Abstract

We demonstrate sidewall gratings in an ultra-low-loss Si3N4 planar waveguide platform. Through proper geometrical design we can achieve coupling constant values between 13 and 310 cm−1. The TE waveguide propagation loss over the range of 1540 to 1570 nm is below 5.5 dB/m.

©2013 Optical Society of America

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References

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  • Publication
  1. L. A. Coldren, “Monolithic tunable diode lasers,” IEEE J. Sel. Top. Quantum Electron. 12(12), 988–999 (2000).
    [Crossref]
  2. A. W. Fang, M. N. Sysak, B. R. Koch, R. Jones, E. Lively, Ying-Hao Kuo, O. Di Liang, Raday, and J. E. Bowers, “Single-wavelength silicon evanescent lasers,” IEEE J. Sel. Top. Quantum Electron. 15(3), 535–544 (2009).
    [Crossref]
  3. D. Pastor, J. Capmany, D. Ortega, V. Tatay, and J. Marti, “Design of apodized linearly chirped fiber gratings for dispersion compensation,” J. Lightwave Technol. 14(11), 2581–2588 (1996).
    [Crossref]
  4. A. Carballar, M. A. Muriel, and J. Azaña, “Fiber grating filter for WDM systems: an improved design,” IEEE Photon. Technol. Lett. 11(6), 694–696 (1999).
    [Crossref]
  5. K. N. Nguyen, P. J. Skahan, J. M. Garcia, E. Lively, H. N. Poulsen, D. M. Baney, and D. J. Blumenthal, “Monolithically integrated dual-quadrature receiver on InP with 30 nm tunable local oscillator,” Opt. Express 19(26), B716–B721 (2011).
    [Crossref] [PubMed]
  6. V. V. Wong, W. Y. Choi, J. M. Carter, C. G. Fonstad, H. I. Smith, Y. Chung, and N. Dagli, “Ridge-waveguide sidewall-grating distributed feedback structures fabricated by x-ray lithography,” J. Vac. Sci. Technol. B 11(6), 2621–2624 (1993).
    [Crossref]
  7. J. T. Hastings, M. H. Lim, J. G. Goodberlet, and H. I. Smith, “Optical waveguides with apodized sidewall gratings via spatial-phase-locked electron-beam lithography,” J. Vac. Sci. Technol. B 20(6), 2753–2757 (2002).
    [Crossref]
  8. E. H. Bernhardi, Q. Lu, H. A. G. M. van Wolferen, K. Wörhoff, R. M. de Ridder, and M. Pollnau, “Monolithic distributed Bragg reflector cavities in Al2O3 with quality factors exceeding 106,” Photonics and Nanostructures – Fundamentals and Applications 9(3), 225–234 (2011).
    [Crossref]
  9. C. Lin, J. R. Adleman, E. W. Jacobs, J. S. Rodgers, R. Liang, T. Chen, and A. Fitting, “Higher order planar-waveguide Bragg grating on curved waveguide,” in Proceedings of the IEEE Photonics Conference, (Arlington, VA, 2011).
  10. T. E. Murphy, J. T. Hastings, and H. I. Smith, “Fabrication and characterization of narrow-band Bragg-refection filters in silicon-on-insulator ridge waveguides,” J. Lightwave Technol. 19(12), 1938–1942 (2001).
    [Crossref]
  11. J. F. Bauters, M. J. R. Heck, D. D. John, J. S. Barton, C. M. Bruinink, A. Leinse, R. G. Heideman, D. J. Blumenthal, and J. E. Bowers, “Planar waveguides with less than 0.1 dB/m propagation loss fabricated with wafer bonding,” Opt. Express 19(24), 24090–24101 (2011).
    [Crossref] [PubMed]
  12. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communication (Oxford, 2007), Chap. 12.
  13. L. A. Coldren, S. W. Corzine, and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits (Wiley, 2012), Chap. 6.
  14. K. O. Hill and G. Meltz, “Fiber Bragg grating technology fundamentals and overview,” J. Lightwave Technol. 15(8), 1263–1276 (1997).
    [Crossref]

2011 (3)

2009 (1)

A. W. Fang, M. N. Sysak, B. R. Koch, R. Jones, E. Lively, Ying-Hao Kuo, O. Di Liang, Raday, and J. E. Bowers, “Single-wavelength silicon evanescent lasers,” IEEE J. Sel. Top. Quantum Electron. 15(3), 535–544 (2009).
[Crossref]

2002 (1)

J. T. Hastings, M. H. Lim, J. G. Goodberlet, and H. I. Smith, “Optical waveguides with apodized sidewall gratings via spatial-phase-locked electron-beam lithography,” J. Vac. Sci. Technol. B 20(6), 2753–2757 (2002).
[Crossref]

2001 (1)

2000 (1)

L. A. Coldren, “Monolithic tunable diode lasers,” IEEE J. Sel. Top. Quantum Electron. 12(12), 988–999 (2000).
[Crossref]

1999 (1)

A. Carballar, M. A. Muriel, and J. Azaña, “Fiber grating filter for WDM systems: an improved design,” IEEE Photon. Technol. Lett. 11(6), 694–696 (1999).
[Crossref]

1997 (1)

K. O. Hill and G. Meltz, “Fiber Bragg grating technology fundamentals and overview,” J. Lightwave Technol. 15(8), 1263–1276 (1997).
[Crossref]

1996 (1)

D. Pastor, J. Capmany, D. Ortega, V. Tatay, and J. Marti, “Design of apodized linearly chirped fiber gratings for dispersion compensation,” J. Lightwave Technol. 14(11), 2581–2588 (1996).
[Crossref]

1993 (1)

V. V. Wong, W. Y. Choi, J. M. Carter, C. G. Fonstad, H. I. Smith, Y. Chung, and N. Dagli, “Ridge-waveguide sidewall-grating distributed feedback structures fabricated by x-ray lithography,” J. Vac. Sci. Technol. B 11(6), 2621–2624 (1993).
[Crossref]

Azaña, J.

A. Carballar, M. A. Muriel, and J. Azaña, “Fiber grating filter for WDM systems: an improved design,” IEEE Photon. Technol. Lett. 11(6), 694–696 (1999).
[Crossref]

Baney, D. M.

Barton, J. S.

Bauters, J. F.

Bernhardi, E. H.

E. H. Bernhardi, Q. Lu, H. A. G. M. van Wolferen, K. Wörhoff, R. M. de Ridder, and M. Pollnau, “Monolithic distributed Bragg reflector cavities in Al2O3 with quality factors exceeding 106,” Photonics and Nanostructures – Fundamentals and Applications 9(3), 225–234 (2011).
[Crossref]

Blumenthal, D. J.

Bowers, J. E.

J. F. Bauters, M. J. R. Heck, D. D. John, J. S. Barton, C. M. Bruinink, A. Leinse, R. G. Heideman, D. J. Blumenthal, and J. E. Bowers, “Planar waveguides with less than 0.1 dB/m propagation loss fabricated with wafer bonding,” Opt. Express 19(24), 24090–24101 (2011).
[Crossref] [PubMed]

A. W. Fang, M. N. Sysak, B. R. Koch, R. Jones, E. Lively, Ying-Hao Kuo, O. Di Liang, Raday, and J. E. Bowers, “Single-wavelength silicon evanescent lasers,” IEEE J. Sel. Top. Quantum Electron. 15(3), 535–544 (2009).
[Crossref]

Bruinink, C. M.

Capmany, J.

D. Pastor, J. Capmany, D. Ortega, V. Tatay, and J. Marti, “Design of apodized linearly chirped fiber gratings for dispersion compensation,” J. Lightwave Technol. 14(11), 2581–2588 (1996).
[Crossref]

Carballar, A.

A. Carballar, M. A. Muriel, and J. Azaña, “Fiber grating filter for WDM systems: an improved design,” IEEE Photon. Technol. Lett. 11(6), 694–696 (1999).
[Crossref]

Carter, J. M.

V. V. Wong, W. Y. Choi, J. M. Carter, C. G. Fonstad, H. I. Smith, Y. Chung, and N. Dagli, “Ridge-waveguide sidewall-grating distributed feedback structures fabricated by x-ray lithography,” J. Vac. Sci. Technol. B 11(6), 2621–2624 (1993).
[Crossref]

Choi, W. Y.

V. V. Wong, W. Y. Choi, J. M. Carter, C. G. Fonstad, H. I. Smith, Y. Chung, and N. Dagli, “Ridge-waveguide sidewall-grating distributed feedback structures fabricated by x-ray lithography,” J. Vac. Sci. Technol. B 11(6), 2621–2624 (1993).
[Crossref]

Chung, Y.

V. V. Wong, W. Y. Choi, J. M. Carter, C. G. Fonstad, H. I. Smith, Y. Chung, and N. Dagli, “Ridge-waveguide sidewall-grating distributed feedback structures fabricated by x-ray lithography,” J. Vac. Sci. Technol. B 11(6), 2621–2624 (1993).
[Crossref]

Coldren, L. A.

L. A. Coldren, “Monolithic tunable diode lasers,” IEEE J. Sel. Top. Quantum Electron. 12(12), 988–999 (2000).
[Crossref]

Dagli, N.

V. V. Wong, W. Y. Choi, J. M. Carter, C. G. Fonstad, H. I. Smith, Y. Chung, and N. Dagli, “Ridge-waveguide sidewall-grating distributed feedback structures fabricated by x-ray lithography,” J. Vac. Sci. Technol. B 11(6), 2621–2624 (1993).
[Crossref]

de Ridder, R. M.

E. H. Bernhardi, Q. Lu, H. A. G. M. van Wolferen, K. Wörhoff, R. M. de Ridder, and M. Pollnau, “Monolithic distributed Bragg reflector cavities in Al2O3 with quality factors exceeding 106,” Photonics and Nanostructures – Fundamentals and Applications 9(3), 225–234 (2011).
[Crossref]

Di Liang, O.

A. W. Fang, M. N. Sysak, B. R. Koch, R. Jones, E. Lively, Ying-Hao Kuo, O. Di Liang, Raday, and J. E. Bowers, “Single-wavelength silicon evanescent lasers,” IEEE J. Sel. Top. Quantum Electron. 15(3), 535–544 (2009).
[Crossref]

Fang, A. W.

A. W. Fang, M. N. Sysak, B. R. Koch, R. Jones, E. Lively, Ying-Hao Kuo, O. Di Liang, Raday, and J. E. Bowers, “Single-wavelength silicon evanescent lasers,” IEEE J. Sel. Top. Quantum Electron. 15(3), 535–544 (2009).
[Crossref]

Fonstad, C. G.

V. V. Wong, W. Y. Choi, J. M. Carter, C. G. Fonstad, H. I. Smith, Y. Chung, and N. Dagli, “Ridge-waveguide sidewall-grating distributed feedback structures fabricated by x-ray lithography,” J. Vac. Sci. Technol. B 11(6), 2621–2624 (1993).
[Crossref]

Garcia, J. M.

Goodberlet, J. G.

J. T. Hastings, M. H. Lim, J. G. Goodberlet, and H. I. Smith, “Optical waveguides with apodized sidewall gratings via spatial-phase-locked electron-beam lithography,” J. Vac. Sci. Technol. B 20(6), 2753–2757 (2002).
[Crossref]

Hastings, J. T.

J. T. Hastings, M. H. Lim, J. G. Goodberlet, and H. I. Smith, “Optical waveguides with apodized sidewall gratings via spatial-phase-locked electron-beam lithography,” J. Vac. Sci. Technol. B 20(6), 2753–2757 (2002).
[Crossref]

T. E. Murphy, J. T. Hastings, and H. I. Smith, “Fabrication and characterization of narrow-band Bragg-refection filters in silicon-on-insulator ridge waveguides,” J. Lightwave Technol. 19(12), 1938–1942 (2001).
[Crossref]

Heck, M. J. R.

Heideman, R. G.

Hill, K. O.

K. O. Hill and G. Meltz, “Fiber Bragg grating technology fundamentals and overview,” J. Lightwave Technol. 15(8), 1263–1276 (1997).
[Crossref]

John, D. D.

Jones, R.

A. W. Fang, M. N. Sysak, B. R. Koch, R. Jones, E. Lively, Ying-Hao Kuo, O. Di Liang, Raday, and J. E. Bowers, “Single-wavelength silicon evanescent lasers,” IEEE J. Sel. Top. Quantum Electron. 15(3), 535–544 (2009).
[Crossref]

Koch, B. R.

A. W. Fang, M. N. Sysak, B. R. Koch, R. Jones, E. Lively, Ying-Hao Kuo, O. Di Liang, Raday, and J. E. Bowers, “Single-wavelength silicon evanescent lasers,” IEEE J. Sel. Top. Quantum Electron. 15(3), 535–544 (2009).
[Crossref]

Leinse, A.

Lim, M. H.

J. T. Hastings, M. H. Lim, J. G. Goodberlet, and H. I. Smith, “Optical waveguides with apodized sidewall gratings via spatial-phase-locked electron-beam lithography,” J. Vac. Sci. Technol. B 20(6), 2753–2757 (2002).
[Crossref]

Lively, E.

K. N. Nguyen, P. J. Skahan, J. M. Garcia, E. Lively, H. N. Poulsen, D. M. Baney, and D. J. Blumenthal, “Monolithically integrated dual-quadrature receiver on InP with 30 nm tunable local oscillator,” Opt. Express 19(26), B716–B721 (2011).
[Crossref] [PubMed]

A. W. Fang, M. N. Sysak, B. R. Koch, R. Jones, E. Lively, Ying-Hao Kuo, O. Di Liang, Raday, and J. E. Bowers, “Single-wavelength silicon evanescent lasers,” IEEE J. Sel. Top. Quantum Electron. 15(3), 535–544 (2009).
[Crossref]

Lu, Q.

E. H. Bernhardi, Q. Lu, H. A. G. M. van Wolferen, K. Wörhoff, R. M. de Ridder, and M. Pollnau, “Monolithic distributed Bragg reflector cavities in Al2O3 with quality factors exceeding 106,” Photonics and Nanostructures – Fundamentals and Applications 9(3), 225–234 (2011).
[Crossref]

Marti, J.

D. Pastor, J. Capmany, D. Ortega, V. Tatay, and J. Marti, “Design of apodized linearly chirped fiber gratings for dispersion compensation,” J. Lightwave Technol. 14(11), 2581–2588 (1996).
[Crossref]

Meltz, G.

K. O. Hill and G. Meltz, “Fiber Bragg grating technology fundamentals and overview,” J. Lightwave Technol. 15(8), 1263–1276 (1997).
[Crossref]

Muriel, M. A.

A. Carballar, M. A. Muriel, and J. Azaña, “Fiber grating filter for WDM systems: an improved design,” IEEE Photon. Technol. Lett. 11(6), 694–696 (1999).
[Crossref]

Murphy, T. E.

Nguyen, K. N.

Ortega, D.

D. Pastor, J. Capmany, D. Ortega, V. Tatay, and J. Marti, “Design of apodized linearly chirped fiber gratings for dispersion compensation,” J. Lightwave Technol. 14(11), 2581–2588 (1996).
[Crossref]

Pastor, D.

D. Pastor, J. Capmany, D. Ortega, V. Tatay, and J. Marti, “Design of apodized linearly chirped fiber gratings for dispersion compensation,” J. Lightwave Technol. 14(11), 2581–2588 (1996).
[Crossref]

Pollnau, M.

E. H. Bernhardi, Q. Lu, H. A. G. M. van Wolferen, K. Wörhoff, R. M. de Ridder, and M. Pollnau, “Monolithic distributed Bragg reflector cavities in Al2O3 with quality factors exceeding 106,” Photonics and Nanostructures – Fundamentals and Applications 9(3), 225–234 (2011).
[Crossref]

Poulsen, H. N.

Raday,

A. W. Fang, M. N. Sysak, B. R. Koch, R. Jones, E. Lively, Ying-Hao Kuo, O. Di Liang, Raday, and J. E. Bowers, “Single-wavelength silicon evanescent lasers,” IEEE J. Sel. Top. Quantum Electron. 15(3), 535–544 (2009).
[Crossref]

Skahan, P. J.

Smith, H. I.

J. T. Hastings, M. H. Lim, J. G. Goodberlet, and H. I. Smith, “Optical waveguides with apodized sidewall gratings via spatial-phase-locked electron-beam lithography,” J. Vac. Sci. Technol. B 20(6), 2753–2757 (2002).
[Crossref]

T. E. Murphy, J. T. Hastings, and H. I. Smith, “Fabrication and characterization of narrow-band Bragg-refection filters in silicon-on-insulator ridge waveguides,” J. Lightwave Technol. 19(12), 1938–1942 (2001).
[Crossref]

V. V. Wong, W. Y. Choi, J. M. Carter, C. G. Fonstad, H. I. Smith, Y. Chung, and N. Dagli, “Ridge-waveguide sidewall-grating distributed feedback structures fabricated by x-ray lithography,” J. Vac. Sci. Technol. B 11(6), 2621–2624 (1993).
[Crossref]

Sysak, M. N.

A. W. Fang, M. N. Sysak, B. R. Koch, R. Jones, E. Lively, Ying-Hao Kuo, O. Di Liang, Raday, and J. E. Bowers, “Single-wavelength silicon evanescent lasers,” IEEE J. Sel. Top. Quantum Electron. 15(3), 535–544 (2009).
[Crossref]

Tatay, V.

D. Pastor, J. Capmany, D. Ortega, V. Tatay, and J. Marti, “Design of apodized linearly chirped fiber gratings for dispersion compensation,” J. Lightwave Technol. 14(11), 2581–2588 (1996).
[Crossref]

van Wolferen, H. A. G. M.

E. H. Bernhardi, Q. Lu, H. A. G. M. van Wolferen, K. Wörhoff, R. M. de Ridder, and M. Pollnau, “Monolithic distributed Bragg reflector cavities in Al2O3 with quality factors exceeding 106,” Photonics and Nanostructures – Fundamentals and Applications 9(3), 225–234 (2011).
[Crossref]

Wong, V. V.

V. V. Wong, W. Y. Choi, J. M. Carter, C. G. Fonstad, H. I. Smith, Y. Chung, and N. Dagli, “Ridge-waveguide sidewall-grating distributed feedback structures fabricated by x-ray lithography,” J. Vac. Sci. Technol. B 11(6), 2621–2624 (1993).
[Crossref]

Wörhoff, K.

E. H. Bernhardi, Q. Lu, H. A. G. M. van Wolferen, K. Wörhoff, R. M. de Ridder, and M. Pollnau, “Monolithic distributed Bragg reflector cavities in Al2O3 with quality factors exceeding 106,” Photonics and Nanostructures – Fundamentals and Applications 9(3), 225–234 (2011).
[Crossref]

Ying-Hao Kuo,

A. W. Fang, M. N. Sysak, B. R. Koch, R. Jones, E. Lively, Ying-Hao Kuo, O. Di Liang, Raday, and J. E. Bowers, “Single-wavelength silicon evanescent lasers,” IEEE J. Sel. Top. Quantum Electron. 15(3), 535–544 (2009).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (2)

L. A. Coldren, “Monolithic tunable diode lasers,” IEEE J. Sel. Top. Quantum Electron. 12(12), 988–999 (2000).
[Crossref]

A. W. Fang, M. N. Sysak, B. R. Koch, R. Jones, E. Lively, Ying-Hao Kuo, O. Di Liang, Raday, and J. E. Bowers, “Single-wavelength silicon evanescent lasers,” IEEE J. Sel. Top. Quantum Electron. 15(3), 535–544 (2009).
[Crossref]

IEEE Photon. Technol. Lett. (1)

A. Carballar, M. A. Muriel, and J. Azaña, “Fiber grating filter for WDM systems: an improved design,” IEEE Photon. Technol. Lett. 11(6), 694–696 (1999).
[Crossref]

J. Lightwave Technol. (3)

K. O. Hill and G. Meltz, “Fiber Bragg grating technology fundamentals and overview,” J. Lightwave Technol. 15(8), 1263–1276 (1997).
[Crossref]

T. E. Murphy, J. T. Hastings, and H. I. Smith, “Fabrication and characterization of narrow-band Bragg-refection filters in silicon-on-insulator ridge waveguides,” J. Lightwave Technol. 19(12), 1938–1942 (2001).
[Crossref]

D. Pastor, J. Capmany, D. Ortega, V. Tatay, and J. Marti, “Design of apodized linearly chirped fiber gratings for dispersion compensation,” J. Lightwave Technol. 14(11), 2581–2588 (1996).
[Crossref]

J. Vac. Sci. Technol. B (2)

V. V. Wong, W. Y. Choi, J. M. Carter, C. G. Fonstad, H. I. Smith, Y. Chung, and N. Dagli, “Ridge-waveguide sidewall-grating distributed feedback structures fabricated by x-ray lithography,” J. Vac. Sci. Technol. B 11(6), 2621–2624 (1993).
[Crossref]

J. T. Hastings, M. H. Lim, J. G. Goodberlet, and H. I. Smith, “Optical waveguides with apodized sidewall gratings via spatial-phase-locked electron-beam lithography,” J. Vac. Sci. Technol. B 20(6), 2753–2757 (2002).
[Crossref]

Opt. Express (2)

Photonics and Nanostructures – Fundamentals and Applications (1)

E. H. Bernhardi, Q. Lu, H. A. G. M. van Wolferen, K. Wörhoff, R. M. de Ridder, and M. Pollnau, “Monolithic distributed Bragg reflector cavities in Al2O3 with quality factors exceeding 106,” Photonics and Nanostructures – Fundamentals and Applications 9(3), 225–234 (2011).
[Crossref]

Other (3)

C. Lin, J. R. Adleman, E. W. Jacobs, J. S. Rodgers, R. Liang, T. Chen, and A. Fitting, “Higher order planar-waveguide Bragg grating on curved waveguide,” in Proceedings of the IEEE Photonics Conference, (Arlington, VA, 2011).

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communication (Oxford, 2007), Chap. 12.

L. A. Coldren, S. W. Corzine, and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits (Wiley, 2012), Chap. 6.

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Figures (7)

Fig. 1
Fig. 1 (a) Schematic image of the waveguide cross section. (b) Top down SEM image of fabricated Si3N4 sidewall grating. Here the wide sections measure 4.6 μm in width and the narrow sections measure 1.0 μm in width. This gives a width difference of 3.6 μm. The difference of the two waveguide widths labeled by the red arrows illustrates this width difference. The period of the structure is 520 nm.

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Fig. 2
Fig. 2 Experimental setup. The tunable laser (Agilent 81680A) and power sensor (Agilent 81633A) are both housed in the same Agilent 8164A Lightwave Measurement System. The polarization controller (PC) is a 3-paddle manual polarization controller from FiberControl. The polarization splitter (PS) is a Thorlabs model CM1-PBS254 cube-mounted polarizing beamsplitter. All fibers connections are made using Corning SMF-28 fibers. The lens at the output of the grating under test before the PS is a standard 20x microscope objective.

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Fig. 3
Fig. 3 Measured reflectivity spectra of four 200 μm long gratings under TE excitation. The fine oscillations are due to the 3 mm Fabry-Perot cavity created by the diced input facet and the gratings found in the center of the 7 mm chip.

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Fig. 4
Fig. 4 Measured and theoretically predicted spectra of a 1000 μm long grating under TE excitation. Here the measured reflectivity spectrum is normalized to a maximum value of 0 dB.

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Fig. 5
Fig. 5 Measured and simulated grating bandwidth for 1000 μm long gratings under TE excitation. The nominal waveguide width is 2.8 μm. The plot also gives fitted and simulated coupling constant values for the same set of gratings.

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Fig. 6
Fig. 6 Measured and predicted Bragg wavelength for different duty cycles of the sidewall geometry.

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Fig. 7
Fig. 7 (a) Cross-section, (b) top-down, and (c) on-axis view of the waveguide geometry used when performing the 2-D finite difference method simulations. The fraction of the power of the mode found within the areas defined by the green boxes (Γ) was used in the calculation of the grating coupling constant (κ) for each grating. Within the simulations it was assumed that the LPCVD SiO2 and the thermal SiO2 were of equivalent index value (nclad).

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Equations (6)

Equations on this page are rendered with MathJax. Learn more.

δ=β− β o = ω− ω o ω o ⋅ π Λ ⋅ n ¯ ( λ ) n ¯ Bragg
Λ= λ o 2 n ¯ .
σ ˜ 2 = κ ˜ 2 − δ ˜ 2
r ˜ g =−j κ ˜ tanh σ ˜ L g σ ˜ +j δ ˜ tanh σ ˜ L g
n grating = n core 2 ⋅( DC 2 )+ n clad 2 ⋅( 1−DC 2 ) ,
κ= ( n grating n ¯ Bragg ) 2 2 λ sin(π⋅DC)⋅Γ⋅( n core − n clad )

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