STRAKE et al.:GUIDED MODES OF Ti:LiNbO,CHANNEL WAVEGUIDES 1127 TLiNbO3 LNb03 Fig.1.Ti:LiNbO,waveguide configuration. 00.20406Q8 10 Co[102 cm3] with Fig.2.Ordinary and extraordinary refractive index dependence on the Ti concentration,Comparison between the experimental(●，×)and the f(u)=exp(-u), analytical dependences used in our model. 8(s) expressions were adopted: erf ((W/2D,)(1 s))erf ((W/2D,)(1 -s)) 0.839(入/um)月 d(入)= 2 (入/um）-0.0645 (2) d,(入)= 0.67(入/um)月 (6) where u =y/D,and s 2x/W are normalized coordi- (入/um)-0.13 nates,D.and D.are the diffusion depths along the two A comparison between measured values [9]and the transverse directions,W is the Ti stripe width before dif- previous analytical expressions for on.(c)is presented fusion,and co is a parameter connected to the total Ti-ion in Fig.2.A small discrepancy can be noted for on at low content per unit length along z,given by coD,W/2.values of ci for on,using F=1.3 x 10-25 cm'and y= By introducing the atomic weight and bulk density of 0.55,discrepancies arise forc>0.9x102 cm-3.The Ti.we can relate co to the Ti stripe thickness 7 before waveguides considered in this paper belong to co=0.5 diffusion [7]: x 1021 cm3 where the difference between on,and on is T=acoD. (3) well approximated by (5). with a=1.57×1023cm3. B.Wave Equation The variation of the bulk crystal refractive indices with Assuming a properly oriented crystal exhibiting a di- temperature and wavelength is considered using Sellmeier agonal permittivity tensor,introducing the usual approx- equations given in [8]. imation of quasi-TE and quasi-TM modes [13],and ne- The local increase of the extraordinary (on(x,y))and glecting the transverse refractive index gradient as ordinary (on (x,y))refractive indices with respect to the compared with the field gradient,a scalar,two-dimen- bulk values (n and n respectively)is related to c by sional wave equation for the dominant field component can be obtained [14]: 6n,(入.c）=d(入)f(c),i=o,e (4) where f gives the dependence of the refractive index vari- +a,2 n+knxy)）-g:》 (x,y)=0 ation on c at a given wavelength and d,(A)takes into account the effects of dispersion.Using the results in [9], (7) [10]at A =0.6328 um as reference,we obtained the fol- where ko =2/A is the free-space wavenumber.B lowing approximating expressions: koNerr is the mode propagation constant,and (x.y)is f(c)=Ec the transverse field distribution (E.for quasi-TE and H for quasi-TM modes )The relations between the wave f(c)=(Fc)". (5) equation parameters n and the field for the two polarizations and the elements n.nn characterizing the In (5)a linear relation is assumed between on,and c(E permittivity tensor are reported in Table 1.In Table Il the =1.2 x 10-23 cm3),while a good fitting of the experi- relations between n,n.n:.D..D..and n.n.D.D mental on versus c dependence requires a nonlinear func- (diffusion depths along and perpendicular to the optical tion f(c).The choice of F and y is such to achieve good axis)for the three possible c-axis orientations are pre- agreement with the experimental data in the c-range of sented. interest.The dispersion of the refractive index changes was approximated by means of a simple oscillator model III.APPROXIMATE ANALYTICAL SOLUTION [9]in the infrared region:after a comparison of the results The approximate analytical technique is based on the available in the literature [9],[11],[12]the following well-known effective-refractive-index method [13].[15]STRAKE ('I ul GUIDED MODES OF TI LINbO, CHANNEL WAVEGUIDES 1127 Fig. 1. Ti : LiNbOI waveguide configuration with f(u) = exp (-U?), g(s) - erf((W/2D1)(1 + 4) + erf((W2D.m - 4) - 2 (2) where U = y/D, and s = 2x/W are normalized coordi￾nates, D, and D,. are the diffusion depths along the two transverse directions, W is the Ti stripe width before dif￾fusion, and eo is a parameter connected to the total Ti-ion content per unit length along z, given by &coD,. W/2. By introducing the atomic weight and bulk density of Ti. we can relate co to the Ti stripe thickness T before diffusion [7]: T = UC~D, (3) with a = 1.57 x lopz3 cm3. The variation of the bulk crystal refractive indices with temperature and wavelength is considered using Sellmeier equations given in [8]. The local increase of the extraordinary ( 6nl,(x, y)) and ordinary ( 6n, (x, y ) ) refractive indices with respect to the bulk values ( nch and nob, respectively) is related to c by 6ni(X, c) = dj(X)i(c), i = 0, e (4) where j; gives the dependence of the refractive index vari￾ation on c at a given wavelength and dj (A) takes into account the effects of dispersion. Using the results in [9], [IO] at X = 0.6328 pm as reference, we obtained the fol￾lowing approximating expressions: fc(c) = Ec fo(c> = (Fc)'. (5) In (5) a linear relation is assumed between 6n,, and c( E = 1.2 X cm3), while a good fitting of the experi￾mental 6n, versus c dependence requires a nonlinear func￾tionL,( e). The choice of F and y is such to achieve good agreement with the experimental data in the c-range of interest. The dispersion of the refractive index changes was approximated by means of a simple oscillator model [9] in the infrared region; after a comparison of the results available in the literature [9], [ll], [12] the following m 05 c" 3 0 !:E 0 0.2 0.L 0.6 0.8 1.0 ~~110~~~~1 - Fig. 2. Ordinary and extraordinary retractive index dependence on the Ti concentration. Comparison between the experimental ( 0. X ) and the analytical dependences used in our model. expressions were adopted: 0.839 (A/pm>' (h/prn)? - 0.0645 0.67 (Xipm)' (Xjprn)' - 0.13' d,,(h) = d,,(X) = A comparison between measured values [9] and the previous analytical expressions for 6n,,,,, (c) is presented in Fig. 2. A small discrepancy can be noted for 6n,, at low values of c;-for 6n,, using F = 1.3 x IO-'' cm' and y = 0.55, discrepancies arise for c > 0.9 x IO" cmP3. The wave uides considered in this paper belong to c0 = 0.5 X 10" cmp3 where the difference between 6n,, and 6n,, is well approximated by (5). B. Wave Equation Assuming a properly oriented crystal exhibiting a di￾agonal permittivity tensor, introducing the usual approx￾imation of quasi-TE and quasi-TM modes [ 131, and ne￾glecting the transverse refractive index gradient as compared with the field gradient, a scalar, two-dimen￾sional wave equation for the dominant field component can be obtained [ 141: (7) where ko = 21r/X is the free-space wavenumber, 0 = koN,,, is the mode propagation constant, and $(x, y) is the transverse field distribution (E, for quasi-TE and H, for quasi-TM modes). The relations between the wave equation parameters a,, a\, n and the field $ for the two polarizations and the elements n,, n,, n; characterizing the permittivity tensor are reported in Table 1. In Table I1 the relations between n,, n,, n;, D,, D,, and n,,, n,,, D,,, D, (diffusion depths along and perpendicular to the optical axis) for the three possible c-axis orientations are pre￾sented. 111. APPROXIMATE ANALYTICAL SOLUTION The approximate analytical tcchnique is based on the well-known effective-refractive-index method [ 131, [ 151