STRAKE e 1.:GUIDED MODES OF Ti:LiNbO,CHANNEL WAVEGUIDES 1129 which is of the form striction because the ERI-method is in principle limited to well guided modes,due to the approximate nature of +(g+4')2+'(1-) o'y Y=0.(11) the decoupling of the two-dimensional wave equation (7) sin into (8a)and (8b). Restricting the analysis to guided modes and assuming B.Normalization the approximate boundary condition Y=0 for u=0,we obtain solutions of (11)for odd values of g'(g'=2g Using the previous results,the expression for the trans- I,q=0,l,2.···),that can be expressed as[18: verse electric or magnetic field component (E,for TE,H. for TM)is: Y()=(sin)“C+,(cos) (12) (s,u)=Npoxp(s)Y(u:s) where C are Gegenbauer polynomials as functions of cos). =Npg(cosh ks)"Cr(-i sinh Ks) The assumed boundary condition is a rather good ap- proximation owing to the large refractive index difference (cosh nu)"C(-i sinh nu).(17) between the LiNbO,crystal and the cover material (in Its normalization according to the relation [19] most cases the cover material is air).With this assumption Y,(;s）=(cosh nu)Ci(-i sinh nu）(13) 2P= (x)2.d and,by comparison of(10)and (11),we obtain: leads to the following approximate expression of N u'(s)=(1-V1+4Mp(s)(a,kD,/n)) D.WNer 4PT Jx(s nr(s)=n呢+(1-M)p(s) +(n/akD)(q'+u'(s)) 14) (u:s)du ds (18 In (13).the number g defines the mode order in depth and Yo(u;s)is the depth distribution of the fundamental where T Zo for TE-modes and T =1/Zo for TM-modes: mode.The'effective index profile''ne(s)given by (14) Zo is the free space impedance and P denotes the optical must be used successively in the solution of the lateral power carried by the mode. field equation (8b). Equation (18)can be evaluated analytically by consid- In order to find X(s)from the solution of(8b),a similar ering Y(u:s)=Y(u;0).Then the integral factorizes approximate procedure is applied by fitting ner(s)with and one obtains,e.g.,for the fundamental TE mode [17], the function [181: nr(s）=i.+δnem(1-R tanh2(ks)）(15) INoo2=D.WNem -2-2a+a1-5r2(-μ) where onm=n()-n and with a suitable choice of nKZo T(-2) the parameters R and K.A treatment completely similar to the case of (8a)leads to: T---4-2 (19) T(-2μ'-2)r(-2μ')/ X(s)=(cosh Ks)"Co(-i sinh Ks) where u'='(0)is obtained from (14)and I is the =(11+4R(a,ko Woncto/2K)) gamma function [18].The result(19)is quite useful since it allows one to avoid the numerical integration needed Ne ni onc (1-R)+(2K/askoW)(p +u) for field normalization. (16） IV.FINITE-ELEMENT SOLUTION where p=O,l,2.···gives the mode order in the lateral One of the methods for solving the two-dimensional direction wave equation without the need of additional approxi- For guided modes the field distribution vanishes for mations is the finite-element method,as described,for in- large values of s or u.From (13)and (16)it follows stance,by [2].[3].[14],[20],and [21].This method al- that (p+u)and (q'+')must be negative.If one of lows to solve the wave equation exactly also in the these numbers becomes zero,the mode undergoes'cut- vectorial form.Several authors [3],[22]compared dis- off.Usually this is expressed in terms of the effective persion curves for rectangular dielectric waveguides,ob- index as Ner lcutor =n tained by scalar and vectorial FEM calculations and found The case M>I or R>1,which may occur in order a very good agreement.Ti:LiNbO3 waveguides having to obtain a good index profile approximation near the very small refractive index changes without lateral dis- waveguide axis,leads to cutoff effective index values de- continuities are much more similar to planar waveguides viating from n.This means,however,no additional re- for which the scalar formulation is rigorous.The FEMSTRAKE [’f ol GUIDbD MODES OF TI LiNbO, CHANNEL WAVEGUIDES 1129 which is of the form striction because the ERI-method is in principle limited to well guided modes, due to the approximate nature of the decoupling of the two-dimensional wave equation (7) into (8a) and (8b). Y = 0. ( 11 Restricting the analysis to guided modes and assuming the approximate boundary condition Y = 0 for u = 0, we obtain solutions of (1 1) for odd values of q’( qr = 2 q + 1, q = 0, 1, 2, * * e), that can be expressed as [lS]: ~(4) = (sin 4)’’ cos c;) (12) where C:/v’) are Gegenbauer polynomials (as functions of cos 4). The assumed boundary condition is a rather good ap￾proximation owing to the large refractive index difference between the LiNbO, crystal and the cover material (in most cases the cover material is air). With this assumption Y[,(u; s) = (cosh qu)” Cig’l I (-i sinh vu) (13) and, by comparison of (10) and (1 l), we obtain: B. Normalization Using the previous results, the expression for the trans￾verse electric or magnetic field component (E, for TE, H, for TM) is: 4(s, = N/,yxp(s) Yyb; s) = N,,,(cosh KS)’ CjIp)( -i sinh KS) (cosh vu)” CiG’i I ( -i sinh vu). Its normalization according to the relation [ 191 (17) +m 2P = ST_ 1 (E x li*) . e’;dxdy --m leads to the following approximate expression of In (1 3), the number q defines the mode order in depth and Y,,(u; s) is the depth distribution of the fundamental mode. The “effective index profile” nzrf( s) given by (14) must be used successively in the solution of the lateral field equation (Sb). In order to find X( s) from the solution of (Sb), a similar approximate procedure is applied by fitting n,Zff( s) with the function n$,(s) = 12: + 6rz&,( 1 - R tanh’ (KS)) (15) where 6n& = &(O) - ni and with a suitable choice of the parameters R and K. A treatment completely similar to the case of (Sa) leads to: X,,(s) = (cosh KS)’ CL”( -i sinh KS) p = ~(1 2 - JI + 4~(a,k~~6n,,,/2K)~) 2 N:,, = n; + 6n:fr,,(1 - R) + (2K/a.,kow)’ (p + p) (16) wherep = 0, I, 2, . . * gives the mode order in the lateral direction. For guided modes the field distribution vanishes for large values of 1 s 1 or I U 1. From (13) and (16) it follows that (p + p) and (q’ + p’) must be negative. If one of these numbers becomes zero, the mode undergoes “cut￾off’. Usually this is expressed in terms of the effective index as N& (cutoff = ni. The case M > 1 or R > 1, which may occur in order to obtain a good index profile approximation near the waveguide axis, leads to cutoff effective index values de￾viating from n/,. This means, however, no additional re￾where T = Zo for TE-modes and T = 1 /Zo for TM-modes; Zo is the free space impedance and P denotes the optical power carried by the mode. Equation (1 8) can be evaluated analytically by consid￾ering Y,(u; s) = Y,(u; 0). Then the integral factorizes and one obtains, e.g., for the fundamental TE mode [ 171, where pr = p’(0) is obtained from (14) and r is the gamma function [lS]. The result (19) is quite useful since it allows one to avoid the numerical integration needed for field normalization. IV. FINITE-ELEMENT SOLUTION One of the methods for solving the two-dimensional wave equation without the need of additional approxi￾mations is the finite-element method, as described, for in￾stance, by [2], [3], [14], [20], and [21]. This method al￾lows to solve the wave equation exactly also in the vectorial form. Several authors (31, [22] compared dis￾persion curves for rectangular dielectric waveguides, ob￾tained by scalar and vectorial FEM calculations and found a very good agreement. Ti : LiNb03 waveguides having very small refractive index changes without lateral dis￾continuities are much more similar to planar waveguides, for which the scalar formulation is rigorous. The FEM