ultimately due to the vectoial stuctue of Maxwell s equations. 15 D. Mutual Inteaction Between the Outemost Region and The Inne Regions We tun now to the mutual inteactions between two diffeent egions, i.e., to an examination of the second sums in the RHS of (17) and (18). We fist evaluate hee the inteaction between the fa field and an inne egion with index n. Fom (19), (43), (44), and (45), we compute A 0, A n = η2 k + η2 k 2 l=n m= l l=n 1 m= l g 1 n (l, m) a E (l, m) 2 g 2 n (l, m) a M (l, m) 2, n 1. Fom (19), (40), (41), and (45), we also each to B 0, B n = 1 k gn 1 (l, m) a M (l, m) k 2 l=n m= l l=n 1 m= l Fom (29), we calculate { } gn 1 (l, m) Re ( i) l+1 b l n = 15 Cf. Section IV-D. g 2 n (l, m) a E (l, m) 2, n 1. (50) (51) { 0, n odd, ( 1) 3n/2 n!2 n k n+1 (l+n)! (l n)!, n even. (52) Similaly, we use (34) to calculate { } gn 2 (l, m) Re ( i) l+1 id l n { kg 1 = n (l, m) (n 1) gn 3 (l, m), 1 n l, lgl+1 3 (l, m), n = l + 1. (53) Hee, we define g 3 n (l, m) { 0, n odd, ( 1) 3n/2 1 (n 1)!2 n 1 k n (l+n 1)! (l n 1)!, n even. (54) Theefoe, it follows that the inteaction between the fa field zone and any inne egion R n, with odd index n is exactly zeo. This supising esult means that half of the mutual inteactions between the egions compising the coe of the antenna nea field on one side, and the fa field on the othe side, is exactly zeo. Moeove, the non-zeo inteactions, i.e., when n is even, ae evaluated exactly in simple analytical fom. We also notice that this nonzeo inteaction with the nth egion R n involves only TM lm and TE lm modes with l n and l n 1. The appeaance of tems with l = n 1 is again due to the polaization stuctue of the adiation field. 16 E. Mutual Inteactions Between Diffeent Inne Regions We continue the examination of the mutual inteactions appeaing in the second tem of the RHS of (17) and (18), but this time we focus on mutual inteactions of only inne egions, i.e., inteaction between egion R n and R n whee both n 1 and n 1. Fom (19), (44), and (45), we aive to A n, A n = η 2 gn,n 4 (l, m) a E (l, m) 2 + η2 k 2 + η2 k 2 l=ϑ n 1 n 1 l=ϑ n n m= l l=ϑ m n m= l m= l g 5 n,n (l, m) a M (l, m) 2 g 6 n,n (l, m) a M (l, m) 2, n, n 1. Similaly, fom (19), (41), and (45), we each to B n, B n = + 1 k k 2 l=ϑ n 1 n 1 l=ϑ n n l=ϑ n 1 n 1 m= l m= l m= l g 4 n,n (l, m) a M (l, m) 2 g 5 n,n (l, m) a E (l, m) 2 g 6 n,n (l, m) a E (l, m) 2, n, n 1. (55) (56) Hee we define ϑ m n max (n, m). Finally, fomulas fo gn,n 4, gn,n 5, and g6 n,n ae deived in Appendix B. Now, it is easy to see that if n + n is even (odd), then n 1 + n 1 is also even (odd). Theefoe, we conclude fom the above and Appendix B that the mutual inteaction between two inne egions R n and R n is exactly zeo if n + n is odd. Fo the case when the inteaction is not zeo, the esult is evaluated in simple analytical fom. This 16 Cf. Section IV-D.

10 10 nonzeo tem involves only TM lm and TE lm modes with l max(n, n ) and l max(n 1, n 1 ). Theefoe, thee exists modes satisfying min(n, n ) l < max(n, n ) and min(n 1, n 1) l < max(n 1, n 1 ) that simply do not contibute to the electomagnetic inteaction between egions R n and R n. The appeaance of tems with l = n 1 is again a consequence of coupling though diffeent modal polaization in the electomagnetic field unde consideation. 17 F. Summay and Conclusion In this Section, we managed to expess all the inteaction integals appeaing in the geneal expession of the antenna adial enegy density (17) and (18) in the exteio egion in closed analytical fom involving only the TM lm and TE lm modes excitation amplitudes a M (l, m) and a E (l, m). The esults tuned out to be intuitive and compehensible if the entie space of the exteio egion is divided into spheical egions undestood in the asymptotic sense as shown in Figue 2. In this case, the adial enegy densities (17) and (18) ae simple powe seies in 1/, whee the amplitude of each tem is nothing but the mutual inteaction between two egions. Fom the basic behavio of such expansions, we now see that the close we appoach the exclusion sphee that diectly encloses the antenna cuent distibution, i.e., what we called egion R, the moe tems we need to include in the enegy density seies. Howeve, the logic of constucting those highe-ode tems clealy shows that only highe-ode fa-field modes ente into the fomation of such inceasing powes of 1/, confiming the intuitive fact that the complexity of the nea field is an expession of iche modal content whee moe (highe-ode) modes ae needed in ode to descibe the inticate details of electomagnetic field spatial vaiation. As a bonus we also find that the complex behavio of the nea field, i.e., that associated with highe-ode fafield modes, is localized in the egions close to the antenna cuent distibution, so in geneal the neae the obsevation to the limit egion R, the moe complex becomes the nea-field spatial vaiation. Finally. it is inteesting to note that almost half of the inteactions giving ise to the amplitudes of the adial enegy density seies (17) and (18) ae exactly zeo i.e., the inteactions between egions R n and R n when n + n is odd. Thee is no immediate apioi eason why this should be the case o even obvious, the logic of the veification pesented hee being afte all essentially computational. We believe that futhe theoetical eseach is needed to shed light on this conclusion fom the conceptual point of view, not meely the computational one. VI. THE CONCEPT OF REACTIVE ENERGY: THE CIRCUIT POINT OF VIEW OF ANTENNA SYSTEMS A. Intoduction In the common liteatue on antennas, the elation (9) has been taken as an indication that the so-called eactive field is esponsible of the imaginay pat of the complex Poynting 17 Cf. Section IV-D. vecto. Since it is this tem that entes into the imaginay pat of the input impedance of the antenna system, and since fom cicuit theoy we usually associate the enegy stoed in the cicuit with the imaginay pat of the impedance, a tend developed in egading the convegent integal (7) as an expession of the enegy stoed in the antenna s suounding fields, and even sometimes call it evanescent field. Hence, thee is a confusion esulting fom the uncitical use of the fomula: eactive enegy = stoed enegy = evanescent enegy. Howeve, thee is nothing in (9) that speaks about such pofound conclusion! The equation, ead at its face value, is an enegy balance deived based on cetain convenient definitions of time-aveaged enegy and powe densities. The fact that the integal of the enegy diffeence appeas as the imaginay pat of the complex Poynting vecto is quite accidental and elates to the contingent utilization of time-hamonic excitation condition. Howeve, the concepts of stoed and evanescent field ae, fist of all, spatial concepts, and, secondly, ae thematically boad; ightly put, these concepts ae fundamental to the field point of view of geneal antenna systems. The conclusion that the stoed enegy is the sole contibuto to the eactive pat of the input impedance of the antenna system is an exaggeation of the cicuit model that was oiginally advanced to study the antenna though its input pot. The field stuctue of the antenna is iche and moe involved than the limited teminal-like point of view implied by cicuit theoy. The concept of eactance is not isomophic to neithe stoed no evanescent enegy. In this section, we will fist caefully constuct the conventional eactive enegy and show that its natual definition emeges only afte the use of the Wilcox expansion in witing the adiated electomagnetic fields. In paticula, we show that the geneal theoem we poved above about the null esult of the inteaction between the fa field and inne layes with odd index is one of the main easons why a finite eactive enegy thoughout the entie exteio egion is possible. Moeove, we show that such eactive enegy is evaluated diectly in closed fom and that no numeical infinite integal is involved in its computation. We then end this section be demonstating the existence of cetain ambiguity in the achieved definition of the eactive enegy when attempts to extend its use beyond the cicuit model of the antenna system ae made. B. Constuction of the Reactive Enegy Densities We will call any enegy density calculated with the point of view of those quantities appeaing in the imaginay pat of (9) eactive densities. 18 When someone ties to calculate the total electomagnetic enegies in the egion V V, the esult is divegent integals. In geneal, we have dv (w h + w e ) =. (57) V V Howeve, condition (8) clealy suggests that thee is a common tem between w e and w h which is the souce of the touble 18 The question of the eactive field is usually ignoed in liteatue unde the claim of having difficulty teating the coss tems [2].

11 11 in calculating the total enegy of the antenna system. We postulate then that w e w 1 e + w ad, w h w 1 h + w ad. (58) Hee we 1 and wh 1 ae taken as eactive enegy densities we hope to pove them to be finite. The common tem w ad is divegent in the sense dvw ad =. (59) V V Theefoe, it is obvious that w h w e = wh 1 w1 e, and theefoe we conclude fom (8) that dv ( wm 1 we) 1 <. (60) V V Next, we obseve that the asymptotic analysis of the complex Poynting theoem allows us to pedict that the enegy diffeence w h w e appoaches zeo in the fa-field zone. This is consistent with (58) only if we assume that w h () w ad (), w e () w ad (). (61) That is, in the asymptotic limit, the postulated quantities wh,e 1 can be neglected in compaison with w ad. In othe wods, the common tem w ad is easily identified as the adiation density at the fa-field zone. 19 It is well-known that the integal of this density is not convegent and hence ou assumption in (59) is confimed. Moeove, this choice fo the common tem in (58) has the meit of making the enegy diffeence, the imaginay pat of (9), devoid of adiation, and hence the common belief in the indistinguishability between the eactive enegy and the stoed enegy. As we will show late, this conclusion cannot be coect, at least not in tems of field concepts. The final step consists in showing that the total enegy is finite. Witing the appopiate sum with the help of (58), we find Wh 1 + W e 1 V V dv ( wh 1 + ) w1 e = lim V ( ) V dv [w h () + w e () 2w ad ]. (62) To pove that this integal is finite, we make use of the Wilcox expansion of the vectoial wavefunction. Fist, we notice that the fa-field adiation pattens ae elated to each othes by B 0 (θ, ϕ) = (1/η)ˆ A 0 (θ, ϕ), (63) This elation is the oigin of the equality of the adiation density of the electic and magnetic types when evaluated in the fa-field zone. That is, we have w ad () = (ε/4)(a 0 A 0)/ 2 = (µ/4)(b 0 B 0)/ 2. (64) Employing the expansion (11) in the enegy densities (2), it is found that w e () = w ad () + ε A 0,A ε 4 n=1 A n,a n 2n+2 + ε 2 n,n =1 n>n A n,a n n+n +2, (65) 19 As will be seen shotly, it is meaningless to speak of a adiation density in the nea-field zone. w h () = w ad () + µ 2 + µ 4 n=1 B 0,B 1 3 B n,b n 2n+2 + µ 2 n,n =1 n>n B n,b n n+n +2. (66) By caefully examining the adial behavio of the total enegies, we notice that the divegence of thei volume integal ove the exteio egion aises fom two types of tems: 1) The fist type is that associated with the adiation density w ad, which takes a functional fom like A 0, A 0 / 2 and B 0, B 0 / 2. The volume integal of such tems will give ise to linealy divegent enegy. 2) The second type is that associated with functional foms like A 0, A 1 / 3 and B 0, B 1 / 3. The volume integal of these tems will esult in enegy contibution that is logaithmically divegent. Howeve, we make use of the fact poved in Section V-D stating that the inteactions A 0, A 1 and B 0, B 1 ae identically zeo. Theefoe, only singulaities of the fist type will contibute to the total enegy. Making use of the equality (64) and the definitions (58), those emaining singulaities can be eliminated and we ae then justified in eaching the following seies expansions fo the eactive adial enegy densities w 1 e () = ε 4 w 1 h () = µ 4 n=1 n=1 A n, A n 2n+2 + ε 2 B n, B n 2n+2 + µ 2 n,n =1 n>n n,n =1 n>n A n, A n n+n +2, (67) B n, B n n+n +2. (68) Fo the pupose of demonstation, let us take a hypothetical spheical suface that encloses the souce egion V 0. Denote by a the adius of smallest such sphee, i.e., R = {(, θ, ϕ) : a}. The evaluation of the total eactive enegy poceeds then in the following way. The expansions (67) and (68) ae unifomly convegent in and theefoe we can intechange the ode of summation and integation in (62). Afte integating the esulting seies tem by tem, we finally aive to the following esults W 1 e = W 1 h = n=1 n=1 (ε/4) A n, A n (2n 1) a 2n 1 + n,n =1 n>n (µ/4) B n, B n (2n 1) a 2n 1 + n,n =1 n>n (ε/2) A n, A n (n + n 1) a n+n 1, (69) (µ/2) B n, B n (n + n 1) a n+n 1. (70) Theefoe, the total eactive enegy is finite. It follows then that the definitions postulated above fo the eactive enegy densities w 1 h and w1 e ae consistent. Moeove, fom the esults of Section V, we now see that total eactive enegies (69) and (70) ae evaluated completely in analytical fom and that in pinciple no computation of infinite numeical integals is needed hee Special cases of (69) and (70) appeaed thoughout liteatue. Fo example, see [2], [3], [5], [6].

12 12 We stess hee that the contibution of the expessions (69) and (70) is not meely having at hand a means to calculate the eactive enegy of the antenna. The main insight hee is the fact that the same fomulas contain infomation about the mutual dependence of 1) the quality facto Q (though the eactive enegy), 2) the size of the antenna (though the dependence on a), and 3) the fa-field adiation patten (though the inteaction tems and the esults of Section V.) The deivation above points to the elational stuctue of the antenna fom the engineeing point of view in the sense that the quantitative and qualitative inteelations of pefomance measues like diectivity and polaization (fa field), matching bandwidth (the quality facto), and the size become all united within one look. 21 The being of the antenna is not undestood by computing few numbes, but athe by the inteconnection of all measues within an integal whole. The elational stuctue of the antenna system will be futhe developed with inceasing sophistication in [9] and [10]. C. The Ambiguity of the Concept of Reactive Field Enegy It is often agued in liteatue that the pocedue outlined hee is a deivation of the enegy stoed in antenna systems. Unfotunately, this matte is questionable. The confusion aises fom the bold intepetation of the tem w ad as a adiation enegy density eveywhee. This cannot be tue fo the following eason. When we wite w ad = (ε/4)e ad E ad = (µ/4)h ad H ad, the esulted quantity is function of the adial distance. Howeve, the expession loses its meaning when the obsevation is at the nea-field zone. Indeed, if one applies the complex Poynting theoem thee, he still gets the same value of the net eal powe flow, but the whole field expession must now be taken into account, not just the fa-field tems. Such field tems, whose amplitudes squaed wee used to calculate w ad, simply don t satisfy Maxwell s equations in the nea-field zone. Fo this eason, it is incoheent to state that since enegy is summable quantity, then we can split the total enegy into adiation density and non-adiation density as we aleady did in (58). These two equations ae definitions fo the quantities wh 1 and w1 e, not deivations of them by a physical agument. 22 To make this agument tanspaent, let us imagine the following scenaio. Scientist X has aleady gone though all the steps of the pevious pocedue and ended up with mathematically sound definitions fo the quantities wh 1 and we, 1 which he duped eactive enegy densities. Now, anothe peson, say Scientist Y, is tying to solve the same poblem. Howeve, fo some eason he does not hit diectly on the tem w ad found by Scientist X, but instead consides the positive tem Υ appeaing in the equation w ad = α + Υ, (71) 21 Extensive numeical analysis of the content of (69) and (70) will be caied out elsewhee [11]. 22 One has always to emembe that the concept of enegy in electomagnetism is not staightfowad. All enegy elations must be viewed as igoous mathematical popositions deived fom the calculus of Maxwell s equations, and aftewads intepeted as enegies and powe in the usual mechanical sense. whee we assume and V V V V dvυ = (72) dvα <. (73) That is, the divegent density w ad is composed of two tems, one convegent and the othe divegent. We futhe equie that w ad () = α () + Υ () Υ (). (74) That is, the asymptotic behavio of w ad is dominated by the tem Υ. The equations of the total enegy density now become and whee w e = w 1 e + w ad = ( w 1 e + α ) + Υ = w 2 e + Υ (75) w h = w 1 h + w ad = ( w 1 h + α ) + Υ = w 2 h + Υ, (76) w 2 e = w 1 e + α, w 2 h = w 1 h + α. (77) Now, it is easily seen that the conditions equied fo the deivation of w 1 h and w1 e ae aleady satisfied fo the new quantities w 2 h and w2 e. That is, we have w h,e () = wh,e 1 () + α () + Υ () Υ () w ad (), (78) which states that the lage agument appoximation of Υ() coincides with the adiation density w ad () at the fa-field zone. Futhemoe, it is obvious that V dv (w V h w e ) = V dv ( V wh 1 ) w1 e = V V dv ( wh 2 ) (79) w2 e, and hence is convegent. Also, V dv ( V wh 2 + ) w2 e = V V dv [(w h w e ) 2w ad ] 2 V V dvα (80) and hence is also convegent. Theefoe, the quantities w 2 h and w 2 e will be identified by Scientist Y as legitimate stoed enegy in his quest fo calculating the eactive enegy density of the antenna. This clealy shows that the eactive enegy calculated this way cannot be a legitimate physical quantity in the sense that it is not unique. In ou opinion, the pocedue of computing the eactive enegy is atificial since it is tailoed to fit an atificial equiement, the engineeing cicuit desciption of the antenna pot impedance. Subtacting the adiation enegy fom the total enegy is not a unique ecipe of emoving infinities. As should be clea by now, nobody seems to have thought that maybe the subtacted tem w ad itself contains a non-divegent tem that is pat of a physically genuine stoed enegy density defined though a non-cicuit appoach, i.e., field fomalism pe se In Pat II [9], we will show explicitly that this is indeed the case.

13 13 D. Citical Reexamination of the Nea-Field Shell We tun now to a qualitative and quantitative analysis of the magnitude of the ambiguity in the identification of the stoed enegy with the eactive enegy. Let a be the minimum size of the hypothetical sphee enclosing the souce egion V 0. Denote by b the adial distance b > a at which the tem w ad dominates asymptotically the eactive enegy densities w 1 h and w 1 e. It is the contibution of w ad to the enegy density lying in the inteval a < < b which is ambiguous in the sense that it can be abitaily decomposed into the sum of two positive functions α() + Υ() in the indicated inteval. Howeve, if the total contibution of the splitable enegy density within this inteval is small compaed with the oveall contibutions of the highe-ode tems, then the ambiguity in the definition of the eactive enegy densities does not lead to seious poblems in pactice. The evaluation of all the integals with espect to gives an expession in the fom 24 We 1 + We 1 = ( ε 4 A 0, A 0 + µ 4 B 0, B 0 ) (b a) + ( n=1 n =1 ε A n,a n +µ B n,b n 4(n+n 1) 1 a n+n 1 1 b n+n 1 (81) The integation with espect to the solid angle yields quantities with the same ode of magnitude. Theefoe, we focus in ou qualitative examination on the adial dependance. It is clea that when a becomes vey small, i.e., a 1, the highe-ode tems dominate the sum and the contibution of the lowest-ode tem can be safely neglected, with all its ambiguities. On the othe hand, when a appoaches the antenna opeating wavelength and beyond, the highe-ode tems apidly decay and the lowest-ode tem dominates the contibution to the total enegy in the inteval a < < b. Since it is in this vey inteval that we find the ambiguity in defining the eactive enegy, we conclude that the eactive enegy as defined in cicuit theoy cannot coespond to a physically meaningful definition of stoed field enegy, and that the esults calculated in liteatue as fundamental limit to antenna Q ae incoheent when the electical size of the exclusion volume appoaches unity and beyond. One moe point that need to be examined in the above agument elates to the choice of b. Of couse, b cannot be fixed abitaily because it is elated to the behavio of the highe-ode tems, i.e., b is the adius of the adiation sphee, the sphee though which most of the field is conveted into adiation field. 25 Theefoe, in ou agument above a eaches the citical value of unit wavelength but cannot incease significantly because it is bounded fom above by b, which is not feely vaying like a. The upshot of the agument is that the vagueness in the pecise value of b is nothing but the vagueness in any asymptotic expansion in geneal whee accuacy is closely tied to the physical conditions of the paticula situation unde consideation. In this situation, 24 In witing (81), we explicitly dopped the zeo tems involving A 0, A 1 and B 0, B 1 in ode simplify the notation. 25 Radiation field does not mean hee popagating wave, but fields that contibute to the eal pat of the complex Poynting vecto. Stictly speaking, the popagating field is close to the adiation field but not exactly the same because the nonpopagating field contibutes to the fa field. See also Pat II [9]. ). the one coesponding to computing the eactive enegy as defined above, the value of the eactive field enegy W 1 h +W 1 e becomes vey small with inceasing a fo the obvious eason that eactive enegy is mostly localized in the nea field close to the antenna. Howeve, it is not clea at what pecise value b one should switch fom nea field into adiation field. Indeed, it is exactly in this way that the entie agument of this pat of the pape was motivated: The cicuit appoach to antennas cannot give coheent pictue of genuine field poblems. All what the common appoach equies is that at a distance lage enough the enegy density conveges (asymptotically) to the adiation density. Howeve, while the total enegy density is appoaching this pomised limit, the eactive enegy is apidly decaying in magnitude, and in such case any ambiguity o eo in the definition of the sepaation of the two densities (which, again, we believe to be non-physical) may poduce vey lage eo, o at least ende the esults of the Q facto not so meaningful. 26 VII. CONCLUSION In this pape, we stated the fomulation of a compehensive theoetical pogam fo the analysis of the antenna electomagnetic field in geneal, and without estiction to a paticula o specific configuation in the souce egions. The study in Pat I, the pesent pape, dealt with the analysis conducted in the spatial domain, that is, by mapping out the vaious spatial egions in the antenna exteio domain and explicating thei electomagnetic behavio. We studied the phenomena of enegy tansfe between these egions and deived exact expessions fo all types of such enegy exchange in closed analytical fom in tems of the antenna TE and TM modes. The fomulation shows that this detailed desciption can be obtained nonecusively meely fom knowledge of the antenna fa-field adiation patten. The esulted constuction shows explicitly the contibution of each mode in the vaious spatial egions of the exteio domain, and also the coupling between diffeent polaization. Of special inteest is the discovey that the mutual inteaction between egions with odd sum of indices is exactly zeo, egadless to the antenna unde study. Such geneal esult appeas to be the eason why the infinite integal of the adial enegy density giving ise to the antenna eactive enegy is finite. The final pats of the pape eexamined the concept of eactive enegy when extended to study the field stuctue of the antenna. We showed how ambiguities in the definition of this cicuit quantity ende it of limited use in antenna nea field theoy pope (matching consideations put aside.) This pepaes fo the tansition to 26 One can even each this conclusion without any evaluation of total enegy. The enegy density itself is assumed to be a physically meaningful quantity. At aound a = 1, all the adial factos in the tems appeaing in (65) and (66) become oughly compaable in magnitude (assuming nomalization to wavelength, i.e., a = 1 is taken hee to be the intemediate-field zone bounday.) Howeve, the lowest-ode tem has an ambiguity in its definition that can be vaied feely up to its full positive level. Thus, thee seems to be a seious poblem beginning in the intemediate-field zone. Even fo lage a, since the oveall eactive enegy density becomes vey small, slight changes in the value of the contibution of the adiation density esulting fom the afoementioned ambiguity ende, in ou opinion, the Q factos cuves epoted in liteatue of limited physical elevance as indicatos of the size of the actually stoed field.

14 14 Pat II of this pape, which is concened with the analysis of the antenna nea field in the spectal domain. APPENDIX A PROOF OF THE UNIFORM CONVERGENCE OF THE ENERGY SERIES USING WILCOX EXPANSION Fom [12], we know that the single seies conveges both absolutely and unifomly in all its vaiables. We pove that the enegy (double) seies is unifomly convegent in the following way. Fist, convet the double sum into a single sum by intoducing a map (n, n ) l. Fom a basic theoem in eal analysis, the multiplication of two absolutely convegent seies can be eaanged without changing its value. This guaantee that ou new single seies will give the same value egadless to the map l = l(n, n ). Finally, we apply the Cauchy citeion of unifom convegence [16] to deduce that the enegy seies, i.e., the oiginal double sum, is unifomly convegent in all its vaiables. APPENDIX B COMPUTATION OF THE FUNCTIONS gn,n 4 (l, m), gn,n 5 (l, m), AND g6 n,n (l, m) Fom (29), we calculate gn,n 4 (l, m) Re { b l nb l { 0, n + n odd, = ( 1) (n+3n )/2 A 1 (n, n ; k), n + n even, (82) whee A 1 (n, n ; k) = n } Fom (33), we also compute (l + n)! (l + n )! (n!2 n k n+1 ) (n!2 n k n +1 ) (l n)! (l n )!. (83) g 5 n,n (l, m) Re { c l nc l n } = l (l + 1) g 4 n 1,n 1 (l, m), 1 n, n l + 1. (84) Fom (34) we find gn,n 6 (l, m) Re { d l nd l (n 1) (n 1) Re { } b l n 1b l n 1 +k 2 Re { } b l nb l n = k (n 1) Re { } b l nib l n +k (n 1) Re { b l n n 1} ibl, 1 n l, l 2 Re { } b l l bl l, n = l + 1. Fom (29), we compute Re { b l nib l } n 1 = Similaly, we have Re { b l nib l } n 1 = n } (85) 0, n + n odd, ( 1) (n+3n )/2 1 A 2 (n, n ; k), n + n even. (86) 0, n + n odd, ( 1) (n +3n)/2 1 A 2 (n, n; k), n + n even. (87) Hee we define A 2 (n, n ; k) (l+n)! (n!2 n k n+1 )(l n)! (l+n 1)!. (88) (n 1)!2 n 1 k n (l n +1)! We have used in obtaining (16) and (17), and also all simila calculations in Section V, the manipulation (i n ) = (i ) n = ( i) n = i n ( 1) n. REFERENCES [1] L. J. Chu, Physical limitations of omni-diectional antennas, J. Appl. Phys., vol. 19, pp , Decembe [2] R. E. Collin and S. Rothschild, Evaluation of antenna Q, IEEE Tans. Antennas Popagat., vol. AP-12, pp , Januay [3] Ronald L. Fante, Quality facto of geneal ideal antennas, IEEE Tans. Antennas Popagat., vol. AP-17, no. 2, pp , Mach [4] David M. Kens, Plane-wave scatteing-matix theoy of antennas and antenna-antenna inteactions: fomulation and applications, Jounal of Reseach of the National Bueau of Standads B. Mathematica Scineces, vol. 80B, no. 1, pp. 5-51, Januay-Mach, [5] D. R. Rhodes, A eactance thoeem, Poc. R. Soc. Lond. A., vol. 353, pp.1-10, Feb [6] Athu D. Yaghjian and Steve. R. Best, Impedance, bandwidth, and Q of antennas, IEEE Tans. Antennas Popagat., vol. 53, no. 4, pp , Apil [7] Said Mikki and Yahia M. Anta, Genealized analysis of the elationship between polaization, matching Q facto, and size of abitay antennas, Poceedings of IEEE APS-URSI Intenational Symposium, Toonto, July 11 17, [8] Said Mikki and Yahia M. Anta, Citique of antenna fundamental limitations, Poceedings of URSI-EMTS Intenational Confeence, Belin, August 16-19, [9] Said M. Mikki and Yahia Anta, Foundation of antenna electomagnetic field theoy Pat II, (submitted). [10] Said M. Mikki and Yahia M. Anta, Mophogenesis of electomagnetic adiation in the nea-field zone, to be submitted. [11] Said M. Mikki and Yahia M. Anta, Genealzied analysis of antenna fundamental measues: A fa-field pespective, to be sumitted to IEEE Tans. Antennas Popagat. [12] C. H. Wilcox, An expansion theoem fo the electomagnetic fields, Communications on Pue and Appl. Math., vol. 9, pp , [13] O. D. Kellogg, Foundations of Potential Theoy, Spinge, [14] Philip Mose and Heman Fesbach, Methods of Theotical Physics II, McGaw-Hill, [15] David John Jackson, Classical Electodynamics, John Wiley & Sons, [16] David Bessoud, A Radical Appoach to Real Analysis, The Mathematical Ameican Society of Ameica (AMS), [17] Hubet Kalf, On the expansion of a function in tems of spheical hamonics in abitay dimensions, Bull. Belg. Math. Soc. Simon Stevin, vol. 2, no. 4, pp , [18] M. Abamowitz and I. A. Stegunn, Handbook of Mathematical Functions, Dove Publications, 1965.

15 1 A Theoy of Antenna Electomagnetic Nea Field Pat II Said M. Mikki and Yahia M. Anta Abstact We continue in this pape a compehensive theoy of antenna nea fields stated in Pat I. The concept of nea-field steamlines is intoduced using the Weyl expansion in which the total field is decomposed into popagating and nonpopagating pats. This pocess involves a beaking of the otational symmety of the scala Geens function that oiginally facilitated the deivation of the Weyl expansion. Such symmety beaking is taken hee to epesent a key to undestanding the stuctue of the nea fields and how antennas wok in geneal. A suitable mathematical machiney fo dealing with the symmety beaking pocedue fom the souce point of view is developed in details and the final esults ae expessed in clea and compact fom susceptible to diect intepetation. We then investigate the concept of enegy in the nea field whee the localized enegy (especially the adial localized enegy) and the stoed enegy ae singled out as the most impotant types of enegy pocesses in the nea-field zone. A new devolvement is subsequently undetaken by genealizing the Weyl expansion in ode to analyze the stuctue of the nea field but this time fom the fa-field point of view. A hybid seies combining the Weyl and Wilcox expansions is deived afte which only the adial steamline pictue tuns out to be compatible with the fa-field desciption via Wilcox seies. We end up with an explication of the geneal mechanism of fa field fomation fom the souce point of view. It is found that the main pocesses in the antenna nea field zone ae educible to simple geometical and filteing opeations. I. INTRODUCTION The esults of the fist pat of this pape [1] have povided us with an insight into the stuctue of what we called the nea-field shell in the spatial domain. This concept has been impotant paticulay in connection with the computation of the eactive enegy of the antenna system, the quantity needed in the estimation of the quality facto and hence the input impedance bandwidth. We have shown, howeve, that since the concept of eactive enegy is mainly a cicuit concept, it is incapable of descibing adequately the moe toublesome concept of stoed field enegy. In this pape, we popose a new look into the stuctue of the nea fields by examining the evanescent pat of the electomagnetic adiation in the vicinity of the antenna. The mathematical teatment will be fundamentally based on the Weyl expansion [6], and hence this will be essentially a spectal method. Such appoach, in ou opinion, is convenient fom both the mathematical and physical point of view. Fo the fome, the availability of the geneal fom of the adiated field via the dyadic Geens function theoem allows the applicability of the Weyl expansion to Fouie-analyze any field fom into its spectal components. Fom the physical point of view, we notice that in pactice the the focus is mainly on moving enegy aound fom once location to anothe. Theefoe, it appeas to us natual to look fo a geneal mathematical desciption of the antenna nea fields in tems of, speaking infomally, pats that do not move (nonpopagating field), and pats that do move (popagating field.) As we will see shotly, the Weyl expansion is well suited to exactly this; it combines both the mathematical and physical pespectives in one step. Such a field decomposition into two pats can theefoe be seen as a logical step towad a fundamental insight into the natue of the electomagnetic nea field. Because of the complexity involved in the agument pesented in this pape, we eview hee the basic ideas and motivations behind each section. In Section II, we povide a moe sophisticated analysis of the nea field that goes beyond the customay (cicuit) view of eactive fields and enegies. To stat with, we ecuit the Weyl expansion in expanding the scala Geens function into popagating and nonpopagating (evanescent) pats. By substituting this expansion into the dyadic Geens function theoem, an expansion of the total fields into popagating and nonpopagating pats becomes feasible. We then beak the otational symmety by intoducing two coodinate system, once is fixed (the global fame), while the othe can otate feely with espect to the fixed fame (the local fame.) We then systematically develop the mathematical machiney that allows us to descibe the decomposition of the electic field into the two modes above along the local fame. It tuns out that an additional otation of the local fame aound its z-axis does not change the decomposition into total popagating and nonpopagating pats along this axis. This cucial obsevation, which can be poved in a staightfowad manne, is utilized to intoduce the concept of adial steamlines. This concept is a desciption of how the electomagnetic fields split into popagating and nonpopagating modes along adial steamlines, like the situation in hydodynamics, but defined hee only in tems of fields. The concept of adial steamlines will appea with the pogess of ou study to be the most impotant stuctue of the antenna nea field fom the engineeing point of view. We also show that the popagating and nonpopagating pats both satisfy Maxwell s equations individually. This impotant obsevation will be needed late in building the enegy intepetation. The Section ends with a geneal flow chat illustating how the spectal composition of the electic field is constucted. This is indeed the essence of the fomation of the antenna nea field, which we associate hee with the nonpopagating pat. In Section III, we futhe study the nea-field steamlines by systematically investigating the enegy associated with ou pevious field decomposition. The fact that the popagating and nonpopagating pats ae Maxwellian fields is exploited to genealize the Poynting theoem to accommodate fo the thee diffeent contibutions to the total enegy, the self enegy of

16 2 the popagating field, the self enegy of the nonpopagating field, and the inteaction enegy between the two fields, which may be positive o negative, while the fist two self enegies ae always positive. We then investigate vaious types of nea field enegies. It appeas that two impotant classes of enegies can be singled out fo futhe consideation, the localization enegy and the stoed enegy. We notice that the latte may not be within the each of the time-hamonic theoy we develop in this pape, but povide expessions to compute the fome enegy type. One conclusion hee is that the adial steamline nonpopagating enegy is convegent in the antenna exteio egion, anothe positive evidence of its impotance. In Section IV, we investigate the nea field stuctue fom the fa-field point of view, i.e., using the Wilcox expansion. To achieve this, a genealization of the Weyl expansion is needed, which we deive and then use to devise a hybid Wilcox- Weyl expansion. The advantage of the hybid expansion is this. While the ecusive stuctue of the Wilcox expansion, and the diect constuction outlined in Pat I [1], allow us to obtain all the tems in the seies by stating fom a given fafield adiation patten, the genealized Weyl expansion pemits a spectal analysis of each tem in ten into popagating and nonpopagating steamlines. We notice that only adial steamlines ae possible hee, which can be intepeted as a stong elation between the the fa field and the nea field of antennas that was not suspected peviously. A moe thoough study of this last obsevation will be conducted in sepaate publication. Finally, in Section V we go back to the analysis of the antenna fom the souce point of view whee we povide a vey geneal explication of the way in which the fa field of antennas is poduced stating fom a given cuent distibution. The theoy explains natually why some antennas like linea wies and patch antennas possess boadside adiation pattens. It tuns out that the whole pocess of the fa field fomation can be descibed in tems of geometical tansfomations and spatial filteing, two easy-to-undestand pocesses. We end the pape by conclusion and oveall assessment of the two-pat pape. II. SPECTRAL ANALYSIS OF ANTENNA NEAR FIELDS: THE CONCEPT OF RADIAL STREAMLINES A. Spectal Decomposition Using the Weyl Expansion We stat by assuming that the cuent distibution of an abitay antenna is given by a continuous electic cuent volume density J() defined on a compact suppot (finite and bounded volume) V. Let the antenna be suounded by an infinite, isotopic, and homogenous space with electic pemittivity ε and magnetic pemeability µ. The electic field adiated by this cuent distibution is given by the dyadic Geens function theoem [9] E () = iωµ d 3 Ḡ (, ) J ( ), (1) V whee the dyadic Geens function is given by [ Ḡ (, ) = I + ] k 2 g (, ), (2) while the scala Geens function is defined as eik g (, ) = 4π. (3) Theefoe, the electomagnetic fields adiated by the antenna 1 can be totally detemined by knowledge of the dyadic Geens function and the cuent distibution on the antenna. We would like to futhe decompose the fome into two pats, one pue popagating and the othe evanescent. This task can be accomplished by using the Weyl expansion [6],[9] whee 2 e ik = ik 2π m(p, q) = dpdq 1 m eik(px+qy+m z ), (4) { 1 p2 q 2, p 2 + q 2 1 i p 2 + q 2 1, p 2 + q 2 > 1. (5) Ou mathematical devolvement has been constained to the condition of time-hamonic excitation, i.e., all time vaiations take the fom exp( iωt). Fom the basic definition of waves [8], we know that wave popagation occus only if the mathematical solution of the poblem can be expessed in the fom Ψ( ct), whee c is a constant and Ψ is some function. 3 Since the time vaiation and the spatial vaiation ae sepaable, it is not difficult to see that the only spatial vaiation that can lead to a total spatio-tempoal solution that confoms to the expession of a popagating wave mentioned above is the exponential fom exp(im), whee m is a eal constant. The pat of the field that can not be put in this fom is taken simply as the nonpopagating potion of the total field. 4 Indeed, the Weyl expansion shows that the total scala Geens function can be divided into the sum of two pats, one as pue popagating waves and the othe as evanescent, hence nonpopagating pat. Explicitly, we wite g (, ) = g ev (, ) + g p (, ), (6) whee the popagating and nonpopagating (evanescent) pats ae given, espectively, by the expessions g ev (, ) = ik 8π 2 p 2 +q 2 >1 dpdq 1 )+q(y y )] m eik[p(x x e im z z, (7) g p (, ) = ik 8π 2 p 2 +q 2 <1 dpdq 1 )+q(y y )] m eik[p(x x e im z z. (8) The Weyl expansion can be significantly simplified by tansfoming the double integals into cylindical coodinates and then making use of the integal epesentation of the Bessel function [9]. The final esults ae 5 g ev (, ) = k ) duj 0 (kρ s 1 + u 2 e k z z u, (9) 4π 0 1 The magnetic field can be easily obtained fom Maxwell s equations. 2 Thoughout this pape, the explicit dependance of m on p and q will be suppessed fo simplicity. 3 Hee, a one-dimensional poblem is assumed fo simplicity. 4 This convention supplies the incentive fo ou whole teatment of the concept of enegies localized and stoed in the antenna fields as pesented in this pape. 5 The details of simila tansfomation will be given explicitly in Section II-F.

17 3 g p (, ) = ik 4π 1 0 ) duj 0 (kρ s 1 + u 2 e ik z z u, (10) (x x ) 2 + (y y ) 2. A outine but impotant whee ρ s = obsevation is that the integal (9), which gives the total evanescent pat of the electic field, is both unifomly and absolutely convegent fo z z. 6 By substituting the Weyl identity (4) into (1) and using (3), we obtain easily the following expansion fo the dyadic Geens function 7 Ḡ (, ) = ik 8π 2 dpdq Īk2 KK k 2 m e ik[p(x x )+q(y y )+m z z ] (11), whee the spectal vaiable (wavevecto) is given by K = ˆxkp + ŷkq + ẑsgn (z z ) km. (12) Hee, sgn stands fo the signum function. 8 Thoughout this pape, we will be concened only with the exteio egion of the antenna, i.e., we don t investigate the fields within the souce egion. Fo this eason, the singula pat that should appea explicitly in the Fouie expansion of the dyadic Geens function (11) in the fom of a delta function was dopped. The dyadic Geens function can be decomposed into two pats, evanescent and popagating, and the coesponding expessions ae given by 9 Ḡ ev (, ) = ik 8π 2 p 2 +q 2 >1 dpdq Īk2 KK k 2 m e ik[p(x x )+q(y y )+m z z ], (13) Ḡ p (, ) = ik 8π 2 p 2 +q 2 <1 dpdq Īk2 KK k 2 m e ik[p(x x )+q(y y )+m z z ]. (14) Substituting the spectal expansion of the dyadic Geens function as given by (11) into (1), we obtain afte intechanging the ode of integation E () = ωkµ 8π 2 dpdq Īk2 KK k 2 m J (k) e ik, (15) whee J (K) is the spatial Fouie tansfom of the souce distibution J (K) = d 3 J ( ) e ik. (16) V The expansion (15) is valid only in the egion z > L and z < L, i.e., the egion exteio to the infinite slab L z L. The eason is that in the integal epesentation of the dyadic Geens function (11), the integation contou is actually 6 See Appendix A. 7 Fist, we bing the diffeentiation opeatos into the integal (see Appendix B fo justification.) Next, the vecto identities exp (A ) = A exp (A ) and B exp (A ) = A B exp (A ) ae used. 8 The signum function is defined as { z, z 0 sgn (z) = z, z < 0 9 Fo the pupose of numeical evaluation, the eade must obseve that the expessions of the dyadic Geens function decomposition (13) and (14) contain moe than two basic integals because of the dependence of K on p and q as indicated by (12). Fig. 1. The geometical desciption of the antenna souce distibution (shaded volume V ) suitable fo the application of Weyl expansion. (a) Global obsevation coodinate system. The spectal epesentation of the adiated field given by (15) is valid only in the egion z > L. (b) Global and local coodinate system. Hee, fo any oientation of the local fame descibed by θ and φ, L will be geate than the maximum dimension of the souce egion V in that diection. dependent nonsmoothly on the souce vaiables. Howeve, fo the egion z > L, it is possible to justify this exchange of ode. 10 B. The Concept of Popagation in the Antenna Nea Field Zone As can be seen fom equation (13) fo the antenna fields expessed in tems of evanescent modes, the expansion itself depends on the choice of the coodinate system while the total field does not. Actually, thee ae two types of coodinates to be taken into account hee, those needed fo the mathematical desciption of the antenna cuent distibution J ( ), i.e., the point, and those associated with the obsevation point. In Figue 1(a), we show only the obsevation fame since the souce fame is absobed into the dummy vaiables of the integal defining the Fouie tansfom of the antenna cuent distibution (16). In the Weyl expansion as oiginally given in (4), the oientation of the obsevation fame of efeence is unspecified. This is nothing but the mathematical expession of the fact that scala electomagnetic souces possess otational symmety, i.e., the field geneated by a point souce located at the oigin depends only on the distance of the obsevation point fom the oigin. At a deepe level, we may take this symmety condition as an integal tait of the undelying spacetime stuctue upon which the electomagnetic field is 10 See Appendix C.

18 4 defined. 11 What is elevant to ou pesent discussion, which is concened with the natue of the antenna nea field, is that the obsevation fame of efeence can be otated in an abitay manne aound a fixed oigin. Let us stat then by fixing the choice fo the oigin of the souce fame x, y, and z. Next, we define a global fame of efeence and label its axis by x, y, and z. Without loss of geneality, we assume that the souce fame is coincident with the global fame. We then intoduce anothe coodinate system with the same oigin of the both the global and souce fames and label its coodinates by x, y, and z. This last fame will act as ou local obsevation fame. It can be oientated in an abitay manne as is evident fom the feedom of choice of the coodinate system in the Weyl expansion (4). We allow the z -axis of ou local obsevation fame to be diected at an abitay diection specified by the spheical angles θ and ϕ, i.e., the z -axis will coincide with the unit vecto ˆ in tems of the global fame. The situation is geometically descibed in Figue 1(b). Thee, the Weyl expansion will be witten in tems of the local fame x, y, and z with egion of validity given by z > L, whee L = L (θ, ϕ) is chosen such that it will be geate than the maximum size of the antenna in the diection specified by θ and ϕ. It can be seen then that ou adiated electic fields witten in tems of the global fame but spectally expanded using the (otating) local fame ae given 12 E () = ωkµ 8π 2 dpdq Īk2 K K k 2 m V d3 J ( ) e ik[px +qy +sgn(z L )mz ] e ik[ px s qy s sgn(z L )mz s], (17) whee the new spectal vecto is given by K = ˆx kp + ŷ kq + ẑ sgn (z L ) km. (18) The catesian coodinates s = x s, y s, z s in (17) epesent the souce coodinates = x, y, z afte being tansfomed into the language of the new fame = x, y, z. 13 In tems of this notation, equation (17) is ewitten in the moe compact fom E () = ωkµ 8π 2 dpdq Īk2 K K k 2 m V d3 J ( ) e ik s e ik. (19) To poceed futhe, we need to wite down the local fame coodinates explicitly in tems of the global fame. To do this, 11 This obsevation can be futhe fomalized in the following way. The field concept is defined at the most pimitive level as a function on space and time. Now what is called space and time is descibed mathematically as a manifold, which is nothing but the pecise way of saying that space and time ae topological spaces that admit diffeentiable coodinate chats (fames of efeences.) We find then that the electomagnetic fields ae functions defined on manifolds. The manifold itself may possess cetain symmety popeties, which in the case of ou Euclidean space ae a otational and tanslational symmety. Although only the otational symmety is evident in the fom of Weyl expansion given by (4), the eade should bea in mind that the tanslational invaiance of the adiated fields has been aleady used implicitly in moving fom (4) to expessions like (13) and (14), whee the souce is located at instead of the oigin. 12 That is, we expand the dyadic Geens function (2) in tems of the local fame and then substitute the esult into (1). 13 These ae equied only in the agument of the dyadic Geens function. the following otation matix is employed 14 R (θ, ϕ) R 11 R 12 R 13 R 21 R 22 R 23, (20) R 31 R 32 R 33 whee the elements ae given by R 11 = sin 2 ϕ + cos 2 ϕ cos θ, R 12 = sin ϕ cos ϕ (1 cos θ), R 13 = cos ϕ sin θ, R 21 = sin ϕ cos ϕ (1 cos θ), R 22 = cos 2 ϕ + sin 2 ϕ cos θ, R 23 = sin ϕ sin θ, R 31 = cos ϕ sin θ, R 32 = sin ϕ sin θ, R 33 = cos θ. (21) In tems of this matix, we can expess the local fame coodinates in tems of the global fame s using the following elations = R (θ, ϕ), s = R (θ, ϕ). (22) It should be immediately stated that this otation matix will also otate the x y -plane aound the z -axis with some angle. We can futhe contol this additional otation by multiplying (20) by the following matix R α cos α sin α 0 sin α cos α , (23) whee α hee epesents some angle though which we otate the x y -plane aound the z -axis. Howeve, as will be shown in Section II-D, a emakable chaacteistic of the field decomposition based on Weyl expansion is that it does not depend on the angle α if we estict ou attention to the total popagating pat and the total evanescent pat of the electomagnetic field adiated by the antenna. Fom (18) and (22), it is found that K = R T K and theefoe K K = ( R T K ) ( K R ) R T KK R, whee T denotes matix tanspose opeation. Moeove, it is easy to show that K ( R ) = ( R T K ). Using these two elation, equation (19) can be put in the fom E () = ωkµ 8π 2 dpdq Īk2 R T KK R k 2 m V d3 J ( ) e i( R T K) e ik ( R ). (24) Theefoe, fom the definition of the spatial Fouie tansfom of the antenna cuent as given by (16), equation (24) can be educed into the fom E () = ωkµ 8π 2 dpdq Īk2 R T KK R k 2 m J ( R T K ) e ik ( R ). (25) Sepaating this integal into nonpopagating (evanescent) and popagating pats, we obtain, espectively, E ev (; û) = ωkµ 8π 2 p 2 +q 2 >1 dpdq Īk2 R T (û) KK R(û) k 2 m J [ R T (û) K ] e ik [ R(û) ], E p (; û) = ωkµ 8π 2 (26) p 2 +q 2 <1 dpdq Īk2 R T (û) KK R(û) k 2 m J [ R T (û) K ] e ik [ R(û) ]. (27) 14 See Appendix D fo the deivation of the matix elements (21).

19 5 We will efe to the expansions (26) and (27) as the geneal decomposition theoem of the antenna fields. They expess the decomposition of the field at location into total evanescent and popagating pats measued along the diection specified by the unit vecto û = ˆx sin θ cos ϕ + ŷ sin θ sin ϕ + ẑ cos θ, i.e., when the z -axis of the local obsevation fame is oiented along the diection of û. Moeove, since it can be poved that this decomposition is independent of an abitay otation of the local fame aound û (see Section II-D), it follows that the quantities appeaing in (26) and (27) ae unique. Howeve, it must be noticed that the expansions (26) and (27) ae valid only in an exteio egion, fo example z > L, whee hee L is taken as the maximum dimension of the antenna cuent distibution. Using the explicit fom of the otation matix (20) given in (21), we find that the geneal decomposition theoem is valid in the egion exteio to the infinite slab enclosed between the two planes [ sin 2 ϕ + cos 2 ϕ cos θ ] x [sin ϕ cos ϕ (1 cos θ)] y cos ϕ sin θz = ±L. (28) This egion will be efeeed to in this pape as the antenna hoizon, meaning the hoizontal ange inside which the simple expessions in (26) and 27) ae not valid. 15 We immediately notice that the antenna hoizon is changing in oientation with evey angles θ and ϕ. This will estict the usefulness of the expansions (26) and (27) in many poblems in field theoy as we will see late. Howeve, a paticulay attactive field stuctue, the adial steamline concept, will not suffe fom this estiction. Towad this fom we now tun. C. The Concept of Antenna Nea-Field Radial Steamlines We focus ou attention on the desciption of the adiated field suounding the antenna physical body using spheical coodinates. In paticula, notice that by inseting = ˆx sin θ cos ϕ + ŷ sin θ sin ϕ + ẑ cos θ into (22), and using the fom of the otation matix given by (20) and (21), one can easily calculate R (θ, ϕ) = 0, 0,. 16 Theefoe, the expansion (25) becomes E () = ωkµ 8π 2 dpdq Īk2 R T KK R k 2 m J ( R T K ) e isgn( L)km, (29) whee L max θ,ϕ L (θ, ϕ). Since the obsevation is of the field popagating o nonpopagating away fom the antenna, we ae always on the banch > L. Futhemoe, by dividing the expansion (29) into popagating and nonpopagating pats, it is finally obtained E ev () = ωkµ 8π p 2 2 +q 2 >1 dpdq R T (θ, ϕ) Ω (p, q) R (θ, ϕ) J [ R T (θ, ϕ) K ] e k p 2 +q 2 1, (30) E p () = ωkµ 8π p 2 2 +q 2 <1 dpdq R T (θ, ϕ) Ω (p, q) R (θ, ϕ) J [ R T (θ, ϕ) K ] e ik 1 p 2 +q 2, (31) 15 Fo an example of calculations made inside the antenna hoizon, see Appendix F. 16 This computation can be consideed as an altenative deivation of the otation matix compaed with the one pesented in Appendix D. whee we have intoduced the spectal polaization dyad defined as 17 Ω (p, q) Īk2 KK k 2 m. (32) We notice that in this way the geneal decomposition theoems (26) and (27) ae alaways satisfied since fo each diection specified by θ and ϕ, the slab enclosed between the two planes given by (28) will also otate such that the obsevation point is always in the exteio egion. This desiable fact is behind the geat utility of the adial steamline concept (to be defined momentaily) in the antenna theoy we ae poposing in this wok. The expansions (30) and (31) can be intepeted as the decomposition of the electomagnetic fields into popagating and nonpopagating waves in the adial diections descibed by the spheical angles θ and ϕ. That is, we do not obtain a plane wave spectum in this fomulation, but instead, what we pefe to call adial steamlines emanating fom the oigin of the coodinate system (conveniently chosen at the cente of the actual adiating stuctue). The physical meaning of steamlines hee is analogous to the situation encounteed in hydodynamics, whee mateial paticles move in tajectoies embedded within continuous fluids. In the case consideed hee, steamlines have the mathematical fom Ψ( ct) fo a popagating mode with constant phase speed c, and hence ae defined completely in tems of fields. As explained ealie, it is only such solutions that epesent a genuine popagating mode; the emaining pat, the evanescent mode in the electomagnetic poblem, epesents clealy the nonpopagating pat of the adiated field. The concept of electomagnetic field steamlines developed above is a logical deduction fom a peculiaity in the Weyl expansion, namely the symmety beaking of the otational invaiance of the scala Geens function, a mathematical tait we popose to elevate to the level of a genuine physical pocess at the heat of the dynamics of the antenna nea fields. 18 It is this fom of adial steamlines that appeas to the authos to be the most natual epesentation of the inne stuctue of the antenna nea fields since it is viewed fom the pespective of the fa fields, which in tun is most conventionally expessed in tems of spheical coodinates. Since antenna enginees almost always descibe the antenna in the fa-field zone (among othe measues like the input impedance), and since such mathematical desciption necessitates a choice of a spheical coodinate system, we take ou global fame intoduced in the pevious pats to coincide with the spheical coodinate system employed by enginees in the chaacteization of antennas. Theefoe, ou nea field pictue, although it stats fom a given cuent distibution in the antenna egion, still patially eflects the pespective of the fa field. In Section IV, we will develop the nea field pictue completely fom the fa field pespective by employing the Wilcox expansion. 17 Fo a discussion of the physical meaning of this dyad, and hence a justification of the poposed name, see Section V. 18 The genealized concept of non-adial steamlines will be developed by the authos in sepaate publications. Fo example, see [3].

20 6 D. Independence of the Spectal Expansion to Abitay otation Aound the Main Axis of Popagation/Nonpopagation We now tun to the issue of the effect of otation aound the main axis chosen to pefom the spectal expansion. As we have aleady seen, the majo idea behind the nea field theoy is the intepetation of the otational invaiance of the scala Geens function in tems of its Weyl expansion. It tuned out that with espect to a given antenna cuent distibution, as long as one is concened with the exteio egion, the obsevation fame of efeence can be abitaily chosen in ode to enact a Weyl expansion with espect to this fame. It is ou opinion that such feedom of choice is not an abitay consequence of the mathematical identity pe se, but athe the deepe expession of the being of electomagnetic adiation as such. Indeed, the vey essence of how antenna woks is the scientific explication of a definite mechanism though which the nea field genetically gives ise to the fa field; in othe wods, the genesis of electomagnetic adiation out fom the nea field shell. Although the full analysis of this poblem will be addessed in futue publications by the authos, we have intoduced so fa the concept of adial steamlines to descibe the convesion mechanism above mentioned in pecise tems. It was found that we can oient the z-axis of the obsevation fame along the unit adial vecto ˆ of the global fame in ode to obtain a decomposition of the total fields popagating and nonpopagating away fom the antenna oigin along the diection of ˆ. 19 It emains to see how ou spectal expansion is affected by a otation of the local fame xy-plane aound the adial diection axis. Moe pecisely, the poblem is stated in the following manne. Conside a point in space descibed by the position vecto in the language of the global fame of obsevation. Assume futhe that the expansion of the electic field into popagating and nonpopagating modes along the diection of the z-axis of this fame was achieved, with values E ev () and E p () giving the evanescent and popagating pats, espectively. Now, keeping the the diection of the z- axis fixed, we meely otate the xy-plane by an angle α aound the z-axis. The electic field is now expanded into evanescent and popagating modes again along the same z-axis, and the esults ae E ev () and E p (), espectively. The question we now investigate is the elation between these two sets of fields. To accomplish this, let us stat fom the oiginal expansion (24) but eplace R (θ, ϕ) by a otation aound the z-axis though an angle α, which can be used to obtain the tansfomed spatial and spectal vaiables though the equations = R α and K = R T α K, whee R α is given by (23). By diect calculation, we obtain K = k (p cos α + sin α) x + k ( p sin α + q cos α) y + sgn (z L) kmz and K = ˆxk (p cos α + q sin α) + ŷk ( p sin α + q cos α) + ẑ sgn (z L ) km. These esults suggest intoducing the substitutions p = p cos α + q sin α and q = p sin α + q cos α, which ae effectively a otation of the pq-plane by and angle α aound the oigin. Being a otation, the Jacobian of this tansfomation is one, i.e., J ( ) R α = 1, whee J( ) denotes the Jacobian of the 19 Cf. equations (30) and (31). tansfomation matix applied to its agument. Also, it is evident that m = 1 p 2 + q 2 = 1 p 2 + q 2 = m. Moeove, this implies that the two egions p 2 + q 2 < 1 and p 2 + q 2 > 1 tansfom into the egions p 2 + q 2 < 1 and p 2 + q 2 > 1, espectively. Afte dividing (24) into evanescent and popagating pat, then otating the pq-plane and changing the spectal vaiables fom p and q to p and q, we find E ev () = ωkµ 8π 2 E p () = ωkµ 8π 2 p 2 +q 2 >1 dp dq Īk2 K(p,q )K(p,q ) k 2 m J ( ) R α J [K (p, q )] e ik(p,q ) (), (33) p 2 +q 2 <1 dp dq Īk2 K(p,q )K(p,q ) k 2 m J ( ) R α J [K (p, q )] e ik(p,q ) (). (34) Applying the esults of the paagaph peceding the two equations above, we conclude that E ev () = E ev (), E p () = E p (). (35) Theefoe, the total evanescent and total popagating pats of the antenna adiated fields ae invaiant to otation aound the z-axis of the obsevation fame. This esult is tue only when we ae inteested in field decomposition into egions in the spectal pq-plane that do not change though otation. Fo example, if we ae inteested in studying pat of the adiated field such that it contains the modes popagating along the z- diection, but with spectal content in the pq-plane inside, say, a squae, then since not evey otation is a symmety opeation fo a squae, we conclude that the quantity of inteest above does vay with otation of the obsevation fame aound the z-axis fo this special case. In this pape, howeve, ou inteest will focus on the total popagating and nonpopagating pats since these ae the quantities that help ationalize the oveall behavio of antennas in geneal. Howeve, it should be kept in mind that fo moe geneal and sophisticated undestanding of nea-field inteactions, it is bette to etain a geneal egion in the pq-plane as the basis fo a boad spectal analysis of the electomagnetic fields (see Figue 2.) E. The Popagating and Nonpopagating Pats ae Maxwellian Ou fomalism concening the expansion of the electomagnetic field poduced by a given antenna cuent distibution into popagating and evanescent modes is still that diectly eflecting the physics of the phenomena unde consideation, which is the laws dictated by Maxwell s equations. We will show now that both the popagating and nonpopagating pats obeys individually Maxwell s equations. The fequency-domain Maxwell s equations in souce-fee homogenous space descibed by electic pemittivity ε and magnetic pemeability µ ae given by E = iωµh, H = iωεe E = 0, H = 0. (36) The fist cul equation in (36) can be used to compute the magnetic field if the electic field is known. We assume that the latte can be expessed by the geneal decomposition theoem as stated in (26) and (27). Noticing the vecto identity

21 7 opeato to the evanescent (popagating) pat of the electic field. That is, H ev = (1/iωµ) E ev, H p = (1/iωµ) E p. (44) Fig. 2. Regions in the spectal pq-plane in tems of which the decomposition of the electomagnetic field into popagating and nonpopagating modes is conducted. The cicle p 2 + q 2 = 1 maks the bounday between the socalled invisible egion p 2 + q 2 > 1 and the visible egion p 2 + q 2 < 1 (a cicula disk.) In geneal, the mathematical desciption of the field can be accomplished with any egion in the spectal plane, not necessay the total egions inside and outside the cicle p 2 + q 2 = 1. In paticula, we show an abitay egion D located inside the popagating modes disk p 2 +q 2 < 1. In geneal, D need not be a pope subset of the egion p 2 + q 2 < 1, but may include abitay potions of both this disk and its complement in the plane. (ψa) = ψ A + ψ A and the elation exp (A ) = A exp (A ), which ae tue in paticula fo constant vecto A and a scala field ψ(), we easily obtain H () = ik 8π 2 dpdq Īk2 R T KK R k 2 m J ( R T K ) R T Ke ik ( R ), (37) whee the cul opeato was bought inside the spectal integal. Next, fom the dyadic identity ab c = a (b c), we wite R T KK R J ( R T K ) = R T K [ (K R ) J ( R T K )]. (38) This allows us to conclude that R T KK R J ( R T K ) R T K = 0. (39) Theefoe, afte sepaating the integal into popagating and evanescent pats, equation (37) becomes H ev (; û) = ik 8π 2 p 2 +q 2 >1 dpdq 1 J [ m R T (û) K ] R T (û) Ke ik ( R ), (40) H p (; û) = ik 8π 2 p 2 +q 2 <1 dpdq 1 J [ m R T (û) K ] R T (û) Ke ik ( R ). (41) The adial steamline magnetic fields coesponding to those given fo the electic field in (30) and (31) ae H ev () = ik 8π 2 p 2 +q 2 >1 dpdq 1 J [ m R T (θ, ϕ) K ] R T (θ, ϕ) Ke k p 2 +q 2 1, (42) H p () = ik 8π 2 p 2 +q 2 <1 dpdq 1 J [ m R T (θ, ϕ) K ] R T (θ, ϕ) Ke ik 1 p 2 +q 2. (43) It is evident fom the oiginal equation (37) that the evanescent (popagating) magnetic field is found by applying the cul Moeove, the divegence of the evanescent and popagating pats of both the electic and magnetic fields is identically zeo. To see this, take the divegence of (26), intechange the ode of integation and diffeentiation, and apply the identity B exp (A ) = A B exp (A ). It follows that E ev (; û) = ωkµ 8π 2 p 2 +q 2 >1 dpdq Īk2 R T (û) KK R(û) k 2 m J [ R T (û) K ] [ R T (û) K ] e ik ( R ). (45) We calculate by ab c = a (b c) and obtain { R T KK R J [ R T K ]} ( R T K) = ( R T K ) ( R T K ) { ( K R ) J [ R T K ]} (46). Howeve, since the otation matix is othogonal, i.e., R T R = Ī, we have ( R T K ) ( R T K ) = k 2 and equation (46) becomes { R T KK R J [ R T K ]} ( R T K) = k 2 J [ R T K ] ( R T K ) (47). Substituting this esult into (45), we find that E ev (; û) = 0. A simila pocedue can now be applied to all othe field pats and the divegence is also zeo. We conclude fom this togethe with equation (44) that E ev = iωµh ev, H ev = iωεe ev E ev = 0, H ev = 0. E p = iωµh p, H p = iωεe p E p = 0, H p = 0. (48) (49) These ae the main esults of this section. They show that each field pat satisfies individually Maxwell s equations. In othe wods, whateve is the diection of decomposition, the esultant fields ae always Maxwellian. Fo the case when the obsevation point lies within the antenna hoizon, it is still possible to apply the same pocedue of this section but to the most geneal expessions given by (114) and (115). It follows again the the popagating and nonpopagating pats ae still Maxwellian. F. Summay and Intepetation By now we know that ou expansion of the electomagnetic field into popagating and nonpopagating modes along a changing diection is well justified by the esult of Section II-D, namely that such expansion along a given diection is independent of an abitay otation of the local obsevation fame aound this diection. This impotant conclusion significantly simplifies the analysis of the antenna nea fields. The eason is that the full otation goup equies thee independent paametes in ode to specify an abitay 3D oientation of the otated obsevation fame. Instead, ou fomulation depends only on two independent paametes, namely θ and ϕ, which ae the same paametes used to chaacteize the degees

22 8 of feedom of the antenna fa field. This step then indicates an intimate connection between the antenna nea and fa fields, which is, elatively speaking, not quite obvious. Howeve, ou knowledge of the stuctue of the nea field, as can be discened fom the expansions (30) and (31), is enhanced by the ecod of the exact manne, as we pogess away fom the antenna along the adial diection ˆ, in which the evanescent field, i.e., the nonpopagating pat, is being continually conveted into popagating modes. As we each the fa-field zone, most of the field contents educe to popagating modes, although the evanescent pat still contibutes asymptotically to the fa field. Fo each diection θ and ϕ, the functional fom of the integands in (30) and (31) will be diffeent, indicating the how of the convesion mechanism we ae concened with. Since close to the antenna most of the nea field content is nonpopagating, we focus now ou attention on the evanescent mode expansion of the electic magnetic field as given by (30). 20 Let us intoduce the cylindical vaiables v and α such that p = v cos α and q = v sin α. Theefoe in the egion p 2 + q 2 > 1, K (v, α) = ˆxkv cos α + ŷkv sin α + ẑik v 2 1. (50) The integal (30) then becomes E ev () = ωkµ 8π vdv 2π dα F (θ, ϕ, v, α) J [ R T (θ, ϕ) K (v, α) ] e k v 2 1, (51) whee F (θ, ϕ, v, α) = Īk 2 R T (θ,ϕ) K(v,α)K(v,α) R(θ,ϕ) ik 2. v 2 1 (52) Next, pefom anothe substitution u = v 2 1. Since du = v / v 2 1dv, it follows that the integal (51) educes to whee E ev () = ωkµ 8π 2 0 du G (θ, ϕ, u) e ku, (53) G (θ, ϕ, u) = 2π 0 dα F ( θ, ϕ, 1 + u 2, α ) J [ R T (θ, ϕ) K ( 1 + u 2, α )]. (54) Theefoe, fo a fixed adial diection θ and ϕ, the functional fom of the evanescent pat of the field along this diection takes the expession of a Laplace tansfom in which the adial position plays the ole of fequency. This fact is inteesting, and suggests that cetain economy in the epesentation of the field decomposition along the adial diection has been aleady achieved by the expansions (30) and (31). To appeciate bette this point, we notice that since R (θ, ϕ) is a otation matix, it satisfies R 1 = R T. In light of this, the change in the integands of (30) and (31) with the oientation of the decomposition axis given by θ and ϕ can be viewed as, fistly, a otation of the spatial Fouie tansfom of the cuent by the invese otation oiginally applied to the local obsevation fame, and, secondly, as applying a similaity 20 The subsequent fomulation in this section can be also developed fo the evanescent pat of the magnetic field (42). tansfomation to tansfom the spectal polaization dyad Ω (p, q) to R 1 (θ, ϕ) Ω (p, q) R (θ, ϕ), that is, the spectal matix Ω (p, q) is undegoing a similaity tansfomation unde the tansfomation R 1, the invese otation. These esults indicate that thee is a simple geometical tansfomation at the coe of the change of the spectal content of the electomagnetic fields, 21 which enacts the decomposition of the electomagnetic fields into nonpopagating and popagating modes. These tansfomations ae simple to undestand and easy to visualize. We summaize the entie pocess in the following manne 1) Calculate the spatial Fouie tansfom of the antenna cuent distibution in a the global obsevation fame. 2) Rotate this Fouie tansfom by the invese otation R 1. 3) Tansfom the spectal polaization dyad by the similaity tansfomation geneated by the invese otation R 1. 4) Multiply the otated Fouie tansfom by the tansfomed spectal polaization dyad. Convet the esult fom catesian coodinates p and q to cylindical coodinates v and α and evaluate the angula (finite) integal with espect to α. That is, aveage out the angula vaiations α. 5) Tansfom as v = 1 + u 2 and compute the Laplace tansfom of the emanning function of u. This will give the functional dependence of the antenna evanescent field on the adial position whee will play the ole of fequency in the Laplace tansfom. The oveall pocess is summaized in the flowchat of Figue 3. The significance of this pictue is that it povides us with a detailed explication of the actual oute to the fa field. Indeed, since the adiation obseved away fom the antenna emeges fom the concete way in which the nonpopagating pat is being tansfomed into popagating modes that escape to the fa field zone, it follows that all of the adiation chaacteistics of antennas, like the fomation of single beams, multiple beams and nulls, polaization, etc, can be taced back into the paticula functional fom of the spectal function appeaing in the Laplace tansfom expession (53). Moeove, we now see that the geneatos of the vaiation of this key functional fom ae basically geometical tansfomation associated with the otation matix R (θ, ϕ) though which we oient the local obsevation fame of efeence. In Section V, the theoetical naative of the fa field fomation stated hee will be futhe illuminated. III. THE CONCEPT OF LOCALIZED AND STORED ENERGIES IN THE ANTENNA ELECTROMAGNETIC FIELD A. Intoduction Amed with the concete but geneal esults of the pevious pats of this pape, we now tun ou attention to a systematic investigation of the phenomena usually associated with the enegy stoed in the antenna suounding field. We have aleady encounteed the tem enegy in ou geneal 21 The functional fom of the integands of (30) and (31)

23 9 Poynting theoem that is moe geneal than the customay one (whee the latte esults fom teating only the total fields.) Stat by expanding both the electic and magnetic fields as E () = E ev ()+E p (), H () = H ev ()+H p (). (55) The complex Poynting vecto is given by [7] Substituting (55) into (56), we find S () = 1 2 E () H (). (56) S () = 1 2 E ev H ev E p H p E ev H p E p H ev. (57) Since it has been poved in Section II-E that each of the popagating and nonpopagating pat of the electomagnetic field is Maxwellian, it follows immediately that the fist and the second tems of the RHS of (57) can be identified with complex Poynting vectos S ev () = 1 2 E ev () H ev (), (58) S p () = 1 2 E p () H p (). (59) Fig. 3. The pocess of foming the nea field fo geneal antenna system. The flowchat descibes the details in which the mechanism of convesion fom evanescent mode to popagating mode unfolds. This is delimited by the vaiation of the nonpopagating pat along the adial diection θ and ϕ, with distance. The flowchat indicates that the changes in the spectal functions can be undestood in tems of simple geometical tansfomations applied to basic antenna quantities like the spatial Fouie tansfom of the antenna cuent distibution and the spectal polaization tenso of the dyadic fee space Geens function. investigation of the antenna cicuit model in [1], whee an effective eactive enegy was defined in conjunction with the cicuit intepetation of the complex Poynting theoem. We have seen that this concept is not adequate when attempts to extend it beyond the confines of the cicuit appoach ae made, pointing to the need to develop a deepe geneal undestanding of antenna nea fields befoe tuning to an examination of vaious candidates fo a physically meaningful definition of stoed enegy. In this section, we employ the undestanding of the nea field stuctue attained in tems of the Weyl expansion of the fee space Geens function in ode to build a solid foundation fo the phenomenon of enegy localization in geneal antenna systems. The upshot of this agument will be ou poposal that thee is a subtle distinction between localization enegy and stoed enegy. The fome is within the each of the time-hamonic theoy developed in this pape, while the latte may equie in geneal an extension to tansient phenomena. B. Genealization of the Complex Poynting Theoem Since we know at this stage how to decompose a given electomagnetic field into popagating and nonpopagating pats, the natual next step is to examine the powe flow in a closed egion. Ou investigation will lead to a fom of the Fom the complex Poynting theoem [7] applied to a soucefee egion, we also find S ev () = 2iω ( w e ev w h ev), (60) S p () = 2iω ( w e p w h p), (61) with electic and magnetic enegy densities defined as w e ev () = ε 4 E ev E ev, w h ev () = µ 4 H ev H ev, (62) w e p () = ε 4 E p E p, w h p () = µ 4 H p H p. (63) It emains to deal with the coss tems (thid and fouth tem) appeaing in the RHS of (57). To achieve this, we need to deive additional Poynting-like theoems. Take the dot poduct of the fist cul equation in (48) with H p. The esult is H p E ev = iωµh p H ev. (64) Next, take the dot poduct of the complex conjugate of the second cul equation in (49) with E ev. The esult is E ev H p = iωεe ev E p. (65) Subtacting (65) and (64), we obtain H p E ev E ev H p = iω ( εe p E ev µh p H ev ). (66) Using the vecto identity (A B) = B ( A) A ( B), equations (66) finally becomes (E ev H p) = iω ( εeev E p µh ev H p). (67) By exactly the same pocedue, the following dual equation can also be deived (E p H ev) = iω (εe p E ev µh p H ev). (68)

24 10 Adding (67) and (68), the following esult is obtained S int = 2iω ( w e int w h int), (69) whee we defined the complex inteaction Poynting vecto by S int 1 ( Eev H p + E p H ) ev, (70) 2 and the time-aveaged inteaction electic and magnetic enegy densities by espectively. It is immediate that w e int ε 2 Re {E p E ev}, (71) w h int µ 2 Re {H p H ev}, (72) w e = w e p + w e ev + w e int, (73) w h = w h p + w h ev + w h int, (74) S = S ev + S p + S int. (75) The justification fo calling the quantities appeaing in (71) and (72) enegy densities is the following. Maxwell s equations fo the evanescent and popagating pats, namely (48) and (49), can be ewitten in the oiginal time-dependent fom. By epeating the pocedue that led to equation (69) but now in the time domain, it is possible to deive the following continuity equation 22 S int + ( u e t int + u h ) int = 0. (76) Hee, we match the time-dependent inteaction Poynting vecto S int = Ēp H ev + Ēev H p (77) with the time-dependent electic and magnetic enegy densities u e int = εēp Ēev, u h int = µ H p H ev, (78) whee Ē and H stand fo the time-dependent (eal) fields. We follow in this teatment the convention of electomagnetic theoy in intepeting the quantities (78) as enegy densities. It is easy now to veify that the expessions (71) and (72) give the time-aveage of the coesponding densities appeaing in (78). Moeove, it follows that the time-aveage of the instantaneous Poynting vecto (77) is given by Re {S int }. Theefoe, the complex Poynting theoem can be genealized in the following manne. In each souce-fee space egion, the total powe flow outside the volume can be sepaated into thee pats, S ev, S p, and S int. Each tem individually is intepeted as a Poynting vecto fo the coesponding field. The evidence fo this intepetation is the fact that a continuitytype equation Poynting theoem can be poved fo each individual Poynting vecto with the appopiate coesponding enegy density See Appendix E fo the deivation of (76). 23 Fo example, conside the enegy theoem (76). This esults states the following. Inside any souce-fee egion of space, the amount of the inteaction powe flowing outside the suface enclosing the egion is equal to negative the time ate decease of the inteaction enegy located inside the suface. This inteaction enegy itself can be eithe positive o negative, but its quantity, is always conseved as stated by (69) o (76). C. The Multifaious Aspects of the Enegy Flux in the Nea Field Accoding to the fundamental expansion given in the geneal decomposition theoem of (26) and (27), at each spatial location, the field can be split into total nonpopagating and popagating pats along a diection given by the unit vecto û. 24 Most geneally, this indicates that if the nea field stoed enegy is to be associated with that potion of the total electomagnetic field that is not popagating, then it follows immediately that the definition of stoed enegy in this way cannot be unique. The eason, obviously, is that along diffeent diections û, the evanescent pat will have diffeent expansions, giving ise to diffeent total enegies. Summaizing this mathematically, we find that the enegy of the evanescent pat of the fields is given by Wev e (û) = ε d 3 E ev [; û ()] 2, (79) 4 V ext whee V ext denotes a volume exteio to the antenna (and possibly the powe supply.) In witing down this expession, we made the assumption that the diections along which the geneal decomposition theoem (26) is applied fom a vecto field û = û (). The fist poblem we encounte with the expession (79) is that it need not convege if the volume V ext is infinite. This can be most easily seen when the vecto field û() is taken as the constant vecto û 0. That is, we fix the obsevation fame fo all points in space, sepaate the evanescent pat, and integate the amplitude squae of this quantity thoughout all space points exteio to the antenna cuent distibution. It is eadily seen that since the field decays exponentially only in one diection (away fom the antenna cuent along û 0 ), then the esulting expession will divege along the pependicula diections. The divegence of the total evanescent enegy in this special case is discussed mathematically in Appendix F. Thee, we poved that the total evanescent enegy will divege unless cetain volumes aound the antenna ae excluded. Caying the analysis in spheical coodinates, we discove that the exteio egion can be divided into fou egions as shown in Figue 5, in which the total enegy conveges only in the uppe and lowe egions. D. The Concept of Localized Enegy in the Electomagnetic Field We now define the localized enegy as the enegy that is not popagating along cetain diections of space. Notice that the tem localized enegy is 1) not necessaily isomophic to stoed enegy and 2) is dependent on cetain vecto field û = û(). The fist obsevation will be discussed in details late. 25 The second obsevation is elated to the fundamental insight gained fom the feedom of choosing the obsevation fame in the Weyl expansion. It seems then that the mathematical 24 Although the paticula mathematical expession given in (26) and (27) ae not valid if the point at which this decomposition is consideed lies within the antenna hoizon, the sepaation into popagating and nonpopagating emains coect in pinciple but the appopiate expession is moe complicated. 25 Cf. Section III-G.

25 11 desciption of the wave stuctue of the electomagnetic field adiated by an antenna cannot be attained without efeence to a paticula local obsevation fame. We have now leaned that only the oientation of the z-axis of this local fame is necessay, educing the additional degees of feedom needed in explicating the wave stuctue of the nea field into two paametes, e.g., the spheical angles θ and ϕ. This insight can be genealized by extending it to the enegy concept. Localization hee liteally means to estict o confine something into a limited volume. The electomagnetic nea field possesses a ich and complex stuctue in the sense that it epesents a latent potential of localization into vaious foms depending on the local obsevation fame chosen to enact the mathematical desciption of the poblem. It is clea then that the localized enegy will be a function of such diections and hence inheently not unique. 26 The oveall pictue boils down to this: to localize o confine the electomagnetic enegy aound the antenna, you fist sepaate the nonpopagating field along the diections in which the potential localization is to be actualized, and then the amplitude squae of this field is taken as a measue of the enegy density of the localized field in question. By integating the esulted enegy density along the volume of inteest, the total localized enegy is obtained. The uncitical appoach to the enegy of the antenna fields confuses the stoed enegy with the localized enegy, and then postulates without justification that this enegy must be independent of the obsevation fame. One may hope that although the enegy density of the evanescent pat is not unique, the total enegy, i.e., the volume integal of the density, may tun out to be unique. Unfotunately, this is not tue in geneal, as can be seen fom the esults of Appendix F. The total convegent evanescent enegy in a give volume depends in geneal on the oientation of the decomposition axis û. The nea-field patten 27 is the quantity of inteest that antenna enginees may conside in studying the local field stuctue. Such new measue descibes the localization of electomagnetic enegy aound the antenna in a way that fomally esembles the concept of diectivity in the fa field. Moeove, based on the geneal mathematical expession of the nea-field patten (127), it is possible to seach fo antenna cuent distibutions J() with paticula oientations of û in which the obtained evanescent enegy density is invaiant. In othe wods, concepts like omnidiectionality, which is a fa-field concept, can be analogously invented and applied to the analysis of the antenna nea field. Due to the obvious complexity of the nea-field enegy expession (127), one expects that a iche symmety patten may develop with no staightfowad connection with the physical geomety of the antenna body. It is because the fa-field pespective involves an integation opeation that the ich subwavelength effects of the antenna spatial cuent distibution 26 The eade should compae this with the definition of quantities like potential and kinetic enegies in mechanics. These quantities will vay accoding to the fame of efeence chosen fo the poblem. This does not invalidate the physical aspect of these enegies since elative to any coodinate system, the total enegy must emain fixed in a (consevative) closed system. Similaly, elative to any local obsevation fame, the sum of the total popagating and nonpopagating fields yields the same actually obseved electomagnetic field. 27 Cf. equation (127) in Appendix F. on the geneated field tend to be smoothed out when viewed fom the pespective of the antenna adiation patten. In the efined appoach of this pape, the cucial infomation of the antenna nea zone coesponds to the shot-wavelength components, i.e., the spectal components p 2 + q 2 > 1, which ae esponsible of giving the field its inticate teain of fine details. These components dominate the field as we appoach the antenna cuent distibution and may be taken as the main object of physical inteest at this localized level. E. The Radial Evanescent Field Enegy in the Nea-Field Shell We now eexamine the concept of the nea-field shell at a geate depth. The idea was intoduced in Pat I [1] in the context of the eactive enegy, i.e., the enegy associated with the cicuit model of the antenna input impedance. As it has been concluded thee, this cicuit concept was not devised based on the field vantage point, but mainly to fit the cicuit pespective elated to the input impedance expessed in tems of the antenna fields as explicated by the complex Poynting theoem. We now have the efined model of the adial evanescent field developed in Section II-C. We define the localized enegy in the nea-field spheical shell as the self enegy of the nonpopagating modes along the adial steamlines enclosed in the egion a < < b. The total local enegy then is the limit of the pevious expession when b. To deive an expession fo the localized electic 28 adial enegy defined this way, substitute (30) to (79) with the identification û = ˆ. It is obtained 29 e, d Wev = ω2 k 2 µ 2 ε 256π V 4 ext d 3 p 2 +q 2 >1 dpdq p 2 +q 2 >1 dp dq R T (θ, ϕ) Ω (p, q) R (θ, ϕ) R T (θ, ϕ) Ω (p, q ) R (θ, ϕ) J [ R T (θ, ϕ) K ] J [ R T (θ, ϕ) K ] ( e k q 2 +p 2 1+ q 2 +p 2 1. (80) By conveting the space integal in (80) into spheical coodinates, and using identity (122) to evaluate the adial integal in the egion a < < b, we end up with the following expession W e, d ev (a b) = ω2 k 2 µ 2 ε { e ik(m+m )b ik(m+m ) eik(m+m )a ik(m+m ) 2π π π dθdϕ sin θ 4 p 2 +q 2 >1 dpdq p 2 +q 2 >1 dp dq R T (θ, ϕ) Ω (p, q) R (θ, ϕ) R T (θ, ϕ) Ω (p, q ) R (θ, ϕ) J [ R T (θ, ϕ) K ] J [ [ R T (θ, ϕ) ] K ] b 2 2b ik(m+m ) 2 [ k 2 (m+m )]} 2 a 2 2a ik(m+m ) 2, k 2 (m+m ) 2 ) (81) 28 Fo easons of economy, thoughout this section we give only the expessions of the electic enegy. The magnetic enegy is obtained in the same way. 29 Thoughout this pape, the convesion of the multiplication of two integals into a double integal, intechange of ode of integation, and simila opeations ae all justified by the esults of the appendices concening the convegence of the Weyl expansion.

26 12 whee m = i q 2 + p 2 1 and m = i q 2 + p 2 1. In paticula, by taking the limit b, it is found that the total adial enegy is finite and is given by W e, d ev (a ) = ω2 k 2 µ 2 ε 2π π π dθdϕ sin θ 4 p 2 +q 2 >1 dpdq p 2 +q 2 >1 dp dq R T (θ, ϕ) Ω (p, q) R (θ, ϕ) R T (θ, ϕ) Ω (p, q ) R (θ, ϕ) J [ R T (θ, ϕ) K ] J [ R T (θ, ϕ) K ] [ ] eik(m+m )a 2a ik(m+m ) ik(m+m ) + 2 a 2. k 2 (m+m ) 2 (82) This final expession shows that in contast to the scenaio of fixed decomposition axis investigated in Appendix F, the total enegy of the adial evanescent enegy in the entie space outside the exclusion sphee = a is convegent. Moeove, it was possible to analytically evaluate the infinite adial space integal. Indeed, expession (82) contains only finite space integals along the angula dependence of the spectal expansion of the adial evanescent mode field enegy density. It appeas to the authos that the adial evanescent mode expansion is the simplest type of nea-field decomposition that will give finite total enegy. The conclusion encoached by (82) stongly suggests that the concept of adial steamlines intoduced in Section II-C is the most natual way to mathematically descibe the nea field of antennas in geneal, especially fom the engineeing point of view. F. Electomagnetic Inteactions Between Popagating and Nonpopagating Fields We tun ou attention now to a close examination of the inteaction electomagnetic field enegy in the nea-field shell of a geneal antenna system. The electic field will again be decomposed into popagating and evanescent pats as E () = E ev () + E p (). The enegy density becomes then w e = ε 4 E ev () 2 + ε 4 E p () 2 + ε 2 Re {E ev () E p ()}. (83) The fist tem is identified with the self enegy density of the evanescent field, the second with the self enegy of the pue popagating pat. The thid tem is a new event in the nea field shell: it epesents a measue of inteaction between the popagating and nonpopagating pats of the antenna electomagnetic fields. While it is elatively easy to intepet the fist two tems as enegies, the thid tem, that which we duped the inteaction link between the fist two types of fields, pesents some poblems. We fist notice that contay to the two self enegies, it can be eithe positive o negative. Hence, this tem cannot be undestood as a epesentative of an entity standing alone by itself like the self enegy, but, instead, it must be viewed as a elative enegy, a elational component in the desciption of the total enegy of the electomagnetic system. To undestand bette this point, we imagine that the two positive enegies standing fo the self inteaction of both the popagating and nonpopagating pats subsist individually as physically existing enegies associated with the coesponding field in the way usually depicted in Maxwell s theoy. The thid tem, howeve, is a mutual inteaction that elates the two self enegies to each othe such that the total enegy will be eithe be lage than the sum of the two self-subsisting enegies (positive inteaction tem) o smalle than this sum (negative inteaction tem.) In othe wods, although we imagine the self enegy density to be a eflection of an actually existing physical entity, i.e., the coesponding field, the two fields nevetheless exists in a state of mutual intedependence on each othe in a way that affects the actual total enegy of the system. Conside now the total enegy in the nea field shell. This will be given by the volume integal of the tems of equation (83). In paticula, we have fo the inteaction tem the following total inteaction enegy W e,d int = ω2 k 2 µ 2 ε 256π 4 Re { V ext d3 p 2 +q 2 <1 dpdq p 2 +q 2 >1 dp dq R T (θ, ϕ) Ω (p, q) R (θ, ϕ) R T (θ, ϕ) Ω (p, q ) R (θ, ϕ) J [ R T (θ, ϕ) K ] J [ R T (θ, ϕ) K ] ( )} e ik 1 p 2 +q 2 +i p 2 +q 2 1. (84) Fo a paticula spheical shell, expessions coesponding to (81) and (82) can be easily obtained. Again, the total inteaction enegy (84) may be negative. Notice that fom the Weyl expansion, most of the field vey close to the antenna cuent distibution is evanescent. On the othe hand, most of the field in the fa-field zone is popagating. It tuns out that the inteaction density is vey small in those two limiting cases. Theefoe, most of the contibution to the total inteaction enegy in (84) comes fom the intemediate-field zone, i.e., the cucial zone in any theoy stiving to descibe the fomation of the antenna adiated fields. It is the opinion of the pesent authos that the existence of the inteaction tem in (83) is not an accidental o side phenomenon, but instead lies at the heat of the genesis of electomagnetic adiation out of the nea-field shell. The theoetical teatment we have been developing so fa is based on the fact that the antenna nea field consists of steamlines along which the field flows not in a metaphoical sense, but in the mathematically pecise manne though which the evanescent mode is being conveted to a popagating modes, and vice vesa. The two modes tansfom into each othe accoding to the diection of the steamlines unde consideation. This indicates that effectively thee is an enegy exchange between the popagating and nonpopagating pats within the neafield shell. Expession (84) is nothing but an evaluation of the net inteaction enegy tansfe in the case of adial steamlines. Since this quantity is a single numbe, it only epesents the oveall aveage of an othewise extemely complex pocess. A detailed theoy analyzing the exact inteaction mechanism is beyond the scope of this pape and will be addessed elsewhee. G. The Notoious Concept of Stoed Enegy Thee exists a long histoy of investigations in the antenna theoy liteatue concening the topic of stoed enegy in adiating systems, both fo concete paticula antennas and

27 13 geneal electomagnetic systems. 30 The quality facto Q is the most widely cited quantity of inteest in the chaacteization of antennas. As we have aleady seen in [1], all these calculations of Q ae essentially those elated to an equivalent RLC cicuit model fo the antenna input impedance. In such simple case, the stoed enegy can be immediately undestood as the enegy stoed in the inducto and capacito appeaing in the cicuit epesentation. In the case of esonance, both ae equal so one type of enegy is usually equied. Mathematically speaking, undelying the RLC cicuit thee is a second-ode odinay diffeential equation that is fomally identical to the govening equation of a hamonic oscillato with damping tem. It is well-known that a mechanical analogy exists fo the electical cicuit model in which the mechanical kinetic and potential enegies will coespond to the magnetic and electic enegies. The stoed mechanical enegy can be shown to be the sum of the two mechanical enegies mentioned above, while the fiction tem will then coespond to the esistive loss in the oscillato [10]. Now, when attempting to extend this basic undestanding beyond the cicuit model towad the antenna as a field oscillato, we immediately face the difficult task of identifying what stands fo the stoed enegy in the field poblem. The fist obsevation we make is that the concept of Q is well-defined and clealy undestood in the context of hamonic oscillatos, which ae mainly physical systems govened by odinay diffeential equations. The antenna poblem, on the othe hand, is most geneally govened by patial diffeential equations. This implies that the numbe of degees of feedom in the field poblem is infinitely lage than the numbe of degees of feedom in the cicuit case. While it is enough to chaacteize the cicuit poblem by only measuing o computing the input impedance as seen when looking into the antenna teminals, the field oscillato poblem equies geneally the detemination of the spatio-tempoal vaiation of six field components thoughout the entie domain of inteest. In ode to bing this enomous complexity into the simple level of second-ode oscillatoy systems, we need to seach fo odinay diffeential equations that summaily encapsulate the most elevant paametes of inteest. We will not attempt such appoach hee, but instead endeavo to claify the geneal equiements fo such study. We stat fom the following quote by Feynman made as pepaation fo his intoduction of the concept of quality facto [10]: Now, when an oscillato is vey efficient... the stoed enegy is vey high we can get a lage stoed enegy fom a elatively small foce. The foce does a geat deal of wok in getting the oscillato going, but then to keep it steady, all it has to do is to fight the fiction. The oscillato can have a geat deal of enegy if the fiction is vey low, and even though it is oscillating stongly, not much enegy is being lost. The efficiency of an oscillato can be measued by how much enegy is stoed, compaed with how 30 Fo a compehensive view on the topic of antenna eactive enegy and the associated quantities like quality facto and input impedance, see [4]. much wok the foce does pe oscillation. The efficiency of the oscillato is what Feynman will immediately identify as the conventional quality facto. Although his discussion focused mainly on mechanical and electic (cicuit) oscillato, i.e., simple systems that can be descibed accuately enough by second-ode odinay diffeential equations, we notice that the above quote is a fine elucidation of the geneal phenomenon of stoed enegy in oscillatoy systems. To see this, let us jump diectly to ou main object of study, the antenna as a field oscillato. Hee, we ae woking in the time-hamonic egime, which means that the poblem is an oscillatoy one. Moeove, we can identify mechanical fiction with adiation loss, o the powe of the adiation escaping into the fa-field zone. In such case, the antenna system can be viewed as an oscillato diven by extenal foce, which is nothing but the powe supplied to the antenna though its input teminal, such that a constant amount of enegy pe cycle is being injected in ode to keep the oscillato unning. Now this oscillato, ou antenna, will geneate a nea-field shell, i.e., a localized field suounding the souce, which will pesist in existence as long as the antenna is unning, an opeation that we can insue by continuing to supply the input teminal with steady powe. The oscillato function, as is wellknown, is inveted: in antenna systems the adiation loss is the main object of inteest that has to be maximized, while the stoed enegy (whateve that be) has to be minimized. The stoed enegy in the field oscillato poblem epesents then an inevitable side effect of the system: a nonpopagating field has to exist in the nea field. We say nonpopagating because anything that is popagating is associated automatically with the oscillato loss; what we ae left with belongs only to the enegy stoed in the fields and which aveages to zeo in the long un. The next step then is to find a means to calculate this stoed enegy. In the hamonic oscillato poblem, this is an extemely easy task. Howeve, in ou case, in which we ae not in possession of such a simple second-ode diffeential equations govening the poblem, one has to esot to indiect method. We suggest that the quantitative detemination of the antenna stoed enegy must evet back to the basic definition of enegy as such. We define the enegy stoed in the antenna suounding fields as the latent capacity to pefom wok when the powe supply of the system is switched off. To undestand the motivation behind this definition, let us make anothe compaison with the time evolution of damped oscillatos. Tansient phenomena can be viewed as a dischage of initial enegy stoed in the system. 31 When the antenna powe supply is on, the adiation loss is completely compensated fo by the powe emoved by the antenna teminals fom the souce geneato, while the antenna stoed enegy emains the same. Now, when the powe supply is switched off, the adiation loss can no longe by accounted fo by the enegy flux though the antenna pot. The question hee is about what happens to the stoed enegy. In ode to answe this question, we need to be moe specific about the desciption of the poblem. It 31 By a tansient is meant a solution of the diffeential equation when thee is no foce pesent, but when the system is not simply at est. [10].

28 14 will be assumed that a load is immediately connected acos the antenna input teminals afte switching off the geneato. The new poblem is still govened by Maxwell s equations and hence can be solved unde the appopiate initial and bounday conditions. It is expected that a complicated pocess will occu, in which pat of the stoed enegy will be conveted to electomagnetic adiation, while anothe potion will be absobed by the load. We define then the actual stoed enegy as the total amount of adiated powe and the powe supplied to the load afte switching off the souce geneato. In this case, the answe to the question about the quantity of the stoed enegy can in pinciple be answeed. Based on this fomulation of the poblem, we find that ou nea field theoy can not definitely answe the quantitative question concening the amount of enegy stoed in the nea field since it is essentially a time-hamonic theoy. A tansient solution of the poblem is possible but vey complicated. Howeve, ou deivations have demonstated a phenomenon that is closely connected with the cuent poblem. This is the enegy exchange between the evanescent and popagating modes. As could be seen fom equation (84), the two pats of the electomagnetic field inteact with each othe. Moeove, by examining the field expession of the inteaction enegy density, we discove that this function ove space extends in a localized fashion in a way simila to the localization of the self evanescent field enegy. This stongly suggests that the inteaction enegy density is pat of the non-moving field enegy, and hence should be included with the self evanescent field enegy as one of the main constituents of the total enegy stoed in the antenna suounding fields. Unfotunately, such poposal faces the difficulty that this total sum of the two enegies may vey well tun out to be negative, in which its physical intepetation becomes poblematic. One way out of this difficulty is to put things in thei appopiate level: the time-hamonic theoy is incapable of giving the fine details of the tempoal evolution of the system; instead, it only gives aveaged steady state quantities. The inteaction between the popagating and nonpopagating field, howeve, is a genuine electomagnetic pocess and is an expession of the essence of the antenna as a device that helps conveting a nonpopagating enegy into a popagating one. In this sense, the inteaction enegy tem pedicted by the time-hamonic theoy measues the net aveage enegy exchange pocess that occus between popagating and nonpopagating modes while the antenna is unning, i.e., supplied by steady powe though its input teminals. The existence of this time-aveaged hamonic inteaction indicates the possibility of enegy convesion between the two modes in geneal. When the geneato is switched off, anothe enegy convesion pocess (the tansient pocess) will take place, which might not be elated in a simple manne to the steady-state quantity The eade may obseve that the situation in cicuit theoy is extemely simple compaed with the field poblem. Thee, the tansient question of the cicuit can be answeed by paametes fom the time-hamonic theoy itself. Fo example, in an RLC cicuit, the Q facto is a simple function of the capacitance, inductance, and esistance, all ae basic paametes appeaing thoughout the steady state and the tansient equations. It is not obvious that such simple paallelism will emain the case in the tansient field poblem. Fig. 4. Geometic illustation fo the pocess of foming the adial localized enegy with espect to diffeent oigins. H. Dependence of the Radial Localized Enegy on the Choice of the Oigin In this section, we investigate the effect of changing the location of the oigin of the local obsevation fame used to compute the adial localized enegy in antenna systems. In equation (80), we pesented the expession of such enegy in tems of a local coodinate system with an oigin fixed in advance. If the location of this oigin is shifted to the position 0, then it follows fom (16) that the only effect will be to multiply the spatial Fouie tansfom of the antenna cuent distibution by exp (ik 0 ). Theefoe, the new total adial localized enegy will become W e,d ev ( 0 ) = ω2 k 2 µ 2 ε 256π V 4 ext d 3 p 2 +q 2 >1 dpdq p 2 +q 2 >1 dp dq R T (θ, ϕ) Ω (p, q) R (θ, ϕ) R T (θ, ϕ) Ω (p, q ) R (θ, ϕ) J [ R T (θ, ϕ) K ] J [ R T (θ, ϕ) K ] ( e i(k K ) 0 e k q 2 +p 2 1+ q 2 +p 2 1. (85) It is obvious that in geneal Wev ad ( 0 ) Wev ad (0), that is, the new localized enegy coesponding to the shifted oigin with espect to the antenna is not unique. This nonuniqueness, howeve, has nothing alaming o even peculia about it. It is a logical consequence fom the Weyl expansion. To see this, conside Figue 4 whee we show the old oigin O, the new oigin located at 0, and an abitay obsevation point outside the antenna cuent egion. With espect to the fame O, the actually computed field at the location is the evanescent pat along the unit vecto û 1 = /. On the othe hand, fo the computation of the contibution at the vey same point but with espect to the fame at 0, the field added thee is the evanescent pat along the diection of the unit vecto û 2 = ( 0 )/ 0. Clealy then the two localized enegies cannot be exactly the same in geneal. The eade is invited to eflect on this conclusion in ode to emove any potential misundestanding. If two diffeent coodinate systems ae used to descibe the adial enegy localized aound the same oigin, i.e., an oigin with the same elative position compaed to the antenna, then the two esults will be exactly the same. The situation illustated in Figue 4 does not efe to two coodinate systems pe se, but to two diffeent choices of the oigin of the adial diections utilized in computing the localized enegy of the antenna unde consideation. Thee is no known law of physics necessitating )

29 15 that the localized enegy has to be the same egadless to the obsevation fame. The vey tem localization is a puely spatial concept, which must make use of a paticula fame of efeence in ode to daw mathematically specific conclusion. In ou paticula example, by changing the elative position of the oigin with espect to the antenna, what is meant by the expession adial localization has also to undego cetain change. Equation (85) gives the exact quantitative modification of this meaning. 33 IV. THE NEAR-FIELD RADIAL STREAMLINES FROM THE FAR FIELD POINT OF VIEW A. Intoduction In this section, we synthesize the knowledge that has been achieved in [1], concening the nea field in the spatial domain, and Section II, which focused mainly on the concept of adial steamlines developed fom the spectal domain pespective. The main mathematical device utilized in pobing the spatial stuctue of the nea field was the Wilcox expansion E () = eik A n (θ, ϕ) n, H () = eik B n (θ, ϕ) n, (86) On the othe hand, the Weyl expansion (4) epesented the majo mathematical tool used to analyze the nea field into its constituting spectal components. Thee is, howeve, a deepe way to look into the poblem. The view of the antenna pesented in [1] is essentially an exteio egion desciption. Indeed, inside the sphee = a, which encloses the antenna physical body, thee is an infinite numbe of cuent distibutions that can be compatible with the Wilcox expansion in the exteio egion. Put diffeently, we ae actually descibing the antenna system fom the fa-field point of view. Indeed, as was aleady shown by Wilcox [5], it is possible to ecusively compute all the highe-ode tems in the expansion (86) stating fom a given fa field. Now, the appoach pesented in Section II is diffeent essentially fo the opposite eason. Thee, the mathematical desciption of the poblem stats fom an actual antenna cuent distibution using the dyadic Geens function as shown in (1). This means that even when inquiing about the fields adiated outside some sphee enclosing the antenna body, the fields themselves ae detemined uniquely by the cuent distibution. It is fo this eason that the analysis following Section II is inevitably moe difficult than [1]. Ou pupose in the pesent section is to each fo a kind of compomise between the two appoaches. Fom the engineeing point of view, the Wilcox seies appoach is moe convenient since it elates diectly to familia antenna measues like fa field and minimum Q. On the othe hand, as we have aleady demonstated in details, the eactive enegy concept is inadequate when extensions beyond the antenna cicuit models ae attempted. The Weyl expansion supplied us 33 An example illustating this elativity can be found in the aea of igidbody dynamics. Thee, the fundamental equations of motion involve the moment of inetia aound cetain axes of otation. It is a well-known fact that this moment of inetial, which plays a ole simila to mass in tanslational motion, does depend on the choice of the axis of otation, and vaies even if the new axis is paallel to the oiginal one. with a much deepe undestanding of the nea field stuctue by decomposing electomagnetic adiation into popagating and nonpopagating pats. What is equied is an appoach that diectly combines the Wilcox seies with the deepe pespective of the Weyl expansion. This we poceed now to achieve in the pesent section. We fist genealize the classical Weyl expansion to handle the special fom appeaing in the Wilcox seies. This allows us then to deive new Wilcox-Weyl expansion, a hybid seies that combines the best of the two appoaches. The final esult is a sequence of highe-ode tems explicating how the adial steamlines split into popagating and nonpopagating modes as we pogessively appoach the antenna physical body, all computed stating fom a given fafield patten, B. Genealization of the Weyl Expansion We stat by obseving the following fom the poduct ule eik = ik n n eik, (87) n+1 which is valid fo n 1. We will be inteested in deiving a spectal epesentation fo e ik/ n+1 since it is pecisly this facto that appeas in the Wilcox expansion (86). Fom (87) wite e ik n e ik n+1 = 1 n (ik eik n e ik n ). (88) The Weyl expansion (4) witten in spheical coodinates educes to e ik = ik dpdq 1 eik, (89) 2π m whee ˆK = ˆxp + ŷq + ẑsgn (cos θ) m, (90) ˆ = ˆx cos ϕ sin θ + ŷ sin ϕ sin θ + ẑ cos θ. (91) By binging the diffeentiation inside the integal, it is possible to achieve e ik = (ik)2 2π dpdq ˆ ˆK m eik. (92) Substituting (89) and (92) into (88), it is found that e ik 2 = (ik)2 dpdq 1 ( ) 1 ˆ ˆK e ik. (93) 2π m Iteating, the following geneal expansion is attained e ik 3 = 1 (ik) 3 2 2π dpdq 1 m ( 1 ˆ ˆK) 2 e ik. (94) Obseving the epeated patten, we aive to the genealized Weyl expansion 34 e ik n+1 = 1 (ik) n n! 2π ( 1 ˆ ˆK) n e ik. dpdq 1 m (95) In eaching into this esult, the diffeentiation and integation wee feely intechanged. The justification fo this is vey close 34 This esult can be igouously poved by applying the pinciple of mathematical induction.

30 16 to the agument in Appendix B and will not be epeated hee. On a diffeent notice, the singulaity θ = π/2 (i.e., z = 0) is avoided in this deivation because ou main inteest is in the antenna exteio egion. C. The Hybid Wilcox-Weyl Expansion We now substitute the genealized Weyl expansion (95) into the wilcox expansion (86) to obtain E () = dpdq 1 (ik) n n! 2πm A n (θ, ϕ) [ n 1 ˆ (θ, ϕ) ˆK (p, q)] e ik(p,q) ˆ(θ,ϕ), (96) H () = dpdq 1 (ik) n n! 2πm B n (θ, ϕ) e ik [ n 1 ˆ (θ, ϕ) ˆK (p, q)] e ik(p,q) ˆ(θ,ϕ). (97) By sepaating the spectal integal into popagating and evanescent pats, we finally aive to ou main esults E ev () = Ξ e n (), (98) whee H ev () = Ξ e n (), (99) Ξ e n () = p 2 +q 2 >1 dpdq 1 (ik) n n! 2πm A n (θ, ϕ) [ n 1 ˆ (θ, ϕ) ˆK (p, q)] e ik(p,q) ˆ(θ,ϕ), (100) Ξ h n () = p 2 +q 2 >1 dpdq 1 (ik) n n! 2πm B n (θ, ϕ) [ n 1 ˆ (θ, ϕ) ˆK (p, q)] e ik(p,q) ˆ(θ,ϕ). (101) Also, we have E p () = P e n (), (102) whee H p () = P e n (), (103) P e n () = p 2 +q 2 <1 dpdq 1 (ik) n n! 2πm A n (θ, ϕ) [ n 1 ˆ (θ, ϕ) ˆK (p, q)] e ik(p,q) ˆ(θ,ϕ), (104) P h n () = p 2 +q 2 <1 dpdq 1 (ik) n n! 2πm B n (θ, ϕ) [ n 1 ˆ (θ, ϕ) ˆK (p, q)] e ik(p,q) ˆ(θ,ϕ). (105) The expansion electic and magnetic functions (100) and (101) can be intepeted in the following manne. The facto ik (p, q) ˆ (θ, ϕ) appeaing in exp [ik (p, q) ˆ (θ, ϕ) ] has an attenuating pat m cos θ = p 2 + q 2 1 cos θ. Theefoe, the field descibed hee consists of evanescent modes along the adial diection specified by the spheical angles θ and ϕ. Similaly, the expansion electic and magnetic functions (104) and (105) ae pue popagating modes along the same adial diection. Thus, we have achieved a mathematical desciption simila to the adial steamline in Section II-C, mainly equations (30) and (31). In the new expansion, the ich infomation encompassing the nea-field spectal stuctue ae given by the functions ( i n k 2/ n!2πm ) A n (θ, ϕ) [ 1 ˆ (θ, ϕ) ˆK (p, q) ] n [ 1 ˆ (θ, ϕ) ˆK (p, q)] n fo and ( i n k 2/ n!2πm ) B n (θ, ϕ) the electic and magnetic fields, espectively. We immediately notice that this spectal function consists of diect multiplication of two easily identified contibutions, the fist is the Wilcox-type expansion given by the angula functions A n and B n, and the second is a common Weyl-type spectal facto given by ( i n k 2/ n!2πm ) [ n. 1 ˆ (θ, ϕ) ˆK (p, q)] This latte is function of both the spectal vaiable p and q, and the spheical angles θ and ϕ. We can now undestand the stuctue of the antenna nea field fom the point of view of the fa field in the following manne. Stat fom a given fa field patten fo a class of antennas of inteest. Stictly speaking, an infinite numbe of actually ealized antennas can be built such that they all agee on the supposed fa field. Mathematically, this is equivalent to stating that the hybid Wilcox-Weyl expansions above ae valid only in the exteio egion > a. We then poceed by computing (ecusively as in [5] o diectly as in [1]) all the vectoial angula functions A n and B n stating fom the adiation patten. With espect to this basic step, a adial steamline spectal desciption of the nea field stuctue can be be constucted by just multiplying the obtained angula vecto field A n and B n by ( i n k 2/ n!2πm ) [ n. 1 ˆ (θ, ϕ) ˆK (p, q)] This will geneate the dependence of the spectal content of the nea field on the adial steamline oientation specified by θ and ϕ. The actual spatial dependence of the popagating and nonpopagating fields can be ecoveed by integating the esult of multiplying the above obtained spectum with the adial steamline functions exp [ik (p, q) ˆ (θ, ϕ) ] ove the egions p 2 + q 2 < 1 and p 2 + q 2 > 1, espectively. A stiking featue in this pictue is its simplicity. Fo abitay antennas, it seems that the spectal effect of including highe-ode tems in the hybid Wilcox-Weyl expansion is nothing but multiplication by highe-ode polynomials of p, q, and m, 35 with coefficients diectly detemined univesally by the diection cosines of the adial vecto along which a neafield steamline is consideed. On the othe hand, antennaspecific details of the adial steamline desciption seem to be supplied diectly by the angula vecto fields A n and B n, which ae functions of the (fa-field) adiation patten. It appeas then that the expansions (98), (102),(99), (103), povide futhe infomation about the antenna, namely the impotance of size. Indeed, the smalle the sphee = a (inside whee the antenna is located), the moe tems in those expansions ae needed in ode to convege to accuate values of the electomagnetic fields. Taking into consideation that the 35 This is intuitively clea since, as we have found in Pat I [1], highe-ode tems in the Wilcox-type expansion coespond to moe complex nea-field adial stuctue as we descend fom the fa zone towad the souce egion, which in tuns necessities the need to include significant shot-wavelength components (i.e., lage p and q components.)

31 17 angula vecto fields A n and B n ae functions of the fa-field adiation patten, we can see now how the hybid Wilcox- Weyl expansion actually elates many paametes of inteest in a unified whole pictue: the fa-field adiation patten, the nea-field stuctue as given by the adial steamlines, the size of the antenna, and the minimum Q (fo matching bandwidth consideation.) It is fo these easons that the authos believe the esults of this pape to be of diect inteest to the antenna engineeing community. Moe extensive analysis of specific antenna types within the lines sketched above will be consideed elsewhee. D. Geneal Remaks We end this section by few emaks on the Wilcox-Weyl expansion. Notice fist that the eactive enegy, as defined in [1], is the fom of the total enegy expessed though the Wilcox seies with the 1/ 2 tem excluded. It is vey clea fom the esults of this section that this eactive enegy includes both nonpopagating and popagating modes. This may povide an insight into the explanations and analysis nomally attached to the elationship between eactive enegy, localized enegy, and stoed enegy. 36 The second emak is about the natue of the new steamline hee. Notice that although we ended up in the hybid Wilcox- Weyl expansion with a adial steamline pictue of the nea field, thee is still a maked diffeence between this paticula steamline and those intoduced in Section II-C fom the souce point of view. The diffeence is that the nonpopagating fields in (100) and (101) ae damped sinusoidal functions while those appeaing, fo example, in (30), ae pue evanescent modes. This is elated to a deepe diffeence between the two appoaches of Section II-C and the pesent one. In using the Wilcox expansion fo the mathematical desciption of the antenna electomagnetic fields, we ae asseting a fa-field point of view and hence ou obtained nea-field insight is aleady biased. This appeas behind the fact that the genealized Weyl integal (95), when sepaated into the two egions inside and outside the cicle p 2 + q 2 = 1, will not give a decomposition into popagating and nonpopagating modes in geneal. The eason is that thee exists in the integand spatial vaiables, mainly the spheical angles θ and ϕ. Only when these two angles ae fixed can we intepet the esulting quantity as popagating and nonpopagating modes with espect to the emaining spatial vaiable, namely. It follows then that fom the fa-field point of view, the only possible meaningful decomposition of the nea field into popagating and nonpopagating pats is the adial steamline pictue. V. THE MECHANISM OF FAR FIELD FORMATION We ae now in a position to put togethe the theoy developed thoughout this pape into a moe concete pesentation by employing it to explain the stuctual fomation of the fa field adiation. This we aim to achieve by elying on the insight into the spectal composition of the nea field povided by the 36 Cf. Section III and [1]. Weyl expansion. In the emaining pats of this section, ou focus will be on applying the souce point of view developed in Section II. The theoy of Section IV, i.e., the fa-field point of view, will be taken up in sepaate wok. Let us assume that the cuent distibution on the antenna physical body was obtained by a numeical solution of Maxwell s equations, ideally using an accuate, pefeably highe-ode, method of moment. 37 We will now explicate the details of how the fa-field patten is ceated stating fom this infomation. We focus on the electic field. Since the fa-field patten is a function of the angula vaiables θ and ϕ, the most natual choice of the appopiate mathematical tool fo studying this poblem is the concept of adial steamlines as developed in Section II-C. A glance at equations (30) and (31) shows that the quantity petinent to the antenna cuent distibution is the spatial Fouie tansfom of this cuent J (K) as defined in (16). Now, to stat with, we choose a global catesian fame of efeence xyz. Relative to this fame we fix the spheical angles θ and ϕ used in the desciption of both the fa-field patten and the adial steamline pictue of the nea field. The global fame is chosen such that the z-axis points in the diection of the boadside adiation. Fo example, if we ae analyzing a linea wie antenna o a planne patch, the global fame is chosen such the the z-axis is pependicula to the wie in the fome case and to the plane containing the patch in the latte case. Although we don t pove this hee, it can be shown that unde these condition the Fouie tansfom of the cuent distibution in the pevious two special cases, as a function of the spectal vaiables p and q, has its maximum value aound the oigin of the pq-plane as shown in Figue 2. Since the majoity of the contibution to the fa field comes fom the popagating modes appeaing in (31), the est being attenuated exponentially as shown in (30), we can pictue the antenna opeation as a two-dimensional low-pass spatial file in the following manne. All spectal components within the unit cicle p 2 + q 2 = 1 (the visible domain) will pass to the fa field, while components outside this egion will be filteed out. Let us call this filte the visible domain filte. 38 Now, the fact that when the global fame is chosen such that its z-axis is oiented in the diection along which the spatial Fouie tansfom of the cuent distibution J (K), as a function of p and q, will have most of its values concentated aound the egion p = q = 0 immediately explains why some antennas, such as linea wies and planne patches, have boadside adiation patten to begin with. We unpack this point by fist noticing how the nea field splits into popagating and nonpopagating steamlines. The mechanism hee, as deived in (30) and (31), is puely geometical. To see this, let us call the egion aound which J (K) is maximum D(p, q); e.g., in the case of planne patch this egion will be centeed aound p = q = 0. What happens is that fo vaying spheical angles θ and ϕ, we 37 It is evident that the poblem fomulated this way is not exact. Howeve, since the integal opeato of the poblem is bounded, the appoximate finite dimensional matix epesentation of this opeato will appoach the coect exact solution in the limit when N. 38 Simila constuction of this filte exists in optics.

32 18 have to otate the spatial Fouie tansfom J (K) by the matix R T (θ, ϕ). This will tanslates into the intoduction of new nonlinea tansfomation of p and q as given by K = R T (θ, ϕ) K. 39 The egion D(p, q) is now tansfomed into D(p, q ). Since we ae viewing the antenna opeation in poducing the fa field patten as a global two-dimensional spatial filte, we must tansfom back into the language of the global fame. The newly tansfomed egion D(p, q ) will be witten in the old language as D (p, q). Theefoe, vaying the obsevation angles θ and ϕ is effectively equivalent to a nonlinea stetching of the oiginal domain D(p, q) given by D (p, q) K = R T (θ,ϕ) K D (p, q). (106) This implies that a e-shaping of the domain D(p, q) is the main cause fo the fomation of the fa-field patten. Indeed, by elocating points within the pq-plane, the effect of the visible domain filte will geneate the fa-field patten. Howeve, thee is also a univesal pat of the filteing pocess that does not depend on the antenna cuent distibution. This is the spectal polaization dyad Ω (p, q) defined by (32). The multiplication of this dyad with m, i.e., the spectal quantity m Ω (p, q), is the outcome of the fact that the electomagnetic field has polaization, o that the poblem is vecto in natue. 40 It is common to all adiation pocesses. We now see that the oveall effect of vaying the obsevation angles can be summaized in the tetiay pocess 1) Rotate the spatial Fouie tansfom by R T (θ, ϕ). 2) Multiply (filte) the otated Fouie tansfom by the spectal polaization dyad Ω (p, q) afte applying to the latte a similaity tansfomation. 3) Filte the esult by the visible domain filte of the antenna. This pocess fully explicates the fomation of the fa-field patten of any antenna fom the souce point of view. As it can be seen, ou theoetical naative utilizes only two types of easy-to-undestand opeations: 1) geometical tansfomations (otation, stetching, similaity tansfomation), and 2) spatial filteing (spectal polaization filteing, visible domain filteing). VI. CONCLUSION This pape povided a boad outline fo the undestanding of the electomagnetic nea fields of geneal antenna systems in the spectal domain. The concept of steamlines was intoduced using the Weyl expansion in ode to pictue the field dynamically as a pocess of continuous decomposition into popagating and nonpopagating steamlines viewed hee fom the souce point of view. We then used the new insight to eexamine the topic of the antenna enegy, suggesting that thee ae multiple possible views of what best chaacteizes the nea field stuctue fom the enegy point of view. The concept of the nea-field adial steamlines was then developed but this time fom the fa-field point of view by deiving 39 This tansfomation is nonlinea because m depends nonlinealy on p and q via the elation m = 1 p 2 + q Cf. Section II-C. a hybid Wilcox-Weyl expansion to mathematically descibe the splitting of the nea field into adial popagating and nonpopagating steamlines constucted ecusively o diectly fom a given fa field adiation patten. The souce point of view was finally used to povide an explanation fo why and how antennas poduce fa-field adiation pattens. It seems fom the oveall consideation of this wok that thee exists a deep connection between the nea and fa fields diffeent fom what is seen in the fist look. Indeed, the esults of Section II-D suggested that only two degees of feedom ae needed to descibe the splitting of the electomagnetic field into popagating and nonpopagating pats, which supplied the theoetical motivation to investigate the adial steamline stuctue of the nea field. Futhemoe, the esults of Section IV showed that the only nea-field decomposition into popagating and nonpopagating modes possible fom the fa-field point of view is the adial steamline pictue intoduced peviously fom the souce point of view. This shows that thee exists an intimate elation between the fa and nea field stuctues, and we suggest that futhe eseach in this diection is needed in ode to undestand the deep implications of this connection fo electomagnetic adiation in geneal. On the side of antenna pactice, we believe that the poposed theoy will play a ole in futue advanced eseach and devolvement of antenna systems. Indeed, Pat I has povided a fomalism suitable fo the visualization of the impotant spatial egions suounding the antenna and the details of enegy exchange pocesses taking place thee. It has been found duing the long histoy of electomagnetic theoy and pactice that the best intuitive but also igoous way fo undestanding the opeation and pefomance of actual devices and systems is the enegy point of view. Fo this eason, the theoy poposed did not stop at the field fomalism, but also went ahead to investigate how this fomalism can be used to povide geneal concete esults concening the pathways of enegy tansfe between vaious egions in the antenna suounding domain of inteest. Fo example, we mention the inteaction theoems developed in Pat I, which povide a quantitative measue of the field modal content passing fom one spatial egion to anothe. As we emphasized epeatedly befoe, this poved to be a natual way in undestanding bette the eactive enegy, the quantity of fundamental impotance in the detemining the behavio of the antenna input impedance. Futhemoe, the specification of all these desciptions in tems of the antenna physical TE and TM modes is continuous with the established tadition in the electomagnetic community in which basic well-undestood solutions of Maxwell s equations ae used to detemine and undestand the complex behavio of the most geneal field. We believe that the geneality of the fomalism developed hee will help futue eseaches to investigate special cases aising fom paticula applications within thei ange of inteest to the community. The moe fundamental teatment pesented in Pat II aims to povide foundations fo the fomalism of Pat I. The stategy we followed hee was the classical Fouie analysis of mathematical physics and engineeing in which complex abitay field foms ae developed in a seies of well-behaved basic solution, i.e., the sinusoidal o hamonic functions. This

33 19 not only povide a solid gounding fo the esults obtained in the diect study conducted in the spatial domain, but also opens the doo fo new windows that may be needed in chaacteizing the field stuctue in emeging advanced applications and expeimental setups. The spectal theoy, which decomposes the fields into evanescent and popagating modes togethe with a fundamental undestanding of thei mutual inteelation, can be elated to the ongoing eseach in nanooptics, imaging, and othe aeas elevant to nanostuctues and atificial mateials. Indeed, the cux of this new devolvement is the manipulation of the inticate way in which the electomagnetic fields inteact with subwavelength (nano) objects. Mathematically and physically, the esonance of such subwavelength stuctues occus upon inteaction with evanescent modes, because the latte coespond to the high-wavenumbe k-components. Theefoe, ou wok in Pat II egading the fine details of the pocess in which the total field is being continually split into popagating and evanescent modes appeas as a natual appoach fo studying the inteaction of a nanoantenna o any adiating stuctue with complex suounding envionments. What is even moe inteesting is to see how such kind of applications (inteaction with complex envionments) can be studied by the same mathematical fomalism used to undestand how the fa field of any antenna (in fee space) is fomed, as suggested paticulaly in Section V. The advantage of having one coheent fomalism that can deal with a wide vaiety of both theoetical and applied issues is one of main incentives that stimulated us in caying out this pogam of antenna nea field theoy eseach. On the moe conventional side, the design and devolvement of antennas adiating in fee space, we have tied to illuminate the nea field stuctue fom both the souce point of view and the fa field pespective at the same time. Both views ae impotant in the actual design pocess. Fo the souce point of view, ou analysis in Pat II, especially Section II-C, elates in a fundamental way the exact vaiation in the antenna cuent distibution to the details of how the nea field convets continually fom evanescent to popagating modes. This can help antenna enginees in devising clues about how to modify the antenna cuent distibution in ode to meet some desiable design o pefomance goals. The advantage gained fom such outcome is educing the dependence on educated guess, andom tial and eo, and expensive optimization tasks, by poviding a solid base fo caying the antenna devolvement pocess in a systematic fashion. The fa field pespective, which was developed in Pat I and continued in Section V of Pat II, could povides a diffeent kind of valuable infomation fo the antenna enginee. Hee, one stats with a specification of a class of antennas compatible with a given fa field adiation patten, and then poceeds in constucting the nea field of all antennas belonging to this class, in both the spatial and spectal domain, in ode to elate fa field pefomance measues, such as diectivity, polaization, null fomation, etc, to nea field chaacteistics, such as input impedance and antenna size. A set of fundamental elations, undestood in this sense, can be geneated using ou fomalism fo any set of objectives of inteest found in a paticula application, and hence guide the design pocess by deciding what kind of inheent conflicts and tadeoffs exist between vaious antagonistic measues. In this way, one can avoid cumbesome effots to enfoce a cetain design goal that can not be achieved in pinciple with any configuation whatsoeve because it happens to violate one of the fundamental limitations mentioned above. APPENDIX A ABSOLUTE AND UNIFORM CONVERGENCE OF THE WEYL EXPANSION We pove this obsevation by using the integal epesentation (9). Fist, notice that fom the definition of the Bessel function, ( u 2 J ) 0 ρ 1 + u 2 e k z u u 2 e k z u. Next, by L Hopital ule, we have lim u 2 e k z u = 0 fo z 0. We u ( conclude then that lim u 2 J ) 0 ρ 1 + u 2 e k z u = 0 fo u z 0. This allows as to wite ( J ) 0 ρ 1 + u 2 e k z u < 1 u 2 fo sufficiently lage u, say u u 0. Notice that this is valid fo any ρ 0 and fo any z z 0 > 0, which is the case hee because we ae woking in the exteio egion of the antenna system. We now apply the Weiestass-M [12] test fo unifom convegence. Specifically, identify M (u) = 1 u 2 and notice that u 0 M (u) du <. It follows then that the integal is absolutely convegent and unifomly convegent in all its vaiables. APPENDIX B INTERCHANGE OF INTEGRATION AND DIFFERENTIATION IN WEYL EXPANSION Hee we intechange the ode of integation and diffeentiation. To pove this, we make use of the following theoem [12]: If f(x, α) is continuous and has continuous patial deivatives with espect to α fo x a and α 1 α α 2, and if a αf (x, α) dx conveges unifomly in the inteval α 1 α α 2, and if a dose not depend on α, then α a f (x, α) dx = We notice that Moeove, it can be easily shown that a f (x, α) dx. α We now conside the deivative of the Weyl expansion (9) with espect to x, y, and z. The last case gives du 0 z J ( ) 0 kρ 1 + u 2 e k z u = sgn (z) k ( du uj ) 0 0 kρ 1 + u 2 e k z u. ( uj ) 0 kρ 1 + u 2 e k z u ue k z u. lim u u2 ue k z u = 0 which implies ( uj ) 0 kρ 1 + u 2 e k z u ue k z u < M (u) = 1 u fo sufficiently lage u. Theefoe, du 2 0 z is unifomly convegent. Also, the integand is continuous. All these equiement ae valid fo ρ 0 and z 0. We conclude then by the theoem stated above that du 0 z = z du. 0 We now conside the deivatives with espect to x (the case with espect to y is essentially the same.) It is possible to wite du x J ( ) 0 kρ 1 + u 2 e k z u 0 = k cos ϕ du ( 1 + u 0 2 J ) 1 kρ 1 + u 2 e k z u, whee the ecuence elation of the deivative of the bessel function was used. Again, fom the popeties of bessel

34 20 functions that, J 1 (x) < 1 fo all positive eal x, so we can wite ( 1 + u 2 J ) 1 kρ 1 + u 2 e k z u < 1 + u 2 e k z u. Fom L Hopital ule, we compute lim u2 1 + u 2 e k z u = u 0. It follows that ( 1 + u 2 J ) 1 kρ 1 + u 2 e k z u < 1 + u2 e k z u < M (u) = 1 u fo sufficiently lage u and the 2 Weiestass-M test guaantee that the integal of the deivative is absolutely and unifomly convegent [12]. Fom the theoem stated ealie on the exchange of the deivative and integal opeatos, it follows that x 0 du = 0 du x. APPENDIX C EXCHANGE OF ORDER OF INTEGRATIONS IN THE RADIATED FIELD FORMULA VIA THE SPECTRAL REPRESENTATION OF THE DYADIC GREENS FUNCTION We can exchange the ode of integations by using the following theoem fom eal analysis [12]: If f(x, α) is continuous fo x a, and α 1 α α 2, and if f (x, α) dx is a unifomly convegent fo α 1 α α 2, we conclude that α2 α 1 a f (x, α) dxdα = α2 a α 1 f (x, α) dαdx. Now, we aleady poved that the Weyl expansion conveges unifomly. In addition, since the antenna cuent distibution is confined to a finite egion it immediately follows by epeated application of the theoem above that we can bing the integation with espect to the souce elements inside the spectal integal. APPENDIX D DERIVATION OF THE ROTATION MATRIX We know that the matix descibing 3D otation by an angle θ aound an axis descibed by the unit vecto û is given by u 2 x + e x c u x u y d u z s u x u z d + u y s u x u y d + u z s u 2 y + e y c u y u z d u x s u x u z d u y s u y u z d + u x s u 2 z + e z c with c = cos θ, s = sin θ, d = 1 cos θ, and e x = 1 u 2 x, e y = 1 u 2 y, e z = 1 u 2 z. In ode to otate the z-axis into the location descibed by the adial vecto ˆ, we imagine the equivalent pocess of otating the oiginal coodinate system by an angle θ aound an axis pependicula to the unit vecto ˆρ and contained within the xy-plane. Such axis of otation is descibed by the unit vecto û = ˆx sin ϕ ŷ cos ϕ. Substituting these values to the otation matix above, the fom given by (20) and (21) follows eadily. APPENDIX E THE TIME-DEPENDENT INTERACTION POYNTING THEOREM Taking the invese Fouie tansfom of equations (48) and (49), the following sets ae obtained Ēev = µ t H ev, H ev = ε tēev, Ēev = 0, H ev = 0, Ēp = µ t H p, H p = ε tēp, Ēp = 0, H p = 0 (107) (108) Take the dot poduct of the fist cul equation in (107) by H p and the second cul equation in (108) by Ēev, subtact the esults. It is found that H p Ēev Ēp H p = εēp tēp µ H p H (109) t ev. Similaly, by taking the dot poduct of the second cul equation in (107) by Ēp and the fist cul equation in (108) by H ev, subtacting the esults, we obtain H ev Ēp Ēev H ev = εēev tēev µ H ev t H p. (110) Applying the vecto identity, (A B) = B ( A) A ( B), equations (109) and (110) become (Ē ev H p ) = ε Ē ev tēev µ H ev t H p, (111) (Ē p H ev ) = ε Ē ev tēev µ H ev t H p. (112) Adding (111) and (112), and obseving the Leibniz poduct ule in handling contibutions of the RHS, equation (76) immediately follows. APPENDIX F ON THE DIVERGENCE OF THE TOTAL EVANESCENT FIELD ENERGY WITH FIXED AXIS OF DECOMPOSITION Expand the dyadic Geens function into evanescent mode along the z-diection by using (11) and then substituting the esult into (1). The following genealized electomagnetic field expansion can be obtained E () = ωkµ 8π 2 V d 3 dpdq Ω (K) J ( ) e ik[p(x x )+q(y y )+m z z ]. (113) That is, we don t hee intechange the ode of the spectal and souce integals because the exteio egion will geneally contain points within the antenna hoizon. By decomposing the field into evanescent and popagating pats, it is found that E ev () = ωkµ 8π V 2 d3 p 2 +q 2 >1 dpdq Ω (K) J ( ) e ik[p(x x )+q(y y )+m z z ], (114) E p () = ωkµ 8π V 2 d3 p 2 +q 2 <1 dpdq Ω (K) J ( ) e ik[p(x x )+q(y y )+m z z ]. (115) Next, a spheical egion enclosing the antenna is intoduced and denoted by V ( 0 ), whee 0 is the adius of the sphee. The total evanescent (nonpopagating) enegy is calculated using (79) with fixed diection of decomposition chosen along the z-axis, which gives afte using (114) Wev e = ω2 k 2 µ 2 ε 256π 4 V ext d 3 V d3 V d3 p 2 +q 2 >1 dpdq p 2 +q 2 >1 dp dq Ω (K) J ( ) Ω (K ) J ( ) e ik[p(x x )+q(y y )+m z z ] e ik[p (x x )+q (y y )+m z z ], (116)

35 21 whee V ext = V V ( 0 ) is the egion exteio to the sphee V ( 0 ). We still don t know if this integal will convege, so expession (116) should be consideed a tentative fomula. Fom physical gounds, it is expected that the calculation will face the poblem of dealing with waves along a plane pependicula to the z-axis. In such domains, the electomagnetic field expansion into evanescent modes along the z-axis consists actually of only pue popagating modes. As will be seen below, when the spheical coodinate system is employed in pefoming the space integal, thee is indeed a convegence poblem when the evaluation of the total enegy appoaches the citical xy-plane. In explicating this difficulty, it will be explicitly shown now that the limit of the total enegy when θ π/2 ± does not exist. Assuming that the ode of integations in (116) can be intechanged (a justification of this assumption will be given late), we wite afte expessing the space catesian coodinates in tems of spheical coodinates whee d 3 d 3 V V p 2 +q 2 >1 dpdq p 2 +q 2 >1 dp dq Ω (K) J ( ) Ω (K ) J ( ) e ik(p x +q y px qy ) W e ev = ω2 k 2 µ 2 ε 256π 4 2π 2π 2 ddθdϕ sin θ e ik[ζ sin θ+m cos θ z +m cos θ z ], (117) ζ = (p p ) cos ϕ + (q q ) sin ϕ. (118) We focus ou attention now on the the integal with espect to, i.e., the integal I = 0 2 de ik[ζ sin θ+m cos θ z +m cos θ z ]. (119) In Figue 5(a), we illustate the geomety of the poblem needed in computing this integal. Hee, two souce points z and z ae equied in the evaluation aound which a change in the definition of the integand occus. The angle θ will detemine the exact location of z and z with espect to 0. Also, implicit hee is the angle ϕ which will geneate the 3D patten out of this plane. To simplify the calculation, the integal (119) will be evaluated fo the special case z = z. Also, it will be assumed that 0 cos θ < z. The motivation behind these assumptions is the anticipation of the esult that the limit θ π/2 does not exist. In this case, it is evident that in such limit the adial vecto ˆ will meet the cicle = 0 befoe any z. Expanding the absolute values appeaing in the integand of (119), we obtain I = I 1 + I 2, whee and I 1 = I 2 = z /cos θ 0 z /cos θ d 2 ik[ζ sin θ υ( cos θ z e )] (120) d 2 e ik[ζ sin θ+υ( cos θ z )], (121) Fig. 5. (a) The geomety behind the calculation of the space integal in (116). Hee, the shaded egion V efes to an abitay antenna cuent distibution enclosed within a fictitious sphee with adius 0. In the figue, the two souce points z and z ae chosen andomly. (b) The diffeentiation of the space V ext exteio to sphee V ( 0 ). The uppe and lowe egions coespond to convegent evanescent enegy integals while the left and ight egions contain divegent evanescent enegy. The z-axis can be feely otated and hence the esulting total evanescent mode enegy in the convegent two egions can acquied fo the pupose of attaining a deepe analysis of the antenna nea field stuctue. In both figues we show only the zy-plane section of the poblem. with υ = m + m. Using the integal identity [ x x 2 e cx dx = e cx 2 c 2x c ] c 3, (122) it is found that [ ] I 1 = e ikζz tan θ z 2 /cos 2 θ A 2z /cos θ A A 3 e ik[ζ 0 sin θ υ( 0 cos θ z )] [ ] 2 0 A 2 0 A A, 3 (123) whee A = ζ sin θ υ cos θ. Similaly, we find [ / ] I 2 = e ikζz tan θ z 2 cos 2 θ 2z /cos θ B B B 3, (124) whee B = ζ sin θ + υ cos θ. And finally, lim θ π 2 (I 1 + I 2 ) = e ik(ζ 0+υz ) [ 2 0 ζ [ + lim e ikζ tan θz z 2 θ π cos 2 θ 2 ( 1 A 1 B ) 2z ] 20 ζ ζ 3 ( 1 ) ] cos θ A 1 2 B. 2 (125) By futhe substituting the values of A and B in tems of θ to the RHS of (125), we discove by diect calculation that this limit does not exist. Actually, it behaves like ( lim eikζz tan θ 1 θ π 2 ± cos θ + 1 ). (126)

36 22 Theefoe, the integal with espect to θ in the tentative enegy expansion (117) is ill-defined. The best we can do is to intoduce an exclusion egion π/2 δ < θ < π/2+δ, and compute the evanescent field enegy in the exteio egions, that is, the uppe and lowe egions 0 θ π/2 δ and π/2+δ θ π, both with 0. In such case, which is depicted in Figue 5(b), it is easy to pove that the enegies computed in the uppe and lowe egions ae finite. This follows fom the fact that the fields in such egions ae exponentially decaying with espect to. Using (122), the coesponding infinite adial integal (119) is convegent. Moeove, by using an agument simila to Appendix A, the same integal can be shown to be unifomly convegent. It follows then that the ode of integations with espect to the souce and space vaiables can be intechanged because the fome is finite. Also, since the Weyl expansion is unifomly convegent fo z z = 0, the integals with espect to the space vaiables and the spectal vaiables can be intechanged except at the plane θ = π/2, which we have aleady excluded. 41 This fomally justifies the geneal expession fo the evanescent field enegy, which now can be witten as Wev e (û, δ) = ω2 k 2 µ 2 ε 256π d 3 d 3 4 V V p 2 +q 2 >1 dpdq p 2 +q 2 >1 dp dq Ω (K) J ( ) Ω (K ) J ( ) e ik(p x +q y px qy ) ( 2π π/2 δ ddθdϕ sin θ + 2π π 0 π/2+δ 0 ddθdϕ sin θ ) [2] Said Mikki and Yahia Anta, Citique of antenna fundamental limitations, Poceedings of URSI-EMTS Intenational Confeence, Belin, August 16-19, [3] Said M. Mikki and Yahia M. Anta, Mophogenesis of electomagnetic adiation in the nea-field zone, to be submitted. [4] Athu D. Yaghjian and Steve. R. Best, Impedance, bandwidth, and Q of antennas, IEEE Tans. Antennas Popagat., vol. 53, no. 4, pp , Apil [5] C. H. Wilcox, An expansion theoem fo the electomagnetic fields, Communications on Pue and Appl. Math., vol. 9, pp , [6] Hemann Weyl, Ausbeitung elektomagnetische Wellen übe einem ebenen Leite, Ann. d. Physik vol. 60, pp , [7] David John Jackson, Classical Electodynamics, John Wiley & Sons, [8] Roge Knobel, An Intoduction to the Mathematical Theoy of Waves, Ameican Mathematical Socity (AMS), [9] Weng Cho Chew, Waves and Fields in Inhomogenous Media, New Yok: Van Nostand Reinhold, [10] Richad P. Feynman, Lectues on Physics I, Addison-Wesley, [11] David Bessoud, A Radical Appoach to Real Analysis, The Mathematical Ameican Society of Ameica (AMS), [12] Sege Lang, Undegaduate Analysis, Spinge-Velag, e ik(ζ sin θ+m cos θ z +m cos θ z ). (127) Hee, we have emphasized the dependence of this enegy expession on the exclusion angle δ. Also, since this enegy depends on the diection of the axis of decomposition (in this paticula example, it was chosen as the z-axis fo simplicity), the dependance on this oientation is etained explicitly. The stuctue of an antenna nea field can be analyzed by calculating the total evanescent enegy fo full azimuthal and elevation angle scan, with a suitable choice fo δ. In this way, we have intoduced what looks like a nea-field patten, in analogy with the fa-field adiation patten. 42 REFERENCES [1] Said M. Mikki and Yahia Anta, Foundation of antenna electomagnetic field theoy Pat I, (submitted). 41 The shewd eade will obseve that in evaluating the integal (119), the integand will meet with the singulaities z z = 0 and z z = 0, at which the Weyl expansion is not unifomly convegent. Howeve, since the adial integal clealy exists, its value is unchanged by the actual value of the integand at the two discete locations mentioned above. This is in contast to the situation of adial integation at the plane θ = π/2. In the latte case, the singulaity z z = 0 is enfoced at a continuum of points and so the intechange of integations, togethe with all subsequent evaluations, ae not justified. 42 The expession (127) is complicated by the fact that the souce and spectal integals cannot be intechanged. In paticula, otation of the axis of decomposition û by a matix R cannot be simplified by effectively otating the spectal vecto K though the invese opeation. Fo this eason, it does not appea possible to gain futhe quick insight into the otation effect on the evanescent enegy as given above.

37 Said M. Mikki (M 08) eceived the Bachelo s and Maste s degees fom Jodan Univesity of Science & Technology, Ibid, Jodan, in 2001 and 2004, espectively, and the Ph.D. degee fom the Univesity of Mississippi, Univesity, in 2008, all in electical engineeing. He is cuently a Reseach Fellow with the Electical and Compute Engineeing Depatment, Royal Militay College of Canada, Kingston, ON, Canada. He woked in the aeas of computational techniques in electomagnetics, evolutionay computing, nanoelectodynamics, and the development of atificial mateials fo electomagnetic applications. His pesent eseach inteest is focused on foundational aspects in electomagnetic theoy. D. Yahia Anta eceived the B.Sc. (Hons.) degee in 1966 fom Alexandia Univesity, and the M.Sc. and Ph.D. degees fom the Univesity of Manitoba, in 1971 and 1975, espectively, all in electical engineeing. In 1977, he was a waded a Govenment of Canada Visiting Fellowship at the Communications Reseach Cente in Ottawa whee he woked with the Space Technology Diectoate on communications antennas fo satellite systems. In May 1979, he joined the Division of Electical Engineeing, National Reseach Council of Canada, Ottawa, whee he woked on polaization ada applications in emote sensing of pecipitation, adio wave popagation, electomagnetic scatteing and ada coss section investigations. In Novembe 1987, he joined the staff of the Depatment of Electical and Compute Engineeing at the Royal Militay College of Canada in Kingston, whee he has held the position of pofesso since He has authoed o co-authoed ove 170 jounal papes and 300 efeeed confeence papes, holds seveal patents, chaied seveal national and intenational confeences and given plenay talks at confeences in many counties. He has supevised o co-supevised ove 80 Ph.D. and M.Sc. theses at the Royal Militay College and at Queen s Univesity, of which seveal have eceived the Goveno Geneal of Canada Gold Medal, the outstanding PhD thesis of the Division of Applied Science as well as many best pape awads in majo symposia. He was elected and seved as the Chaiman of the Canadian National Commission fo Radio Science (CNC, URSI, ), Commission B National Chai ( ),holds adjunct appointment at the Univesity of Manitoba, and, has a coss appointment at Queen's Univesity in Kingston. He also seves, since Novembe 2008, as Associate Diecto of the Defence and Secuity Reseach Institute (DSRI).