Abstract

We propose a design method for silicon ring resonators (RRs) with a free spectral range (FSR) insensitive to fabrication variations. Two waveguide-core widths are used in the RR, with opposite signs of the group-index derivative with respect to the width. This results in cancellation of the width-dependent FSR changes. The systematic deviation of the realized width from the design width is determined and is used for calibrating the calculated relation of group index versus width. This enables a more accurate FSR value and well-aimed robust performance. We present two robust design examples. Experimental results match well with the predictions. For the deliberately introduced ±10 nm core-width change, the FSR variation of the robust designs is only about 30% of the value measured from the RR with a single core width. This design method can be used to improve the performance of photonic integrated circuits using multiple RRs. As the FSR of a RR is not easily tunable, the robust design is beneficial to applications where an accurate FSR is required, such as in microwave photonics.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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References

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  1. W. Bogaerts, Y. Xing, and U. Khan, “Layout-aware variability analysis, yield prediction, and optimization in photonic integrated circuits,” IEEE J. Sel. Top. Quantum Electron. 25(5), 1–13 (2019).
    [Crossref]
  2. Z. Lu, J. Jhoja, J. Klein, X. Wang, A. Liu, J. Flueckiger, J. Pond, and L. Chrostowski, “Performance prediction for silicon photonics integrated circuits with layout-dependent correlated manufacturing variability,” Opt. Express 25(9), 9712–9733 (2017).
    [Crossref]
  3. Y. Xing, J. Dong, S. Dwivedi, U. Khan, and W. Bogaerts, “Accurate extraction of fabricated geometry using optical measurement,” Photonics Res. 6(11), 1008–1020 (2018).
    [Crossref]
  4. Y. Xing, M. Wang, A. Ruocco, J. Geessels, U. Khan, and W. Bogaerts, “Extracting multiple parameters from a compact circuit for performance evaluation,” in European Conference on Integrated Optics, Belgium, (2019).
  5. W. Bogaerts and L. Chrostowski, “Silicon photonics circuit design: methods, tools and challenges,” Laser Photonics Rev. 12(4), 1700237 (2018).
    [Crossref]
  6. S. Dwivedi, H. D’heer, and W. Bogaerts, “Maximizing fabrication and thermal tolerances of all-silicon FIR wavelength filters,” IEEE Photonics Technol. Lett. 27(8), 871–874 (2015).
    [Crossref]
  7. M. Uenuma and T. Motooka, “Temperature-independent silicon waveguide optical filter,” Opt. Lett. 34(5), 599–601 (2009).
    [Crossref]
  8. B. Guha, A. Gondarenko, and M. Lipson, “Minimizing temperature sensitivity of silicon Mach-Zehnder interferometers,” Opt. Express 18(3), 1879–1887 (2010).
    [Crossref]
  9. S. Dwivedi, H. D’heer, and W. Bogaerts, “A compact all-silicon temperature insensitive filter for WDM and bio-sensing applications,” IEEE Photonics Technol. Lett. 25(22), 2167–2170 (2013).
    [Crossref]
  10. P. Xing and J. Viegas, “Broadband CMOS-compatible SOI temperature insensitive Mach-Zehnder interferometer,” Opt. Express 23(19), 24098–24107 (2015).
    [Crossref]
  11. P. Dong, W. Qian, H. Liang, R. Shafiiha, D. Feng, G. Li, J. E. Cunningham, A. V. Krishnamoorthy, and M. Asghari, “Thermally tunable silicon racetrack resonators with ultralow tuning power,” Opt. Express 18(19), 20298–20304 (2010).
    [Crossref]
  12. Y. Yanagase, S. Suzuki, Y. Kokubun, and Sai Tak Chu, “Box-like filter response and expansion of FSR by a vertically triple coupled microring resonator filter,” J. Lightwave Technol. 20(8), 1525–1529 (2002).
    [Crossref]
  13. F. Xia, M. Rooks, L. Sekaric, and Y. Vlasov, “Ultra-compact high order ring resonator filters using submicron silicon photonic wires for on-chip optical interconnects,” Opt. Express 15(19), 11934–11941 (2007).
    [Crossref]
  14. M. S. Dahlem, C. W. Holzwarth, A. Khilo, F. X. Kärtner, H. I. Smith, and E. P. Ippen, “Reconfigurable multi-channel second-order silicon microring-resonator filterbanks for on-chip WDM systems,” Opt. Express 19(1), 306–316 (2011).
    [Crossref]
  15. S. Feng, T. Lei, H. Chen, H. Cai, X. Luo, and A. W. Poon, “Silicon photonics: from a microresonator perspective,” Laser Photonics Rev. 6(2), 145–177 (2012).
    [Crossref]
  16. J. Lloret, J. Sancho, M. Pu, I. Gasulla, K. Yvind, S. Sales, and J. Capmany, “Tunable complex-valued multi-tap microwave photonic filter based on single silicon-on-insulator microring resonator,” Opt. Express 19(13), 12402–12407 (2011).
    [Crossref]
  17. D. Pérez, I. Gasulla, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Multipurpose silicon photonics signal processor core,” Nat. Commun. 8(1), 636 (2017).
    [Crossref]
  18. A. Li, T. Van Vaerenbergh, P. De Heyn, P. Bienstman, and W. Bogaerts, “Backscattering in silicon microring resonators: a quantitative analysis,” Laser Photonics Rev. 10(3), 420–431 (2016).
    [Crossref]
  19. F. G. Peternella, B. Ouyang, R. Horsten, M. Haverdings, P. Kat, and J. Caro, “Interrogation of a ring-resonator ultrasound sensor using a fiber Mach-Zehnder interferometer,” Opt. Express 25(25), 31622–31639 (2017).
    [Crossref]

2019 (1)

W. Bogaerts, Y. Xing, and U. Khan, “Layout-aware variability analysis, yield prediction, and optimization in photonic integrated circuits,” IEEE J. Sel. Top. Quantum Electron. 25(5), 1–13 (2019).
[Crossref]

2018 (2)

Y. Xing, J. Dong, S. Dwivedi, U. Khan, and W. Bogaerts, “Accurate extraction of fabricated geometry using optical measurement,” Photonics Res. 6(11), 1008–1020 (2018).
[Crossref]

W. Bogaerts and L. Chrostowski, “Silicon photonics circuit design: methods, tools and challenges,” Laser Photonics Rev. 12(4), 1700237 (2018).
[Crossref]

2017 (3)

2016 (1)

A. Li, T. Van Vaerenbergh, P. De Heyn, P. Bienstman, and W. Bogaerts, “Backscattering in silicon microring resonators: a quantitative analysis,” Laser Photonics Rev. 10(3), 420–431 (2016).
[Crossref]

2015 (2)

P. Xing and J. Viegas, “Broadband CMOS-compatible SOI temperature insensitive Mach-Zehnder interferometer,” Opt. Express 23(19), 24098–24107 (2015).
[Crossref]

S. Dwivedi, H. D’heer, and W. Bogaerts, “Maximizing fabrication and thermal tolerances of all-silicon FIR wavelength filters,” IEEE Photonics Technol. Lett. 27(8), 871–874 (2015).
[Crossref]

2013 (1)

S. Dwivedi, H. D’heer, and W. Bogaerts, “A compact all-silicon temperature insensitive filter for WDM and bio-sensing applications,” IEEE Photonics Technol. Lett. 25(22), 2167–2170 (2013).
[Crossref]

2012 (1)

S. Feng, T. Lei, H. Chen, H. Cai, X. Luo, and A. W. Poon, “Silicon photonics: from a microresonator perspective,” Laser Photonics Rev. 6(2), 145–177 (2012).
[Crossref]

2011 (2)

2010 (2)

2009 (1)

2007 (1)

2002 (1)

Asghari, M.

Bienstman, P.

A. Li, T. Van Vaerenbergh, P. De Heyn, P. Bienstman, and W. Bogaerts, “Backscattering in silicon microring resonators: a quantitative analysis,” Laser Photonics Rev. 10(3), 420–431 (2016).
[Crossref]

Bogaerts, W.

W. Bogaerts, Y. Xing, and U. Khan, “Layout-aware variability analysis, yield prediction, and optimization in photonic integrated circuits,” IEEE J. Sel. Top. Quantum Electron. 25(5), 1–13 (2019).
[Crossref]

Y. Xing, J. Dong, S. Dwivedi, U. Khan, and W. Bogaerts, “Accurate extraction of fabricated geometry using optical measurement,” Photonics Res. 6(11), 1008–1020 (2018).
[Crossref]

W. Bogaerts and L. Chrostowski, “Silicon photonics circuit design: methods, tools and challenges,” Laser Photonics Rev. 12(4), 1700237 (2018).
[Crossref]

A. Li, T. Van Vaerenbergh, P. De Heyn, P. Bienstman, and W. Bogaerts, “Backscattering in silicon microring resonators: a quantitative analysis,” Laser Photonics Rev. 10(3), 420–431 (2016).
[Crossref]

S. Dwivedi, H. D’heer, and W. Bogaerts, “Maximizing fabrication and thermal tolerances of all-silicon FIR wavelength filters,” IEEE Photonics Technol. Lett. 27(8), 871–874 (2015).
[Crossref]

S. Dwivedi, H. D’heer, and W. Bogaerts, “A compact all-silicon temperature insensitive filter for WDM and bio-sensing applications,” IEEE Photonics Technol. Lett. 25(22), 2167–2170 (2013).
[Crossref]

Y. Xing, M. Wang, A. Ruocco, J. Geessels, U. Khan, and W. Bogaerts, “Extracting multiple parameters from a compact circuit for performance evaluation,” in European Conference on Integrated Optics, Belgium, (2019).

Cai, H.

S. Feng, T. Lei, H. Chen, H. Cai, X. Luo, and A. W. Poon, “Silicon photonics: from a microresonator perspective,” Laser Photonics Rev. 6(2), 145–177 (2012).
[Crossref]

Cao, W.

D. Pérez, I. Gasulla, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Multipurpose silicon photonics signal processor core,” Nat. Commun. 8(1), 636 (2017).
[Crossref]

Capmany, J.

D. Pérez, I. Gasulla, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Multipurpose silicon photonics signal processor core,” Nat. Commun. 8(1), 636 (2017).
[Crossref]

J. Lloret, J. Sancho, M. Pu, I. Gasulla, K. Yvind, S. Sales, and J. Capmany, “Tunable complex-valued multi-tap microwave photonic filter based on single silicon-on-insulator microring resonator,” Opt. Express 19(13), 12402–12407 (2011).
[Crossref]

Caro, J.

Chen, H.

S. Feng, T. Lei, H. Chen, H. Cai, X. Luo, and A. W. Poon, “Silicon photonics: from a microresonator perspective,” Laser Photonics Rev. 6(2), 145–177 (2012).
[Crossref]

Chrostowski, L.

Crudgington, L.

D. Pérez, I. Gasulla, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Multipurpose silicon photonics signal processor core,” Nat. Commun. 8(1), 636 (2017).
[Crossref]

Cunningham, J. E.

D’heer, H.

S. Dwivedi, H. D’heer, and W. Bogaerts, “Maximizing fabrication and thermal tolerances of all-silicon FIR wavelength filters,” IEEE Photonics Technol. Lett. 27(8), 871–874 (2015).
[Crossref]

S. Dwivedi, H. D’heer, and W. Bogaerts, “A compact all-silicon temperature insensitive filter for WDM and bio-sensing applications,” IEEE Photonics Technol. Lett. 25(22), 2167–2170 (2013).
[Crossref]

Dahlem, M. S.

De Heyn, P.

A. Li, T. Van Vaerenbergh, P. De Heyn, P. Bienstman, and W. Bogaerts, “Backscattering in silicon microring resonators: a quantitative analysis,” Laser Photonics Rev. 10(3), 420–431 (2016).
[Crossref]

Dong, J.

Y. Xing, J. Dong, S. Dwivedi, U. Khan, and W. Bogaerts, “Accurate extraction of fabricated geometry using optical measurement,” Photonics Res. 6(11), 1008–1020 (2018).
[Crossref]

Dong, P.

Dwivedi, S.

Y. Xing, J. Dong, S. Dwivedi, U. Khan, and W. Bogaerts, “Accurate extraction of fabricated geometry using optical measurement,” Photonics Res. 6(11), 1008–1020 (2018).
[Crossref]

S. Dwivedi, H. D’heer, and W. Bogaerts, “Maximizing fabrication and thermal tolerances of all-silicon FIR wavelength filters,” IEEE Photonics Technol. Lett. 27(8), 871–874 (2015).
[Crossref]

S. Dwivedi, H. D’heer, and W. Bogaerts, “A compact all-silicon temperature insensitive filter for WDM and bio-sensing applications,” IEEE Photonics Technol. Lett. 25(22), 2167–2170 (2013).
[Crossref]

Feng, D.

Feng, S.

S. Feng, T. Lei, H. Chen, H. Cai, X. Luo, and A. W. Poon, “Silicon photonics: from a microresonator perspective,” Laser Photonics Rev. 6(2), 145–177 (2012).
[Crossref]

Flueckiger, J.

Gasulla, I.

D. Pérez, I. Gasulla, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Multipurpose silicon photonics signal processor core,” Nat. Commun. 8(1), 636 (2017).
[Crossref]

J. Lloret, J. Sancho, M. Pu, I. Gasulla, K. Yvind, S. Sales, and J. Capmany, “Tunable complex-valued multi-tap microwave photonic filter based on single silicon-on-insulator microring resonator,” Opt. Express 19(13), 12402–12407 (2011).
[Crossref]

Geessels, J.

Y. Xing, M. Wang, A. Ruocco, J. Geessels, U. Khan, and W. Bogaerts, “Extracting multiple parameters from a compact circuit for performance evaluation,” in European Conference on Integrated Optics, Belgium, (2019).

Gondarenko, A.

Guha, B.

Haverdings, M.

Holzwarth, C. W.

Horsten, R.

Ippen, E. P.

Jhoja, J.

Kärtner, F. X.

Kat, P.

Khan, U.

W. Bogaerts, Y. Xing, and U. Khan, “Layout-aware variability analysis, yield prediction, and optimization in photonic integrated circuits,” IEEE J. Sel. Top. Quantum Electron. 25(5), 1–13 (2019).
[Crossref]

Y. Xing, J. Dong, S. Dwivedi, U. Khan, and W. Bogaerts, “Accurate extraction of fabricated geometry using optical measurement,” Photonics Res. 6(11), 1008–1020 (2018).
[Crossref]

Y. Xing, M. Wang, A. Ruocco, J. Geessels, U. Khan, and W. Bogaerts, “Extracting multiple parameters from a compact circuit for performance evaluation,” in European Conference on Integrated Optics, Belgium, (2019).

Khilo, A.

Khokhar, A. Z.

D. Pérez, I. Gasulla, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Multipurpose silicon photonics signal processor core,” Nat. Commun. 8(1), 636 (2017).
[Crossref]

Klein, J.

Kokubun, Y.

Krishnamoorthy, A. V.

Lei, T.

S. Feng, T. Lei, H. Chen, H. Cai, X. Luo, and A. W. Poon, “Silicon photonics: from a microresonator perspective,” Laser Photonics Rev. 6(2), 145–177 (2012).
[Crossref]

Li, A.

A. Li, T. Van Vaerenbergh, P. De Heyn, P. Bienstman, and W. Bogaerts, “Backscattering in silicon microring resonators: a quantitative analysis,” Laser Photonics Rev. 10(3), 420–431 (2016).
[Crossref]

Li, G.

Li, K.

D. Pérez, I. Gasulla, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Multipurpose silicon photonics signal processor core,” Nat. Commun. 8(1), 636 (2017).
[Crossref]

Liang, H.

Lipson, M.

Liu, A.

Lloret, J.

Lu, Z.

Luo, X.

S. Feng, T. Lei, H. Chen, H. Cai, X. Luo, and A. W. Poon, “Silicon photonics: from a microresonator perspective,” Laser Photonics Rev. 6(2), 145–177 (2012).
[Crossref]

Mashanovich, G. Z.

D. Pérez, I. Gasulla, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Multipurpose silicon photonics signal processor core,” Nat. Commun. 8(1), 636 (2017).
[Crossref]

Motooka, T.

Ouyang, B.

Pérez, D.

D. Pérez, I. Gasulla, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Multipurpose silicon photonics signal processor core,” Nat. Commun. 8(1), 636 (2017).
[Crossref]

Peternella, F. G.

Pond, J.

Poon, A. W.

S. Feng, T. Lei, H. Chen, H. Cai, X. Luo, and A. W. Poon, “Silicon photonics: from a microresonator perspective,” Laser Photonics Rev. 6(2), 145–177 (2012).
[Crossref]

Pu, M.

Qian, W.

Rooks, M.

Ruocco, A.

Y. Xing, M. Wang, A. Ruocco, J. Geessels, U. Khan, and W. Bogaerts, “Extracting multiple parameters from a compact circuit for performance evaluation,” in European Conference on Integrated Optics, Belgium, (2019).

Sai Tak Chu,

Sales, S.

Sancho, J.

Sekaric, L.

Shafiiha, R.

Smith, H. I.

Suzuki, S.

Thomson, D. J.

D. Pérez, I. Gasulla, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Multipurpose silicon photonics signal processor core,” Nat. Commun. 8(1), 636 (2017).
[Crossref]

Uenuma, M.

Van Vaerenbergh, T.

A. Li, T. Van Vaerenbergh, P. De Heyn, P. Bienstman, and W. Bogaerts, “Backscattering in silicon microring resonators: a quantitative analysis,” Laser Photonics Rev. 10(3), 420–431 (2016).
[Crossref]

Viegas, J.

Vlasov, Y.

Wang, M.

Y. Xing, M. Wang, A. Ruocco, J. Geessels, U. Khan, and W. Bogaerts, “Extracting multiple parameters from a compact circuit for performance evaluation,” in European Conference on Integrated Optics, Belgium, (2019).

Wang, X.

Xia, F.

Xing, P.

Xing, Y.

W. Bogaerts, Y. Xing, and U. Khan, “Layout-aware variability analysis, yield prediction, and optimization in photonic integrated circuits,” IEEE J. Sel. Top. Quantum Electron. 25(5), 1–13 (2019).
[Crossref]

Y. Xing, J. Dong, S. Dwivedi, U. Khan, and W. Bogaerts, “Accurate extraction of fabricated geometry using optical measurement,” Photonics Res. 6(11), 1008–1020 (2018).
[Crossref]

Y. Xing, M. Wang, A. Ruocco, J. Geessels, U. Khan, and W. Bogaerts, “Extracting multiple parameters from a compact circuit for performance evaluation,” in European Conference on Integrated Optics, Belgium, (2019).

Yanagase, Y.

Yvind, K.

IEEE J. Sel. Top. Quantum Electron. (1)

W. Bogaerts, Y. Xing, and U. Khan, “Layout-aware variability analysis, yield prediction, and optimization in photonic integrated circuits,” IEEE J. Sel. Top. Quantum Electron. 25(5), 1–13 (2019).
[Crossref]

IEEE Photonics Technol. Lett. (2)

S. Dwivedi, H. D’heer, and W. Bogaerts, “Maximizing fabrication and thermal tolerances of all-silicon FIR wavelength filters,” IEEE Photonics Technol. Lett. 27(8), 871–874 (2015).
[Crossref]

S. Dwivedi, H. D’heer, and W. Bogaerts, “A compact all-silicon temperature insensitive filter for WDM and bio-sensing applications,” IEEE Photonics Technol. Lett. 25(22), 2167–2170 (2013).
[Crossref]

J. Lightwave Technol. (1)

Laser Photonics Rev. (3)

S. Feng, T. Lei, H. Chen, H. Cai, X. Luo, and A. W. Poon, “Silicon photonics: from a microresonator perspective,” Laser Photonics Rev. 6(2), 145–177 (2012).
[Crossref]

A. Li, T. Van Vaerenbergh, P. De Heyn, P. Bienstman, and W. Bogaerts, “Backscattering in silicon microring resonators: a quantitative analysis,” Laser Photonics Rev. 10(3), 420–431 (2016).
[Crossref]

W. Bogaerts and L. Chrostowski, “Silicon photonics circuit design: methods, tools and challenges,” Laser Photonics Rev. 12(4), 1700237 (2018).
[Crossref]

Nat. Commun. (1)

D. Pérez, I. Gasulla, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Multipurpose silicon photonics signal processor core,” Nat. Commun. 8(1), 636 (2017).
[Crossref]

Opt. Express (8)

F. Xia, M. Rooks, L. Sekaric, and Y. Vlasov, “Ultra-compact high order ring resonator filters using submicron silicon photonic wires for on-chip optical interconnects,” Opt. Express 15(19), 11934–11941 (2007).
[Crossref]

P. Dong, W. Qian, H. Liang, R. Shafiiha, D. Feng, G. Li, J. E. Cunningham, A. V. Krishnamoorthy, and M. Asghari, “Thermally tunable silicon racetrack resonators with ultralow tuning power,” Opt. Express 18(19), 20298–20304 (2010).
[Crossref]

P. Xing and J. Viegas, “Broadband CMOS-compatible SOI temperature insensitive Mach-Zehnder interferometer,” Opt. Express 23(19), 24098–24107 (2015).
[Crossref]

Z. Lu, J. Jhoja, J. Klein, X. Wang, A. Liu, J. Flueckiger, J. Pond, and L. Chrostowski, “Performance prediction for silicon photonics integrated circuits with layout-dependent correlated manufacturing variability,” Opt. Express 25(9), 9712–9733 (2017).
[Crossref]

B. Guha, A. Gondarenko, and M. Lipson, “Minimizing temperature sensitivity of silicon Mach-Zehnder interferometers,” Opt. Express 18(3), 1879–1887 (2010).
[Crossref]

M. S. Dahlem, C. W. Holzwarth, A. Khilo, F. X. Kärtner, H. I. Smith, and E. P. Ippen, “Reconfigurable multi-channel second-order silicon microring-resonator filterbanks for on-chip WDM systems,” Opt. Express 19(1), 306–316 (2011).
[Crossref]

J. Lloret, J. Sancho, M. Pu, I. Gasulla, K. Yvind, S. Sales, and J. Capmany, “Tunable complex-valued multi-tap microwave photonic filter based on single silicon-on-insulator microring resonator,” Opt. Express 19(13), 12402–12407 (2011).
[Crossref]

F. G. Peternella, B. Ouyang, R. Horsten, M. Haverdings, P. Kat, and J. Caro, “Interrogation of a ring-resonator ultrasound sensor using a fiber Mach-Zehnder interferometer,” Opt. Express 25(25), 31622–31639 (2017).
[Crossref]

Opt. Lett. (1)

Photonics Res. (1)

Y. Xing, J. Dong, S. Dwivedi, U. Khan, and W. Bogaerts, “Accurate extraction of fabricated geometry using optical measurement,” Photonics Res. 6(11), 1008–1020 (2018).
[Crossref]

Other (1)

Y. Xing, M. Wang, A. Ruocco, J. Geessels, U. Khan, and W. Bogaerts, “Extracting multiple parameters from a compact circuit for performance evaluation,” in European Conference on Integrated Optics, Belgium, (2019).

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Figures (7)

Fig. 1.
Fig. 1. Calculated group index $n_g$ of air-cladding silicon waveguides ($h = {220}$ nm) as a function of waveguide width $w$ at $\lambda = {1550}$ nm. As an example, the inset shows the profile of the $\mathrm {TE}_{0}$ mode with $w = {450}$ nm and $h = {220}$ nm, for which $n_g$ is calculated.
Fig. 2.
Fig. 2. Schematic of the RR design with two waveguide widths, giving an FSR robust to waveguide-width variations.
Fig. 3.
Fig. 3. (a) Measured transmission of a test MZI with a design width of 450 nm, together with the fitted cosine. (b) Estimated real waveguide width as a function of design width.
Fig. 4.
Fig. 4. (a) $n_g$ versus width. The blue curve is obtained by calibrating the red curve from Fig. 1 using the linear function in Fig. 3(b). (b) Derivative curves $\partial n_{g} / \partial w$ calculated from the curves in (a).
Fig. 5.
Fig. 5. Predicted FSR as a function of waveguide-width variation, for devices robust 1 and robust 2. For comparison, the behavior for a normal design is also included.
Fig. 6.
Fig. 6. (a) Measured normalized transmission spectrum of RR robust 1 from die 3. (b) Zoom-in of the dip near 1552 nm in (a) (highlighted by the red ellipse). The red curve is a fit of Eq. (5) to the data points.
Fig. 7.
Fig. 7. (a) Measured FSRs from die 1 to die 4, for RRs without intentional width variations. The inset shows the die locations on the wafer. (b) The measured FSR from die 3 as a function of intentional waveguide-width change.

Tables (1)

Tables Icon

Table 1. Measured FSR deviations from the design value ( 124.783 GHz ) and range-dependent FSR variations Δ F S R for the measured devices on die 3. The uncertainty of each number results from the uncertainties of the FSR values derived from the transmission data. The uncertainties of the FSR values result from the confidence bounds of the two λ r values (see Section 4), using uncertainty propagation.

Equations (5)

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

F S R f = c n g L .
F S R λ = λ 2 n g L ,
F S R f = c n g ( w 1 ) L w 1 + n g ( w 2 ) L w 2 + 2 n g , t ¯ L t .
n g ( w ) w | w 1 L w 1 + n g ( w ) w | w 2 L w 2 = 0 .
T p a s s ( λ ) = a ( λ λ r ) 2 + ε ( γ r / 2 ) 2 ( λ λ r ) 2 + ( γ r / 2 ) 2 .

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