Abstract

A computational approach to evaluate the bit-error ratio (BER) in silicon photonic systems employing high-order phase-shift keying (PSK) modulation formats is presented. Specifically, the investigated systems contain a silicon based optical interconnect, namely a strip silicon photonic waveguide or a silicon photonic crystal waveguide, and direct-detection receivers suitable to detect PSK and amplitude-shaped PSK signals. The superposition of a PSK signal and complex additive white Gaussian noise passes through the optical interconnect and subsequently through two detection-branch receivers. To model the signal propagation in the silicon optical interconnects we used a modified nonlinear Schrödinger equation, which incorporates all relevant linear and nonlinear optical effects and the mutual interaction between free-carriers and the optical field. Finally, the BER is calculated by applying a frequency-domain approach based on the Karhunen-Loève series expansion method. Our computational studies of the BER reveal that the optical power, type of PSK modulation, waveguide length, and group-velocity are key factors characterizing the system BER, their influence on BER being more significant in a photonic system with larger nonlinearity. In particular, our analysis shows that the system performance is affected to a much larger extent when the signal propagates in the slow-light regime, despite the fact that this regime allows for a significantly reduced length of optical interconnects.

© 2017 Optical Society of America

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References

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2016 (2)

S. Lavdas and N. C. Panoiu, “Theory of pulsed four-wave mixing in one-dimensional silicon photonic crystal slab waveguides,” Phys. Rev. B 93, 115435 (2016).
[Crossref]

J. You, S. Lavdas, and N. C. Panoiu, “Theoretical comparative analysis of BER in multi-channel systems with strip and photonic crystal silicon waveguides,” IEEE J. Sel. Top. Quantum Electron. 22, 4400810 (2016).
[Crossref]

2015 (1)

J. You and N. C. Panoiu, “Calculation of bit error rates in optical systems with silicon photonic wires,” IEEE J. Quantum Electron. 51, 8400108 (2015).

2012 (1)

Y. A. Vlasov, “Si CMOS-integrated nano-photonics for computer and data communications beyond 100G,” IEEE. Commun. Mag. 50, S67–S72 (2012).
[Crossref]

2011 (1)

A. Mafi and S. Raghavan, “Nonlinear phase noise in optical communication systems using eigenfunction expansion method,” Opt. Eng. 50, 055003 (2011).
[Crossref]

2010 (2)

S. Assefa, F. Xia, W. M. J. Green, C. L. Schow, A. V. Rylyakov, and Y. A. Vlasov, “CMOS-Integrated Optical Receivers for On-Chip Interconnects,” IEEE J. Sel. Top. Quantum Electron. 16, 1376–1385 (2010).
[Crossref]

N. C. Panoiu, J. F. McMillan, and C. W. Wong, “Theoretical analysis of pulse dynamics in silicon photonic crystal wire waveguides,” IEEE J. Sel. Top. Quantum Electron. 16, 257–266 (2010).
[Crossref]

2009 (4)

2008 (2)

2007 (5)

2006 (10)

C. Gunn, “CMOS photonics for high-speed interconnects,” IEEE Micro 26, 58–66 (2006).
[Crossref]

X. Chen, N. C. Panoiu, and R. M. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42, 160–170 (2006).
[Crossref]

J. P. Winzer and R. J. Essiambre, “Advanced optical modulation formats,” Proc. IEEE 94, 952–985 (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] [PubMed]

R. A. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 1678–1687 (2006).
[Crossref]

E. Agrell and M. Karlsson, “Performance comparison of optical 8-ary differential phase-shift keying systems with different decision schemes: Comment,” Opt. Express 14, 1700–1701 (2006).
[Crossref] [PubMed]

A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguides,” Opt. Express 14, 4357–4362 (2006).
[Crossref] [PubMed]

E. Dulkeith, Y. A. Vlasov, X. Chen, N. C. Panoiu, and R. M. Osgood, “Self-phase-modulation in submicron silicon-on-insulator photonic wires,” Opt. Express 14, 5524–5534 (2006).
[Crossref] [PubMed]

R. Dekker, A. Driessen, T. Wahlbrink, C. Moormann, J. Niehusmann, and M. Forst, “Ultrafast Kerr-induced alloptical wavelength conversion in silicon waveguides using 1.5 μm femtosecond pulses,” Opt. Express 14, 8336–8346 (2006).
[Crossref] [PubMed]

N. C. Panoiu, X. Chen, and R. M. Osgood, “Modulation instability in silicon photonic nanowires,” Opt. Lett. 31, 3609–3611 (2006).
[Crossref] [PubMed]

2005 (7)

2004 (3)

2003 (1)

2002 (1)

R. U. Ahmad, F. Pizzuto, G. S. Camarda, R. L. Espinola, H. Rao, and R. M. Osgood, “Ultracompact corner-mirrors and T-branches in silicon-on-insulator,” IEEE Photon. Technol. Lett. 14, 65–67 (2002).
[Crossref]

2001 (1)

R. Ho, K. W. Mai, and M. A. Horowitz, “The future of wires,” Proc. IEEE 89, 490–504 (2001).
[Crossref]

2000 (2)

K. K. Lee, D. R. Lim, H. C. Luan, A. Agarwal, J. Foresi, and L. C. Kimerling, “Effect of size and roughness on light transmission in a Si/SiO2 waveguide: experiments and model,” Appl. Phys. Lett. 77, 1617–1619 (2000).
[Crossref]

M. Rohde, C. Caspar, N. Heimes, M. Konitzer, E. J. Bachus, and N. Hanik, “Robustness of DPSK direct detection transmission format in standard fibre WDM systems,” Electron. Lett. 36, 1483–1484 (2000).
[Crossref]

1987 (1)

R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23, 123–129 (1987).
[Crossref]

Agarwal, A.

K. K. Lee, D. R. Lim, H. C. Luan, A. Agarwal, J. Foresi, and L. C. Kimerling, “Effect of size and roughness on light transmission in a Si/SiO2 waveguide: experiments and model,” Appl. Phys. Lett. 77, 1617–1619 (2000).
[Crossref]

Agrawal, G. P.

Agrell, E.

Ahmad, R. U.

R. U. Ahmad, F. Pizzuto, G. S. Camarda, R. L. Espinola, H. Rao, and R. M. Osgood, “Ultracompact corner-mirrors and T-branches in silicon-on-insulator,” IEEE Photon. Technol. Lett. 14, 65–67 (2002).
[Crossref]

Assefa, S.

S. Assefa, F. Xia, W. M. J. Green, C. L. Schow, A. V. Rylyakov, and Y. A. Vlasov, “CMOS-Integrated Optical Receivers for On-Chip Interconnects,” IEEE J. Sel. Top. Quantum Electron. 16, 1376–1385 (2010).
[Crossref]

Bachus, E. J.

M. Rohde, C. Caspar, N. Heimes, M. Konitzer, E. J. Bachus, and N. Hanik, “Robustness of DPSK direct detection transmission format in standard fibre WDM systems,” Electron. Lett. 36, 1483–1484 (2000).
[Crossref]

Baets, R.

Barclay, P. E.

Barwicz, T.

Beckx, S.

Benner, A. F.

J. A. Kash, A. F. Benner, F. E. Doany, D. M. Kuchta, B. G. Lee, P. K. Pepeljugoski, L. Schares, C. L. Schow, and M. Taubenblatt, “Optical Interconnects in Exascale Supercomputers,” in 23rd Annual Meeting of the IEEE Photonics Society (2010), 483–484.

Benner, F.

F. Benner, M. Ignatowski, J. A. Kash, D. M. Kuchta, and M. B. Ritter, “Exploitation of optical interconnects in future server architectures,” IBM J. Res. Dev. 49, 755–775 (2005).
[Crossref]

Bennett, B. R.

R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23, 123–129 (1987).
[Crossref]

Bergman, K.

K. Shacham, K. Bergman, and L. P. Carloni, “Photonic networks-on-chip for future generations of chip multiprocessors,” IEEE Trans. Comput. 57, 1246–1260 (2008).
[Crossref]

Bienstman, P.

Bogaerts, W.

Boyraz, O.

Byun, H.

Camarda, G. S.

R. U. Ahmad, F. Pizzuto, G. S. Camarda, R. L. Espinola, H. Rao, and R. M. Osgood, “Ultracompact corner-mirrors and T-branches in silicon-on-insulator,” IEEE Photon. Technol. Lett. 14, 65–67 (2002).
[Crossref]

Carloni, L. P.

K. Shacham, K. Bergman, and L. P. Carloni, “Photonic networks-on-chip for future generations of chip multiprocessors,” IEEE Trans. Comput. 57, 1246–1260 (2008).
[Crossref]

Caspar, C.

M. Rohde, C. Caspar, N. Heimes, M. Konitzer, E. J. Bachus, and N. Hanik, “Robustness of DPSK direct detection transmission format in standard fibre WDM systems,” Electron. Lett. 36, 1483–1484 (2000).
[Crossref]

Chen, X.

R. M. Osgood, N. C. Panoiu, J. I. Dadap, X. Liu, X. Chen, I-W. Hsieh, E. Dulkeith, W. M. J. Green, and Y. A. Vlassov, “Engineering nonlinearities in nanoscale optical systems: physics and applications in dispersion-engineered silicon nanophotonic wires,” Adv. Opt. Photonics 1, 162–235 (2009).
[Crossref]

I-W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, R. M. Osgood, S. J. McNab, and Y. A. Vlasov, “Crossphase modulation-induced spectral and temporal effects on co-propagating femtosecond pulses in silicon photonic wires,“ Opt. Express 15, 1135–1146 (2007).
[Crossref] [PubMed]

N. C. Panoiu, X. Chen, and R. M. Osgood, “Modulation instability in silicon photonic nanowires,” Opt. Lett. 31, 3609–3611 (2006).
[Crossref] [PubMed]

E. Dulkeith, Y. A. Vlasov, X. Chen, N. C. Panoiu, and R. M. Osgood, “Self-phase-modulation in submicron silicon-on-insulator photonic wires,” Opt. Express 14, 5524–5534 (2006).
[Crossref] [PubMed]

X. Chen, N. C. Panoiu, and R. M. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42, 160–170 (2006).
[Crossref]

Chen, X. G.

Chou, C. Y.

Claps, R.

Cohen, O.

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

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004).
[Crossref] [PubMed]

Dadap, J. I.

Dekker, R.

Dimitropoulos, D.

Doany, F. E.

J. A. Kash, A. F. Benner, F. E. Doany, D. M. Kuchta, B. G. Lee, P. K. Pepeljugoski, L. Schares, C. L. Schow, and M. Taubenblatt, “Optical Interconnects in Exascale Supercomputers,” in 23rd Annual Meeting of the IEEE Photonics Society (2010), 483–484.

Driessen, A.

Dulkeith, E.

Dumon, P.

Espinola, R. L.

R. U. Ahmad, F. Pizzuto, G. S. Camarda, R. L. Espinola, H. Rao, and R. M. Osgood, “Ultracompact corner-mirrors and T-branches in silicon-on-insulator,” IEEE Photon. Technol. Lett. 14, 65–67 (2002).
[Crossref]

Essiambre, R. J.

J. P. Winzer and R. J. Essiambre, “Advanced optical modulation formats,” Proc. IEEE 94, 952–985 (2006).
[Crossref]

Fang, A.

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

Foresi, J.

K. K. Lee, D. R. Lim, H. C. Luan, A. Agarwal, J. Foresi, and L. C. Kimerling, “Effect of size and roughness on light transmission in a Si/SiO2 waveguide: experiments and model,” Appl. Phys. Lett. 77, 1617–1619 (2000).
[Crossref]

Forestieri, E.

Forst, M.

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] [PubMed]

A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguides,” Opt. Express 14, 4357–4362 (2006).
[Crossref] [PubMed]

Freude, W.

Fukuda, H.

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] [PubMed]

A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguides,” Opt. Express 14, 4357–4362 (2006).
[Crossref] [PubMed]

Gan, F.

Geis, M.

Green, W. M. J.

S. Assefa, F. Xia, W. M. J. Green, C. L. Schow, A. V. Rylyakov, and Y. A. Vlasov, “CMOS-Integrated Optical Receivers for On-Chip Interconnects,” IEEE J. Sel. Top. Quantum Electron. 16, 1376–1385 (2010).
[Crossref]

R. M. Osgood, N. C. Panoiu, J. I. Dadap, X. Liu, X. Chen, I-W. Hsieh, E. Dulkeith, W. M. J. Green, and Y. A. Vlassov, “Engineering nonlinearities in nanoscale optical systems: physics and applications in dispersion-engineered silicon nanophotonic wires,” Adv. Opt. Photonics 1, 162–235 (2009).
[Crossref]

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R. M. Osgood, N. C. Panoiu, J. I. Dadap, X. Liu, X. Chen, I-W. Hsieh, E. Dulkeith, W. M. J. Green, and Y. A. Vlassov, “Engineering nonlinearities in nanoscale optical systems: physics and applications in dispersion-engineered silicon nanophotonic wires,” Adv. Opt. Photonics 1, 162–235 (2009).
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I-W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, R. M. Osgood, S. J. McNab, and Y. A. Vlasov, “Crossphase modulation-induced spectral and temporal effects on co-propagating femtosecond pulses in silicon photonic wires,“ Opt. Express 15, 1135–1146 (2007).
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F. Benner, M. Ignatowski, J. A. Kash, D. M. Kuchta, and M. B. Ritter, “Exploitation of optical interconnects in future server architectures,” IBM J. Res. Dev. 49, 755–775 (2005).
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Itabashi, S.

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Jalali, B.

Jones, R.

H. S. Rong, R. Jones, A. S. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
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A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004).
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Karlsson, M.

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Kash, J. A.

F. Benner, M. Ignatowski, J. A. Kash, D. M. Kuchta, and M. B. Ritter, “Exploitation of optical interconnects in future server architectures,” IBM J. Res. Dev. 49, 755–775 (2005).
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K. K. Lee, D. R. Lim, H. C. Luan, A. Agarwal, J. Foresi, and L. C. Kimerling, “Effect of size and roughness on light transmission in a Si/SiO2 waveguide: experiments and model,” Appl. Phys. Lett. 77, 1617–1619 (2000).
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M. Rohde, C. Caspar, N. Heimes, M. Konitzer, E. J. Bachus, and N. Hanik, “Robustness of DPSK direct detection transmission format in standard fibre WDM systems,” Electron. Lett. 36, 1483–1484 (2000).
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Koonath, P.

Koos, C.

Kuchta, D. M.

F. Benner, M. Ignatowski, J. A. Kash, D. M. Kuchta, and M. B. Ritter, “Exploitation of optical interconnects in future server architectures,” IBM J. Res. Dev. 49, 755–775 (2005).
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J. A. Kash, A. F. Benner, F. E. Doany, D. M. Kuchta, B. G. Lee, P. K. Pepeljugoski, L. Schares, C. L. Schow, and M. Taubenblatt, “Optical Interconnects in Exascale Supercomputers,” in 23rd Annual Meeting of the IEEE Photonics Society (2010), 483–484.

Lavdas, S.

S. Lavdas and N. C. Panoiu, “Theory of pulsed four-wave mixing in one-dimensional silicon photonic crystal slab waveguides,” Phys. Rev. B 93, 115435 (2016).
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J. You, S. Lavdas, and N. C. Panoiu, “Theoretical comparative analysis of BER in multi-channel systems with strip and photonic crystal silicon waveguides,” IEEE J. Sel. Top. Quantum Electron. 22, 4400810 (2016).
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J. A. Kash, A. F. Benner, F. E. Doany, D. M. Kuchta, B. G. Lee, P. K. Pepeljugoski, L. Schares, C. L. Schow, and M. Taubenblatt, “Optical Interconnects in Exascale Supercomputers,” in 23rd Annual Meeting of the IEEE Photonics Society (2010), 483–484.

Lee, D.

Lee, K. K.

K. K. Lee, D. R. Lim, H. C. Luan, A. Agarwal, J. Foresi, and L. C. Kimerling, “Effect of size and roughness on light transmission in a Si/SiO2 waveguide: experiments and model,” Appl. Phys. Lett. 77, 1617–1619 (2000).
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A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004).
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K. K. Lee, D. R. Lim, H. C. Luan, A. Agarwal, J. Foresi, and L. C. Kimerling, “Effect of size and roughness on light transmission in a Si/SiO2 waveguide: experiments and model,” Appl. Phys. Lett. 77, 1617–1619 (2000).
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Lipson, M.

A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguides,” Opt. Express 14, 4357–4362 (2006).
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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] [PubMed]

Q. Xu, B. Shmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325–327 (2005).
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A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004).
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H. S. Rong, R. Jones, A. S. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
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N. C. Panoiu, X. Liu, and R. M. Osgood, “Self-steepening of ultrashort pulses in silicon photonic nanowires,” Opt. Lett. 34, 947–949 (2009).
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R. M. Osgood, N. C. Panoiu, J. I. Dadap, X. Liu, X. Chen, I-W. Hsieh, E. Dulkeith, W. M. J. Green, and Y. A. Vlassov, “Engineering nonlinearities in nanoscale optical systems: physics and applications in dispersion-engineered silicon nanophotonic wires,” Adv. Opt. Photonics 1, 162–235 (2009).
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K. K. Lee, D. R. Lim, H. C. Luan, A. Agarwal, J. Foresi, and L. C. Kimerling, “Effect of size and roughness on light transmission in a Si/SiO2 waveguide: experiments and model,” Appl. Phys. Lett. 77, 1617–1619 (2000).
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R. Ho, K. W. Mai, and M. A. Horowitz, “The future of wires,” Proc. IEEE 89, 490–504 (2001).
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McMillan, J. F.

N. C. Panoiu, J. F. McMillan, and C. W. Wong, “Theoretical analysis of pulse dynamics in silicon photonic crystal wire waveguides,” IEEE J. Sel. Top. Quantum Electron. 16, 257–266 (2010).
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Moormann, C.

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A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004).
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J. I. Dadap, N. C. Panoiu, X. G. Chen, I. W. Hsieh, X. P. Liu, C. Y. Chou, E. Dulkeith, S. J. McNab, F. N. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, and R. M. Osgood, “Nonlinear-optical phase modification in dispersion-engineered Si photonic wires,” Opt. Express 16, 1280–1299 (2008).
[Crossref] [PubMed]

I-W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, R. M. Osgood, S. J. McNab, and Y. A. Vlasov, “Crossphase modulation-induced spectral and temporal effects on co-propagating femtosecond pulses in silicon photonic wires,“ Opt. Express 15, 1135–1146 (2007).
[Crossref] [PubMed]

N. C. Panoiu, X. Chen, and R. M. Osgood, “Modulation instability in silicon photonic nanowires,” Opt. Lett. 31, 3609–3611 (2006).
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E. Dulkeith, Y. A. Vlasov, X. Chen, N. C. Panoiu, and R. M. Osgood, “Self-phase-modulation in submicron silicon-on-insulator photonic wires,” Opt. Express 14, 5524–5534 (2006).
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Painter, O. J.

Paniccia, M.

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

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004).
[Crossref] [PubMed]

Panoiu, N. C.

J. You, S. Lavdas, and N. C. Panoiu, “Theoretical comparative analysis of BER in multi-channel systems with strip and photonic crystal silicon waveguides,” IEEE J. Sel. Top. Quantum Electron. 22, 4400810 (2016).
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S. Lavdas and N. C. Panoiu, “Theory of pulsed four-wave mixing in one-dimensional silicon photonic crystal slab waveguides,” Phys. Rev. B 93, 115435 (2016).
[Crossref]

J. You and N. C. Panoiu, “Calculation of bit error rates in optical systems with silicon photonic wires,” IEEE J. Quantum Electron. 51, 8400108 (2015).

N. C. Panoiu, J. F. McMillan, and C. W. Wong, “Theoretical analysis of pulse dynamics in silicon photonic crystal wire waveguides,” IEEE J. Sel. Top. Quantum Electron. 16, 257–266 (2010).
[Crossref]

R. M. Osgood, N. C. Panoiu, J. I. Dadap, X. Liu, X. Chen, I-W. Hsieh, E. Dulkeith, W. M. J. Green, and Y. A. Vlassov, “Engineering nonlinearities in nanoscale optical systems: physics and applications in dispersion-engineered silicon nanophotonic wires,” Adv. Opt. Photonics 1, 162–235 (2009).
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[Crossref] [PubMed]

J. I. Dadap, N. C. Panoiu, X. G. Chen, I. W. Hsieh, X. P. Liu, C. Y. Chou, E. Dulkeith, S. J. McNab, F. N. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, and R. M. Osgood, “Nonlinear-optical phase modification in dispersion-engineered Si photonic wires,” Opt. Express 16, 1280–1299 (2008).
[Crossref] [PubMed]

I-W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, R. M. Osgood, S. J. McNab, and Y. A. Vlasov, “Crossphase modulation-induced spectral and temporal effects on co-propagating femtosecond pulses in silicon photonic wires,“ Opt. Express 15, 1135–1146 (2007).
[Crossref] [PubMed]

N. C. Panoiu, X. Chen, and R. M. Osgood, “Modulation instability in silicon photonic nanowires,” Opt. Lett. 31, 3609–3611 (2006).
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E. Dulkeith, Y. A. Vlasov, X. Chen, N. C. Panoiu, and R. M. Osgood, “Self-phase-modulation in submicron silicon-on-insulator photonic wires,” Opt. Express 14, 5524–5534 (2006).
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[Crossref]

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Pepeljugoski, P. K.

J. A. Kash, A. F. Benner, F. E. Doany, D. M. Kuchta, B. G. Lee, P. K. Pepeljugoski, L. Schares, C. L. Schow, and M. Taubenblatt, “Optical Interconnects in Exascale Supercomputers,” in 23rd Annual Meeting of the IEEE Photonics Society (2010), 483–484.

Pizzuto, F.

R. U. Ahmad, F. Pizzuto, G. S. Camarda, R. L. Espinola, H. Rao, and R. M. Osgood, “Ultracompact corner-mirrors and T-branches in silicon-on-insulator,” IEEE Photon. Technol. Lett. 14, 65–67 (2002).
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Poulton, C.

Pradhan, S.

Q. Xu, B. Shmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325–327 (2005).
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A. Mafi and S. Raghavan, “Nonlinear phase noise in optical communication systems using eigenfunction expansion method,” Opt. Eng. 50, 055003 (2011).
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Rakich, P. T.

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Rao, H.

R. U. Ahmad, F. Pizzuto, G. S. Camarda, R. L. Espinola, H. Rao, and R. M. Osgood, “Ultracompact corner-mirrors and T-branches in silicon-on-insulator,” IEEE Photon. Technol. Lett. 14, 65–67 (2002).
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F. Benner, M. Ignatowski, J. A. Kash, D. M. Kuchta, and M. B. Ritter, “Exploitation of optical interconnects in future server architectures,” IBM J. Res. Dev. 49, 755–775 (2005).
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M. Rohde, C. Caspar, N. Heimes, M. Konitzer, E. J. Bachus, and N. Hanik, “Robustness of DPSK direct detection transmission format in standard fibre WDM systems,” Electron. Lett. 36, 1483–1484 (2000).
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H. S. Rong, R. Jones, A. S. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
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A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004).
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S. Assefa, F. Xia, W. M. J. Green, C. L. Schow, A. V. Rylyakov, and Y. A. Vlasov, “CMOS-Integrated Optical Receivers for On-Chip Interconnects,” IEEE J. Sel. Top. Quantum Electron. 16, 1376–1385 (2010).
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A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004).
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J. A. Kash, A. F. Benner, F. E. Doany, D. M. Kuchta, B. G. Lee, P. K. Pepeljugoski, L. Schares, C. L. Schow, and M. Taubenblatt, “Optical Interconnects in Exascale Supercomputers,” in 23rd Annual Meeting of the IEEE Photonics Society (2010), 483–484.

Schmidt, B. S.

A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguides,” Opt. Express 14, 4357–4362 (2006).
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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).
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S. Assefa, F. Xia, W. M. J. Green, C. L. Schow, A. V. Rylyakov, and Y. A. Vlasov, “CMOS-Integrated Optical Receivers for On-Chip Interconnects,” IEEE J. Sel. Top. Quantum Electron. 16, 1376–1385 (2010).
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J. A. Kash, A. F. Benner, F. E. Doany, D. M. Kuchta, B. G. Lee, P. K. Pepeljugoski, L. Schares, C. L. Schow, and M. Taubenblatt, “Optical Interconnects in Exascale Supercomputers,” in 23rd Annual Meeting of the IEEE Photonics Society (2010), 483–484.

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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).
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Q. Xu, B. Shmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325–327 (2005).
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J. A. Kash, A. F. Benner, F. E. Doany, D. M. Kuchta, B. G. Lee, P. K. Pepeljugoski, L. Schares, C. L. Schow, and M. Taubenblatt, “Optical Interconnects in Exascale Supercomputers,” in 23rd Annual Meeting of the IEEE Photonics Society (2010), 483–484.

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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).
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A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguides,” Opt. Express 14, 4357–4362 (2006).
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N. C. Panoiu, J. F. McMillan, and C. W. Wong, “Theoretical analysis of pulse dynamics in silicon photonic crystal wire waveguides,” IEEE J. Sel. Top. Quantum Electron. 16, 257–266 (2010).
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Xia, F.

S. Assefa, F. Xia, W. M. J. Green, C. L. Schow, A. V. Rylyakov, and Y. A. Vlasov, “CMOS-Integrated Optical Receivers for On-Chip Interconnects,” IEEE J. Sel. Top. Quantum Electron. 16, 1376–1385 (2010).
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Xia, F. N.

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Q. Xu, B. Shmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325–327 (2005).
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J. You, S. Lavdas, and N. C. Panoiu, “Theoretical comparative analysis of BER in multi-channel systems with strip and photonic crystal silicon waveguides,” IEEE J. Sel. Top. Quantum Electron. 22, 4400810 (2016).
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J. You and N. C. Panoiu, “Calculation of bit error rates in optical systems with silicon photonic wires,” IEEE J. Quantum Electron. 51, 8400108 (2015).

Adv. Opt. Photonics (1)

R. M. Osgood, N. C. Panoiu, J. I. Dadap, X. Liu, X. Chen, I-W. Hsieh, E. Dulkeith, W. M. J. Green, and Y. A. Vlassov, “Engineering nonlinearities in nanoscale optical systems: physics and applications in dispersion-engineered silicon nanophotonic wires,” Adv. Opt. Photonics 1, 162–235 (2009).
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Figures (8)

Fig. 1
Fig. 1

Schematics of the Si photonic system investigated in this work. It contains a Si waveguide and a direct-detection receiver with bi-level electrical decisions. The receiver has two branches, an intensity-detection and a phase-detection branch, with the latter consisting of N Mach-Zehnder interferometers. Two types of waveguides are investigated: one is a strip waveguide with uniform cross-section with height, h = 250 nm, and width, w = 900 nm and the other one is a PhC waveguide with lattice constant, a = 412 nm, hole radius, r = 0.22a, and slab thickness, h = 0.6a.

Fig. 2
Fig. 2

Constellation diagrams of the investigated signal modulation formats. (a), (b), (c), (d), (e), (f), are for 2PSK, 4PSK, 8PSK, 16PSK, A2PSK, and A4PSK modulation, respectively. (g), (h), (i) The decision boundaries for 4PSK, 8PSK, and 16PSK modulation formats.

Fig. 3
Fig. 3

(a), (b), (c) Signal constellation of 8PSK signals with SNR = 25 dB and P = 10 mW, at the output of a Si-PhW, Si-PhCW-FL, and Si-PhCW-SL, respectively. The dots indicate the noisy signals and the asterisks represent the ideal output signal without noise and phase shift.

Fig. 4
Fig. 4

Top and bottom panels show the eye diagrams of real and imaginary part of received 8PSK signals after fifth-order Butterworth optical filter, respectively. From left to right, the panels correspond to the Si-PhW, Si-PhCW-FL, and Si-PhCW-SL. The input power P = 10 mW, SNR = 25 dB, and length of Si-PhW and Si-PhCW is 5 cm and 500 μm, respectively.

Fig. 5
Fig. 5

System BER of various modulation formats for direct-detection receivers with bi-level decison. From left to right, the panels correspond to a Si-PhW, a Si-PhCW operated in the FL regime, and a Si-PhCW operated in the SL regime.

Fig. 6
Fig. 6

System BER vs. SNR determined for different input power. From left to right, the panels correspond to a Si-PhW, a Si-PhCW operated in the FL regime, and Si-PhCW operated in the SL regime. The dashed lines, solid lines and dash-dot lines correspond to A2PSK, 4PSK, and 2PSK modulated signals, respectively.

Fig. 7
Fig. 7

System BER vs SNR, calculated for different waveguide lengths. From left to right, the panels correspond to a Si-PhW, a Si-PhCW operated in the FL regime, and Si-PhCW operated in the SL regime. The dashed lines, solid lines, and dash-dot lines correspond to A2PSK, 4PSK, and 2PSK modulated signals, respectively.

Fig. 8
Fig. 8

BER calculated for several system receiver configurations. From left to right, the panels correspond to a Si-PhW, a Si-PhCW operated in the FL regime, and Si-PhCW operated in the SL regime. In all cases, an 8PSK modulation format is considered. The electrical filter is a fifth-order Bessel filter, wheres the optical filter is a Lorenzen filter (red line), Gaussian filter (blue line), super-Gaussian filter (black line), and sixth-order Butterworth filter (purple line).

Tables (1)

Tables Icon

Table 1 The optical parameters of silicon waveguides used in numerical simulations.

Equations (25)

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i u ( z , t ) z β 2 2 2 u t 2 + i c κ 2 n v g ( α i + α fc ) u ( z , t ) + ω 0 κ n v g δ n fc u ( z , t ) + γ | u ( z , t ) | 2 u ( z , t ) = 0 ,
N t = N t c + γ ω 0 A nl | u ( z , t ) | 4 ,
γ = { 3 ω 0 0 16 v g 2 Γ 𝒲 2 , for Si PhWs , 3 ω 0 0 a 16 v g 2 Γ W 2 , for Si PhcWs ,
u ( z , t ) = [ P ( z ) e i Φ 0 + a ( z , t ) ] e i Φ ( z ) ,
N s ( z ) = γ t c ω 0 A nl P 2 ( z ) ξ P 2 ( z ) .
d P d z = c κ n v g α i P c κ n v g σ α ξ P 3 2 γ P 2 ,
d Φ d z = ω 0 κ n v g σ n ξ P 2 γ P ,
a z = i β 2 2 2 a t 2 c κ 2 n v g α i a c κ 2 n v g σ α ξ P 2 [ a + 4 e i Φ 0 ( cos Φ a + sin Φ 0 a ) ] + i 4 ω 0 κ n v g σ n × ξ P 2 e i Φ 0 ( cos Φ 0 a + sin Φ 0 a ) γ P a + 2 i γ P e i Φ 0 ( cos Φ 0 a + sin Φ 0 a ) .
d A d z = β 2 2 Ω 2 A c κ 2 n v g α i A c κ 2 n v g σ α ξ P 2 [ A + 4 cos Φ 0 ( cos Φ 0 A + sin Φ 0 A ) ] 4 ω 0 κ n v g σ n ξ P 2 sin Φ 0 ( cos Φ 0 A + sin Φ 0 A ) γ P A 2 P ( γ sin Φ 0 + γ cos Φ 0 ) ( cos Φ 0 A + sin Φ 0 A ) ,
d A d z = β 2 2 Ω 2 A c κ 2 n v g α i A c κ 2 n v g σ α ξ P 2 [ A + 4 sin Φ 0 ( cos Φ 0 A + sin Φ 0 A ) ] + 4 ω 0 κ n v g σ n ξ P 2 cos Φ 0 ( cos Φ 0 A + sin Φ 0 A ) γ P A + 2 P ( γ cos Φ 0 γ sin Φ 0 ) ( cos Φ 0 A + sin Φ 0 A ) ,
y n ( t ) = X * ( f 1 ) K n ( f 1 , f 2 ) X ( f 2 ) e 2 π i ( f 2 f 1 ) t d f 1 d f 2 ,
K n ( f 1 , f 2 ) = H o * ( f 2 ) H o ( f 1 ) H e ( f 1 f 2 ) [ H n , U * ( f 2 ) H n , U ( f 1 ) H n , L * ( f 2 ) H n , L ( f 1 ) ] .
H n , U ( f ) = 1 2 C r ( e 2 π i f T s + e i ϕ n ) ,
H n , L ( f ) = 1 2 C r ( e 2 π i f T s e i ϕ n ) ,
y ( t ) = α = 1 2 M + 1 β = 1 2 M + 1 x α * K α β x β ,
x α = X ( f α ) e 2 π i f α t Δ f ,
K α β = K ( f α , f β ) Δ f .
y ( t ) = x T 𝒦 x .
{ Λ T Σ T 𝒦 Σ Λ } α β = η α δ α β ,
y ( t ) = α = 1 4 M + 1 η α w α 2 .
𝔼 { w } = Λ T Σ 1 𝔼 { s + n } = Λ T Σ 1 𝔼 { s } σ ,
𝔼 { ww T } = 𝔼 { Λ T Σ 1 x x T Σ 1 T Λ } = I ,
Ψ y ( s ) = 𝔼 { e s y } = α = 1 4 M + 2 exp ( η α σ σ 2 s 1 2 η α s ) 1 2 η α s .
𝒫 = 1 2 [ 𝒫 ( y > y th | s = 0 ) + 𝒫 ( y < y th | s = P ) ] .
BER = 1 n = 1 4 ( 1 BER n ) 3 .

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