Open Access
Add to BibSonomyAbstract
The objective of this paper is the detailed study of polarization conversion in deformed high index contrast (HIC) waveguides. The type of deformation considered here is the slanted sidewalls of buried channel waveguides. Polarization conversion of HIC waveguides are investigated for possible core refractive indices ranging from 2 (SiNx) to 3.5 (Si), by using numerical schemes based on the finite-element and beam propagation methods. The numerical results show that polarization conversion can be greatly magnified in HIC channel waveguides. For example, in Si-wire waveguides, complete polarization conversions can occur within just tens of micrometers.
©2006 Optical Society of America
1. Introduction
The realization of highly integrated optical circuits, or nanophotonic devices, has been the subject of intense investigation in the new technological era. In particular, high index contrast (HIC) waveguides, such as Si-wire waveguides [1–3], have attracted considerable attention. Due to the very high refractive index contrast Δ between Si (≈3.5) and SiO2 (≈1.45) in Si-wire waveguides, light can be strongly confined in the core region, and it is therefore possible to reduce the bending radii to several micrometers [4, 5], leading to ultra-small optical integrated circuits. With this feature, various ultra-small functional optical devices based on Si-wire waveguides have been proposed and fabricated [6–10]. Among possible candidate material platforms, SiNx can be alternatively used as a core medium of HIC waveguides [10–13], due to the fact that it has a high refractive index. Additionally, flexible circuit designs are possible, because SiNx offers wide refractive-index tunability in the range between n = 2.0 and 3.5 [14, 15].
A serious and unavoidable problem in HIC waveguides is their polarization dependence. Due to the large refractive index difference between the core and cladding, the guided modes of HIC waveguides are found to be hybrid, and thus the birefringence of the guided modes increases. This problem of birefringence can be resolved by employing square channel waveguides with the geometrical symmetry. For this purpose, Si-wire waveguides with ultra-small dimensions of 300 nm × 300 nm have been fabricated successfully [6].
However, there are always fabrication errors caused by stress, sidewall etching processes, etc. It has been shown that such deformations induce undesirable polarization conversion for silica waveguides [16–18 ] and for semiconductor waveguides [19, 20]. In HIC waveguides, this effect is expected to be further magnified since the value of Δ(in the range between 20 and 40 %) is over ten times larger than that of typical silica waveguides, including silica waveguides with relatively high index contrasts (1 % < Δ < 1.5 %) [17, 18].
For practical implementations of HIC waveguide circuits, it is important to take into account the impact of possible deformations on polarization conversion. However adequate study of polarization conversion in deformed HIC waveguides, particularly in the buried channel structures (so-called photonic wires) in which light can be tightly confined, is missing from the international literature. We therefore, in this paper, investigate the impact of deformations, slanted sidewalls, of the channel waveguides. The polarization conversion of the HIC waveguides are examined for various possible values of the core refractive index ranging from 2 (SiNx) to 3.5 (Si), using numerical schemes based on the finite-element method [21, 22]. The numerical results presented here show that polarization conversion can be greatly magnified in the HIC channel waveguides. Especially, for the Si-wire waveguides, a complete polarization conversion can occur within just tens of micrometers.
Fig. 1. Cross-sectional view of a HIC waveguide with slanted sidewalls, where d, a, and θ are the waveguide height, width, and slant angle, respectively. n core and n clad denote the refractive indices of the core and cladding, respectively.
2. Finite-element analysis of deformed HIC waveguides
In three-dimensional single-mode channel waveguides, there are two orthogonal guided modes, i.e. quasi-TE and quasi-TM modes. In the quasi-TE modes, the horizontal, x component of the electric field (Ex) is dominant. On the other hand, the vertical, y component of the electric field (Ey) is dominant in the quasi-TM modes. These modes can be clearly distinguished in rectangular waveguides, while in square waveguides they are degenerated. However, if the waveguides become structurally asymmetric, the electric fields in the minor components increase significantly. In this condition, Ex and Ey can exchange their power at regular intervals along the propagation direction, resulting in polarization conversion. In particular, if the magnitudes in both components are comparable, |Ex| ≈ |Ey|, a complete power transfer can occur. Due to the significantly high values of A in HIC waveguides, in comparison to silica waveguides, polarization conversion stemming from unexpected structural asymmetries will be further magnified. These effects, therefore, can have unpredictable consequences for the performance of designed HIC waveguide structures.
We consider a buried channel waveguide as shown in Fig. 1, uniform along the propagation direction z. a and d represent the waveguide width and height, respectively. The operating wavelength is λ = 1.55 μm. A cladding refractive index of n clad = 1.45 (SiO2) is maintained through this paper. Assuming Si or SiNx, the refractive index of the core is set as n core = 3.5, 3.0, 2.5, or 2.0. The waveguide dimension is chosen to ensure single-mode operations, to eliminate losses caused by higher-order modes. Figure 2 shows the normalized cut-off wavelength, λc/a, for the square channel waveguides (a = d) as a function of n core. In the case of Si-wire waveguides (n core = 3.5) with the dimension of 300 nm × 300 nm, shown to be practically feasible [6], the cut-off wavelength is calculated as λc = 1.45 μm. In the case of n core = 2.0, 2.5, and 3.0, the values of d are set as 700 nm, 470 nm, and 360 nm, respectively, to produce cut-off wavelengths close to the value of the wavelength for the n core = 3.5, i.e. to be λc ≈ 1.45 μm.
Fig. 2. Normalized cutoff wavelength λc/a of the square channel waveguide as a function of refractive index of the core, n core.
Fig. 3. Rotation parameter R as a function of waveguide width a for (a) n core = 3.5 and d = 300 nm, (b) n core = 3.0 and d = 360 nm, (c) n core = 2.5 and d = 470 nm, and (d) n core = 2.0 and d = 700 nm.
In terms of the type of deformation in the waveguide, we recognize the following situation: a waveguide with vertical sidewalls (θ = 0°, z < 0) is deformed at a certain transverse plane (z = 0), and the sidewalls are slanted with angle 0, as shown in Fig. 1, in the waveguide beyond this plane (z > 0). Here, we introduce rotation parameter R [22] defined as
where n(x,y) is the transverse refractive index distribution. The integration is carried out for the quasi-TE modes.
Figure 3 shows rotation parameter R as a function of waveguide width a for (a) n core = 3.5 and d = 300 nm, (b) n core = 3.0 and d = 360 nm, (c) n core = 2.5 and d = 470 nm, and (d) n core = 2.0 and d = 700 nm. In the case of 0 = 0° (in the square and rectangular waveguides), R is almost zero. This means the two orthogonal guided modes are well separated. It is helpful to recognize that the value of R increases slightly, even in the case of θ = 0°, for higher n core. Especially, in the case of n core = 3.5 (Si-wire), R is greater than 1 % meaning that the modes are strongly hybrid.
When θ differs from 0°, the value of R increases significantly around d ≈ a. This means that in the square channel waveguides, which exhibit no birefringence, complete polarization conversions can occur with slight structural deformations. The polarization conversions can be avoided by choosing a value of a different from d (rectangular waveguides), although the birefringence will be higher in this case. If 0 is further extended (θ = 5° or 10°), R increases for all the widths a and complete polarization conversions can occur when the value of a is smaller than d. In such cases, significant polarization conversion is unavoidable even in the rectangular waveguides.
The polarization conversions can increase for higher values of R. Theoretically, the complete polarization conversions occur only when R = 1. These conditions can be understood from the vector magnetic field distributions on the transverse xy plane, shown in Fig. 4. The orthogonal fundamental modes in Figs. 4(a) and 4(b) correspond to the quasi-TE and quasi-TM modes, respectively, in a rectangular waveguide with vertical sidewalls, d = 300 nm, a = 280 nm, and θ = 0°. The two modes in Figs. 4(c) and 4(d) are for the deformed rectangular waveguide with slanted sidewalls, θ= 1°, and Figs. 4(e) and 4(f) show those for a deformed square waveguide, d = a = 300 nm and θ = 5°. If a waveguide is deformed instantly, the quasi-TE mode in (a), which has hitherto propagated without any coupling to the other orthogonal quasi-TM mode shown in (b), couples to both fundamental modes in the deformed waveguide. In the case of the rectangular waveguide shown in Figs. 4(c) and 4(d), the quasi-TE mode couples mostly to the mode shown in (c), since the dominant magnetic components of both fields are in the y axis; the power coupling to the mode shown in (d) is much smaller, resulting in a low polarization conversion. In contrast, if the deformed square waveguide with d = a = 300 nm and θ = 5° is excited by the quasi-TE mode of a non-deformed waveguide, the dominant y component of the excited field is almost equally split into both fundamental modes in Figs. 4(e) and 4(f), with R ≈ 1, in the deformed square waveguide; therefore, the two modes couple almost completely, resulting in a complete polarization conversion.
Next, we discuss the half-beat length, Lπ, calculated by
Where β1 and β2 are the propagation constants of the two orthogonal fundamental modes in the deformed waveguides. L π corresponds to the coupling length of the two fundamental modes, where the maximum polarization conversion can occur. Figure 5 shows, for example, the effect of width a on the propagation constants in the Si-wire waveguide (n core = 3.5). It is observed that the separation between the two propagation constants, |β1 - β2|, increases for greater slant angles θ; this tendency corresponds to the theory for weekly guiding structures [24].
Fig. 4. Vector magnetic field distributions of the two fundamental modes in the Si-wire with n core = 3.5 and d = 300 nm. (a), (b) for a = 280 nm and θ = 0°; (c), (d) for a = 280 nm and θ = 1°; (e), (f) for a = 300 nm and θ = 5°.
Fig. 5. Propagation constants in the Si-wire waveguide (n core = 3.5) as a function of waveguide width a.
Fig. 6. Half-beat length L π as a function of waveguide width a for (a) n core = 3.5 and d = 300 nm, (b) n core = 3.0 and d = 360 nm, (c) n core = 2.5 and d = 470 nm, and (d) n core = 2.0 and d = 700 nm.
Figures 6(a), 6(b), 6(c), and 6(d) show half-beat length L π, as a function of waveguide width a for (a) n core = 3.5 and d = 300 nm, (b) n core = 3.0 and d = 360 nm, (c) n core = 2.5 and d = 470 nm, and (d) n core = 2.0 and d= 700 nm. The value of L π itself is maximized when a = d (the square channel waveguides). Alternatively, L π reduces for larger angles θ in all the figures. This can be understood in terms of the change in the separation between the two propagation constants, |β1 - β2|, as explained in the above paragraph.
One remarkable tendency is that Lπ dramatically reduces for higher refractive indices of the core, n core. In particular, in the Si-wire waveguide (n core = 3.5), the complete polarization conversion can occur within just tens of micrometers. This significant reduction of L π, corresponding to the stronger coupling between the two fundamental modes, can be understood from the magnetic field distributions, shown in Figs. 7(a) and 7(b), and the mode profiles on the waveguide center, y = 0, shown in Fig. 8. The fields in Figs. 7(a) and 7(b) are for n core = 3.5 and a = d = 300 nm and for n core = 2.0 and a = d = 700 nm, respectively. In both cases, the fields are confined well within the core region. However, as can be observed in the mode profiles in Fig. 8, where the amplitudes are normalized by the power, the mode confinement in the Si-wire (n core = 3.5) is much stronger than that in the HIC waveguides with lower n core, resulting in stronger coupling and reduction of L π.
Fig. 7. Magnetic field distributions of the horizontal (x) component for (a) n core = 3.5, a = d = 300 nm, and θ = 1° and for (b) n core = 2.0, a = d = 700 nm, and θ = 1°.
Fig. 8. Normalized field profiles in the deformed square waveguides (θ = 1°) on the waveguide center, y = 0, for n core = 3.5 (a = d = 300 nm), n core = 3.0 (a = d = 360 nm), n core = 2.5 (a = d = 470 nm), and n core = 2.0 (a = d = 700 nm).
Fig. 9. Polarization conversions in Si-wire waveguides as a function of propagation distance with n core = 3.5 and d = 300 nm for (a) θ = 1° and (b) θ = 5°.
Fig. 10. (a) |Ex| and (b) |Ey| distributions of slanted Si-wire waveguides with n core = 3.5, a = 300 nm, and θ = 5° in the xz plane at the center of the waveguide in the y direction.
3. Beam propagation simulation of polarization conversion
In this section, the effect of polarization conversions in the deformed HIC waveguides is investigated through the beam propagation analysis [22]. The slanted waveguide is excited by the quasi-TE mode of a non-deformed (square or rectangular) waveguide. Figures 9(a) and 9(b) show, in the case of Si-wire waveguides, the rate of polarization conversions as a function of propagation distance z, for θ = 1° and 5°, respectively. The polarization conversion is obtained by
where E(z) is the electric field at the distance z obtained with the beam propagation method. E0 and H 0 are the electric fields and magnetic fields, respectively, of the fundamental eigenmodes in the non-deformed waveguide. The subscripts TE and TM represent the quasi-TE and quasi-TM modes, respectively. iz is the unit vector in the z direction. P TM(z) corresponds to the TM-polarized power of the propagating light and P TE(0) corresponds to the power of input TE-polarized light. It is clear from Fig. 9 that the polarization conversion occurs at regular intervals. These intervals are in a good agreement with the half-beat length L π evaluated in the previous section. It is also clear that the polarization conversion increases for larger values of R.
The fact that in the case of θ = 5° shown in Fig. 9(b), the polarization conversion occurs within just micrometers is remarkable. Even when the slant angle is as small as θ = 1°, nonnegligible polarization rotations, greater than 2.4 %, which can cause undesirable effects in practical devices, are observed within the propagation distance of micrometers, for all the widths (a = 270 to 300 nm) in Fig. 9 (a). These ultra-short half-beat lengths make it difficult to avoid the influence of unexpected polarization conversions by trying to increase the half-beat length, even taking into account the size reduction of HIC devices due to the small bending radii. It is, therefore, advisable that the waveguides be designed so that the maximum polarization conversion reduces to a permissible level. For this purpose, one effective approach is to increase the ratio of height d and width a. For example, the maximum conversion becomes less than 1 % in the case of d = 300 nm and a = 250 nm. In contrast, in the design of waveguides with zero birefringence, i.e. d = a, the tolerance for the slant angle of the sidewalls are very severe.
Figures 10(a) and 10(b) show |Ex| and |Ey| distributions of slanted Si-wire waveguides with n core = 3.5, a = 300 nm, and θ = 5° in the xz plane at the center of the waveguide (across y direction). Periodical power transfer between Ex and Ey is clearly observed.
Fig. 11. Polarization conversions in the HIC waveguides as a function of propagation distance with n core = 3.0 and d = 360 nm for (a) θ = 1° and (b) θ = 5°.
Fig. 12. Polarization conversions in the HIC waveguides as a function of propagation distance with n core = 2.5 and d = 470 nm for (a) θ = 1° and (b) θ = 5°.
Fig. 13. Polarization conversions in the HIC waveguides as a function of propagation distance with n core = 2.0 and d = 700 nm for (a) θ = 1° and (b) θ = 5°.
Finally, Figs. 11, 12, and 13 show the evaluation of power fraction Py(z) in the HIC waveguides with different refractive indices of the cores, n core = 3.0, 2.5, and 2.0, respectively. The half-beat lengths and the maximum polarization conversions observed in the figures are in firm agreement with those evaluated in the previous section. Although, as expected, there is a tendency for the half-beat lengths to increase for smaller n core, these half-beat lengths are still much shorter, even for n core = 2.0, than typical waveguide lengths used in practical devices. For example, in the case of n core = 2.0 and θ = 1°, shown in Fig. 13(a), the half-beat length for a = 700 nm is still as short as L π = 575 μm. Moreover, even at a shorter propagation distance (e.g. z = 100 μm), the polarization conversions are significantly high (greater than 7 %). It is, therefore, advisable to reduce the maximum polarization conversion down to a permissible level, rather than to try to increase the half-beat length, for both material systems discussed in this paper, Si-wires and HIC waveguides consisting of SiNx.
4. Conclusions
We have investigated in detail the polarization conversion in deformed HIC waveguides. The core material was assumed to be Si or SiNx, with various refractive indices in the wide range between 2 and 3.5. Our analysis has shown that in HIC channel waveguides, if the shape of the waveguide approaches that of a square and the refractive index contrast between core and cladding increases, the polarization conversion can be greatly magnified due to fabrication errors caused by stress, sidewall etching processes, etc. According to our literature research, the observation is reported for the first time.
The half-beat length is so short, e.g. tens of micrometers in the case of the Si-wire waveguides, that it is difficult to avoid the damage caused by polarization conversion to practical devices, such as directional couplers, waveguide gratings, etc. To reduce the effect of polarization conversion to a permissible level, one effective approach is to sufficiently increase the ratio for the waveguide width and height. Although the waveguides with such high ratio for the width and height exhibit high birefringence, this problem can be avoided through the implementation of a polarization diversity scheme [25–27]. This scheme has been originally proposed to circumvent the significantly high polarization dependence in HIC waveguides. The numerical results presented in this paper have also supported the usefulness of such an approach, from a separate source, to overcome the problem of polarization conversion.
Acknowledgments
The authors would like to sincerely thank Dr. Nikolaos John Florous from Hokkaido University for fruitful discussions.
References and links
1. J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390, 143–145 (1997). [CrossRef]
2. B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998). [CrossRef]
3. K. K. Lee, D. R. Lim, L C. Kimerling, J. Shin, and F. Cerrina, “Fabrication of ultralow-loss Si/SiO2 waveguides by roughness reduction,” Opt. Lett. 26, 1888–1890 (2001). [CrossRef]
4. A. Sakai, G. Hara, and T. Baba, “Propagation characteristics of ultrahigh-Δ optical waveguide on silicon-on-insulator substrate,” Jpn. J. Appl. Phys. 40, L383–L385 (2001). [CrossRef]
5. Y. A. Vlasov and S. J. McNab, “Losses in single-mode silicon-on-insulator strip waveguides and bends,” Opt. Express 12, 1622–1631 (2004). [CrossRef] [PubMed]
6. T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, “Microphotonics devices based on silicon microfabrication technology,” IEEE J. Sel. Topics Quantum Electron. 11, 232–240 (2005). [CrossRef]
7. K. Sasaki, F. Ohno, A. Motegi, and T. Baba, “Arrayed waveguide grating of 70×60 μm2 size based on Si photonic wire waveguides,” Electron. Lett. 41, 801–802 (2005). [CrossRef]
8. R. L. Espinola, J. I. Dadap, R. M. Osgood Jr., S. J. McNab, and Y. A. Vlasov, “C-band wavelength conversion in silicon photonic wire waveguides,” Opt. Express 13, 4341–4349 (2005). [CrossRef] [PubMed]
9. G. R. Roelkens, J. Brouckaert, D. Taillaert, P. Dumon, W. Bogaerts, D. Van Thourhout, R. Baets, R. Nötzel, and M. Smit, “Integration of InP/InGaAsP photodetectors onto silicon-on-insulator waveguide circuits,” Opt. Express 13, 10102–10108 (2005). [CrossRef] [PubMed]
10. C. Li, N. Ma, and A. W. Poon, “Waveguide-coupled octagonal microdisk channel add-drop filters,” Opt. Lett. 29, 471–473 (2004). [CrossRef] [PubMed]
11. N. Daldosso, M. Melchiorri, F. Riboli, M. Girardini, G. Pucker, M. Crivellari, P. Bellutii, A. Lui, and L. Pavesi, “Comparison among various Si3N4 waveguide geometries grown within a CMOS fabrication pilot line,” J. Lightwave Technol. 22, 1734–1740 (2004). [CrossRef]
12. M. Melchiorri, N. Daldosso, F. Sbrana, L. Pavesi, G. Pucker, C. Kompocholis, P. Bellutii, and A. Lui, “Propagation losses of silicon nitride waveguides in the near-infrared range,” Appl. Phys. Lett. 86, 121111 (2005). [CrossRef]
13. T. Chu, H. Yamada, S. Ishida, and Y. Arakawa, “Compact 1 × N thermo-optic switches based on silicon photonic wire waveguides,” Opt. Express 13, 10109–10114 (2005). [CrossRef] [PubMed]
14. N. Ibaraki and H. Fritzsche, “Properties of amorphous semiconducting a-Si:H/a-SiNx:H multilayer films and of a-SiNx:H alloys,” Phys. Rev. B 30, 5791–5799 (1984). [CrossRef]
15. C.-C. Lee, H.-L. Chen, J.-C. Hsu, and C.-L. Tien, “Interference coatings based on synthesized silicon nitride,” Appl. Opt. 38, 2078–2082 (1999). [CrossRef]
16. K. Takada and S. Mitachi, “Polarization crosstalk dependence on length in silica-based waveguides measured by using optical low coherence interference,” J. Lightwave Technol. 16, 1413–1422 (1998). [CrossRef]
17. B. M. A. Rahman, N. Somasiri, and M. Windmann, “Polarization crosstalk in high index contrast planar silica waveguides,” IEEE Photon. Technol. Lett. 14, 1109–1111 (2002). [CrossRef]
18. N. Somasiri and B. M. A. Rahman, “Polarization crosstalk in high index contrast planar silica waveguides with slanted sidewalls,” J. Lightwave Technol., 21, 54–60 (2003). [CrossRef]
19. S. S. A. Obayya, S. Haxha, B. M. A. Rahman, and K. T. V. Grattan, “Numerical modeling of polarization conversion in semiconductor electro-optic modulators,” Appl. Opt. 44, 1032–1038 (2005). [CrossRef] [PubMed]
20. E. D. Finlayson, J. M. Heaton, B. M. A. Rahman, and S. S. A. Obayya, “Polarization conversion in passive deep-etched GaAs/AlGaAs waveguides,” J. Lightwave Technol. 24, 1425–1432 (2006). [CrossRef]
21. K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers,” IEEE J. Quantum Electron. 38, 927–933 (2002). [CrossRef]
22. K. Saitoh and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. 19, 405–413 (2001). [CrossRef]
23. V. P. Tzolov and M. Fontaine, “A passive polarization converter free of longitudinally-periodic structure,” Opt. Commun. 127, 7–13 (1996). [CrossRef]
24. M. Fontaine, “Theoretical approach to investigating cross-phase modulation phenomena in waveguides with arbitrary cross sections,” J. Opt. Soc. Am. B 14, 1444–1452 (1997). [CrossRef]
25. C. K. Madsen, “Optical all-pass filters for polarization mode dispersion compensation,” Opt. Lett. 25, 878–880 (2000). [CrossRef]
26. D. Taillaert, H. Chong, P. I. Borel, L. H. Frandsen, R. M. De La Rue, and R. Baets, “A compact two-dimensional grating coupler used as a polarization splitter,” IEEE Photon. Technol. Lett. 15, 1249–1251 (2003). [CrossRef]
27. M. R. Watts and H. A. Haus, “Integrated mode-evolution-based polarization rotators,” Opt. Lett. 30, 138–140 (2005). [CrossRef] [PubMed]
