Abstract

We investigated the high-sensitivity interferometric autocorrelation of ultrafast optical pulses utilizing two-photon absorption in sub-micrometer silicon p-i-n waveguides. The autocorrelation sensitivities were evaluated to be about 0.5 and 4.5 × 10−8 W2 for 1- and 0.5-mm devices, respectively. Such sensitivities are about 100 times higher than the traditional two-photon conductivity photodetectors in commercial autocorrelators; thus favor weak pulse characterization. We comprehensively studied the interferometric autocorrelation performances by the experiment and FDTD (finite-difference time-domain) simulation. The pulse energy dependences of measured autocorrelation photocurrents and pulse widths were well explained by the simulation with the free carrier absorption and free carrier plasma effect considered. The autocorrelation error tends to occur if the pulse energy is high enough to cause strong free carrier effects and the threshold pulse energy for error occurrence is increased for shorter devices, but accurate autocorrelation measurement was achieved for sub-Watts pulses at which the influences of free carrier effects on interferometric autocorrelation was negligible. The minimum applicable range of pulse widths was estimated from waveguide dispersion analysis to be ~0.09 and 0.13 ps with a 10% target error for 0.5-mm and 1-mm devices, respectively. The interferometric autocorrelation in sub-micrometer silicon p-i-n waveguides is promising as a monolithic photonic device for on-chip monitor and diagnostics of weak ultrafast pulses.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (2)

2016 (2)

2014 (2)

E. Z. Chong, T. F. Watson, and F. Festy, “Autocorrelation measurement of femtosecond laser pulses based on two-photon absorption in GaP photodiode,” Appl. Phys. Lett. 105(6), 062111 (2014).
[Crossref]

C. Monat, C. Grillet, M. Collins, A. Clark, J. Schroeder, C. Xiong, J. Li, L. O’Faolain, T. F. Krauss, B. J. Eggleton, and D. J. Moss, “Integrated optical auto-correlator based on third-harmonic generation in a silicon photonic crystal waveguide,” Nat. Commun. 5(1), 3246 (2014).
[Crossref] [PubMed]

2013 (2)

J. An, K. Pyun, O. Kwon, and D. E. Kim, “An autocorrelator based on a Fabry-Perot interferometer,” Opt. Express 21(1), 70–78 (2013).
[Crossref] [PubMed]

R. Hayakawa, N. Ishikura, H. C. Nguyen, and T. Baba, “Two-photon-absorption photodiodes in Si photonic-crystal slow-light waveguides,” Appl. Phys. Lett. 102(3), 031114 (2013).
[Crossref]

2009 (1)

E. K. Tien, X. Z. Sang, F. Qing, Q. Song, and O. Boyraz, “Ultrafast pulse characterization using cross phase modulation in silicon,” Appl. Phys. Lett. 95(5), 051101 (2009).
[Crossref]

2007 (3)

N. Suzuki, “FDTD Analysis of Two-Photon Absorption and Free-Carrier Absorption in Si High-Index-Contrast Waveguides,” J. Lightwave Technol. 25(9), 2495–2501 (2007).
[Crossref]

Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91(2), 021111 (2007).
[Crossref]

A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. 90(19), 191104 (2007).
[Crossref]

2006 (3)

2004 (2)

T. Krug, M. Lynch, A. L. Bradley, J. F. Donegan, L. P. Barry, H. Folliot, J. S. Roberts, and G. Hill, “High-Sensitivity Two-Photon Absorption Microcavity Autocorrelator,” IEEE Photonics Technol. Lett. 16(6), 1543–1545 (2004).
[Crossref]

A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, “Net optical gain in a low loss silicon-on-insulator waveguide by stimulated Raman scattering,” Opt. Express 12(18), 4261–4268 (2004).
[Crossref] [PubMed]

2002 (3)

A. J. Sabbah and D. M. Riffe, “Femtosecond pump-probe reflectivity study of silicon carrier dynamics,” Phys. Rev. B 66(16), 165217 (2002).
[Crossref]

T. Hirayama and M. Sheik-Bahae, “Real-time chirp diagnostic for ultrashort laser pulses,” Opt. Lett. 27(10), 860–862 (2002).
[Crossref] [PubMed]

T. K. Liang, H. K. Tsang, I. E. Day, J. Drake, A. P. Knights, and M. Asghari, “Silicon waveguide two-photon absorption detector at 1.5 μm wavelength for autocorrelation measurements,” Appl. Phys. Lett. 81(7), 1323–1325 (2002).
[Crossref]

1997 (1)

M. M. Karkhanehchi, C. J. Hamilton, and J. H. Marsh, “Autocorrelation Measurements of Modelocked Nd:YLF Laser Pulses Using Two-Photon Absorption Waveguide Autocorrelator,” IEEE. Photonics Tech. Lett. 9(5), 645–647 (1997).
[Crossref]

1987 (1)

R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
[Crossref]

1985 (1)

1976 (1)

M. M. Choy and R. L. Byer, “Accurate second-order susceptibility measurements of visible and infrared nonlinear crystals,” Phys. Rev. B 14(4), 1693–1706 (1976).
[Crossref]

Agrawal, G. P.

Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91(2), 021111 (2007).
[Crossref]

An, J.

Asghari, M.

T. K. Liang, H. K. Tsang, I. E. Day, J. Drake, A. P. Knights, and M. Asghari, “Silicon waveguide two-photon absorption detector at 1.5 μm wavelength for autocorrelation measurements,” Appl. Phys. Lett. 81(7), 1323–1325 (2002).
[Crossref]

Baba, T.

K. Kondo and T. Baba, “On-chip autocorrelator using counter-propagating slow light in a photonic crystal with two-photon absorption photodiodes,” Optica 4(9), 1109–1112 (2017).
[Crossref]

R. Hayakawa, N. Ishikura, H. C. Nguyen, and T. Baba, “Two-photon-absorption photodiodes in Si photonic-crystal slow-light waveguides,” Appl. Phys. Lett. 102(3), 031114 (2013).
[Crossref]

Barry, L. P.

T. Krug, M. Lynch, A. L. Bradley, J. F. Donegan, L. P. Barry, H. Folliot, J. S. Roberts, and G. Hill, “High-Sensitivity Two-Photon Absorption Microcavity Autocorrelator,” IEEE Photonics Technol. Lett. 16(6), 1543–1545 (2004).
[Crossref]

Bender, D. A.

Bennett, B. R.

R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
[Crossref]

Boyd, R. W.

Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91(2), 021111 (2007).
[Crossref]

Boyraz, O.

E. K. Tien, X. Z. Sang, F. Qing, Q. Song, and O. Boyraz, “Ultrafast pulse characterization using cross phase modulation in silicon,” Appl. Phys. Lett. 95(5), 051101 (2009).
[Crossref]

Bradley, A. L.

T. Krug, M. Lynch, A. L. Bradley, J. F. Donegan, L. P. Barry, H. Folliot, J. S. Roberts, and G. Hill, “High-Sensitivity Two-Photon Absorption Microcavity Autocorrelator,” IEEE Photonics Technol. Lett. 16(6), 1543–1545 (2004).
[Crossref]

Bristow, A. D.

A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. 90(19), 191104 (2007).
[Crossref]

Byer, R. L.

M. M. Choy and R. L. Byer, “Accurate second-order susceptibility measurements of visible and infrared nonlinear crystals,” Phys. Rev. B 14(4), 1693–1706 (1976).
[Crossref]

Caughey, D. M.

D. M. Caughey and R. E. Thomas, “Carrier Mobilities in Silicon Empirically Related to Doping and Field,” in Proceedings of the IEEE (1967), pp. 2192–2193.
[Crossref]

Chen, X.

Chong, E. Z.

E. Z. Chong, T. F. Watson, and F. Festy, “Autocorrelation measurement of femtosecond laser pulses based on two-photon absorption in GaP photodiode,” Appl. Phys. Lett. 105(6), 062111 (2014).
[Crossref]

Choy, M. M.

M. M. Choy and R. L. Byer, “Accurate second-order susceptibility measurements of visible and infrared nonlinear crystals,” Phys. Rev. B 14(4), 1693–1706 (1976).
[Crossref]

Clark, A.

C. Monat, C. Grillet, M. Collins, A. Clark, J. Schroeder, C. Xiong, J. Li, L. O’Faolain, T. F. Krauss, B. J. Eggleton, and D. J. Moss, “Integrated optical auto-correlator based on third-harmonic generation in a silicon photonic crystal waveguide,” Nat. Commun. 5(1), 3246 (2014).
[Crossref] [PubMed]

Cohen, O.

Collins, M.

C. Monat, C. Grillet, M. Collins, A. Clark, J. Schroeder, C. Xiong, J. Li, L. O’Faolain, T. F. Krauss, B. J. Eggleton, and D. J. Moss, “Integrated optical auto-correlator based on third-harmonic generation in a silicon photonic crystal waveguide,” Nat. Commun. 5(1), 3246 (2014).
[Crossref] [PubMed]

Cong, G.

Day, I. E.

T. K. Liang, H. K. Tsang, I. E. Day, J. Drake, A. P. Knights, and M. Asghari, “Silicon waveguide two-photon absorption detector at 1.5 μm wavelength for autocorrelation measurements,” Appl. Phys. Lett. 81(7), 1323–1325 (2002).
[Crossref]

Dinu, M.

Donegan, J. F.

T. Krug, M. Lynch, A. L. Bradley, J. F. Donegan, L. P. Barry, H. Folliot, J. S. Roberts, and G. Hill, “High-Sensitivity Two-Photon Absorption Microcavity Autocorrelator,” IEEE Photonics Technol. Lett. 16(6), 1543–1545 (2004).
[Crossref]

Drake, J.

T. K. Liang, H. K. Tsang, I. E. Day, J. Drake, A. P. Knights, and M. Asghari, “Silicon waveguide two-photon absorption detector at 1.5 μm wavelength for autocorrelation measurements,” Appl. Phys. Lett. 81(7), 1323–1325 (2002).
[Crossref]

Dulkeith, E.

Eggleton, B. J.

Y. Zhang, C. Husko, S. Lefrancois, I. H. Rey, T. F. Krauss, J. Schröder, and B. J. Eggleton, “Cross-phase modulation-induced spectral broadening in silicon waveguides,” Opt. Express 24(1), 443–451 (2016).
[Crossref] [PubMed]

C. Monat, C. Grillet, M. Collins, A. Clark, J. Schroeder, C. Xiong, J. Li, L. O’Faolain, T. F. Krauss, B. J. Eggleton, and D. J. Moss, “Integrated optical auto-correlator based on third-harmonic generation in a silicon photonic crystal waveguide,” Nat. Commun. 5(1), 3246 (2014).
[Crossref] [PubMed]

Fauchet, P. M.

Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91(2), 021111 (2007).
[Crossref]

Festy, F.

E. Z. Chong, T. F. Watson, and F. Festy, “Autocorrelation measurement of femtosecond laser pulses based on two-photon absorption in GaP photodiode,” Appl. Phys. Lett. 105(6), 062111 (2014).
[Crossref]

Folliot, H.

T. Krug, M. Lynch, A. L. Bradley, J. F. Donegan, L. P. Barry, H. Folliot, J. S. Roberts, and G. Hill, “High-Sensitivity Two-Photon Absorption Microcavity Autocorrelator,” IEEE Photonics Technol. Lett. 16(6), 1543–1545 (2004).
[Crossref]

Grillet, C.

C. Monat, C. Grillet, M. Collins, A. Clark, J. Schroeder, C. Xiong, J. Li, L. O’Faolain, T. F. Krauss, B. J. Eggleton, and D. J. Moss, “Integrated optical auto-correlator based on third-harmonic generation in a silicon photonic crystal waveguide,” Nat. Commun. 5(1), 3246 (2014).
[Crossref] [PubMed]

Hak, D.

Hamilton, C. J.

M. M. Karkhanehchi, C. J. Hamilton, and J. H. Marsh, “Autocorrelation Measurements of Modelocked Nd:YLF Laser Pulses Using Two-Photon Absorption Waveguide Autocorrelator,” IEEE. Photonics Tech. Lett. 9(5), 645–647 (1997).
[Crossref]

Hasselbeck, M. P.

Hayakawa, R.

R. Hayakawa, N. Ishikura, H. C. Nguyen, and T. Baba, “Two-photon-absorption photodiodes in Si photonic-crystal slow-light waveguides,” Appl. Phys. Lett. 102(3), 031114 (2013).
[Crossref]

Hill, G.

T. Krug, M. Lynch, A. L. Bradley, J. F. Donegan, L. P. Barry, H. Folliot, J. S. Roberts, and G. Hill, “High-Sensitivity Two-Photon Absorption Microcavity Autocorrelator,” IEEE Photonics Technol. Lett. 16(6), 1543–1545 (2004).
[Crossref]

Hirayama, T.

Husko, C.

Imasaka, T.

Ishikura, N.

R. Hayakawa, N. Ishikura, H. C. Nguyen, and T. Baba, “Two-photon-absorption photodiodes in Si photonic-crystal slow-light waveguides,” Appl. Phys. Lett. 102(3), 031114 (2013).
[Crossref]

Karkhanehchi, M. M.

M. M. Karkhanehchi, C. J. Hamilton, and J. H. Marsh, “Autocorrelation Measurements of Modelocked Nd:YLF Laser Pulses Using Two-Photon Absorption Waveguide Autocorrelator,” IEEE. Photonics Tech. Lett. 9(5), 645–647 (1997).
[Crossref]

Kida, Y.

Kilper, D. C.

Kim, D. E.

Knights, A. P.

T. K. Liang, H. K. Tsang, I. E. Day, J. Drake, A. P. Knights, and M. Asghari, “Silicon waveguide two-photon absorption detector at 1.5 μm wavelength for autocorrelation measurements,” Appl. Phys. Lett. 81(7), 1323–1325 (2002).
[Crossref]

Kondo, K.

Krauss, T. F.

Y. Zhang, C. Husko, S. Lefrancois, I. H. Rey, T. F. Krauss, J. Schröder, and B. J. Eggleton, “Cross-phase modulation-induced spectral broadening in silicon waveguides,” Opt. Express 24(1), 443–451 (2016).
[Crossref] [PubMed]

C. Monat, C. Grillet, M. Collins, A. Clark, J. Schroeder, C. Xiong, J. Li, L. O’Faolain, T. F. Krauss, B. J. Eggleton, and D. J. Moss, “Integrated optical auto-correlator based on third-harmonic generation in a silicon photonic crystal waveguide,” Nat. Commun. 5(1), 3246 (2014).
[Crossref] [PubMed]

Krug, T.

T. Krug, M. Lynch, A. L. Bradley, J. F. Donegan, L. P. Barry, H. Folliot, J. S. Roberts, and G. Hill, “High-Sensitivity Two-Photon Absorption Microcavity Autocorrelator,” IEEE Photonics Technol. Lett. 16(6), 1543–1545 (2004).
[Crossref]

Kwon, O.

Lefrancois, S.

Li, J.

C. Monat, C. Grillet, M. Collins, A. Clark, J. Schroeder, C. Xiong, J. Li, L. O’Faolain, T. F. Krauss, B. J. Eggleton, and D. J. Moss, “Integrated optical auto-correlator based on third-harmonic generation in a silicon photonic crystal waveguide,” Nat. Commun. 5(1), 3246 (2014).
[Crossref] [PubMed]

Liang, T. K.

T. K. Liang, H. K. Tsang, I. E. Day, J. Drake, A. P. Knights, and M. Asghari, “Silicon waveguide two-photon absorption detector at 1.5 μm wavelength for autocorrelation measurements,” Appl. Phys. Lett. 81(7), 1323–1325 (2002).
[Crossref]

Lin, Q.

Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91(2), 021111 (2007).
[Crossref]

Liu, A.

Lynch, M.

T. Krug, M. Lynch, A. L. Bradley, J. F. Donegan, L. P. Barry, H. Folliot, J. S. Roberts, and G. Hill, “High-Sensitivity Two-Photon Absorption Microcavity Autocorrelator,” IEEE Photonics Technol. Lett. 16(6), 1543–1545 (2004).
[Crossref]

Maegami, Y.

Marsh, J. H.

M. M. Karkhanehchi, C. J. Hamilton, and J. H. Marsh, “Autocorrelation Measurements of Modelocked Nd:YLF Laser Pulses Using Two-Photon Absorption Waveguide Autocorrelator,” IEEE. Photonics Tech. Lett. 9(5), 645–647 (1997).
[Crossref]

Monat, C.

C. Monat, C. Grillet, M. Collins, A. Clark, J. Schroeder, C. Xiong, J. Li, L. O’Faolain, T. F. Krauss, B. J. Eggleton, and D. J. Moss, “Integrated optical auto-correlator based on third-harmonic generation in a silicon photonic crystal waveguide,” Nat. Commun. 5(1), 3246 (2014).
[Crossref] [PubMed]

Moss, D. J.

C. Monat, C. Grillet, M. Collins, A. Clark, J. Schroeder, C. Xiong, J. Li, L. O’Faolain, T. F. Krauss, B. J. Eggleton, and D. J. Moss, “Integrated optical auto-correlator based on third-harmonic generation in a silicon photonic crystal waveguide,” Nat. Commun. 5(1), 3246 (2014).
[Crossref] [PubMed]

Nguyen, H. C.

R. Hayakawa, N. Ishikura, H. C. Nguyen, and T. Baba, “Two-photon-absorption photodiodes in Si photonic-crystal slow-light waveguides,” Appl. Phys. Lett. 102(3), 031114 (2013).
[Crossref]

Niigaki, R.

O’Faolain, L.

C. Monat, C. Grillet, M. Collins, A. Clark, J. Schroeder, C. Xiong, J. Li, L. O’Faolain, T. F. Krauss, B. J. Eggleton, and D. J. Moss, “Integrated optical auto-correlator based on third-harmonic generation in a silicon photonic crystal waveguide,” Nat. Commun. 5(1), 3246 (2014).
[Crossref] [PubMed]

Ohno, M.

Okano, M.

Osgood, R. M.

Paniccia, M.

Panoiu, N. C.

Pinault, S. C.

Piredda, G.

Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91(2), 021111 (2007).
[Crossref]

Potasek, M. J.

Pyun, K.

Qing, F.

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[Crossref]

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[Crossref]

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T. Krug, M. Lynch, A. L. Bradley, J. F. Donegan, L. P. Barry, H. Folliot, J. S. Roberts, and G. Hill, “High-Sensitivity Two-Photon Absorption Microcavity Autocorrelator,” IEEE Photonics Technol. Lett. 16(6), 1543–1545 (2004).
[Crossref]

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A. J. Sabbah and D. M. Riffe, “Femtosecond pump-probe reflectivity study of silicon carrier dynamics,” Phys. Rev. B 66(16), 165217 (2002).
[Crossref]

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E. K. Tien, X. Z. Sang, F. Qing, Q. Song, and O. Boyraz, “Ultrafast pulse characterization using cross phase modulation in silicon,” Appl. Phys. Lett. 95(5), 051101 (2009).
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[Crossref]

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[Crossref]

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E. K. Tien, X. Z. Sang, F. Qing, Q. Song, and O. Boyraz, “Ultrafast pulse characterization using cross phase modulation in silicon,” Appl. Phys. Lett. 95(5), 051101 (2009).
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Appl. Opt. (1)

Appl. Phys. Lett. (6)

E. Z. Chong, T. F. Watson, and F. Festy, “Autocorrelation measurement of femtosecond laser pulses based on two-photon absorption in GaP photodiode,” Appl. Phys. Lett. 105(6), 062111 (2014).
[Crossref]

T. K. Liang, H. K. Tsang, I. E. Day, J. Drake, A. P. Knights, and M. Asghari, “Silicon waveguide two-photon absorption detector at 1.5 μm wavelength for autocorrelation measurements,” Appl. Phys. Lett. 81(7), 1323–1325 (2002).
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[Crossref]

Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91(2), 021111 (2007).
[Crossref]

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[Crossref]

IEEE J. Quantum Electron. (1)

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M. M. Karkhanehchi, C. J. Hamilton, and J. H. Marsh, “Autocorrelation Measurements of Modelocked Nd:YLF Laser Pulses Using Two-Photon Absorption Waveguide Autocorrelator,” IEEE. Photonics Tech. Lett. 9(5), 645–647 (1997).
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Figures (5)

Fig. 1
Fig. 1 (a) Principle schematic of interferometric autocorrelation using the silicon p-i-n waveguide as a photocurrent detector. (b) Fabricated device and its cross-section scanning capacitance microscope indicating carrier distribution in the rib waveguide. (c) Measurement system of interferometric autocorrelation. FL: fiber mode-lock laser. TDL: tunable delay line. PM: power meter. SM: source meter. PC: computer.
Fig. 2
Fig. 2 Experimental interferometric autocorrelation waveforms of the device lengths (a) L = 1 mm and (b) L = 0.5 mm. FDTD simulated autocorrelation waveforms without phase noise for (c) L = 1 mm and (d) L = 0.5 mm, and with phase noise (e) L = 1 mm and (f) L = 0.5 mm.
Fig. 3
Fig. 3 Pulse energy dependences of (a) peak and swing photocurrents of L = 1 mm and (b) pulse widths measured from autocorrelation waveforms as explained in section 3.1. The unit of βTPA is cm/GW. To be noted that the pulse energy in this figure is the calibrated one using the 4.2-dB loss.
Fig. 4
Fig. 4 (a) Electric field (Ex) of two autocorrelated pulses with a time delay Δt = 1 ps before and after propagation along the 1 mm p-i-n waveguide for the pulse energies of 1 and 10 pJ. x denotes the TE polarization. (b) Corresponding temporal response of the index change Δn due to the TPA-induced free carrier effect monitored at z = 0.5 mm, the center of waveguide. z denotes the optical propagation direction along waveguide.
Fig. 5
Fig. 5 Wavelength dependences of (a) group index and (b) dispersion parameter D. (c) Pulse broadening due to GVD. (c) Error ratio in a relation to the input pulse width. Error ratio is defined as 10 × log(Δw/w0) where w0 and Δw are the input pulse width and pulse broadening, respectively.

Equations (10)

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μ 0 H t =×E
D t =×H
D= ε r E+ P i
P α = c 0 ε 0 n 0 α iω E
P FCA,e,h = c 0 ε 0 n 0 σ e,h N e,h iω E
P plasma =2 ε 0 n 0 Δn( N e , N h )E
P TPA = c 0 2 ε 0 2 n 0 2 β TPA 2iω | E | 2
σ e,h = e 3 λ 2 4 π 2 c 0 3 ε 0 n 0 1 m * e,h 2 μ e,h
Δn= e 2 λ 2 8 π 2 c 0 2 ε 0 n 0 Δ N e,h m * e,h
d N e,h dt = c 0 2 ε 0 2 n 0 2 β TPA 8ω | E | 4 N e,h τ

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