## Integrated Bragg gratings in spiral waveguides |

Optics Express, Vol. 21, Issue 7, pp. 8953-8963 (2013)

http://dx.doi.org/10.1364/OE.21.008953

Acrobat PDF (4038 KB)

### Abstract

Over the last two decades, many filters requiring custom spectral responses were obtained from photo-inscribed fiber Bragg gratings because of the flexibility inherent to this technology. However, Bragg gratings in silicon waveguides have the potential to provide faster and more efficient tuning capabilities when compared to optical fiber devices. One drawback is that Bragg gratings filters with elaborate spectral amplitude and phase responses often require a long interaction length, which is not compatible with current integration trends in CMOS compatible photonic circuits. In this paper, we propose to make Bragg gratings in spiral-shaped waveguides in order to increase their lengths while making them more compact. The approach preserves the flexibility of regular straight grating structures. More specifically, we demonstrate 2-mm long gratings wrapped in an area of 200 µm x 190 µm without any spectral degradation due to waveguide curvature. Furthermore, we interleave three spiral waveguides with integrated gratings thereby tripling the density and demonstrate good phase compensation for each of them. Finally, we show that this approach is compatible with phase-apodization of the grating coupling coefficient.

© 2013 OSA

## 1. Introduction

2. K. Ikeda, M. Nezhad, and Y. Fainman, “Wavelength selective coupler with vertical gratings on silicon chip,” Appl. Phys. Lett. **92**(20), 201111 (2008). [CrossRef]

3. S. Honda, Z. Wu, J. Matsui, K. Utaka, T. Edura, M. Tokuda, K. Tsutsui, and Y. Wada, “Largely-tunable wideband Bragg gratings fabricated on SOI rib waveguides employed by deep-RIE,” Electron. Lett. **43**(11), 630–631 (2007). [CrossRef]

4. W. Shi, X. Wang, C. Lin, H. Yun, Y. Liu, T. Baehr-Jones, M. Hochberg, N. A. F. Jaeger, and L. Chrostowski, “Electrically tunable resonant filters in phase-shifted contra-directional couplers,” in 2012 9th International Conference on Group IV Photonics (GFP)78–80 (2012). [CrossRef]

6. W. A. Zortman, D. C. Trotter, and M. R. Watts, “Silicon photonics manufacturing,” Opt. Express **18**(23), 23598–23607 (2010). [CrossRef] [PubMed]

7. W. Bogaerts, P. Dumon, D. V. Thourhout, D. Taillaert, P. Jaenen, J. Wouters, S. Beckx, V. Wiaux, and R. G. Baets, “Compact wavelength-selective functions in silicon-on-insulator photonic wires,” IEEE J. Sel. Top. Quantum Electron. **12**(6), 1394–1401 (2006). [CrossRef]

9. C. Koos, C. G. Poulton, L. Zimmermann, L. Jacome, J. Leuthold, and W. Freude, “Ideal bend contour trajectories for single-mode operation of low-loss overmoded waveguides,” IEEE Photon. Technol. Lett. **19**(11), 819–821 (2007). [CrossRef]

7. W. Bogaerts, P. Dumon, D. V. Thourhout, D. Taillaert, P. Jaenen, J. Wouters, S. Beckx, V. Wiaux, and R. G. Baets, “Compact wavelength-selective functions in silicon-on-insulator photonic wires,” IEEE J. Sel. Top. Quantum Electron. **12**(6), 1394–1401 (2006). [CrossRef]

9. C. Koos, C. G. Poulton, L. Zimmermann, L. Jacome, J. Leuthold, and W. Freude, “Ideal bend contour trajectories for single-mode operation of low-loss overmoded waveguides,” IEEE Photon. Technol. Lett. **19**(11), 819–821 (2007). [CrossRef]

10. M. Heiblum and J. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Sel. Top. Quantum Electron. **11**(2), 75–83 (1975). [CrossRef]

12. W. W. Lui, C.-L. Xu, T. Hirono, K. Yokoyama, and W.-P. Huang, “Full-vectorial wave propagation in semiconductor optical bending waveguides and equivalent straight waveguide approximations,” J. Lightwave Technol. **16**(5), 910–914 (1998). [CrossRef]

14. B. M. A. Rahman, D. M. H. Leung, S. S. A. Obayya, and K. T. V. Grattan, “Numerical analysis of bent waveguides: bending loss, transmission loss, mode coupling, and polarization coupling,” Appl. Opt. **47**(16), 2961–2970 (2008). [CrossRef] [PubMed]

*R*> ~2 μm [15

15. K. Kakihara, N. Kono, K. Saitoh, and M. Koshiba, “Full-vectorial finite element method in a cylindrical coordinate system for loss analysis of photonic wire bends,” Opt. Express **14**(23), 11128–11141 (2006). [CrossRef] [PubMed]

16. A. D. Simard, N. Belhadj, Y. Painchaud, and S. LaRochelle, “Apodized silicon-on-insulator Bragg gratings,” IEEE Photon. Technol. Lett. **24**(12), 1033–1035 (2012). [CrossRef]

## 2. Waveguide and grating design

17. S. Zamek, D. T. Tan, M. Khajavikhan, M. Ayache, M. P. Nezhad, and Y. Fainman, “Compact chip-scale filter based on curved waveguide Bragg gratings,” Opt. Lett. **35**(20), 3477–3479 (2010). [CrossRef] [PubMed]

18. A. D. Simard, N. Ayotte, Y. Painchaud, S. Bedard, and S. LaRochelle, “Impact of sidewall roughness on integrated Bragg gratings,” J. Lightwave Technol. **29**(24), 3693–3704 (2011). [CrossRef]

*w*) of 1200 nm since it does not have more than 3 transverse electric (TE) modes and its sensitivity to sidewall roughness is strongly decreased. Input/output light coupling is achieved with compact focusing grating couplers [19

19. F. Van Laere, T. Claes, J. Schrauwen, S. Scheerlinck, W. Bogaerts, D. Taillaert, L. O’Faolain, D. Van Thourhout, and R. Baets, “Compact focusing grating couplers for silicon-on-insulator integrated circuits,” IEEE Photon. Technol. Lett. **19**(23), 1919–1921 (2007). [CrossRef]

20. X. Wang, W. Shi, M. Hochberg, K. Adam, E. Schelew, J. F. Young, N. A. F. Jaeger, and L. Chrostowski, “Lithography simulation for the fabrication of silicon photonic devices with deep-ultraviolet lithography,” in 2012 IEEE 9th International Conference on Group IV Photonics (GFP), 288–290 (2012). [CrossRef]

21. T. Baehr-Jones, R. Ding, Y. Liu, A. Ayazi, T. Pinguet, N. C. Harris, M. Streshinsky, P. Lee, Y. Zhang, A. E.-J. Lim, T.-Y. Liow, S. H.-G. Teo, G.-Q. Lo, and M. Hochberg, “Ultralow drive voltage silicon traveling-wave modulator,” Opt. Express **20**(11), 12014–12020 (2012). [CrossRef] [PubMed]

### 2.1 Spiral waveguide definition

*x*and

*y*coordinates of the spiral are given by the real and imaginary parts of

*S*defined bywhere “sgn” is the sign function and

*ρ*is the angle of rotation that increases along the spiral waveguide (

*ρ*= 0 at the center). The term

*R*(

*ρ*)e

^{i}^{|}

^{ρ}^{|}is the spiral itself with a radius of curvature that is changing linearly with the angle of rotation. Without the Δ

*x*term in Eq. (1), the profile of the radius of curvature would be given by the black dotted line in Fig. 2(a) having a minimum value equal to

*R*at

_{0}*ρ*= 0. A simple spiral could be designed by using an s-shaped waveguide having a radius of curvature of

*R*

_{0}/2 to connect a spiral waveguide defined by

*R*(

*ρ*)e

*, with*

^{iρ}*ρ*> 0, to another one defined by

*R*(

*ρ*)e

*e*

^{iρ}*[22*

^{iπ}22. A. D. Simard, Y. Painchaud, and S. LaRochelle, “Integrated Bragg gratings in curved waveguides,” in the 23rd Annual Meeting of the Photonics Society Denver, USA, paper ThU3 (2010). [CrossRef]

*R*to

_{0}*R*

_{0}/2). The addition of the term Δ

*x*in Eq. (1) avoids this discontinuity by smoothly shifting the center of rotation between the two spirals. Figure 2(a) shows the numerically calculated profile of the radius of curvature,

*R(z),*as a function of the position on the spiral path from the input to the output of the waveguide (refer to by “

*z*-position”). When Δ

*x*becomes negligible, outside the red dots in Fig. 1(a) (located in this case at

*ρ = ±*5

*π*/4), the waveguide behaves as a simple spiral with a radius of curvature that increases linearly with

*ρ.*

*R*= 46.85 µm and Δ

_{0}*w*= 12.5 µm for each spirals, while the

*α*parameter, which controls the rate of change of the center of rotation as function of

*ρ*, is respectively 0.3355, 0.671 and 1.0065. On one hand, when

*α*is reduced, the minimum value of the radius of curvature of the spiral becomes smaller and on the other hand, when

*α*is increased, the radius of curvature increases significantly in the central portion of the spiral (around the location [

*z*-position around ± 0.2 mm in Fig. 2(a)) which decreases the waveguide spacing in this area. Consequently,

*α*must be chosen carefully.

### 2.2 Grating design

*θ*is a phase term that can be used to incorporate an arbitrary chirp function, Λ is the design grating period,

*n*(

*λ*) is the effective index of the straight waveguide, λ is the optical wavelength, Δ

*n*is the grating index modulation amplitude and δ

*n*(

*R*(

*z*)) is the effective index perturbation caused by the curvature, as shown in Fig. 2(b) for a waveguide of 1200 nm x 220 nm. The dependency of the waveguide effective index as a function of curvature was obtained using a finite elements mode solver simulator combined with the ESW approximation. The calculations take into account the specific profile of the radius of curvature calculated as a function of the

*z*-position for the given spiral. The phase term Ω(

*z*) in Eq. (4) is added to compensate for the effective index distortion

*δn*(

*R*(

*z*)) caused by the spiral and is given by

### 2.3 Mapping of the grating on the spiral

22. A. D. Simard, Y. Painchaud, and S. LaRochelle, “Integrated Bragg gratings in curved waveguides,” in the 23rd Annual Meeting of the Photonics Society Denver, USA, paper ThU3 (2010). [CrossRef]

## 3. Grating characterization

*λ*(

_{B}*z*)), which is equal to 2

*n*(

*λ*)Λ(z), and the index modulation amplitude profiles, Δ

*n*(

*z*), of the IBGs using an inverse scattering algorithm, namely the integral layer peeling algorithm proposed in [23

23. A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron. **39**(8), 1018–1026 (2003). [CrossRef]

*λ*and Δ

_{B}*n*profiles were retrieved with sufficient precision. More details on the post-processing procedure can be found in [24]. The designed grating spectral responses are shown in blue in Fig. 3(a), Fig. 4(a), Fig. 5(d) and Fig. 6(a). The grating spectra displayed in this paper are typical results. Over thirty gratings have been characterized with similar responses.

## 4. Experimental results and discussion

### 4.1 Phase correction effect

*R*

_{0}= 46.85 µm, Δ

*w*= 12.5 µm and

*α*= 0.671 which results in spirals having a minimal radius of curvature of 20 µm, a length of 2 mm and an area of 200 µm x 190 µm. The spirals have the same parameters as those shown in Fig. 1(a) with

*α*= 0.671.

16. A. D. Simard, N. Belhadj, Y. Painchaud, and S. LaRochelle, “Apodized silicon-on-insulator Bragg gratings,” IEEE Photon. Technol. Lett. **24**(12), 1033–1035 (2012). [CrossRef]

18. A. D. Simard, N. Ayotte, Y. Painchaud, S. Bedard, and S. LaRochelle, “Impact of sidewall roughness on integrated Bragg gratings,” J. Lightwave Technol. **29**(24), 3693–3704 (2011). [CrossRef]

### 4.2 Highly integrated gratings

*S*) and

*R*

_{0}= 59 µm, Δ

*w*= 15 µm,

*α*= 0.671 and

*N*= 1.345 which gives three spirals spaced by 5 µm, having a minimal radius of curvature of 20 µm for the side-spirals and 25 µm for the central spiral, a path length of 2 mm (each spiral has the same total length) and a total area of 230 µm x 215 µm. It can be noticed that the total footprint of these three IBGs is almost the same as the single spiral grating described in the previous section, thereby tripling the integration efficiency. The radius of curvature of the three gratings and their associated phase correction terms are shown in Fig. 5(b) and Fig. 5(c) respectively. As can be seen in Fig. 5(b), the central spiral has a symmetric radius of curvature profile. However, even if the side spirals experienced asymmetric profiles, their physical structure can be as easily compensated. Figure 5(d) shows that the interleaved spiral does not affect significantly the grating spectrum as well as the fact that each waveguides are much closer than in the previous section. As it was the case for single-spiral, the correction phase term compensates well the effective index perturbation caused by the curvature; at least, the phase error has amplitude in the center portion of the grating that is smaller than the random phase noise observed near its input and output ends.

### 4.3 Apodized gratings

16. A. D. Simard, N. Belhadj, Y. Painchaud, and S. LaRochelle, “Apodized silicon-on-insulator Bragg gratings,” IEEE Photon. Technol. Lett. **24**(12), 1033–1035 (2012). [CrossRef]

*z*-dependent amplitude (

*ϕ*(

*z*)). Consequently, the last term of Eq. (4) representing the grating can be written in the form

*results in amplitude apodized grating with a spectral response having satellite resonances out of the band of interest when Λ*

_{m}*is sufficiently small (in the present case Λ*

_{m}*= 17.5 μm). Figure 6(c) shows that the effective index modulation of the grating now follows the designed Gaussian profile with a 1 mm full-width half-maximum indicating that the correction applied to the grating does not affect the apodization profile. Unfortunately, IBGs in SOI usually experience a fair amount of phase noise, which prevents the apodization to properly reduce the sidelobe amplitude. To confirm this point, the green curve of Fig. 6(a) presents the spectral response obtained using the grating apodization profile shown in Fig. 6(c), but with an ideal Bragg wavelength profile (no phase noise). As a result, the sidelobes suppression ratio is decreased by more than 20 dB, which corresponds closely to the design (in blue). Consequently, the variation of the apodization profile compared to the ideal design is not a major source of spectral distortion and we thus conclude that spiral IBGs are compatible with phase-apodization.*

_{m}## 5. Conclusion

*w*, in order to improve the compactness of the spiral-gratings. This work shows that as the quality of SOI wafers improves, long IBGs with high quality spectral characteristics will be achievable. This design approach can be used for various grating types and strengths thereby giving increased flexibility for the layout of photonic circuits.

## Acknowledgments

## References and links

1. | Y. Painchaud, M. Poulin, C. Latrasse, N. Ayotte, M.-J. Picard, and M. Morin, “Bragg grating notch filters in silicon-on-insulator waveguides,” in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides paper BW2E.3, Optical Society of America (2012). |

2. | K. Ikeda, M. Nezhad, and Y. Fainman, “Wavelength selective coupler with vertical gratings on silicon chip,” Appl. Phys. Lett. |

3. | S. Honda, Z. Wu, J. Matsui, K. Utaka, T. Edura, M. Tokuda, K. Tsutsui, and Y. Wada, “Largely-tunable wideband Bragg gratings fabricated on SOI rib waveguides employed by deep-RIE,” Electron. Lett. |

4. | W. Shi, X. Wang, C. Lin, H. Yun, Y. Liu, T. Baehr-Jones, M. Hochberg, N. A. F. Jaeger, and L. Chrostowski, “Electrically tunable resonant filters in phase-shifted contra-directional couplers,” in 2012 9th International Conference on Group IV Photonics (GFP)78–80 (2012). [CrossRef] |

5. | W. A. Zortman, M. R. Watts, and D. C. Trotter, “Determination of wafer and process induced resonant frequency variation in silicon microdisk-resonators,” in Integrated Photonics and Nanophotonics Research and Applications (2009). |

6. | W. A. Zortman, D. C. Trotter, and M. R. Watts, “Silicon photonics manufacturing,” Opt. Express |

7. | W. Bogaerts, P. Dumon, D. V. Thourhout, D. Taillaert, P. Jaenen, J. Wouters, S. Beckx, V. Wiaux, and R. G. Baets, “Compact wavelength-selective functions in silicon-on-insulator photonic wires,” IEEE J. Sel. Top. Quantum Electron. |

8. | S. Spector, M. W. Geis, D. Lennon, R. C. Williamson, and T. M. Lyszczarz, “Hybrid multi-mode/single-mode waveguides for low loss,” in Integrated Photonics Research, Optical Society of America (2004). |

9. | C. Koos, C. G. Poulton, L. Zimmermann, L. Jacome, J. Leuthold, and W. Freude, “Ideal bend contour trajectories for single-mode operation of low-loss overmoded waveguides,” IEEE Photon. Technol. Lett. |

10. | M. Heiblum and J. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Sel. Top. Quantum Electron. |

11. | T. E. Murphy, “Integrated optical grating-based matched filters for fiber-optic communications,” Massachusetts Institute of Technology (1996). |

12. | W. W. Lui, C.-L. Xu, T. Hirono, K. Yokoyama, and W.-P. Huang, “Full-vectorial wave propagation in semiconductor optical bending waveguides and equivalent straight waveguide approximations,” J. Lightwave Technol. |

13. | S. S. A. Obayya, B. M. A. Rahman, and K. T. V. Grattan, “Full vectorial finite element modal solution of curved optical waveguides,” Laser Phys. Lett. |

14. | B. M. A. Rahman, D. M. H. Leung, S. S. A. Obayya, and K. T. V. Grattan, “Numerical analysis of bent waveguides: bending loss, transmission loss, mode coupling, and polarization coupling,” Appl. Opt. |

15. | K. Kakihara, N. Kono, K. Saitoh, and M. Koshiba, “Full-vectorial finite element method in a cylindrical coordinate system for loss analysis of photonic wire bends,” Opt. Express |

16. | A. D. Simard, N. Belhadj, Y. Painchaud, and S. LaRochelle, “Apodized silicon-on-insulator Bragg gratings,” IEEE Photon. Technol. Lett. |

17. | S. Zamek, D. T. Tan, M. Khajavikhan, M. Ayache, M. P. Nezhad, and Y. Fainman, “Compact chip-scale filter based on curved waveguide Bragg gratings,” Opt. Lett. |

18. | A. D. Simard, N. Ayotte, Y. Painchaud, S. Bedard, and S. LaRochelle, “Impact of sidewall roughness on integrated Bragg gratings,” J. Lightwave Technol. |

19. | F. Van Laere, T. Claes, J. Schrauwen, S. Scheerlinck, W. Bogaerts, D. Taillaert, L. O’Faolain, D. Van Thourhout, and R. Baets, “Compact focusing grating couplers for silicon-on-insulator integrated circuits,” IEEE Photon. Technol. Lett. |

20. | X. Wang, W. Shi, M. Hochberg, K. Adam, E. Schelew, J. F. Young, N. A. F. Jaeger, and L. Chrostowski, “Lithography simulation for the fabrication of silicon photonic devices with deep-ultraviolet lithography,” in 2012 IEEE 9th International Conference on Group IV Photonics (GFP), 288–290 (2012). [CrossRef] |

21. | T. Baehr-Jones, R. Ding, Y. Liu, A. Ayazi, T. Pinguet, N. C. Harris, M. Streshinsky, P. Lee, Y. Zhang, A. E.-J. Lim, T.-Y. Liow, S. H.-G. Teo, G.-Q. Lo, and M. Hochberg, “Ultralow drive voltage silicon traveling-wave modulator,” Opt. Express |

22. | A. D. Simard, Y. Painchaud, and S. LaRochelle, “Integrated Bragg gratings in curved waveguides,” in the 23rd Annual Meeting of the Photonics Society Denver, USA, paper ThU3 (2010). [CrossRef] |

23. | A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron. |

24. | A. D. Simard, Y. Painchaud, and S. LaRochelle, “Characterization of integrated Bragg grating profiles,” in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides paper BM3D.7 (2012). |

**OCIS Codes**

(050.2770) Diffraction and gratings : Gratings

(130.3120) Integrated optics : Integrated optics devices

(140.4780) Lasers and laser optics : Optical resonators

(130.7408) Integrated optics : Wavelength filtering devices

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: February 7, 2013

Revised Manuscript: March 15, 2013

Manuscript Accepted: March 28, 2013

Published: April 4, 2013

**Citation**

Alexandre D. Simard, Yves Painchaud, and Sophie LaRochelle, "Integrated Bragg gratings in spiral waveguides," Opt. Express **21**, 8953-8963 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-7-8953

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### References

- Y. Painchaud, M. Poulin, C. Latrasse, N. Ayotte, M.-J. Picard, and M. Morin, “Bragg grating notch filters in silicon-on-insulator waveguides,” in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides paper BW2E.3, Optical Society of America (2012).
- K. Ikeda, M. Nezhad, and Y. Fainman, “Wavelength selective coupler with vertical gratings on silicon chip,” Appl. Phys. Lett.92(20), 201111 (2008). [CrossRef]
- S. Honda, Z. Wu, J. Matsui, K. Utaka, T. Edura, M. Tokuda, K. Tsutsui, and Y. Wada, “Largely-tunable wideband Bragg gratings fabricated on SOI rib waveguides employed by deep-RIE,” Electron. Lett.43(11), 630–631 (2007). [CrossRef]
- W. Shi, X. Wang, C. Lin, H. Yun, Y. Liu, T. Baehr-Jones, M. Hochberg, N. A. F. Jaeger, and L. Chrostowski, “Electrically tunable resonant filters in phase-shifted contra-directional couplers,” in 2012 9th International Conference on Group IV Photonics (GFP)78–80 (2012). [CrossRef]
- W. A. Zortman, M. R. Watts, and D. C. Trotter, “Determination of wafer and process induced resonant frequency variation in silicon microdisk-resonators,” in Integrated Photonics and Nanophotonics Research and Applications (2009).
- W. A. Zortman, D. C. Trotter, and M. R. Watts, “Silicon photonics manufacturing,” Opt. Express18(23), 23598–23607 (2010). [CrossRef] [PubMed]
- W. Bogaerts, P. Dumon, D. V. Thourhout, D. Taillaert, P. Jaenen, J. Wouters, S. Beckx, V. Wiaux, and R. G. Baets, “Compact wavelength-selective functions in silicon-on-insulator photonic wires,” IEEE J. Sel. Top. Quantum Electron.12(6), 1394–1401 (2006). [CrossRef]
- S. Spector, M. W. Geis, D. Lennon, R. C. Williamson, and T. M. Lyszczarz, “Hybrid multi-mode/single-mode waveguides for low loss,” in Integrated Photonics Research, Optical Society of America (2004).
- C. Koos, C. G. Poulton, L. Zimmermann, L. Jacome, J. Leuthold, and W. Freude, “Ideal bend contour trajectories for single-mode operation of low-loss overmoded waveguides,” IEEE Photon. Technol. Lett.19(11), 819–821 (2007). [CrossRef]
- M. Heiblum and J. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Sel. Top. Quantum Electron.11(2), 75–83 (1975). [CrossRef]
- T. E. Murphy, “Integrated optical grating-based matched filters for fiber-optic communications,” Massachusetts Institute of Technology (1996).
- W. W. Lui, C.-L. Xu, T. Hirono, K. Yokoyama, and W.-P. Huang, “Full-vectorial wave propagation in semiconductor optical bending waveguides and equivalent straight waveguide approximations,” J. Lightwave Technol.16(5), 910–914 (1998). [CrossRef]
- S. S. A. Obayya, B. M. A. Rahman, and K. T. V. Grattan, “Full vectorial finite element modal solution of curved optical waveguides,” Laser Phys. Lett.2(3), 131–136 (2005). [CrossRef]
- B. M. A. Rahman, D. M. H. Leung, S. S. A. Obayya, and K. T. V. Grattan, “Numerical analysis of bent waveguides: bending loss, transmission loss, mode coupling, and polarization coupling,” Appl. Opt.47(16), 2961–2970 (2008). [CrossRef] [PubMed]
- K. Kakihara, N. Kono, K. Saitoh, and M. Koshiba, “Full-vectorial finite element method in a cylindrical coordinate system for loss analysis of photonic wire bends,” Opt. Express14(23), 11128–11141 (2006). [CrossRef] [PubMed]
- A. D. Simard, N. Belhadj, Y. Painchaud, and S. LaRochelle, “Apodized silicon-on-insulator Bragg gratings,” IEEE Photon. Technol. Lett.24(12), 1033–1035 (2012). [CrossRef]
- S. Zamek, D. T. Tan, M. Khajavikhan, M. Ayache, M. P. Nezhad, and Y. Fainman, “Compact chip-scale filter based on curved waveguide Bragg gratings,” Opt. Lett.35(20), 3477–3479 (2010). [CrossRef] [PubMed]
- A. D. Simard, N. Ayotte, Y. Painchaud, S. Bedard, and S. LaRochelle, “Impact of sidewall roughness on integrated Bragg gratings,” J. Lightwave Technol.29(24), 3693–3704 (2011). [CrossRef]
- F. Van Laere, T. Claes, J. Schrauwen, S. Scheerlinck, W. Bogaerts, D. Taillaert, L. O’Faolain, D. Van Thourhout, and R. Baets, “Compact focusing grating couplers for silicon-on-insulator integrated circuits,” IEEE Photon. Technol. Lett.19(23), 1919–1921 (2007). [CrossRef]
- X. Wang, W. Shi, M. Hochberg, K. Adam, E. Schelew, J. F. Young, N. A. F. Jaeger, and L. Chrostowski, “Lithography simulation for the fabrication of silicon photonic devices with deep-ultraviolet lithography,” in 2012 IEEE 9th International Conference on Group IV Photonics (GFP), 288–290 (2012). [CrossRef]
- T. Baehr-Jones, R. Ding, Y. Liu, A. Ayazi, T. Pinguet, N. C. Harris, M. Streshinsky, P. Lee, Y. Zhang, A. E.-J. Lim, T.-Y. Liow, S. H.-G. Teo, G.-Q. Lo, and M. Hochberg, “Ultralow drive voltage silicon traveling-wave modulator,” Opt. Express20(11), 12014–12020 (2012). [CrossRef] [PubMed]
- A. D. Simard, Y. Painchaud, and S. LaRochelle, “Integrated Bragg gratings in curved waveguides,” in the 23rd Annual Meeting of the Photonics Society Denver, USA, paper ThU3 (2010). [CrossRef]
- A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron.39(8), 1018–1026 (2003). [CrossRef]
- A. D. Simard, Y. Painchaud, and S. LaRochelle, “Characterization of integrated Bragg grating profiles,” in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides paper (2012).

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