General Solution of EM Wave Propagation in Anisotropic Media

Transcription

1 Jounal of the Koean Physical Society, Vol. 57, No. 1, July 2010, pp Geneal Solution of EM Wave Popagation in Anisotopic Media Jinyoung Lee Electical and Electonic Engineeing Depatment, Koea Advanced Institute of Science and Technology, Dajeon Seoktae Lee Electonic Engineeing Depatment, Semyung Univesity, Jecheon (Received 20 Apil 2010, in final fom 22 June 2010) When anisotopy is involved, the wave equation becomes simultaneous patial diffeential equations that ae not easily solved. Moeove, when the anisotopy occus due to both pemittivity and pemeability, these equations ae insolvable without a numeical o an appoximate method. The poblem is essentially due to the fact neithe ɛ no µ can be extacted fom the cul tem, when they ae in it. The tems E (o H) and ɛe (o µh) ae pactically independent vaiables, and E and H ae coupled to each othe. Howeve, if Maxwell s equations ae manipulated in a diffeent way, new wave equations ae obtained. The obtained equations can be applied in anisotopic, as well as isotopic, cases. In addition, E and H ae decoupled in the new equations, so the equations can be solved analytically by using tenso Geen s functions. PACS numbes: Jb Keywods: Anisotopy, Dyadic Geen s functions DOI: /jkps I. INTRODUCTION As the impotance of anisotopic devices has inceased in many fields of optics and micowaves, wave popagation in anisotopic media has been widely studied ove the last decades [1]. The anisotopic natue basically stems fom the polaization o magnetization that can occu in mateials when extenal fields pass by. Geneally, cetain axis components of the E and the H fields ae influenced by othe axis components and by those of the same axis. This is why matices ae involved in ɛ and µ. Theefoe, Maxwell s equations and the wave equations ae also epesented in matix fom. Mathematically the electic field E and the magnetic field H ae not only vectos, but also ank-1 tensos, which implies that they obey the set of ules of coodinates tansfomation. This also implies that ɛ and µ ae tensos of ank-2. In odinay homogeneous isotopic media, a wave equation with a souce tem is solved with an odinay adaptation of Geen s functions. Howeve, in anisotopic media, the equations become linea simultaneous patial diffeential equations (PDE). These equations contain all infomation petaining to the anisotopic popeties. As they ae linea, Geen s functions can be consideed eas- aameas@kaist.ac.k stlee@semyung.ac.k -55- ily with the souce tems. These types of solutions with Geen s functions have been studied unde the name of dyadic Geen s functions ove the last few decades. Most of published papes pesented the special popeties of applications o methods of obtaining Geen s functions in a specified set of coodinates. This pape pesents a geneal methodology of solving the wave popagation equation in an anisotopic envionment to obtain the E and the H fields. The tenso Geen s functions GE ij and GH ij fo an electic and a magnetic field ae used, as these equations ae linea. Geneally fo an unbounded case, a Fouie-tansfomed Geen s function is useful. It changes diffeential equations into algebaic equations in k-space. Theefoe, finding the Fouie-tansfomed Geen s functions is the coe pocedue in solving the wave equations. In Section II, the anisotopic chaacteistics ae only concened with the pemittivity ɛ, which becomes a matix. The coesponding Fouie-tansfomed Geen s functions fo this case ae easily obtained. Theefoe the equation system is solvable. The Section III addesses a case in which anisotopy is descibed only with the matix pemeability µ. The mathematical pocedue in this case is simila to that discussed in Section II. The subsequent section, Section IV, is an abitay case. The anisotopy comes fom both the pemittivity and the pemeability. The equations appea to be insolvable in an analytical sense. Othe algebaic manipulations

2 -56- Jounal of the Koean Physical Society, Vol. 57, No. 1, July 2010 of Maxwell s equations lead to new wave equations, in which E and H ae decoupled. Analytic solutions ae also possible with the use of Geen s functions. The last section contains the discussion and conclusion. II. ANISOTROPY FROM PERMITTIVITY The typical method fo deiving of the wave equation stats fom E = ( µh), (1) H = J + ( ɛe). (2) In an odinay homogeneous isotopic case, the constants ɛ and µ ae extacted fom the cul tem, and the equations lead to wave equations. In this section, it is assumed that anisotopy exists due to the pemittivity. Additionally, ɛ is a matix, but µ is a scala numbe. Thus, Eq. (1) is consideed as an isotopic case. On the othe hand, Eq. (2) has a poblem in the cul tem on the ight side. The matix ɛ cannot come out of the cul tem, and Maxwell s 3d equation cannot be applied. Theefoe, the equations become E + 1 c 2 µ ɛ 2 E 2 = µ 0µ J, (3) 2 H = J ɛ 0 ( ɛ E). (4) Hee the fist tem of Eq. (3) was not expanded to ( E) 2 E, as the divegence of E cannot be witten as ɛ 1 ρ if using Maxwell s equation. When ɛ is a matix, the coect fom of the fist Maxwell s equation is ɛe = ρ, instead of E = ɛ 1 ρ. The ɛ cannot move out fom the inside of divegence. Except fo the fact that ɛ is matix, Eq. (3) esembles an isotopic case. It is and equation fo E, but it is impotant to note that the ɛ in the cul of Eq. (4) cannot come to the font of the cul opeato, as a constant numbe does. If the matix satisfying A E = ɛ E is found, Maxwell s equation becomes applicable. Howeve thee is unfotunately no matix like A. Theefoe, ɛe in the cul tem seves as an independent vaiable in a pactical sense. Thus, E and H ae coupled in the equation, which is not solvable by itself. Nevetheless Eq. (3) can be analytically solved using tenso Geen s functions. The tenso Geen s function is explained in the appendix to check if it satisfies the Geen s function condition and whethe o not it solves the poblem. Afte obtaining the solution E, the esult is substituted into Eq. (4). A Solution fo H then becomes possible with the Geen s function technique as well. In this case, the tem ( ɛ E) on the ight side of Eq. (4) becomes a pat of the souce tems, as E has aleady been solved fom Eq. (3). To solve the equations, it is convenient to assume that the time dependency is hamonic, o e iωt, and to use the following notations fo the souce tems of Eqs. (3) and (4): U() = iωµ 0 µ J, (5) V() = J iωɛ 0 (ɛ E). (6) The coesponding Geen s functions satisfies the following conditions: GE (1) (, ) + k 2 0 GE (1) (, ) = δ( ), (7) 2 GH (1) (, ) = δ( ). (8) GE and GH ae the Geen s functions fo an electic and a magnetic field, espectively. The supescipt (1) in GE and GH seves to distinguish the anisotopy that occus. 1 denotes this fo ɛ, 2 fo µ, and 3 denotes this fo both ɛ and µ. The Fouie tansfom is effective fo an unbound case: GE (1) (, ) = ge (1) (k) e ik ( ) d 3 k. (9) Inseting the Fouie-tansfomed function in Eq. (9) into Eq. (7), we obtain the algebaic equations ge (1) (k) = 1 ((k 2 I k k) k 2 0 µ ɛ ). (10) Hee, efes to the diect poduct, and I is an identity matix. The above equation fo ge (1) (k) is the main step involved in the solution. Cottis et al. calculated this Geen s functions in cylindical coodinates [2]. The GE fo the Geen s functions could be calculated by inseting Eq. (10) into Eq. (9). The fom of Eq. (8) is identical to the Poisson equation in electostatics. The following equation is, thus, applicable: GH (1) (, I ) = 4π. (11) The solutions can then be witten as follows accoding to the usual Geen s functions method: E() = H() = GE (1) (, )U( )d 3, (12) GH (1) (, )V( )d 3. (13) Looking at Eq. (3), ( 2 E) is concealed in ( E), and the equation is a type of wave equation. As ɛ

3 Geneal Solution of EM Wave Popagation in Anisotopic Media Jinyoung Lee and Seoktae Lee -57- is a matix, E 1, E 2 and E 3 ae coupled to each othe. Theefoe, this equation is fundamentally simultaneous PDE. If ɛ is diagonalized, E i s becomes decoupled afte eplacing ( E) by ( E) 2 E. If the eigenvalues ae λ 1, λ 2, and λ 3, the wave vecto changes fom k i to k i λi on the pincipal axis. Theefoe, we can compute its efactive index o change of the wave velocity along the coesponding axis. H() = GH (2) (, ) V( )d 3. (22) The souce tem in the above equation is V() = J. The next step is to find the solution. As the function fo H, has aleady been solved, the souce tem of Eq. (14) becomes U() = iωµ 0 ( (µ H)). The solution E is then III. ANISOTROPY FROM PERMEABILITY E() = GE (2) (, ) U( )d 3. (23) When anisotopy comes only fom µ, µ becomes a matix, and ɛ emains a scala numbe. The basic equations ae as follows: E = iωµ 0 ( (µ H)), (14) 2 H + k0ɛ 2 µ H = J. (15) As in the pevious section, (µ H) in Eq. (14) makes the poblem complex. This equation cannot be solved diectly. Howeve, Eq. (15) is meely Helmholtz-type equations with the souce tem, J. The calculation is caied out by using a Fouie tansfom, as in the peceding section. The Geen s functions fo Eqs. (14) and (15) ae as follows: GE (2) (, ) = δ( ), (16) 2 GH (2) (, ) + k 2 0ɛ µ GH (2) (, ) = δ( ). (17) The intoduction of the Geen s functions GE and GH poceeds identically as it did befoe. Eq. (17) is a geneal Helmholtz-type equation of the type studied by many authos [3]. The Fouie-tansfomed Geen s function ge (2) (k) and gh (2) (k) ae defined as follows: GE (2) (, ) = ge (2) (k)e ik ( ) d 3 k, (18) GH (2) (, ) = gh (2) (k)e ik ( ) d 3 k. (19) Hence, the algebaic fom of ge (2) (k), gh (2) (k) is obtained though insetions into Eqs. (16) and (17): ge (2) (k) = 1 (k 2 I k k), (20) gh (2) (k, ) = 1 (ɛ µ k0 2 Ik2 ). (21) whee the denominato is also a matix showing a tenso popety. At this point, the solution fo H is possible using GH (2) (, ): Now, the fields E and H ae solved when the anisotopic chaacteistic comes fom eithe ɛ o µ. Poblems wee noted in dealing with the cul tem including an anisotopic facto. Howeve, this difficulty is cicumvented by solving the othe equation fist and by placing the esult into the cul tem, which causes poblem. The solutions can be obtained, but the equations loose thei oiginal wave shapes. This does not mean they ae not a wave, but the analytical wave natue is not diectly obsevable. IV. ANISOTROPY FROM PERMITTIVITY AND PERMEABILITY When anisotopy occus due to both ɛ and µ, the wave equations become moe complicated to solve. This difficulty aises in the same way. The tems ɛ E and µ H ae meely a linea combination of the oiginal E and H, but when contained inside the cul tem ( ), they act as independent vaiables. Fo example, when thee is a tem ( E), we can eplace it by ( ) B using Maxwell s equations. Howeve, this eplacement cannot be applied to the tem ( ɛ E). It is pactically anothe unknown to extent that E is solved. The basic wave equations take the following foms: ( E) = iωµ 0 (µ H), (24) 2 H = J iωɛ 0 (ɛ E). (25) The goal hee is clealy to find E and H, but thee ae two moe tems that invoke poblems, ( µ H) and ( ɛ E). Theefoe, it is impossible to obtain the analytical solution simultaneously fom the wave equations. A bette appoach is to go back to the oiginal Maxwell s equations and deive new equations instead of elying on the oiginal wave equations: (µ H) E = µ 0, (26) (ɛ E) H = J + ɛ 0. (27)

4 -58- Jounal of the Koean Physical Society, Vol. 57, No. 1, July 2010 The magnetic field H is witten fom Eq. (26) as 1 H = µ 1 ( E) dt. (28) µ 0 Inseting the above equations into Eq. (27) and diffeentiating with espect to time gives (µ 1 ( E)) + 1 c 2 ɛ 2 E J. (29) 2 = µ 0 This equation estoes the oiginal wave equations when the media is isotopic. Moeove, it is an equation fo E that is decoupled fom H. In fact, this equation holds in any case, egadless of the existence of anisotopy. If the time dependency in E and J ae assumed to be hamonic o e iωt, the above equations ead. (µ 1 ( E)) k 2 0 ɛ E = iωµ 0 J (30) As the equation is linea, the tenso Geen s function (µ 1 ( GE (3) (, ))) k 2 0 ɛ GE (3) (, ) = δ( ) (31) is also possible. Assuming that ge (3) (, ) is Fouietansfomed in the same way to the Eq. (18) in the last section, the spatial Geen s function GE (3) (, ) = ge (3) (k)e ik ( ) d 3 k. (32) Equation (31) povides the algebaic elationship fo ge (3) : k µ 1 (k ge (3) ) k 2 0ɛ ge (3) = 1. (33) By applying the following matix k epesenting the wave vecto k [4], k = 0 k z k y k z 0 k x, (34) k y k x 0 Eq. (33) above becomes ge (3) (k) = 1 k µ 1 k + k0 2 ɛ. (35) Hee, k is defined in ectangula coodinates. This implies that k is coodinates dependent. Now, ge (3) (k) is obtained, and the calculation of GE (3) and E() ae staightfowad. GE (3) (, ) = E() = ge (3) (k) e ik ( ) d 3 k, (36) GE (3) (, ) U( )d 3, (37) whee U() is the ight tem of Eq. (30) as a souce. The solution of H() is obtained diectly fom Eq. (28) by inseting E. In this section, a classical wave equation itself was not consideed. Instead, a deivation of new solvable wave equations fom Maxwell s equations was pesented. V. DISCUSSION AND CONCLUSION When anisotopy occus due to eithe pemittivity o pemeability, the wave equation is solved by using Geen s functions. Howeve, in an abitay case in which anisotopy occus in both ɛ and µ, the equation is not substantially solvable in an analytic sense. The usual means is to esot to a numeical calculation. Howeve, thee is anothe method that gadually appoaches solutions by iteation. This does not need to be a numeical calculation. It can be pefomed analytically. Eithe ɛ o µ becomes a scala numbe; then, the whole system can be solved as befoe. Theefoe, the solution of the equations becomes feasible by eplacing ɛ with a scala numbe. The solution obtained in this way is the fist tial solution. The equations ae then solved again by inseting the fist solutions into the cul tem, but fom this time, the oiginal matix value fo ɛ is used instead of the fist numbe that was chosen at the beginning. An impoved second solution fo E and H is then calculated. If this iteation continues, the esults appoach some conveging functions, which ae supposed to be solutions to the equations. This pocedue can be caied out until diffeence between the (n 1) th solution and the n th solution becomes less than a pescibed level. This pocess can be done by using a symbolic calculation [5]. Howeve, it is meely an appoximate method, egadless of whethe it is a numeic o a symbolic calculation. Instead of solving the oiginal equations, we deived new wave equations that wee especially useful fo an anisotopic poblem via a slight manipulation of Maxwell s equations. The new equation is identical to the oiginal wave equation when thee is no anisotopic chaacteistic. The fundamental advantage of this new equation is the fact that E and H ae decoupled, unlike in the oiginal equations, making it possible to find analytical popagatos. In an actual calculation, thee is one facto to conside; i.e. Eq. (35) becomes singula in the isotopic case. The deteminant of the denominato becomes zeo in this special case when ɛ = µ = scala numbe. In such an isotopic case, the solution can be obtained in an odinay manne. Equation (29) was deived fo an electic field. An equation fo the magnetic field can also be obtained in a simila way. The esult is as follows: ɛ 1 ( H) + 1 c 2 µ 2 H 2 = (ɛ 1 J). (38) Solving this equation is identical to the pocedue fo E given in the last section. The new wave equations can be justified in seveal ways. The best appoach is to compae the expeimental values with the calculated esult fo the new equations. Howeve, many aspects can be examined analytically by compaing the esults of the two systems of the

5 Geneal Solution of EM Wave Popagation in Anisotopic Media Jinyoung Lee and Seoktae Lee -59- equations. The fist compaison is that the two equation sets ae identical in an isotopic case. these can be seen immediately by inspection, because thee is no matix. Fo anisotopic cases, the Fouie-tansfomed Geen s functions can be checked whethe they ae identical o not. Cottis, Vazouas and Spyou calculated the Fouie-tansfomed dyadic Geen s functions in a dielectic anisotopy, and thei esult is the same as Eq. (10). The esult of the new equation fo an identical case is given below accoding to Eq. (35): ge (3) 1 (k) = k k + k0 2 ɛ. (39) Given that thee is no anisotopy in µ, the identity matix is substituted in place of µ 1. The same esult is obtained using the elationship k k = k k k 2 I. The othe compaison is fo the case of magnetic anisotopy. In this case ɛ = I. The esult of the Fouie-tansfomed Geen s functions of the oiginal equation is given in Eq. (21). It can be compaed with the esult of Eq. (38), which gives gh (3) 1 = k ɛ 1 k + k0 2 µ 1 = k I k + k0 2 µ 1 = k k k 2 I + k0 2 µ. (40) Howeve, the k k tem becomes zeo when it is applied to a magnetic field. Hence, the esult of the oiginal equation is identical to that of the new equations. Fo moe veification, the plane wave E = E 0 exp[i(k ωt)] can be consideed in an anisotopic medium. When the elative pemittivity tenso is given in tems of efactive indices as ɛ = n2 x n 2 y 0, (41) 0 0 n 2 z the efactive indices then satisfy the following wellknown condition: n 2 x n 2 (ˆk yˆk 2 z) 2 n 2ˆk xˆky n 2ˆk xˆkz n 2ˆk xˆky n 2 y n 2 (ˆk x 2 + ˆk z) 2 n 2ˆk yˆkz = 0, (42) n 2ˆk xˆky n 2ˆk yˆkz n 2 z n 2 (ˆk xˆk 2 y) 2 whee the conventional notations n = c ω k and ˆk i = ki k ae used. Exactly the same esult is also obtained fom the new wave equation, Eq. (29). This is actually a natual consequences, because they ae based on the same mathematical gound. As fo advantages of the new equations, it is clea that oiginal equations do not allow an analytical appoach in the geneal case wheeas the new equations give analytical popagatos (Fouie-tansfomed Geen s functions). The point is whethe o not they ae decoupled. This is a consideable diffeence between the two systems, although the new equations need numeical calculations, fo example integation to get GE(, ), GH(, ) o the final answes E and H. The existence of popagatos is supposed to bing a non-negligible amount of code-saving effect. As a esult, an equation set Eqs. (29) and (38), is obtained and descibes wave popagation in anisotopic media. This shows that the two systems ae commutable and equivalent. The new wave equation is expected to be useful in analytical eseach of anisotopic popeties beyond what a numeical appoach can addess. APPENDIX A: PROOF OF THE TENSOR GREEN S FUNCTION As Eq. (3) is linea, it is natual to conside Geen s functions. Howeve, the equation is a multi-linea simultaneous PDE, and can veify if the use of Geen s functions is pope. If Geen s functions wok, the following type of solutions can be assumed as valid: E i = GE ij U j d. (A1) The functions GE ij ae devised to play the oles of influence functions to geneate the fields due to ρ and J. At this stage, it is unknown whethe o not they ae Geen s functions. The poof should be pefomed as to whethe they satisfy the conditions of Geen s functions befoe adapt ion of Geen s functions. Eq. (3) can be witten in component fom as ɛ ikq ɛ klm 2 E 1 x m x q + 1 c 2 µ (ɛ ) ij E j = iωµ 0 µ J i. (A2)

6 -60- Jounal of the Koean Physical Society, Vol. 57, No. 1, July 2010 By inseting Eq. (A1) into Eq. (A2), one can calculate the next equation: { 2 GE 1s ɛ ikq ɛ klm + 1 } x m x q c 2 µ (ɛ ) ij GE js U s d = U i, (A3) whee U i = iωµ 0 µ J i fo the souce tem. Inside the backet is GE k 2 0µ ɛ GE, and to maintain the equality, as expected, the equation aives at the definition of the Geen s functions: GE k 2 0µ ɛ 2 GE 2 = Iδ( ). (A4) GE and I ae 3-by-3 matices and the above equation is simply a definition of the Geen s function. The othe tenso Geen s functions used in this aticle all have simila foms. Moeove, it is easy to pove that they meet the definition of the Geen s function. The equation in Eq. (A4) is essentially nine equations fo GE ij that ae coupled to each othe. Howeve, the equations become decoupled and much simple to solve though diagonalization of ɛ. REFERENCES [1] N. Macuvitz, J. Schwinge, J. Appl. Phys. 22, 806 (1951); F. V. Bunkin, J. Exp. Theo. Phys. 32, 338 (1957); C. T. Tai, Dyadic Geen s Functions in Electomagnetic Theoy (IEEE Pess, New Yok, 1971); L. B. Felsen and Nathan Macuvitz, Radiation and Scatteing of Waves (Pentice Hall, 1972). [2] P. G. Cottis, C. N. Vazouas and C. Spyou, IEEE Tans. Antennas Popag. 47, 154 (1995); P. G. Cottis, C. N. Vazouas and C. Spyou, ibid. 47, 195 (1999). [3] A. B. Gnilenko and A. B. Yakovlev, IEE Poc-H. 146, 111 (1999); D. Van Oden, V. Lomakin, IEEE Tans. Antennas Popag. 57, 1973 (2009); R. C. Wittmann, IEEE Tans. Antennas Popag. 36, 1078 (1988). [4] A. Eoglu and J. K. Lee, IEEE Antennas and Popagation Society Intenational Symposium (9-14 July 2006, Albuguegue, NM), p [5] Le-Wei Li, Mook-Seng Leong and Tat-Soon Yeo, IEEE Antennas Popag. 43, 118 (2001).