Abstract

Interferometric behaviours in radiation generation process of a one-dimensional photonic-crystal (PhC) reflector are investigated on a silica waveguide. An analytical model, which can calculate total radiation losses, is presented after analysing eigen-mode properties and radiation pattern characteristics. Our model takes into account the interference of component radiated waves generated at interfaces between different waveguide sections and is verified by comparing with numerical simulations of periodic and non-periodic one-dimensional PhC reflectors. Its capability to quickly find minimum-radiation tapering geometries helps circumvent time-consuming numerical simulations. Our model negates the conventional knowledge that a gradually tapered transition from uniform waveguide to PhC waveguide yields lower radiation loss. In one example, the gradually tapered transition produces 2.3dB more radiation than the optimum geometry.

© 2012 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003).
    [CrossRef] [PubMed]
  2. H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in context,” Science 298(5597), 1372–1377 (2002).
    [CrossRef] [PubMed]
  3. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987).
    [CrossRef] [PubMed]
  4. S. T. Chu, B. E. Little, W. Pan, T. Kaneko, S. Sato, and Y. Kokubun, “An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid,” IEEE Photon. Technol. Lett. 11(6), 691–693 (1999).
    [CrossRef]
  5. A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317(5839), 783–787 (2007).
    [CrossRef] [PubMed]
  6. M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459(7246), 550–555 (2009).
    [CrossRef] [PubMed]
  7. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
    [CrossRef] [PubMed]
  8. S. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305(5681), 227–229 (2004).
    [CrossRef] [PubMed]
  9. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-Gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
    [CrossRef] [PubMed]
  10. P. R. Villeneuve, J. S. Foresi, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390(6656), 143–145 (1997).
    [CrossRef]
  11. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
    [CrossRef] [PubMed]
  12. B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005).
    [CrossRef]
  13. P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Loncar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009).
    [CrossRef]
  14. M. Notomi, E. Kuramochi, and H. Taniyama, “Ultrahigh-Q nanocavity with 1D photonic gap,” Opt. Express 16(15), 11095–11102 (2008).
    [CrossRef] [PubMed]
  15. M. W. McCutcheon and M. Loncar, “Design of a silicon nitride photonic crystal nanocavity with a Quality factor of one million for coupling to a diamond nanocrystal,” Opt. Express 16(23), 19136–19145 (2008).
    [CrossRef] [PubMed]
  16. A. R. Md Zain, N. P. Johnson, M. Sorel, and R. M. De La Rue, “Ultra high quality factor one dimensional photonic crystal/photonic wire micro-cavities in silicon-on-insulator (SOI),” Opt. Express 16(16), 12084–12089 (2008).
    [CrossRef] [PubMed]
  17. Q. Quan, P. B. Deotare, and M. Loncar, “Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide,” Appl. Phys. Lett. 96(20), 203102 (2010).
    [CrossRef]
  18. Y. Gong and J. Vu?kovi?, “Photonic crystal cavities in silicon dioxide,” Appl. Phys. Lett. 96(3), 031107 (2010).
    [CrossRef]
  19. Q. Quan, I. B. Burgess, S. K. Y. Tang, D. L. Floyd, and M. Loncar, “High-Q, low index-contrast polymeric photonic crystal nanobeam cavities,” Opt. Express 19(22), 22191–22197 (2011).
    [CrossRef] [PubMed]
  20. J. Goh, I. Fushman, D. Englund, and J. Vu?kovi? “Genetic optimization of photonic bandgap structures,” Opt. Express 15(13), 8218 (2007).
    [CrossRef] [PubMed]
  21. O. Dorn and D. Lesselier, “Level set methods for inverse scattering,” Inverse Probl. 22(4), R67–R131 (2006).
    [CrossRef]
  22. S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78(22), 3388–3390 (2001).
    [CrossRef]
  23. K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express 10(15), 670–684 (2002).
    [CrossRef] [PubMed]
  24. P. Lalanne and J. P. Hugonin, “Bloch-wave engineering for high-Q, small-V microcavities,” IEEE J. Quantum Electron. 39(11), 1430–1438 (2003).
    [CrossRef]
  25. K. O. Hill, Y. Fujii, D. C. Johnson, and B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett. 32(10), 647–649 (1978).
    [CrossRef]
  26. A. W. Snyder and J. E. Love, Optical Waveguide Theory (Chapman and Hall, 1983).
  27. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Second Edition) (Princeton University Press, 2008).
  28. Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4), 046607 (2003).
    [CrossRef] [PubMed]
  29. JrP. G. Suchoski and V. Ramaswamy, “Exact numerical technique for the analysis of step discontinuities and tapers in optical dielectric waveguides,” J. Opt. Soc. Am. A 3(2), 194–203 (1986).
  30. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
    [CrossRef]
  31. A. Yariv, Optical Electronics in Modern Communications (Oxford University Press, 1997).
  32. A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
    [CrossRef] [PubMed]
  33. P. Lalanne, S. Mias, and J. P. Hugonin, “Two physical mechanisms for boosting the quality factor to cavity volume ratio of photonic crystal microcavities,” Opt. Express 12(3), 458–467 (2004).
    [CrossRef] [PubMed]

2011 (1)

2010 (2)

Q. Quan, P. B. Deotare, and M. Loncar, “Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide,” Appl. Phys. Lett. 96(20), 203102 (2010).
[CrossRef]

Y. Gong and J. Vu?kovi?, “Photonic crystal cavities in silicon dioxide,” Appl. Phys. Lett. 96(3), 031107 (2010).
[CrossRef]

2009 (2)

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Loncar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009).
[CrossRef]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459(7246), 550–555 (2009).
[CrossRef] [PubMed]

2008 (3)

2007 (2)

J. Goh, I. Fushman, D. Englund, and J. Vu?kovi? “Genetic optimization of photonic bandgap structures,” Opt. Express 15(13), 8218 (2007).
[CrossRef] [PubMed]

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317(5839), 783–787 (2007).
[CrossRef] [PubMed]

2006 (1)

O. Dorn and D. Lesselier, “Level set methods for inverse scattering,” Inverse Probl. 22(4), R67–R131 (2006).
[CrossRef]

2005 (1)

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005).
[CrossRef]

2004 (2)

2003 (5)

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
[CrossRef] [PubMed]

K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003).
[CrossRef] [PubMed]

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
[CrossRef] [PubMed]

Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4), 046607 (2003).
[CrossRef] [PubMed]

P. Lalanne and J. P. Hugonin, “Bloch-wave engineering for high-Q, small-V microcavities,” IEEE J. Quantum Electron. 39(11), 1430–1438 (2003).
[CrossRef]

2002 (2)

H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in context,” Science 298(5597), 1372–1377 (2002).
[CrossRef] [PubMed]

K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express 10(15), 670–684 (2002).
[CrossRef] [PubMed]

2001 (1)

S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78(22), 3388–3390 (2001).
[CrossRef]

1999 (2)

S. T. Chu, B. E. Little, W. Pan, T. Kaneko, S. Sato, and Y. Kokubun, “An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid,” IEEE Photon. Technol. Lett. 11(6), 691–693 (1999).
[CrossRef]

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-Gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
[CrossRef] [PubMed]

1997 (2)

P. R. Villeneuve, J. S. Foresi, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390(6656), 143–145 (1997).
[CrossRef]

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[CrossRef]

1996 (1)

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[CrossRef] [PubMed]

1987 (1)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987).
[CrossRef] [PubMed]

1986 (1)

1978 (1)

K. O. Hill, Y. Fujii, D. C. Johnson, and B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett. 32(10), 647–649 (1978).
[CrossRef]

Akahane, Y.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005).
[CrossRef]

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
[CrossRef] [PubMed]

Armani, A. M.

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317(5839), 783–787 (2007).
[CrossRef] [PubMed]

Armani, D. K.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
[CrossRef] [PubMed]

Asano, T.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005).
[CrossRef]

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
[CrossRef] [PubMed]

Burgess, I. B.

Camacho, R.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459(7246), 550–555 (2009).
[CrossRef] [PubMed]

Chan, J.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459(7246), 550–555 (2009).
[CrossRef] [PubMed]

Chen, J. C.

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[CrossRef] [PubMed]

Chu, S. T.

S. T. Chu, B. E. Little, W. Pan, T. Kaneko, S. Sato, and Y. Kokubun, “An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid,” IEEE Photon. Technol. Lett. 11(6), 691–693 (1999).
[CrossRef]

Dapkus, P. D.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-Gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
[CrossRef] [PubMed]

De La Rue, R. M.

Deotare, P. B.

Q. Quan, P. B. Deotare, and M. Loncar, “Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide,” Appl. Phys. Lett. 96(20), 203102 (2010).
[CrossRef]

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Loncar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009).
[CrossRef]

Doherty, A. C.

H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in context,” Science 298(5597), 1372–1377 (2002).
[CrossRef] [PubMed]

Dorn, O.

O. Dorn and D. Lesselier, “Level set methods for inverse scattering,” Inverse Probl. 22(4), R67–R131 (2006).
[CrossRef]

Eichenfield, M.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459(7246), 550–555 (2009).
[CrossRef] [PubMed]

Englund, D.

Erdogan, T.

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[CrossRef]

Fan, S.

S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78(22), 3388–3390 (2001).
[CrossRef]

P. R. Villeneuve, J. S. Foresi, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390(6656), 143–145 (1997).
[CrossRef]

Fan, S. H.

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[CrossRef] [PubMed]

Ferrera, J.

P. R. Villeneuve, J. S. Foresi, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390(6656), 143–145 (1997).
[CrossRef]

Flagan, R. C.

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317(5839), 783–787 (2007).
[CrossRef] [PubMed]

Floyd, D. L.

Foresi, J. S.

P. R. Villeneuve, J. S. Foresi, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390(6656), 143–145 (1997).
[CrossRef]

Frank, I. W.

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Loncar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009).
[CrossRef]

Fraser, S. E.

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317(5839), 783–787 (2007).
[CrossRef] [PubMed]

Fujii, Y.

K. O. Hill, Y. Fujii, D. C. Johnson, and B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett. 32(10), 647–649 (1978).
[CrossRef]

Fushman, I.

Goh, J.

Gong, Y.

Y. Gong and J. Vu?kovi?, “Photonic crystal cavities in silicon dioxide,” Appl. Phys. Lett. 96(3), 031107 (2010).
[CrossRef]

Hill, K. O.

K. O. Hill, Y. Fujii, D. C. Johnson, and B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett. 32(10), 647–649 (1978).
[CrossRef]

Hugonin, J. P.

Imada, M.

S. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305(5681), 227–229 (2004).
[CrossRef] [PubMed]

Ippen, E. P.

P. R. Villeneuve, J. S. Foresi, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390(6656), 143–145 (1997).
[CrossRef]

Joannopoulos, J. D.

S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78(22), 3388–3390 (2001).
[CrossRef]

P. R. Villeneuve, J. S. Foresi, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390(6656), 143–145 (1997).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[CrossRef] [PubMed]

Johnson, D. C.

K. O. Hill, Y. Fujii, D. C. Johnson, and B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett. 32(10), 647–649 (1978).
[CrossRef]

Johnson, N. P.

Johnson, S. G.

S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78(22), 3388–3390 (2001).
[CrossRef]

Kaneko, T.

S. T. Chu, B. E. Little, W. Pan, T. Kaneko, S. Sato, and Y. Kokubun, “An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid,” IEEE Photon. Technol. Lett. 11(6), 691–693 (1999).
[CrossRef]

Kawasaki, B. S.

K. O. Hill, Y. Fujii, D. C. Johnson, and B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett. 32(10), 647–649 (1978).
[CrossRef]

Khan, M.

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Loncar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009).
[CrossRef]

Kim, I.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-Gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
[CrossRef] [PubMed]

Kimerling, L. C.

P. R. Villeneuve, J. S. Foresi, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390(6656), 143–145 (1997).
[CrossRef]

Kippenberg, T. J.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
[CrossRef] [PubMed]

Kokubun, Y.

S. T. Chu, B. E. Little, W. Pan, T. Kaneko, S. Sato, and Y. Kokubun, “An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid,” IEEE Photon. Technol. Lett. 11(6), 691–693 (1999).
[CrossRef]

Kulkarni, R. P.

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317(5839), 783–787 (2007).
[CrossRef] [PubMed]

Kuramochi, E.

Kurland, I.

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[CrossRef] [PubMed]

Lalanne, P.

Lee, R. K.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-Gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
[CrossRef] [PubMed]

Lesselier, D.

O. Dorn and D. Lesselier, “Level set methods for inverse scattering,” Inverse Probl. 22(4), R67–R131 (2006).
[CrossRef]

Li, Z. Y.

Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4), 046607 (2003).
[CrossRef] [PubMed]

Lin, L. L.

Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4), 046607 (2003).
[CrossRef] [PubMed]

Little, B. E.

S. T. Chu, B. E. Little, W. Pan, T. Kaneko, S. Sato, and Y. Kokubun, “An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid,” IEEE Photon. Technol. Lett. 11(6), 691–693 (1999).
[CrossRef]

Loncar, M.

Q. Quan, I. B. Burgess, S. K. Y. Tang, D. L. Floyd, and M. Loncar, “High-Q, low index-contrast polymeric photonic crystal nanobeam cavities,” Opt. Express 19(22), 22191–22197 (2011).
[CrossRef] [PubMed]

Q. Quan, P. B. Deotare, and M. Loncar, “Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide,” Appl. Phys. Lett. 96(20), 203102 (2010).
[CrossRef]

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Loncar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009).
[CrossRef]

M. W. McCutcheon and M. Loncar, “Design of a silicon nitride photonic crystal nanocavity with a Quality factor of one million for coupling to a diamond nanocrystal,” Opt. Express 16(23), 19136–19145 (2008).
[CrossRef] [PubMed]

Mabuchi, H.

H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in context,” Science 298(5597), 1372–1377 (2002).
[CrossRef] [PubMed]

McCutcheon, M. W.

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Loncar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009).
[CrossRef]

M. W. McCutcheon and M. Loncar, “Design of a silicon nitride photonic crystal nanocavity with a Quality factor of one million for coupling to a diamond nanocrystal,” Opt. Express 16(23), 19136–19145 (2008).
[CrossRef] [PubMed]

Md Zain, A. R.

Mekis, A.

S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78(22), 3388–3390 (2001).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[CrossRef] [PubMed]

Mias, S.

Noda, S.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005).
[CrossRef]

S. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305(5681), 227–229 (2004).
[CrossRef] [PubMed]

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
[CrossRef] [PubMed]

Notomi, M.

O’Brien, J. D.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-Gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
[CrossRef] [PubMed]

Ogawa, S.

S. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305(5681), 227–229 (2004).
[CrossRef] [PubMed]

Okano, M.

S. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305(5681), 227–229 (2004).
[CrossRef] [PubMed]

Painter, O.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459(7246), 550–555 (2009).
[CrossRef] [PubMed]

K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express 10(15), 670–684 (2002).
[CrossRef] [PubMed]

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-Gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
[CrossRef] [PubMed]

Pan, W.

S. T. Chu, B. E. Little, W. Pan, T. Kaneko, S. Sato, and Y. Kokubun, “An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid,” IEEE Photon. Technol. Lett. 11(6), 691–693 (1999).
[CrossRef]

Quan, Q.

Q. Quan, I. B. Burgess, S. K. Y. Tang, D. L. Floyd, and M. Loncar, “High-Q, low index-contrast polymeric photonic crystal nanobeam cavities,” Opt. Express 19(22), 22191–22197 (2011).
[CrossRef] [PubMed]

Q. Quan, P. B. Deotare, and M. Loncar, “Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide,” Appl. Phys. Lett. 96(20), 203102 (2010).
[CrossRef]

Ramaswamy, V.

Sato, S.

S. T. Chu, B. E. Little, W. Pan, T. Kaneko, S. Sato, and Y. Kokubun, “An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid,” IEEE Photon. Technol. Lett. 11(6), 691–693 (1999).
[CrossRef]

Scherer, A.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-Gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
[CrossRef] [PubMed]

Smith, H. I.

P. R. Villeneuve, J. S. Foresi, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390(6656), 143–145 (1997).
[CrossRef]

Song, B. S.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005).
[CrossRef]

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
[CrossRef] [PubMed]

Sorel, M.

Spillane, S. M.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
[CrossRef] [PubMed]

Srinivasan, K.

Steinmeyer, G.

P. R. Villeneuve, J. S. Foresi, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390(6656), 143–145 (1997).
[CrossRef]

Suchoski, P. G.

Tang, S. K. Y.

Taniyama, H.

Thoen, E. R.

P. R. Villeneuve, J. S. Foresi, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390(6656), 143–145 (1997).
[CrossRef]

Vahala, K. J.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459(7246), 550–555 (2009).
[CrossRef] [PubMed]

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317(5839), 783–787 (2007).
[CrossRef] [PubMed]

K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003).
[CrossRef] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
[CrossRef] [PubMed]

Villeneuve, P. R.

P. R. Villeneuve, J. S. Foresi, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390(6656), 143–145 (1997).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[CrossRef] [PubMed]

Vuckovic, J.

Y. Gong and J. Vu?kovi?, “Photonic crystal cavities in silicon dioxide,” Appl. Phys. Lett. 96(3), 031107 (2010).
[CrossRef]

J. Goh, I. Fushman, D. Englund, and J. Vu?kovi? “Genetic optimization of photonic bandgap structures,” Opt. Express 15(13), 8218 (2007).
[CrossRef] [PubMed]

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987).
[CrossRef] [PubMed]

Yariv, A.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-Gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
[CrossRef] [PubMed]

Yoshimoto, S.

S. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305(5681), 227–229 (2004).
[CrossRef] [PubMed]

Appl. Phys. Lett. (5)

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Loncar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009).
[CrossRef]

Q. Quan, P. B. Deotare, and M. Loncar, “Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide,” Appl. Phys. Lett. 96(20), 203102 (2010).
[CrossRef]

Y. Gong and J. Vu?kovi?, “Photonic crystal cavities in silicon dioxide,” Appl. Phys. Lett. 96(3), 031107 (2010).
[CrossRef]

S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78(22), 3388–3390 (2001).
[CrossRef]

K. O. Hill, Y. Fujii, D. C. Johnson, and B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett. 32(10), 647–649 (1978).
[CrossRef]

IEEE J. Quantum Electron. (1)

P. Lalanne and J. P. Hugonin, “Bloch-wave engineering for high-Q, small-V microcavities,” IEEE J. Quantum Electron. 39(11), 1430–1438 (2003).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

S. T. Chu, B. E. Little, W. Pan, T. Kaneko, S. Sato, and Y. Kokubun, “An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid,” IEEE Photon. Technol. Lett. 11(6), 691–693 (1999).
[CrossRef]

Inverse Probl. (1)

O. Dorn and D. Lesselier, “Level set methods for inverse scattering,” Inverse Probl. 22(4), R67–R131 (2006).
[CrossRef]

J. Lightwave Technol. (1)

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[CrossRef]

J. Opt. Soc. Am. A (1)

Nat. Mater. (1)

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005).
[CrossRef]

Nature (5)

P. R. Villeneuve, J. S. Foresi, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390(6656), 143–145 (1997).
[CrossRef]

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
[CrossRef] [PubMed]

K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003).
[CrossRef] [PubMed]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459(7246), 550–555 (2009).
[CrossRef] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
[CrossRef] [PubMed]

Opt. Express (7)

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4), 046607 (2003).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987).
[CrossRef] [PubMed]

Science (4)

S. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305(5681), 227–229 (2004).
[CrossRef] [PubMed]

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-Gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
[CrossRef] [PubMed]

H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in context,” Science 298(5597), 1372–1377 (2002).
[CrossRef] [PubMed]

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317(5839), 783–787 (2007).
[CrossRef] [PubMed]

Other (3)

A. Yariv, Optical Electronics in Modern Communications (Oxford University Press, 1997).

A. W. Snyder and J. E. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Second Edition) (Princeton University Press, 2008).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Schematic of a PhC cavity which is surrounded by two PhC reflectors. The presence of the PhC reflectors not only leads to light recirculation inside the cavity but also results in light radiation out of the waveguide.

Fig. 2
Fig. 2

Photonic band structures of (a) a silica/air quarter wave plate stack and (b) a silicon/air quarter wave plate stack. The real [Re(k)] and the imaginary (α) parts of the wave vectors are plotted in different colors (black and red respectively). The gray shadow areas represent the PBGs. The inserts are the schematics of the wave plate stacks. nSilica = 1.46, nSilicon = 3.46.

Fig. 3
Fig. 3

(a) Schematic of a 1D PhC reflector in a silica slab waveguide. The geometrical parameters, the coordinate, and the polarization direction are defined in the picture. (b) Wave vector diagram of a periodic PhC segment with w = 1.3μm, t = 0.4μm, and Λ = 0.664μm. The gray area is the air light cone, and the dashed line represents the working wavelength. (c) The upper (red squares) and the lower (blue squares) edges of the PBGs as a function of the parameters w and t. The values of w are 0.1/0.3/0.5/0.7/0.9/1.1/1.3μm respectively.

Fig. 4
Fig. 4

Schematics of two 9-period PhC reflectors having (a) prolate (1.3μm × 0.2μm) and (b) oblate (0.7μm × 0.4μm) elliptical holes respectively. These two PhC reflectors have the same PBG width as the black circles in Fig. 3(c) indicate. The red arrows represent the input and the reflected light rays. (c) Simulated transmission (blue) and reflection (green) curves of the PhC reflector in (a). The black and the red curves are the derived radiation losses of the two PhC reflectors. The vertical gray line is the working wavelength.

Fig. 5
Fig. 5

(a) Simulated and (b) calculated total radiation losses of periodic 1D PhC reflectors as a function of the number of periods. w = 1.3μm, t = 0.05/0.1/0.15/0.2/0.25/0.3/0.35/0.4μm, and λ = 1.55μm. The y-axes in (a) and (b) are both logarithmic and span 22.2dB. (c) Simulated radiation losses at three different working wavelengths for the 1D PhC reflector with w = 1.3μm and t = 0.1μm. The gray vertical lines in all the plots denote the periods of the oscillations.

Fig. 6
Fig. 6

Transverse coupling coefficients from the guided wave to the forward-radiated waves in a slab silica waveguide (2a = 1.4μm). The coupling is caused by an infinitely thin elliptical hole with w = 1.3μm. The inset shows the schematic.

Fig. 7
Fig. 7

Schematics of radiation generation processes occurring in a periodic 1D PhC reflector.

Fig. 8
Fig. 8

FDTD simulated radiation pattern of a 1D PhC reflector with w = 1.3μm, t = 0.1μm, and N = 4. The inset shows the schematic.

Fig. 9
Fig. 9

(a) Calculated (solid line) and simulated (square symbols) radiation losses of 11-layer 1D PhC reflectors with the variation of t1. The other PhC segments have t = 0.4μm. The minimums of the two curves are labeled by the vertical red lines. (b) Calculated and (c) Simulated radiation losses as a function of ( t 1 , t 2 )[0,0.4]μm×[0,0.4]μm . Same color bars are used in (b) and (c). The non-periodic PhC reflector, the five uniform PhC waveguide sections, and the definitions of t 1,2 are denoted in the schematic. The white circles in (b) and (c) label the minimum-radiation points. The red crosses represent the progressly-tapered point.

Equations (9)

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

K t = ω 4 w 2 w 2 Δε(x) e x (G) (x) [ e x (R) (x)] * dx ,
{ d A f dz =κ A b d A b dz = κ * A f ,
{ A f (z)= A f (0)| κ |cosh[| κ |(zL)]+ A b (L) κ * sinh(| κ |z) | κ |cosh(| κ |L) A b (z)= A f (0)κsinh[| κ |(zL)]+ A b (L)| κ |cosh(| κ |z) | κ |cosh(| κ |L) ,
[ A f (z) A b (z) ]=[ cosh(αz) sinh(αz) sinh(αz) cosh(αz) ][ A f (0) A b (0) ]=M[ A f (0) A b (0) ],
H y (x) e i(βzωt) h y (x)={ Acos( k 1 a) e k 2 (xa) , x>a Acos( k 1 x), a<x<a Acos( k 1 a) e k 2 (x+a) , x<a , e x (x)= β ω ε 0 ε(x) h y (x), e z (x)= i ω ε 0 ε(x) d h y (x) dx .
h y (x)={ B{cos( k 1 'a)cos[ k 2 '(xa)] k 1 ' ε 2 k 2 ' ε 1 sin( k 1 'a)sin[ k 2 '(xa)]}, x>a Bcos( k 1 'x), a<x<a B{cos( k 1 'a)cos[ k 2 '(x+a)]+ k 1 ' ε 2 k 2 ' ε 1 sin( k 1 'a)sin[ k 2 '(x+a)]}, x<a , e x (x)= β' ω ε 0 ε(x) h y (x), e z (x)= i ω ε 0 ε(x) d h y (x) dx .
β | A | 2 ω ε 0 [ a 2 ε 1 + sin(2 k 1 a) 4 ε 1 k 1 + 1+cos(2 k 1 a) 4 ε 2 k 2 ]=1,
β' | B | 2 ω ε 0 ε 2 [ cos 2 ( k 1 'a)+ ( k 1 ' ε 2 k 2 ' ε 1 ) 2 sin 2 ( k 1 'a)]= 2 π .
K t = Δεββ'A B * 4ω ε 0 2 ε 1 2 { sin[ ( k 1 + k 1 ')w /2 ] k 1 + k 1 ' + sin[ ( k 1 k 1 ')w /2 ] k 1 k 1 ' }.

Metrics