Y.-F.Li and J.W.Y.Lit Vol.4,No.4/April 1987/J.Opt.Soc.Am.A 675 where N is the effective index: E±i(x)=Texp[年(w±/do)(x-ti-] N=B/k. (35) (t:i=5t+-)<x<+m From Egs.(34)the relations can be obtained as follows: -:i=m;-m<x<t-m-以，(41c u42=02-2c品 (36a) where E+o and E-o are given by Eqs.(6). If we denote the time-averaged power in each layer by Pi, w:2=-u2=u2c2-402, (36b) then we obtain for the TE modes from Eqs.(41) where Po=Ro[2+(sin 2+0+sin 2-0)/2uol, (42a) c2=,2-n (37) no-nm For the TE modes,we use the above normalizations to [=-(m-1,.,-1,+1,,+(-1小，(42b) write the eigenvalue Eq.(8)in the form Pi=R±wi(t:i=-:i=m, (42c) 20=φ+0+中-0+9π (q=0,1,2,…, (38) where where the half-phase shifts to and -0are (43a) 中±0=tan- 39a) R=4() T ittanh±i+ R=((2 do [i=-(m-1),.,-1,+1,.,+(0-1小，(43b) (:i=1,2,..,l-2-:=1,2,,m-2), (39b) (+:i=5-:i=m), (43c) Vi= W±(+1) do where T:is given by Egs.(40) W±i (+:i=l-1;-:i=m-1).(39c SOME APPLICATIONS OF MULTILAYER With the aid of WAVEGUIDES T0=E+0, (40a) Multilayer waveguides have been widely used recently in many optical devices,as we mentioned before.We present T±1=E0cosp0: (40b) some new results in this section to show that they have important application potentials in integrated optics. The applications of five-layer symmetrical slab wave- guides in lasers have been discussed by Adams,6 and we know that the five-layer symmetrical slab waveguides offer some advantages as compared with the symmetrical three- -(小a] layer slab waveguides.We show now that if we appropriate- ly add some more layers,there are additional advantages. For a symmetrical structure we can rewrite the eigenvalue (+:i=2,3…,-i=2,3,,m), Eq.(38)as (40c) 40=0+(q/2)π (g=0,1,2,), (44) Eqs.(5)can be rewritten in the form where the half-phase shift o is Eo(x)=Tocos (t-o<x<t+0), o=tan- (45a) (41a) )Tcosh[(wadld )( d cosh := i+tanh-1 do witi tanh (+:i=1,2,,l-1;t+-)<x<t+d (位=1,2，.，m-2)(45b) -:i=1,2,,m-1t-<x<t--, (45c) (41b) d+tanhVol. 4, No. 4/April 1987/J. Opt. Soc. Am. A 675 where N is the effective index: N = 1/k. (35) From Eqs. (34) the relations can be obtained as follows: 2= 2- v2 ci2 Ui 2 = V2 Ci2 , Wi2 =Ui2 = VCi Uo , E+E(x) = Tj exp[LF(w+i/do)(x -t+(i-))] (+: i = I; t+(-)<x X X)< + 41; -: i = m; -X < x < tml)(41c) (36a) where E+o and E-0 are given by Eqs. (6). If we denote the time-averaged power in each layer by Pi, (36b) then we obtain for the TE modes from Eqs. (41) PO = Ro[2 + (sin 2 0+o + sin 2'Pa)/2uo], no 2 -ni2 - Ci 2 2-=2 no2 -n ,2 (37) Pi = Ri {(di/do) + [sinh 2(d Wi - C + sinh 24j/2w,} For the TE modes, we use the above normalizations to [i = -(m - 1,..,-1, +1, .. ., + (I - 1)], (42b) write the eigenvalue Eq. (8) in the form 2uo='0+o+'0Po+qir (q =0,1,2,.... ), (38) where the half-phase shifts 0+0 and 'Po are +0 = tan-' (- tanh iPl (39a) di d= w~j + tanh'1 (i, ) tanh ik(i+,) (+:i = 1,2,...,1-2; -:= 1,2,... ,m- 2), (39b) 4 i d - w~ + tanh'1 (3(i+l) do ~ / (+: i= I-1; -: i= m-1). (39c) With the aid of =To = E+os T+1 = E+o cos O)J.Os T-j = Eo cos 'P0 n cosh (d-k W~k) X [1 - tanh (dk W~k) tanh ¢Plk (+:i=2,3,...,1; -:i=2,3,. where Ro=dao (2 k2wAk 2 / Ri= do Wl (Ll) (2 cosh 2 i) [i = -(m -1) .- I, +1, .. ., + (I -1)], R+j = do ( 11T 2 ) (+: i = ; -:i =m), where Ti is given by Eqs. (40). (43a) (43b) (43c) SOME APPLICATIONS OF MULTILAYER WAVEGUIDES (40a) Multilayer waveguides have been widely used recently in many optical devices, as we mentioned before. We present (40b) some new results in this section to show that they have important application potentials in integrated optics. The applications of five-layer symmetrical slab wave￾guides in lasers have been discussed by Adams, 6 and we know that the five-layer symmetrical slab waveguides offer some advantages as compared with the symmetrical three￾layer slab waveguides. We show now that if we appropriate￾ly add some more layers, there are additional advantages. For a symmetrical structure we can rewrite the eigenvalue nm), Eq. (38) as (40c) Eqs. (5) can be rewritten in the form Eo(x) = To cos [d (x-do) +'P+oJ (t 0o < x < t 0 ), (41a) E i(x) E~~~(x) = T~i cosh[(w~i/do)(x - t±(i-l)) T 4+i] = ~ cosh 4 ,~j (:=12..I1 t+(i)< x< 4t+i; -i1, 2,...m-1; t-i < x< t(i-1)), (41b) uo = 00 + (q/2)7r (q = 0,1, 2 ,.. .), where the half-phase shift 'P is 0' = tan 1 tanh do ( tanh (i = 1,2,. . ., m - 2), (45b) Om-, = dn Wmi + tanh'1 Wm (45c) where (42a) (42c) (44) (45a) Y.-F. Li and J. W. Y. Lit P+i = R±ilw±i (+: i = 1; -: i = M),