Abstract

Precise and reliable apodization of silicon integrated Bragg gratings (IBGs) is the key to realizing their spectral tailoring for many optical applications such as optical signal processing and wavelength-division multiplexing systems. However, apodization in a silicon IBG that is typically realized by modifying the physical waveguide grating structure can also introduce unwanted grating phase variations that can affect the grating response. In this paper, we present a model to characterize apodized silicon IBGs which can take such apodization phase noise (APN) into account, based on direct synthesis of the physical grating structure. The model is used to characterize a set of different silicon IBGs apodized by lateral misalignment (ΔL) and duty-cycle (DC) modulations and designed with different responses, and the results show that the APN can greatly distort the complex responses of the gratings. Then, we develop a methodology to compensate the APN and thus to correct the distorted grating responses. The designed silicon IBGs were fabricated and tested experimentally. The accuracy of the model is examined by comparing the measured grating spectra with those predicted by the model. Spectral corrections are then demonstrated in Gaussian-apodized gratings based on ΔL- and DC-modulated silicon IBGs and a square-shaped filter developed on a ΔL-modulated IBG. Finally, a complex spectral correction of a photonic Hilbert transformer developed on a ΔL-modulated silicon IBG is achieved.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. A. D. Simard, N. Belhadj, Y. Painchaud, and S. LaRochelle, “Apodized silicon-on-insulator Bragg gratings,” IEEE Photon. Technol. Lett. 24, 1033–1035 (2012).
    [Crossref]
  2. K. Bédard, A. D. Simard, B. Filion, Y. Painchaud, L. A. Rusch, and S. LaRochelle, “Dual phase-shift Bragg grating silicon photonic modulator operating up to 60 Gb/s,” Opt. Express 24, 2413–2419 (2016).
    [Crossref] [PubMed]
  3. W. Shi, V. Veerasubramanian, D. Patel, and D. V. Plant, “Tunable nanophotonic delay lines using linearly chirped contradirectional couplers with uniform Bragg gratings,” Opt. Lett. 39, 701–703 (2014).
    [Crossref] [PubMed]
  4. N. C. Harris, D. Grassani, A. Simbula, M. Pant, M. Galli, T. Baehr-Jones, M. Hochberg, D. Englund, D. Bajoni, and C. Galland, “Integrated source of spectrally filtered correlated photons for large-scale quantum photonic systems,” Phys. Rev. X 4, 041047 (2014).
  5. J. W. Silverstone, D. Bonneau, J. L. O’Brien, and M. G. Thompson, “Silicon quantum photonics,” IEEE J. Sel. Topics Quantum Electron. 22, 390–402 (2016).
    [Crossref]
  6. X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophotonics 6, 821–828 (2013).
    [PubMed]
  7. N. N. Klimov, S. Mittal, M. Berger, and Z. Ahmed, “On-chip silicon waveguide Bragg grating photonic temperature sensor,” Opt. Lett. 40, 3934–3936 (2015).
    [Crossref] [PubMed]
  8. L. Chrostowski and M. Hochberg, Silicon photonics design: from devices to systems (Cambridge University Press, 2015).
    [Crossref]
  9. X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, “Uniform and sampled Bragg gratings in SOI strip waveguides with sidewall corrugations,” IEEE Photon. Technol. Lett. 23, 290–292 (2011).
  10. A. D. Simard, M. J. Strain, L. Meriggi, M. Sorel, and S. LaRochelle, “Bandpass integrated Bragg gratings in silicon-on-insulator with well-controlled amplitude and phase responses,” Opt. Lett. 40, 736–739 (2015).
    [Crossref] [PubMed]
  11. R. Cheng and L. Chrostowski, “Multichannel photonic Hilbert transformers based on complex modulated integrated Bragg gratings,” Opt. Lett. 43, 1031–1034 (2018).
    [Crossref] [PubMed]
  12. S. Kaushal, R. Cheng, M. Ma, A. Mistry, M. Burla, L. Chrostowski, and J. Azaña, “Optical signal processing based on silicon photonics waveguide Bragg gratings,” Frontiers of Optoelectronics 11 (2),163–188 (2018).
  13. M. Burla, L. R. Cortés, M. Li, X. Wang, L. Chrostowski, and J. Azaña, “Integrated waveguide Bragg gratings for microwave photonics signal processing,” Opt. Express 21, 25120–25147 (2013).
    [Crossref] [PubMed]
  14. H. Qiu, J. Jiang, P. Yu, D. Mu, J. Yang, X. Jiang, H. Yu, R. Cheng, and L. Chrostowski, “Narrow-band add-drop filter based on phase-modulated grating-assisted contra-directional couplers,” J. Lightwave Technol. 36, 3760–3764 (2018).
    [Crossref]
  15. M. J. Strain, S. Thoms, D. S. MacIntyre, and M. Sorel, “Multi-wavelength filters in silicon using superposition sidewall Bragg grating devices,” Opt. Lett. 39, 413–416 (2014).
    [Crossref] [PubMed]
  16. S. Paul, T. Saastamoinen, S. Honkanen, M. Roussey, and M. Kuittinen, “Multi-wavelength filtering with a waveguide integrated phase-modulated Bragg grating,” Opt. Lett. 42, 4635–4638 (2017).
    [Crossref] [PubMed]
  17. T. Erdogan, “Fiber grating spectra,” J. lightwave technology 15, 1277–1294 (1997).
    [Crossref]
  18. M. Ma, Z. Chen, H. Yun, Y. Wang, X. Wang, N. A. F. Jaeger, and L. Chrostowski, “Apodized spiral Bragg grating waveguides in silicon-on-insulator,” IEEE Photon. Technol. Lett. 30, 111–114 (2018).
    [Crossref]
  19. D. Tan, K. Ikeda, and Y. Fainman, “Cladding-modulated Bragg gratings in silicon waveguides,” Opt. Lett. 34, 1357–1359 (2009).
    [Crossref] [PubMed]
  20. Y.-J. Hung, Y.-C. Liang, C.-W. Huang, J.-F. Shih, S. Hu, T.-H. Yen, C.-W. Kao, and C.-H. Chen, “Narrowband silicon waveguide Bragg reflector achieved by highly ordered graphene oxide gratings,” Opt. Lett. 42, 4768–4771 (2017).
    [Crossref] [PubMed]
  21. H. Sakata, “Sidelobe suppression in grating-assisted wavelength-selective couplers,” Opt. Lett. 17, 463–465 (1992).
    [Crossref] [PubMed]
  22. D. Wiesmann, C. David, R. Germann, D. Emi, and G. Bona, “Apodized surface-corrugated gratings with varying duty cycles,” IEEE Photon. Technol. Lett. 12, 639–641 (2000).
    [Crossref]
  23. X. Wang, Y. Wang, J. Flueckiger, R. Bojko, A. Liu, A. Reid, J. Pond, N. A. Jaeger, and L. Chrostowski, “Precise control of the coupling coefficient through destructive interference in silicon waveguide Bragg gratings,” Opt. Lett. 39, 5519–5522 (2014).
    [Crossref] [PubMed]
  24. H. P. Bazargani, M. Burla, L. Chrostowski, and J. Azaña, “Photonic Hilbert transformers based on laterally apodized integrated waveguide Bragg gratings on a SOI wafer,” Opt. Lett. 41, 5039–5042 (2016).
    [Crossref] [PubMed]
  25. M. J. Strain and M. Sorel, “Integrated III–V Bragg gratings for arbitrary control over chirp and coupling coefficient,” IEEE Photon. Technol. Lett. 20, 1863–1865 (2008).
    [Crossref]
  26. D. Oser, D. Pérez-Galacho, C. Alonso-Ramos, X. L. Roux, S. Tanzilli, L. Vivien, and L. Labonté, and Éric Cassan, “Subwavelength engineering and asymmetry: two efficient tools for sub-nanometer-bandwidth silicon Bragg filters,” Opt. Lett. 43, 3208–3211 (2018).
    [Crossref] [PubMed]
  27. R. Oliveira, P. Neves, J. Pereira, and A. Pohl, “Numerical approach for designing a Bragg grating acousto-optic modulator using the finite element and the transfer matrix methods,” Opt. Commun. 281, 4899 – 4905 (2008).
    [Crossref]
  28. 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 The 9th International Conference on Group IV Photonics (GFP), (2012), pp. 288–290.
  29. M. C. Troparevsky, A. S. Sabau, A. R. Lupini, and Z. Zhang, “Transfer-matrix formalism for the calculation of optical response in multilayer systems: from coherent to incoherent interference,” Opt. Express 18, 24715–24721 (2010).
    [Crossref] [PubMed]
  30. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (The Oxford Series in Electrical and Computer Engineering) (Oxford University Press, 2006).
  31. J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
    [Crossref]
  32. C. Sima, J. C. Gates, H. L. Rogers, P. L. Mennea, C. Holmes, M. N. Zervas, and P. G. R. Smith, “Phase controlled integrated interferometric single-sideband filter based on planar Bragg gratings implementing photonic Hilbert transform,” Opt. Lett. 38, 727–729 (2013).
    [Crossref] [PubMed]
  33. Z. Lu, H. Yun, Y. Wang, Z. Chen, F. Zhang, N. A. Jaeger, and L. Chrostowski, “Broadband silicon photonic directional coupler using asymmetric-waveguide based phase control,” Opt. Express 23, 3795–3808 (2015).
    [Crossref] [PubMed]
  34. Y. Wang, X. Wang, J. Flueckiger, H. Yun, W. Shi, R. Bojko, N. A. F. Jaeger, and L. Chrostowski, “Focusing sub-wavelength grating couplers with low back reflections for rapid prototyping of silicon photonic circuits,” Opt. Express 22, 20652–20662 (2014).
    [Crossref] [PubMed]
  35. J. Jiang, H. Qiu, G. Wang, Y. Li, T. Dai, D. Mu, H. Yu, J. Yang, and X. Jiang, “Silicon lateral-apodized add–drop filter for on-chip optical interconnection,” Appl. Opt. 56, 8425–8429 (2017).
    [Crossref] [PubMed]
  36. C. Sima, J. Gates, C. Holmes, P. Mennea, M. Zervas, and P. Smith, “Terahertz bandwidth photonic Hilbert transformers based on synthesized planar Bragg grating fabrication,” Opt. Lett. 38, 3448–3451 (2013).
    [Crossref] [PubMed]
  37. X. Wang, W. Shi, H. Yun, S. Grist, N. A. Jaeger, and L. Chrostowski, “Narrow-band waveguide Bragg gratings on SOI wafers with CMOS-compatible fabrication process,” Opt. Express 20, 15547–15558 (2012).
    [Crossref] [PubMed]
  38. G. B. Hocker and W. K. Burns, “Mode dispersion in diffused channel waveguides by the effective index method,” Appl. Opt. 16, 113–118 (1977).
    [Crossref] [PubMed]

2018 (5)

2017 (3)

2016 (3)

2015 (3)

2014 (5)

2013 (4)

2012 (2)

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

X. Wang, W. Shi, H. Yun, S. Grist, N. A. Jaeger, and L. Chrostowski, “Narrow-band waveguide Bragg gratings on SOI wafers with CMOS-compatible fabrication process,” Opt. Express 20, 15547–15558 (2012).
[Crossref] [PubMed]

2011 (1)

X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, “Uniform and sampled Bragg gratings in SOI strip waveguides with sidewall corrugations,” IEEE Photon. Technol. Lett. 23, 290–292 (2011).

2010 (1)

2009 (1)

2008 (2)

M. J. Strain and M. Sorel, “Integrated III–V Bragg gratings for arbitrary control over chirp and coupling coefficient,” IEEE Photon. Technol. Lett. 20, 1863–1865 (2008).
[Crossref]

R. Oliveira, P. Neves, J. Pereira, and A. Pohl, “Numerical approach for designing a Bragg grating acousto-optic modulator using the finite element and the transfer matrix methods,” Opt. Commun. 281, 4899 – 4905 (2008).
[Crossref]

2001 (1)

J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
[Crossref]

2000 (1)

D. Wiesmann, C. David, R. Germann, D. Emi, and G. Bona, “Apodized surface-corrugated gratings with varying duty cycles,” IEEE Photon. Technol. Lett. 12, 639–641 (2000).
[Crossref]

1997 (1)

T. Erdogan, “Fiber grating spectra,” J. lightwave technology 15, 1277–1294 (1997).
[Crossref]

1992 (1)

1977 (1)

Adam, K.

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 The 9th International Conference on Group IV Photonics (GFP), (2012), pp. 288–290.

Ahmed, Z.

Alonso-Ramos, C.

Azaña, J.

Baehr-Jones, T.

N. C. Harris, D. Grassani, A. Simbula, M. Pant, M. Galli, T. Baehr-Jones, M. Hochberg, D. Englund, D. Bajoni, and C. Galland, “Integrated source of spectrally filtered correlated photons for large-scale quantum photonic systems,” Phys. Rev. X 4, 041047 (2014).

Bajoni, D.

N. C. Harris, D. Grassani, A. Simbula, M. Pant, M. Galli, T. Baehr-Jones, M. Hochberg, D. Englund, D. Bajoni, and C. Galland, “Integrated source of spectrally filtered correlated photons for large-scale quantum photonic systems,” Phys. Rev. X 4, 041047 (2014).

Bazargani, H. P.

Bédard, K.

Belhadj, N.

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

Berger, M.

Bojko, R.

Bona, G.

D. Wiesmann, C. David, R. Germann, D. Emi, and G. Bona, “Apodized surface-corrugated gratings with varying duty cycles,” IEEE Photon. Technol. Lett. 12, 639–641 (2000).
[Crossref]

Bonneau, D.

J. W. Silverstone, D. Bonneau, J. L. O’Brien, and M. G. Thompson, “Silicon quantum photonics,” IEEE J. Sel. Topics Quantum Electron. 22, 390–402 (2016).
[Crossref]

Burla, M.

Burns, W. K.

Chen, C.-H.

Chen, Z.

M. Ma, Z. Chen, H. Yun, Y. Wang, X. Wang, N. A. F. Jaeger, and L. Chrostowski, “Apodized spiral Bragg grating waveguides in silicon-on-insulator,” IEEE Photon. Technol. Lett. 30, 111–114 (2018).
[Crossref]

Z. Lu, H. Yun, Y. Wang, Z. Chen, F. Zhang, N. A. Jaeger, and L. Chrostowski, “Broadband silicon photonic directional coupler using asymmetric-waveguide based phase control,” Opt. Express 23, 3795–3808 (2015).
[Crossref] [PubMed]

Cheng, R.

Cheung, K. C.

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophotonics 6, 821–828 (2013).
[PubMed]

Chrostowski, L.

S. Kaushal, R. Cheng, M. Ma, A. Mistry, M. Burla, L. Chrostowski, and J. Azaña, “Optical signal processing based on silicon photonics waveguide Bragg gratings,” Frontiers of Optoelectronics 11 (2),163–188 (2018).

M. Ma, Z. Chen, H. Yun, Y. Wang, X. Wang, N. A. F. Jaeger, and L. Chrostowski, “Apodized spiral Bragg grating waveguides in silicon-on-insulator,” IEEE Photon. Technol. Lett. 30, 111–114 (2018).
[Crossref]

H. Qiu, J. Jiang, P. Yu, D. Mu, J. Yang, X. Jiang, H. Yu, R. Cheng, and L. Chrostowski, “Narrow-band add-drop filter based on phase-modulated grating-assisted contra-directional couplers,” J. Lightwave Technol. 36, 3760–3764 (2018).
[Crossref]

R. Cheng and L. Chrostowski, “Multichannel photonic Hilbert transformers based on complex modulated integrated Bragg gratings,” Opt. Lett. 43, 1031–1034 (2018).
[Crossref] [PubMed]

H. P. Bazargani, M. Burla, L. Chrostowski, and J. Azaña, “Photonic Hilbert transformers based on laterally apodized integrated waveguide Bragg gratings on a SOI wafer,” Opt. Lett. 41, 5039–5042 (2016).
[Crossref] [PubMed]

Z. Lu, H. Yun, Y. Wang, Z. Chen, F. Zhang, N. A. Jaeger, and L. Chrostowski, “Broadband silicon photonic directional coupler using asymmetric-waveguide based phase control,” Opt. Express 23, 3795–3808 (2015).
[Crossref] [PubMed]

X. Wang, Y. Wang, J. Flueckiger, R. Bojko, A. Liu, A. Reid, J. Pond, N. A. Jaeger, and L. Chrostowski, “Precise control of the coupling coefficient through destructive interference in silicon waveguide Bragg gratings,” Opt. Lett. 39, 5519–5522 (2014).
[Crossref] [PubMed]

Y. Wang, X. Wang, J. Flueckiger, H. Yun, W. Shi, R. Bojko, N. A. F. Jaeger, and L. Chrostowski, “Focusing sub-wavelength grating couplers with low back reflections for rapid prototyping of silicon photonic circuits,” Opt. Express 22, 20652–20662 (2014).
[Crossref] [PubMed]

M. Burla, L. R. Cortés, M. Li, X. Wang, L. Chrostowski, and J. Azaña, “Integrated waveguide Bragg gratings for microwave photonics signal processing,” Opt. Express 21, 25120–25147 (2013).
[Crossref] [PubMed]

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophotonics 6, 821–828 (2013).
[PubMed]

X. Wang, W. Shi, H. Yun, S. Grist, N. A. Jaeger, and L. Chrostowski, “Narrow-band waveguide Bragg gratings on SOI wafers with CMOS-compatible fabrication process,” Opt. Express 20, 15547–15558 (2012).
[Crossref] [PubMed]

X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, “Uniform and sampled Bragg gratings in SOI strip waveguides with sidewall corrugations,” IEEE Photon. Technol. Lett. 23, 290–292 (2011).

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 The 9th International Conference on Group IV Photonics (GFP), (2012), pp. 288–290.

L. Chrostowski and M. Hochberg, Silicon photonics design: from devices to systems (Cambridge University Press, 2015).
[Crossref]

Cortés, L. R.

Dai, T.

David, C.

D. Wiesmann, C. David, R. Germann, D. Emi, and G. Bona, “Apodized surface-corrugated gratings with varying duty cycles,” IEEE Photon. Technol. Lett. 12, 639–641 (2000).
[Crossref]

Doerfler, M.

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophotonics 6, 821–828 (2013).
[PubMed]

Emi, D.

D. Wiesmann, C. David, R. Germann, D. Emi, and G. Bona, “Apodized surface-corrugated gratings with varying duty cycles,” IEEE Photon. Technol. Lett. 12, 639–641 (2000).
[Crossref]

Englund, D.

N. C. Harris, D. Grassani, A. Simbula, M. Pant, M. Galli, T. Baehr-Jones, M. Hochberg, D. Englund, D. Bajoni, and C. Galland, “Integrated source of spectrally filtered correlated photons for large-scale quantum photonic systems,” Phys. Rev. X 4, 041047 (2014).

Erdogan, T.

J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
[Crossref]

T. Erdogan, “Fiber grating spectra,” J. lightwave technology 15, 1277–1294 (1997).
[Crossref]

Fainman, Y.

Fard, S. T.

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophotonics 6, 821–828 (2013).
[PubMed]

Filion, B.

Flueckiger, J.

Galland, C.

N. C. Harris, D. Grassani, A. Simbula, M. Pant, M. Galli, T. Baehr-Jones, M. Hochberg, D. Englund, D. Bajoni, and C. Galland, “Integrated source of spectrally filtered correlated photons for large-scale quantum photonic systems,” Phys. Rev. X 4, 041047 (2014).

Galli, M.

N. C. Harris, D. Grassani, A. Simbula, M. Pant, M. Galli, T. Baehr-Jones, M. Hochberg, D. Englund, D. Bajoni, and C. Galland, “Integrated source of spectrally filtered correlated photons for large-scale quantum photonic systems,” Phys. Rev. X 4, 041047 (2014).

Gates, J.

Gates, J. C.

Germann, R.

D. Wiesmann, C. David, R. Germann, D. Emi, and G. Bona, “Apodized surface-corrugated gratings with varying duty cycles,” IEEE Photon. Technol. Lett. 12, 639–641 (2000).
[Crossref]

Grassani, D.

N. C. Harris, D. Grassani, A. Simbula, M. Pant, M. Galli, T. Baehr-Jones, M. Hochberg, D. Englund, D. Bajoni, and C. Galland, “Integrated source of spectrally filtered correlated photons for large-scale quantum photonic systems,” Phys. Rev. X 4, 041047 (2014).

Grist, S.

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophotonics 6, 821–828 (2013).
[PubMed]

X. Wang, W. Shi, H. Yun, S. Grist, N. A. Jaeger, and L. Chrostowski, “Narrow-band waveguide Bragg gratings on SOI wafers with CMOS-compatible fabrication process,” Opt. Express 20, 15547–15558 (2012).
[Crossref] [PubMed]

Harris, N. C.

N. C. Harris, D. Grassani, A. Simbula, M. Pant, M. Galli, T. Baehr-Jones, M. Hochberg, D. Englund, D. Bajoni, and C. Galland, “Integrated source of spectrally filtered correlated photons for large-scale quantum photonic systems,” Phys. Rev. X 4, 041047 (2014).

Hochberg, M.

N. C. Harris, D. Grassani, A. Simbula, M. Pant, M. Galli, T. Baehr-Jones, M. Hochberg, D. Englund, D. Bajoni, and C. Galland, “Integrated source of spectrally filtered correlated photons for large-scale quantum photonic systems,” Phys. Rev. X 4, 041047 (2014).

L. Chrostowski and M. Hochberg, Silicon photonics design: from devices to systems (Cambridge University Press, 2015).
[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 The 9th International Conference on Group IV Photonics (GFP), (2012), pp. 288–290.

Hocker, G. B.

Holmes, C.

Honkanen, S.

Hu, S.

Huang, C.-W.

Hung, Y.-J.

Ikeda, K.

Jaeger, N. A.

Jaeger, N. A. F.

M. Ma, Z. Chen, H. Yun, Y. Wang, X. Wang, N. A. F. Jaeger, and L. Chrostowski, “Apodized spiral Bragg grating waveguides in silicon-on-insulator,” IEEE Photon. Technol. Lett. 30, 111–114 (2018).
[Crossref]

Y. Wang, X. Wang, J. Flueckiger, H. Yun, W. Shi, R. Bojko, N. A. F. Jaeger, and L. Chrostowski, “Focusing sub-wavelength grating couplers with low back reflections for rapid prototyping of silicon photonic circuits,” Opt. Express 22, 20652–20662 (2014).
[Crossref] [PubMed]

X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, “Uniform and sampled Bragg gratings in SOI strip waveguides with sidewall corrugations,” IEEE Photon. Technol. Lett. 23, 290–292 (2011).

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 The 9th International Conference on Group IV Photonics (GFP), (2012), pp. 288–290.

Jiang, J.

Jiang, X.

Kao, C.-W.

Kaushal, S.

S. Kaushal, R. Cheng, M. Ma, A. Mistry, M. Burla, L. Chrostowski, and J. Azaña, “Optical signal processing based on silicon photonics waveguide Bragg gratings,” Frontiers of Optoelectronics 11 (2),163–188 (2018).

Kirk, J.

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophotonics 6, 821–828 (2013).
[PubMed]

Klimov, N. N.

Kuittinen, M.

Labonté, L.

LaRochelle, S.

Li, M.

Li, Y.

Liang, Y.-C.

Liu, A.

Lu, Z.

Lupini, A. R.

Ma, M.

S. Kaushal, R. Cheng, M. Ma, A. Mistry, M. Burla, L. Chrostowski, and J. Azaña, “Optical signal processing based on silicon photonics waveguide Bragg gratings,” Frontiers of Optoelectronics 11 (2),163–188 (2018).

M. Ma, Z. Chen, H. Yun, Y. Wang, X. Wang, N. A. F. Jaeger, and L. Chrostowski, “Apodized spiral Bragg grating waveguides in silicon-on-insulator,” IEEE Photon. Technol. Lett. 30, 111–114 (2018).
[Crossref]

MacIntyre, D. S.

Mennea, P.

Mennea, P. L.

Meriggi, L.

Mistry, A.

S. Kaushal, R. Cheng, M. Ma, A. Mistry, M. Burla, L. Chrostowski, and J. Azaña, “Optical signal processing based on silicon photonics waveguide Bragg gratings,” Frontiers of Optoelectronics 11 (2),163–188 (2018).

Mittal, S.

Mu, D.

Neves, P.

R. Oliveira, P. Neves, J. Pereira, and A. Pohl, “Numerical approach for designing a Bragg grating acousto-optic modulator using the finite element and the transfer matrix methods,” Opt. Commun. 281, 4899 – 4905 (2008).
[Crossref]

O’Brien, J. L.

J. W. Silverstone, D. Bonneau, J. L. O’Brien, and M. G. Thompson, “Silicon quantum photonics,” IEEE J. Sel. Topics Quantum Electron. 22, 390–402 (2016).
[Crossref]

Oliveira, R.

R. Oliveira, P. Neves, J. Pereira, and A. Pohl, “Numerical approach for designing a Bragg grating acousto-optic modulator using the finite element and the transfer matrix methods,” Opt. Commun. 281, 4899 – 4905 (2008).
[Crossref]

Oser, D.

Painchaud, Y.

K. Bédard, A. D. Simard, B. Filion, Y. Painchaud, L. A. Rusch, and S. LaRochelle, “Dual phase-shift Bragg grating silicon photonic modulator operating up to 60 Gb/s,” Opt. Express 24, 2413–2419 (2016).
[Crossref] [PubMed]

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

Pant, M.

N. C. Harris, D. Grassani, A. Simbula, M. Pant, M. Galli, T. Baehr-Jones, M. Hochberg, D. Englund, D. Bajoni, and C. Galland, “Integrated source of spectrally filtered correlated photons for large-scale quantum photonic systems,” Phys. Rev. X 4, 041047 (2014).

Patel, D.

Paul, S.

Pereira, J.

R. Oliveira, P. Neves, J. Pereira, and A. Pohl, “Numerical approach for designing a Bragg grating acousto-optic modulator using the finite element and the transfer matrix methods,” Opt. Commun. 281, 4899 – 4905 (2008).
[Crossref]

Pérez-Galacho, D.

Plant, D. V.

Pohl, A.

R. Oliveira, P. Neves, J. Pereira, and A. Pohl, “Numerical approach for designing a Bragg grating acousto-optic modulator using the finite element and the transfer matrix methods,” Opt. Commun. 281, 4899 – 4905 (2008).
[Crossref]

Pond, J.

Qiu, H.

Ratner, D. M.

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophotonics 6, 821–828 (2013).
[PubMed]

Reid, A.

Rogers, H. L.

Roussey, M.

Roux, X. L.

Rusch, L. A.

Saastamoinen, T.

Sabau, A. S.

Sakata, H.

Schelew, E.

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 The 9th International Conference on Group IV Photonics (GFP), (2012), pp. 288–290.

Schmidt, S.

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophotonics 6, 821–828 (2013).
[PubMed]

Shi, W.

Y. Wang, X. Wang, J. Flueckiger, H. Yun, W. Shi, R. Bojko, N. A. F. Jaeger, and L. Chrostowski, “Focusing sub-wavelength grating couplers with low back reflections for rapid prototyping of silicon photonic circuits,” Opt. Express 22, 20652–20662 (2014).
[Crossref] [PubMed]

W. Shi, V. Veerasubramanian, D. Patel, and D. V. Plant, “Tunable nanophotonic delay lines using linearly chirped contradirectional couplers with uniform Bragg gratings,” Opt. Lett. 39, 701–703 (2014).
[Crossref] [PubMed]

X. Wang, W. Shi, H. Yun, S. Grist, N. A. Jaeger, and L. Chrostowski, “Narrow-band waveguide Bragg gratings on SOI wafers with CMOS-compatible fabrication process,” Opt. Express 20, 15547–15558 (2012).
[Crossref] [PubMed]

X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, “Uniform and sampled Bragg gratings in SOI strip waveguides with sidewall corrugations,” IEEE Photon. Technol. Lett. 23, 290–292 (2011).

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 The 9th International Conference on Group IV Photonics (GFP), (2012), pp. 288–290.

Shih, J.-F.

Silverstone, J. W.

J. W. Silverstone, D. Bonneau, J. L. O’Brien, and M. G. Thompson, “Silicon quantum photonics,” IEEE J. Sel. Topics Quantum Electron. 22, 390–402 (2016).
[Crossref]

Sima, C.

Simard, A. D.

Simbula, A.

N. C. Harris, D. Grassani, A. Simbula, M. Pant, M. Galli, T. Baehr-Jones, M. Hochberg, D. Englund, D. Bajoni, and C. Galland, “Integrated source of spectrally filtered correlated photons for large-scale quantum photonic systems,” Phys. Rev. X 4, 041047 (2014).

Skaar, J.

J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
[Crossref]

Smith, P.

Smith, P. G. R.

Sorel, M.

Strain, M. J.

Tan, D.

Tanzilli, S.

Thompson, M. G.

J. W. Silverstone, D. Bonneau, J. L. O’Brien, and M. G. Thompson, “Silicon quantum photonics,” IEEE J. Sel. Topics Quantum Electron. 22, 390–402 (2016).
[Crossref]

Thoms, S.

Troparevsky, M. C.

Vafaei, R.

X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, “Uniform and sampled Bragg gratings in SOI strip waveguides with sidewall corrugations,” IEEE Photon. Technol. Lett. 23, 290–292 (2011).

Veerasubramanian, V.

Vivien, L.

Wang, G.

Wang, L.

J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
[Crossref]

Wang, X.

M. Ma, Z. Chen, H. Yun, Y. Wang, X. Wang, N. A. F. Jaeger, and L. Chrostowski, “Apodized spiral Bragg grating waveguides in silicon-on-insulator,” IEEE Photon. Technol. Lett. 30, 111–114 (2018).
[Crossref]

Y. Wang, X. Wang, J. Flueckiger, H. Yun, W. Shi, R. Bojko, N. A. F. Jaeger, and L. Chrostowski, “Focusing sub-wavelength grating couplers with low back reflections for rapid prototyping of silicon photonic circuits,” Opt. Express 22, 20652–20662 (2014).
[Crossref] [PubMed]

X. Wang, Y. Wang, J. Flueckiger, R. Bojko, A. Liu, A. Reid, J. Pond, N. A. Jaeger, and L. Chrostowski, “Precise control of the coupling coefficient through destructive interference in silicon waveguide Bragg gratings,” Opt. Lett. 39, 5519–5522 (2014).
[Crossref] [PubMed]

M. Burla, L. R. Cortés, M. Li, X. Wang, L. Chrostowski, and J. Azaña, “Integrated waveguide Bragg gratings for microwave photonics signal processing,” Opt. Express 21, 25120–25147 (2013).
[Crossref] [PubMed]

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophotonics 6, 821–828 (2013).
[PubMed]

X. Wang, W. Shi, H. Yun, S. Grist, N. A. Jaeger, and L. Chrostowski, “Narrow-band waveguide Bragg gratings on SOI wafers with CMOS-compatible fabrication process,” Opt. Express 20, 15547–15558 (2012).
[Crossref] [PubMed]

X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, “Uniform and sampled Bragg gratings in SOI strip waveguides with sidewall corrugations,” IEEE Photon. Technol. Lett. 23, 290–292 (2011).

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 The 9th International Conference on Group IV Photonics (GFP), (2012), pp. 288–290.

Wang, Y.

Wiesmann, D.

D. Wiesmann, C. David, R. Germann, D. Emi, and G. Bona, “Apodized surface-corrugated gratings with varying duty cycles,” IEEE Photon. Technol. Lett. 12, 639–641 (2000).
[Crossref]

Yang, J.

Yariv, A.

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (The Oxford Series in Electrical and Computer Engineering) (Oxford University Press, 2006).

Yeh, P.

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (The Oxford Series in Electrical and Computer Engineering) (Oxford University Press, 2006).

Yen, T.-H.

Young, J. F.

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 The 9th International Conference on Group IV Photonics (GFP), (2012), pp. 288–290.

Yu, H.

Yu, P.

Yun, H.

Zervas, M.

Zervas, M. N.

Zhang, F.

Zhang, Z.

Appl. Opt. (2)

Frontiers of Optoelectronics (1)

S. Kaushal, R. Cheng, M. Ma, A. Mistry, M. Burla, L. Chrostowski, and J. Azaña, “Optical signal processing based on silicon photonics waveguide Bragg gratings,” Frontiers of Optoelectronics 11 (2),163–188 (2018).

IEEE J. Quantum Electron. (1)

J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
[Crossref]

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

J. W. Silverstone, D. Bonneau, J. L. O’Brien, and M. G. Thompson, “Silicon quantum photonics,” IEEE J. Sel. Topics Quantum Electron. 22, 390–402 (2016).
[Crossref]

IEEE Photon. Technol. Lett. (5)

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

X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, “Uniform and sampled Bragg gratings in SOI strip waveguides with sidewall corrugations,” IEEE Photon. Technol. Lett. 23, 290–292 (2011).

M. Ma, Z. Chen, H. Yun, Y. Wang, X. Wang, N. A. F. Jaeger, and L. Chrostowski, “Apodized spiral Bragg grating waveguides in silicon-on-insulator,” IEEE Photon. Technol. Lett. 30, 111–114 (2018).
[Crossref]

D. Wiesmann, C. David, R. Germann, D. Emi, and G. Bona, “Apodized surface-corrugated gratings with varying duty cycles,” IEEE Photon. Technol. Lett. 12, 639–641 (2000).
[Crossref]

M. J. Strain and M. Sorel, “Integrated III–V Bragg gratings for arbitrary control over chirp and coupling coefficient,” IEEE Photon. Technol. Lett. 20, 1863–1865 (2008).
[Crossref]

J. Biophotonics (1)

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophotonics 6, 821–828 (2013).
[PubMed]

J. Lightwave Technol. (1)

J. lightwave technology (1)

T. Erdogan, “Fiber grating spectra,” J. lightwave technology 15, 1277–1294 (1997).
[Crossref]

Opt. Commun. (1)

R. Oliveira, P. Neves, J. Pereira, and A. Pohl, “Numerical approach for designing a Bragg grating acousto-optic modulator using the finite element and the transfer matrix methods,” Opt. Commun. 281, 4899 – 4905 (2008).
[Crossref]

Opt. Express (6)

Opt. Lett. (14)

A. D. Simard, M. J. Strain, L. Meriggi, M. Sorel, and S. LaRochelle, “Bandpass integrated Bragg gratings in silicon-on-insulator with well-controlled amplitude and phase responses,” Opt. Lett. 40, 736–739 (2015).
[Crossref] [PubMed]

R. Cheng and L. Chrostowski, “Multichannel photonic Hilbert transformers based on complex modulated integrated Bragg gratings,” Opt. Lett. 43, 1031–1034 (2018).
[Crossref] [PubMed]

D. Tan, K. Ikeda, and Y. Fainman, “Cladding-modulated Bragg gratings in silicon waveguides,” Opt. Lett. 34, 1357–1359 (2009).
[Crossref] [PubMed]

Y.-J. Hung, Y.-C. Liang, C.-W. Huang, J.-F. Shih, S. Hu, T.-H. Yen, C.-W. Kao, and C.-H. Chen, “Narrowband silicon waveguide Bragg reflector achieved by highly ordered graphene oxide gratings,” Opt. Lett. 42, 4768–4771 (2017).
[Crossref] [PubMed]

H. Sakata, “Sidelobe suppression in grating-assisted wavelength-selective couplers,” Opt. Lett. 17, 463–465 (1992).
[Crossref] [PubMed]

W. Shi, V. Veerasubramanian, D. Patel, and D. V. Plant, “Tunable nanophotonic delay lines using linearly chirped contradirectional couplers with uniform Bragg gratings,” Opt. Lett. 39, 701–703 (2014).
[Crossref] [PubMed]

N. N. Klimov, S. Mittal, M. Berger, and Z. Ahmed, “On-chip silicon waveguide Bragg grating photonic temperature sensor,” Opt. Lett. 40, 3934–3936 (2015).
[Crossref] [PubMed]

C. Sima, J. Gates, C. Holmes, P. Mennea, M. Zervas, and P. Smith, “Terahertz bandwidth photonic Hilbert transformers based on synthesized planar Bragg grating fabrication,” Opt. Lett. 38, 3448–3451 (2013).
[Crossref] [PubMed]

C. Sima, J. C. Gates, H. L. Rogers, P. L. Mennea, C. Holmes, M. N. Zervas, and P. G. R. Smith, “Phase controlled integrated interferometric single-sideband filter based on planar Bragg gratings implementing photonic Hilbert transform,” Opt. Lett. 38, 727–729 (2013).
[Crossref] [PubMed]

D. Oser, D. Pérez-Galacho, C. Alonso-Ramos, X. L. Roux, S. Tanzilli, L. Vivien, and L. Labonté, and Éric Cassan, “Subwavelength engineering and asymmetry: two efficient tools for sub-nanometer-bandwidth silicon Bragg filters,” Opt. Lett. 43, 3208–3211 (2018).
[Crossref] [PubMed]

M. J. Strain, S. Thoms, D. S. MacIntyre, and M. Sorel, “Multi-wavelength filters in silicon using superposition sidewall Bragg grating devices,” Opt. Lett. 39, 413–416 (2014).
[Crossref] [PubMed]

S. Paul, T. Saastamoinen, S. Honkanen, M. Roussey, and M. Kuittinen, “Multi-wavelength filtering with a waveguide integrated phase-modulated Bragg grating,” Opt. Lett. 42, 4635–4638 (2017).
[Crossref] [PubMed]

X. Wang, Y. Wang, J. Flueckiger, R. Bojko, A. Liu, A. Reid, J. Pond, N. A. Jaeger, and L. Chrostowski, “Precise control of the coupling coefficient through destructive interference in silicon waveguide Bragg gratings,” Opt. Lett. 39, 5519–5522 (2014).
[Crossref] [PubMed]

H. P. Bazargani, M. Burla, L. Chrostowski, and J. Azaña, “Photonic Hilbert transformers based on laterally apodized integrated waveguide Bragg gratings on a SOI wafer,” Opt. Lett. 41, 5039–5042 (2016).
[Crossref] [PubMed]

Phys. Rev. X (1)

N. C. Harris, D. Grassani, A. Simbula, M. Pant, M. Galli, T. Baehr-Jones, M. Hochberg, D. Englund, D. Bajoni, and C. Galland, “Integrated source of spectrally filtered correlated photons for large-scale quantum photonic systems,” Phys. Rev. X 4, 041047 (2014).

Other (3)

L. Chrostowski and M. Hochberg, Silicon photonics design: from devices to systems (Cambridge University Press, 2015).
[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 The 9th International Conference on Group IV Photonics (GFP), (2012), pp. 288–290.

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (The Oxford Series in Electrical and Computer Engineering) (Oxford University Press, 2006).

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Figures (14)

Fig. 1
Fig. 1 (a) Schematic flow showing the process of the SS-TMM modeling. (b) The upper figure plots the normalized Gaussian apodization profile (blue, left axis) and the translated lateral misalignment-to-period ratio Δ L / Λ (red, right axis) along the grating; the bottom diagrams illustrate the grating structures at the different positions (1-3), whose locations are indicated in the upper plot. (c) Schematic illustrations of spatial sampling of the cell structure of a lateral misalignment-modulated IBG in different cases of (i) rectangular and (ii) sinusoidal grating shapes, where the bottom figures plot the corresponding Δ W ( k ) profiles; Δ W ( k ) is defined as the width variation of the kth segment from the unperturbed waveguide width [denoted as W in Fig. 1(c)]. (d) Illustration of the transfer matrices describing the wave propagation through an interface (left) and through a uniform section (right); the upper diagrams show the original grating structures while the bottom ones illustrate the equivalent multiplayer structures used in the SS-TMM modeling.
Fig. 2
Fig. 2 Comparison of the implementation process in conventional CMT-TMM and SS-TMM.
Fig. 3
Fig. 3 (a) Calibrated model for Δ n e f f -versus- Δ W used in the SS-TMM for IBGs developed on 500 × 220 nm SOI strip waveguides. (b) Reflection and transmission spectra of the Δ L -modulated Gaussian-apodized IBG calculated by the SS-TMM using a sampling interval of 6 nm. (c) Comparison of the calculated reflection spectra of the IBG using different sampling intervals. (d) Comparison of the calculated reflection spectra of the IBGs with different grating shapes of rectangular and sinusoidal under the same sampling interval of 6 nm. The black curves in (c)-(d) represent the ideal reflection spectrum calculated via the CMT-TMM.
Fig. 4
Fig. 4 (a) Schematic illustration of the DC modulation in an IBG (left) and the relationship between the DC and grating strength (right). (b) Normalized apodization profile (blue, left axis) and the converted DC distribution along the grating (red, right axis). (c) Spectrum of the DC-modulated Gaussian-apodized IBG calculated by the SS-TMM (blue), and the ideal spectrum (black).
Fig. 5
Fig. 5 (a) Spectrum of the designed square filter, which has been modified to be physically realizable. (b) Grating coupling coefficient κ (upper) and phase φG (bottom) profiles required by the designed spectral response, calculated via LPA. The blue curves in (c) and (d) are the SS-TMM predicted spectra of the Δ L- and DC-modulated IBGs, respectively, where the ideal spectrum (black) calculated by the CMT-TMM is also included in each figure for comparison.
Fig. 6
Fig. 6 (a) Complex spectral response of the designed photonic Hilbert transformer, which has been modified to be physically realizable. (b) Grating coupling coefficient κ (upper) and phase φG (bottom) profiles required by the design. (c) Complex spectral response of the Δ L-modulated IBG calculated by the SS-TMM.
Fig. 7
Fig. 7 Design process of an IBG with the APN compensation included; the procedures enclosed by the red dashed line are for the extraction of the APN distribution. LPA: layer peeling algorithm [31].
Fig. 8
Fig. 8 (a) Coupling coefficient (blue, left axis) and phase (red, right axis) profiles used for creating the temporary IBG structure that is dedicated for the APN extraction purpose. (b) Amplitude response of the temporary IBG calculated bythe SS-TMM. (c) Coupling coefficient (blue, left axis) and phase (red, right axis) profiles synthesized from the complex response of the temporary IBG using LPA. (d) Phase differences between neighboring periods.
Fig. 9
Fig. 9 Schematic illustrations of the correction of (a) grating period Λ and (b) distance between adjacent grating corrugations d according to different values of Δ ϕ C ( i ) in a uniform IBG.
Fig. 10
Fig. 10 (a) and (c) show the Λ and d correction profiles, respectively, for the designed square filter based on the Δ L-modulated IBG. (b) and (d) present the SS-TMM calculated spectra of the Λ- and d-corrected IBGs, respectively, with the ideal spectrum (black) included in each figure for comparison.
Fig. 11
Fig. 11 (a)-(c) are scanning electron microscope (SEM) images of the testing circuit of a grating and the structures of fabricated Δ L-modulated and DC-modulated silicon IBGs, respectively; BDC: broadband directional coupler [33].
Fig. 12
Fig. 12 (a)-(c) Experimental data of the Δ L-modulated Gaussain-apodized IBG. (a) Measured (blue) and SS-TMM predicted (red) spectra of the originally designed IBG. (b) Λ-correction profile of the IBG. (c) Measured spectrum of the Λ-corrected IBG (blue). (d)-(f) Experimental data of the DC-modulated Gaussain-apodized IBG. (d) Measured (blue) and SS-TMM predicted (red) spectra of the originally designed IBG. (e) Λ-correction profile of the IBG. (f) Measured spectrum of the Λ-corrected IBG (blue). The black curves in (a), (c), (d) and (f) are the ideal spectra for comparison.
Fig. 13
Fig. 13 Experimental data of the square filter based on the Δ L-modulated IBG. (a) Measured (blue) and SS-TMM predicted (red) spectra of the originally designed IBG. (b) Λ-correction profile of the IBG. (c) Measured spectrum of the Λ-corrected IBG (blue). The black curves in (a) and (c) are the ideal spectra for comparison.
Fig. 14
Fig. 14 (a) Measured (blue) and SS-TMM predicted (red) amplitude and phase responses of the photonic Hilbert transformer based on the original Δ L-modulated IBG. (b) Λ-correction profile of the IBG. (c) Measured complex response of the Λ-corrected IBG (blue), and the ideal response (black). (d) Building block used in the numerical analysis for achieving a single side-band filtering response. (e) Calculated transfer functions of the building block when using the experimental S parameters of the original (red) and corrected (blue) IBGs.

Tables (2)

Tables Icon

Table 1 Spectral parameters of the Gaussian-apodized IBGs

Tables Icon

Table 2 Spectral parameters of the square filters based on Δ L-modulated IBGs

Equations (12)

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

Δ L ( i ) = Λ cos 1 ( κ n ( i ) ) π
[ A 1 B 1 ] = [ T 11 T 12 T 21 T 22 ] [ A 2 B 2 ]
T k ( k + 1 ) = [ n k + n k + 1 2 n k n k + 1 n k n k + 1 2 n k n k + 1 n k n k + 1 2 n k n k + 1 n k + n k + 1 2 n k n k + 1 ]
T k = [ e β d 0 0 e β d ]
T t o t = [ T t o t 11 T t o t 12 T t o t 21 T t o t 22 ] = T 1 T 12 T 2 ... T ( N 1 ) T ( N 1 ) N
r = T t o t 21 T t o t 11
t = ( T t o t 11 ) 1
M m = [ cosh  ( s m l m ) + i Δ β 2 s m sinh  ( s m l m ) i κ m s m sinh  ( s m l m ) i κ m * s m sinh  ( s m l m ) cosh  ( s m l m ) i Δ β 2 s m sinh  ( s m l m ) ]
κ n = sin  ( π × D C )
Δ ϕ C ( i ) = Δ ϕ ( i )
ΔΛ ( i ) = Λ × Δ ϕ C ( i ) 2 π
Δ d ( i ) = Λ × Δ ϕ C ( i ) 2 π

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