Indian Journal of Pure & Applied Physics

Vol. 39, March 200 1 , pp. 137-148

Propagation losses in silica-on-silicon optical wave2uid� - Theoretical

analysis and comparison with expermental results

�su, �r & ifshattacharyya

· .

�n Film Technology Group, National Physical Laboratory, Dr. K.S. Krishnan Road, New Delhi - 11000 Received 17 May 2000; revised 20 October 2000; accepted 30 October 2000

L Silica-on-silicon optical waveguides are being increasingly used for the design and fabrication of a large variety of optical

integrated circuits for applications in optical communication systems. One of the crucial factors in the configuration of these waveguides is the minimum thickness of the buffer layer of silica [ between the guiding core layer of doped silica and the silicon

substrate 1 that is required in order to keep the propagation loss of a guided mode, launched in the core and leaking into the substrate; below a specified limit. An earlier analysis by Stutius et at. involved the solution of a complex eigenvalue equation

and a perturbation method to calculate the propagation loss in this waveguide structure. Ghatak et ai. developed a novel 'leaky mode' analysis and a matrix method of numerical analysis to study the propagation of a mode in leaky waveguide structures. The

method of Ghatak et ai. has b�en applied iRotitis !,sIIer to calculate the propagation losses in silica-on-si licon waveguides, and the results are found to be identical to those obtained by the earlier approach of Stutius et al. The results of calculations by

different methods are also compared with experimental results for some typical waveguid:,J 1 Intfodu�t.ion

Dielectric optical waveguides have been studied e(C.tensively over the past 20 years and more, especially after the advent of fibre and integrated optics 1.2". With the increasing applications of optical integrated circuits (OIC) in optical communication systems, OIC's of greater complexity are being designed, fabricated, tested in the field, packaged and made available commercially .. Silica-on-silicon optical waveguides are predominantly being used for the design and fabrication of these OIC's. One of the crucial factors in designing the configuration of these waveguides is the minimum thickness of tl)e buffer layer of silica [between the guiding core layer of doped silica and the silicon substrate] that is required in order to keep. the. propagation loss of a guided mode, launched in the core and leaking into the substrate, below a specified limit. An earlier analysis by Stutius and Streifer4 involved the solution of a complex eigenvalue equation [also reported by Aarnio et ai. 5] and a perturbation method to calculate the propagation loss in this waveguide structure. Ghatak et al. have developed a novel leaky mode analysis6,7 and a matrix method of numerical analysis8 to study the 'propagation of a guided mode in leaky waveguide structures. The leaky mode analysis of Ghatak et al.has been applied to calculate the

propagation losses in silica-on-silicon waveguides, and the results are found to be identical to those obtained by the earlier approach of Stutius et ai. The results of calculations for typical waveguide structures are also compared using the analytical and numerical methods. Finally, the results of calculations by the different methods have been compared with the experimental results for some typical waveguides.

2 Waveguide Configuration and Parameters The basic configuration of a three medium slab

waveguide is shown in Fig. 1 . It consists of a core guiding layer of refractive index nf and thickness tf, with a semi-infinite cover medium (refractive index nc) above and a semi-infinite medium (refractive index nb) below it. The media are assumed to have,no absorption. Only the case n,. > nb 2 nc, has been considered where a certain number of guided modes can propagate in the waveguide without any loss. The parameters are defined as: y/ = n/ e - � 2 = - Kc 2 , y/ = n/ e - � 2 = K/ ,

2 2 e A2 2 . Yb = nb - I-' = - Kb . . . (I )

where k = 2 1t / A , � is the wavelength of light in vacuum and � is the propagation constant of a guided mode. The modal transverse electric fields Ey (for TE

138 INDIAN J PURE & APPL PHYS VOL 39, MARCH 200 1

+ 'lie 'lie ne �/

°rz "' / � + 'IIf 'IIf

Fig. I - Basic configuration of a three medium slab waveguide

+ \jIc \jIc

nc �/ Or Z � � + �x nf \jIf \jIf tt

/ � +

tb nb \jib \ \jib

\ / �+

n. "'. \jI.

Fig. 2 - Basic configuration of a four medium slab waveguide

modes) and transverse magnetic fields Hy (for TM modes) will be oscillatory functions of x in the core (0 > x > tf ), and will be exponentially decaying functions of x away from the core, in the upper and lower media (x < o and x > tr). The field in each medium is, in general, the sum of the fields of upward propagating (-) and downward propagating (+) waves, shown as 'Jf � in Fig. I . The expressions for the transverse E and H fields in the various medial ,2,3 are given in the Appendix. By applying the appropriate boundary conditions at the interfaces between the media, eigenvalue equations are obtained, which can be solved to obtain the propagation constants of the various guided modes of the waveguide. The eigenvalue equationsl,2,3, as well as the expressions for the transverse E and H field amplitudes obtained by applying the mode power normalisation conditionl ,2,3, are also given in the Appendix.

Next the authors consider the four medium slab waveguide structure shown in Fig. 2. Here, a fourth medium (called the substrate) of refractive index ns, larger than nr, is introduced below the medium of refractive index nb, which is now called the buffer layer and has a thickness tb. Since ns > nr, some of the power in a guided mode launched in the core layer (nf) will leak into the substrate (ns), resulting in a propagation loss for the guided mode. We introduce the parameter

2 2 k2 (.t2 2 Ys = ns - p = Ks . . . (2) which, along with the parameters introduced in Eq.

( 1 ), characterises the four medium waveguide. Now the propagation constant � is complex, with the imaginary part related to the propagation loss of the mode, as will be seen later. The transverse Eand H fields will be oscillatory functions of x in the core, and exponentially growing and / or decaying functions of x in the cover, buffer and substrate regions. In each medium, the field is the sum of the fields of upward (-) and downward (+) propagating waves, shown as " in Fig. 2. The expressions for the transverse E and H fields in the different media, the eigenvalue equation to be solved for �4, as well as the orthonormality condition for the radiation modes of the waveguide3, are given in the Appendix.

For the present study of silica-on-sil icon waveguides, the various media of the four medium waveguide are chosen as follows: Cover [ ne] . . . . . . Air ( n = 1.0 ] Buffer [ nb] . . . . . . . Si lica [ n about 1.46 at 1550 nm ], thickness 5 - 1 5 p,m Core [ nd . . . . . . . . . . Doped silica [ n is 0.5 - 2 % greater than buffer ], thickness 1 - 3 Ilm Substrate [ ns] . . . . Silicon [ n about '3.48 - i . 1O·6 at 1550 nm ]

3 Perturbation Analysis of Stutius et al.4

Stutius et al.4 studied the four medium waveguide structure and derived the eigenvalue equation given in Eqs. (A. 1 0) and (A. I 2). The solution of this complex eigenvalue equation in the complex plane is tricky and difficult, although it has been done by Aamio et al. 5 as well. Stutius et al.also carried out a perturbation analysis in which the propagation constant of the four medium waveguide is taken to be

. .. (3)

BASU et al.: SILICON OPTICAL WAVEGUIDES 1 39

where �g is the propagation constant of the three medium structure (without substrate, with buffer as lower medium), as given in the Appendix, and �B is found after substantial manipulation to b� given by

[ Ks - i. Kb ] exp [ - 2 % tb ] �� = 2 K/ Kb -.----------

where

W = �g [ tr. ( K/ + Kb 2 ) + ( Kb I %2 ) . ( K/ + Kb 2 )

+ (Kc I Ke 2 ) . ( K/ + Kb2 ) . ( Ke 2 + K/) I ( Kc 2 + K/) ]

and Kc = Ke [ TE modes ], (nr Inc )2 . Ke [ TM modes ] Kb = % [ TE modes ], (nr I nb f. Kb [ TM modes ]

Ks= Ks [ TE modes ], (nb I ns i. Ks [ TM modes ] ... (4)

The real part of �� is the change in the propagation constant �g of the 3 medium waveguide, while the imaginary part of �� is related to the propagation loss of the mode in the four medium waveguide caused by leakage of power from the core into the substrate. It is reported4 that the result of the perturbation analysis closely agrees with the solution of the complex eigenvalue equation.

4 Leaky Mode Analysis of Ghatak et al.3,6,7

It is assumed that a guided mode is launched in the core of a three layer waveguide at z = 0, and study the subsequent propagation of the mode in the four layer waveguide. First the TE modes are studied. The fractional power that remains in the core is given bl

where

. . . (5)

The only significant contribution to 1 $(�) 12 comes in the region around � "" �g 3. Using Eq. (A.8) one can get:

with U and V given by Eq. (A.9). Around � "" �g, using Eqs. (A.2) and (A.2a) one can get

Further, from Eq. (A.8), one can get

which vanishes around � "" �g. So expanding Eb-in a Taylor series around � "" �g one can obtain

Eb - "" ( � -�g) [ d Eb - I d� ] � = �g

= (� -�g ) [ (d Eb- I d Kd .(d Krl d�] � = �g . . . (7)

But from Eqs (1) and (2), one can have

[d Krl d �] �=�g=-�gl Kfg

[ d Ke I d Kd � = �g = -Ktg I Keg

[ d Kb I dKd � = �g = -Ktg I %g . . . (8)

Using the expression for Eb- in Eqs. (A.8) and (A.9) in Eq. (7), and using Eq. (8), one can get

Eb-= [Ec- 12]. [ �g (�-�g) / (Krg2 %g) ] . [(Krg2 + Keg2)(Krg2 + %g2)] 112. [tf + 11 %g + 1IKeg ] . . . (9)

Using the expressions for E/, Es-, E/ and Ef - in Eq. (A.8), and using Eqs. (A.3) and (A. 14), after considerable algebraic manipulations one can finally obtain

.

where

4 2 2 KCg Ksg %g exp ( - 2 %g tb ) r = ---------------

�g (Krg2+ Kbg2)(Ksg2 + %g2) [tc + 11 Kbg + l/Keg ]

1 40 INDIAN J PURE & APPL PHYS VOL 39, MARCH 2001

... (II)

From Eq. (5), one can finally see that the power in the core is given by

W(z) = exp [ - 2 r z ] . . . ( 1 2)

so that the propagation loss of a TE mode in the four-layer slab waveguide is 7

Loss [ dB / cm ] = 8.6843 r (cm·l) . . . ( 1 3)

On carrying out a similar analysis for the TM modes, one can get

r

where Mb and Me are given by Eq. (A.6).

On comparing the expressions for L\ � and r in Eq. ( 1 1 ) and Eq. ( 1 4) with Eq. (4), one can see that the real part of L\ � in Eq. (4) is identical with the expression for - L\ � in Eqs ( 1 1 ) and ( 1 4), and the expression for the imaginary part of L\ � in Eq. (4) is identical with the expressions for r in Eqs. ( 1 1 ) and ( 1 4). Thus one can find that the methods of calculation developed by Stutius et ai.4 and by Ghatak et ai.3,6,7 yield identical results.

.

5 Numerical Matrix Method of Ghatak et al.3,8

Ghatak et at. 3.8 developed a novel matrix method for the calculation of the propagation constant and propagation loss of modes in a multi-media slab waveguide structure. This method involves calculation of the elements of a 2 x 2 matrix for each interface between the media, which are functions of the propagation constant � and the refractive indices and thicknesses of the layers on either side of the interface. The matrices are multiplied in sequence, and using the elements of the product matrix and the individual matrices one can calculate the fractional power in the guiding core layer as a function of �. At the actual value of the propagation constant of the mode, the fractional

Fig. 3(a) - Basic configuration of a five medium slab waveguide, used for the numerical matrix method

power has a peak. The position of the maximum of the peak yields the propagation constant of the mode, while the ful l width at half maximum (FWHM) of the peak yields 2r and thence the propagation loss of the mode. More accurately, one can fit the peak to a Lorentzian function l ike in Eq. (10), to obtain L\ � and r. The elegance of this method is that one does not need to carry out complicated algebraic manipulations as in the methods of Sec. 3 and 4, and is also spared the task of carrying out complex algebraic manipulations in cases where some media are absorbing, with complex refractive indices.

To outline the procedure, let us consider a five medium structure shown in Fig, 3(a), with a wave in medium 1 incident on the interface between media I and 2 . The refractive indices, layer thicknesses and fields of the upward and downward propagating waves in the various media are shown. For each interface, a 2 x 2 matrix is introduced as fol lows :

Let � be the propagation constant, an invariant for all the media. For the jth interface between mediaj and j + 1 , introduce the matrix

exp(iDj) Rj exp(i�) ) R exp( -i 8) exp( -i 8) J J J

h s:: . I ( 2 k 2 A2 w ere U j = t j " n j 0 - I-' ) = t j Y j

and Rj = [ Y j - Y j + I ] / [ Y j + Y j + I ]

. . . ( 1 5a)

BASU et al. : SILICON OPTICAL WAVEGUIDES 1 4 1

Tj = 2 Y j / [ Y j + Y j + 1 ] for TE modes . . . (I5b)

and 2 2 ] /[ 2 2 Rj = [nj Y j + 1 - n j + 1 Y j nj Y j + 1 + n j + 1 )'j]

2 2 2 Tj = 2 n j n j + 1 Y j / [ n j Y j + 1 + n j + 1 Y j ] for

TM modes . . . ( l 5c)

. . . (J 6)

For a wave incident from medium 1 , there will be no upward propagating wave in medium 5, i.e. \f 5-= O. Suppose that medium 3 is the guiding layer, then one can find the quantity 1 \f , + / \f 1

+ 1 2 which represents the fractional power in the core layer. This quantity is calculated as a function of �, as shown in Fig. 3(b). From this plot, the propagation constant �g' and FWHM 2 r are determined.

This method is applied to the study of the four­layer silica-an-silicon waveguide structure. As the substrate has the highest refractive index among the four media, the sequence of media is chosen as follows:

Substrate . . . 1 , buffer . . . 2, core . . . 3, cover . . . 4 and it is assumed that a wave in medium 1 is incident on the interface between media I and 2. Then the matrices

1 :::1'

�g Fig. 3(b) - Variation of fractional power in the core with the propagation constant, showing the determination of the mode

propagation constant and propagation loss

for the three interfaces have th(1 following parameters:

S2 ... n2= nb, n, = nf, t2= tb

S, ... n, = nf, 114 = nc, t, = tr . . . ( 1 7)

Let the product SI S2 S, = P. Since one must now have \f 4-= 0, one can have

. . . ( 1 8)

and

. . . ( l 8a)

and so

. . . ( 1 9)

One therefore has to calculate 1 \f,+ / \f1+ 12 as a function of �, and find the location and FWHM of each peak, corresponding to a mode of the waveguide.

6 Results and Comparison of the Different Methods

In order to calculate the mode propagation constant and propagation loss for a silica-on-silicon waveguide, and compare the results obtained by the different methods described in Sec. 3, 4 and 5, a waveguide is chosen with the fol lowing parameters:

A = 1 .550 /lm nc = 1 .0 nf= 1 .457 tr = 2.5 /lm nb= 1 .443 tb = 3 - 1 2 /lm ns= 3 .475 - i . 1 0-6

. . . (20)

Calculations are carried out by the analytical methods described in Sec. 3 and 4, which yield identical results. The waveguide supports two modes, TEo and TMo. The results are presented in Table 1 :

It is seen that as the thickness of the buffer layer increases, there is less leakage of power from the core into the substrate and so the propagation loss of the mode decreases. If, for example, a target value is set for the maximum propagation loss of the TEo mode as 0. 1 dB/cm, then the minimum required thickness of the

142 I NDIAN J PURE & APPL PHYS VOL 39, MARCH 2001

Table I - Mode propagation constant and propagation loss for three medium slab waveguide [with semi-infinite buffer, no silicon

substrate

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

1 1 .0

12.0

Mode

TEo

™o

TEo

™o

TEo

™o

TEo

™o

TEo

™o

TEo

™o

TEo

™o

TEo

™o

TEo

™o

TEo

™o

Pg (em-I) = 58599.961 [ TEo ] 58577.897 [ TMo ]

21.221

23.216

10.506

12.424

5.201

6.649

2.575

3 .558

1.275

1.904

0.631

1.019

0.312

0.545

0.155

0.292

0.077

0.156

0.038

0.084

58578.740

58554.681

58589.455

58565.473

58594.760

58571.248

58597.386

58574.339

58598.686

58575.993

58599.330

58576.878

58599.649

58577.352

58599.807

5R577.605

58599.885

58577.741

58599.923

5857isI3

1.445080

1.444486

1.445344

1.444752

1.445475

1.444895

1 .445540

1.444971

1.445572

1.445012

1.445588

1.445034

1.445596

1.445046

1 .445599

1.445052

1.445601

1.445055

1.445602

1.445057

1 . 1 656

6.7046

0.5770

3.5881

0.2857

1.9202

0.1414

1.0276

0.0700

0.5500

0.0347

0.2943

0.0172

0.1575

0.0085

O.OS43

0.0042

0.0451

0.0021

0.0241

Loss

(dB/em)

1 0.122

5 8.225

5 .011

3 I. I 60

2.481

16.676

1.228

8.924

0.60S

4.776

0.301

2.556

0.149

1.368

0.074

0.732

0.037

0.392

0.018

0.210

buffer layer is about 9 -10 11m. Further, the TEo mode always has less propagation loss than the TMo mode.

shows the curve obtained by calculations using the numerical matrix method, and the results of the Lorentzian fit to the calculated curve. It is seen that the values of the propagation constant of the mode and of r, as read from the calculated curve (position of peak and half of FWHM) and as obtained from the Lorentzian fit to the curve, agree very well with each other.

Next, the results of calculations are presented by the numerical matrix method described in Sec. 5. The waveguide parameters specified in Eq. (20) above, are taken with a buffer layer thickness tb of 7 11m. Fig. 4

BASU et al.: SILICON OPTICAL WAVEGUIDES 143

6000

5000 'iii § .D 4DOU e � .E 3000

i U ! 2000

u..

1000

NUMERICAL MATRIX METHOD

TMomode n.==1.0 n,=1.457 nb=1.443 n.=3475,1.0E-6 r.=2.6 microns G,=7.0 microns Wavelength = 1.55 microns

--- Calculated results ...... .. Loremian fit

AS RE.AD FROM CI.JRVE Peak at 58575.93 cm·1 FWHM:l.193 cm·1 r=O.597 cm·1

LORENTZIAN FfT: Peak at 58576.00 cm·1 FWHM=1.216 cm-1 r:::0.608cm·1 o�������������

58585 58510 58575 58580 58585

Propagation constant (em")

Fig. 4 - Comparison of the results read from the calculated curve obtained by the numerical matrix method, and as obtained

by the Lorentzian fit to the calculated curve. The waveguide parameters and the results are shown in the tigure

The results obtained are also compared by the analytical methods described in Sec. 3 and 4 with the results of the numerical matrix method described in Sec. 5. The waveguide parameters specified in Eq. (20)

above are taken with a buffer layer thickness of I 0 �m. The results obtained are presented in Table 2:

It is seen that the results of the analytical and numerical methods agree to within I part in 106 for the propagation constant, and to within I part in 104 for the propagation loss.

In Fig. 5, are shown the variation of propagation loss with buffer layer thickness for the TEo mode, calculated by both the analytical and the numerical methods. The curves are almost identical for large buffer layer thicknesses and correspondingly low values of the propagation loss, where the waveguide is expected to be of practical use. Deviations occur only at lower buffer layer thicknesses, when the propagation losses are quite large and the waveguide would not be of much practical use.

16

\ 14 \ � 12 \ �10 \ � \ c .2 "Iii \ :il' 6 � Q. 4

\ \. \, '. " . ' .. ' .

TEo mode --- Analytical method ..... Numerical method

Buffer layer thickness (mcrons) 10 12

Fig. 5 - Variation of propagation loss of the TEo mode with

thickness of the buffer layer. calculated by the analytical method

(bold l ine) and the numerical matrix method (dashed l ine), The

waveguide parameters are given in the text [Eq. (20)].

The results of the analytical and numerical methods are also compared for the same waveguide operating in the visible region, where the absorption loss of silicon is appreciably high. The following waveguide parameters are now chosen:

"A = 0.6328 �m nc = 1.0 nf= 1.472 tf = 2.5 �m

nb=1.457 tb=2.5�m ns=3.882-i. 0.019

... (21)

The dispersion of silica has been taken into account in assigning values for nb and nf., This waveguide supports four modes : TEo. TMo, TEl and TMI. The results are compared in Table 3:

It is seen that the agreement is excellent for the TEo and TMo modes, while it is not so good for the TEl and TMI modes, especially in the case of the propagation losses, which are considerably higher than for the TEo and TMo modes.

7 Comparison of Theoretical values with Experimental Results

There are very few published experimental results for silica-an-silicon waveguides which specifically mention the measured waveguide parameters (refractive indices and layer thicknesses) and the measured mode propagation constants (or mode indices) and the mode propagation losses for a particular waveguide. Usually, only the ranges of values of these parameters and the

-'

144 INDIAN J PURE & APPL PHYS VOL 39, MARCH 2001

Table 2 - Propagation constant and loss values by analytical and matrix methods

Method of calculation Mode Pg' (cm-') Pg'/ k r (cm-') Loss (dB / em)

Analytical method

[Sec. 3 and 4 ] TEo 58599.807 1.445599 0.0085 0.074

™o 58577.605 1.445052 0.0843 0.732 Matrix method [Sec. 5]

TEo 58599.808 1.445599 0.0085 0.074

™o 58577.604 1.445052 0.0845 0.734

Table 3 - Propagation constant and loss values by analytical and matrix methods

Method of calculation Mode Pg' (cm-')

TEo 145804.692 Analytical method

[Sec. 3 and 4] ™o 145791.140

TE, 144840.966

TM, 144804.147

TEo 145804.699 Matrix method [Sec. 5 ]

™o 145791.145

TE, 144840.527

TM, 144803.385

best results for the mode propagation loss are quoted. Stutius et al.

4 are perhaps the only authors who have

published experimental data for their waveguides and also compared the experimental data with the results of their perturbation analysi s described in Sec. 3. They have fabricated and studied waveguides with a core layer of silicon oxynitride and a buffer layer of silicon dioxide on a silicon substrate. Refer to the identical results of the perturbation analysis (Sec. 3) and the leaky mode analysis (Sec. 4) as the results of the analytical method, and compare these as well as the results of the numerical method (Sec. 5) with the experimental results.

Pg' / k r (cm-') Loss (dB / cm)

1.468446 0.0022 0.019

1.468310 0.0149 0.130

1.458740 0.2422 2.103

1.458369 1.9925 17.304

1.468446 0.0021 0.018

1.468310 0.0150 0.130

1.458736 0.2740 2.379

1.458362 2.3800 20.668

Consider a waveguide with the following parameters:

A = 0.6328 �m nc = 1.0 nr= 2.030 (SiON) tr= 0.32 12 �m nb = 1.458 (Si02) tb = 0.8165 �m ns= 3.882 - i . 0.019

The experimentally measured data and the calculated results are tabulated below:

It is seen that the mode propagation constants and the mode indices for all three modes, calculated by the

BASU et al.: SILICON OPTICAL WAVEGUIDES 145

Table 4 - Values of�, N and propagation loss by experimental, analytical and numerical methods

Gg Mode propagation constant Mode refractive index Propagation loss

� (cm -I) N = �/ k (dB/cm)

TEo ™o TEl TEo ™o TEl TEo ™o TEl

Experimental Results 188500 182300 154400 1.898 1.836 1.555 <0.1 <0.1 6.0

Analytical method 188403 182270 154422 1.89747 1.83570 1.55523 0.0005 0.0008 3.939

Numerical method 188403 182270 154422 1.89747 1.83570 1.55523 0.0006 0.0010 4.254

Table 5 - Values of�, N and propagation loss by experimental, analytical and numerical methods.

Mode propagation constant � (em -I)

TEo ™o

Experimental Results 188500 182300

Analytical method 188403 182269

Numerical method 188403 182269

analytical method and by the numerical method, agree very well with the experimental results, to within 0.025 %. The mode propagation losses for the TEo and TMo modes, calculated by both methods, agree well with each other and with the experimental results. For the TEl mode, the agreement among the two calculated results is not so good, with the numerical method predicting a slightly higher loss, but they both agree fairly well with the experimental result.

Consider a waveguide with the same parameters as before, except that the buffer layer thickness is now tb = 0.400 /..lm. Because of the smaller buffer layer thickness, the modes are not as well confined to the core, and only the TEo and TMo modes were observed and their characteristics measured by Stutius et el.4. The results are presented in Table 5.

While the results for the mode propagation constants and mode indices agree very well as before,

Mode refractive index Propagation loss N = � / k (dB/cm)

TEo ™o TEo ™o

1.898 1.836 3.0 8.3

1 .89747 1.83569 2.646 7.516

1.89747 1.83569 2.850 9.760

the results for the mode propagation loss agree well for the TEo mode only. For the TMo mode, the analytical method predicts a propagation loss in good agreement with the experimental result, but the numerical method predicts a higher loss than the experimentally measured value. This indicates that the analytical method works well even when the modes are getting close to cut-off and the propagation losses are increasing. But the numerical method may not yield reliable values for the propagation loss in such cases, rather its value may be overestimated.

As further confirmation of the above conclusion, another waveguide with the same parameters as before, is considered,except that the buffer layer thickness is now tb = 0.590 /..lm. In this case, the TEl mode is observed but it is quite close to cut-off, with a measured propagation loss of about 50 dB/cm. In this case, the analytical method predicts a mode propagation loss of 44.96 dB/cm, while the numerical method predicts a

1 46 INDIAN J PURE & APPL PHYS VOL 39, MARCH 2001

loss of 78.56 dB/cm. Thus the numerical method again overestimates the propagation loss in this case.

8 Conclusions The mode propagation constant or mode index and

the mode propagation loss have been calculated for typical silica-on-silicon waveguides, using three different methods. The results of the two analytical methods are found to be in exact agreement with each other. The numerical method agrees with the analytical methods as far as the mode propagation constants or mode indices are concerned. However, for modes approaching cut-off, whose propagation losses are increasing, the numerical method overestimates the propagation loss. The results of calculation by these methods have been compared with experimental results on some waveguides, and excellent agreement between the calculated and experimental resu lts has been obtained, except for overestimation of the propagation loss by the numerical method for modes approaching cut-off.

Acknowledgements The authors gratefully acknowledge the

stimulating and helpful discussions with Prof. K. Thyagarajan of lIT, Delhi , who very kindly acquainted the authors with the analytical and matrix methods of

calculation used in this paper. This work was carried out under a project funded by the Ministry of Information Technology, Govt. of India, for which the authors are most thankful. Finally, the authors are grateful to the Director, National Physical Laboratory, for his kind permission to publish this paper.

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(Academic Press, New York), 1974. 2 Kogelnik H, in Integrated Optics. ed. T. Tamir (Springer

Verlag, New York), 1975. 3 Ghatak A & Thyagarajan K, Optical electronics (Cambridge

Univ. Press, UK), (Indian edition from Foundation Books, New Delhi), 1993.

4 Stutius W & Streifer W, Silicon nitride films on silicon for

optical waveguides, (Appl. Opt., 16(12), 1977 p. 3218 - 3222. 5 Aarnio J, Kersten P & Lauckner J, Determination of reji"(/ctive

indices and thicknesses of silica double layer slab waveguides

on silicon (lEE Proc.-Optoelectron., 142(5) Oct. 1995, p. 241

- 247. 6 Ghatak A K, Leaky modes in optical waveguides (Optical and

Quantum Eleclronics, 17, 1985 p. 311 - 321. 7 Thyagarajan K, Diggavi S & Ghatak A K, Analytical

investigations of leaky and absorbing planar structures

(Optical and Quantum Electronics, 19, 1987 p. 131 - 137. 8 Ghatak A K, Thyagarajan K & Shenoy M R, Numerical

analysis of planar optical waveguides using matrix approach (IEEE 1. Lightwave Technology, LT-5(5), May 1987, p. 660-667.

APPENDIX

Three �edium slab waveguide, nr > nb � n/,2,3

The z and time dependence of the transverse y component of the E or H field of a guided mode is exp [i.(m t - � z)], where m = k I c. In the upper and lower media are semi-infinite, the fields must have a decaying exponential x dependence for waves propagating away from the boundary with the core medium, while there are no waves propagating towards the core, i.e. \fI/ = \fIb-= O.

TE modes The fields Ey are given by (subscripts g signify guided modes):

.

\fIeg = \fIcg-= Ecg- exp [ Keg x ] x < 0 \fIfg = \fIrt + \fIrg - = Eft exp [ - i Kfg X ] + Efg - exp [i Kfg x] 0 < x < tr

\fIbg= \fIbt = Ebt exp [ - Kbg( x - tr)]] x > tr . . . (A.I)

By applying the boundary condition that these fields must be continuous across the boundaries at x = 0 and x = tr, expressions for the field amplitudes are obtained in terms of each other. Also the eigenvalue equation is obtained, whose solution(s) yield the propagation constant(s) � of the guided mode(s):

Krg tr - tan -J [ Keg I Kfg] - tan -J [ Kbg I Ktg] = V n, v = 0, 1,2, . . . (A .2)

or Ug + Vg = 0,

where Ug= cos Krgtr+ (Keg I Ktg) sin Krgtr

Vg = (Keg I Kbg) cos Krg tr B (Kfl Kb ) sin Kfg tr . . . (A. 2a)

On normalising the power in a guided mode to unity, one can get

I Efg ± I 2 = 1 I [ 2 ( tr + 1 I Kbg + ) I Keg ) ]

BASU et al.: SILICON OPTICAL WAVEGUIDES 1 47

± 2 ± 2 2 / 2)] 1 Ecg 1 = 41 Erg 1 / [ 1 + (Keg Krg

+ 2 ±121U 12 1 Ebg -1 = 1 Ecg g

TM modes The fields Hy are given by

'¥cg = Hcg - exp [ Kegx] x < 0

... (A.3)

'I'rg = Hrg + exp [-i Krg x] + Hrg - exp [ i Ktg x] 0 < x < tf '¥bg = Hbg + exp [ -Kbg (x - tr)]"] x > tf . . . (A.4)

The eigenvalue equation is

... (A.S)

or Ug + Vg = 0, where

Ug= cos Krg tr+ (nr/ nc)2 (Keg / Krg ) sin Krg tr,

Vg = (nb / nc)2 (Keg / Kbg) cos Kr g tr B (l1b / nr)2 (Kr/ Kb ) sin Krg tr ... (A.Sa)

On normalising the power in a guided mode to unity, one can get

2 2 2 2 N2/ 2 N2/ where Mb = N / nr + N / nb -·1, Me = nr +

n/ -I, N = � / k ... (A.6)

and similar expressions for the other field amplitudes.

Four medium slab waveguide, ns > nf > nh � nc3,4

In this case, the modes are no longer guided modes but are 'leaky modes'6, in which some power leaks from the core through the buffer into the substrate. For these modes, the fields in the cover and substrate regions will have decaying exponential x dependence for the waves propagating away from the boundaries. In the core region, the fields will have oscillatory x dependence. In the buffer region, the fields will have growing exponential x dependence for the wave propagating towards the core - buffer boundary, and decaying

exponential x dependence for the wave propagating away from the core - buffer boundary.

TE modes The fields Ey are given by:

'¥c = '¥c- = Ec- exp [ Kex] x < 0 '¥r='I't + '¥r-=Etexp [-i Krx]+Er-exp [iKtx ] 0 < x < tf '¥b= '¥b+ + '¥b- = Eb+ exp [ -%( x - tr)] +

Eb - exp [ % ( x - tr)] (tb + tf) > x > tf '¥s= '1'/ + '¥s-= E/ exp [ -Ks { X -(tb + tr) }] +

Es -exp [ Ks { x -(tb + tr) }] x > (tb + tf) ... (A.7)

By applying the boundary condition that these fields must be continuous across the boundaries at x = 0,

x = tr and x = (tb + tr), and we get expressions for the field amplitudes in terms of each other as follows:

Et = Ec - [I + i Kc / Kf] / 2 , Ef· = Ec - . [I - i Kc / Kr] / 2

Eb+ = Ec-. [ U - V] / 2, Eb-= Ec-. [ U + V] /2

E/ = [ Eb + / 2 ] [I -i Kb / Ks ] exp ( -Kb tb ) + [ Eb -/ 2 ] [I + i Kb / Ks] exp ( Kb tb )

Es -=[Eb+/2][I + i Kb/Ks]exp ( -Kbtb) + [Eb-/2] [1-iKb/Ks]exp (% tb) ... (A.8)

where

U = cos Kftf+ (Ke/ Kr) sin Krtr,

... (A.9)

By assuming that the field in the substrate is only a decaying field propagating away from the buffer B substrate boundary, i.e. '¥s-= 0, the eigenvalue equation is obtained whose solution(s) yield the propagation constant(s) � of the mode(s):

( Ks + i Kb ) ( U + V) + ( Ks -i Kb ) ( U - V) exp[ -2Kbtb]=0 ... (A.IO)

TM modes The fields Hy are given by

'¥c = '¥c-= Hc- exp [ Ke x ] x < 0 '¥r= '¥t + '¥r- = Ht exp [ - i Kr x] +

Hr- exp [ i Kr x] 0 < x < tf '¥b = '¥b+ + '¥b- = Hb+ exp [ -Kb (x - tr)] + Hb- exp [ Kb (x - tr) ] (t" + t}) > x> tf

148 INDIAN J PURE & APPL PHYS VOL 39, MARCH 2001

'I's= 'I'/ + 'I's-= H/ exp [-Ks { x -(tb + tf) }] + Hs - exp [ Ks { X -(tb + tf) }] X > (tb + tl) . . . (A. I I )

In the same way as for the TE modes, one can get expressions for the field amplitudes in terms of each other, as well as the eigenvalue equation whose solution(s) yield the propagation constant(s) � of the mode(s) :

[ (nb I ns )2 Ks + i % ] [ U + V] + [ (nb I ns )2 Ks -i % ] [ u - V] exp [ - 2 % tb] = 0,

where

U = cos Krtf+ [ ( nfl nc )2KcI Kr] sin Kftr,

V = [ Kc I { (nf I nb ) 2 Kb } ] cos Kr tf­[ Kfl ( nfl nb i% } ] sin Kftf . . . (A. 12)

The orthonormality condition for the radiation modes requires that -'

00

. . . (A. 13)

-00

Following the same procedure as in Ghatak et at.", one can apply this condition to the fields of the modes given in Eqs (A.7) and (A.9) and get

. . . (A. 14)