Weighted-index beam-propagation method for analysis of three-dimensional optical structures

D.A.M.Khalil T.A.Amer

btdexing terms: Three-dimensional optical structures, Beam-propagation method, WeiEhted-index method

Abstract: The three-dimensional beam propagation method (3DBPM) is very slow and requires a huge amount of computer memory. A new beam-propagation technique for the analysis of three-dimensional integrated optical structures is presented. This technique depends on the use of the weighted-index method to transform the three-dimensional BPM into two coupled two- dimensional BPM techniques. Numerical results show good agreement with the standard 3DBPM while both the computational time and the memory required are greatly reduced.

1 introduction

The beam-propagation method (BPM), first introduced by Feit and Fleck [1-31, is now well known as a powerful technique for the analysis of integrated optical structures [4]. It has been widely used to analyse different passive and active components such as tapers [ti], directional couplers, electrooptic modulators [6] etc. fabricated on both semiconductor and dielectric substrates. Since all the integrated optical structures are three-dimensional, having a refractive index which varies with x, y and z, it is necessary to use a three- dimensional version of the beam-propagation method (3DBPM). The disadvantages of this method come from the large computer memory it needs and the long calculation time. To obtain some idea of magnitudes involved in the discussion, consider a two-dimensional beam-propagation method showing a time complexity zLD in comparison with an equivalent three-dimensional BPM with a time complexity z3D. If it is assumed that both algorithms apply the same technique (FFT or FD), the relation [7]

(1) 7 3 D

T2D - 2 Ny

is obtained, where Ny is the number of discretisation points in the other transverse direction ('J direction). The equality in eqn. 1 corresponds to a linear time complexity. This means that the CPU-time of a three-

@ IEE, 1997 IEE Proceedings online no. 19971274 Paper first received 9th December 1996 and in revised form 1st April 1997 The authors are with the Ain Shams University, Faculty of Engineering, Electronics and Communications Engineering Department, 1 El-Sarayat St., Abbasia (11517), Cairo, Egypt

dimensional BPM simulation is at least N times that for the two-dimensional BPM simulations !or the same problem. Thus it is strongly recommended that two- dimensional BPM be used whenever possible.

From a theoretical point of view, the two- dimensional BPM can be only applied to beam propagation in slab-waveguide structures, i.e. when the structure is homogeneous in one transverse direction. However, very few interesting applications of this type are known. For practical applications, the two- dimensional BPM is usually used on a two-dimensional effective-refractive-index profile which is a projection of the entire three-dimensional device [7]. This projection can be carried out using the effective-index method. Such a technique, consisting of using both the effective-index method to convert the three-dimensional structure into a two-dimensional one and then the two- dimensional BPM, is called here the EIBPM. However, this technique is limited to structures which are z- invariant in one transverse dimension where z is the propagation direction, and in addition the effective index method has its inherent limitations 181.

For these reasons a new technique is proposed in this paper for the analysis of three-dimensional structures. The technique is based on the use of the weighted- index method (WIM) [9-111 with the beam- propagation method to form an efficient three- dimensional-propagation technique. In this technique, a two-dimensional BPM is used in two perpendicular planes and in each of these planes a weighted index (a refractive index weighted by the field distribution in the other plane) is used as an effective index seen in the BPM. The beam propagation in the two perpendicular planes is thus coupled through the weighted index which is recalculated at each propagation step. This method is referred to as the WIBPM, Beginning with the theoretical base of the proposed method, explained through its mathematical formulation, this paper considers in detail the application of the method in different situations. Both fast-Fourier-transform (FFT) BPM and finite-difference (FD) BPM may be used for the two-dimensional propagation in the WIBPM. In this work the classical FFT-based BPM is used to test the method.

2 Theoretical analysis

The weighted-index beam-propagation method (WIBPM) is based on the use of separation of variables such that the field propagation, assumed to be in the z direction, can be treated in the xz and y z planes, independently. The concept used in the modal-

197 IEE Proc.-Optoelectron.. Vol. 144, No. 4, August 1997

weighted-index method [9-111 is then employed to calculate the effective refractive index of a waveguide. However, instead of solving, by iteration, two modal equations to obtain the modal propagation coefficient, two propagation equations are solved which are coupled to each other through the field distribution. To explain this idea, the scalar paraxial Helmholtz equation is written as:

, d E d2E d2E 2jkon,- = ~ + - + k,{n2(x, y, z) - n:)E dz ax2 ay2

( 2 ) where, for the paraxial approximation, it is assumed that

( 3 )

Using the separation of variables, one can write the field E(x, y , z ) in the form

E(? Y, 4 = F ( x , 4 G ( Y , .) (4) which, when used in the paraxial Helmholtz equation, gives

d 2 F d2G - - = G + -F dy2 + k o { n 2 ( x , y , z ) - n2)FG

(5) While separation of variables is used, eqn. 5 still cannot be represented in the form of two separate equations in x and y . This is due to the functional dependence of the refractive-index distribution n(x, y , z). To separate eqn. 5 into two equations, the equation is multiplied by G(y) and integrated on y , giving

Then using the paraxial approximation (eqn. 3) again on the function Gb), the term dG/az with respect to nokoG may be neglected, giving

dF d2F 2jk,n,- = ~

d z 8x2 s n 2 ( x , y, z)G2dy + $ G e d y

J G2dY

(7 ) Eqn. 7 can then be rewritten as a paraxial-wave equa- tion for F(x):

where

and a similar expression could be obtained for GCy) by multiplying eqn. 5 by F(x) and integrating with respect to x to obtain

. dG d2G 2 j k n - = ~ + k;{n:(y, z) - n2)G (10) dz dy2

where

k: J n2(x , y, z )F2dz - J ( dz

Now, eqn. 8 represents a Helmholtz wave equation in two dimensions, which may be solved using the FFT- based BPM or the finite-difference BPM in its two- dimensional version to obtain F(x, z, + Az) if F(x, zo) is known. However to solve this equation it is necessary to know the function G(y, z) to calculate the weighted index nl(x) using eqn. 9. On the other hand, G(y, z) could be obtained from eqn. 10 which is also a two- dimensional Helmholtz wave equation in GCy) with a refractive index n2@) weighted by F(x, z). Thus we have two beam-propagation equations coupled through the weighted indices nl(x) and nzb) . Using the field at each propagation step to modify the weighted refrac- tive-index distribution at the next step, the field could be advanced mutually in two planes. Begining with the field GO, zo)F(x, z,), Gb, zo) is then used to calculate a weighted refractive index nl(x) as given by eqn. 9. A 2DBPM technique could then be used to solve eqn. 8 to obtain F(x, z, + Az) which can be used to calculate the weighted refractive index n2b) using eqn. 10. Once the index is calculated, it can be employed to find the field Gb, z, + Az) by using again the 2DBPM tech- nique. The total field after AZ propagation is then

E(x , y, z,+Az) = F ( x , z ,+Az)G(y, z,+Az) (12) and the solution could be advanced to the end of the structure.

Note that, for each propagation step Az, the 2DBPM is applied twice: in the xz plane and in the yz plane. This would result in a calculation time (and a memory size) which is double that required for a two-dimen- sional BPM. However, because of the use of numerical integration in each propagation step, the time required is expected to be five or six times that of the two- dimensional BPM, but still much less than the time required for the 3DBPM. On the other hand, if a com- putational window consists of M points in the y direc- tion and N points in the x direction, in the 3DBPM it is necessary to analyse N x A4 points of the field while in the WIBPM only N + A4 points need to be analysed, which means an important reduction in the memory requirements by a factor of ( M x N)/(M + N). This basic difference is due to the separation-of-variables approximation.

3 Application of the WIBPM

3.7 Rectangular directional coupler To compare the method presented in this paper with the conventional 3DBPM and EIBPM, it was applied to the directional-coupler structure whose cross-section

IEE Pvoc -0ptoelectvon Vol 144, No 4, August 1997 198

is shown in Fig. 1. This is a hypothetical buried- waveguide structure which can be used as a test structure, since it avoids the numerical problems encountered in the conventional BPM when dealing with strong step-index distribution. It also enables the calculation time in a three-dimensional BPM to be reduced. For such a structure, the coupling length is quite sensitive to parameter variations and thus it can be used as a test value. A study of a directional coupler with a strong step-index distribution is given in Section 3.3

d d ' Y A

*i-----? - z=c

4 S

, ...

X

z

Fig. 1 lure with n, = 3.405, n2 = 3.4, d = 4 pn, t = 3 p

Cross-section of the considered buried directional coupler struc-

To calculate the coupling length L, of the structure it was excited by a Gaussian beam centred at the left- hand guide (I) described by

With propagation, the field E(x, y , zo) is calculated at ;any z = z , and projected on the input field Ein(x, y , 0) = Fiu(x, 0) Gin(y, 0). The projection coefficient defined ;1S

d Z 0 ) z 1 pi&, O N x , z0)dz J G d y , O)G(y, 4 d Y

l 2 p i n ~ z , 0 ) 1 2 d z J 0 IGzn(y,0)12~y

DY

0 - -

(14)

DY

is then calculated, where D, and Dy are the dimensions of the calculation window in the x and y directions. When this projection coefficient is plotted as a function of z , it can be used to determine the coupling length of the directional coupler. This is done as follows. When q is maximum, the field is mainly in the first guide (left-hand guide). With propagation, the field maxi- inum is displaced horizontally as shown in Figs. 2 and 3 . Finally after a coupling length of propagation, the field is totally transferred to the other guide (right-hand guide) and q is minimum. Thus, a plot of the projec- tion coefficient showing maxima and minima may be used to calculate the coupling length of the coupler.

A typical plot of q as a function of z , calculated using the WIBPM, for a coupler having a spacing s = 4 1x11 between the two guides, is shown in Fig. 4. It shows successive maximum and minimum values, as expected. The propagation distance between a

I'EE Proc.-Optoelectron.. Vol. 144, No. 4, August 1997

maximum and a minimum could then be considered as the coupling length. This technique has been used previously for planar structures and comparison with the theoretical coupling lengths has shown perfect agreement [12]. It has the advantage that it is not necessary to calculate the eigenmodes of the two coupled guides, which is not an easy task for two- dimensional dielectric waveguides.

4

. . Fig. 2 buried directional-coupler structure calculated with WIBPM

Quasi-3-dimensional plot of the field as it propgates through the

*, P.m Fig.3 directional-coupler structure calculated with WIBPM

Contour plot of the field as it propagates through the buried

0 10 20 30 LO 50

Projection coeficient with propagation distance calculated by the propagation distance z, mm

Fig. 4 WIBPM for a buried directional coupler

199

A comparison between the projection coefficient q calculated using the WIBPM, the 3DBPM and the EIBPM as a function of z , for a coupler having a spacing s = 4pm, is shown in Fig. 5. An 80pm x 6 0 p n computational window in the x and y directions is used. The operating wavelength is 1.3 pm and the axial discretisation is taken to be Az = 4 q . The number of mesh points used is 128 x 128 for the 3DBPM, 256 for the EIBPM and 256 f 256 for the WIBPM. The numerical parameters are all typical parameters used in BPM calculations for such a step-index guide. The calculation time per propagation step is found to be 40s for the 3DBPM, 1.5s for the WIBPM, and 0.25s for the EIBPM on a 486DX-33 personal computer and using the fast-Fourier-transform technique. At small propagation distances a sharp decrease in the value of q is noticeable. This is because part of the field energy is transferred by the radiation modes leaving the structure while only the power carried by the guided modes remains in it. Notice also that, at small propagation distances, a large difference exists between the value of q calculated by the EIBPM and that calculated by the WIBPM or the 3DBPM. This is because the radiation in two dimensions greatly differs from that into three dimensions. Good agreement between the 3DBPM and WIBPM can also be noted.

0 1 2 3 L 5 6 7 propagation direction z,mm

Projection coeflcient 7 or guide se aratwn s Fig.5 directional coupler using the 3DB$Mj the W h P i and the EIBPM (i) EIBPM (ii) WIBPM (iii) 3DBPM

4 pm of a buried

1 i

8 7 + - - - 7 - - 1 - o ~ " " l " . . ~ " . . ~ . ~ . . I

2 3 4 5 6 spacing S, pm

Coupling length of the buried directional coupler against the sepa- Fig. 6 ration between guides calculated by 3DBPM, WIBPM and EIBPM -0- WIBPM -X- EIBPM -0- 3DBPM

Fig. 6 shows the coupling length of the directional coupler calculated by these three techniques for differ- ent separations. The results shows that the WIBPM fits better with the 3DBPM, especially at low separations. On the other hand, the EIBPM gives an underestima- tion of the coupling length.

200

3.2 Rectangular taper structure Consider now a real three-dimensional structure in which the cross-section of the guiding structure has variations in both transverse directions (x and y ) with propagation. This is the typical case of the rectangular taper shown in Fig. 7. For such structures the EIBPM cannot be used. The structure is assumed to be a buried one with a core refractive index of 3.405 and a back- ground refractive index of 3.4. The dimensions of the input guide are t l = 3 p and dl = 4pm while for the output guide t2 = 6 pn and d2 = 8 pm.

......................... ....., , t2;1 tl/p z = o Ix z

2000 ~ ,

2 00 Schematic diagram of the rectangular taper structure studied Fig.7

The input guide is excited by a Gaussian field distri- bution, at 1.3 pm wavelength, described by

The WIBPM and the 3DBPM are then used to calcu- late the maximum field at the centre of the taper as it propagates down through the structure. This could be used as an indicator to characterise the field propaga- tion. The variations of the field maximum with the propagation distance calculated by the 3DBPM and the WIBPM are shown in Fig. 8. Good agreement can be observed.

0- ...............

: i

0 500 1000 1500 2000 2500 Dropaaation distance z.um . .~

Fig.8 (i) 3DBPM, (ii) WIBPM

Maximum field at the centre of the taper structure

These results are obtained using a window of 80pm x 6 0 q in the x and y directions, respectively. The step size in the propagation direction is Az = 4 ~ . The number of points for the WIBPM is 256 x 256 and for the 3DBPM is 128 x: 128. This shows that the WIBPM could be used to predict the performance of real three- dimensional structures with good accuracy.

3.3 Rib directional coupler The BPM is known to be an accurate tool for modelling propagation in waveguides with small refractive index variation. Intuition suggests, on the other hand, that the BPM is inappropriate for waveguides such as rib waveguides which exhibit large refractive-index discontinuities. Here the applicability of the WIBPM for analysing such structures is tested.

IEE Proc -0ptoelectron , Val 144, No 4, August 1997

The method is applied on two semiconductor-rib directional couplers to calculate their coupling lengths. These structures were presented by Robertson et al. [13] as test structures and have been studied by the effective-index method [ 131, the finite-difference method [13] and the three-dimensional BPM [14]. Fig. 9 together with Table 1 shows the structure and the parameters of the directional-coupler devices to be analysed here.

w s , I . y ” w ) “ 3 4

Fig. 9 Semiconductor-rib directional-coupler structure to be analysed

Table 1: Parameters of the rib directional couplers stud- ied

Structure n, n2 n3 t, tz w s

1 3.44 3.435 1 2.5 3.5 4 2

2 3.44 3.36 1 0.1 0.9 3 2

Table 2: Comparison of the coupling length of both the directional-coupler structures used using WIBPM, finite- difference, EIM and 3DBPM

Structure 1 Structure 2 (mm) (mm)

Method

WIBPM 1.112 0.794

FD(1) [ I l l 1.273 0.797

W 2 ) [ I l l 1.347 0.81 1

EIM [ I l l 1.152 0.807

3DBPM [I21 0.930 0.710

Both structures are designed for operation at a wave- length A = 1.55 pn, and the refractive-index values used correspond to those in the GaAsiGaAlAs material sys- tem. Structure 1 is used to provide a low insertion loss for coupling to an optical fibre due to the large dimen- sions of the guide. The guiding layer is relatively thick and the stripe width and height are adjusted to give a more symmetric mode shape. Structure 2 has a very small rib height, allowing the mode to extend laterally, which is useful for directional-coupler structures since strong coupling between adjacent guides will result in short coupling lengths.

Both structures are excited with a Gaussian field, centred at the left-hand guide, with width w in the x- direction and ( t l + t2) in the y direction. The projection coefficient is evaluated at each step of propagation to calculate the coupling length, as explained above. The calculations for both structures are performed using a step size of 0.1 pn and a number of points of 256 x 256 in the transverse direction. The window for structure 1 is taken to be 8 0 ~ x 8 0 ~ , while for structure 2 it is 80pm x 12pm in the x and y directions, respectively.

Table 2 shows the values of the coupling lengths cal- culated by the WIBPM together with those calculated

IEE Proc.-Optoelectron., Vol. 144, No. 4, August 1997

by the effective-index method, finite difference method and 3DBPM, as reported in the literature.

FD( 1) and FD(2) are both finite-difference techniques but they differ in the application of the boundary conditions [13]. The 3DBPM is applied with a very small step size Az = 0.0625 and very small Ax and Ay [14]. The WIBPM gives comparable results with large step sizes. This is due to the effect of smoothing of the refractive index by the optical field distribution. From Table 2, it can be seen that the results of the WIBPM are the nearest to those calculated by the 3DBPM in the two cases. However, this should be accepted with caution because the BPM may not be the most suitable technique for such a structure with very strong step index. This ensures that, if there is an inaccuracy in the results, it may be due to the BPM limitations rather than the WIBPM algorithm.

4 Conclusion

A new BPM technique for the analysis of three- dimensional optical integrated structures has been presented. This technique, called here the weighted- index beam-propagation method (WIBPM), is based on the application of the two-dimensional BPM in two perpendicular planes with the refractive index weighted by the field in the other plane. First, the technique is tested on a directional coupler to calculate its coupling length and results are compared with those obtained by standard 3DBPM and EIBPM. Results of the WIBPM are more accurate than those of the EIBPM where the 3DBPM results are taken as a reference.

In addition, this technique is used for the analysis of a real three-dimensional structure in which the EIBPM cannot be applied. A buried rectangular taper is thus considered as a test structure to calculate the field max- imum with propagation. The results are very close to those of the three-dimensional BPM.

Finally, the WIBPM is applied to a semiconductor- rib directional coupler and compared with other numerical results in the literature. Again, good agree- ment with the 3DBPM is obtained. In addition, a larger step size may be used in the WIBPM due to weighting of the refractive index by the field.

Compared with the 3DBPM, the WIBPM provides accurate results with a much smaller memory requirement and time consumption. If N x M points are used in the analysis, then the memory required for the 3DBPM is proportional to N x M , while for the WIBPM it is only N + M. Optimisation techniques in the programming could also reduce the memory required for WIBPM much more. In addition, the WIBPM is much faster than the 3DBPM. For a practical case of 256 x 256 points used for a buried directional coupler, the time required in the 3DBPM is about 140 times that required for the WIBPM.

5 Acknowledgment

We record our grateful thanks for many helpful and comments from Prof. Dr. Hani F. Ragaie, Ain Shams University.

6 References

1 FEIT, M.D., and FLECK, J.A.: ‘Light propagation in graded- index optical fibers’, Appl. Opt., 1978, 17, (24), pp. 3990-3998

201

2 FEIT, M.D., and FLECK, J.A.: ‘Computation of mode proper- ties in optical fiber waveguides by a propagating beam method’,

FEIT, M.D., and FLECK, J.A.: ‘Computation of mode eigen- functions in graded index optical fibres by the propagating beam method’, Appl. Opt., 1980, 19, (13), pp. 2240-2246

4 BEATS, R., KACZMARSKI, P., and VANKWIKELBERGE, P.: ‘Design and modeling of passive and active optical waveguide devices’ in MARSH, J.H., and DE LA RUE, R.M. (Eds.): ‘Waveguide optoelectronics’ (Kluwer Academic Publishers, 1992) KHALIL, D., and TEDJINI, S.: ‘Coherent coupling of radiation modes in Mach-Zehnder electrooptic modulators’, ZEEE J. Quan- tum Electron., 1992, 28, (5), pp. 1236-1239

6 DUPORT, I., BENECH, P., KHALIL, D., and RIMET, R.: ‘Study of linear taper waveguides made by ion-exchange in glass’,

7 MARZ, R.: ‘Integrated optics: design and modeling’ (Artech House, 1995)

8 KUMER, A., CLARK, D.F., and CULSHAW, B.: ‘Explanation of errors inherent in the effective index method for analyzing rec- tangular core waveguides’, Opt. Lett., 1988, 13, (12), pp. 1129- 1131

AppZ. Opt., 1980, 19, (7), pp. 1154-1164 3

5

J. PhyS.-D: Appl. Phys., 1992, 25, (6), pp. 913-918

9 KENDALL, P.C., ADAMS, M.J., RITCHIE, S., and ROBERT- SON, M.J.: ‘Theory for calculating approximate values for the propagation constants of an optical rib waveguide by weighting the refractive indices’, TEE Proc. A, 1987, 134, (8), pp. 699-702

ROY, P.W., and ADAMS, M.J.: ‘The weighted index method, a new technique for analyzing planar optical waveguides’, J. Light- wave Technol., 1989, 7, (12), pp. 2105-2110

11 BENSON, T.M., and KENDALL, P.C.: ‘Variational techniques including effective and weighted index methods’ in KONG, J.A., and HUANG, W.P. (Eds.): ‘Progress in electromagnetic research’ (EMW Publishing, 1995), vol. 10, pp. 1-40

12 KHALIL, D.: ‘Les modes rayonnes en optique integree: analyse et applications’. PhD thesis, Institut National Polytechnique de Grenoble INPG, France, 1993

13 ROBERTSON, M.J., RITCHIE, S., and DAYAN, P.: ‘Semicon- ductor waveguides: analysis of optical propagation in single rib structures and directional couplers’, ZEE Proc. J , 1985, 132, (6), pp. 33G-342

14 FEIT, M.D., and FLECK, J.A.: ‘Analysis of rib waveguides and couplers by a propagating beam method’, J. Opt. Soc. Am., A, 1990, 7, (l), pp. 73-79

10 ROBERTSON, M.J., KENDALL, P.C., RITCHIE, S., MCIL-

202 IEE Puoc.-Optoelectron., Vol. 144, No. 4, August 1997