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Polarization-independent single-mode rib waveguides on silicon-on-insulator for telecommunication wavelengths
1 September 2002
Optics Communications 210 (2002) 43–49 www.elsevier.com/locate/optcom
Polarization-independent single-mode rib waveguides on silicon-on-insulator for telecommunication wavelengths L. Vivien, S. Laval *, B. Dumont, S. Lardenois, A. Koster, E. Cassan Institut d’Electronique Fondamentale, CNRS UMR 8622, Bat. 220, Universit e Paris Sud, 91405 Orsay Cedex, France Received 14 February 2002; received in revised form 31 May 2002; accepted 5 June 2002
Abstract A theoretical analysis of the influence of geometrical parameters on the polarization sensitivity properties is reported for single-mode silicon-on-insulator rib waveguides. The SOI rib-waveguide dimensions (height, width and etching depth) leading simultaneously to single mode propagation and polarization independence have been calculated for waveguide heights ranging from 0.75 to 2 lm and for telecommunication wavelengths from 1.53 to 1.61 lm. The minimum etching depth is obtained for a nearly constant value of the ratio of the rib width over the waveguide height, within the considered height range, and is close to the rib width. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Rib waveguides; Silicon-on-insulator; Single-mode waveguides; Polarization independence
1. Introduction Silicon-on-insulator (SOI) material is of interest for integrated optoelectronic circuits since it offers the potentiality of monolithic integration of optical and electronic functions on a single substrate. Moreover, silicon is transparent at telecommunication wavelengths, in particular from 1.53 to 1.61 lm, and the silicon film of SOI substrates can be used as a low-loss waveguide [1]. The main advantages of the SOI technology arise from the strong light confinement in very small waveguides, due to the large refractive index difference between silicon and silicon oxide, and from the possibility
*
Corresponding author. Fax: +33-1-6915-4050. E-mail address: [email protected] (S. Laval).
of using established silicon microelectronics technology. In addition to the low cost of silicon technology and the availability of large wafers, SOI allows to significantly miniaturize the optical devices. Wavelength division multiplexing (WDM) is one of the technologies that is rapidly developing to increase transmission capacity and flexibility in broadband optical fiber telecommunication networks. An important point is that most of the devices for telecommunication applications, in particular arrayed waveguide gratings (AWG) which are widely used in WDM systems, require polarisation insensitivity at wavelengths ranging from 1.53 to 1.61 lm [2]. Single-mode waveguides are also required in order to reduce propagation loss [3,5]. These two conditions are not simultaneously fulfilled up to now in published results.
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 6 8 1 - 4
44
L. Vivien et al. / Optics Communications 210 (2002) 43–49
Single mode SOI waveguides have been reported for large rib structures, whose height is larger than several microns [3] which are birefringent. Only few studies [4] have debated polarization insensitivity in small SOI waveguides (waveguide height 61:5 lm), but they are related to multimode structures. Waveguide birefringence can be defined either from the effective index difference or from the group index difference between the TE and TM modes. The first one corresponds to a phase delay and is the relevant parameter for most of the applications, in which both TE and TM modes must propagate at the same velocity. The group index difference introduces distortions of pulsed signals after propagation and has also an influence on the performances of devices such AWG. So both have to be considered, even if the emphasis is put on phase birefringence. The aim of this study is to determine waveguide geometry for heights lower than few microns, to take benefits from the high field confinement in SOI waveguides to reduce the global size of the devices, taking into account simultaneously polarization insensitivity and singlemode condition. This has not been carried out yet for SOI waveguides at telecommunication wavelengths. This article reports on numerical simulations of single-mode and polarization-independent rib waveguides with various geometries (height, width and etching depth). Polarization insensitivity and single-mode conditions are studied for waveguide height ranging from 0.75 to 2 lm. Effective indices are determined for both TE and TM polarizations, for wavelength ranging from 1.53 to 1.61 lm, using a film mode-matching method.
2. Rib waveguide geometry The rib waveguide cross-section is shown in Fig. 1. The two dielectric materials, SiO2 and Si, have n0 and n1 refractive indices, respectively, taking into account the material dispersion at the wavelength of interest. For large rib waveguides, Soref et al. have shown that it is possible to get a single-mode propagation condition in a rib waveguide, even if
Fig. 1. Cross-section of rib waveguide in SOI structure. Representation of slices and layers used in numerical simulation.
the planar waveguide with the same thickness is multi-modal. They gave a relation between the geometrical parameters of the waveguide [6,7] W r 6 0:3 þ pffiffiffiffiffiffiffiffiffiffiffiffi H 1 r2
for
r P 0:5;
ð1Þ
where W is the rib width, H is the inner rib height, r is the fractional height of the side regions compared to the rib centre (the outer–inner ratio) as defined in Fig. 1. For a better understanding, we will also consider the etching depth P ¼ H ð1 rÞ which directly gives the edge height of the rib waveguide. In the published studies, waveguides that fulfil this relation have very broad sections, several micron of width and height, and the sensitivity to light polarization have not been considered. In SOI waveguides, the silicon layer (guiding layer) is not birefringent, only modal birefringence between TE and TM modes exists, strongly related to waveguide geometry. The reduction of the latter must be done while the single-mode character of the waveguide is preserved, in order to reduce propagation loss. Polarization insensitivity can be obtained for ridge waveguides with a square section but the monomode character then imposes edge sizes smaller than 0.35 lm. Rib waveguides considered in this study have heights and widths
L. Vivien et al. / Optics Communications 210 (2002) 43–49
lower than 2 lm, in order to keep a good compromise between system compacity and light coupling easiness. The conditions for such a rib waveguide to be at the same time polarization independent and singlemode are derived hereafter.
3. Numerical simulations A program based on the film mode matching (FMM) method due to Sudbø [8–11] has been used to study the influence of the geometrical parameters for the waveguide described above. The waveguide cross-section is considered as a sandwich of slices. Each slice corresponds to a planar multi-layer structure (Fig. 1). The FMM method involves finding the TE and TM modes of the planar waveguide in each slice, collecting the modes that have the same modal wavevector component in the propagation direction, and matching field distributions at the slice boundaries by adjusting the modal amplitudes in each layered structure. For very simple waveguide geometries like rectangular guides or symmetric ribs the method is known to produce accurate mode field distributions with very small computational effort. The method lives up to its promises when generalized to more practical geometries and high index difference systems. All the modes propagating in the rib waveguide are calculated for both TE and TM polarizations and the corresponding effective index (neff ) is determined. The waveguide birefringence ðDneff Þ is the difference of the effective indices for each polarization defined as: Dneff ¼ neff;TE neff;TM : From neff values at various wavelengths k, it is possible to calculate the group index (ng ) by applying the following equation: oneff ; ð2Þ ok oneff =ok is determined from neff values calculated at wavelengths ranging from 1.53 to 1.61 lm, and by fitting the results by a linear curve. The curve slope corresponds to oneff =ok and thus allows to determine ng . ng ¼ neff k
45
For simulations, the heights of the layers number 1, 2 and 3 are, respectively, 1, 1.5 (depending of silicon height) and 1 lm. The horizontal size of calculation windows is larger than 10 lm and the boundary conditions are electric or magnetic walls (the same results are obtained for both boundary conditions). The indices of silica (n ¼ 1.444 at k ¼ 1:55 lm) and silicon (n ¼ 3.475 at k ¼ 1:55 lm) are adjusted for each wavelength [12].
4. Influence of geometrical parameters This section presents the influence of each parameter of the rib waveguide geometry on the polarisation sensitivity in order to design singlemode and polarization independent structures. 4.1. Width and etching depth influences At first, the H and W parameters of the rib waveguide are fixed, and the effective indices of the different modes, for both TE and TM polarizations are determined for a wide range of the r parameter, i.e., from 0.3 to 0.55. The first result to point out is that the Eq. (1) validity range is expanded for r values fairly smaller than 0.5: monomode propagation conditions exist for waveguide widths limited by Eq. (1) even for deeply etched structures. The birefringence has been calculated for r values from 0.37 to 0.55, corresponding to etching depths from 0.675 to 0.945 lm. As an example, the evolutions of neff and ng index birefringence are plotted in Fig. 2 as a function of the parameter r, for H ¼ 1:5 lm and W ¼ 1 lm at the wavelength 1.53 lm. The variation of r in the figure corresponds to an etching depth ranging from 0.87 to 0.945 lm. The birefringence of effective or group index can be cancelled for given values of the etching depth (parameter r). However, for a given etching depth, it is not possible to obtained simultaneously Dneff ¼ 0 and Dng ¼ 0. Simulations have been carried out for various wavelengths in the telecom band, from 1.53 to 1.61 lm. The results are qualitatively similar. For
46
L. Vivien et al. / Optics Communications 210 (2002) 43–49
Fig. 2. Birefringence evolution of effective and group index as a function of the parameter rs for H ¼ 1:5 lm and W ¼ 1 lm at wavelength k ¼ 1:53 lm.
wavelengths between 1.53 and 1.61 lm and waveguide height smaller than 2 lm, the cancellation of the effective index birefringence always occurs for r parameters larger than the ones for which the group index birefringence cancels. This etching depth difference is smaller for the largest waveguides compatible with the monomode condition. According to the applications which are aimed at, two possibilities to realize a single-mode and polarization-independent rib waveguide can be selected: the etching depth for Dneff ¼ 0 or for Dng ¼ 0. A compromise with Dneff ¼ Dng ¼ Dn can also be chosen, with Dn values as small as possible. To determine the waveguide width influence, simulations have been carried out by varying W from 0.8 to 1.1 lm, for H ¼ 1:5 lm and for wavelengths from 1.53 to 1.61 lm. For W larger than 1.1 lm, the rib waveguide becomes multimode. The influence of the waveguide width on the etching depth corresponding to Dneff ¼ 0 for a guide height of 1.5 lm is shown in Fig. 3, at k ¼ 1:53 lm and k ¼ 1:61 lm. The calculated points are fitted by a polynomial of degree 2. The etching depth is minimum for a waveguide width of 0.97 lm. The smallest the etching depth, the easiest the realization. Assuming fabrication tolerances of 10 nm on the etching depth leads to an accuracy of about 0.1 lm on the waveguide width,
Fig. 3. Waveguide width influence on etching depth to cancel the effective index birefringence, H ¼ 1:5 lm at k ¼ 1:53 lm and k ¼ 1:61 lm.
which is quite acceptable. The wavelength sensitivity also lies in this etching tolerance range. It is worth noting that an inaccuracy of about 10 nm on etching depth introduces a birefringence difference as small as 6 104 . The evolution of the group index birefringence versus guide width for zero effective index birefringence is plotted in Fig. 4 for both wavelengths 1.53 lm and 1.61 lm. In order to minimize the group index birefringence, the waveguide width has to be increased. However, single-mode property is only conserved if this width is smaller than 1.1 lm for a height H ¼ 1:5 lm. Indeed, for a width equal to 1.2 lm, a second mode appears. The residual group index birefringence is of the order of 2 103 .
Fig. 4. Evolution of group index birefringence with the waveguide width W to cancel the effective index birefringence, H ¼ 1:5 lm at k ¼ 1:53 lm and k ¼ 1:61 lm.
L. Vivien et al. / Optics Communications 210 (2002) 43–49
47
Fig. 5. Confinement of intensity in rib waveguide for TE and TM polarizations for guide width W ¼ 0:8, 1, 1.1 lm, and H ¼ 1:5 lm at k ¼ 1:53 lm.
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L. Vivien et al. / Optics Communications 210 (2002) 43–49
The field distribution in the rib waveguide for the TE and TM polarizations is shown in Fig. 5 for three different widths (0.8; 1; 1.1 lm). The TE field slightly expands in the thinner silicon film region, and this effect is more important as the width is reduced. The TM mode appears to be better confined in the central part of the rib waveguide. Increasing the waveguide width yields a more similar field profiles of the TE and TM modes in the guide. This behaviour can be explained by the fact that effective index increases with the width, implying a stronger confinement. 4.2. Height influence Simulations have also been carried out for several waveguide heights, from 0.75 to 2 lm. For each height value, the effective index and the etching depth for both TE and TM polarizations are determined for different waveguide widths in order to determine the conditions to obtain singlemode and polarization-independent rib waveguide. Some results are given in Fig. 6 for k ¼ 1:53 lm. The r parameter corresponding to zero effective index birefringence, is plotted versus the W/H ratio for the various height values. As in Fig. 2, a minimum of the etching depth (corresponding to a maximum of the parameter r) is observed for
Fig. 6. Parameter r as a function of the ratio W/H to cancel the effective index birefringence, for waveguide height H from 0.75 lm to 2 mm at 1.53 lm.
each curve. What is worth noting is that the r maximum occurs for a quasi-constant value of the ratio W/H. Indeed, W/H varies from 0.643 to 0.648 when H is, respectively, changed from 0.75 to 2 lm. Thus W/H increases only very slightly with the waveguide height. The parameter r corresponding to zero effective index birefringence increases with the waveguide height. For the various height values, the evolution of the parameter r (or the etching depth) with the W/H ratio is very similar. However, the maximum ratio W/H to keep a single-mode waveguide decreases as H increases. The last point of each curve determines the limit to have single-mode waveguide. For example, for H ¼ 2 lm, the maximum width for a monomode waveguide is 1.3 lm, which corresponds to W =H ¼ 065. So, for thicker waveguides, the etching depth would have to be a larger fraction of the silicon layer thickness. The ratio P/H, etching depth per waveguide height, calculated for the width value corresponding to the minimum etching depth (W =H 0; 645) varies from 0.64 to 0.59 for H varying from 0.75 to 2 lm. The more the height is reduced, the more the structure of the guide tends towards a square strip waveguide.
5. Conclusion Investigation of small size rib SOI waveguides exhibiting simultaneously single-mode and polarization insensitive characters have been presented. Using a mode solver program, the etching depth, width and height values have been determined to design waveguides which fulfil these two conditions, for wavelengths ranging from 1.53 to 1.61 lm. It has been shown that the single-mode condition limits the waveguide width, but is valid even for deeply etched waveguides. The width to height ratio of rib waveguides with a minimum etching depth is nearly constant to whatever the waveguide height is within the considered range. The minimum etching depth is of the same order than the rib width. The more critical parameter for the fabrication of such waveguides is the etching depth. The sen-
L. Vivien et al. / Optics Communications 210 (2002) 43–49
sitivity to width fluctuations and to the wavelength lies inside the etching depth reliability. The residual birefringence is then less than a few 104 . In future works, the birefringence and loss characteristics [13] of bends in such SOI waveguides will be considered.
Acknowledgements This work is financially supported by AlcatelOpto+. The authors acknowledge A. Carenco for helpful advices. The authors also thank Daniel Pascal for fruitful discussions.
References [1] A. Layadi, A. Vonsovici, R. Orobtchouk, D. Pascal, A. Koster, Opt. Commun. 146 (1998) 31.
49
[2] P.D. Trinh, S. Yegnanarayanan, F. Coppinger, B. Jalali, IEEE Photon. Technol. Lett. 9 (1997) 940. [3] J. Schmidtchen, A. Splett, B. Sch€ uppert, K. Petermann, Electron. Lett. 27 (1991) 1486. [4] M.R.T. Pearson, A. Bezinger, A. Del^age, J.W. Fraser, S. Jaz, P.E. Jessop, D.-X. Xu, in: Proc. SPIE Int. Soc. Opt. Eng., 3953, 2001, p. 1938. [5] T. Zinke, U. Fischer, A. Splett, B. Sch€ uppert, K. Petermann, Electron. Lett. 29 (1993) 2031. [6] R.A. Soref, J. Schmidtchen, K. Petermann, IEEE J. Quantum Electron. 27 (1991) 1971. [7] U. Fischer, T. Zinke, J.R. Kropp, F. Arndt, K. Petermann, IEEE Photon. Technol. Lett. 8 (1996) 647. [8] A.Sv. Sudbø, Pure Appl. Opt. 2 (1993) 211. [9] A.Sv. Sudbø, IEEE Photon. Technol. Lett. 5 (1993) 342. [10] A.Sv. Sudbø, Pure Appl. Opt. 3 (1994) 381. [11] FIMMWAVE program, a Photon Design Product. [12] E.D. Palik, Handbook of Optical Constants of Solids, Academic Press Inc., New York, 1985. [13] H.Z. Wei, J.Z. Yu, X.F. Zhang, Z.L. Liu, Q.M. Wang, W. Shi, C.S. Fang, J. Infrared Millim. Waves 20 (2001) 398.
Optics Communications 210 (2002) 43–49 www.elsevier.com/locate/optcom
Polarization-independent single-mode rib waveguides on silicon-on-insulator for telecommunication wavelengths L. Vivien, S. Laval *, B. Dumont, S. Lardenois, A. Koster, E. Cassan Institut d’Electronique Fondamentale, CNRS UMR 8622, Bat. 220, Universit e Paris Sud, 91405 Orsay Cedex, France Received 14 February 2002; received in revised form 31 May 2002; accepted 5 June 2002
Abstract A theoretical analysis of the influence of geometrical parameters on the polarization sensitivity properties is reported for single-mode silicon-on-insulator rib waveguides. The SOI rib-waveguide dimensions (height, width and etching depth) leading simultaneously to single mode propagation and polarization independence have been calculated for waveguide heights ranging from 0.75 to 2 lm and for telecommunication wavelengths from 1.53 to 1.61 lm. The minimum etching depth is obtained for a nearly constant value of the ratio of the rib width over the waveguide height, within the considered height range, and is close to the rib width. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Rib waveguides; Silicon-on-insulator; Single-mode waveguides; Polarization independence
1. Introduction Silicon-on-insulator (SOI) material is of interest for integrated optoelectronic circuits since it offers the potentiality of monolithic integration of optical and electronic functions on a single substrate. Moreover, silicon is transparent at telecommunication wavelengths, in particular from 1.53 to 1.61 lm, and the silicon film of SOI substrates can be used as a low-loss waveguide [1]. The main advantages of the SOI technology arise from the strong light confinement in very small waveguides, due to the large refractive index difference between silicon and silicon oxide, and from the possibility
*
Corresponding author. Fax: +33-1-6915-4050. E-mail address: [email protected] (S. Laval).
of using established silicon microelectronics technology. In addition to the low cost of silicon technology and the availability of large wafers, SOI allows to significantly miniaturize the optical devices. Wavelength division multiplexing (WDM) is one of the technologies that is rapidly developing to increase transmission capacity and flexibility in broadband optical fiber telecommunication networks. An important point is that most of the devices for telecommunication applications, in particular arrayed waveguide gratings (AWG) which are widely used in WDM systems, require polarisation insensitivity at wavelengths ranging from 1.53 to 1.61 lm [2]. Single-mode waveguides are also required in order to reduce propagation loss [3,5]. These two conditions are not simultaneously fulfilled up to now in published results.
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 6 8 1 - 4
44
L. Vivien et al. / Optics Communications 210 (2002) 43–49
Single mode SOI waveguides have been reported for large rib structures, whose height is larger than several microns [3] which are birefringent. Only few studies [4] have debated polarization insensitivity in small SOI waveguides (waveguide height 61:5 lm), but they are related to multimode structures. Waveguide birefringence can be defined either from the effective index difference or from the group index difference between the TE and TM modes. The first one corresponds to a phase delay and is the relevant parameter for most of the applications, in which both TE and TM modes must propagate at the same velocity. The group index difference introduces distortions of pulsed signals after propagation and has also an influence on the performances of devices such AWG. So both have to be considered, even if the emphasis is put on phase birefringence. The aim of this study is to determine waveguide geometry for heights lower than few microns, to take benefits from the high field confinement in SOI waveguides to reduce the global size of the devices, taking into account simultaneously polarization insensitivity and singlemode condition. This has not been carried out yet for SOI waveguides at telecommunication wavelengths. This article reports on numerical simulations of single-mode and polarization-independent rib waveguides with various geometries (height, width and etching depth). Polarization insensitivity and single-mode conditions are studied for waveguide height ranging from 0.75 to 2 lm. Effective indices are determined for both TE and TM polarizations, for wavelength ranging from 1.53 to 1.61 lm, using a film mode-matching method.
2. Rib waveguide geometry The rib waveguide cross-section is shown in Fig. 1. The two dielectric materials, SiO2 and Si, have n0 and n1 refractive indices, respectively, taking into account the material dispersion at the wavelength of interest. For large rib waveguides, Soref et al. have shown that it is possible to get a single-mode propagation condition in a rib waveguide, even if
Fig. 1. Cross-section of rib waveguide in SOI structure. Representation of slices and layers used in numerical simulation.
the planar waveguide with the same thickness is multi-modal. They gave a relation between the geometrical parameters of the waveguide [6,7] W r 6 0:3 þ pffiffiffiffiffiffiffiffiffiffiffiffi H 1 r2
for
r P 0:5;
ð1Þ
where W is the rib width, H is the inner rib height, r is the fractional height of the side regions compared to the rib centre (the outer–inner ratio) as defined in Fig. 1. For a better understanding, we will also consider the etching depth P ¼ H ð1 rÞ which directly gives the edge height of the rib waveguide. In the published studies, waveguides that fulfil this relation have very broad sections, several micron of width and height, and the sensitivity to light polarization have not been considered. In SOI waveguides, the silicon layer (guiding layer) is not birefringent, only modal birefringence between TE and TM modes exists, strongly related to waveguide geometry. The reduction of the latter must be done while the single-mode character of the waveguide is preserved, in order to reduce propagation loss. Polarization insensitivity can be obtained for ridge waveguides with a square section but the monomode character then imposes edge sizes smaller than 0.35 lm. Rib waveguides considered in this study have heights and widths
L. Vivien et al. / Optics Communications 210 (2002) 43–49
lower than 2 lm, in order to keep a good compromise between system compacity and light coupling easiness. The conditions for such a rib waveguide to be at the same time polarization independent and singlemode are derived hereafter.
3. Numerical simulations A program based on the film mode matching (FMM) method due to Sudbø [8–11] has been used to study the influence of the geometrical parameters for the waveguide described above. The waveguide cross-section is considered as a sandwich of slices. Each slice corresponds to a planar multi-layer structure (Fig. 1). The FMM method involves finding the TE and TM modes of the planar waveguide in each slice, collecting the modes that have the same modal wavevector component in the propagation direction, and matching field distributions at the slice boundaries by adjusting the modal amplitudes in each layered structure. For very simple waveguide geometries like rectangular guides or symmetric ribs the method is known to produce accurate mode field distributions with very small computational effort. The method lives up to its promises when generalized to more practical geometries and high index difference systems. All the modes propagating in the rib waveguide are calculated for both TE and TM polarizations and the corresponding effective index (neff ) is determined. The waveguide birefringence ðDneff Þ is the difference of the effective indices for each polarization defined as: Dneff ¼ neff;TE neff;TM : From neff values at various wavelengths k, it is possible to calculate the group index (ng ) by applying the following equation: oneff ; ð2Þ ok oneff =ok is determined from neff values calculated at wavelengths ranging from 1.53 to 1.61 lm, and by fitting the results by a linear curve. The curve slope corresponds to oneff =ok and thus allows to determine ng . ng ¼ neff k
45
For simulations, the heights of the layers number 1, 2 and 3 are, respectively, 1, 1.5 (depending of silicon height) and 1 lm. The horizontal size of calculation windows is larger than 10 lm and the boundary conditions are electric or magnetic walls (the same results are obtained for both boundary conditions). The indices of silica (n ¼ 1.444 at k ¼ 1:55 lm) and silicon (n ¼ 3.475 at k ¼ 1:55 lm) are adjusted for each wavelength [12].
4. Influence of geometrical parameters This section presents the influence of each parameter of the rib waveguide geometry on the polarisation sensitivity in order to design singlemode and polarization independent structures. 4.1. Width and etching depth influences At first, the H and W parameters of the rib waveguide are fixed, and the effective indices of the different modes, for both TE and TM polarizations are determined for a wide range of the r parameter, i.e., from 0.3 to 0.55. The first result to point out is that the Eq. (1) validity range is expanded for r values fairly smaller than 0.5: monomode propagation conditions exist for waveguide widths limited by Eq. (1) even for deeply etched structures. The birefringence has been calculated for r values from 0.37 to 0.55, corresponding to etching depths from 0.675 to 0.945 lm. As an example, the evolutions of neff and ng index birefringence are plotted in Fig. 2 as a function of the parameter r, for H ¼ 1:5 lm and W ¼ 1 lm at the wavelength 1.53 lm. The variation of r in the figure corresponds to an etching depth ranging from 0.87 to 0.945 lm. The birefringence of effective or group index can be cancelled for given values of the etching depth (parameter r). However, for a given etching depth, it is not possible to obtained simultaneously Dneff ¼ 0 and Dng ¼ 0. Simulations have been carried out for various wavelengths in the telecom band, from 1.53 to 1.61 lm. The results are qualitatively similar. For
46
L. Vivien et al. / Optics Communications 210 (2002) 43–49
Fig. 2. Birefringence evolution of effective and group index as a function of the parameter rs for H ¼ 1:5 lm and W ¼ 1 lm at wavelength k ¼ 1:53 lm.
wavelengths between 1.53 and 1.61 lm and waveguide height smaller than 2 lm, the cancellation of the effective index birefringence always occurs for r parameters larger than the ones for which the group index birefringence cancels. This etching depth difference is smaller for the largest waveguides compatible with the monomode condition. According to the applications which are aimed at, two possibilities to realize a single-mode and polarization-independent rib waveguide can be selected: the etching depth for Dneff ¼ 0 or for Dng ¼ 0. A compromise with Dneff ¼ Dng ¼ Dn can also be chosen, with Dn values as small as possible. To determine the waveguide width influence, simulations have been carried out by varying W from 0.8 to 1.1 lm, for H ¼ 1:5 lm and for wavelengths from 1.53 to 1.61 lm. For W larger than 1.1 lm, the rib waveguide becomes multimode. The influence of the waveguide width on the etching depth corresponding to Dneff ¼ 0 for a guide height of 1.5 lm is shown in Fig. 3, at k ¼ 1:53 lm and k ¼ 1:61 lm. The calculated points are fitted by a polynomial of degree 2. The etching depth is minimum for a waveguide width of 0.97 lm. The smallest the etching depth, the easiest the realization. Assuming fabrication tolerances of 10 nm on the etching depth leads to an accuracy of about 0.1 lm on the waveguide width,
Fig. 3. Waveguide width influence on etching depth to cancel the effective index birefringence, H ¼ 1:5 lm at k ¼ 1:53 lm and k ¼ 1:61 lm.
which is quite acceptable. The wavelength sensitivity also lies in this etching tolerance range. It is worth noting that an inaccuracy of about 10 nm on etching depth introduces a birefringence difference as small as 6 104 . The evolution of the group index birefringence versus guide width for zero effective index birefringence is plotted in Fig. 4 for both wavelengths 1.53 lm and 1.61 lm. In order to minimize the group index birefringence, the waveguide width has to be increased. However, single-mode property is only conserved if this width is smaller than 1.1 lm for a height H ¼ 1:5 lm. Indeed, for a width equal to 1.2 lm, a second mode appears. The residual group index birefringence is of the order of 2 103 .
Fig. 4. Evolution of group index birefringence with the waveguide width W to cancel the effective index birefringence, H ¼ 1:5 lm at k ¼ 1:53 lm and k ¼ 1:61 lm.
L. Vivien et al. / Optics Communications 210 (2002) 43–49
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Fig. 5. Confinement of intensity in rib waveguide for TE and TM polarizations for guide width W ¼ 0:8, 1, 1.1 lm, and H ¼ 1:5 lm at k ¼ 1:53 lm.
48
L. Vivien et al. / Optics Communications 210 (2002) 43–49
The field distribution in the rib waveguide for the TE and TM polarizations is shown in Fig. 5 for three different widths (0.8; 1; 1.1 lm). The TE field slightly expands in the thinner silicon film region, and this effect is more important as the width is reduced. The TM mode appears to be better confined in the central part of the rib waveguide. Increasing the waveguide width yields a more similar field profiles of the TE and TM modes in the guide. This behaviour can be explained by the fact that effective index increases with the width, implying a stronger confinement. 4.2. Height influence Simulations have also been carried out for several waveguide heights, from 0.75 to 2 lm. For each height value, the effective index and the etching depth for both TE and TM polarizations are determined for different waveguide widths in order to determine the conditions to obtain singlemode and polarization-independent rib waveguide. Some results are given in Fig. 6 for k ¼ 1:53 lm. The r parameter corresponding to zero effective index birefringence, is plotted versus the W/H ratio for the various height values. As in Fig. 2, a minimum of the etching depth (corresponding to a maximum of the parameter r) is observed for
Fig. 6. Parameter r as a function of the ratio W/H to cancel the effective index birefringence, for waveguide height H from 0.75 lm to 2 mm at 1.53 lm.
each curve. What is worth noting is that the r maximum occurs for a quasi-constant value of the ratio W/H. Indeed, W/H varies from 0.643 to 0.648 when H is, respectively, changed from 0.75 to 2 lm. Thus W/H increases only very slightly with the waveguide height. The parameter r corresponding to zero effective index birefringence increases with the waveguide height. For the various height values, the evolution of the parameter r (or the etching depth) with the W/H ratio is very similar. However, the maximum ratio W/H to keep a single-mode waveguide decreases as H increases. The last point of each curve determines the limit to have single-mode waveguide. For example, for H ¼ 2 lm, the maximum width for a monomode waveguide is 1.3 lm, which corresponds to W =H ¼ 065. So, for thicker waveguides, the etching depth would have to be a larger fraction of the silicon layer thickness. The ratio P/H, etching depth per waveguide height, calculated for the width value corresponding to the minimum etching depth (W =H 0; 645) varies from 0.64 to 0.59 for H varying from 0.75 to 2 lm. The more the height is reduced, the more the structure of the guide tends towards a square strip waveguide.
5. Conclusion Investigation of small size rib SOI waveguides exhibiting simultaneously single-mode and polarization insensitive characters have been presented. Using a mode solver program, the etching depth, width and height values have been determined to design waveguides which fulfil these two conditions, for wavelengths ranging from 1.53 to 1.61 lm. It has been shown that the single-mode condition limits the waveguide width, but is valid even for deeply etched waveguides. The width to height ratio of rib waveguides with a minimum etching depth is nearly constant to whatever the waveguide height is within the considered range. The minimum etching depth is of the same order than the rib width. The more critical parameter for the fabrication of such waveguides is the etching depth. The sen-
L. Vivien et al. / Optics Communications 210 (2002) 43–49
sitivity to width fluctuations and to the wavelength lies inside the etching depth reliability. The residual birefringence is then less than a few 104 . In future works, the birefringence and loss characteristics [13] of bends in such SOI waveguides will be considered.
Acknowledgements This work is financially supported by AlcatelOpto+. The authors acknowledge A. Carenco for helpful advices. The authors also thank Daniel Pascal for fruitful discussions.
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