Abstract

The potential for on-chip optical amplification at λs=1530nm in silicon photonics using erbium-doped rich polymers integrated in silicon-slot waveguides is investigated by using a four-level spectroscopic model and a pumping wavelength of 1.48 μm. It is shown that the key parameter driving the whole amplification efficiency is the slot waveguide linear loss level, while optimization of the hollow core waveguide cross section leads to a slot width around 130 nm. Our investigations show that an on-chip optical gain of about 10 dB can be obtained with 7dB/cm loss slot waveguides. Such a propagation loss is directly sustained by experimental results obtained for photonic structures fabricated using silicon technology. Due to the likelihood of improved technological processes in the near future, the slot waveguide loss level was swept in the 310dB/cm range, showing that on-chip optical gain up to 30 dB can be seriously envisaged for slot waveguides with optimized optical losses around 3dB/cm.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. Vivien and L. Pavesi, Handbook of Silicon Photonics, Series in Optics and Optoelectronics (Taylor & Francis, 2013).
  2. J. E. Bowers, H. Park, Y.-H. Kuo, A. W. Fang, R. Jones, M. J. Paniccia, O. Cohen, and O. Raday, “Integrated optical amplifiers on silicon waveguides,” IPNRA Conference, Hybrid Integration Symposium, Salt Lake City, Utah, 2007.
  3. H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature 433, 292–294 (2005).
    [CrossRef]
  4. H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
    [CrossRef]
  5. M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
    [CrossRef]
  6. A. J. Kenyon, “Recent developments in rare-earth doped materials for optoelectronics,” Prog. Quantum Electron. 26, 225–284 (2002).
    [CrossRef]
  7. C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express 15, 5976–5980 (2007).
    [CrossRef]
  8. J. D. B. Bradley, M. Costa e Silva, M. Gay, L. Bramerie, A. Driessen, K. Wörhoff, J. C. Simon, and M. Pollnau, “170  Gbits/s transmission in an erbium-doped waveguide amplifier,” Opt. Express 17, 22201–22208 (2009).
    [CrossRef]
  9. J. D. B. Bradley, L. Agazzi, D. Geskus, F. Ay, K. Wörhoff, and M. Pollnau, “Gain bandwidth of 80  nm and 2  dB/cm peak gain in Al2O3:Er3+ optical amplifiers on silicon,” J. Opt. Soc. Am. B 27, 187–196 (2010).
    [CrossRef]
  10. J. D. B. Bradley, R. Stoffer, A. Bakker, L. Agazzi, F. Ay, K. Worhoff, and M. Pollnau, “Integrated Al2O3:Er3+ zero-loss optical amplifier and power splitter with 40  nm bandwidth,” IEEE Photon. Technol. Lett. 22, 278–280 (2010).
    [CrossRef]
  11. V. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29, 1209–1211 (2004).
    [CrossRef]
  12. C. A. Barrios and M. Lipson, “Electrically driven silicon resonant light emitting device based on slot-waveguide,” Opt. Express 13, 10092–10101 (2005).
    [CrossRef]
  13. C. Chen, D. Zhang, T. Li, D. Zhang, L. Song, and Z. Zhen, “Erbium-ytterbium codoped waveguide amplifier fabricated with solution-processable complex,” Appl. Phys. Lett. 94, 041119 (2009).
    [CrossRef]
  14. A. Spott, T. Baehr-Jones, R. Ding, Y. Liu, R. Bojko, T. O’Malley, A. Pomerene, C. Hill, W. Reinhardt, and M. Hochberg, “Photolithographically fabricated low-loss asymmetric silicon slot waveguides,” Opt. Express 19, 10950–10958 (2011).
    [CrossRef]
  15. D. Zhang, C. Chen, F. Wang, and D. M. Zhang, “Optical gain and upconversion luminescence in LaF3:Er, Yb nanoparticles-doped organic–inorganic hybrid materials waveguide amplifiers,” Appl. Phys. B 98, 791–795 (2010).
    [CrossRef]
  16. A. E. Siegman, Lasers (University Science Books, 1986).
  17. T. Baehr-Jones, M. Hochberg, C. Walker, and A. Scherer, “High-Q optical resonators in silicon-on-insulator-based slot waveguides,” Appl. Phys. Lett. 86, 081101 (2005).
    [CrossRef]
  18. H. Sun, A. Chen, D. Abeysinghe, A. Szep, and R. S. Kim, “Reduction of scattering loss of silicon slot waveguides by RCA smoothing,” Opt. Lett. 37, 13–15 (2012).
    [CrossRef]
  19. R. Ding, T. Baehr-Jones, W. J. Kim, X. Xiong, R. Bojko, J. M. Fedeli, M. Fournier, and M. Hochberg, “Low-loss strip-loaded slot waveguides in silicon-on-insulator,” Opt. Express 18, 25061–25067 (2010).
    [CrossRef]
  20. T. Alasaarela, D. Korn, L. Alloatti, A. Säynätjoki, A. Tervonen, R. Palmer, J. Leuthold, W. Freude, and S. Honkanen, “Reduced propagation loss in silicon strip and slot waveguides coated by atomic layer deposition,” Opt. Express 19, 11529–11538 (2011).
    [CrossRef]
  21. A. Säynätjoki, L. Karvonen, T. Alasaarela, X. Tu, T. Liow, M. Hiltunen, A. Tervonen, G. Lo, and S. Honkanen, “Low-loss silicon slot waveguides and couplers fabricated with optical lithography and atomic layer deposition,” Opt. Express 19, 26275–26282 (2011).
    [CrossRef]
  22. Z. Wang, N. Zhu, Y. Tang, L. Wosinski, D. Dai, and S. He, “Ultacompact low-loss coupler between strip and slot waveguides,” Opt. Lett. 34, 1498–1500 (2009).
    [CrossRef]
  23. R. Guo, B. Wang, X. Wang, L. Wang, L. Jiang, and Z. Zhou, “Optical amplification in Er/Yb silicate slot waveguide,” Opt. Lett. 37, 1427–1429 (2012).
    [CrossRef]
  24. S. K. Selvaraja, P. De Heyn, G. Winroth, P. Ong, G. Lepage, C. Cailler, A. Rigny, K. Bourdelle, W. Bogaerts, D. VanThourhout, J. Van Campenhout, and P. Absil, “Highly uniform and low-loss passive silicon photonics devices using a 300  mm CMOS platform,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper Th2A.33.
  25. P. A. Anderson, B. S. Schmidt, and M. Lipson, “High confinement in silicon slot waveguides with sharp bends,” Opt. Express 14, 9197–9202 (2006).
    [CrossRef]
  26. J. H. Ferziger, Numerical Methods for Engineering Application (Wiley-Interscience, 1998).
  27. L. Cognolato, C. S. De Bernardi, M. Ferraris, A. Gnazzo, S. Morasca, and D. Scarano, “Spectroscopic properties of Er3+-doped glasses for the realization of active waveguides by ion-exchange technique,” CSELT Technical Reports XIX, 277–281 (1991).
  28. W. J. Miniscalco, “Erbium-doped glasses for fiber amplifiers at 1500 nm,” J. Lightwave Technol. 9, 234–250 (1991), (soda-lime silicate glass).
    [CrossRef]
  29. G. N. Van den Hoven, R. J. I. M. Koper, A. Polman, C. van Dam, J. W. M. van Uffelen, and M. K. Smit, “Net optical gain at 1.53  μm in Er-doped Al2O3 waveguides on silicon,” Appl. Phys. Lett. 68, 1886–1888 (1996).
    [CrossRef]
  30. J. D. B. Bradley and M. Pollnau, “Erbium-doped integrated waveguide amplifiers and lasers,” Laser Photon. Rev. 5, 368–403 (2011).
    [CrossRef]

2012

2011

2010

J. D. B. Bradley, R. Stoffer, A. Bakker, L. Agazzi, F. Ay, K. Worhoff, and M. Pollnau, “Integrated Al2O3:Er3+ zero-loss optical amplifier and power splitter with 40  nm bandwidth,” IEEE Photon. Technol. Lett. 22, 278–280 (2010).
[CrossRef]

D. Zhang, C. Chen, F. Wang, and D. M. Zhang, “Optical gain and upconversion luminescence in LaF3:Er, Yb nanoparticles-doped organic–inorganic hybrid materials waveguide amplifiers,” Appl. Phys. B 98, 791–795 (2010).
[CrossRef]

J. D. B. Bradley, L. Agazzi, D. Geskus, F. Ay, K. Wörhoff, and M. Pollnau, “Gain bandwidth of 80  nm and 2  dB/cm peak gain in Al2O3:Er3+ optical amplifiers on silicon,” J. Opt. Soc. Am. B 27, 187–196 (2010).
[CrossRef]

R. Ding, T. Baehr-Jones, W. J. Kim, X. Xiong, R. Bojko, J. M. Fedeli, M. Fournier, and M. Hochberg, “Low-loss strip-loaded slot waveguides in silicon-on-insulator,” Opt. Express 18, 25061–25067 (2010).
[CrossRef]

2009

2007

2006

P. A. Anderson, B. S. Schmidt, and M. Lipson, “High confinement in silicon slot waveguides with sharp bends,” Opt. Express 14, 9197–9202 (2006).
[CrossRef]

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[CrossRef]

2005

H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature 433, 292–294 (2005).
[CrossRef]

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
[CrossRef]

T. Baehr-Jones, M. Hochberg, C. Walker, and A. Scherer, “High-Q optical resonators in silicon-on-insulator-based slot waveguides,” Appl. Phys. Lett. 86, 081101 (2005).
[CrossRef]

C. A. Barrios and M. Lipson, “Electrically driven silicon resonant light emitting device based on slot-waveguide,” Opt. Express 13, 10092–10101 (2005).
[CrossRef]

2004

2002

A. J. Kenyon, “Recent developments in rare-earth doped materials for optoelectronics,” Prog. Quantum Electron. 26, 225–284 (2002).
[CrossRef]

1996

G. N. Van den Hoven, R. J. I. M. Koper, A. Polman, C. van Dam, J. W. M. van Uffelen, and M. K. Smit, “Net optical gain at 1.53  μm in Er-doped Al2O3 waveguides on silicon,” Appl. Phys. Lett. 68, 1886–1888 (1996).
[CrossRef]

1991

L. Cognolato, C. S. De Bernardi, M. Ferraris, A. Gnazzo, S. Morasca, and D. Scarano, “Spectroscopic properties of Er3+-doped glasses for the realization of active waveguides by ion-exchange technique,” CSELT Technical Reports XIX, 277–281 (1991).

W. J. Miniscalco, “Erbium-doped glasses for fiber amplifiers at 1500 nm,” J. Lightwave Technol. 9, 234–250 (1991), (soda-lime silicate glass).
[CrossRef]

Abeysinghe, D.

Absil, P.

S. K. Selvaraja, P. De Heyn, G. Winroth, P. Ong, G. Lepage, C. Cailler, A. Rigny, K. Bourdelle, W. Bogaerts, D. VanThourhout, J. Van Campenhout, and P. Absil, “Highly uniform and low-loss passive silicon photonics devices using a 300  mm CMOS platform,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper Th2A.33.

Agazzi, L.

J. D. B. Bradley, L. Agazzi, D. Geskus, F. Ay, K. Wörhoff, and M. Pollnau, “Gain bandwidth of 80  nm and 2  dB/cm peak gain in Al2O3:Er3+ optical amplifiers on silicon,” J. Opt. Soc. Am. B 27, 187–196 (2010).
[CrossRef]

J. D. B. Bradley, R. Stoffer, A. Bakker, L. Agazzi, F. Ay, K. Worhoff, and M. Pollnau, “Integrated Al2O3:Er3+ zero-loss optical amplifier and power splitter with 40  nm bandwidth,” IEEE Photon. Technol. Lett. 22, 278–280 (2010).
[CrossRef]

Alasaarela, T.

Alloatti, L.

Almeida, V.

Anderson, P. A.

Ay, F.

J. D. B. Bradley, R. Stoffer, A. Bakker, L. Agazzi, F. Ay, K. Worhoff, and M. Pollnau, “Integrated Al2O3:Er3+ zero-loss optical amplifier and power splitter with 40  nm bandwidth,” IEEE Photon. Technol. Lett. 22, 278–280 (2010).
[CrossRef]

J. D. B. Bradley, L. Agazzi, D. Geskus, F. Ay, K. Wörhoff, and M. Pollnau, “Gain bandwidth of 80  nm and 2  dB/cm peak gain in Al2O3:Er3+ optical amplifiers on silicon,” J. Opt. Soc. Am. B 27, 187–196 (2010).
[CrossRef]

Baehr-Jones, T.

Bakker, A.

J. D. B. Bradley, R. Stoffer, A. Bakker, L. Agazzi, F. Ay, K. Worhoff, and M. Pollnau, “Integrated Al2O3:Er3+ zero-loss optical amplifier and power splitter with 40  nm bandwidth,” IEEE Photon. Technol. Lett. 22, 278–280 (2010).
[CrossRef]

Barrios, C. A.

Bogaerts, W.

S. K. Selvaraja, P. De Heyn, G. Winroth, P. Ong, G. Lepage, C. Cailler, A. Rigny, K. Bourdelle, W. Bogaerts, D. VanThourhout, J. Van Campenhout, and P. Absil, “Highly uniform and low-loss passive silicon photonics devices using a 300  mm CMOS platform,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper Th2A.33.

Bojko, R.

Bourdelle, K.

S. K. Selvaraja, P. De Heyn, G. Winroth, P. Ong, G. Lepage, C. Cailler, A. Rigny, K. Bourdelle, W. Bogaerts, D. VanThourhout, J. Van Campenhout, and P. Absil, “Highly uniform and low-loss passive silicon photonics devices using a 300  mm CMOS platform,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper Th2A.33.

Bowers, J. E.

J. E. Bowers, H. Park, Y.-H. Kuo, A. W. Fang, R. Jones, M. J. Paniccia, O. Cohen, and O. Raday, “Integrated optical amplifiers on silicon waveguides,” IPNRA Conference, Hybrid Integration Symposium, Salt Lake City, Utah, 2007.

Bradley, J. D. B.

J. D. B. Bradley and M. Pollnau, “Erbium-doped integrated waveguide amplifiers and lasers,” Laser Photon. Rev. 5, 368–403 (2011).
[CrossRef]

J. D. B. Bradley, L. Agazzi, D. Geskus, F. Ay, K. Wörhoff, and M. Pollnau, “Gain bandwidth of 80  nm and 2  dB/cm peak gain in Al2O3:Er3+ optical amplifiers on silicon,” J. Opt. Soc. Am. B 27, 187–196 (2010).
[CrossRef]

J. D. B. Bradley, R. Stoffer, A. Bakker, L. Agazzi, F. Ay, K. Worhoff, and M. Pollnau, “Integrated Al2O3:Er3+ zero-loss optical amplifier and power splitter with 40  nm bandwidth,” IEEE Photon. Technol. Lett. 22, 278–280 (2010).
[CrossRef]

J. D. B. Bradley, M. Costa e Silva, M. Gay, L. Bramerie, A. Driessen, K. Wörhoff, J. C. Simon, and M. Pollnau, “170  Gbits/s transmission in an erbium-doped waveguide amplifier,” Opt. Express 17, 22201–22208 (2009).
[CrossRef]

Bramerie, L.

Cailler, C.

S. K. Selvaraja, P. De Heyn, G. Winroth, P. Ong, G. Lepage, C. Cailler, A. Rigny, K. Bourdelle, W. Bogaerts, D. VanThourhout, J. Van Campenhout, and P. Absil, “Highly uniform and low-loss passive silicon photonics devices using a 300  mm CMOS platform,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper Th2A.33.

Chen, A.

Chen, C.

D. Zhang, C. Chen, F. Wang, and D. M. Zhang, “Optical gain and upconversion luminescence in LaF3:Er, Yb nanoparticles-doped organic–inorganic hybrid materials waveguide amplifiers,” Appl. Phys. B 98, 791–795 (2010).
[CrossRef]

C. Chen, D. Zhang, T. Li, D. Zhang, L. Song, and Z. Zhen, “Erbium-ytterbium codoped waveguide amplifier fabricated with solution-processable complex,” Appl. Phys. Lett. 94, 041119 (2009).
[CrossRef]

Cognolato, L.

L. Cognolato, C. S. De Bernardi, M. Ferraris, A. Gnazzo, S. Morasca, and D. Scarano, “Spectroscopic properties of Er3+-doped glasses for the realization of active waveguides by ion-exchange technique,” CSELT Technical Reports XIX, 277–281 (1991).

Cohen, O.

H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature 433, 292–294 (2005).
[CrossRef]

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
[CrossRef]

J. E. Bowers, H. Park, Y.-H. Kuo, A. W. Fang, R. Jones, M. J. Paniccia, O. Cohen, and O. Raday, “Integrated optical amplifiers on silicon waveguides,” IPNRA Conference, Hybrid Integration Symposium, Salt Lake City, Utah, 2007.

Costa e Silva, M.

Dai, D.

De Bernardi, C. S.

L. Cognolato, C. S. De Bernardi, M. Ferraris, A. Gnazzo, S. Morasca, and D. Scarano, “Spectroscopic properties of Er3+-doped glasses for the realization of active waveguides by ion-exchange technique,” CSELT Technical Reports XIX, 277–281 (1991).

De Heyn, P.

S. K. Selvaraja, P. De Heyn, G. Winroth, P. Ong, G. Lepage, C. Cailler, A. Rigny, K. Bourdelle, W. Bogaerts, D. VanThourhout, J. Van Campenhout, and P. Absil, “Highly uniform and low-loss passive silicon photonics devices using a 300  mm CMOS platform,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper Th2A.33.

Ding, R.

Driessen, A.

Fang, A.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
[CrossRef]

H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature 433, 292–294 (2005).
[CrossRef]

Fang, A. W.

J. E. Bowers, H. Park, Y.-H. Kuo, A. W. Fang, R. Jones, M. J. Paniccia, O. Cohen, and O. Raday, “Integrated optical amplifiers on silicon waveguides,” IPNRA Conference, Hybrid Integration Symposium, Salt Lake City, Utah, 2007.

Fedeli, J. M.

Ferraris, M.

L. Cognolato, C. S. De Bernardi, M. Ferraris, A. Gnazzo, S. Morasca, and D. Scarano, “Spectroscopic properties of Er3+-doped glasses for the realization of active waveguides by ion-exchange technique,” CSELT Technical Reports XIX, 277–281 (1991).

Ferziger, J. H.

J. H. Ferziger, Numerical Methods for Engineering Application (Wiley-Interscience, 1998).

Foster, M. A.

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[CrossRef]

Fournier, M.

Freude, W.

Gaeta, A. L.

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[CrossRef]

Gay, M.

Geskus, D.

Gnazzo, A.

L. Cognolato, C. S. De Bernardi, M. Ferraris, A. Gnazzo, S. Morasca, and D. Scarano, “Spectroscopic properties of Er3+-doped glasses for the realization of active waveguides by ion-exchange technique,” CSELT Technical Reports XIX, 277–281 (1991).

Guo, R.

Hak, D.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
[CrossRef]

H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature 433, 292–294 (2005).
[CrossRef]

He, S.

Hill, C.

Hiltunen, M.

Hochberg, M.

Honkanen, S.

Jacome, L.

Jiang, L.

Jones, R.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
[CrossRef]

H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature 433, 292–294 (2005).
[CrossRef]

J. E. Bowers, H. Park, Y.-H. Kuo, A. W. Fang, R. Jones, M. J. Paniccia, O. Cohen, and O. Raday, “Integrated optical amplifiers on silicon waveguides,” IPNRA Conference, Hybrid Integration Symposium, Salt Lake City, Utah, 2007.

Karvonen, L.

Kenyon, A. J.

A. J. Kenyon, “Recent developments in rare-earth doped materials for optoelectronics,” Prog. Quantum Electron. 26, 225–284 (2002).
[CrossRef]

Kim, R. S.

Kim, W. J.

Koos, C.

Koper, R. J. I. M.

G. N. Van den Hoven, R. J. I. M. Koper, A. Polman, C. van Dam, J. W. M. van Uffelen, and M. K. Smit, “Net optical gain at 1.53  μm in Er-doped Al2O3 waveguides on silicon,” Appl. Phys. Lett. 68, 1886–1888 (1996).
[CrossRef]

Korn, D.

Kuo, Y.-H.

J. E. Bowers, H. Park, Y.-H. Kuo, A. W. Fang, R. Jones, M. J. Paniccia, O. Cohen, and O. Raday, “Integrated optical amplifiers on silicon waveguides,” IPNRA Conference, Hybrid Integration Symposium, Salt Lake City, Utah, 2007.

Lepage, G.

S. K. Selvaraja, P. De Heyn, G. Winroth, P. Ong, G. Lepage, C. Cailler, A. Rigny, K. Bourdelle, W. Bogaerts, D. VanThourhout, J. Van Campenhout, and P. Absil, “Highly uniform and low-loss passive silicon photonics devices using a 300  mm CMOS platform,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper Th2A.33.

Leuthold, J.

Li, T.

C. Chen, D. Zhang, T. Li, D. Zhang, L. Song, and Z. Zhen, “Erbium-ytterbium codoped waveguide amplifier fabricated with solution-processable complex,” Appl. Phys. Lett. 94, 041119 (2009).
[CrossRef]

Liow, T.

Lipson, M.

Liu, A.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
[CrossRef]

H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature 433, 292–294 (2005).
[CrossRef]

Liu, Y.

Lo, G.

Miniscalco, W. J.

W. J. Miniscalco, “Erbium-doped glasses for fiber amplifiers at 1500 nm,” J. Lightwave Technol. 9, 234–250 (1991), (soda-lime silicate glass).
[CrossRef]

Morasca, S.

L. Cognolato, C. S. De Bernardi, M. Ferraris, A. Gnazzo, S. Morasca, and D. Scarano, “Spectroscopic properties of Er3+-doped glasses for the realization of active waveguides by ion-exchange technique,” CSELT Technical Reports XIX, 277–281 (1991).

Nicolaescu, R.

H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature 433, 292–294 (2005).
[CrossRef]

O’Malley, T.

Ong, P.

S. K. Selvaraja, P. De Heyn, G. Winroth, P. Ong, G. Lepage, C. Cailler, A. Rigny, K. Bourdelle, W. Bogaerts, D. VanThourhout, J. Van Campenhout, and P. Absil, “Highly uniform and low-loss passive silicon photonics devices using a 300  mm CMOS platform,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper Th2A.33.

Palmer, R.

Paniccia, M.

H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature 433, 292–294 (2005).
[CrossRef]

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
[CrossRef]

Paniccia, M. J.

J. E. Bowers, H. Park, Y.-H. Kuo, A. W. Fang, R. Jones, M. J. Paniccia, O. Cohen, and O. Raday, “Integrated optical amplifiers on silicon waveguides,” IPNRA Conference, Hybrid Integration Symposium, Salt Lake City, Utah, 2007.

Park, H.

J. E. Bowers, H. Park, Y.-H. Kuo, A. W. Fang, R. Jones, M. J. Paniccia, O. Cohen, and O. Raday, “Integrated optical amplifiers on silicon waveguides,” IPNRA Conference, Hybrid Integration Symposium, Salt Lake City, Utah, 2007.

Pavesi, L.

L. Vivien and L. Pavesi, Handbook of Silicon Photonics, Series in Optics and Optoelectronics (Taylor & Francis, 2013).

Pollnau, M.

J. D. B. Bradley and M. Pollnau, “Erbium-doped integrated waveguide amplifiers and lasers,” Laser Photon. Rev. 5, 368–403 (2011).
[CrossRef]

J. D. B. Bradley, R. Stoffer, A. Bakker, L. Agazzi, F. Ay, K. Worhoff, and M. Pollnau, “Integrated Al2O3:Er3+ zero-loss optical amplifier and power splitter with 40  nm bandwidth,” IEEE Photon. Technol. Lett. 22, 278–280 (2010).
[CrossRef]

J. D. B. Bradley, L. Agazzi, D. Geskus, F. Ay, K. Wörhoff, and M. Pollnau, “Gain bandwidth of 80  nm and 2  dB/cm peak gain in Al2O3:Er3+ optical amplifiers on silicon,” J. Opt. Soc. Am. B 27, 187–196 (2010).
[CrossRef]

J. D. B. Bradley, M. Costa e Silva, M. Gay, L. Bramerie, A. Driessen, K. Wörhoff, J. C. Simon, and M. Pollnau, “170  Gbits/s transmission in an erbium-doped waveguide amplifier,” Opt. Express 17, 22201–22208 (2009).
[CrossRef]

Polman, A.

G. N. Van den Hoven, R. J. I. M. Koper, A. Polman, C. van Dam, J. W. M. van Uffelen, and M. K. Smit, “Net optical gain at 1.53  μm in Er-doped Al2O3 waveguides on silicon,” Appl. Phys. Lett. 68, 1886–1888 (1996).
[CrossRef]

Pomerene, A.

Poulton, C.

Raday, O.

J. E. Bowers, H. Park, Y.-H. Kuo, A. W. Fang, R. Jones, M. J. Paniccia, O. Cohen, and O. Raday, “Integrated optical amplifiers on silicon waveguides,” IPNRA Conference, Hybrid Integration Symposium, Salt Lake City, Utah, 2007.

Reinhardt, W.

Rigny, A.

S. K. Selvaraja, P. De Heyn, G. Winroth, P. Ong, G. Lepage, C. Cailler, A. Rigny, K. Bourdelle, W. Bogaerts, D. VanThourhout, J. Van Campenhout, and P. Absil, “Highly uniform and low-loss passive silicon photonics devices using a 300  mm CMOS platform,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper Th2A.33.

Rong, H.

H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature 433, 292–294 (2005).
[CrossRef]

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
[CrossRef]

Säynätjoki, A.

Scarano, D.

L. Cognolato, C. S. De Bernardi, M. Ferraris, A. Gnazzo, S. Morasca, and D. Scarano, “Spectroscopic properties of Er3+-doped glasses for the realization of active waveguides by ion-exchange technique,” CSELT Technical Reports XIX, 277–281 (1991).

Scherer, A.

T. Baehr-Jones, M. Hochberg, C. Walker, and A. Scherer, “High-Q optical resonators in silicon-on-insulator-based slot waveguides,” Appl. Phys. Lett. 86, 081101 (2005).
[CrossRef]

Schmidt, B. S.

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[CrossRef]

P. A. Anderson, B. S. Schmidt, and M. Lipson, “High confinement in silicon slot waveguides with sharp bends,” Opt. Express 14, 9197–9202 (2006).
[CrossRef]

Selvaraja, S. K.

S. K. Selvaraja, P. De Heyn, G. Winroth, P. Ong, G. Lepage, C. Cailler, A. Rigny, K. Bourdelle, W. Bogaerts, D. VanThourhout, J. Van Campenhout, and P. Absil, “Highly uniform and low-loss passive silicon photonics devices using a 300  mm CMOS platform,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper Th2A.33.

Sharping, J. E.

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Simon, J. C.

Smit, M. K.

G. N. Van den Hoven, R. J. I. M. Koper, A. Polman, C. van Dam, J. W. M. van Uffelen, and M. K. Smit, “Net optical gain at 1.53  μm in Er-doped Al2O3 waveguides on silicon,” Appl. Phys. Lett. 68, 1886–1888 (1996).
[CrossRef]

Song, L.

C. Chen, D. Zhang, T. Li, D. Zhang, L. Song, and Z. Zhen, “Erbium-ytterbium codoped waveguide amplifier fabricated with solution-processable complex,” Appl. Phys. Lett. 94, 041119 (2009).
[CrossRef]

Spott, A.

Stoffer, R.

J. D. B. Bradley, R. Stoffer, A. Bakker, L. Agazzi, F. Ay, K. Worhoff, and M. Pollnau, “Integrated Al2O3:Er3+ zero-loss optical amplifier and power splitter with 40  nm bandwidth,” IEEE Photon. Technol. Lett. 22, 278–280 (2010).
[CrossRef]

Sun, H.

Szep, A.

Tang, Y.

Tervonen, A.

Tu, X.

Turner, A. C.

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[CrossRef]

Van Campenhout, J.

S. K. Selvaraja, P. De Heyn, G. Winroth, P. Ong, G. Lepage, C. Cailler, A. Rigny, K. Bourdelle, W. Bogaerts, D. VanThourhout, J. Van Campenhout, and P. Absil, “Highly uniform and low-loss passive silicon photonics devices using a 300  mm CMOS platform,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper Th2A.33.

van Dam, C.

G. N. Van den Hoven, R. J. I. M. Koper, A. Polman, C. van Dam, J. W. M. van Uffelen, and M. K. Smit, “Net optical gain at 1.53  μm in Er-doped Al2O3 waveguides on silicon,” Appl. Phys. Lett. 68, 1886–1888 (1996).
[CrossRef]

Van den Hoven, G. N.

G. N. Van den Hoven, R. J. I. M. Koper, A. Polman, C. van Dam, J. W. M. van Uffelen, and M. K. Smit, “Net optical gain at 1.53  μm in Er-doped Al2O3 waveguides on silicon,” Appl. Phys. Lett. 68, 1886–1888 (1996).
[CrossRef]

van Uffelen, J. W. M.

G. N. Van den Hoven, R. J. I. M. Koper, A. Polman, C. van Dam, J. W. M. van Uffelen, and M. K. Smit, “Net optical gain at 1.53  μm in Er-doped Al2O3 waveguides on silicon,” Appl. Phys. Lett. 68, 1886–1888 (1996).
[CrossRef]

VanThourhout, D.

S. K. Selvaraja, P. De Heyn, G. Winroth, P. Ong, G. Lepage, C. Cailler, A. Rigny, K. Bourdelle, W. Bogaerts, D. VanThourhout, J. Van Campenhout, and P. Absil, “Highly uniform and low-loss passive silicon photonics devices using a 300  mm CMOS platform,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper Th2A.33.

Vivien, L.

L. Vivien and L. Pavesi, Handbook of Silicon Photonics, Series in Optics and Optoelectronics (Taylor & Francis, 2013).

Walker, C.

T. Baehr-Jones, M. Hochberg, C. Walker, and A. Scherer, “High-Q optical resonators in silicon-on-insulator-based slot waveguides,” Appl. Phys. Lett. 86, 081101 (2005).
[CrossRef]

Wang, B.

Wang, F.

D. Zhang, C. Chen, F. Wang, and D. M. Zhang, “Optical gain and upconversion luminescence in LaF3:Er, Yb nanoparticles-doped organic–inorganic hybrid materials waveguide amplifiers,” Appl. Phys. B 98, 791–795 (2010).
[CrossRef]

Wang, L.

Wang, X.

Wang, Z.

Winroth, G.

S. K. Selvaraja, P. De Heyn, G. Winroth, P. Ong, G. Lepage, C. Cailler, A. Rigny, K. Bourdelle, W. Bogaerts, D. VanThourhout, J. Van Campenhout, and P. Absil, “Highly uniform and low-loss passive silicon photonics devices using a 300  mm CMOS platform,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper Th2A.33.

Worhoff, K.

J. D. B. Bradley, R. Stoffer, A. Bakker, L. Agazzi, F. Ay, K. Worhoff, and M. Pollnau, “Integrated Al2O3:Er3+ zero-loss optical amplifier and power splitter with 40  nm bandwidth,” IEEE Photon. Technol. Lett. 22, 278–280 (2010).
[CrossRef]

Wörhoff, K.

Wosinski, L.

Xiong, X.

Xu, Q.

Zhang, D.

D. Zhang, C. Chen, F. Wang, and D. M. Zhang, “Optical gain and upconversion luminescence in LaF3:Er, Yb nanoparticles-doped organic–inorganic hybrid materials waveguide amplifiers,” Appl. Phys. B 98, 791–795 (2010).
[CrossRef]

C. Chen, D. Zhang, T. Li, D. Zhang, L. Song, and Z. Zhen, “Erbium-ytterbium codoped waveguide amplifier fabricated with solution-processable complex,” Appl. Phys. Lett. 94, 041119 (2009).
[CrossRef]

C. Chen, D. Zhang, T. Li, D. Zhang, L. Song, and Z. Zhen, “Erbium-ytterbium codoped waveguide amplifier fabricated with solution-processable complex,” Appl. Phys. Lett. 94, 041119 (2009).
[CrossRef]

Zhang, D. M.

D. Zhang, C. Chen, F. Wang, and D. M. Zhang, “Optical gain and upconversion luminescence in LaF3:Er, Yb nanoparticles-doped organic–inorganic hybrid materials waveguide amplifiers,” Appl. Phys. B 98, 791–795 (2010).
[CrossRef]

Zhen, Z.

C. Chen, D. Zhang, T. Li, D. Zhang, L. Song, and Z. Zhen, “Erbium-ytterbium codoped waveguide amplifier fabricated with solution-processable complex,” Appl. Phys. Lett. 94, 041119 (2009).
[CrossRef]

Zhou, Z.

Zhu, N.

Appl. Phys. B

D. Zhang, C. Chen, F. Wang, and D. M. Zhang, “Optical gain and upconversion luminescence in LaF3:Er, Yb nanoparticles-doped organic–inorganic hybrid materials waveguide amplifiers,” Appl. Phys. B 98, 791–795 (2010).
[CrossRef]

Appl. Phys. Lett.

T. Baehr-Jones, M. Hochberg, C. Walker, and A. Scherer, “High-Q optical resonators in silicon-on-insulator-based slot waveguides,” Appl. Phys. Lett. 86, 081101 (2005).
[CrossRef]

G. N. Van den Hoven, R. J. I. M. Koper, A. Polman, C. van Dam, J. W. M. van Uffelen, and M. K. Smit, “Net optical gain at 1.53  μm in Er-doped Al2O3 waveguides on silicon,” Appl. Phys. Lett. 68, 1886–1888 (1996).
[CrossRef]

C. Chen, D. Zhang, T. Li, D. Zhang, L. Song, and Z. Zhen, “Erbium-ytterbium codoped waveguide amplifier fabricated with solution-processable complex,” Appl. Phys. Lett. 94, 041119 (2009).
[CrossRef]

CSELT Technical Reports

L. Cognolato, C. S. De Bernardi, M. Ferraris, A. Gnazzo, S. Morasca, and D. Scarano, “Spectroscopic properties of Er3+-doped glasses for the realization of active waveguides by ion-exchange technique,” CSELT Technical Reports XIX, 277–281 (1991).

IEEE Photon. Technol. Lett.

J. D. B. Bradley, R. Stoffer, A. Bakker, L. Agazzi, F. Ay, K. Worhoff, and M. Pollnau, “Integrated Al2O3:Er3+ zero-loss optical amplifier and power splitter with 40  nm bandwidth,” IEEE Photon. Technol. Lett. 22, 278–280 (2010).
[CrossRef]

J. Lightwave Technol.

W. J. Miniscalco, “Erbium-doped glasses for fiber amplifiers at 1500 nm,” J. Lightwave Technol. 9, 234–250 (1991), (soda-lime silicate glass).
[CrossRef]

J. Opt. Soc. Am. B

Laser Photon. Rev.

J. D. B. Bradley and M. Pollnau, “Erbium-doped integrated waveguide amplifiers and lasers,” Laser Photon. Rev. 5, 368–403 (2011).
[CrossRef]

Nature

H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature 433, 292–294 (2005).
[CrossRef]

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
[CrossRef]

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[CrossRef]

Opt. Express

R. Ding, T. Baehr-Jones, W. J. Kim, X. Xiong, R. Bojko, J. M. Fedeli, M. Fournier, and M. Hochberg, “Low-loss strip-loaded slot waveguides in silicon-on-insulator,” Opt. Express 18, 25061–25067 (2010).
[CrossRef]

A. Spott, T. Baehr-Jones, R. Ding, Y. Liu, R. Bojko, T. O’Malley, A. Pomerene, C. Hill, W. Reinhardt, and M. Hochberg, “Photolithographically fabricated low-loss asymmetric silicon slot waveguides,” Opt. Express 19, 10950–10958 (2011).
[CrossRef]

T. Alasaarela, D. Korn, L. Alloatti, A. Säynätjoki, A. Tervonen, R. Palmer, J. Leuthold, W. Freude, and S. Honkanen, “Reduced propagation loss in silicon strip and slot waveguides coated by atomic layer deposition,” Opt. Express 19, 11529–11538 (2011).
[CrossRef]

A. Säynätjoki, L. Karvonen, T. Alasaarela, X. Tu, T. Liow, M. Hiltunen, A. Tervonen, G. Lo, and S. Honkanen, “Low-loss silicon slot waveguides and couplers fabricated with optical lithography and atomic layer deposition,” Opt. Express 19, 26275–26282 (2011).
[CrossRef]

C. A. Barrios and M. Lipson, “Electrically driven silicon resonant light emitting device based on slot-waveguide,” Opt. Express 13, 10092–10101 (2005).
[CrossRef]

P. A. Anderson, B. S. Schmidt, and M. Lipson, “High confinement in silicon slot waveguides with sharp bends,” Opt. Express 14, 9197–9202 (2006).
[CrossRef]

C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express 15, 5976–5980 (2007).
[CrossRef]

J. D. B. Bradley, M. Costa e Silva, M. Gay, L. Bramerie, A. Driessen, K. Wörhoff, J. C. Simon, and M. Pollnau, “170  Gbits/s transmission in an erbium-doped waveguide amplifier,” Opt. Express 17, 22201–22208 (2009).
[CrossRef]

Opt. Lett.

Prog. Quantum Electron.

A. J. Kenyon, “Recent developments in rare-earth doped materials for optoelectronics,” Prog. Quantum Electron. 26, 225–284 (2002).
[CrossRef]

Other

L. Vivien and L. Pavesi, Handbook of Silicon Photonics, Series in Optics and Optoelectronics (Taylor & Francis, 2013).

J. E. Bowers, H. Park, Y.-H. Kuo, A. W. Fang, R. Jones, M. J. Paniccia, O. Cohen, and O. Raday, “Integrated optical amplifiers on silicon waveguides,” IPNRA Conference, Hybrid Integration Symposium, Salt Lake City, Utah, 2007.

S. K. Selvaraja, P. De Heyn, G. Winroth, P. Ong, G. Lepage, C. Cailler, A. Rigny, K. Bourdelle, W. Bogaerts, D. VanThourhout, J. Van Campenhout, and P. Absil, “Highly uniform and low-loss passive silicon photonics devices using a 300  mm CMOS platform,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper Th2A.33.

J. H. Ferziger, Numerical Methods for Engineering Application (Wiley-Interscience, 1998).

A. E. Siegman, Lasers (University Science Books, 1986).

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 (8)

Fig. 1.
Fig. 1.

(a) Normalized electric field energy for a cross section of silicon-slot waveguide filled by a Er3+-doped polymer for a 1530 nm wavelength signal amplification. (b) Mode effective index as a function of the slot width where the waveguide remains TE monomode. (c) Fraction of electric field energy n2|E|2dA in the polymer cladding region (slot and surroundings).

Fig. 2.
Fig. 2.

Signal gain as a function of the slot width for equal pump and signal losses (αp and αs) for typical silicon-slot waveguide linear loss values around 10dB/cm. The amplifier length for the blue curve is 15 mm, 18 mm for the green curve, and 21 mm for the red one, respectively. A 1 μW input signal power was further considered. The considered optical index values are also given in the table.

Fig. 3.
Fig. 3.

SEM images of the silicon-slot waveguides’ fabricated structures: (a) cross section of a 80 nm slot width SOI waveguide; (b) cross section of a 180 nm slot width waveguide; and (c), (d) views along the propagation direction highlighting the etching and low roughness.

Fig. 4.
Fig. 4.

Propagation losses of filled silicon-slot waveguides: (a) transmission of the 80±10nm slot waveguide for different lengths. (b) Linear regression of optical transmission at the signal wavelength (1530 nm) versus slot waveguide lengths for a 80±10nm slot width silicon waveguide. (c) Summary of waveguide loss for pump and signal wavelengths in different slot widths

Fig. 5.
Fig. 5.

Evolution of the optical gain (colormap) as a function of the input pump power and the waveguide length (optical losses, 8dB/cm; input signal power, 1 μW).

Fig. 6.
Fig. 6.

Optimized waveguide amplifier. (a) Optimum gain versus propagation losses from 3 to 10dB/cm for different slot widths. (b) Optimal pump power versus propagation losses for the same kind of waveguides. (c) Optimal lengths versus propagation losses.

Fig. 7.
Fig. 7.

Hybrid Er3+-doped polymer filling the cladding of a slot waveguide. A cross section is shown for the normalized dielectric energy for the pump beam (λp=1480nm), where Wr and Hr are 220 nm and Ws=100nm.

Fig. 8.
Fig. 8.

Er3+ four-level transitions’ considered system at the pumping wavelength λP=1480nm; the lifetime and energy in each level are specifically shown.

Tables (1)

Tables Icon

Table 1. Spectroscopic Parameters Used in the EDWA Optical Gain Model

Equations (8)

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

dn1(x,y,z)dt=(W12+R12)n1+(W21+R21+A21R)n2+Cupn22C14n1n4,
dn2dt=(W12+R12)n1(W21+R21+A21R+R24ESA)n2+A32NRn32(Cupn22C14n1n4),
dn4dt=R24ESAn2+Cupn22C14n1n4A43NRn4,
NEr=n1+n2+n3+n4,
Wijk=λkhcσij(λk)Ik(x,y)Pk(z),
dPs(z)dz=Ps(z)Is(x,y)[σ12s(λs)n1(x,y,z)σ21s(λs)n2(x,y,z)]dxdyPs(z)αs,
dPp(z)dz=Pp(z)Ip(x,y)[σ12p(λp)n1(x,y,z)σ21p(λp)n2(x,y,z)+σ24p(λp)n2(x,y,z)]dxdyPp(z)αp,
R24ESA=λphcσ24p(λp)Ip(x,y)Pp(z).

Metrics