## Abstract

We determine group index and group velocity dispersion (GVD) of SOI single-mode strip waveguides (photonic wires) with 525×226nm cross-section over the entire telecommunication bandwidth by employing an integrated Mach-Zehnder interferometer. The measured GVD yields 4400 ps/(nm∙km) at 1550 nm and exceeds that of standard single-mode fibers by almost three orders of magnitude. In the photonic wires the GVD is mainly determined by strong light confinement rather than by material dispersion. Our results indicate that despite this high GVD, dispersion-induced signal impairment is negligible in photonic circuits for data rates up to 100-Gb/s and total waveguide lengths as long as about 1 meter. The measured group index and GVD are used as benchmarks to compare model calculations originating from four different theoretical methods.

©2006 Optical Society of America

## Corrections

Eric Dulkeith, Fengnian Xia, Laurent Schares, William M. J. Green, Lidija Sekaric, and Yurii A. Vlasov, "Group index and group velocity dispersion in silicon-on-insulator photonic wires: errata," Opt. Express**14**, 6372-6372 (2006)

https://www.osapublishing.org/oe/abstract.cfm?uri=oe-14-13-6372

## 1. Introduction

The high refractive index contrast achievable in silicon-on-insulator (SOI) structures provides strong confinement of light and allows aggressive scaling of photonic components close to the diffraction limit. Moreover, due to compatibility with CMOS fabrication SOI is considered as a promising platform for ultra-dense on-chip integration of photonic and electronic circuitry. Recently, much attention has been drawn to photonic devices such as rings, modulators, attenuators, wavelength demultiplexers and switches [1–9]. It has also been found that already at moderate input powers the efficient mode confinement leads to an enhanced nonlinear response of the waveguide, thus providing a platform for nonlinear all-optical devices as well. For instance, wavelength conversion based on four-wave mixing and broadband light sources due to continuum generation were recently reported [10–12]. On the other hand, the large index contrast causes the waveguide dispersion to dominate over intrinsic material dispersion. Therefore, designing photonic components demands very precise knowledge of the dispersion properties. However, due to a lack of reported experimental data the first generation of devices are solely based on model calculations and the accuracy of applied theoretical methods becomes an important issue.

Scalability is one of the major advantages of SOI structures and allows designing high density photonic circuits. Single mode propagation is highly desirable to avoid excessive bending losses and polarization conversion losses. Curved waveguides (an indispensable component for such circuits) with radii of only a few µm have been recently demonstrated [13, 14]. However, it was also shown that higher order modes leak out of the bends thus introducing additional propagation losses. Moreover, even a straight photonic wire already exhibits large differences in group index and dispersion between the optical modes of different polarization [15]. This makes the design of polarization-insensitive components almost impossible (e.g. directional couplers with a 3-dB splitting efficiency for all modes at the very same wavelength). To avoid the influence of potential polarization mode conversion and to simplify the design rules, the TM cutoff must be shifted to shorter wavelengths in order to achieve a *truly* single-mode propagation supported solely by the fundamental TE_{01} mode.

Figure 1 illustrates the phase-space of SOI strip waveguides for the two lowest TE and TM modes as a function of waveguide width for three different Si slab thicknesses. The results are based on 3D planewave calculations using the MIT photonic-band code [16]. Black lines present the mode cutoff wavelengths.

For a slab height of e.g. *h* = 226 nm [Fig. 1(b)] and a waveguide width of *w* = 525 nm, TE_{01} and TM_{01} modes are both supported between 1400 and 1700 nm (grey region) whereas at longer wavelengths the waveguide become *truly* TE_{01} single moded (grey hatched region).

The presented mode-map suggests reducing height and width to achieve single TE01 moded waveguides at around 1550 nm. However, decreasing the waveguide dimensions strongly enhances propagation losses. It is known that the smaller the cross-section of the waveguide, the higher the modes’ field intensity becomes at the waveguide edges, leading to enhanced light scattering due to sidewall surface roughness [17]. On account of this, the geometry of the waveguides investigated in this paper presents a trade-off between a minimal number of supported modes (TE_{01} and TM_{01}) and surface roughness introduced losses.

## 2. Experimental set-up and device design

For transmission measurements either a broadband LED source (1200 – 1700 nm) or a tunable diode laser (1470 - 1631 nm) is utilized. The light is launched into and collected from the waveguide by using tapered, micro-lensed polarization maintaining fibers. Polarization is controlled by a set of polarizer, λ/2 and λ/4 plates with a total rejection ratio of over 30 dB between TE and TM polarizations. The transmitted light is characterized by an optical spectrum analyzer with a resolution of 60 pm (applying the LED source) or by an InGaAs detector combined with an integrating sphere (applying the diode laser). The diode laser is tuned in 20-pm steps to collect high resolution transmission data. The eye diagram in Fig. 5 is measured with a 1528-nm DFB laser, followed by a 40-Gb/s external electro-optic modulator driven by a pattern generator (PG). The average power at the input port of the waveguide is 10 mW, and the total optical losses of the experimental setup including the waveguide are determined as 11 dB. The signal is detected by a 40-GHz photodiode on a 20-GHz scope head. The frequency response of the waveguide is determined by replacing the PG and the scope with a 50-GHz scalar network analyzer.

Strip waveguides and Mach-Zehnder interferometers (MZI) are fabricated on lightly *p*-doped 200-mm SOI wafers manufactured by SOITEC. The wafer has a silicon device layer with a thickness *h* of 226 nm on top of a 1-μm buried oxide layer (BOX). Wafer processing is carried out on a standard CMOS line at the IBM Watson Research Center as described in reference [13]. A 50-nm-thick oxide is deposited via chemical vapor deposition to serve as a hard mask for etching. Wafers are then coated with photoresist and exposed by electron beam lithography. The obtained resist pattern is transferred to the oxide to act as a mask to further pattern the Si layer. To achieve high coupling efficiency, polymer-based spot-size converters with inverted taper geometry are defined on each end of the waveguide [18]. Waveguide width *w* and Si slab thickness are determined from scanning electron microscope (SEM) images as *w* = 525 nm and *h* = 226 nm, respectively. According to the mode map in Fig. 1, these values ensure that for TE polarization only the lowest order mode is supported for wavelengths longer than 1400 nm. Figure 2 shows a SEM image of the MZI. 3-dB directional couplers are used to split and combine the light. The MZI is folded such that both arms contain the very same number of bends (radius = 6 μm). According to Ref 14, losses for bends with radii larger than 2 μm are negligible, suggesting a minimal impact of the bends on the waveguide dispersion. To achieve a free spectral range (FSR) of only a few nm between the interference peaks, the straight section of the two arms of the MZI is set to differ by a length of *ΔL* = 80 μm.

## 3. Experimental results

#### 3.1 Group index

In order to measure the waveguide dispersion we have designed an integrated Mach-Zehnder-Interferometer (MZI). This integrated geometry provides a high signal-to-noise ratio (SNR) since externally induced phase distortions are absent. Figure 2 shows the transmission spectrum of the MZI, measured with tunable laser diodes from 1470 to 1630 nm. The length difference between the two arms causes the laser signals to be phase-shifted with respect to each other and a strong cosine-like interference pattern is generated once the light recombines again at the output directional coupler. The relative phase shift between two adjacent extrema is equal to *π*. The formula *n _{g}* =

*(λ*∙

_{min}*λ*/

_{max})*(2 ∙ ΔL ∙ Δλ)*allows us to determine the spectral dependence of the group index ng from the interference fringes [19]. The spectral positions of minima and maxima are denoted as

*λ*,

_{min}*λ*, where

_{max}*ΔL*and

*Δλ*is the length difference of the MZI arms and the spectral distance between adjacent minima and maxima, respectively. The directional couplers are designed to perform a 3-dB splitting efficiency of the TE

_{01}mode around 1550 nm. This agrees well with the modulation depth of the interference fringes in Fig. 2 exceeding more than 25 dB between 1550 - 1600 nm. A more detailed description of the directional coupler design and performance can be found elsewhere [20].

Figure 3 shows the group indices for TE and TM polarization deduced from the measured transmission spectra. Red and blue dots correspond to measurements employing the laser diode or the broadband LED light source, respectively. For wavelengths far away from the mode cutoff (Fig. 1), between 1270 and 1631 nm, the experimentally determined *n _{g,TE}* increases nearly monotonically from

*n*~ 4.16 to 4.37. As expected, the group index

_{g,TE}*n*exceeds the refractive index of bulk silicon, clearly indicating that in case of photonic wires with strong optical confinement the propagation constant of the waveguide is dominated by the waveguides’ geometry and not by material properties. The observed increase of

_{g,TE}*n*can be explained considering the relationship

_{g,TE}*n*=

_{g}*n*-

_{eff}*λ*∙(

*dn*/

_{eff}*dλ*), with

*n*as the effective index of the waveguide. Confinement decreases for longer wavelengths and as a result

_{eff}*n*does as well. Hence, the second term including

_{eff}*dn*/

_{eff}*dλ*becomes positive and leads to an overall increase of

*n*. Note the weak upwards slope of

_{g,TE}*n*at shorter wavelengths around 1250 nm. This is caused by the intrinsic material dispersion of silicon which rises while approaching the absorption edge at 1100 nm [21].

_{g,TE}The group index for TM polarized light exhibits values of about *n _{g,TM}* ~ 5 at 1250 nm, remains nearly constant up to ~ 1350 nm, and then declines significantly to

*n*~ 4. The asymmetric geometry of the waveguide imposes the TM mode to be less confined,

_{g,TM}*n*more strongly wavelength dependent and the TM cutoff to be redshifted. Consequently, the second term in

_{eff}*n*=

_{g}*n*-

_{eff}*λ*∙ (

*dn*/

_{eff}*dλ*) is even larger than in the case of TE and lifts

*n*above the values of

_{g,TM}*n*. This is in good agreement with the experimental results in Fig. 3. However, for longer wavelengths the mode cutoff for TM

_{g,TE}_{01}at ~ 1700 nm is already very close [Fig. 1(b)] and the confinement of the TM mode becomes even less. Due to the asymptotic convergence of

*n*towards the flat effective index of the oxide layer

_{g,TM}*n*the contribution of (

_{SiOx}*dn*/

_{eff}*dλ*) vanishes and causes a sharp decrease of

*n*.

_{g,TM}The larger experimental error for *n _{g,TM}* stems from a lower SNR and fringe visibility for TM polarized light and results in a less accurate determination of the interference peaks. The low transmission intensity originates first, from the polymer spot-size converters known to possess higher losses for TM [18] and second, from the directional couplers which are not designed for 3dB splitting of TM polarized light. More interestingly, at wavelengths above
1500 nm no further evaluation of

*n*is possible at all. This is surprising given that the mode-map in Fig. 1(b) suggests a TM cutoff at 1700 nm rather than 1500 nm. Experimentally measured mode cutoffs for TE and TM polarizations for waveguides of different widths are shown in Fig. 1(b) (blue squares and red circles, respectively). The blueshift of experimental data with respect to the calculations is seen for all waveguide widths. We believe that substrate leakage can be accounted for this. Close to the mode cutoff, the effective index of the mode approaches the index of the cladding and confinement vanishes. As a result, the mode volume expands heavily and efficient substrate coupling becomes possible. From a technological point of view, substrate-coupling induced TM mode stripping can in fact be beneficial as it enables

_{g,TM}*truly*single-TE-mode propagation without the necessity of changing the waveguide geometry (Fig. 1, grey hatched area).

## 3.2. GVD and third-order dispersion

From the measured group index we can deduce the dispersion parameter *D* by using *D* = *1/c* ∙ (*dn _{g}*/

*dλ*). While the experimental noise of

*n*is small enough to accomplish accurate comparison with model calculations, the required derivation of

_{g}*n*to evaluate

_{g}*D*results in a very large noise-induced error. One way of minimizing the error is statistical averaging. The formula

*Δψ*= arccos

*{[2I(λ)*-

*(I*+

_{max}*I*/

_{min})]*(I*-

_{max}*I*describes the phase difference at a given wavelength

_{min})}*λ*. between two adjacent extema caused by the phase shift between the MZI arms [22].

*I*is the intensity and

*I*,

_{max}*I*are the maximum and minimum fringe intensities within one half-cycle. This formula permits the analysis of not only the fringe maxima and minima (FSR ~ 3 nm) but also all intermediate data points (resolution 20 pm). The inset of Fig. 4 presents the extracted phase change derived from the spectrum of a single interference fringe (1563 - 1566 nm). It progresses strictly linearly from 0 to

_{min}*π*and demonstrates the highly cosine-like behavior of the oscillation. The group index at a given wavelength

*λ*is calculated again by

*n*=

_{g}*{(λ*∙

_{max}∙ λ)*Δψ}*/

*{(2π ∙ ΔL ∙ (λ*providing more than 100 data points for

_{max}- λ)}*n*over a bandwidth of only 3 nm (note that the phase difference

_{g}*Δψ*in the formula of

*n*in Sec. 3.1 is equal to

_{g}*Δψ*=

*π*). To derive the dispersion parameter

*D*the mean value

*n*of each max-to-min half-cycle is taken.

_{g,mean}Figure 4 presents the experimental dispersion parameter *D* of the 525×226nm SOI waveguide for TE polarization (red dots). The results are extracted from the low noise laser diode measurements, as shown in the inset of Fig. 3. *D* ranges from 3700 to 4900 ps/(nm∙km) and, within this experimental scattering, remains constant throughout the shown spectral region. The SOI waveguide exhibits positive dispersion owning a mean value of *D _{mean}* ~ 4400 ps/(nm∙km). Furthermore, the negligible experimental slope of

*D*(Fig. 4) already implicates relatively small third order dispersion (TOD) in the experimentally accessible wavelength range.

Within the scope of experimental accuracy, we can estimate TOD lying within large noise-induced standard deviation of 0 ± 50 ps/nm^{2}∙km. Note that the measured dispersion *D _{mean}* is about three orders of magnitude larger than the dispersion of standard single-mode fibers. Moreover, using the Sellmeier equation [23, 24] we can calculate the material dispersion of silicon as

*D*~ - 830 ps/(nm∙km), illustrating that

_{mat}*D*exceeds

_{mean}*D*more than fivefold. We can conclude that, for the SOI waveguides investigated in this paper, the group velocity and higher order dispersion terms are primarily determined by the geometry of the waveguide. Opposed to this, second order dispersion with

_{mat}*D*~ - 910 ps/(nm∙km) was reported for Si rib waveguides [25]. This value agrees very well with

*D*and demonstrates that if the dimensions of a waveguide are large (e.g. μm-range), the light confinement weakens and the total dispersion of the waveguide is then primarily governed by material dispersion.

_{mat}Although the experimental results demonstrate large absolute GVD values for the SOI waveguides, their implementation into photonic integrated circuits should cause only negligible signal distortions. For example, if we assume transmission of optical pulses with a width τ = 2 ps (corresponding to data rates of 100-Gb/s or above), the dispersion length *L _{D}* can be estimated from the relationship

*L*= (2

_{D}*πc*/

*λ*) ∙ (

^{2}*τ*) to be in the meter range [26]. This is more than sufficient with respect to photonic integrated circuits on a length scale of some mm or even cm. Therefore, at this length data transmission is more likely to be impaired by waveguide losses or optical nonlinearities [27] rather than GVD and TOD.

^{2}/DFigure 5(a) shows a measured 25-Gb/s eye diagram with a PRBS 2^{31}-1 non-return-to-zero (NRZ) pattern. As expected, hardly any distortions are observed in the eye diagram since the physical length of the SOI waveguide (4 mm) is two orders of magnitude shorter than the dispersion length.

The distortion-free transmission through the SOI waveguide is also confirmed by the flat frequency response, shown in Fig. 5 (b). For frequencies up to at least 35 GHz (limit is set by the network analyzer sensitivity) the amplitude has a variation of less than 1 dB which is within the precision of the measurement system. Detectable distortion may be observable for sub-picosecond pulses (e.g. for *τ* = 100 fs the dispersion length *L _{D}* is reduced to ~ 2 mm).

## 4. Theoretical models

To investigate the accuracy of different numerical approaches (a crucial issue regarding the design of photonic devices) the experimental results for *n _{g}* and GVD are used as a benchmark for the calculated results obtained from the following four methods:

- A fully-vectorial, 3D planewave expansion method (PW) [16]. It solves the eigenproblem of Maxwell’s equation in each defined unit cell to evaluate the eigenfrequencies
*w(k)*. A super-cell contains the photonic wire waveguide on top of oxide substrate. The accuracy of this method is defined by number of planewaves per unit cell. This so-called grid resolution is set to 16×16×32 to achieve a reasonable trade-off between the error in eigenvalue convergence (< 2 %) and computing time [16]. The MPB software package, version 1.4.2, is used [28]. - A fully-vectorial, 3D Finite-Difference-Time-Domain (FDTD) method [29]. The computational cell is discretized over a spatial grid. At each grid point Maxwell’s equations are numerically simulated in time. The accuracy of this method is defined by the spatial-grid size. The grid resolution is set to 20×20×20nm. The group index extracted from Fabry-Perot interference fringes in the calculated transmission spectrum of waveguides with lengths ranging from 5 to 50 μm. The Full Wave software package, version 4, is used [30].
- A fully-vectorial, 3D mode-matching method [31]. This method uses the eigenmode expansion (EME) technique to model light propagation in optical waveguides. EME expresses solutions to Maxwell’s equations for a given structure in terms of forward and backward-propagating modes of the local refractive index profile. The accuracy depends on the applied mode basis set and the total number of modes used. 100 one dimensional modes are used to approximate the final mode profile. The grid resolution is set to 13×13 nm. The FimmWave software package, version 4.3.4, is used [32].
- A semi-vectorial, 3D time-domain Beam Propagation method (BPM) [33]. It requires the generation of correlation functions from the numerical solutions of a wave equation. These correlation functions are in turn Fourier-transformed. The resulting spectra display sharp resonances corresponding to mode groups, and the positions and heights of these resonances determine the mode properties. The grid resolution is set to 10×10×20 nm. Full transparent boundary conditions are used. The BeamPROP software package, version 5.1.1, is used [34].

In all four methods the material dispersion of Si is calculated by the Sellmeier equation [23, 24] and implemented into the propagations constants. The dielectric permittivity of the oxide layer is assumed to be constant at ε_{Si02} = 2.13. The input values used for the Si slab height and the waveguide are taken from SEM measurements. It is noted that all applied model calculations assume an infinitely thick oxide layer beneath the silicon waveguide and hence do not account for potential substrate leakage and shifts in mode cutoffs due to waveguide-substrate coupling.

The experimental results are now compared to the model calculations. The slope and the absolute values of the measured group index *n _{g,TE}* are in excellent agreement with the EME method and the PW calculations (Fig. 3). However, BPM yields lower group indices with a nearly constant, flat slope over the entire range from 1250–1650 nm whereas the FDTD method implies slightly higher group indices with an almost identical slope to the PW method. Similar agreement is found for TM polarized light. The constant group index of

*n*~ 5 between 1250 and 1350 nm, followed by the strong drop to

_{g,TE}*n*~ 4 is reproduced by EME and PW. BPM yields again results which are slightly below the experimental ones.

_{g,TM}Figure 4 compares the calculated GVD (EME method, 3 different waveguide widths) with the experimental GVD. Both show nearly constant values between 1475 and 1630 nm. However, depending on the waveguide width *D* ranges from ~ 3750, 2800 and 2000 ps/(nm∙km) at 1550nm for *w* = 400, 470 and 525 nm, respectively. The deviation between experiment with *D _{mean}* ~ 4400 ps/(nm∙km) and the corresponding calculation with

*w*= 525 nm is about a factor of ~ 2. This discrepancy can be explained by the steeper slope of

*n*compared to the models (inset of Fig. 3). The estimated experimental value of 0 ps/nm

_{g,TE}^{2}∙km for TOD agrees with model calculations as well. The results of EME method for

*w*= 525 nm are close and yield 2 ps/nm

^{2}∙km within the spectral range from 1475 to 1630 nm.

The presented calculations of the dispersion parameter D in Fig. 4 demonstrate the possibility to tune the dispersion via the geometry of the waveguide. Decreasing the waveguide width from 525 to 400 nm not only facilitates single mode propagation (Fig. 1) but also shifts the zero-dispersion point by ~ 300 nm. Simultaneously, the dispersion increases by less than a factor of 2. Depending on specific applications, a carefully designed geometry of the waveguide should enable achieving dispersion-free single mode propagation.

## 5. Conclusion

We have determined, to our knowledge for the first time, group index (*n _{g}*), group velocity dispersion (GVD) and third order dispersion (TOD) of 525×226nm cross-section SOI single-mode strip waveguides (photonic wires) over the entire telecommunication bandwidth. We find very high GVD at 1550 nm (~ 4400 ps/nm∙km), about three orders of magnitude larger than the GVD of standard single-mode optical fibers. More important, the GVD is primarily determined by the waveguides’ geometry. In addition, we measure the frequency response of the photonic wires for signal bandwidths up to 35 GHz. The results indicate that GVD-induced signal distortions in high speed data transmission of 100 Gb/s are negligible for waveguides with a length up to about 1 meter.

The experimentally measured propagation constants are used to benchmark the accuracy of model calculations originating from four different numerical methods. Depending on the used method (plane-wave, beam-propagation, eigenmode-expansion, FDTD) we find good agreement with the experiment within 2 % for *n _{g}* and a factor of 2 for GVD.

## Acknowledgments

The authors gratefully acknowledge the contributions of Dr. Sharee McNab and Dr. Lidija Sekaric (IBM T. J. Watson Research Center). The authors also want to thank Prof. Richard M. Osgood Jr., Dr. Nicolae C. Panoiu and Xiaogang Chen (Microelectronics Sciences Laboratories, Columbia University, New York) for useful discussions. This work was supported in part by the DARPA Slow Light project (J. Lowell, DSO), under grant N00014-04-C-0455.

## References and links

**1. **R. U. Ahmad, F. Pizzuto, G. S. Camarda, R. L. Espinola, H. Rao, and R. M. Osgood, “Ultracompact corner-mirrors and T-branches in silicon-on-insulator,” IEEE Photon. Technol. Lett. **14**, 65–67 (2002). [CrossRef]

**2. **V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature **431**, 1081–1084 (2004). [CrossRef] [PubMed]

**3. **V. R. Almeida, Q. F. Xu, and M. Lipson, “Ultrafast integrated semiconductor optical modulator based on the plasma-dispersion effect,” Opt. Lett. **30**, 2403–2405 (2005). [CrossRef] [PubMed]

**4. **T. Fukazawa, F. Ohno, and T. Baba, “Very compact arrayed-waveguide-grating demultiplexer using Si photonic wire waveguides,” Jpn. J. Appl. Phys. Part 2 **43**, L673–L675 (2004). [CrossRef]

**5. **L. C. Kimerling, “Silicon microphotonics,” Appl. Surf. Sci. **159**, 8–13 (2000). [CrossRef]

**6. **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]

**7. **C. Manolatou, S. G. Johnson, S. H. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “High-density integrated optics,” J. Lightwave Technol. **17**, 1682–1692 (1999). [CrossRef]

**8. **A. Sakai, T. Fukazawa, and T. Baba, “Low loss ultra-small branches in a silicon photonic wire waveguide,” IEICE Trans. Electron. **E85C**, 1033–1038 (2002).

**9. **K. Yamada, T. Tsuchizawa, T. Watanabe, J. I. Takahashi, E. Tamechika, M. Takahashi, S. Uchiyama, H. Fukuda, T. Shoji, S. I. Itabashi, and H. Morita, “Microphotonics devices based on silicon wire waveguiding system,” IEICE Trans. Electron. **E87C**, 351–358 (2004).

**10. **O. Boyraz, P. Koonath, V. Raghunathan, and B. Jalali, “All optical switching and continuum generation in silicon waveguides,” Opt. Express **12**, 4094–4102 (2004). [CrossRef] [PubMed]

**11. **R. L. Espinola, J. I. Dadap, R. M. Osgood, S. J. McNab, and Y. A. Vlasov, “C-band wavelength conversion in silicon photonic wire waveguides,” Opt. Express **13**, 4341–4349 (2005). [CrossRef] [PubMed]

**12. **H. Fukuda, K. Yamada, T. Shoji, M. Takahashi, T. Tsuchizawa, T. Watanabe, J. Takahashi, and S. Itabashi, “Four-wave mixing in silicon wire waveguides,” Opt. Express **13**, 4629–4637 (2005). [CrossRef] [PubMed]

**13. **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]

**14. **F. Ohno, T. Fukazawa, and T. Baba, “Mach-Zehnder interferometers composed of mu-bends and mu-branches in a Si photonic wire waveguide,” Jpn. J. Appl. Phys. Part 1 **44**, 5322–5323 (2005). [CrossRef]

**15. **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]

**16. **S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173–190 (2001). [CrossRef] [PubMed]

**17. **K. K. Lee, D. R. Lim, H. C. Luan, A. Agarwal, J. Foresi, and L. C. Kimerling, “Effect of size and roughness on light transmission in a Si/SiO2 waveguide: Experiments and model (vol 77, pg 1617, 2000),” Appl. Phys. Lett. **77**, 2258–2258 (2000). [CrossRef]

**18. **S. J. McNab, N. Moll, and Y. A. Vlasov, “Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides,” Opt. Express **11**, 2927–2939 (2003). [CrossRef] [PubMed]

**19. **Y. A. Vlasov, M. OBoyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature **438**, 65–69 (2005). [CrossRef] [PubMed]

**20. **F. Xia, L. Sekaric, and Y. A. Vlasov, “Mode conversion losses in silicon-on-insulator photonic wire based racetrack resonators,” Opt. Express, accepted, manuscript number 10415 (2006).

**21. **B. Tatian, “Fitting Refractive-Index Data With The Sellmeier Dispersion Formula,” Appl. Opt. **23**, 4477–4485 (1984). [CrossRef] [PubMed]

**22. **Y. Liang and C. P. Grover, “Modified white-light Mach-Zehnder interferometer for direct group-delay measurements,” Appl. Opt. **37**, 4105–4111 (1998). [CrossRef]

**23. **D. Marcuse, *Light Transmission Optics* (Van Nostrand Reinhold, New York, 1982).

**24. **X. G. Chen, N. C. Panoiu, and R. M. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE Journal Of Quantum Electronics **42**, 160–170 (2006). [CrossRef]

**25. **H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 mu m wavelength,” Appl. Phys. Lett. **80**, 416–418 (2002). [CrossRef]

**26. **G. P. Agrawal, *Fiber-Optic Communication Systems* (Wiley-Interscience, 2002). [CrossRef]

**27. **G. W. Rieger, K. S. Virk, and J. F. Young, “Nonlinear propagation of ultrafast 1.5 mu m pulses in high-index-contrast silicon-on-insulator waveguides,” Appl. Phys. Lett. **84**, 900–902 (2004). [CrossRef]

**28. **G. Johnson Steven, “MIT Photonic Bands (MPB), Manual 1.42, http://ab-initio.mit.edu/mpb/,” (2004).

**29. **A. Taflove and S. C. Hagness, *Computational Electrodynamics: The Finite-Difference Time-Domain Method* (2005).

**30. **
Rsoft Design Group, “FullWAVE 4.0 Manual, http://www.rsoftdesign.com” (2005).

**31. **D. F. G. Gallagher and T. P. Felici, “Eigenmode expansion methods for simulation of optical propagation in photonics: Pros and cons,” in *Integrated Optics: Devices, Materials and Technologies VII*,
Y. S. Sidorin and A. Tervonen; Eds. Proc. SPIE **4987**, 69–82 (2003). [CrossRef]

**32. **
Photon Design, “FimmWave Manual 4.3.4, http://www.photond.com” (2005).

**33. **M. D. Feit and J. A. Fleck, “Computation of mode properties in optical fiber wave-guides by a propagating beam method,” Appl. Opt. **19**, 1154–1164 (1980). [CrossRef] [PubMed]

**34. **
Rsoft Design Group, “BeamPROP 5.1.1 Manual, http://www.rsoftdesign.com,” (2005).