YARIV:COUPLED-MODE THEORY 921 power flow of 2W/m.The normalization condition is thus n2 propagation (20) -x-t n3 where the symbolm denotes the mth confined TE mode cor- Fig.3.The basic configuration of a slab dielectric waveguide. responding to mth eigenvalue of (19). Using (17)in (20)we determine finite number of confined TE modes with field components Ey,Hx,and H:,andTM modes with components Hy,Ex,and 71/2 E2.The"radiation"modes of this structure which are not Cn=2hm ⊙ (21) confined to the inner layer are not considered in this paper 18-1+ and will be ignored.The field component Ey of the TE modes,as an example,obeys the wave equation Since the modes 8,(m)are orthogonal we have 6-g0,1=23 (15) 8,8,md= 201.m B (22) J-o We take E(x,z,t)in the form B.TM Modes E(x,z,)=8(x)et-. (16) The field components are The transverse function &(x)is taken as H,(c,2,)=50(xeu-a [C exp (-qx), 0≤x<m E.,,)=84=足5现，x2-m we Oz e C[cos (hx)-(q/h)sin (hx)1, 主0H 8,(x)= -t≤x≤0 E,(x,z,t)=- (23) we dx C[cos (ht)+(q/h)sin (ht)]exp [p(x 1)], -0<x≤-1(17) The transverse function C,(x)is taken as which,applying(15)to regions 1,2,3,yields cos (ht)+sin (hr) p+1) x<一t h=(n22k2-B2)1va 3C,(x)= cos (hx)+sin (hx) 一1<x<0 9=(82-m:k)a p=(82-n32k32 h Ce x>0.(24) kw/c. (18) From the requirement that E,and H be continuous atx=0 The continuity of Hy and Ez at the interfaces requires that and x =-t,we obtain! the various propagation constants obey theeigenvalueequa- tion tan (ht)=- 9十p (19) 1-) tan(h创)=+到 -pq (25) where This equation in conjunction with(18)is used to obtain the eigenvalues B of the confined TE modes. The constant Cappearing in(17)isarbitrary.Wechooseit ng3 D 9 in such a way that the field 8(x)in(17)corresponds to a power flow of I W(per unit width in the y direction)in the The normalization constant C is chosen so that the field mode.A mode whose E=48(x)will thus correspond to a represented by(23)and(24)carries 1 W per unit width in the y direction. The assumed form of E,in (17)is such that 8 and 3C=(i/wu) a8,/ax are continuous atx =0 and that 8,is continuous at x =-f.All that is left is to require continuity of as/ax at x =-t.This leads to (19). A,e*x=是四k=1 2w J-o EYARIV: COUPLED-MODE THEORY 92 1 power flow of 1 A I W/m. The normalization condition is nl thus n2 - propagation n3 x=-t where the symbol m denotes themth confined TE mode cor￾Fig. 3. The basic configuration of a slab dielectric waveguide. responding to mth eigenvalue of (19). Using (1 7) in (20) we determine finite number of confined TE modes with field components E,, H,, and Hz, andTMmodeswithcomponents H,, E,, and E,. The "radiation" modes of this structure which are not cM = 2hm y2. (21) and will be ignored. The field component E, of the TE modes, as an example, obeys the wave equation Since the modes are orthogonal we have confined to the inner layer are not considered in this paper [P., (t + - 11 + --)(hm~ + qmz), 4m Pm We take E,(x, z, t) in the form B. TM Modes Ey(x,z,l) =&y(x)e"wt-flz'. (16) I The field components are The transverse function &,,(x) is taken as H,(x, z, t) = Xy(x)ei(Wt-iBZ) COS (hx) - (q/h) sin (hx)], &"(X) = -t<xIO which, applying (15) to regions 1, 2, 3, yields The continuity of H, and E, at the interfaces requires that From the requirement that and Hz becontinuous at x = thevariouspropagationconstantsobeytheeigenvalueequa￾and x = -t, we obtain' tion tan (ht) = 4+P . (1 9) tan (hi) = htP + 4) h(l - y) ha - (25) where This equation in conjunction with (18) is used to obtain the eigenvalues p of the confined TE modes. 2 2 Theconstant Cappearingin (17) isarbitrary. Wechooseit ji G -sp, n2 n q -4j 4. in such a way that the field &,(x) in (17) corresponds to a n3 nl power flow of 1 W (per unit width in they direction) in the The normalization constant C is chosen so that the field mode. A mode whose EN = A& .(x) will thus correspond to a represented by (23) and (24) carries 1 W per unit width in the y direction. The assumed form of E, in (17) is such that E, and X, = (i/wp) a o,/ax are continuous at x = 0 and that E, is continuous at x = -I. All that is left is to require continuity of aE,/ax at x = -I. This leads to (1 9). 1 HUEx* dx = !.-/rn x,"o dx = 1 2 -m 2u -m E