1380 JOURNAL OF LIGHTWAVE TECHNOLOGY,VOL 10,NO.10,OCTOBER 1992 Coupled-Zigzag-Wave Theory for Guided Waves in Slab Waveguide Arrays Kin Seng Chiang Abstract-The guided wave in a slab waveguide array is treated In this paper,a new normal-mode analysis for a general as a set of coupled zigzag waves that propagate in the individual slab waveguide array is presented.The present analysis has waveguides.This treatment leads to an exact dispersion relation several advantages.First,it leads naturally to a simple physical for the TE and TM modes of the array,which can be expressed in a recurrence form and is easy to evaluate.Approximations for description of the guided modes of the array,which turns out weakly coupled waveguides are discussed and compared with the to be the familiar zigzag-wave description that was originally coupled-mode theory.It is shown that the present theory can be applied to a single slab waveguide [16].Second,it results in cast in a form that closely resembles the coupled-mode theory. a particularly simple dispersion relation that can be evaluated efficiently.Third,under the weak-coupling assumption,the I.INTRODUCTION solution from the present analysis reduces to a form that modm ica evitam pe closely resembles the coupled-mode solution,and therefore provides a bridge between the normal-mode analysis and the coupled-mode analysis.All these aspects are of both ulators,Bragg deflectors,high-power semiconductor lasers, theoretical and practical interest. and multiple-quantum-well devices.In general,there are two approaches-the coupled-mode analysis and the normal-mode analysis-that can be used for the study of slab waveguide II.THEORY arrays. We consider a waveguiding structure that consists of N In a coupled-mode analysis,the mode field in the array high-index slabs sandwiched between N+1 low-index slabs. is approximated by a superposition of the mode fields in The coordinate system is so chosen that the refractive index the individual waveguides.The propagation constants of the of the structure varies only in the z direction and the optical array modes can be found by solving a system of coupled- wave propagates in the z direction.The refractive-index profile mode equations relating the amplitudes of the waveguide of this structure is shown in Fig.1,where ni(for i= modes [1]-[5].Analytical results have been obtained for some 1,2,...,N)is the refractive index of the high-index layer uniform [1]-[3]or nearly uniform [4],[5]slab waveguide iand ni-1.i (for i=2,3,...,N)is the refractive index of arrays.For general arrays,the coupled-mode equation system the low-index layer that connects the two neighboring high- must be solved numerically.Because the coupled-mode theory index layers i-1 and i.no.1 and nN.N+1 are the indexes has a simple physical interpretation and can be conveniently of the substrates,which are assumed to extend to infinity. applied to two-dimensional problems such as optical fiber It should be noted that a single subscript has been used to arrays (see,for example,[6]-[8)),it is more widely used denote the refractive index of a high-index layer while double for array analysis.However,the coupled-mode theory is subscripts have been used for a low-index layer.The same inherently approximate and accurate only for weakly coupled convention will be used for other parameters throughout this waveguides. paper to distinguish between the high-index and low-index In a normal-mode analysis,a waveguide array is treated layers.The boundaries of the high-index layer i are defined as a single composite waveguiding structure and the guided by the coordinatesL(the Left side)and(the Right modes(the normal modes)of the array are studied directly.In side).The thickness of the layer with indexnis given by the case of slab waveguide arrays,the fields in the slabs can di=xRi-Li and that of the layer with index n-1.is be represented exactly and the application of the boundary given by di-1,i=Li-R:-1.The structure shown in Fig.1 conditions at all slab interfaces can lead,in principle,to can be regarded as an array of N slab waveguides. the dispersion relation (or eigenvalue equation)for the array. To facilitate discussion,we define Although it appears to be a standard boundary-value problem the form and the complexity of the solution depend heavily on =(nk2-g2)2i=1,2,…,N (1) how the fields are represented [9]-[15].Nevertheless,several methods have been developed for dispersion calculation for general multilayer structures [9]-[12]. 9-1i=(92-n-1.k2)/2 i=1,2,…,N+1(2) Manuscript received November 21991;revised May 8,1992. The author is with the Division of Applied Physics,Commonwealth where k 2/A is the free-space wavenumber (A is the Scientific and Industrial Research Organization,Lindfield 2070.Australia. free-space wavelength)and is the propagation constant IEEE Log Number 9202327. of the guided mode.gi and g-can be identified as the 0733-8724/92$03.00@1992IEEE
1380 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 10, NO. 10, OCTOBER 1992 Coupled-Zigzag-Wave Theory for Guided Waves in Slab Waveguide Arrays Kin Seng Chiang Abstract-The guided wave in a slab waveguide array is treated as a set of coupled zigzag waves that propagate in the individual waveguides. This treatment leads to an exact dispersion relation for the TE and TM modes of the array, which can be expressed in a recurrence form and is easy to evaluate. Approximations for weakly coupled waveguides are discussed and compared with the coupled-mode theory. It is shown that the present theory can be cast in a form that closely resembles the coupled-mode theory. I. INTRODUCTION slab waveguide array is the basic structure of many A modern optical devices, such as beam splitters, modulators, Bragg deflectors, high-power semiconductor lasers, and multiple-quantum-well devices. In general, there are two approaches-the coupled-mode analysis and the normal-mode analysis-that can be used for the study of slab waveguide arrays. In a coupled-mode analysis, the mode field in the array is approximated by a superposition of the mode fields in the individual waveguides. The propagation constants of the array modes can be found by solving a system of coupledmode equations relating the amplitudes of the waveguide modes [1]-[5]. Analytical results have been obtained for some uniform [1]-[3] or nearly uniform [4], [5] slab waveguide arrays. For general arrays, the coupled-mode equation system must be solved numerically. Because the coupled-mode theory has a simple physical interpretation and can be conveniently applied to two-dimensional problems such as optical fiber arrays (see, for example, [6]-[8]), it is more widely used for array analysis. However, the coupled-mode theory is inherently approximate and accurate only for weakly coupled waveguides. In a normal-mode analysis, a waveguide array is treated as a single composite waveguiding structure and the guided modes (the normal modes) of the array are studied directly. In the case of slab waveguide arrays, the fields in the slabs can be represented exactly and the application of the boundary conditions at all slab interfaces can lead, in principle, to the dispersion relation (or eigenvalue equation) for the array. Although it appears to be a standard boundary-value problem, the form and the complexity of the solution depend heavily on how the fields are represented [9] -[ 151. Nevertheless, several methods have been developed for dispersion calculation for general multilayer structures [9]-[12]. Manuscript received November 2, 1991; revised May 8, 1992. The author is with the Division of Applied Physics, Commonwealth IEEE Log Number 9202327. Scientific and Industrial Research Organization, Lindfield 2070, Australia. In this paper, a new normal-mode analysis for a general slab waveguide array is presented. The present analysis has several advantages. First, it leads naturally to a simple physical description of the guided modes of the array, which turns out to be the familiar zigzag-wave description that was originally applied to a single slab waveguide [16]. Second, it results in a particularly simple dispersion relation that can be evaluated efficiently. Third, under the weak-coupling assumption, the solution from the present analysis reduces to a form that closely resembles the coupled-mode solution, and therefore provides a bridge between the normal-mode analysis and the coupled-mode analysis. All these aspects are of both theoretical and practical interest. 11. THEORY We consider a waveguiding structure that consists of N high-index slabs sandwiched between N + 1 low-index slabs. The coordinate system is so chosen that the refractive index of the structure varies only in the x direction and the optical wave propagates in the z direction. The refractive-index profile of this structure is shown in Fig. 1, where ni (for z = 1, 2, . . . , N) is the refractive index of the high-index layer i and nz-l,i (for i = 2, 3, . . . , N) is the refractive index of the low-index layer that connects the two neighboring highindex layers i - 1 and z. no,^ and n~,~+1 are the indexes of the substrates, which are assumed to extend to infinity. It should be noted that a single subscript has been used to denote the refractive index of a high-index layer while double subscripts have been used for a low-index layer. The same convention will be used for other parameters throughout this paper to distinguish between the high-index and low-index layers. The boundaries of the high-index layer i are defined by the x coordinates x~i (the Left side) and XR~ (the Right side). The thickness of the layer with index ni is given by di = XR~ - x~i and that of the layer with index ni-l,i is given by di-1,i = x~i - x&iPl. The structure shown in Fig. 1 can be regarded as an array of N slab waveguides. To facilitate discussion, we define qi = (n:lc2 - /?2)1/2 i = 1, 2, . . . , N (1) where IC = 2r/A is the free-space wavenumber (A is the free-space wavelength) and p is the propagation constant of the guided mode. qi and qiPl,i can be identified as the 0733-8724/92$03.00 0 1992 IEEE
CHIANG:COUPLED-ZIGZAG-WAVE THEORY FOR GUIDED WAVES 1381 n0, +1 X XLI XRI XLi-1 XRi-1 XRi XLi+1 XRi+1 XLN XRN Fig.I.Refractive-index profile of a general slab waveguide array. transverse components of the wave vectors in various layers. and,for the boundary =Ri,as The derivation and the physical interpretation to be described are particularly simple and clear if we restrict ourselves to =ERi cos[qi(ERi-)-Ri] the solution of the mode index B/k that is not larger than xLi≤x≤xR:(6) the lowest refractive index among the high-index layers and yet not smaller than the highest refractive index among the p=CRiexp-g,+1(E-xR引 low-index layers including the substrates.That is, +DRi exp+g.i+1(x-工RaJ maxo-1ws是≤mia时 (3) rri≤r≤ELi+1 (7) where CLi,DLi,ELi,CRi,DRi,and ERi are the field forj=1,2,...,N+1 and 1,2,...,N.The condition given amplitudes,and Li and Ri are the phase angles,respectively. by (3)is equivalent to the requirement that both i and q-1. In the following,the derivation for the TE modes is outlined are real and nonnegative.Physically,this means that the mode and the necessary modifications for the TM modes are given. field is oscillatory in all the high-index layers and evanescent in all the low-index layers.Apart from this assumption,which A.Guided TE Modes limits our discussion to an array of N waveguides (instead of an arbitrary multilayer structure),no other restriction is For the TE modes,both and du/d are continuous at the imposed on the profile.Waveguide arrays in most practical boundaries.Applying these continuity conditions at =L applications do satisfy (3).It should be pointed out,however, and r Ri,respectively,leads to that the present analysis can be readily generalized to remove this assumption. tan Li=i-1. 1-: The major feature of the present approach is that the fields 1+T1 (8) at the two sides of each boundary are expressed independently. Because each layer has two boundaries,the field within each tan oRi= 9.i+11-「 （1+TrE (9) layer (except for the first and the last layers that extend to infinity)is represented twice.In this way the application where ILi=DLi/CLi and TRi=DRi/CRi are the amplitude of the boundary conditions can be greatly simplified.The ratios. requirement that the two representations of the field within The field must be continuous everywhere in the high-index each layer are identical can lead to a system of resonance layer i.This implies that (5)and (6)represent the same field. conditions,which describes the dispersion characteristics of This is true only when the arguments of the cosine functions the whole structure. in these expressions differ by an integral number of x.This In general,the field can be expressed,for the boundary leads to T TLi,as min +oLi+Ri-qidi=0 i=1,2,...,N (10) 的=CLi exp[-q:-l,i(xLi一x] +DLi exp+qi-1.i(Li-)] where mi is an integer(>0)associated with the high-index layer i.Equation (10)has the same form as the resonance ERi-1≤E≤ELi (4) condition for a single slab waveguide [16]and has a simple zigzag-wave interpretation.The term gid;is the phase shift ψ=ELi cosg:(x-xLi）-中Li, that the zigzag wave acquires when it traverses the guiding rLi≤E≤tR (5)layer once,since qi is the transverse component of the wave
CHIANG: COUPLED-ZIGZAG-WAVE THEORY FOR GUIDED WAVES 1381 Fig. 1. Refractive-index profile of a general slab waveguide array. transverse components of the wave vectors in various layers. The derivation and the physical interpretation to be described and, for the boundary z = XR~, as are particularly simple and clear if we restrict ourselves to the solution of the mode index P/k that is not larger than @ = ER, cos[qz(zRt - x) - 4Rz] ~Lz 5 x 5 zRt (6) the lowest refractive index among the high-index layers and yet not smaller than the highest refractive index among the low-index layers including the substrates. That is, @ = CRz exp[-%,z+l(z - ~RZ)] + DRz exp[fqt z+l(z - ~Rz)] (3) P Ic- max {TL-~,~} 5 - < min {n,} for3 = 1, 2, .”, N+landl. 2, ..., N.Theconditiongiven by (3) is equivalent to the requirement that both q2 and qz-l,t are real and nonnegative. Physically, this means that the mode field is oscillatory in all the high-index layers and evanescent in all the low-index layers. Apart from this assumption, which limits our discussion to an array of N waveguides (instead of an arbitrary multilayer structure), no other restriction is imposed on the profile. Waveguide arrays in most practical applications do satisfy (3). It should be pointed out, however, that the present analysis can be readily generalized to remove this assumption. The major feature of the present approach is that the fields at the two sides of each boundary are expressed independently. Because each layer has two boundaries, the field within each layer (except for the first and the last layers that extend to infinity) is represented twice. In this way the application of the boundary conditions can be greatly simplified. The requirement that the two representations of the field within each layer are identical can lead to a system of resonance conditions, which describes the dispersion characteristics of the whole structure. In general, the field $ can be expressed, for the boundary x = XL~, as * = CLZ exP[-q,-l,t(xLz - .)I + DL~ exp[+ qz-l.t(zLz - z)] zR2-1 5 x 5 XLz (4) XRz 5 z 5 %Lz+l (7) where CL^, DLt, EL%, CR~, DR~, and ER% are the field amplitudes, and q5Lz and &R, are the phase angles, respectively. In the following, the derivation for the TE modes is outlined and the necessary modifications for the TM modes are given. A. Guided TE Modes For the TE modes, both $ and d$/dx are continuous at the boundaries. Applying these continuity conditions at x = XL~ and r = XR~, respectively, leads to (9) where rLz = DL2/CLt and rRz = DR2/CRz are the amplitude ratios. The field must be continuous everywhere in the high-index layer 2. This implies that (5) and (6) represent the same field. This is true only when the arguments of the cosine functions in these expressions differ by an integral number of T. This leads to mzr + 4~~ + &R~ - yzdz = 0 z = 1, 2, . . . , N (10) where m, is an integer (2 0) associated with the high-index layer 2. Equation (10) has the same form as the resonance condition for a single slab waveguide [16] and has a simple zigzag-wave interpretation. The term q2dz is the phase shift that the zigzag wave acquires when it traverses the guiding layer once, since y2 is the transverse component of the wave
111 1382 JOURNAL OF LIGHTWAVE TECHNOLOGY,VOL.10,NO.10,OCTOBER 1992 vector.Quantities 2Li and 2Ri can be interpreted as the qf-9经.+1 phase shifts acquired by the wave when they are refected PRi= (22) at the two boundaries.Equation (10),when multiplied by 2 2q9,+1 says that the round-trip phase shift acquired by the ray is and equal to a multiple of 2.Integer mi can be identified as q+1-92.+1 the mode order for the individual waveguide and gives the PLitl= (23) number of nulls in the mode field in that waveguide.It should 2q+19,i+1 be emphasized that(10)applies to every high-index layer,Equation(20)shows that only the phase shifts associated with and therefore,represents N zigzag waves.Our next step is to two neighboring guiding layers are explicitly coupled. determine the phase shifts Li and Ri By substituting (11)and (12)into the resonance condition We can write (10)and using the following definitions, pLi=中gLi-△pLi (11) F=tan(中sLf+pri-qd） (24) a=中Ri-△pRa (12) &Li=tan△pLi (25) with 中sLi=tan】经-l,d (13) qi iBi=tan△pra (26) Ri=tan-1 i+1 we obtain (14) q 6Li+6Ri F:=1-6LORi =1,2,…N (27) We thus find from (8)and (9)that 2TLiqi-1.i9i The left-hand side of(27),Fi,contains only the parameters for △pLe=tanl (15) an isolated waveguide,whereas the right-hand side contains 9g(1+TL)+q-1,(1-Ta】 the phase shifts that couple to those of the neighboring waveguides.There are N equations in (27)representing N △pri=tan 2TR9i,+19 zigzag waves.Additionally,there are N-1 auxiliary equations (16) 2(1+TRi)+qi+(1-TRi) given by (20): It can be seen that L andR are the phase shifts when the 6Ri6Li+1=ci.i+1(1-PRi8Ri)(1 -PLi+16Li+1) guiding layers are isolated,ie.,infinitely separated in distance. i=1,2,…,N-1 (28) △Li and△中thus represent the changes in these phase shifts as a result of reducing the waveguide separations from which couple the phase shifts of adjacent zigzag waves. infinity. Because the field of a guided mode vanishes at r =-oo According to(4),the field in the layer bounded by rRi and and x =+o0,we have DLI DRN 0,and hence, Li+i can also be expressed as 6L1 6RN 0.There are 2N-1 unknowns,i.e.,B,6L =CLi+1 exp[-qi.i+1(Li+1-)] (fori=2,3,…,N)and6i(fori=1,2.…,N-1月 they can be found from (27)and (28)for a given free- +DLi+1 exp[+qii+1(Li+1-)] space wavenumber k.As (27)is a system of transcendental E≤x≤xLi+I: (17) equations,there can be many sets of solutions,or many sets of zigzag waves,corresponding to different modes of the Equation (17)must be identical to (7),since both expressions represent the same field.Equating(17)and(7),we find that array.For each set of solutions,the amplitude ratios can be determined from (15)and (16),the mode orders m;for TRiTLi+1 exp(-20i,i+1) (18) the individual waveguides can be found from (10),and the field pattern can be constructed from(4)-(7),where one field where amplitude is allowed to take an arbitrary value (say,CL=1). 0,i+1=,i+1d,i+1 (19) In summary,the guided wave in a slab waveguide can be regarded as a zigzag wave that resembles a ray bouncing back substituting (18)into (15)and (16)leads to and forth in the guiding slab that acquires phase shifts at the tan△bRi tan.△pLi+1=c,i+1(1-PRitan.△pra) boundaries.For an isolated waveguide,the phase shifts at the ·(1-PLi+1tan△pLi+1) (20) two boundaries are given by 2L and 2Ri,respectively. When the waveguides are brought together to form an array, where the phase shifts are reduced,respectively,by the amounts 449i,i+19+1exp(-2ai,i+1） 2ALi and 2ARi,as illustrated in Fig.2.When a guided Ci,i+1 (21) mode exists,the zigzag waves in all waveguides satisfy the (q+匠+1）(q+1+q驻.+1) transverse resonance conditions (27)simultaneously,and the
1382 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 10, NO. 10, OCTOBER 1992 22 (22) vector. Quantities 24~i and 24Ri can be interpreted as the phase shifts acquired by the wave when they are reflected Yi - 4i,i+l 2qi Yi , i + 1 PRi = at the two boundaries. Equation (lo), when multiplied by 2, says that the round-trip phase shift acquired by the ray is equal to a multiple of 27r. Integer mi can be identified as number of nulls in the mode field in that waveguide. It should be emphasized that (10) applies to every high-index layer, and therefore, represents N zigzag waves. Our next step is to determine the phase shifts 4~i and 4~i. and (23) Y?+l - 4$+l the mode order for the individual waveguide and gives the PLi+l = 2qi+ 1 Yi ,i+ 1 Equation (20) shows that only the phase shifts associated with two neighboring guiding layers are explicitly coupled. By substituting (11) and (12) into the resonance condition We can write (10) and using the following definitions, 4~i = (11) Fi = tan(4sLz + 4s~i - qidi) (24) 4sLi - A4~i with we obtain a = 1, 2, ... N. (27) ~Lz + SRO 4sRz = tan-' 42,2+1. (14) F, = Qz We thus find from (8) and (9) that 1 - 6~~6~~ The left-hand side of (27), F,, contains only the parameters for an isolated waveguide, whereas the right-hand side contains the phase shifts that couple to those of the neighboring waveguides. There are N equations in (27) representing N zigzag waves. Additionally, there are N- 1 auxiliary equations (15) 1 1 2rLzQz-1,Zqz 4,2(1 + rLz) + 4,2_1J1 - FLZ) . (16) given by (20): 2rRz qz ,a+ 1 qt [ A~L, = tan-' + rRz) + $,,,+1(1 - ~Rz) A$R, = tan-' It can be seen that dS~i and $s~i are the phase shifts when the guiding layers are isolated, i.e., infinitely separated in distance. A4~i and A4~i thus represent the changes in these phase shifts as a result of reducing the waveguide separations from infinity. According to (4), the field in the layer bounded by XR~ and zLi+l can also be expressed as II, = CL~+I exp[-qi,i+l(zLi+l - x)] + DLi+l eXP[+Yi,i+l(xLi+l - .)I XRi 5 x 5 XLi+l. (17) Equation (17) must be identical to (7), since both expressions represent the same field. Equating (17) and (7), we find that r~ir~i+l = exp(-2ai,z+l) (18) Qi,i+l = Yi,i+ldi,i+l. (19) where substituting (18) into (15) and (16) leads to tan A~R; tan A4~i+l = Ci,i+l(l - p~i tan A~R;) . (1 - PLi+l tanM~i+l) (20) where (21) 4qiqZ,i+lqi+l exp(-2ai,i+l) (Y? + ~?,i+l) ($+I + d,i+l) Ci,i+l = which couple the phase shifts of adjacent zigzag waves. Because the field of a guided mode vanishes at z = --33 and x = +m, we have DL~ = DRN = 0, and hence, 6~1 = ~RN = 0. There are 2N - 1 unknowns, i.e., p, 6~i (for i = 2, 3. .", N) and SR~ (for i = 1, 2, .... N - 1); they can be found from (27) and (28) for a given freespace wavenumber IC. As (27) is a system of transcendental equations, there can be many sets of solutions, or many sets of zigzag waves, corresponding to different modes of the array. For each set of solutions, the amplitude ratios can be determined from (15) and (16), the mode orders mi for the individual waveguides can be found from (lo), and the field pattern can be constructed from (4)-(7), where one field amplitude is allowed to take an arbitrary value (say, CL1 = 1). In summary, the guided wave in a slab waveguide can be regarded as a zigzag wave that resembles a ray bouncing back and forth in the guiding slab that acquires phase shifts at the boundaries. For an isolated waveguide, the phase shifts at the two boundaries are given by 24s~i and 24s~i, respectively. When the waveguides are brought together to form an array, the phase shifts are reduced, respectively, by the amounts 2A4~i and 2A4~i, as illustrated in Fig. 2. When a guided mode exists, the zigzag waves in all waveguides satisfy the transverse resonance conditions (27) simultaneously, and the
CHIANG:COUPLED-ZIGZAG-WAVE THEORY FOR GUIDED WAVES 1383 III.RECURRENCE FORMULA The system of transcendental equations,(27)and (28), although simple in form,is not easy to solve,especially 2pi=2(中sLi-△pL) n-1,t when the number of waveguides is large.It is desirable ni to obtain a single dispersion equation that relates only the propagation constant and the free-space wavenumber.As 2中=2（中贴-△中m) n,2+1 shown in the Appendix,such a dispersion relation can be derived by elimination of the phase quantities 6Rand Li in(27)and(28). A.General Arrays Fig.2.The wave in each waveguide of the array propagates in a zigzag manner and acquires phase shifts at the boundaries.For a guided mode to We can write the dispersion relation for an array of N exist,the zigzag waves in all waveguides must satisfy the transverse resonance elements as conditions simultaneously. EN-0 (34) phase shifts acquired by the neighboring zigzag waves are where en is an implicit function of B and k.It is shown in the coupled according to (28).This provides a simple coupled- Appendix that +1(for i>1)can be expressed as a simple zigzag-wave description for the guided modes of the array. linear combination of e;and i-1: B.Guided TM Modes Ei+1=Ji+1ei-Ki+1ei-1 ≥1 (35) For the TM modes,apart from the field itself,the field gra- with e1=F and so I where (see bottom of page) dient divided by the square of the index,i.e.,(d/dr)/n() and must be continuous everywhere.When we apply the appro- priate boundary conditions for the TM modes to (4)-(7)in K+1=+1-p+1F+1+F (37) the same manner as described for the TE modes,we obtain 1-PLiFi exactly the same system of equations,(27)and(28),provided Equation (35)is a recurrence formula from which ex- that the relevant parameters are now given by pressions for e2,a,.,EN can be successively obtained. 4n7n+1n2+19g2i+19+1cxp(-2a,i+1） Equation(35),which is effectively a single transcendental Ci,+1= equation,is not only simple in form but also easy to solve. (nq+ni+19匠+i)(n+19i+1+n.+19听+i) Conventional root-searching techniques can be used to find (29) the propagation constant from (35).No matrix manipulations are needed and a large computer memory is not required. Once the propagation constant is found,the phase shifts nq听-n4+19ii+1 PRi= (30) can be calculated successively from (27)and (28).Insofar 2n好n2.i+199i,i+1 as the computational efficiency is concerned,the present method should be comparable to the widely used matrix and method,[9],[10],[14],[15],which involves root searching n+12+1-n+19听+1 for a matrix equation that contains transcendental functions. PLi+1= (31) It should be emphasized,however,that the major contribution 2n3+1n2.i+19+19,i+1 of the present analysis is the provision of the simple explicit The phase shifts for isolated waveguides are now given by dispersion relations(27),(28),and(35),which can facilitate the analytical study of waveguide arrays. Osti=tan-1 ng-L (32)B.Identical Waveguides n-1,9 and In the special case of equally separated identical wave- guides,we have F=F2=…三F,c1,2=c2,3=…=C ORi=tan-1 niqit (33) and PR=p2=,=pL2=pL3=·≡p.Equation(35) n2,i+19a can be simplified to With these modifications,(27)and (28),as well as the coupled-={F[1+c(1-p2)]+2cphe:-c(1+F2)e1i1 zigzag-wave description,apply to both the TE and TM modes. (38) J+1=F+1+4+11-PL+1E+[+p)+L-PpL】 (36) 1-pLiFi
CHIANG: COUPLED-ZIGZAG-WAVE THEORY FOR GUIDED WAVES 1383 111. RECURRENCE FORMULA The system of transcendental equations, (27) and (28), although simple in form, is not easy to solve, especially when the number of waveguides is large. It is desirable to obtain a single dispersion equation that relates only the propagation constant and the free-space wavenumber. As shown in the Appendix, such a dispersion relation can be derived by elimination of the phase quantities 6~2 and SL; in (27) and (28). ni . Fig. 2. The wave in each waveguide of the array propagates in a zigzag manner and acquires phase shifts at the boundaries. For a guided mode to exist, the zigzag waves in all waveguides must satisfy the transverse resonance conditions simultaneously. phase shifts acquired by the neighboring zigzag waves are coupled according to (28). This provides a simple coupledzigzag-wave description for the guided modes of the array. B. Guided TM Modes For the TM modes, apart from the field itself, the field gradient divided by the square of the index, i.e., (d+/d~)/n(x)~, must be continuous everywhere. When we apply the appropriate boundary conditions for the TM modes to (4)-(7) in the same manner as described for the TE modes, we obtain exactly the same system of equations, (27) and (28), provided that the relevant parameters are now given by and (31) n2"+ 1 q2"+ 1 - n2",i+ 1 d,i+ 1 w+ 1 .T, z+ 1 42+ 1 4iI i+ 1 ' PLi+l = The phase shifts for isolated waveguides are now given by and (33) With these modifications, (27) and (28), as well as the coupledzigzag-wave description, apply to both the TE and TM modes. A, General Arrays elements as We can write the dispersion relation for an array of N where EN is an implicit function of p and k. It is shown in the Appendix that (for i 2 1) can be expressed as a simple linear combination of €2 and €2-1: €;+I = Ji+lEi - Ki+lEi-l i 2 1 (35) with €1 = F1 and EO = 1 where (see bottom of page) and Equation (35) is a recurrence formula from which expressions for c2, €3, . . . , EN can be successively obtained. Equation (35), which is effectively a single transcendental equation, is not only simple in form but also easy to solve. Conventional root-searching techniques can be used to find the propagation constant from (35). No matrix manipulations are needed and a large computer memory is not required. Once the propagation constant is found, the phase shifts can be calculated successively from (27) and (28). Insofar as the computational efficiency is concerned, the present method should be comparable to the widely used matrix method, [9], [lo], [14J, [15], which involves root searching for a matrix equation that contains transcendental functions. It should be emphasized, however, that the major contribution of the present analysis is the provision of the simple explicit dispersion relations (27), (28), and (35), which can facilitate the analytical study of waveguide arrays. B. Identical Waveguides In the special case of equally separated identical wave- - guides, we have F1 = F2 = . . . = F, c1,2 = c2,3 = . . . = c, and p~1 = pR2 = . . . = p~2 = p~3 = . . . p. Equation (35) can be simplified to ~ , + 1 = ( F [ l + ~ ( l - p ~ ) ] + 2 ~ p } ~ ; - ~ ( l + F ~ ) ~ i - l ill (38)
1384 JOURNAL OF LIGHTWAVE TECHNOLOGY.VOL 10,NO.10,OCTOBER 1992 with e=F and o 1.It is obvious that,for an array can be regarded as small,i.e.,Li 1 and i1.With of N elements,(38)is an Nth-order polynomial of F and. these approximations,(27)becomes in general,gives N distinct solutions for F.This means that each guided mode of the single waveguide splits into a band Fi=6Li+6Ri (39) of N nondegenerate modes when the separations of the N waveguides are reduced from infinity. with R=ci.+1.The corresponding recurrence formula can be obtained from (35)-(37)by neglecting insignificant terms: IV.COMPARISON WITH A RECENT THEORY It is interesting to compare the present approach with that Ei+1=Fi+1ei-Cii+1Ei-1 ≥1 (40) recently reported for the analysis of multilayer structures [11, with e1 F and Eo =1. (12].In (11]and [12],the field in an arbitrary guiding slab As an example,(40)gives is expressed by a trigonometric function,whereas those in all other slabs are expressed by hyperbolic functions.This FF2-c1,2=0 (41) treatment leads to a single transverse resonance condition similar to (10)but for the chosen guiding slab only.The for two dissimilar waveguides. guided wave in a multilayer structure is thus represented by a Synchronization coupling between the two waveguides oc- single zigzag wave in a chosen layer with phase angles that curs when F and F2 are equal,i.e.,the normal modes of the depend on the properties of all other layers.The resultant array are represented by the equation F=F=.On dispersion relation contains hyperbolic functions.Because the other hand,when the synchronization condition is far from hyperbolic functions are used,which can formally represent being satisfied,F and F2 are very different in magnitude.In both oscillatory and evanescent fields,this approach is general the case of F2F,(41)implies F0,whereas in the enough to handle arbitrary multilayer structures.However. case F F2,F20.In these cases,the normal modes in the special case of a waveguide array that consists of of the array can be well approximated by the normal modes of alternating high-and low-index layers,it seems rather artificial the individual waveguides.This is also a well-known result of that the guided wave in the whole array is represented by just the coupled-mode theory (8. one zigzag wave in an arbitrarily chosen waveguide,since, As another example,for an array consisting of N equally from a physical point of view,there is no reason why only one separated identical waveguides,(40)with F+F and waveguide of the array should receive special treatment.The ci.+=c is an Nth-order polynomial of F and gives N present approach,on the other hand,treats all high-index layers explicit solutions for F.The results are given in Table I equitably so that the guided wave of the array is represented for N up to five,where the coupled-mode solutions for the by N coupled zigzag waves.It appears that the present model propagation constants [3]are also presented for comparison can give a more natural and intuitive physical picture for the This table clearly shows the correspondence between the guided waves in a waveguide array.The physical interpretation present theory and the coupled-mode analysis.It can be seen of the present approach becomes even more explicit and from the table that the constants associated with the factor c/2 appealing in the next section where a close correspondence in the dispersion relations are identical to those associated with between the present theory and the coupled-mode theory is the coupling coefficient in the propagation constants.The found for weakly coupled waveguides. existence of such a close correspondence is rather surprising This comparison does not imply that the present approach is since these two sets of results are derived from two completely superior to that given in [11]and [12],since the latter can dea different approaches,both involving approximations.In fact, with more general multilayer structures than the former.Here it can be shown that in the weak-coupling limit,the square we just emphasize on the more appealing physical picture and root of the factor c arising from the present normal-mode the particularly simple recurrence dispersion formula(35)that theory is proportional to the coupling coefficient arising can be obtained from the present theory for the special case from the coupled-mode theory.It can also be proved from of a waveguide array. (40)that for an odd number of waveguides,F=0 is always a solution,which gives a propagation constant equal to that of a single waveguide.The same result has been obtained from the V.WEAK-COUPLING APPROXIMATIONS coupled-mode theory [3],where only the interactions between In this section,we show how the mode dispersion rela- two neighboring waveguides are considered.To the author's tion can be simplified for weakly coupled waveguides and knowledge,this is the first analytical comparison between expressed in a form that closely resembles the coupled-mode the normal-mode theory and the coupled-mode theory for a theory. waveguide array. The waveguides are weakly coupled when the waveguide To demonstrate the accuracy of the weak-coupling ap- separations (i.e.,the thicknesses of the low-index layers in proximation,the dispersion characteristics for an array of Fig.1)are large,and the mode of the single waveguide of two identical waveguides are calculated.The thickness of concern is not close to cutoff.In our context,this means that the two guiding layers is denoted by 2t and their separa- ci.+given by (21)is much smaller than unity,since ci.+ tion denoted by 2d.The refractive indexes for the guiding is proportional to exp(-2+1)1,where i.is given layers and the cladding layers are denoted by ne and ns, by (19).The changes of the phase shifts at all the boundaries respectively.For a given normalized frequency V,defined by
1384 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 10, NO. 10, OCTOBER 1992 with €1 = F and EO = 1. It is obvious that, for an array of N elements, (38) is an Nth-order polynomial of F and, in general, gives N distinct solutions for F. This means that each guided mode of the single waveguide splits into a band of N nondegenerate modes when the separations of the N waveguides are reduced from infinity. Iv. COMPARISON WITH A RECENT THEORY It is interesting to compare the present approach with that recently reported for the analysis of multilayer structures [ll], [12]. In [ll] and [12], the field in an arbitrary guiding slab is expressed by a trigonometric function, whereas those in all other slabs are expressed by hyperbolic functions. This treatment leads to a single transverse resonance condition similar to (10) but for the chosen guiding slab only. The guided wave in a multilayer structure is thus represented by a single zigzag wave in a chosen layer with phase angles that depend on the properties of all other layers. The resultant dispersion relation contains hyperbolic functions. Because hyperbolic functions are used, which can formally represent both oscillatory and evanescent fields, this approach is general enough to handle arbitrary multilayer structures. However, in the special case of a waveguide array that consists of alternating high- and low-index layers, it seems rather artificial that the guided wave in the whole array is represented by just one zigzag wave in an arbitrarily chosen waveguide, since, from a physical point of view, there is no reason why only one waveguide of the array should receive special treatment. The present approach, on the other hand, treats all high-index layers equitably so that the guided wave of the array is represented by N coupled zigzag waves. It appears that the present model can give a more natural and intuitive physical picture for the guided waves in a waveguide array. The physical interpretation of the present approach becomes even more explicit and appealing in the next section where a close correspondence between the present theory and the coupled-mode theory is found for weakly coupled waveguides. This comparison does not imply that the present approach is superior to that given in [ll] and [12], since the latter can deal with more general multilayer structures than the former. Here we just emphasize on the more appealing physical picture and the particularly simple recurrence dispersion formula (35) that can be obtained from the present theory for the special case of a waveguide array. V. WEAK-COUPLING APPROXIMATIONS In this section, we show how the mode dispersion relation can be simplified for weakly coupled waveguides and expressed in a form that closely resembles the coupled-mode theory. The waveguides are weakly coupled when the waveguide separations (i.e., the thicknesses of the low-index layers in Fig. 1) are large, and the mode of the single waveguide of concern is not close to cutoff. In our context, this means that given by (21) is much smaller than unity, since ~i,i+~ is proportional to exp(-2cui,i+1) << 1, where ai,i+l is given by (19). The changes of the phase shifts at all the boundaries can be regarded as small, i.e., 6~~ << 1 and 6~~ << 1. With these approximations, (27) becomes Fz = SLz f ~RZ (39) with = C,,,+I. The corresponding recurrence formula can be obtained from (35)-(37) by neglecting insignificant terms: Ez+1 = Fz+lEz - Cz,z+lEz-I 2 2 1 (40) with €1 = F1 and EO = 1. As an example, (40) gives F1F2 - ~1,2 = 0 (41) for two dissimilar waveguides. Synchronization coupling between the two waveguides occurs when F1 and F2 are equal, i.e., the normal modes of the array are represented by the equation F1 = F2 = f$;. On the other hand, when the synchronization condition is far from being satisfied, F1 and F2 are very different in magnitude. In the case of IF21 >> IFlI, (41) implies F1 N 0, whereas in the case 1 F1 I >> I F2 1, F2 21 0. In these cases, the normal modes of the array can be well approximated by the normal modes of the individual waveguides. This is also a well-known result of the coupled-mode theory [8]. As another example, for an array consisting of A' equally separated identical waveguides, (40) with F,+1 F and cz,,+l G c is an Nth-order polynomial of F and gives N explicit solutions for F. The results are given in Table I for N up to five, where the coupled-mode solutions for the propagation constants [3] are also presented for comparison. This table clearly shows the correspondence between the present theory and the coupled-mode analysis. It can be seen from the table that the constants associated with the factor c1I2 in the dispersion relations are identical to those associated with the coupling coefficient K. in the propagation constants. The existence of such a close correspondence is rather surprising since these two sets of results are derived from two completely different approaches, both involving approximations. In fact, it can be shown that in the weak-coupling limit, the square root of the factor c arising from the present normal-mode theory is proportional to the coupling coefficient 6 arising from the coupled-mode theory. It can also be proved from (40) that for an odd number of waveguides, F = 0 is always a solution, which gives a propagation constant equal to that of a single waveguide. The same result has been obtained from the coupled-mode theory [3], where only the interactions between two neighboring waveguides are considered. To the author's knowledge, this is the first analytical comparison between the normal-mode theory and the coupled-mode theory for a waveguide array. To demonstrate the accuracy of the weak-coupling approximation, the dispersion characteristics for an array of two identical waveguides are calculated. The thickness of the two guiding layers is denoted by 2t and their separation denoted by 2d. The refractive indexes for the guiding layers and the cladding layers are denoted by nc and ns, respectively. For a given normalized frequency V, defined by