1636 IEEE TRANSACTIONS ON MAGNETICS, VOL. MAG-18, NO. 6 , NOVEMBER 1982

NUMERICAL SOLUTION O F THE INTEGRAL EQUATIONS FOR MSSW*

F. R . Morqenthaler and T . Bhattacharjee ** t

ABSTRACT

Magnetostatic surface wave (MSSWj propagation on one o r more co-planar rectangular f i lms of f ini te width i s solved numericaily using an integral equation formulation. In the case of uniform in-plane bias, the eigenfrequencies and the associated eigenvectors are obtained by solving the coupled equations which determines the frequency dispersion and the spa t ia l form of MSSW potent ia i s . i n add i t ion t o modes h a v i n g the expected localization of energy on the f i lm sur- f aces , t he r e su l t s confirm the existence of certain modes having primary energy concentration a t t h e f i l m edges.

The formulation i s a l so spec ia l ized to the case of two uniformly magnetized rectangular parallel str ips separated by an a i r g a p and coupled v i a the f r inging f i e l d s of magnetostatic waves. The resul ts should prove useful i n model l ing direct ional couplers for MSSW.

INTRODUCTION

Renewed i n t e r e s t [ l ] , [ 2 ] , [3j i n the use of magnetostatic waves ( P E W ) in signal processing devices derives from the f ac t t ha t ope ra t ion i s poss ib l e d i r e c t l y a t microwave r a the r t han a t RF frequencies. However, s t r ingent cont ro i of the frequency dispersion [2] i s requi red and t h i s f a c t has ied us t o reevaluate the d i spers ion charac te r i s t ics o f th in f i lm fe r r i te wave guides. Recently Piorgenthaler [4] has described an integrai equation formulation of the general bound- ary value problem involving thin f i lms w i t h in-plane b ias f ie lds t ransverse to the d i rec t ion of propagation when b o t h the saturation magnetization and the in-plane b i a s f ie ld a re a l lowed to be nonuniform. The formula- t ion has the advantage that the boundary conditions at the edges of f i n i t e w i d t h s t r i p s a r e r i g o r o u s l y s a t i s - f i e d .

In th i s paper , MSSW propagation on rectangular s t r ip of fe r r i te f i lm i s so lved numer ica l iy us ing the above mentioned approach; in the case of multiple s t r ip s , t hey a r e assumed para l le l a n d coplanar. Special emphasis i s given for those modes where X , the diagonal component of the Polder susceptabili ty tensor , i s p iecewise cont inuous and positive because t h a t i s t h e s u r f a c e wave regime. The e f f e c t of ex- change and anisotropy have been neglected for simplic- i ty .

,THE FORNULATION

The geometry under cons ide ra t ion i s shown i n F i g . l pa r t i cu la r i zed t o two f e r r i t e s t r i p s w i t h s a t u r a t i o n magnetization values IT1 and M2 . Each has the same thickness d , b u t arbi t rary widths w1 and w2 ; they are separated by an a i r gap g . Two per fec t ly con- ducting ground planes are assumed t o be placed a t a dis tance D, ahove and D- below the f e r r i t e strips.

**+Department of Electrical Engineering and Computer Science and Research Laboratory of Elec t ronics , Massachusetts Insti tute of Technology, Cambridge, MA. (+ On leave from Jadavpur University, India).

* Research supported i!: p a r t by the Joint Services Electronics Program (JSEP) and the U.S. Army under cont rac t #DAAG29-81-K-0126.

**The Coupled Str ip Analysis was supported by B a t t e l l e Laboratories under STAS 0033.

+ Dr. T . Bhattacharjee was supported by U.S. Air Force under RADC postdoctoral program.

Using the method, as explained in [3j and [4], the magnetostatic potential in the region /zI < d / 2 can be expanded a s

where, for suppressed exp ( j u t ) var ia t ion ,

Because the material parameters and b i a s f i e lds a r e assumed t o be uniform, i t follows that X i s piecewise constant. Therefore X,(@,x) can e a s i l y be found i n e a c h f e r r i t e s t r i p and in the a i r reg ions sur rounding them. These so lu t ions can then be joined together by requiring t h a t Xa(@,x) and dXN(B,x ) /dx be continuous a t t h e s t r i p edges

The p a i r of coupled integral equations that deter- mine the generally complex values of C l ( a ) and $ (a) a re

exp(j ,x) ~ ~ ( $ , x ) {[+:(;X2+g 2 1 tanh ( J ( U ^ 2 + B - 2 ID?:)

When the x a x i s symmetry permits the modes t o be e i t h e r even or odd, the factor ,exp(jax) in E q . ( 3 ) can be replaced by e i the r cos (ax ) o r s in (6x) and the C 1 , 2 ( ~ ) values can be taken as real . Although the upper l i m i t of t he i n t eg ra t ion i s ac tua l ly L = ~0 , the contr ibut ions due to the f r ing ing f ie lds a re general ly negl igible for values of 2L/w on the order o f 2 or l e s s . Because, we approximate the integration over CY by a f i n i t e sum, that approximation i s more accurate in the range of small x when L i s moderate.

U U M E R I C A L SOLUTION Single S t r ip Case

In the case of a uniform f ie ld a long x , ( l t x ) i s cons tan t wi th in the fe r r i te and the x in tegra t ion has been obtained in closed form [ 3 ] . T h u s , the coupled equations reduce t o s ingle integrals over the spat ia l frequency domain (a). The a- in tegra t ion i n E q . (3 ) i s approximated a s a ~ f i n i t e sum over N sampling points within the range of integration. The points can be chosen either uniformly spaced or nonuniformly d i s t r i b - uted (such asAthat used in Gauss-iegendre quadrature). N values of ~1 must a l so be chosen. The r e s u l t a n t 2N x 2N l inear matrix equation is then solved and the eigen frequencies and the associated eigen vectors are obtained.

The major contr ibut ion to E q . (3 ) occurs for

(F1 w) '' < a2-B2 < (F2 2)' and i1 $ < < F2 where the bandwidth fac tors F1 and F, a re ad jus t - .. 9 .

ed so t h a t the N points chosen sample those values of a and B corresponding t o maximum mode energy. For the.

0018-9464/82/1 IOO-l636$00.75G 1982 IEEE

numerical resu l t s p resented below, F, = F, = 0 and F = Fn = F . Too small a value of F wii 1 exclude values of a impor tan t to the f r ing ing f ie lds a t the s t r ip edges ; too l a rge a value will undervalue the pr inc ip le components and cause h i g h r ipp le in the approximate eigen potentials. I t should be noted that adjustments of the values of N , F and possibly L/W fo r d i f f e ren t po r t ions of the n(6) spectrum i s help- fu l . Natura l ly , l a rge va lues o f N and F are expected to provide better approximations; b u t too l a rge a value of N may actually cause reduced accuracy due t o round-off and t runca t ion e r rors incur red i n solving 2N x 2N matrix equations. Values of N and F t h a t emphasize the energy i n t h e s t r i p r e l a t i v e t o t h a t i n the f r i n g i n g f i e l d may predict the frequency spectrum fa i r ly accu ra t e ly , ye t g ive a d i s to r t ed p i c tu re of the e igen po ten t ia l s . F ina l ly , we remark t h a t tapering of the C l (a i ) and C2(ai) c o e f f i c i e n t s can serve to reduce or eliminate eigen potential overshoot and r ipp le t ha t i s a s soc ia t ed w i th t he S ibbs phenomenon [ 5 ] . I n order to emphasize the different types of width modes, we consider the case of a very thick f i im (d/w = . l ) chosen so a s t o s epa ra t e t he mode frequen- c i e s . I n Fig. 2 , the frequency dispersion of both even and odd modes of comparativeiy law order i s given for D, = m , while in Fig, 3 , the r i e igenpotent ia l s o f representa t ive modes a re p lo t t ed . i n addi t ion t o the expected frequency branches, the dispersion diagram exhibits branches with nearly horizontal or even back- ward wave character. Accurate determination of these and the other higher-order modes requires larger values of N and F . Note tha t in F ig . 3, la rge r ipp les i n the regions Ix j /w > 0.5 can occur due t o t h e f i n i t e values of N and F . These a r e , of course, not present in the actual modes.

2 L

-

t

Fig.

F i g . 2

1637

u///////j/u/////////////I///////// z

j H X l 5 I

1 The geometry under considerat ion.

I-

Frequent!! dispei*sioii of a s ingle strip with representat ive modes l a b e l l e d a s a , b , c , d , e & f . Even and odd modes are respectivc-ly denoted by + and 0 . (Notice the appearance of the back- \ward wave type modes in the spectrum, the potential of one such i s sh~~.e!n 1::. Fig. 3 ( d ) . )

Double Strip Case

Sirniiar anaiysis can be car r ied ou t i n the case of two s t r i p s . However, we choose t o deveiop an approxi- mate analysis, based upor: coupiing o f modes, thar: i s .va!.id when the fringing f ields that provide the couplinc. between the two s t r i p s a r e r e l a t i v e l y weak.

The linearized eqv$tions of motion for the rf m a - net iza t ion i n each O f , t h e s t r i p s , a s i n F i g . l , a r e .?.: the form

-

where, the subscripts 1 and 2 lahe l the par t icu lar s t r i p . For simplicity magnetic anisotropy i s e i t h e r ignored or assumed to ac t pa ra l l e l t o t he dc magnetic f i e l d s H 1 , 2 = i x H x ; i n e i t h e r c a s e , t h e s t a t i c rnaqnetization vectors are aligned with R ; so t n a t /Mi are the respect ive saturat ion magnet izat ions and M12Hx1 ,2 > 0 .

1 3 2

Fig. 3 MSSW eigenpotent ia ls of the Fig. 2 modes labeled a , b , c , d , e , & f .

i t i s convenient to expand h 1 , 2 a s , -.

- h l = ii +F and E2 = F21+F22 where, h i i s t h e t o t a l rf magnet ic f ie ld act ing upon s t r i p F1 with nil t h a t p a r t due t o mi and Ki2 t h a t due t o z2 . i f the cross-coupling between t h e s t r i p s i s r e l a t ive ly sma l l , i t i s reasonable to assume t h a t t h e forms of El andm2 a r e unchanged from those calculated from the uncoupled equations .

- 11 12 -

The nex t s t ep i s t he re fo re , t o neg lec t F12 a n d

h21 and solve for the two uncoupled surface modes. For d/w<<l and neglecting w i d t h var ia t ions , the approx- imate complex magnetostatic potentials above, within and below f e r r i t e strip # I , a r e r e spec t ive ly

-

1638

z 2 a (5; As i s well k n o w n , the frequency disperzion ti:2t r e s u l t s f:-om applying continufty of ji a n d m + h Z a t b o t h ,

z = 0 and z = d i s given by

Ll = (ax1 + 1 / 2 W K l ) - 4 UM1 01 ( 6 ) 2 1 2 , - 2 1 0 / d

I

Simi la r r e su l t s ho ld . fo r s t r i p $2. I t follows approximatel)/ that E q . (4)must be express ib le as

+ ( L l - ~ o l ) m l = -wM1 hl;

Then, the final t a sk wi 11 be to express h2f i n terms o f m l and ( h l i in terms o f m2). Since the fringing potential provides the coupling, we need to develop a reasondble estimate for jiZ1 , the potentia3 within s t r i p #2 due t o the mode i n s t r i p # l . The form o f the Laplacian fringing potential ifl present when s t r ip f2 i s removed, can be expressed as

+ +

r+m

Since, I J , ~ ~ , the potential above, within and beiow s t r i p # l due t o mode # l , i s rea.sonably well-approximat- ed by E q . ( 5 ) ever near the s t r ip edge x = 0 , we f ind the coe f f i c i en t s C ( p ) by matching the potentials along the plane x = 0 . This i s done by taking the Fourier transform of E q . ( 8 ) . The component of h f l

which has the form of the mode 82 potent ia l i s defined by $2, where hz l = c~~ 12 ( u 2 k i n q the potent ia l w i th in t he s t r i p ) . Tne coef f ic ien t CZl can be found

by minimizing the means square error a n d , f o r ! ~ l ~ : ~ > > l i s given by approximately

b The coupl ing coef f ic ien ts can be defined as

+ C Z 1 4 - h 2 1 C12/Al = h1+2

%1 = __ + - ~ + + x1 ml x 2 m2

+ ; K12 = - (10)

Equations ( 7a,b ) have nontr ivial solut ions when

( " - m o l ) ( w - w o 2 ) = wMl WM2 K12 K21

When the two strips are identical , , the coupling i s maximally e f f i c i e n t , w = o +W I K - . 1 and power t r a n s f e r between t h e s t r i p s i s p o s s i b l e because the composite normal modes (even and odd functions of x with respect to the midplane between the two s t r i p s ) nave, f o r the same frequency u , s l i g h t l y d i f f e r e n t values of B . The propagation distance for coropiete power t r a n s f e r between l i n e s i s g i v e n by = n / ~ e . blhen fid i s l a r g e enough, so that appreciable exponen- t i a l va r i a t ion o f themode po ten t i a l s i s ev iden t w i th in the s t r ips , the formulas for K 1 2 and KZ1 simnlify because $f can be approximated by an exp(-oi.) Varia-

t i o n . The r e s u l t of t he ana lys i s fo r l a d l > 1 i s aiven as

0 M I2

In Fig. 4, we snow plo ts o f / 3 i w i ~ ~ ~ i vs . ~d for d i f fe ren t va lues 3f t! 2 g / d r a t i o .

.- 9 3

__ REFERENCES

i l l J . H . Col l ins and J . ivi. Owens, "iqagnetostatic Wave a n d SAH Devices - Similar i t ies , Differences a n d Trade o f f s , " 1978 IEEE Internat ional Symposium o n Ci rcu i t s and Systems Proceedings ( I E E E , New York, i978) p . 536.

[2] P,A.DC/EEA Microwave Magnetics Technoiogy Workshop Proceedings, Bedford, MA, June 10-11, i981.

[3] "Magnetostatic and Magnetoelastic Phenomena a n d Devices," - Session in 27th Annual Conference on plagnetism and Magnetic Mater ia ls , At lanta , GA, November 10-1 3, 1981 .

[4] F . R . Morgenthaler, "Magnetostatic Surface Modes in Nonuniform Thin Fiims w i t n In-plane Bias F ie lds ," J . Appl Physics, Vol. 52, No. 3 , Part 11, p p . 2267-2265, 1981.

[5] E . A . Guillemin, "The Mathematics of Ci rcu i t - Analysis," John Wiley & Sons, Inc., New York, pp. 495-535.