IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 41, NO. 11. NOVEMBER 1993 1907

Calculation of the Fundamental Mode Sizes in Optical Channel Waveguides

Using Gaussian Quadrature Nicolas A. F. Jaeger and Benny P. C. Tsou

Abstract-A fast numerical method using Gaussian quadrature, which takes only seconds on a microcomputer, is presented for calculating the fundamental mode sizes in optical chan- nel waveguides. Variational expressions for the square of the propagation constant, 32. of the TE- and TM-like modes are derived using the vector wave equations. For channel waveguides with gradual refractive index distributions, these expressions approach the variational expression obtained using the scalar wave equation. To show the usefulness of our numerical technique we present the results for titanium indiffised lithium niobate channel waveguides, which are commonly used in integrated optical circuits. Since these waveguides have gradual refractive index distributions, both types of expressions give the same results; however, it takes less time to compute the mode sizes when using the variational expression obtained from the scalar wave equation. We find the calculated mode sizes are in good agreement with published measurements. From the comparison process, best fit parameters are obtained, which give mode sizes close to the values published in the literature. For one special case we are able to obtain an analytical variational expression and we use it to test the accuracy of our numerical method. We find that the values of 3’ given by both methods agree to six significant figures.

I. INTRODUCTION

INGLE mode channel waveguides fabricated in optical S substrates ,are used in integrated optical devices such as modulators, sensors, etc.. The optical field distribution in the transverse plane, i.e., the plane perpendicular to the direction of propagation, must often be known, e.g., when calculating the coupling efficiency between an optical fibre and such a waveguide [1]-[3] or the overlap parameter between the applied electric field and the optical field for a phase modulator [4], [5]. Here we present a fast numerical method for calculating the width parameters of the Hermite-Gaussian function that is often used in approximating the optical field distributions in such channel waveguides [ 11-[4]. The width parameters are directly related to the fundamental mode size ( l / e intensity) and are obtained using the variational prin- ciple [6]. In this work, using the vector wave equation for the electric field, variational expressions for the square of

Manuscript received June 30, 1992; revised October 13, 1992. This work was supported by grants from the Rogers’ Cable Labs Fund, the Science Council of British Columbia, and the Natural Sciences and Engineering Research Council of Canada.

The authors are with the Department of Electrical Engineering, University of British Columbia, Vancouver, B.C., Canada.

IEEE Log Number 9208359.

the propagation constant are derived for the TE- and TM- like modes. However, for channel waveguides with gradual refractive index distributions, which are often encountered in integrated optics, these variational expressions become identical to the one derived from the scalar wave equation. Since the variational expression obtained from the scalar wave equation has a much simpler form, and is valid for most cases, it is often used, e.g., in [7] and [8], to estimate the fundamental mode sizes of dielectric channel waveguides. In [7], for the particular case of the TM-like mode in z-cut, y- propagating Ti:LiNb03 channel waveguides, a closed-form variational expression was obtained to compute the width parameters. In’[8], it was mentioned that the integrals in the variational expression could be calculated numerically. Here we show that since the Hermite-Gaussian function is used to approximate the optical field distribution, Gaussian quadrature can be used to rapidly evaluate the integrals in the variational expressions; giving results that are virtually identical to the ones obtained analytically for the abovementioned particular case. However, our numerical method has the advantage that it may be applied to more general cases for which analytic expressions are not easily, or perhaps possibly, obtained.

Our method was implemented on a microcomputer for both the TE- and TM-like modes for y-propagating Ti:LiNbO3 channel waveguides fabricated in t-cut substrates so that we could compare the results with published measurements [9]. The two width parameters, for a particular mode, were determined in under a second for the variational expression obtained from the scalar wave equation and in a few seconds for the more complicated variational expressions obtained from the vector wave equations. We compared the width parameters calculated using both types of variational expres- sions and found them to be identical. However, given a particular refractive index distribution which varies in the two transverse directions the variational expressions obtained from the vector wave equations predicted slightly different propagation constants for the TE- and TM-like modes while giving the same width parameters; whereas the variational expression obtained from the scalar wave equation predicted the same propagation constant for both mode types.

Finally, we also obtained a closed-form variational expres- sion for the TM-like mode, which is similar to the one given in [7], for the type of channel waveguide we used in our simulation, so that we could compare the results obtained analytically with those obtained numerically.

0162-8828/93$03.00 0 1993 IEEE

1908 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 41, NO. 11, NOVEMBER 1993

11. THEORY The exact optical field distribution in the transverse plane

satisfies the vector wave equations [lo]

V:Et + (n2(x , y)k,2 - p2)Et = -v~[v, In n 2 ( z , y) . Et] ( 1 4

v ~ H , + (n2(x , y)k,2 - p 2 ) H t = (Vt x H t ) x V t In n 2 ( z , y)

(1b)

where ko is the free space wavenumber, n(rc,y) is the trans- verse refractive index distribution, Et and Ht are the trans- verse components of the electric and magnetic fields, V, and 02 are, respectively, the transverse grad and Laplacian operators, and p is the propagation constant in the z direction. Here we assume z is the lateral direction (Le., parallel to the surface of the channel waveguide) and y is the depth direction. Hence the mode is TE-like when the electric field is polarized primarily parallel to the x axis, and the mode is TM-like when the electric field is polarized primarily parallel to the y axis. With the above definitions, the vector wave equations for the electric fields can be written as d2E, d2E, a22+ay2

+ { d2j12n2 d l n E, dln n2 +-- a x dx

for the TE-like mode (or E, mode) and

d2E, d2E, d22+=

+ { d2p;2n2 d l n E, d l n n2 +-- dY dY

(3)

for the TM-like mode (or E, mode), where the subscript on E indicates the polarization direction of the dominant electric field component.

Variational expressions can be obtained from ( 2 ) and (3) if an a priori knowledge of the field distribution is assumed. Here we use the Hermite-Gaussian function

4 ( x , y ) = A (:,) - exp [ -- ; ( $ + $ ) I ,

to approximate the transverse optical field distribution of the lowest order mode, where w, and w, are width parameters defining the extent of the field distribution in the x and y directions. Equation (4) provides a reasonable description of the mode profile for Ti:LiNbOs channel waveguides [3], where the refractive index distribution is symmetric in the lateral direction and is strongly asymmetric in the depth direction, which is a result of a large difference in the substrate and superstrate indices. Similar functions have also been used in estimating the fundamental mode sizes in channel waveguides [7], [8], [ll]. The discontinuity of the refractive index distribution at the surface of the channel waveguide, Le., y = 0, does not affect the calculations since 4(xl y) vanishes at that boundary. However, the drawback of the Hermite- Gaussian trial function is that the evanescent fields have the form exp ( - r2 ) , whereas for most types of waveguides the fields actually decay as exp ( - r ) . Therefore, for calculations involving the evanescent fields, a different trial function, such as the one presented in [12], would have to be used.

It is shown later that the use of the Hermite-Gaussian function allows us to employ Gaussian quadrature to evaluate the necessary integrals in the variational expressions, making our method both fast and more general in nature.

Substituting the field approximation given by (4) for E, and E,, the variational expressions for p2 can be shown to be

0.5 1.5 p2= wf w;

F ( x . y ) L x p [-($ + $ ) ] d x d y L m L 4

for the TE-like mode [ l l ] and (see equation (6) at bottom of page) for the TM-like mode (note that since the trial function is an approximation to the actual field distribution, p2 as calculated provides a lower bound to the square of the actual propagation constant). In general, analytical solutions cannot be obtained for the integrals so that one has to resort to numerical integrations. Once ( 5 ) and (6) can be accurately evaluated, an optimization routine can be used to find the values of w, and w, for which p2 is a maximum.

JAEGER AND TSOU: CALCULATION OF THE FUNDAMENTAL MODE SIZES IN OPTICAL CHANNEL WAVEGUIDES 1909

The integrals in (5) and (6) are in precisely the forms which lend themselves to rapid and accurate evaluation using Gaussian quadrature. Consequently, the integrals are evaluated as

where the nodes .rh. and yh. are the roots of the Hermite poly- nomial HTL( . r ) [13] and the generalized Laguerre polynomial L~,$)(y’) (141, respectively. Here k is used as an index. The weights 1Jk and V k are

and

where UI; is taken from [13] and the above expression for Si. was derived for this work for even values of m. The values of V’k were checked against tabulated values in [ 141 calculated for all values of 711. We use even integers for both 7) and 111,

thus the nodes are distributed symmetrically in the :r+y plane, with none lying on the axes. Since the integrands are also even functions of both :I‘ and y. the number of summations in (7a)-(7c) can be reduced by a factor of 4.

The integers 11, and 711, determine the accuracy of the integra- tion which is related to the behavior of n(.c. g) in the integrand. Large integer values of 7) and 711 are required for strongly varying n ( z . g ) while for gradually varying n ( . r . g). which is often encountered in integrated optical circuits, smaller integer values are often sufficient. However, once calculated, for a particular choice of n. and VI,, the nodes and weights can be stored in a file for subsequent evaluations of the integrals. For the applications we are about to describe, setting both n and vi to 16 was sufficient to obtain w, and wy to three significant figures for both the TE- and TM-like modes (in fact, 10 was sufficient for the TM-like mode).

The variational expression derived from the scalar wave equation can be obtained from either equation (5 ) or (6) by setting all partial derivatives to zero, i.e.,

0.5 1.5

+ (.. 14) 211, wy

(9)

where the integral is evaluated as

For the channel waveguides given in the example below, equation (9) gave the same results for the width parameters as equations (5 ) and (6), with the [j*’s differing only in the fifth significant figures. As one would expect, (5) and (6) gave different values for 4’ for the TE-like mode than for the TM- like mode. However, to compute the difference in p2, which was usually very small, large integers, in the range of 30 to 40. had to be used for both n and m..

111. EXAMPLE: Ti:LiNbO3 CHANNEL WAVEGUIDE

Ti:LiNbOs channel waveguides are often used in inte- grated optic devices. Here y-propagating Ti: LiNbO3 channel waveguides fabricated in a z-cut substrate are simulated since the calculated mode sizes can be compared with published measurements [9]. For this work, the simulations were done on a 33-MHz 486 microcomputer.

In Ti:LiNb03 channel waveguides the increase in the re- fractive index is related to the concentration of the indiffused titanium ions. Assuming the deposited titanium has completely diffused into the substrate, a commonly used model for calcu- lating the titanium ion concentration, where we have replaced

by z as the axis perpendicular to the surface, is [15]

where

and where r is the prediffusion titanium thickness, W is the prediffusion titanium strip width, t is the diffusion time, T is the diffusion temperature in Kelvin, Do, and Do, are, respectively, the diffusion constants in the 2 and z directions, Q is the activation energy, and k~ is Boltzmann’s constant.

To be consistent with [9], we set A, = 1.3pm,n0 = 2 . 2 1 9 5 . ~ ~ ~ = 2.1450, the prediffusion Ti strip thickness to 700A for TE-like mode calculations and 750A for TM-like mode calculations, the diffusion temperature to 1025”C, the diffusion time to 6 h, and the Ti strip widths to values ranging from 2.5 to 10 pm.

The ordinary and extraordinary refractive indices of L i m o 3 are functions of the titanium concentration C. If their values are n,, and TI,, when C is zero, the change in ne is linearly

1910 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 41, NO. 11, NOVEMBER 1993

w A 0 TE-like m o d e

0.014 - 8.00 -

0.012 - C

x v)

W C

0 C

W

'- 6.00 - 0.010 - * .-

+- .-

4.00 -

0.008 - c

\ a

7

0.006 - v

W N ._ v) 2.00 - a, - -0 1

- E I 0.00

0.00 0.50 1.00 1.50 2.00 2.50 2.0

Fig. 1. Ordinary refractive index change, Ano. as a function of C(%).

a

- - b - -

\: a 0

- - d - -

Legend:

b full width lower bound calculated ful width c full depth upper bound o full depth in [9] d full depth lower bound calculated full depth

I I I I I

a full width upper bound full width in 191,

2.5 3.0 3.5 4.0 4.5

related to C, i.e., An, = aeC, while the change in no is not [16]. To obtain An, we fitted a fourth order polynomial to the data given in reference [I61 where linear extrapolation of their curve was used for large values of C [17]. The polynomial is

where a0 = -1.65 x 1 0 - 5 , a ~ = 1.05,a2 = -81.6,a3 = 3.88 x lo3, and a4 = -6.64 x lo4 and is plotted in Fig. 1.

Since there were various published data that could be used for the diffusion constants Do, and Do,, and An, and An, as functions of C, we used various combinations of these to obtain upper and lower bounds for our mode size calculations. Here we first assumed isotropic titanium diffusion [18]. Hence for the upper bounds we used the following data: Do, and Do, = 1.350 x 108pm2/h [19], a, = 0.625 [16], and (12) for An,. For the lower bounds we used: Do, and Do, = 7.88 x 107pm2/h [20], and a, = 0.760 [21]; as for An,, we used a scaling factor a, to obtain a least square fit of equation (12) to the piece-wise linear data for An, in [21] giving a, = 1.30. For all the calculations, we used 2.22 eV for Q [19].

The calculations were performed using both types of vari- ational expressions. The width parameters calculated by each respective method were found to be identical to 3 significant figures.

The calculated bounds, along with the published TE- and TM-like mode sizes, are shown in Figs. 2(a) and 2(b). The l / e intensity full widths and full depths were obtained by multiplying w, by 2 and wz by 1.38, respectively. The bounds for the TM-like modes show that there is a set of values for Do,, Do,, and a, within the published ranges for which a close match between the calculated and measured mode sizes can be obtained. However, the bounds for the TE-like modes are consistently lower than the measured mode sizes published, indicating that the values for An, are too large.

._

.- 4.00

W \

5 $ 1 2.00

TM-like mode

Leg end :

(same as figure 2a)

0.00 2, 0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2

Ti str ip width W ( p m )

(b)

Fig. 2. of the Ti strip width. (a) TE-like mode. (b) TM-like mode.

Comparison of the calculated and published mode sizes as functions

Using a least square criterion, the values of Do,, Do,, a,, and a, were varied to obtain a best fit between our calculated and the published TE- and TM-like mode sizes. Here one set of values was obtained for all the strip widths yielding the following results: Do, = 1.013 x 10' pm2/h, Do, = 1.350 x 108pm2/h, a, = 0.625 and a, = 0.82. Mode sizes calculated using these values are also shown in Figs. 2(a) and 2(b). It can be seen that the best agreement occurs for strip widths between 3-6 pm.

Using the above best fit parameters we graphically demon- strate the stationary nature of p2. Figures 3(a) and (b) show the plots of p2 versus w, and w, for the particular case of the 4 pm strip width for the TE- and TM-like modes, respectively.

JAEGER AND TSOU: CALCULATION OF THE FUNDAMENTAL MODE SIZES IN OPTICAL CHANNEL WAVEGUIDES

0.00

T E - I i k e mode

Max i mum ( 2 . 97 , 2 . 9 1 , 1 1 5 . 1 0 9 )

I I I 1 I I I I

(a)

TM-I i k e m o d e

Max i mum ( 2 . 22 , 2 . 16 , 107. 6 3 2 )

Fig. 3. Plots of 13' versus I P ~ and u', . using the parameters obtained from our least square fit procedure, for a 4 p m Ti strip width. (a) TE-like mode. (b) TM-like mode.

It is clear that p2 is a maximum for the indicated width parameters. The time taken to find the width parameters for the scalar expression was 0.33 s.

A similar analysis for the next higher order TM-like mode in the lateral direction indicates that it cuts in when W is about 6 pm. Discarding the data for W = 6 p m and above, due to the possible effect such a mode would have in the measurements, and again performing a least square analysis, better fits to the published measurements can be achieved by

A

5 C

x (I) C a, C

a,

.- 6.00 - w ._

+ .- 4.00 -

1911

e Q @ a

Legend: N

TE-like mode

calculated full width 0 full width in 91 * full width in [9]

calculoted fu l l width o full deoth in 191 A full deDth in r91

Fig. 4. Comparison of the calculated and published TE- and TM-like mode sizes as functions of the Ti strip width. Here, the measured data for strips 6 p m wide and above have been discarded.

using the following parameters: DAz = 1.238 x 1O8pm2/h, Db, = 1.350 x 108pm2/h, a', = 0.679, and ab = 0.84. Fig. 4 shows plots of the improved fits.

Iv. COMPARISON OF NUMERICAL AND ANALYTICAL RESULTS

To test the accuracy of our numerical technique, it is desir- able to find a case for which analytical results are obtainable. Indeed, we can do this comparison for the TM-like mode of the channel waveguide simulated in Section 111. Since the relationship between An, and C is linear, a closed-form variational expression based on the scalar wave equation can be obtained from (9)

"/ 2

rrf ( d z m J where 7 ~ ~ ( . r , z ) = 72: + 2n,n,C(.r. 2 ) .

Using the first set of best fit parameters, we recalculated all the width parameters using (13) for all the strip widths. We found the results were identical to those obtained by our numerical method to three significant figures. We also compared the values of p2 and found them to be identical to six significant figures, which was more than sufficient for the maximization process.

More often than not, an analytic expression, like the one shown above cannot be obtained; an immediate example is the TE-like mode of the Ti:LiNbOa channel waveguide where the relationship between An, and G is nonlinear. On the other hand, our numerical method is ideally suited for solving such cases.

1912 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 41, NO. 11, NOVEMBER 1993

V. CONCLUSION A fast numerical method for calculating the fundamen-

tal mode sizes in optical channel waveguides is presented. Applying it to Ti:LiNbO3 channel waveguides gave results that both fitted the measured data well and exhibited the same behaviour as that exhibited by the measured data as the strip widths varied. We also obtained parameters that allowed better fits to the measured mode sizes than those found in the literature. We showed that the method is accurate by comparing the numerical results with analytical results. We found our numerical method to be more general, allowing i t to be applied to other types of channel waveguides,

REFERENCES

[l] W. K. Burns and G. B. Hocker, “End fire coupling between optical fibres and diffused channel waveguides,” Applied Optics, vol. 16, no. 8, pp. 2048-2050, 1977.

[2] R. C. Alferness, V. R. Ramaswamy, S. K. Korotky, M. D. Divino, and L. L. Buhl, “Efficient single-mode fibre to titanium diffused lithium niobate waveguide coupling for X = 1.32fim,” IEEE J . Quantum Electron., vol. QE-18, no. 10, pp. 1807-1813, 1982.

[3] R. Keil and F. Auracher, “Coupling of single-mode Ti-diffused LiNbO3 waveguides to single-mode fibers,” Optics Commun., vol. 30, no. 1, pp.

[4] D. Marcuse, “Optimal electrode design for integrated optics modula- tors,” IEEE J. Quantum Electron., vol. QE-18, no. 3, pp. 39-3-398, 1982.

[SI R. C. Alfemess, “Waveguide electrooptic modulators,” IEEE Trans. Microwave Theory Tech., vol. MlT-30, no. 8, pp. 1121-1137, 1982.

[6] M. J. Adams, An Introduction to Optical Waveguides. New York: Wiley, 1981.

(71 S. K. Korotky, W. J. Minford, L. L. Buhl, M. D. Divino, and R. C. Alferness, “Mode size and method for estimating the propagation constant of single-mode Ti:LiNbOs strip waveguides,” IEEE J . Quantum Electron., vol. QE-18, no. 10, pp. 1796-1801, 1982.

[8] D. Marcuse, “Trading coupling loss for modulation efficiency in elec- trooptic modulators,” IEEE J. Quantum Electron., vol. QE-23, no. 3,

[9] P. G. Suchoski and R. V. Ramaswamy, “Minimum-mode-size low-loss Ti:LiNbOa channel waveguides for efficient modulator operation at 1.3 pm,” IEEE J. Quantum Electron., vol. QE-23, no. 10, pp. 1673-1679,

23-28, 1979.

pp. 352-358, 1987.

.. 1987. A. W. Snyder and J. D. Lave, Optical Waveguide Theory. New York: Chapman and Hall, 1983. N. A. F. Jaeger and L. Young, “Voltage-induced optical waveguide modulator in lithium niobate,” IEEE J. Quantum Electron., vol. QE-25, no. 4, pp. 720-728, 1989. A. Ankiewicz and G. D. Peng, “Generalized Gaussian approximation for single-mode fibers,” J. Lightwave Technol., vol. LT-10, no. 1, pp. 22-27, 1992. F. B. Hildebrand, Introduction to Numerical Analysis. New York: Dover, 1974.

114) A. H. Stroud and D. Secrest, “Approximate integration formulas for certain spherically symmetric regions,” Math. Computation, vol. 17, pp. 105-133, 1963.

1151 G. B. Hocker and W. K. Bums, “Mode dispersion in diffused channel waveguides by the effective index method,” Appl. Opt., vol. 16, no. 1, pp. 113-118, 1977.

[16] M. Minakata, S. Saito, M. Shibata, and S. Miyazama, “Precise de- termination of refractive-index changes in Ti-diffused LiNbO3 optical waveguides,” J. Appl. Phys., vol. 49, no. 9, pp. 4677-4682, 1978.

(171 M. D. Feit, J. A. Fleck, and L. McCaughan, “Comparison of calculated and measured performance of difFused channel-waveguide couplers,” J . Opt. SOC. Amer., vol. 73, no. 10, pp. 1296-1304, 1983.

[l8] R. J. Holmes and D. M. Smyth, “Titanium diffusion into LiNbO3 as a function of stoichiometry,” J . Appl. Phys., vol. 55, no. 10, pp. 3531-3535, 1984.

[19] S. Fouchet, A. Carenco, C. Daguet, R. Guglielmi, and L. Riviere, “Wave- length dispersion of Ti induced refractive index change in LiNbO3 as a function of diffusion parameters,” J. Lightwave Technol., vol. LT-5, no. 5, pp. 700-708, 1987.

[ZO] K. Sugii, M. Fukuma, and H. Iwasaki, “A study on titanium diffu- sion into L i m o 3 waveguides by electron probe analysis and X-ray diffraction methods,” J. Materials Scc., vol. 13, pp. 52S533, 1978.

[Zl] K. T. Koai and P. L. Liu, “Modeling of Ti:LiNbO3 waveguide devices: Part I-Directional couplers,” J. Lightwave Technol., vol. LT-7, no. 3, pp. 533-539, 1989.

Nicolas A. F. Jaeger was born in New Rochelle, NY, in 1957. He received his B.Sc. degree from the University of the Pacific, Stockton, CA, in 1981, and the M.A.Sc. and Ph.D. degrees from the University of British Columbia, Vancouver, B.C., Canada, in 1986 and 1989, respectively, all in electrical engineering.

Since 1989 he has been an Assistant Professor in the Department of Electrical Engineering, Univer- sity of British Columbia, and since 1991 he has been the Director of the University’s Center for Advanced

He has been a Member of the Science Council of British Columbia’s Peer Technology in Microelectronics.

Review committee since 1991.

Benny P. C. Tsou was born in Taipei, Taiwan, in 1965. He received the B.A.Sc. degree in electrical engineering in 1988 and the B.Sc. (Hon.) degree in physics in 1990, all from the University of British Columbia, Vancouver, B.C., Canada. Presently, he is a candidate for the M.A.Sc. degree in electrical engineering at the University of British Columbia, where his research interests include numerical mod- elling and fabrication of optical waveguides and modulators.