672 J.Opt.Soc.Am.A/Vol.4,No.4/April 1987 Y.-F.Li and J.W.Y.Lit n. or ne-p -x=t+1-1) X=t+(12 E±0=(-1)9E-0, (6b) -X=t+1 h2=h2n02-82, n+1 (7a) -x=t+o =do ---X=0 p2=82-h2n,2亿=-m,,-1,+1,..,+0,(7b) n。 -x=t-0=-do where k is the free-space wave number and g is the mode n1 -X=t-1 order shown in the eigenvalue equation,which can be writ- ten as n.m-p X=t-(m2】 x=t-(m-1) 2h0d0=中+0+中-0+9T(q=0,1,2,…), (8) n-m where the half-phase shifts +oand -o are Fig.1.Geometry of the structure of an L-layer waveguide,where L =l+m+1. ±o=tanl Ptanh± (9a) ho z dependence of a mode is given by the function exp(-jBz), i=P±di+tanh- where B is the propagation constant. /Pit业tanh±+1) P±i Guided Transverse-Electric Modes (+:i=1,2,.,l-2:-:i=1,2,..,m-2),(9b) TE modes have only three field components:Ey,H,and H.. yi=p±idti+tanh P±i+I By solving Maxwell's equations in each layer we can get the solutions,which must satisfy the boundary conditions at each interface.Making use of the fact that the fields of (+:i=l-1;-:i=m-1).(9c) guided modes must vanish atx=,we can get the electric fields Ey in the various layers as follows: Eo in Eq.(6a)is the amplitude of the field in the layer of no. As expected,Egs.(8)and(9)are independent of the choice of the x coordinate. 1.For the layer i=0,where-do=t-o<x<t+o=do, The magnetic fields are given by Eo(x)=E+0 cos[ho(x-do)++ol H,=-8E (10a) E-o cos[ho(X do)-ol. (5a) μ 2.For the layers i=±l, H,=上迟 (10b) cosh[p±i(x-t±o)年±i] wu dx E+1(x)=E±0cos中0 cosh±l Guided Transverse-Magnetic Modes (+:t+o<x<t+i-:t-1<x<t-o.(5b) TM modes have only three field components:Hy,Ex,and E2. Proceeding as outlined above for TE modes,we obtain the 3.For the general layers i=2,3,..., magnetic fields Hy in the various layers as follows: =1 E±：(x)=E±0cos中±0 cosh(p±hd±h-±k) 1.For the layer i=0,where-do=t-o<x<t+o=do, cosh±h Ho(x)=H+o cos[ho(x-do)+0] xcoshp±ix-±i-)年4d cosh±i =H-o cos[ho(x do)--0]. (11a) (t:i=2,3,,l-1;t+-w<x<t+8 2.For the layers i=±l, -:i=2,3,,m-1;ti<x<tt-.(5c) H±(x)=H±0cos中±0 cosh[p±i(x-t0)干±] cosh±i' 4.For the outermost layers i=lor m, (+:t+o<x<t+1-:t-1<x<t-o.(11b) E±：(x)=E0cos中±0 cosh(p±d±h-'±h) 3.For the general layers i=2,3,..., cosh±h H生i(x)=H±0cos中0 cosh(p±d±k-±h) Xexp干p±(x-t±i-] 1 cosh±k (+:i=k,t+-)<x<+o;-:i=m,-o<x<tm-以 xcosh[p±ix-t生-i年] (5d) cosh±i' In the above equations (+:i=2,3,.,1-1;t+i-)<x<t+d E0=Eo exp±jhod。-p】 (6a) -:i=2,3,,m-1;t-<x<t-e-)以.(11c672 J. Opt. Soc. Am. A/Vol. 4, No. 4/April 1987 n., x Y -X= -X= t+I t+( 1-2) -1) -X= t+i -X= tto =do nF1 _ ______ -__ -- X=O no _--- n5. -X= t-1 -X= t-Cm-2) -X= t-CM-1 n-.m-,) Fig. 1. Geometry of the structure of an L-layer waveguide, where L = I + m + 1. z dependence of a mode is given by the function exp(-jfz), where 1 is the propagation constant. Guided Transverse-Electric Modes TE modes have only three field components: Ey, H., and H,. By solving Maxwell's equations in each layer we can get the solutions, which must satisfy the boundary conditions at each interface. Making use of the fact that the fields of guided modes must vanish at x = d -, we can get the electric fields E, in the various layers as follows: 1. For the layer i = 0, where -do = t- 0 < x < t+o = do, Eo(x) = E+o cos[ho(x - do) + 0+o] = E-0 cos[ho(X + do) - '-o]. 2. For the layers i = +1, E~l(x) = E+o cos 0+0 cosh[pl,(x - t±) T ip] cosh 4'i (+: t+o < x < t+j; -: t_1< x < t0). (5b) 3. For the general layers i = 2, 3, . . . i1cosh(pkd~ -k ik E~i(x) = E~o cos (ko0 cosh ipidki- : X cosh[pj(x - t±(il)) T 14i] cosh 4'i (:i = 2, 3_ . .., I-1; t+(i-1) < x < t+j; -:i = 2, 3, . . . , m -1; t-i < X < t_-i-l)). (5c) 4. For the outermost layers i = I or m, Ejx(x) = E~o cos t4-o o sh(Pbd k)] coshi14kh X exp[:Fp~i(x - t±(j-l))] (+ ,t+(l_l) < X < +<;-i=m, -- < x < t(~) (5d) In the above equations (5a) or Eio= (-2)qE_0, ho 2 = k2no 2 - i2, (6b) (7a) pi2 = _2 - 2n2 ( =-m,...,-I, +I,..., +l), (7b) where k is the free-space wave number and q is the mode order shown in the eigenvalue equation, which can be writ￾ten as 2hodo = 0+0 + 0-° + qr (q = 0,1, 2,...), where the half-phase, shifts 0+o and 0-0 are o tan 1 (Ph tanh V1j= piidi + tanh. ( p(i+1) tanh ,(i+l) (+:i=,2,...,I-P2; -:i=1,2,...,m-2), i= Psidi + tanh-1 (P(i+l) (+:i=I-1; -:i=m-1). (8) (9a) (9b) (9c) Eo in Eq. (6a) is the amplitude of the field in the layer of no. As expected, Eqs. (8) and (9) are independent of the choice of the x coordinate. The magnetic fields are given by Hx =--Ey (lOa) cQLL i aEy H2 = - co," ax (lOb) Guided Transverse-Magnetic Modes TM modes have only three field components: Hy, Ex, and E,. Proceeding as outlined above for TE modes, we obtain the magnetic fields Hy in the various layers as follows: 1. For the layer i = 0, where -do = t. 0 < x < t+o = do, Ho(x) = H+o cos[ho(x - do) + 0+0'] = H-0 cos[ho(x + do)- 0-'] 2. For the layers i = 1, H± (x) = H±o cosp0o (Ila) cosh[p, 1 (x -t±) j T i1I] cosh VI,,' (+: t+o < x < t4l; -: t-1 < x < t-0). (llb) 3. For the general layers i = 2, 3, . . ., H~i(x) = H+o cos 0+o' chp cosh - 1 cosh[pi(x - t±(i-)) =F lii] cosh 4,j' (+: i = 2, 3_ .. ., I - 1; t+(i-,) < x < t+j; E~o = Eo -: i = 2, 3_ .., m -1; t-i < X < t_(U-l)). (llc) exp[+j(hodo - qo)]6 Y.-F. Li and J. W. Y. Lit (6a)