Sing u larity-corrected spectra I index met hod

A. Vu kovic I? Sewell S.Sujecki T.M.Benson P.C. Kendall

Indexing tesms Dielectric rib waveguides, Spectral index method

Abstract: The spectral index method (SIM) for analysis of the rib waveguide in air is improved by including the singular behaviour of the principal field component in the vicinity of the re- entrant corners, thereby giving the singular SIM (SSIM). This generally improves the value of the modal propagation constant as compared with more exact solutions. The new approach is successfully extended further to the rib waveguide coupler.

1 Introduction

The dielectric rib waveguide is ubiquitous in modern optical integrated circuits. The typical structure of the rib waveguide is shown in Fig. 1. The refractive indices of the cladding layer, guiding layer and substrate are n l , n2 and n3, respectively, with n2 > n3 > nl . In the present work we assume that all these refractive indices are real, i.e. that the propagation is lossless. As a low- loss structure that is relatively easy to fabricate, the rib waveguide is widely used as a basic signal carrier as well as finding application in devices such as tapers, splitters, lasers and modulators. To realise the full potential that this structure offers, the designer is dependent on finding quickly highly accurate values for the propagation constants and valid field profiles. The spectral index method (SIM) achieves both accuracy and speed with surprising ease and simplicity

"2 I D n3

Fig. 1 lateral crowsection

0 IEE, 1998 ZEE Proceedzngs online no. 19981719 Paper first received 6th May and in revised form 2nd September 1997 The authors are with the Department of Electrical and Electronic Engi- neenng, The University of Nottingham, University Park, Nottingham NG7 2RD, UK

Dimensions and sefiuctive index distribution of sib waveguide's

Other methods available for calculating design parameters may be classified as either semianalytical or purely numerical. The semianalytical methods often achieve excellent results. One such method that is very simple and widely used is the effective index method (EIM) [l, 21. However, for some rib waveguide struc- tures the EIM is found to be inaccurate, especially near cutoff. Furthermore, the field profiles produced are only of limited use. The weighted index (WI) method is an extension of the EIM, which improves accuracy and extends the range of structures that can be analysed to cover most rib waveguide geometries. Nevertheless the general accuracy of this method is limited [3] since it assumes field solutions to be separable in the two transverse directions. Other semianalytical techniques, including approaches based on series expansion and mode matching, are reviewed in [4]. The other class, consisting of purely numerical methods such as the finite difference (FD) [5, 61 and finite element (FE) [7] methods, is computationally and memory intensive. Regarding these two classes of methods, the semiana- lytical methods are very efficient but of limited accu- racy, whereas the numerical methods are accurate but of limited use within an iterative design environment. The SIM is a compromise between these two extremes for the semiconductor rib waveguide in air and pro- duces very accurate results in a matter of seconds [8- 121. As noted in [13] the SIM is one of a number of highly accurate waveguide analysis methods based on Fourier techniques which include the Fourier operator transform (F-OPT) [14, 151 and generalised Fourier variational (GFV) [16] methods.

Certain modern applications such as the meander coupler require a vectorial approach, which in turn demands that particular attention be paid to the behav- iour of fields in the vicinity of dielectric corners of high refractive index contrast where transverse electric field components became singular [ 171. The standard SIM does not include explicitly the singular nature of the field component around the dielectric corners. The solution of electromagnetic problems, such as this, in the presence of a sharp dielectric corner has been widely analysed [ 17-21]. Incorporation of the correct local field behaviour is of great importance in analysing the vector modes of waveguide structures including such features [17]. In the presence of dielectric corners, the electric field exhibits a singular behaviour produced by its transverse components, which results in a more complicated field distribution. In this paper we investi- gate an approach for incorporating this singularity explicitly in the SI algorithm. The proposed method is

59 IEE pro^ -0ptoelettron , Vol 145, No I , February 1998

very simple and efficient and gives values of propaga- tion constants closer to those obtained by the purely numerical methods, without the need to consider the minor field components. The proposed method can be readily implemented and incurs only a minimal increase in run time when compared with the standard SI method.

2 Spectral index method (SIM)

In this Section we give a short overview of the SI method [8] and then proceed in Section 3 to include the singular nature of the field components around dielec- tric corners.

The SIM is a Fourier spectral technique, and it solves the two-dimensional Helmholtz wave equation,

where E(x, y , z) = E(x, y)e-jP’, E(x, y ) is the principal polarised electric field profile, p is the propagation con- stant, and if A is the free space wavelength and n, the local refractive index then ki = 2mi/;l.

Fig.2 new position on which the boundvy condition becomes E = 0

The way in which the osition o j the rib is moved outwur& to a

The SI method replaces the original rib structure by an ‘effective’ structure (Fig. 2) by displacing its actual physical dimensions to new ones on which the optical field is set to zero. In this way, the penetration of the optical field into the cladding is modelled. The concept of effective boundaries is most accurate for structures with small field penetration such as those with semi- conductor-air interfaces. For a waveguide fabricated in a 111-V semiconductor system the refractive index step at the aiddielectric discontinuity is approximately 3: 1.

In Fig. 2, the co-ordinate origin is at the centre of the base of the rib, with the x-axis taken horizontally and the y-axis vertically. The total space in the rib waveguide is divided into two regions: the rib region GI (-H

Eqn. 1 is solved independently in the two regions Ql and Q2. At the boundary between these regions continuity of E is imposed but the derivative JElJy is allowed to be discontinuous. The error in p is minimised by employing a variational boundary condition at the interface between the regions (see Appendix, Section 8).

For the structure shown in Fig. 2 the effective boundaries are [8-121: for the TE mode,

y < 0) and the layered slab region Q2 (y > 0).

W ’ = W + A , D ’ = D + A , H ’ = H (2) while for the TM mode,

where W ’ = W + A , D ’ = D + A , H ’ = H ( 3 )

These penetration depths A , , correspond to the effec- tive penetration depths of tangential ( t ) and normal (n) electric fields, respectively. The advantage of using this effective structure is that the wave eqn. 1 is now sepa- rable in regions RI and Q2. The field in region Ql (inside the rib) is approximated laterally by its funda- mental Fourier coefficient. Thus we write

Y) = F(z)G(y) (5) where, for the fundamental symmetric mode,

(6) 7r F ( z ) = cos(slz) with sl = ~

2W‘ and for the fundamental antisymmetric mode,

(7) 7r F ( z ) = sin(s2z) with s a =

The analysis is similar for other modes. The corresponding functions Gb) are

W

where 7/1,2 = ( k j - s22 - p2)1’2. Note that 7/1,2 may be imaginary, but eqn. 8 remains real.

We now restrict further description to the fundamen- tal symmetric mode and find a transcendental equation for p.

In the region G2 beneath the rib a Fourier transform (FT) with respect to x is used to reduce the dimension- ality of the wave equation. Thus, using @(s, y ) to repre- sent the FT of E(x, y) gives

where

The solution of eqn. 9 is written as

where Q ( S , Y) = f ( s ) g ( s , Y) (11)

and for 0 < y < D g ( s , y) = A sin(r2y) + B cos(r2y)

with r2 = (k,” - s2 - p2)W (13)

g(s ,y) = Cexp(r3g), r3 = ( P ~ + S ~ - ~ F $ ) ~ J ’ ~ (14) while for D < y < 00

The solutions in the regions .RI and Q2 are matched using a variational principle [9-121 derived in the Appendix (Section S), namely

Thereby a transcendental equation for p is formed, namely

60 IEE Pvoc -Optoelectron., Vol. 145, No 1, February 1598

where the normalised gradient function just below the base of the rib is IQ), given by

An.t- , "1

The dispersion eqn. 16 and the field distribution are the principal results of this, the standard SI method, and it is simple to solve it for p using a bisection trap- ping routine or any other real or complex similar root- finding algorithm.

3 Singular SI method (SSIM)

2W'

+ 2w , . - -_. P o .... .

We proceed to generalise the preceding analysis to include the singularity present at the re-entrant corners of the rib. To investigate the order of the singularity we follow [18], which studies the behaviour of fields near dielectric wedges and suggests that the order of singu- larity predicted by static theory can also be used for dynamic problems. It is well known that electric field distribution around the dielectric corner varies as r", where r is the distance from the corner [19]. For the 270" re-entrant dielectric corner at the bottom of the rib, the order a of the singularity can be found as [19]

At,"

which gives a value of a = -0.277 for n2 = 3.44, n1 = 1. Along the base of the effective rib the behaviour of

the principal field component for the quasi-TE mode (E, component) and the quasi-TM mode (E, compo- nent) in the rib for the symmetric case can be given as

;:H /E=o x - _ _

where the subscript s denotes 'singular', and for the fundamental mode considered we put A = At for TE polarised modes and A = A,, for TM polarised modes and take

A 0 D n2 D'

Y I "3

Fig.3 How the true position of the rib waveguide with dielectsic re- entrant corner singulasities in E at A and B is used to reproduce the neur- singulurities in E along the line y = 0, for use in the present investigation (see text)

This is demonstrated in Fig. 3 , where the distances PA and PB are given by PA2 = ( W + x ) ~ + A2 and PB2 = (W - x ) ~ + A*. Thus, in the case A, + 0, F,(x) tends to ( W2 - x2)" cos(7cxRW), which has a correct order of singularity at the re-entrant corners of a rib with E = 0

IEE Proc -Optoelectron, Vol 145, No 1, February 1998

on it and zero penetration depth (i.e. a scalar problem). Also, if A is small, but non-zero, then eqn. 20 satisfac- torily models the nearby singularities at x = -L W.

The dispersion eqn. 16 now generalises to n=cc

n=l m=oo

m=-m

where A(&) and B(S,) are coefficients for the exponen- tial Fourier expansions of F,(x) in eqn. 20 just above and just below y = 0, respectively, and over widths 2 W and 2L, respectively, where L is the lateral half-width (electric or magnetic wall) used to realise the discrete FT in y > 0. L is chosen to be so large that its position does not appreciably affect the values of the propaga- tion constants and field profiles so obtained. In prac- tice the overall program run time for SSIM (i.e. including the singularity) is found to be increased of only marginally over SIM (i.e. the non-singular case).

4 Results

The singular SI method (SSIM) will now be applied to both single rib waveguides and symmetric directional couplers. Where appropriate, both the symmetric and asymmetric modes supported by the structures are analysed.

The first rib waveguide analysed, in the notation of Fig. 1, has width 2 W = 3 . 0 ~ and substrate depth H i- D = lpm, where H is the etch depth. The refractive indices are n1 = 1 (air), n2 = 3.44 (GaAs) and n3 = 3.4 (GaAlAs). The operating wavelength is il = 1 . 1 5 ~ . The structure only supports one quasi-TE and one quasi-TM guided mode.

With the singularity exponent a = -0.277, modal propagation constants for both quasi-TE and quasi- TM modes were obtained. These values are here pre- sented in normalised form b, where

Table 1: INormalised propagation constants for funda- mental quasi-TE mode obtained from present singular method (SSIM)) and compared with standard method (SIM) and 'exact' values

SSIM SIM EXACT

H,pm b b b

0.1 0.3863 0.3844 0.3880

0.2 0.3654 0.3618 0.3668

0.3 0.3503 0.3460 0.3509

0.4 0.3380 0.3337 0.3373

0.5 0.3275 0.3239 0.3267

0.6 0.3185 0.3160 0.3178

0.7 0.3110 0.3098 0.3108

0.8 0.3052 0.3052 0.3055

0.9 0.3010 0.3018 0.3018

1 .o 0.2983 0.2985 0.2992

Table 1 compares values of normalised propagation constants for the fundamental quasi-TE mode for dif- ferent values of etch depths, obtained using the present

61

method (SSIM) with results obtained by SIM [8-121, and Fourier operator methods (F-OPT) [ 131. Normal- ised propagation constants for the quasi-TM mode were also obtained and are presented in Table 2, together with SIM [8] and F-OPT [13] results. We have obtained total agreement to four decimal places between the F-OPT method [I 31 and the semivectorial finite difference method (FD) [22]; in consequence, we refer to these two sets of results as ‘exact’ and combine them in one column.

Table 2: Normalised propagation constants for funda- mental quasi-TM mode obtained from present singular method (SSIM)) and compared with standard method (SIM) and ’exact‘ values

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 .o

SSlM

b

0.3436

0.3239

0.3094

0.2972

0.2864

0.2773

0.2700

0.2649

0.2618

0.2604

SIM

b

0.341 1

0.3198

0.3054

0.2945

0.2859

0.2790

0.2737

0.2698

0.2670

0.2649

EXACT

b

0.3441

0.3244

0.3095

0.2976

0.2880

0.2804

0.2746

0.2703

0.2678

0.2652

As can be seen from the Tables, all the methods are in excellent agreement with each other, especially when it is remembered that the normalised propagation con- stant b of eqn. 22 is a highly sensitive parameter. The inclusion of the singularities into the SIM has most effect for shallow etch depths where it also brings results into closer agreement overall with the estab- lished ‘exact’ semivectorial FD and F-OPT ‘bench- mark’ results. For large rib heights the fields are well confined in the rib region as can be seen in Fig. 4 for the quasi-TE case with H = 0.9pn. The relative field intensity in the vicinity of the corner discontinuity in the dielectric constant is therefore much smaller than for the smaller rib heights. The influence of the dielec- tric corner on the modal propagation constant is found to be much less significant in these structures.

Fig.4 Quasi-TE mode Jield profile ( I O 54 coutows) for deeply etched rib wuveguide structure 2W = 3 . 0 ~ ~ H = 0 . 9 p , D = O.lw, n , = 1, n, = 3 44, n3 = 3.4 and A = LI5pn

It should be noted that, as we are using a variational procedure, and as taking a = 0 reproduces the SIM results, then putting in the singularity a = -0.277 should in principle improve the value of the normalised propagation constant 6. However, as SIM already makes an approximation before invoking the presence of the singularity, one would prefer to see advances in analysis. Heuristic calculations with a view to improv- ing Tables 1 and 2 are in progress. Nevertheless, the b values in Tables 1 and 2 are more accurate than any hitherto obtained by such simple means over such a wide range of rib heights.

Table 3: Showing b for fundamental antisymmetric quasi-TE mode obtained with present singular method (SSIM) and compared with standard method (SIM) and finite difference (FD) values. Dimensions and indices in text

SSlM SIM FD

H,pm b b b

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 .o

0.3478

0.2814

0.2424

0.2147

0.1926

0.1746

0.1599

0.1486

0.1404

0.1351

0.3453

0.2759

0.2335 0.2440

0.2051

0.1841 0.1908

0.1683

0.1563 0.1590

0.1474

0.1411 0.1417

0.1367

Table 4: As Table 3, but for quasi-TM mode

SSlM SIM FD

H,pm B b b

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 .o

0.3007

0.2400

0.2048

0.1778

0.1548

0.1353

0.1195

0.1079

0.1008

0.0977

0.2964

0.2307

0.1930 0.2039

0.1678

0.1494 0.1558

0.1356

0.1252 0.1279

0.1176

0.1122 0.1132

0.1084

For larger rib widths the waveguide may support higher order modes, but the order of singularity remains the same as for the corresponding fundamental mode. The fundamental antisymmetric mode is studied here for a structure having, in the notation of Fig. 1, 2 W = 4pn, H + D = 1 pn, refractive indices n1 = 1, n2 = 3.44 and n3 = 3.4, and operating wavelength ;1 = 1.15 pn. Table 3 studies the fundamental quasi-TE antisymmetric mode, and Table 4 the fundamental antisymmetric quasi-TM mode.

The field profile obtained using the SSIM for the fundamental quasi-TE mode is shown in Fig 5, for a guide with 2W = 4.0pn, H = 0.3pn, D = 0 . 7 ~ and refractive indices n1 = 1, n2 = 3.44 and n3 = 3.4, a t wavelength A = 1 . 1 5 ~ . The field profile for the first antisymmetric quasi-TE mode for this structure is

IEE Pvoc -0ptoelectvon Vol 145 No I , Februuvy 1998 62

shown in Fig. 6. The singular behaviour of the field is highly localised near the dielectric corner and cannot be observed on these plots. However, it is found that the field profiles obtained more closely resemble FDIF- OPT results than those from the standard SIM.

Fig. 5 2W = 4.0ym, H = 0.3pm, D = 0 . 7 ~ ~ nl = 1, n2 = 3.44, n3 = 3.4 and /1 = 1 . 1 5 ~

Field profile for symmetric quasi-TE mode obtained using SSZM

Fig. 6 SSIMfor structure us in Fig. 5

Field profile for antisymmetric quasi-TE mode obtained using

i n3 d I

n4 + Fig. 7 Directional coupler geometry

I

Y

We have also used SSIM to analyse a directional coupler comprised of twin ribs. The coupled waveguide structure is shown in Fig. 7, and we analyse the case with 2W = 2.4pn1, t = 0 . 4 ~ , d = 0 . 2 p , H = 2.0pn1, refractive indices n1 = 1, n2 = n4 = 3.17 and n3 = 3.38, at the operating wavelength A = 1 . 5 5 ~ . The two ribs cannot operate independently, but support two super- modes: symmetric and antisymmetric about x = 0, with propagation constants /?, and PUT, respectively. Com- plete interchange of optical power occurs in a distance L,, known as the coupling length, where

(23 ) 7r L , =

P s - Pas

Table 5 shows values of coupling length L, for the quasi-TE mode for different rib separations T of the inner rib edges in the range T = 1 to 4pn1. Results are presented for the SSIM, standard SIM [23], F-OPT [13], FD 1221, vectorial finite element (VFE) [24] and polarised finite difference (PFD) 1241 methods. Table 6

IEE Proc -0ptoelectron Vol 145, No I , Februury 1998

presents coupling lengths for the quasi-TM mode for the same structure, where SI 1231 results are compared with those obtained using vectorial finite element (VFE) [24] and polarised finite element (PFE) [24] methods. It may be seen that the standard SIM is improved. The comparison of F-OPT and FD for T = 4 p in particular may have been affected by the greater number of Fourier coefficients required in the first instance, and the greater number of mesh points in the second.

Table 5: Directional coupler: coupling lengths in mm compared for quasi-TE mode where T is separation of inner edges of twin ribs (see text)

T;pm SSIM SIM F-OPT FD VFE PFD

1 0.345 0.351 0.340 0.337 0.340 0.340

2 0.696 0.698 0.682 0.669 0.734 0.698

3 1.375 1.360 1.326 1.307 1.427 1.384

4 2.686 2.618 2.541 2.527 3.039 2.768

Table 6: As Table 5, but for quasi-TM mode

T;pm SSIM SIM VFE PFD

1 0.366 0.386 0.369 0.375

2 0.912 0.966 0.941 0.918

3 2.224 2.369 2.570 2.220

4 5.388 5.771 6.410 5.236

5 Conclusions

By incorporating the correct field behaviour of the principal field component around the re-entrant dielec- tric corners of a rib waveguide in air, the standard spectral index method (SIM) is extended to include the singularity (SSIM), whilst maintaining simplicity and efficiency. Results obtained using SSIM agree well with the best results available for both rib waveguides and twin rib directional couplers.

6 Acknowledgment

The authors acknowledge with thanks the financial support of EPSRC under research grant GRiL28753.

References

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A

8 Appendix

The following variational method may be used to find the propagation constant /3 of a scalar wave equation whose wave number k is a function of x and y only. Then

(24) d2E d2E - + __ + ( k 2 - P2)E = 0 a x 2 ay2

Form the Rayleigh quotient for the trial value {p } of D2, where

(25) and

// E2dx = I

Then the correct trial field distribution E(x, y ) will maximise the trial value of p. Thus p2 = [P2lMAX

Suppose that the cross-section of the waveguide is divided into two parts across a contour C (e.g. the rib waveguide with C being the line of the base of the rib). Assume that E is given along C (i.e. E = E, there). Then the following is true: solve eqn. 24 on one side of C with E, given on C. Also solve eqn. 24 similarly on the other side of C with E, given on C. Then we can show that

where t denotes the arc length along the tangent and [aE/an] is the jump in the normal direction n of aEidn. Eqn. 27 therefore constitutes a transcendental equation for finding an approximation to p which is insensitive to small variations in the trial functions chosen.

64 IEE Puoc.-Optoekctron., Vol. 145, No I , February 1998