(C) 2002 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 14, NO. 8, AUGUST 2002
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Polarization Crosstalk in High Index Contrast Planar Silica Waveguides B. M. A. Rahman, N. Somasiri, and M. Windmann
Abstract—Polarization crosstalk and effect of the waveguide sidewall slope angle in a planar silica waveguide with high index contrast is calculated by using rigorous full vectorial numerical approaches. Index Terms—Finite-element methods, optical polarization, optical waveguide theory, polarization crosstalk, silica waveguides.
I. INTRODUCTION
M
ODES IN OPTICAL waveguide with two-dimensional confinement are not strictly TE or TM, but hybrid in nature, which means that all the six components of the vector magnetic and electric fields are always present. For the fundamental quasi-TE ( ) mode the component is dominant and the nondominant component is often very small. Similarly, for the fundamental quasi-TM ( ) mode its component is the dominant and the component is nondominant. In this study, the modal hybridness is defined as the ratio of the nondominant to the dominant field components. Modal hybridness is higher for semiconductor waveguides, such as for rib waveguides, because of their high index contrast . It has also been shown that a waveguide incorporating a curved section [1] or with a slanted sidewall [2], [3], where structural symmetry is broken, modal hybridness is enhanced considerably, and this asymmetry has been exploited to design passive polarization rotators. Hybridness of optical modes in silica waveguide with low index contrast is considerably smaller. Although polarization-mode dispersion has been clearly identified as a major problem for high-data-rate long-distance optical data transmission in silica fibers, however, polarization crosstalk in a centimeter long planar silica waveguide has not yet been regarded as a serious problem. Recently, Takada and Mitachi [4] have reported more than 21-dB polarization crosstalk in a 10-m-long silica waveguide with 0.4% index contrast, which they attributes due to the presence of the curved corners. However, recently some optical component manufacturers are developing plans to use waveguides with even higher index contrast to reduce waveguide bending radius, which in turn may reduce the chip size or can be used to increase functionality of the chip, for example, Inoue and Hida reported [5] that only by using 1.5% , a 400 channels arrayed waveguide (AWG) can be fabricated on a 15-cm wafer. In this letter, our investigation on the possibility of polarization crosstalk for such high silica waveguides and the possible effect of their nonideal slanted sidewall is reported. Manuscript received November 5, 2001; revised April 8, 2002. The authors are with City University, London EC1V 0HB, U.K. (e-mail:
[email protected]). Publisher Item Identifier S 1041-1135(02)06018-4.
II. THEORY To study modal hybridness it is essential to consider a fully vectorial modal solution approach. In this study, the -field based variational formulation [6] is used, which is considered to be one of the most appropriate approach for optical waveguides, since all three components of the magnetic field are naturally continuous across the dielectric interfaces, unlike its dual -field based formulation. To study the effect of the sidewall slope angle it is also essential to represent this feature exactly. In this respect, the finite difference method generally considers a staircase like approximation and needs many steps to adequately represent the slant sidewall, but in the finite-element method, an arbitrary slant angle can be represented exactly with a much smaller number of elements. To calculate power conversion between the two orthogonally polarized modes, it is necessary to apply a rigorous junction analysis approach. In this respect, the simple overlap method may be inadequate and in this study a rigorously convergent fully vectorial least squares boundary residual (LSBR) method [7] is used which can also consider all the guided and radiated modes simultaneously to satisfy the boundary conditions at the junction interface. III. RESULTS In the design of photonic devices and systems, mostly squareshaped waveguides are considered because of their nearly circular mode shapes. However, in this case, two polarized modes also degenerate when the height , and the width of the waveguide are exactly equal. In this study, the height is taken as 6 m and the width is varied to study its effect on the modal degeneration. The core index is assumed as 1.5 and the cladding index is also varied to control the index contrast . In this study, the operating wavelength is considered to be 1.55 m. When the waveguide width is not very close to its height, then for quasi-TM (TE) mode, its dominant ( ) component is nearly circular and the nondominant, ( ) component is antisymmetric along the both - and -axes of the waveguide, with small field singularities at the four waveguide corners. As the overlap integral of the vector modal fields of the two polarized modes would be small, there exist only a small possibility of power conversion between the two modes at the waveguide irregularities. For this case, when m, the longitudinal propagation constant ( ) of the quasi-TM mode is higher than that of the quasi-TE mode. If the waveguide width is increased, propagation constants of both the quasi-TE and quasi-TM modes increase. They are equal when m, which is same as its height, and the two modes degenerate. When is wider than 6 m, propagation constant ( ) of the
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Fig. 1. Variation of the modal hybridness with the waveguide width W for different slant angles.
quasi-TE mode is higher than that of quasi-TM mode. When they degenerate, for both the modes, both of their transverse field components ( and ) are symmetric in shape and also equal in amplitude. However, this mode degeneration occurs only when is exactly equal or very close to its height. However, it has been reported that the modal hybridness is increased when the structure is not symmetrical [1]–[3]. It may also be possible that during the fabrication process, the structure sidewall may not be exactly vertical and the effect of this deviation is studied next. For such a waveguide with slightly slanted sidewall, both the and field profiles for both the polarizations may be nearly symmetric and may also have nearly equal amplitudes when is nearly equal to , but they do not have to be exactly equal. It is expected that for such waveguides, there would be significant overlap between the vector profiles of the two polarized modes and a significant power conversion may take place between them. Variation of the modal hybridness with the waveguide width for different slant angles are shown in Fig. 1 for %. It can be noted that for 0 slant angle modal hybridness reduces to 50% of its maximum value when its width deviates only 20 nm from being a perfectly square waveguide. However, for , when deviates from 6- m by as much as 0.3 m, modal hybridness remains more than 50%. In this case, modal hybridness does not depend strongly on the index contrast. The variation of the half-beat length between the two polarized modes are shown in Fig. 2 for %. The half-beat length is defined as the ), where and are the propagation constants of the two quasi-TE and TM fundamental modes. At a distance equal to half-beat length, 100% polarization conversion can take place whenever modes are phase matched. It can be noted that maximum is nearly 50 m when , hence polarization conversion in a waveguide only a few centimeter long would be negligible. However, even for small slant angle , the maximum value drastically reduced to only 60 cm. This value reduces further to 6 cm when . This signifies that any slight deformity in the waveguide’s cross-section may have a significant effect on the polarization conversion in silica waveguides. value for % and is also shown as a dotted line.
IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 14, NO. 8, AUGUST 2002
Fig. 2. Variation of the half-beat length L with the waveguide width W .
Fig. 3. Variation of the modal coefficients and is incident.
with W , when TE mode
Next, the LSBR method is used to find the modal coefficients and of the quasi-TE and TM modes, respectively, when a pure TE mode is incident from the input guide. To satisfy the boundary conditions at the junction plane, this incident mode excites two hybrid modes. The variation of the and with the width is shown in Fig. 3. It can be noted that when m, reduces to and increases to and both the mode carries equal power. For waveguides with nearly perfect vertical sidewall, , rapidly decreases when waveguide width is different than its height . However, it can also be noted that as the slant angle increases values decrease much more slowly with its . The evolution of mode conversion from incident TE to TM is shown in Fig. 4 for 10 slant angle and %. It can be noticed that at length 60 mm, all the TE power would be converted to TM polarized power for m. Even when the modes are not phase matched, for example, for m, more than 76% power conversion can take place and its effect on the system performance should be considered. It can also be noted that when guide width is smaller than 6 m, although maximum conversion is lower, however, its initial rate of polarization conversion is slightly higher than that of for m.
(C) 2002 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE RAHMAN et al.: POLARIZATION CROSSTALK IN HIGH INDEX CONTRAST PLANAR SILICA WAVEGUIDES
Fig. 4. Evolution of TM polarized power along the axial direction z , when TE mode is incident.
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Fig. 6. Variation of the polarization crosstalk with the silica guide.
W in a 10-mm-long
the polarization crosstalk is better than 33 dB, however, this value would deteriorate rapidly as the waveguide length is increased. On the other hand for and %, crosstalk is worse than 13 dB and this value deteriorates further for % to below 9 dB for a 10-mm-long waveguide. For a waveguide with m, its crosstalk figure is slightly better, however, for a much longer waveguide this may not be true, as maximum polarization conversion only takes place when the modes degenerate. IV. CONCLUSION
Fig. 5. width
W for different slant angles.
Variation of the maximum polarization conversion with the waveguide
Next, maximum power transfer between the modes is calculated from the modal coefficients and modal fields. The variation of maximum power transfer with the waveguide width is shown in Fig. 5. It can be noted that for m, it is always possible to convert completely from one polarization state to another. However, as already shown in Fig. 2, that when , length required ( ) would be too long. It is also shown that when m, although maximum power transfer drops rapidly as the width deviates from 6 m value, however, polarization crosstalk may still be significant. However, when the sidewalls are not vertical, maximum power conversion value increases rapidly with the slant angles. It can be noted that a waveguide with m and , nearly 50% TE power can be converted to TM power if the waveguide length is equal to , which in this case is only 6 cm. A planar silica waveguide could be significantly long, such as in an AWG chip, and polarization-mode conversion may be a serious challenge for such systems. Polarization crosstalk for a moderately long 10-mm waveguide is shown in Fig. 6. It can be seen for a waveguide with m, , and %
It is shown that a significant polarization conversion can take place in a long silica waveguide with high , but this may not be critical for a 1-cm-long waveguide with perfectly vertical sidewalls. However, if its sidewalls are not perfectly vertical then polarization crosstalk may be significant for many silicabased components. It is also known that curvature in waveguide also enhances the polarization conversion [1], hence effect of long curved waveguides, such as inside an AWG chip, also requires further investigation to avoid undesirable polarization crosstalk in the systems. REFERENCES [1] S. S. A. Obayya, B. M. A. Rahman, K. T. V. Grattan, and H. A. El-Mikati, “Beam propagation modeling of polarization rotation in deeply etched semiconductor bent waveguides,” IEEE Photon. Technol. Lett., vol. 13, pp. 681–683, July 2001. [2] V. P. Tuzolov and H. Fontaine, “A passive converter free of longitudinally periodic structure,” Opt. Commun., vol. 127, pp. 7–13, 1996. [3] B. M. A. Rahman, S. S. A. Obayya, N. Somasiri, M. Rajarajan, K. T. V. Grattan, and H. A. El-Mikathi, “Design and characterization of compact single-section passive polarization rotator,” J. Lightwave Technol., vol. 19, pp. 512–519, Apr. 2001. [4] K. Takada and S. Mitachi, “Polarization crosstalk dependence on length in silica-based waveguides measured by using optical low coherence interference,” J. Lightwave Technol., vol. 16, pp. 1413–1422, Aug. 1998. [5] Y. Inoue and Y. Hida, “Large-scale arrayed-waveguide gratings,” in Proc. OECC/IOOC, Sydney, Australia, July 2001, pp. 29–32. [6] B. M. A. Rahman and J. B. Davies, “Finite element solution of integrated optical waveguides,” J. Lightwave Technol., vol. 2, pp. 682–688, May 1984. , “Analysis of optical waveguide discontinuities,” J. Lightwave [7] Technol., vol. 6, pp. 52–57, Jan. 1988.
