Microwave magnetoelectric fields: An analytical study of topological characteristics

Transcription

1 Microwave magnetoelectric field: An analytical tudy of toological characteritic R. Joffe 1,, R. Shavit 1, and E. O. Kamenetkii 1, 1 Microwave Magnetic Laboratory, Deartment of Electrical and Comuter Engineering, Ben Gurion Univerity of the Negev, Beer Sheva, Irael Deartment of Electrical and Electronic Engineering, Shamoon College of Engineering, Beer Sheva, Irael December 10, 015 Abtract The near field originated from a mall quai-two-dimenional ferrite dik with magnetic-diolarmode (MDM) ocillation are the field with broken dual (electric-magnetic) ymmetry. Numerical tudie how that uch field called the magnetoelectric (ME) field are ditinguihed by the ower-flow vortice and helicity arameter [E. O. Kamenetkii, R. Joffe, and R. Shavit, Phy. Rev. E 87, 0301 (013)]. Thee numerical tudie can well exlain recent exerimental reult with MDM ferrite dik. In the reent aer, we obtain analytically toological characteritic of the ME-field mode. For thi uroe, we ue a method of ucceive aroximation. In the econd aroximation we take into account the influence of the edge region of an oen ferrite dik, which are excluded in the firt-aroximation olving of the magnetotatic (MS) ectral roblem. Baed on the analytical method, we obtain a ure tructure of the electric and magnetic field outide the MDM ferrite dik. The analytical tudie can dilay ome fundamental feature that are non-obervable in the numerical reult. While in numerical invetigation, one cannot earate the ME field from the external electromagnetic (EM) radiation, the reent theoretical analyi allow clearly ditinguih the eigen toological tructure of the ME field. Imortantly, thi ME-field tructure give evidence for certain henomena that can be related to the Tellegen and bianiotroic couling effect. We dicu the quetion whether the MDM ferrite dik can exhibit roertie of the cro magnetoelectric olarizabilitie. PACS number(): 41.0.Jb; 4.50.Tx; g I. INTRODUCTION The electric dilacement current in Maxwell equation allow rediction of wave roagation of electromagnetic field. In a mall ferrite amle (with ize much le than the free-ace electromagnetic wavelength), at ferromagnetic-reonance frequencie, one ha negligibly mall variation of electric energy. In thi cae, a dynamical roce in a amle i decribed by three differential equation, without the electric dilacement current, [1 4] B 0, (1) B E, t () H 0. (3)

2 A formal ue of a et of differential Eq. (1) (3), doe not allow conideration of any retardation effect. However, without Eq. (), baed on Eq. (1), (3) and the contitutive relation B H, (4) where i the ermeability tenor, one can obtain olution for roagating wave inide a ferrite amle. Taking into account the temoral-dierion roertie of a ferrite material at a ferromagnetic reonance, one obtain the Walker equation for magnetotatic-otential (MSotential) wave function (introduced by a relation H ) [5]. The ocillation in mall ferrite article, decribed by the Walker equation, are called magnetotatic-wave (MS-wave) or magnetic-diolar-mode (MDM) ocillation [1 5]. Interaction of mall ferrite article with electromagnetic radiation i not a trivial roblem. MDM ocillation in ferrite here excited by external microwave field were exerimentally oberved, for the firt time, by White and Solt in 1956 [6]. Afterward, exeriment with dikform ferrite ecimen revealed unique ectra of ocillation. While in a cae of a ferrite here one oberved only a few wide abortion eak of MDM ocillation [6], for a ferrite dik there wa a multireonance (atomic-like) ectrum with very har reonance eak [7 9]. Analytically, it wa hown [10 1] that, contrary to herical geometry of a ferrite article analyzed in Ref. [5], the Walker equation (together with the homogeneou boundary condition for function and it derivative) for quai-d geometry of a ferrite dik give the Hilbert-ace energy-tate election rule for MDM ectra. There are o called G-mode ectral olution [10 15]. When we aim to obtain the MDM ectral olution taking into account alo the electric field in a ferrite dik [to obtain the olution taking into account Eq. ()], we have to conider the boundary condition for a magnetic flux denity, B. Analytically, it wa hown that in thi cae (becaue of ecific boundary condition for a magnetic flux denity on a lateral urface of a ferrite dik), one ha the helical-mode reonance and the ectral olution are decribed by double-valued function [1, 13]. There are o called L-mode ectral olution. For L mode, the electric field in a vacuum region near a ferrite dik ha two art: E Ec E, where i the curl-field comonent and E c E i the otential-field comonent [15]. While the curl electric field H in vacuum we define from the Maxwell equation Ec 0, the otential electric t field in vacuum i calculated by integration over the ferrite-dik region, where the ource E ( m) m (magnetic current j ) are given. Here m i dynamical magnetization in a ferrite dik. It t wa hown that in vacuum near a ferrite dik, the region with non-zero calar roduct E c E can exit [15]. Thi calar roduct i called the field helicity. For time-harmonic field, the time-averaged helicity factor i exreed a [15] E c F 0 Im E 4. (5) MDM ocillation in a quai-d ferrite dik are macrocoically coherent quantum tate, which exerience broken mirror ymmetry and alo broken time-reveral ymmetry [1, 13]. Free-ace microwave field, emerging from magnetization dynamic in quai-d ferrite dik, E c

3 carry orbital angular momentum and are characterized by ower-flow vortice and non-zero helicity. Symmetry roertie of thee field called magnetoelectric (ME) field are different from ymmetry roertie of free-ace electromagnetic (EM) field. At the MDM frequency MDM i, we have for magnetic induction B E. The ME-field helicity denity i MDM nonzero only at the reonance frequencie of MDM and i exreed a MDM0 MDM0 MDM F Im ie B Re E B Re E H 4 4 4c, (6) where c In thi equation, both the electric and magnetic field are otential field. The arameter defined by Eq. (6) i different from the time-averaged otical (electromagnetic) chirality denity, which i obtained for both, the electric and magnetic, curl field and i exreed a [16 19] 0 Im E. (7) The ME-field helicity denity F wa analyzed in numerical tudie [15]. A numerical analyi 1 how alo that ditribution of the real ower-flow denity Re E H of a ME field contitute the vortex toological tructure in vacuum [15, 0]. Alo, in Ref. [1] it wa hown numerically that together with the real ower-flow denity, a ME field i characterized by the 1 imaginary Im E H ower-flow denity. ME-couling roertie, oberved in the near-field tructure, are originated from magnetization dynamic of MDM in a quai-d ferrite dik. In general, ME-couling effect manifet in numerou macrocoic henomena in olid. Phyic underlying thee henomena become evident through a ymmetry analyi. In iolating crytal material, in which both atial inverion and time-reveral ymmetrie are broken, a magnetic field can induce electric olarization and, converely, an electric field can induce magnetization [1, ]. Without requirement of a ecial kind of a crytal lattice, a ME-couling term aear in magnetic ytem with toological tructure of magnetization. In articular, there can be chiral, toroidal, and vortex tructure of magnetization [3, 4]. Other examle on a role of magnetization toology in the ME-couling effect concern orbital magnetization. A it wa dicued in Ref. [5, 6], an adequate decrition of magnetim in magnetic material hould not only include the in contribution, but alo hould account for effect originating in the orbital magnetim. It wa hown that in the two-dimenional cae, orbital magnetization i exhibited due to exceeding of chiral-edge circulation in one direction over chiral-edge circulation in ooite direction [6]. Recently, it wa hown that ME couling can occur alo in iotroic dielectric due to an effect of orbital ME olarizability toological ME-couling effect [7 9]. In uch a cae, one ha the contribution of orbital current to the ME couling. The orbital ME olarizability i due to the eudocalar art of the ME couling and i equivalent to the addition of a term to the electromagnetic Lagrangian the axion electrodynamic term [30]. That i why the orbital ME reone in iotroic dielectric i referred a the axion orbital ME olarizability [7, 8]. In thi aer, we develo a theoretical analyi of the near field originated from a MDM ferrite dik. The electric and magnetic field in vacuum are obtained baed on the magnetization ditribution of MDM inide a ferrite dik. With ue of thi model, we tudy analytically B 3

4 toological characteritic of the ME-field mode. We how that in a vacuum region very near to a ferrite amle there i a good correondence between the analytical and numerical reult for the field tructure. For an incident electromagnetic field, the MDM ferrite dik look a a tra with focuing to a ring, rather than a oint. While in numerical invetigation, one cannot earate the ME field from the external EM radiation, the theoretical analyi allow clearly ditinguih the eigen toological tructure of ME field. Thi may concern, in articular, an imortant quetion whether the MDM ferrite dik can exhibit roertie of the magnetoelectric (cro) olarizabilitie. In the aer, we examine the analytically obtained ME-field tructure in a view of the Tellegen and bianiotroic couling effect. Baed on the analytical tudy, we invetigate the active and reactive comonent of the comlex ower-flow denity of the ME field. The toological tructure of ME field demontrated by uch ower flow ditribution can exlain tability of the eigentate of MDM ocillation. II. THE MODEL AND ANALYTICAL RESULTS FOR THE ME-FIELD STRUCTURES We ue a method of ucceive aroximation. In the firt-aroximation olving of the MDM ectral roblem, we obtain analytically the RF magnetization inide a ferrite dik. The known magnetization and the magnetic current derived from thi magnetization are conidered a ource for the magnetic and electric field outide a ferrite dik. Baed on the econd aroximation, we find the magnetic and electric comonent of the ME field and analyze toology of the ME field in vacuum. In the theoretical analyi, we ue the ame dik arameter a in Ref. [14, 15]: the yttrium iron garnet (YIG) dik ha a diameter of 3 mm and the dik thickne i d 0.05 mm; the dik i normally magnetized by a bia magnetic field H Oe; the aturation magnetization of the ferrite i 4 M 1880 G. We aume that magnetic loe in a ferrite dik are negligibly mall. A. The firt-aroximation olution for MS-otential wave function In the firt-aroximation olution, we ignore the influence of the edge region outide a quai- D ferrite dik [10] (ee Fig. 1). Fig. 1. An oen quai-d ferrite dik. Baed on thi model, we can ue earation of variable. Analytically, there are two ectral model for the MDM ocillation. Thee model give o-called G- and L-mode. The MSotential wave function for L-mode i written a [10, 1 15] 4

5 C ( z) ( r, ), (8), n, n, n, n where a dimenionle effective membrane function, ( r, ) n i defined by the Beel-function order 1,, 3,... and the number of zero of the Beel function correonding to different radial variation n 1,, 3,... In Eq. (8), i a dimenionle function of the MS-otential, n( z ) ditribution along z axi. C, n QC i an amlitude coefficient where, n Q i a dimenional unit coefficient of 1A and i a normalized dimenionle amlitude. Inide a ferrite dik C,n ( r, d z d ) the MS-otential wave function i rereented a r 1 i r,, z, t C, nj co z in z e e it, (9) where i a roagation contant for MS wave along z axi and i a diagonal comonent of the ermeability tenor. Solution in a form of Eq. (9) how the azimuthally-roagating-wave behavior for MS-otential membrane function. To define the normalized dimenionle amlitude we will ue the Hilbert-ace energyeigentate G-mode ectral olution. Auming that the calar-wave membrane function for the G-mode ectral olution i rereented a [1 15] C,n we can write, (10) C, n, n, n, n, n S C ds. (11) Here integration i made over an entire oen-dik region in the membrane function i exreed a r, lane. Normalization of C, n 1. (1), n Fig. how the calculated coefficient number of radial variation ( n 1,, 3,... ). C,n for the 1 t -order Beel-function ( 1) and the 5

6 Fig.. Normalized amlitude coefficient for MS-otential wave function. The MDM are with the 1 t -order Beel-function ( n ). 1) and the number of radial variation ( 1,, 3,... Baed on olution (9) for every MDM, we can find the firt-aroximation olution for the magnetic field both inide and outide a ferrite dik H MDM. (13) MDM Alo, we can find the ectral magnetization ditribution inide a ferrite dik: m, (14) MDM MDM where i the magnetic ucetibility tenor [4]. For a normally magnetized ferrite dik, we have only the radial and azimuthal comonent of magnetization. Thee comonent are exreed a 1 r m r a i m ( r,, z, t) C, co in r n z z J J e e it, (15) 6

7 where 1 r m r a i m ( r,, z, t) ic, n co z in z J J e e, a it, (16) are diagonal and off-diagonal comonent of the magnetic ucetibility tenor, reectively [4], and J i a derivative (with reect to the argument) of the Beel function of order. Fig. 3 and 4 how MS-otential wave function for the 1 t and nd MDM, reectively. Fig. 3. MS-otential wave function (the 1 t aroximation) for the 1 t MDM. The MDM i decribed with the 1 t -order Beel-function (, correond to the main thickne mode. (a) The function ; (b) the radial ditribution of the MS-otential wave function; (c) the icture of intenity of membrane MS-otential wave function ( r, ), which rotate in the lane erendicular to the dik axi z. () z 1) and the radial variation n 1. The variation along z axi, () z 7

8 Fig. 4. MS-otential wave function (the 1 t aroximation) for the nd MDM. The MDM i decribed with the 1 t -order Beel-function (. The variation along z axi,, correond to the main thickne mode. (a) The function ; (b) the radial ditribution of the MS-otential wave function; (c) the icture of intenity of membrane MSotential wave function ( r, ), which rotate in the lane erendicular to the dik axi z. () z 1) and the radial variation n () z Thee mode are decribed by the 1 t -order Beel-function ( the number of radial variation n = 1 and n =, reectively. The variation along z axi, decribed by the function, correond to the main thickne mode. Baed on the known MS-otential wave function, radial and azimuthal comonent of magnetization were calculated. The in-lane ditribution of the magnetization (at the lane z d ) are hown in Fig. 5 for the 1 t and nd MDM. () z 1). The 1 t and nd MDM have 8

9 Fig. 5. The magnetization ditribution in a quai-d ferrite dik. (a), (b) The 1 t and nd MDM, reectively, for the hae t 0 ; (c), (d) the 1 t and nd MDM, reectively, for the hae t 90. Every elementary magnetic diole rotate in the xy lane. Becaue of the azimuthallyroagating-wave behavior for MS-otential membrane function, the entire icture of the magnetization rotate a well. Direction of the magnetization rotation, clockwie or counterclockwie, deend on a direction of a bia magnetic field. The arrow are unit vector. Every elementary magnetic diole rotate in the xy lane. Becaue of the azimuthallyroagating-wave behavior for MS-otential membrane function, the entire icture of the magnetization rotate a well. Direction of the magnetization rotation, clockwie or counterclockwie, deend on the direction of the bia magnetic field. B. The econd-aroximation olution The known, from Eq. (15), (16), MDM magnetization ditribution inide a ferrite dik allow making an analyi by taking into conideration the edge region, hown in Fig 1. A a reult, one can obtain an entire tructure of the MS-otential wave function and the field outide a ferrite dik. There are the econd-aroximation olution. Uing Eq. (1), (13) and taking into account that B ( H m) 0, we obtain the following Poion equation m, (17) 9

10 where the right-hand-ide term i conidered a an effective magnetic charge. If a ferrite dik ha a volume V and urface S, we ecify m inide V a magnetization of a certain MDM and aume that it fall uddenly to zero at the urface S. For a given MDM, the olution of Eq. (17) for the MS-otential wave function outide a ferrite dik i [31], (18) 1 m( x ') n m( x ') ( x) dv ds 4 x x ' x x ' where n i the outwardly directed normal to urface S. The quantity can be conidered a an effective urface magnetic charge denity. Becaue of non-uniform magnetization ditribution throughout the volume V, both integral in the right-hand-ide of Eq. (18) contribute to the MSotential olution. Since there are only the radial and azimuthal comonent of the MDM magnetization, urface S in Eq. (18) i a lateral urface of a ferrite dik. Baed on Eq. (18), we find the econd-aroximation olution for the otential magnetic field outide the ferrite dik: n m m( x ') x x ' nm( x ') x x ' 1 H ( x) dv ds 4 ' '. (19) 3 3 x x x x We can alo obtain the econd-aroximation olution for the magnetic flux denity outide the ferrite dik: B( x) H( x). (0) 0 Fig. 6 how the normalized magnetic field ditribution H ˆ H / H outide the ferrite dik (on the xz cro-ectional lane) for the 1 t and nd MDM. 10

11 Fig. 6. Normalized magnetic field ditribution Hˆ H / H outide a ferrite dik (on the xz croectional lane) hown for a certain hae, t 0. (a), (b) The 1 t and nd MDM, reectively, for the 1 t aroximation; (c), (d) the 1 t and nd MDM, reectively, for the nd aroximation. The arrow are unit vector. One can comare the magnetic field calculated baed on the 1 t aroximation [Fig. 6 (a) and (b)] with the field calculated baed on the nd aroximation [Fig. 6 (c) and (d)]. Inide a ferrite dik the Faraday equation () i written a ( m) E i0h j, (1) where we introduced a magnetic current ( m) j, exreed a ( m) j i0m. () Outide a ferrite dik, in vacuum, we conider an electric field a comoed by two comonent: E E E. The curl electric field Ec i defined a c E i H. (3) c 0 11

12 Here H i the otential magnetic field in vacuum, found baed on Eq. (19). The otential electric field in vacuum i originated from a ource region: a MDM ferrite dik with a magnetic E current. Baed on an evident duality with the claical-electrodynamic roblem of the magnetotatic magnetic field originated from the electric-current ource region [31], in Ref. [15] it wa hown that the otential electric field outide the ferrite dik i defined a j ( m) For every MDM, the otential electric field E ( m) j x x x. (4) 3 x x 1 E ( x) dv 4 E outide the ferrite dik i obtained baed on Eq. (4). A magnetic current i calculated baed on Eq. (15), (16), (). Fig. 7 how the normalized electric field ditribution E ˆ E/ E outide the ferrite dik ( m) j (on the xz cro-ectional lane) for the 1 t and nd MDM. Fig. 7. Ditribution of the normalized otential electric field outide a ferrite dik (on Eˆ E/ E the xz cro-ectional lane) hown for a certain hae, t 0. (a) The 1 t MDM; (b) the nd MDM. The arrow are unit vector. In Fig. 8 one can ee the calculated ditribution of the magnetic and electric field for the 1 t and nd MDM on a vacuum xy lane 0 um above the ferrite dik. 1

13 Fig. 8. The field ditribution on a vacuum xy lane 0 um above a ferrite dik hown for a certain hae, t 0. (a), (b) The magnetic and electric field, reectively, for the 1 t MDM; (c), (d) the magnetic and electric field, reectively, for the nd MDM. The arrow are unit vector. In thee figure we reent only the ditribution of the otential field E for the 1 t and nd MDM. A comarative analyi of the analytically derived ditribution of the otential field and numerically obtained ditribution of the total field E E Ec, hown below in Fig. 15 and 16, give evidence for a negligibly mall role of the curl electric field Theoretically, the otential electric field integration over a ferrite-dik volume for a known magnetic current E E c E in the field toology. in a vacuum region near a ferrite dik i found by ( m) j [Eq. (4)]. To obtain the curl electric field E c, one ha to olve the differential equation (3) taking into account that the magnetic field in a vacuum region i the evanecent-wave field. For geometry of an oen ferrite dik [with taking into conideration the edge region (ee Fig. 1)], analytical olution of Eq. (3) i aociated with ubtantial difficultie. Becaue of a negligibly mall role of the curl electric field E c in the near-field toology, an analyi of uch a comonent of the electric field i beyond the frame of thi work. 13

14 C. Toological characteritic of ME field Baed on the known analytical olution for the MS-otential wave function and the field outide the ferrite dik, we can obtain toological characteritic of the ME field. The ditribution of the helicity denity and the ower-flow denity comrie thee characteritic. The ME-field helicity denity, exreed by Eq. (6), wa calculated baed on Eq. (19) and (4) for otential magnetic and electric field. Fig. 9 how the calculated ME-field helicity denity ditribution for the 1 t and nd MDM. One can ee that a dimenion of the ME-field helicity denity F i Joule/meter 4. For a otential field, thi i a gradient of energy denity which define field trength. So, in our cae, factor F i a trength of a ME field. For otential electric and magnetic field, magnitude and direction of the field trength are determined by force acting on a tet electric charge or an electric current element. The quetion what kind of a tet element hould be introduced to determine the trength of the ME field i oen. Thi quetion i beyond an analyi in the reent aer. Fig. 9. The ME-field helicity denity analytically calculated baed on Eq. (6). (a) and (b) The ditribution on a vacuum xy lane 0 um above a ferrite dik for the 1 t and nd MDM, reectively; (c) and (d) the ditribution on a cro-ectional xz lane for the 1 t and nd MDM, reectively. In an analyi of toological roertie of ME field, an imortant quetion arie what i a ditribution of an angle between the otential electric and magnetic field in the near-field ace. While for regular EM-field roblem, the vector of electric and magnetic field in vacuum are mutually erendicular in ace, in our cae, a ace angle between the electric and magnetic 14

15 field in vacuum i, definitely, not. An angle between the otential electric and magnetic field we will characterize by the arameter called the normalized ME-field helicity: 90 Re E H co. (5) E H Here ubcrit for an angle mean otential. The ditribution of on a croectional xz lane are hown in Fig. 10 for the 1 t and nd MDM. co Fig. 10. The normalized ME-field helicity ( co ) ditribution on a cro-ectional xz lane. (a) For the 1 t MDM; (b) for the nd MDM. The inertion illutrate mutual orientation of the electric and magnetic field in different oint on the z axi above a ferrite dik. In fact, thi arameter rereent mutual orientation of the electric- and magnetic-field vector hown in Fig. 6 (c, d), 7, and 8. In an analyi of the comlex ower-flow denity of the ME field, we will ue the following conideration. A imle analyi of the energy balance equation for monochromatic MS wave in a magnetic medium with mall loe [1] how that the real ower flow denity for a MDM i exreed a i Re i B B B. (6) Here the mode number include both the Beel-function order and the number of zero of the Beel function correonding to different radial variation n. With ue of the firtaroximation olution for the MS-otential wave function, exreed by Eq. (9), we can calculate the quantity. The. It i eay to how [3] that inide a ferrite dik 0 only non-zero comonent of the real ower flow denity inide a ferrite dik i the azimuth comonent, which i exreed a [1, 3]: r z 15

16 i 1 r z C z i a r r r (, ) ( ) 1 ( r, ) C ( z) ( r, ) a, r r (7) where and The quantitie a are, reectively, the diagonal and off-diagonal comonent of the tenor., circulating around a circle r (where ( r, z) r ), are the MDM owerflow-denity vortice with core at the dik center. At a vortex center the amlitude of equal to zero. In a vacuum region, outide a ferrite dik we have from Eq. (7) 0 1 ( r, z) C ( z) ( r, ). (8) r It i clear that, both inide and outide a ferrite dik. In Ref. [1], an analyi of the energy relation for MDM wa extended by an introduction alo the imaginary ower flow denity for a MDM. For a mode, thi imaginary ower flow denity i exreed a q Im i B B B. (9) Similarly to the above calculation of the imaginary ower flow denity imaginary ower flow denity q i, we can calculate the baed on the firt-aroximation olution (9). In thi cae, we can how that inide a ferrite that q 0, while q 0 and q 0. For q r z r, we obtain: 1 q z C r z ia r r r ( ) ( ) 1 () r C ( z) a ( r). r r (30) Outide the ferrite dik we have q (, z) C ( z) r () r r. (31) For q z, we obtain () z q C( z) ( r, ) z z. (3) The above reult for the comlex ower-flow denity of the ME field, obtained baed on the firt-aroximation olution, can be imroved with ue of the econd-aroximation olution 16

17 for MS-otential wave function, in which an entire tructure of the field outide a ferrite dik i taken into account. For the econd-aroximation olution, quantitie of the real ower flow denity and the imaginary ower flow denity outide a ferrite dik are calculated baed on Eq. (6) and (9) uing Eq. (18) and (0) for the MS-otential wave function and the magnetic flux denity, reectively. Fig. 11 and 1 how the ME-field active and reactive ower-flow ditribution outide a ferrite dik for the 1 t and nd MDM calculated baed on Eq. (6) and (9). q Fig. 11. The ME-field active-ower-flow ditribution outide a ferrite dik calculated baed on Eq. (6). (a) and (b) The ditribution on a vacuum xy lane 0 um above a ferrite dik for the 1 t and nd MDM, reectively; (c) and (d) the ditribution on a cro-ectional xz lane for the 1 t and nd MDM, reectively. The arrow are unit vector. 17

18 Fig. 1. The ME-field reactive-ower-flow ditribution outide a ferrite dik calculated baed on Eq. (9). (a) and (b) The ditribution on a vacuum xy lane 0 um above a ferrite dik for the 1 t and nd MDM, reectively; (c) and (d) the ditribution on a cro-ectional xz lane for the 1 t and nd MDM, reectively. The arrow are unit vector. The real and imaginary ower flow denitie, defined by Eq. (6) and (9), are related to the time averaged real and imaginary art of a vector roduct of the curl electric and otential magnetic field. A imle maniulation (taking into account that B 0 and H ) how that ( E H ) H E i B i B. (33) c c Following Ref. [33], one can conclude that thi equation give E c H i B. (34) For mode, we have Re Re i B E c H (35) and 18

19 Im q Im i B E c H. (36) Imortantly, deite the fact that uch exreion a ReE H and ImE H look like the real and imaginary Poynting vector, the MS-wave ower flow denitie cannot be baically related to the EM-wave ower flow denitie. The Poynting vector i obtained for EM radiation which i decribed by the two curl oerator Maxwell equation for the electric and magnetic field [31]. Thi i not the cae decribed by Eq. (33), where, for the MS wave, we have otential magnetic and curl electric field. While Eq. (35) and (36), are relevant only for a curl electric field in a vacuum region near a ferrite dik, the quetion about a role of the otential electric field E c E in the vector-roduct quadratic relation arie a well. Formally, we can extend our analyi of the active ower flow alo to calculation of the quantity Re E H. Thi quantity, for the 1 t and nd MDM, i hown in Fig. 13. Similar to the reult hown in Fig. 11, in the icture in Fig. 13 we can ee that there are the ower-flow-denity vortice with core at the dik center. At a vortex center the amlitude i equal to zero. Fig. 13. The quantity Re E H outide a ferrite dik. (a) and (b) The ditribution on a vacuum xy lane 0 um above a ferrite dik for the 1 t and nd MDM, reectively; (c) and (d) the ditribution on a cro-ectional xz lane for the 1 t and nd MDM, reectively. The arrow are unit vector. 19

20 In vacuum, near the central region of the MDM ferrite dik, one ha in-lane rotating vector of the otential electric and magnetic field [15]. It wa hown [1] that for thee inning electric- and magnetic-field vector, a time averaged imaginary art of a vector roduct of the electric and magnetic field i related to the reactive ower flow denity. Baed on Eq. (19) and (4), we can find analytically the ditribution of the quantity Im E H outide the ferrite dik. Fig. 14 how the calculated quantity of the reactive ower flow dik for the 1 t and nd MDM. Contrary to amlitude at the dik center. Re E H, the quantity Im E H outide the ferrite Im E H ha a maximal Fig. 14. The quantity of the reactive ower flow Im E H outide a ferrite dik. (a) and (b) The ditribution on a vacuum xy lane 0 um above a ferrite dik for the 1 t and nd MDM, reectively; (c) and (d) the ditribution on a cro-ectional xz lane for the 1 t and nd MDM, reectively. The arrow are unit vector. In Section IV of the aer we dicu more in detail the obtained analytical reult on the MEfield toology. III. VERIFICATION OF THE ANALYTICAL RESULTS BY NUMERICAL STUDIES Our analytical reult of the ME-field tructure are well verified by numerical tudie baed on the HFSS imulation rogram. Comarion between the analytical and numerical tudie of the field comonent, hown in Fig. 15 and 16 for the 1 t and nd MDM, give good correondence between thee reult. In thee figure, the analytically derived electric field are otential field calculated baed on Eq. (4). 0

21 Fig. 15. Comarion between the analytical and numerical reult of the field comonent for the 1 t MDM. The analytically derived electric field are otential field calculated baed on Eq. (4). The field comonent are hown a the quantitie normalized to the modulu of the field vector. (a) The electric-field comonent; (b) the magnetic-field comonent. A comarative analyi i made for the field comonent on a vacuum xy lane 50 um above a ferrite dik. 1

22 Fig. 16. Comarion between the analytical and numerical reult of the field comonent for the nd MDM. The analytically derived electric field are otential field calculated baed on Eq. (4). The field comonent are hown a the quantitie normalized to the modulu of the field vector. (a) The electric-field comonent; (b) the magnetic-field comonent. A comarative analyi i made for the field comonent on a vacuum xy lane 50 um above a ferrite dik. The field comonent are hown a the quantitie normalized to the modulu of the field vector. A an illutrative examle of the ME-field toology obtained baed on both the theoretical and numerical analye, we how in Fig. 17 the normalized ME-field helicity denity ditribution numerically calculated for the 1 t and nd MDM.

23 Fig. 17. The ME-field helicity denity ditribution calculated numerically. (a) and (b) The ditribution on a vacuum xy lane 0 um above a ferrite dik for the 1t and nd MDM, reectively; (c) and (d) the ditribution on a cro-ectional xz lane for the 1t and nd MDM, reectively. One can comare thee ditribution with the analytically calculated ME-field helicity denity ditribution hown in Fig. 9. Evidently, for uch a toological arameter, there i good correondence between the two tye of analye. We retrict now further verification of the obtained analytical reult by roer reference of our reviou numerical tudie. The activeower-flow vortice hown in Fig. 11 and 13 are well verified by uch vortice obtained numerically in Ref. [14, 15, 0, 1, 3]. The reactive ower flow Im E H hown in Fig. 14 are in good correondence with the numerical reult of the reactive ower flow in Ref. [1]. The analytical tudie, however, dilay ome more imortant feature that are non-obervable in the numerical reult. We dicu thi in the next Section of the aer. IV. DISCUSSION The quetion of the active ower flow of ME field i not comrehenively clear. The real ower flow denity for MDM exreed by Eq. (6), i obtained baed on energy balance equation for monochromatic MS wave in a magnetic medium with mall loe [1]. Following a formal analyi made in Ref. [33], one can conclude that uch real ower flow denity for MDM i related to a real art of the vector roduct of the curl electric field and otential magnetic field. At the ame time, a we analyzed above, uch an exreion i not related to the electromagnetic 3

24 Poynting vector. Our analytical reult reveal alo ome more intereting feature. When one comare the active ower flow ditribution in Fig. 11 and 13, one find evident imilarity between thee icture. It mean that there i imilarity between the quantitie Re E H and Re E H. So, the role of the curl electric field and otential electric field in the active ower flow ditribution are almot inditinguihable. Certainly, in the numerical tudie we are dealing with the total electric field, without any earation to the curl and otential art. Another intereting quetion concern the reactive ower flow of ME field. A we have hown above, there are two tye of reactive ower flow. The firt one, defined by Eq. (9), i hown in Fig. 1, while the econd one, defined a Im E H, i hown in Fig. 14. One can ee that thee two tye of the reactive ower flow are localized at different art of the ferrite dik and have different direction with reect to the dik. While in the icture hown in Fig. 1 we have the drain-tye vector orientation, for the icture hown in Fig. 14 there are the ource-tye vector orientation. At the ame time, it i worth noting that the magnitude of the reactive ower flow in Fig. 1 i ufficiently maller than uch a magnitude in Fig. 14. Thi fact can exlain why in the numerical analyi in Ref. [1] we oberve only the reactive ower flow defined a Im E H. A very imortant iue become evident when one analye the analytical reult of the normalized ME-field helicity, co, hown in Fig. 10. In fact, there are the ditribution of an angle between the otential electric and magnetic field. From Fig. 10 (a) one can ee that for the 1 t MDM, the otential electric and magnetic field in an entire vacuum region are mutually arallel (above a ferrite dik) or mutually anti-arallel (below a ferrite dik). For the nd MDM [ee Fig. 10 (b)], above a ferrite dik there are the region both with mutually arallel and antiarallel electric and magnetic field. Between thee two region, there i an intermediate area of mutual orientation of the field vector. Similar ditribution one ha below a ferrite dik. Now the quetion arie: Whether uch ditribution of the otential electric and magnetic field can be conidered a the field originated from a Tellegen article? Tellegen conidered an aembly of electric-magnetic diole twin, all of them lined u in the ame fahion (either arallel or antiarallel) [34]. Since 1948, when Tellegen uggeted uch "glued air" a tructural element for comoite material, electromagnetic roertie of thee comlex media wa a ubject of eriou theoretical tudie (ee, e.g. Ref. [35 37]). Till now, however, the roblem of creation of the Tellegen medium i a ubject of trong dicuion. The quetion, whether the Tellegen article really exit in electromagnetic, i till oen. The electric olarization i arity-odd and timereveral-even. At the ame time, the magnetization i arity-even and time-reveral-odd [31]. Thee ymmetry relationhi make quetionable an idea that a imle combination of two (electric and magnetic) mall diole can give the local cro-olarization (magnetoelectric) effect. In our cae of the ME field we do not have roertie of the cro (magnetoelectric) olarizabilitie. The ME-couling roertie are originated from magnetization dynamic of MDM ocillation in a quai-d ferrite dik. Thee ocillation are macrocoically coherent quantum tate, which exerience broken mirror ymmetry and alo broken time-reveral ymmetry [1, 13]. The otential electric and magnetic field of the ME-field tructure are the field originated from the toological roertie of magnetization [15]. It i alo worth noting here that the normalized ME-field helicity calculated baed on Eq. (5) and hown in Fig. 10, i different from uch a arameter hown in the numerical reult [38 40]. In the numerical invetigation, one cannot earate the ME field from the external EM radiation. Alo, one cannot earate the otential and curl field. It mean that in the numerical c 4

25 tudie, the normalized ME-field helicity i calculated a E Re E H co E H, where the field and contain both the otential and curl comonent of the external-radiation EM and eigen-ocillation ME field. The analytical and numerical reult on the normalized ME-field helicity are eentially different in vacuum region, where a magnitude of the otential electric field i reduced (the vacuum region ufficiently far from a ferrite dik and the vacuum region which are eriheral with reect to the dik center). H V. CONCLUSION Since MDM ocillation are energetically orthogonal (G mode), one ha the ame oition of the ectral eak (with reect to a ignal frequency at a contant bia magnetic field or with reect to a bia magnetic field at a contant ignal frequency) in different microwave tructure with an embedded quai-d ferrite dik. At the ame time, in uch different tructure there are different numerically oberved toological characteritic of the microwave field. While in numerical invetigation, one cannot earate the ME field from the external EM radiation, the theoretical analyi allow clearly ditinguih the eigen toological tructure of ME field. In the reent aer, we obtained analytically toological characteritic of the ME-field mode. For thi uroe, we ued a method of ucceive aroximation. Baed on the analytical method, we have hown a ure tructure of the electric and magnetic field outide a MDM ferrite dik. We analyzed theoretically the fundamental toological characteritic, which are not oberved in the numerical reult. The analytical tudie of toological roertie of the ME field can be ueful for novel near- and far-field microwave alication. Strongly localized ME field oen unique erective for enitive microwave robing of tructural characteritic of chemical and biological object. The reence of a biological amle with chiral roertie will necearily alter the near-field toology which in turn will change the ectral characteritic of the MDM ferrite dik. Reference [1] L. D. Landau and E. M. Lifhitz, Electrodynamic of Continuou Media, nd ed (Pergamon, Oxford, 1984). [] D. C. Matti, The Theory of Magnetim (Harer & Row Publiher, New York, 1965). [3] A. I. Akhiezer, V. G. Bar'yakhtar, and S. V. Peletminkii, Sin Wave (North-Holland, Amterdam, 1968). [4] A. G. Gurevich and G. A. Melkov, Magnetic Ocillation and Wave (CRC Pre, New York, 1996). [5] L. R. Walker, Phy. Rev. 105, 390 (1957). [6] R. L. White and I. H. Solt, Jr., Phy. Rev. 104, 56 (1956). [7] J. F. Dillon, Jr., J. Al. Phy. 31, 1605, (1960). [8] T. Yukawa and K. Abe, J. Al. Phy. 45, 3146 (1974). [9] E. O. Kamenetkii, A. K. Saha, and I. Awai, Phy. Lett. A 33, 303 (004). [10] E. O. Kamenetkii, Phy. Rev. E 63, (001). [11] E. O. Kamenetkii, M. Sigalov, and R. Shavit, J. Phy.: Conden. Matter 17, 11 (005). [1] E. O. Kamenetkii, J. Phy. A: Math. Theor. 40, 6539 (007). [13] E. O. Kamenetkii, J. Phy.: Conden. Matter, (010). [14] E. O. Kamenetkii, R. Joffe, and R. Shavit, Phy. Rev. A 84, (011). [15] E. O. Kamenetkii, R. Joffe, and R. Shavit, Phy. Rev. E 87, 0301 (013). [16] D. M. Likin, J. Math. Phy. 5, 696 (1964). 5

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