Y.-F.Li and J.W.Y.Lit Vol.4,No.4/April 1987/J.Opt.Soc.Am.A 671 General formulas for the guiding properties of a multilayer slab waveguide Yi-Fan Li Guelph-Waterloo Program for Graduate Work in Physics,University of Waterloo,Waterloo,Ontario N2L 3G1. Canada John W.Y.Lit Department of Physics and Computing.Wilfrid Laurier University,Waterloo,Ontario N2L3C5,Canada Received May 26,1986:accepted November 24,1986 General formulas describing field distributions and eigenvalue equations are obtained for both transverse-electric and transverse magnetic modes in a multilayer slab waveguide.New results show that additional multilayers can produce useful effects,such as increasing the cutoff values and the confinement factors of guided modes. INTRODUCTION THEORY Multilayer structured waveguides have been widely used The general structure of an L-layer medium is shown in Fig. recently in many optical devices,such as modulators,switch- 1,where L=l+m 1.ni and di are,respectively,the es,directional couplers,Bragg deflectors,spectrum analyz- refractive index and the thickness of the layer i.We have ers,and semiconductor lasers. chosen no to be the highest refractive index for convenience, A three-layer slab waveguide is the simplest optical wave- but this is not a restriction: guide that has been well studied and documented.1-5 Wave- (1) guides with more than three layers have been studied by no>ni (i=-m,,-1,+1,…,+0. many authors.6-19 The eigenvalue equations for the four- With the choice of the coordinate system in Fig.1,we have layer structure have been derived by the wave theory and the t0=±do, (2a) ray theory.7-11 The five-layer symmetrical guide with aniso- tropic dielectric permittivity has been considered by Nelson and McKenna.12 A special structure of a five-layer wave- d guide,the so-called W waveguide,has interesting properties with respect to mode cutoffs and confinement factors.13 (t:i=1,,..,1-1;-:i=1,..,m-1),(2b) Ruschin and Marom14 have obtained the explicit eigenvalue equations of the symmetrical seven-layer waveguide for both where t+is the x coordinate of the interface between the even and odd modes by using matrix treatment.Multilayer layers +i and +(i+1)above the interfacex =0,and t-iis the waveguides with periodic index distributions have also been x coordinate of the interface between the layers -i and-(i+ studied.15-17 An explicit eigenvalue equation of a periodic 1)below the interface x=0. stratified waveguide has been obtained by Yeh et al.17 By In order to obtain a complete description of the modes of using the matrix method,Walpita and Revelli have studied multilayer waveguides,we begin with Maxwell's equations: the general N-layer waveguide,but their results involved complex matrices.18,19 X E=-udH/ot, (3) In this paper we derive the explicit formulas for the field 7×H=en;2aE/at (i=-m,,-1,0,+1,,+0. distributions and the eigenvalue equations for both trans- (4) verse-electric (TE)and transverse-magnetic (TM)modes in a general multilayer slab waveguide,starting with Maxwell's e and u are,respectively,the dielectric permittivity and the equations.The results are compared with those obtained magnetic permeability of vacuum.We do not consider mag- by some other authors,and some applications of the formu- netic materials in this paper,so the use of the vacuum value las are also considered. u is sufficient. A one-dimensional analysis is presented here.However, We simplify the description of the waveguide by assuming it may be applied to the more general case of two-dimension- that there is no variation in the y direction,which means al guides by using the effective-index approximation20.21 to chat d/dy =0.The time dependence of the field is harmonic, separate a two-dimensional problem into two one-dimen- expressed as exp(jwt).Since we are interested in obtaining sional cases. the normal modes of the waveguide,we assume also that the 0740-3232/87/040671-07$02.00 @1987 Optical Society of America
Vol. 4, No. 4/April 1987/J. Opt. Soc. Am. A 671 General formulas for the guiding properties of a multilayer slab waveguide Yi-Fan Li Guelph-Waterloo Program for Graduate Work in Physics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada John W. Y. Lit Department of Physics and Computing, Wilfrid Laurier University, Waterloo, Ontario N2L 3C5, Canada Received May 26, 1986; accepted November 24, 1986 General formulas describing field distributions and eigenvalue equations are obtained for both transverse-electric and transverse magnetic modes in a multilayer slab waveguide. New results show that additional multilayers can produce useful effects, such as increasing the cutoff values and the confinement factors of guided modes. INTRODUCTION Multilayer structured waveguides have been widely used recently in many optical devices, such as modulators, switches, directional couplers, Bragg deflectors, spectrum analyzers, and semiconductor lasers. A three-layer slab waveguide is the simplest optical waveguide that has been well studied and documented.'- 5 Waveguides with more than three layers have been studied by many authors. 6-19 The eigenvalue equations for the fourlayer structure have been derived by the wave theory and the ray theory. 7 -"1 The five-layer symmetrical guide with anisotropic dielectric permittivity has been considered by Nelson and McKenna.12 A special structure of a five-layer waveguide, the so-called W waveguide, has interesting properties with respect to mode cutoffs and confinement factors.'3 Ruschin and Marom14 have obtained the explicit eigenvalue equations of the symmetrical seven-layer waveguide for both even and odd modes by using matrix treatment. Multilayer waveguides with periodic index distributions have also been studied.' 5 - 17 An explicit eigenvalue equation of a periodic stratified waveguide has been obtained by Yeh et al.17 By using the matrix method, Walpita and Revelli have studied the general N-layer waveguide, but their results involved complex matrices. 18,19 In this paper we derive the explicit formulas for the field distributions and the eigenvalue equations for both transverse-electric (TE) and transverse-magnetic (TM) modes in a general multilayer slab waveguide, starting with Maxwell's equations. The results are compared with those obtained by some other authors, and some applications of the formulas are also considered. A one-dimensional analysis is presented here. However, it may be applied to the more general case of two-dimensional guides by using the effective-index approximation 2 0,21 to separate a two-dimensional problem into two one-dimensional cases. THEORY The general structure of an L-layer medium is shown in Fig. 1, where L = 1 + m + 1. ni and di are, respectively, the refractive index and the thickness of the layer i. We have chosen no to be the highest refractive index for convenience, but this is not a restriction: (1) With the choice of the coordinate system in Fig. 1, we have tlo = +do, (2a) t~i = (do + E d ) k=l (+ =1 . .., I - 1; -: i = J, . ,m - J), (2b) where t+j is the x coordinate of the interface between the layers +i and +(i + 1) above the interface x = 0, and t-i is the x coordinate of the interface between the layers -i and -(i + 1) below the interface x = 0. In order to obtain a complete description of the modes of multilayer waveguides, we begin with Maxwell's equations: V X E = -yH/at, (3) V X H =,Eni 2Ma/t (i =-,...-1, 0, +1, .. ., +1). (4) E and At are, respectively, the dielectric permittivity and the magnetic permeability of vacuum. We do not consider magnetic materials in this paper, so the use of the vacuum value ut is sufficient. We simplify the description of the waveguide by assuming that there is no variation in the y direction, which means that alay = 0. The time dependence of the field is harmonic, expressed as exp(jcot). Since we are interested in obtaining the normal modes of the waveguide, we assume also that the 0740-3232/87/040671-07$02.00 (© 1987 Optical Society of America Y.-F. Li and J. W. Y. Lit no > ni (i = -M,..., -1, + 1, ... , + 1)
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.(11c
672 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 written 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)
Y.-F.Li and J.W.Y.Lit Vol.4,No.4/April 1987/J.Opt.Soc.Am.A.673 4.For the outermost layers i=lor m, SOME SPECIAL CASES OF MULTILAYER WAVEGUIDES cosh(p±kd±k-±k) H(x)=H0cos cosh±k Three-Layer Waveguide Using Egs.(5)with l=m 1 (n+for the superstrate and n-1 for the substrate),we can obtain immediately the electric fields Xexp[干P±(x-t±- E for TE modes in a three-layered waveguide: (+:i=kt+-)<x<+o-:i=m; -o<x<t-(m-1) E-ocos(hod0-中-dexp-p1(x-t+o】(t+o<x<m), (11d) (16a) In the above equations E-o cos[ho(x-t_o)-ol (t_o<x<t+o),(16b) (12a) E0cos中-oexp[p-1(x-t-0】(-o<x<t-小.(16c) Ho=同oexp(h,dg-o] The eigenvalue equation is given by Eq.(8),and ho,p+,and p-1 are given by Eqs.(7).The half-phase shifts are or H+0=(-1)9H-0 (12b) p±o=tan-l(p±i/ho). (17) Similarly,one can get the magnetic fields H for TM modes: The factor 8/B is incorporated into the field amplitude to ensure that the transverse field changes its sign when the H-0cos(hodo-中-0)exp[-p(x-t】(t+0<x<m), propagation direction is reversed,and Ho in Eq.(12a)is the (18a) field amplitude in the layer of no. H-ocos[ho(x-t-o)-0](t_o<x<t+o),(18b) The eigenvalue equation can be written as H-ocos中_oexplp-1(x-t_-】(-m<x<t-oJ.(18C) 2hod0=Φ+0'+中-0+qπ(g=0,1,2,…). (13) The eigenvalue equation is given by Eq.(8).The half-phase The half-phase shifts o'and -0'are shifts are P±1 0=tan (19) o±o'=tan (14a) 2 ho n±i2ho These results are already well known. ±i'=p±dti+tanh-l n±2 P±i+tanh+ Four-Layer Waveguide Using Eqs.(5)with l=2,m =1 (n2 for the superstrate and n-1 (+:i=1,2,,l-2-:i=1,2,,m-2), (14b) for the substrate),we can get for a four-layer waveguide the electric fields E.: ±i=p±d±i+tanh-l n2P±i+D n生i+)2P生 2 cosh(p+id+1-+) E-0cos(2hod0-中-0) xp[-p+2(x-t+1)] cosh+ (+:i=1-1;-:i=m-1),(14c) (t+1<x<o),(20a) where ho and p;are given by Eqs.(7). We can obtain the electric fields as Eo cos(2hod)coshp cosh+ E=AH。 (15a) (t+0≤x<t+),(20b) niwe E-o cos[ho(x-to)-ol (t-0<x<t+),(20c) Ex=、jH E-ocos-oexp[p-1(x-t_o)](-o<x<t_o). nwe ox (20d) (位=-m,,-1,0,+1,,+0.(15b) The eigenvalue equation is still given by Eq.(8),and ho,p+, p+2,and p-1are given by Eqs.(7).The half-phase shiftso It is interesting that the influence of any layer on the eigenval- andΦ-oare ue equation is through the parameter For example,in the TE modes the effect produced when the lth layer is added to -0=tan(p-1/ho), (21a) the structure is,as Egs.(9)show,to add the parameter(1) +o=tan [(p+/ho)tanh+], (21b) at the interface between n+(-1)and n+.The same applies to the TM modes.The eigenvalue equations of the structure are +1=p+1d+1+tanh-l(p+2/p+i). (21c) formally the same even when the structure is changed.How- ever,the half-phase shifts (+o and -0 in the TE mode, These results are the same as those obtained by Wang22 and and in the TM mode)do depend on the structure. Smith.11
Vol. 4, No. 4/April 1987/J. Opt. Soc. Am. A 673 4. For the outermost layers i = 1 or m, H i(x) = Hio cos 514o' [I| ch d - ' 1 +i 11 ~~cosh ~'P J X exp[=Fpi(x -t(i-l))] (+: i = 1; t+(1-1) < X < +cx) -: i = m; -W < x < t_(Mld)). (l1d) In the above equations H~o = 0 Ho exp[+Ej(hodo -Wo')] (12a) SOME SPECIAL CASES OF MULTILAYER WAVEGUIDES Three-Layer Waveguide Using Eqs. (5) with 1 = m = 1 (n+l for the superstrate and n-1 for the substrate), we can obtain immediately the electric fields E, for TE modes in a three-layered waveguide: E-0 cos(hodo - 0_o)exp[-pj(x - t+0)] (t+o < x < -), (16a) E-0cos[ho(x - t 0) - w E-0 cos '- 0 exp[p-,(x - t_0)] (-- < x < t- 0). (16c) The eigenvalue equation is given by Eq. (8), and ho, p+,, and p-1 are given by Eqs. (7). The half-phase shifts are 0,0 = tan-'(p±1 /ho). The factor 3/1013 is incorporated into the field amplitude to ensure that the transverse field changes its sign when the propagation direction is reversed, and Ho in Eq. (12a) is the field amplitude in the layer of no. The eigenvalue equation can be written as 2hodo = 0 +0' + 0-° + qr (q = 0,1, 2,.. .). The half-phase shifts k+o' and 'Po' are 0± = tan-1 (Z h. tanh Awl' , (13) Similarly, one can get the magnetic fields H, for TM modes: H-0 cos(hodo - 0_o')exp[-pj(x - t+0)] (t+o < x < c), (18a) H- 0 cos[ho(x - t 0 ) - 'P-o'] (t-o < x < t+0), (18b) H-0 cos 0-o' exp[p-,(x - t-0 )] (-a) < x <'t_o). (18c) The eigenvalue equation is given by Eq. (8). The half-phase shifts are (19) (14a) Tnon . ) These results are already well known.4 n i 2 ifI+l tanh A(i+,)) n(i+l) PPi (+:i= 1,2,...,1-2; -:i= 1,2, ... ,m-2), (14b) {ii = p~id~i + tanh'l Four-Layer Waveguide Using Eqs. (5) with 1 = 2, m = 1 (n2 for the superstrate and n-1 for the substrate), we can get for a four-layer waveguide the electric fields E,: E-0 cos(2hodo - 0) ni+ P(i+l) / n±(i+,) 2 p±i (+: i= I-1; -: i= m-1), cosh(p+ld+l - + exp[-P+2(X - cosh V+ 1j (t41 < x < a), (20a) (14c) where ho and pi are given by Eqs. (7). We can obtain the electric fields as E-0 cos(2hodo - 0-0) cosh[p+,(x - t+o ) -+1 cosh V'+, (t+o <, x < tjl, (20b) (15a) E. 0 cos[ho(x - to) - P-0] (t.o < x < t+o), (20c) E-0 cos 0-0 exp[p_ 1 (x - t 0 )] (-a) <X <t 0). (20d) (i = -m, .. ., -1, 0, +1,..., +1). (15b) It is interesting that the influence of any layer on the eigenvalue equation is through the parameter 4i. For example, in the TE modes the effect produced when the 1th layer is added to the structure is, as Eqs. (9) show, to add the parameter I+(1_1) at the interface between n+(1-l) and n+I. The same applies to the TM modes. The eigenvalue equations of the structure are formally the same even when the structure is changed. However, the half-phase shifts (0+0 and 0-0 in the TE mode, 4P+o' and 0-o' in the TM mode) do depend on the structure. The eigenvalue equation is still given by Eq. (8), and ho, p+', P+2, and p-' are given by Eqs. (7). The half-phase shifts o+o and '0- are 0-o = tan-'(p-,/ho), 0+0 = tan-'[(p+,/ho) tanh '+ 1], +1= p+ld+l + tanh-'(p+ 2 /p+1). (21a) (21b) (21c) These results are the same as those obtained by Wang22 and Smith.'" or (12b) (17) Ai = p~idsi + tanh-' Ei i= d Hi, ni WE a xHi E2i = - iz ax Y.-F. Li and J. W. Y. Lit (t-o < x < t+o), (16b) H+o = (-1)"H-0
674. J.Opt.Soc.Am.A/Vol.4,No.4/April 1987 Y.-F.Liand J.W.Y.Lit n3 tan(hodo) (for even modes) (27a) 4 -cot(hodo) (for odd modes) n2 (27b) d2 If we set n d no nor n2>B>ni ng (28) 2do and n1 p:=l82-h2n,22 (6=1,2,3), (29) n2 then we have n3 P1=P1, Fig.2.Geometry of a symmetrical seven-layer waveguide D2=jp2, (30) The above verifications show that,for the cases of three P3=p3 and four layers,our general formulas give results that agree with the published results. Equation(26)becomes Symmetrical Seven-Layer Waveguide ho A1+ P3 tanh(pid)+ tanh(P2d2) Ruschin and Marom14obtained the eigenvalue equations for a symmetrical seven-layer waveguide by using matrix treat- ment.In this section we shall get the eigenvalue equations P3+tanh(pidi) P2 P3 tanh(pdi) tanh(p2d2).(31) for the same structure by using our formulas and then com- P2 pare our results with theirs. The symmetrical seven-layer waveguide is shown in Fig.2. By using our parameters and performing lengthy algebraic We have dropped all the pluses and minuses from the sub- manipulations,it can be shown that our eigenvalue equation scripts because the structure is symmetrical.Layers 0 and 2 (31)will lead to the same solution for both the odd and even are the guiding layers with high refractive indices.n is the modes given by Eqs.(21a)and(21b)in the paper by Ruschin index of refraction of the clad,and n3 is that of the surround- and Marom.14 Here we wish to point out that in that paper ing medium there is a typographical error,namely,the minus between the For a symmetrical waveguide,the eigenvalue Eq.(8)can two main terms in Eq.(21a)should be a plus.23 be written as We can check the eigenvalue equations quickly by reducing them to those for the symmetrical five-or three-layer wave- 2hodo=200+q (g=0,1,.), (22) guides. that is, Symmetrical Five-Layer Waveguide tan(hodo)=tan o (23) We put d2=0,so Eq.(31)yields ho for the even modes and A p 1+P3 tanh() P3 tanh(p d). (32) -cot(hodo)=tanφpo (24) Symmetrical Three-Layer Waveguide for the odd modes. The half-phase shift o is given by We put di=d2=0,so Eq.(31)yields tan o=(pi/ho)tanh vi, (25a) A=p3/ho (33) tanh(p:d)+(pi+/p)tanhit which is a well-known result. tanh(pptanh(p:d)tanh (i=1,2,.,l-2), (25b) FORMULAS IN TERMS OF THE NORMALIZED VARIABLES tanh(pi-1d1)+(pilpi-1) tanh1+(p/p-tanh(p-d-1) (25c) To obtain the properties of a multilayer slab waveguide,we have to evaluate the eigenvalue Eq.(8)for TE modes and Eq. By using Eqs.(23)-(25),we can get the two eigenvalue equa- (13)for TM modes to get the propagation constant B.To tions for the symmetrical seven-layer waveguide: make the results of such a numerical evaluation more broadly applicable,we introduce a normalized thickness v and other tanh(pd) normalized parameters: u2=h2d2n02-n-nm2), (34a) +++ae]时 u2=h2d02(n,2-N. (34b) tanh(p2d2),(26) w:2=h2d2W2-n,3)(i=-m,,-1,0,+1,.,+0, where (34c)
674. J. Opt. Soc. Am. A/Vol. 4, No. 4/April 1987 n3 T n, d2 fli~~~~~~~~~~~~~~~~~~ ni di no 2do ni n2 n3 Fig. 2. Geometry of a symmetrical seven-layer waveguide. The above verifications show that, for the cases of three and four layers, our general formulas give results that agree with the published results. Symmetrical Seven-Layer Waveguide Ruschin and Marom14 obtained the eigenvalue equations for a symmetrical seven-layer waveguide by using matrix treatment. In this section we shall get the eigenvalue equations for the same structure by using our formulas and then compare our results with theirs. The symmetrical seven-layer waveguide is shown in Fig. 2. We have dropped all the pluses and minuses from the subscripts because the structure is symmetrical. Layers 0 and 2 are the guiding layers with high refractive indices. nj is the index of refraction of the clad, and n3 is that of the surrounding medium. For a symmetrical waveguide, the eigenvalue Eq. (8) can be written as 2hod0 2='20 +q-zr (q =0,1I....), that is, tan(hodo) = tan 00 for the even modes and -cot(hodo) = tan 'o for the odd modes. The half-phase shift 0' is given by (22) (23) (24) A = an(hodo) -cot(hodo) (for even modes) (for odd modes) (27a) (27b) If we set no n2 > 1 > n,, n3 and Pi= I13-k 2 ni2 1 2 (i = 1, 2, 3), then we have Pi = Pi, P2 = JP2, (28) (29) (30) P3 = P3- Equation (26) becomes °A 11+ 3 tanh(pd,) +JP - tanh(pld,) tanh(p 2A2)r Pi Pi L P J j = + tanh(pldj) - [r = tanh(pidj) tanh(P 2d2 ). (31) Pi ~ ~ L'P2J By using our parameters and performing lengthy algebraic manipulations, it can be shown that our eigenvalue equation (31) will lead to the same solution for both the odd and even modes given by Eqs. (21a) and (21b) in the paper by Ruschin and Marom.14 Here we wish to point out that in that paper there is a typographical error, namely, the minus between the two main terms in Eq. (21a) should be a plus.23 We can check the eigenvalue equations quickly by reducing them to those for the symmetrical five- or three-layer waveguides. Symmetrical Five-Layer Waveguide We put d2 = 0, so Eq. (31) yields ° A [1 + 3 tanh(pldj) = 3 + tanh(pldl). (32) Symmetrical Three-Layer Waveguide We put d1 = d2 = 0, so Eq. (31) yields tan 'P = (p,/ho)tanh 4s,, tanh(pid;) + (Dp:_/p:)tanh tib:- (25a) tanh - = r ti *1 - I " 1-1 i1 + (pi+,/pj)tanh(pjdj)tanh 4i+i (i = 1, 2,...,1 - 2), (25b) tanh(pj_1dj_1 ) + (pl/IP-,) 1 + (pj/p1_I)tanh(p1.. dj..) By using Eqs. (23)-(25), we can get the two eigenvalue equations for the symmetrical seven-layer waveguide: °A 1+ P3 tanh(pdj) + [-+ p2tanh(pldl) utanh(p2A2)| Pi 1 Pi L2 PI J = Pp + tanh(pldl) + +-tanh(pdl)J p2 tanh(pAd2 ), (26) where A = P3/hoX (33) which is a well-known result. FORMULAS IN TERMS OF THE NORMALIZED VARIABLES )To obtain the properties of a multilayer slab waveguide, we have to evaluate the eigenvalue Eq. (8) for TE modes and Eq. (13) for TM modes to get the propagation constant 13. To make the results of such a numerical evaluation more broadly applicable, we introduce a normalized thickness v and other normalized parameters: V2= k 2 d0 2 (n 0 2 -n 2), Ui2 = k 2 d0 2 (n 2 - N2 ). wi2 = k 2 d0 2 (N2 -n 2) (34a) (34b) (i =-m,. .. .,-1, O. +1, ..., +1), (34c) Y.-F. Li and J. W. Y. Lit
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+tanh
Vol. 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 waveguides 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 threelayer slab waveguides. We show now that if we appropriately 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),