Abstract

We present an equivalent circuit model for a silicon carrier-depletion single-drive push–pull Mach–Zehnder modulator (MZM) with its traveling wave electrode made of coplanar strip lines. In particular, the partial-capacitance technique and conformal mapping are used to derive the capacitance associated with each layer. The PN junction is accurately modeled with the fringe capacitances taken into consideration. The circuit model is validated by comparing the calculations with the simulation results. Using this model, we analyze the effect of several key parameters on the modulator performance to optimize the design. Experimental results of MZMs confirm the theoretical analysis. A 56 Gb/s on–off keying modulation and a 40 Gb/s binary phase-shift keying modulation are achieved using the optimized modulator.

© 2016 Chinese Laser Press

Full Article  |  PDF Article

Corrections

27 July 2016: A correction was made to the copyright.


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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  28. K. Goi, N. Ishikura, H. Ishihara, S. Sakamoto, K. Ogawa, T.-Y. Liow, X. Tu, G.-Q. Lo, and D. L. Kwong, “Low-voltage silicon Mach-Zehnder modulator operating at high temperatures without thermo-electric cooling,” in Optical Fiber Communication Conference (Optical Society of America, 2016), paper W2A–23.
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2015 (5)

2014 (4)

K. Goi, A. Oka, H. Kusaka, Y. Terada, K. Ogawa, T.-Y. Liow, X. Tu, G.-Q. Lo, and D.-L. Kwong, “Low-loss high-speed silicon IQ modulator for QPSK/DQPSK in C and L bands,” Opt. Express 22, 10703–10709 (2014).
[Crossref]

R. Ding, Y. Liu, Y. J. Ma, Y. S. Yang, Q. Li, A. E. J. Lim, G. Q. Lo, K. Bergman, T. Baehr-Jones, and M. Hochberg, “High-speed silicon modulator with slow-wave electrodes and fully independent differential drive,” J. Lightwave Technol. 32, 2240–2247 (2014).
[Crossref]

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, M. Nedeljkovic, Y. Hu, D. J. Thomson, K. Li, P. R. Wilson, S.-W. Chen, and S. S. Hsu, “Recent breakthroughs in carrier depletion based silicon optical modulators,” Nanophotonics 3, 229–245 (2014).
[Crossref]

H. Xu, X. Y. Li, X. Xiao, Z. Y. Li, Y. D. Yu, and J. Z. Yu, “Demonstration and characterization of high-speed silicon depletion-mode Mach–Zehnder modulators,” IEEE J. Sel. Top. Quantum Electron. 20, 23–32 (2014).
[Crossref]

2013 (6)

2012 (4)

2011 (1)

2010 (2)

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4, 518–526 (2010).
[Crossref]

T. Y. Liow, K. W. Ang, Q. Fang, J. F. Song, Y. Z. Xiong, M. B. Yu, G. Q. Lo, and D. L. Kwong, “Silicon modulators and germanium photodetectors on SOI: monolithic integration, compatibility, and performance optimization,” IEEE J. Sel. Top. Quantum Electron. 16, 307–315 (2010).
[Crossref]

1998 (1)

V. Milanovic, M. Ozgur, D. C. DeGroot, J. A. Jargon, M. Gaitan, and M. E. Zaghloul, “Characterization of broad-band transmission for coplanar waveguides on CMOS silicon substrates,” IEEE Trans. Microwave Theory Tech. 46, 632–640 (1998).
[Crossref]

1997 (1)

E. L. Chen and S. Y. Chou, “Characteristics of coplanar transmission lines on multilayer substrates: modeling and experiments,” IEEE Trans. Microwave Theory Tech. 45, 939–945 (1997).
[Crossref]

1993 (1)

W. Heinrich, “Quasi-TEM description of MMIC coplanar lines including conductor-loss effects,” IEEE Trans. Microwave Theory Tech. 41, 45–52 (1993).
[Crossref]

1987 (1)

Y. R. Kwon, V. M. Hietala, and K. S. Champlin, “Quasi-TEM analysis of ‘slow-wave’ mode propagation on coplanar microstructure MIS transmission lines,” IEEE Trans. Microwave Theory Tech. 35, 545–551 (1987).
[Crossref]

Ang, K. W.

T. Y. Liow, K. W. Ang, Q. Fang, J. F. Song, Y. Z. Xiong, M. B. Yu, G. Q. Lo, and D. L. Kwong, “Silicon modulators and germanium photodetectors on SOI: monolithic integration, compatibility, and performance optimization,” IEEE J. Sel. Top. Quantum Electron. 16, 307–315 (2010).
[Crossref]

Ayazi, A.

Azadeh, S. S.

Baehr-Jones, T.

Bergman, K.

Bogaerts, W.

Buca, D.

Chagnon, M.

Champlin, K. S.

Y. R. Kwon, V. M. Hietala, and K. S. Champlin, “Quasi-TEM analysis of ‘slow-wave’ mode propagation on coplanar microstructure MIS transmission lines,” IEEE Trans. Microwave Theory Tech. 35, 545–551 (1987).
[Crossref]

Chee, E. K. S.

Chen, E. L.

E. L. Chen and S. Y. Chou, “Characteristics of coplanar transmission lines on multilayer substrates: modeling and experiments,” IEEE Trans. Microwave Theory Tech. 45, 939–945 (1997).
[Crossref]

Chen, J.

Y. Zhou, L. Zhou, F. Su, J. Xie, H. Zhu, X. Li, and J. Chen, “Linearity measurement of a silicon single-drive push-pull Mach-Zehnder modulator,” in Conference on Lasers and Electro-Optics (CLEO): Science and Innovations (Optical Society of America, 2015), paper SW3N-6.

Chen, J. P.

J. T. Wang, L. J. Zhou, H. K. Zhu, R. Yang, Y. Y. Zhou, L. Liu, T. Wang, and J. P. Chen, “Silicon high-speed binary phase-shift keying modulator with a single-drive push-pull high-speed traveling wave electrode,” Photon. Res. 3, 58–62 (2015).
[Crossref]

H. K. Zhu, L. J. Zhou, T. Wang, L. Liu, C. Y. Wong, Y. Y. Zhou, R. Yang, X. W. Li, and J. P. Chen, “Optimized silicon QPSK modulator with 64-Gb/s modulation speed,” IEEE Photon. J. 7, 1–6 (2015).
[Crossref]

Chen, L.

P. Dong, L. Chen, and Y.-k. Chen, “High-speed low-voltage single-drive push-pull silicon Mach-Zehnder modulators,” Opt. Express 20, 6163–6169 (2012).
[Crossref]

L. Chen, P. Dong, and Y. K. Chen, “Chirp and dispersion tolerance of a single-drive push-pull silicon modulator at 28  Gb/s,” IEEE Photon. Technol. Lett. 24, 936–938 (2012).
[Crossref]

Chen, S.-W.

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, M. Nedeljkovic, Y. Hu, D. J. Thomson, K. Li, P. R. Wilson, S.-W. Chen, and S. S. Hsu, “Recent breakthroughs in carrier depletion based silicon optical modulators,” Nanophotonics 3, 229–245 (2014).
[Crossref]

Chen, Y. K.

L. Chen, P. Dong, and Y. K. Chen, “Chirp and dispersion tolerance of a single-drive push-pull silicon modulator at 28  Gb/s,” IEEE Photon. Technol. Lett. 24, 936–938 (2012).
[Crossref]

Chen, Y.-k.

Chou, S. Y.

E. L. Chen and S. Y. Chou, “Characteristics of coplanar transmission lines on multilayer substrates: modeling and experiments,” IEEE Trans. Microwave Theory Tech. 45, 939–945 (1997).
[Crossref]

DeGroot, D. C.

V. Milanovic, M. Ozgur, D. C. DeGroot, J. A. Jargon, M. Gaitan, and M. E. Zaghloul, “Characterization of broad-band transmission for coplanar waveguides on CMOS silicon substrates,” IEEE Trans. Microwave Theory Tech. 46, 632–640 (1998).
[Crossref]

Ding, J. F.

Ding, R.

Dong, P.

L. Chen, P. Dong, and Y. K. Chen, “Chirp and dispersion tolerance of a single-drive push-pull silicon modulator at 28  Gb/s,” IEEE Photon. Technol. Lett. 24, 936–938 (2012).
[Crossref]

P. Dong, L. Chen, and Y.-k. Chen, “High-speed low-voltage single-drive push-pull silicon Mach-Zehnder modulators,” Opt. Express 20, 6163–6169 (2012).
[Crossref]

Fang, Q.

X. Tu, T.-Y. Liow, J. Song, X. Luo, Q. Fang, M. Yu, and G.-Q. Lo, “50-Gb/s silicon optical modulator with traveling-wave electrodes,” Opt. Express 21, 12776–12782 (2013).
[Crossref]

T. Y. Liow, K. W. Ang, Q. Fang, J. F. Song, Y. Z. Xiong, M. B. Yu, G. Q. Lo, and D. L. Kwong, “Silicon modulators and germanium photodetectors on SOI: monolithic integration, compatibility, and performance optimization,” IEEE J. Sel. Top. Quantum Electron. 16, 307–315 (2010).
[Crossref]

Fedeli, J.

Fedeli, J. M.

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. F. Hu, M. Nedeljkovic, X. Yang, P. Petropoulos, and G. Z. Mashanovich, “High performance Mach–Zehnder-based silicon optical modulators,” IEEE J. Sel. Top. Quantum Electron. 19, 85–94 (2013).
[Crossref]

Fournier, M.

Gaitan, M.

V. Milanovic, M. Ozgur, D. C. DeGroot, J. A. Jargon, M. Gaitan, and M. E. Zaghloul, “Characterization of broad-band transmission for coplanar waveguides on CMOS silicon substrates,” IEEE Trans. Microwave Theory Tech. 46, 632–640 (1998).
[Crossref]

Gajda, A.

D. Petousi, L. Zimmermann, A. Gajda, M. Kroh, K. Voigt, G. Winzer, B. Tillack, and K. Petermann, “Analysis of optical and electrical tradeoffs of traveling-wave depletion-type Si Mach–Zehnder modulators for high-speed operation,” IEEE J. Sel. Top. Quantum Electron. 21, 199–206 (2015).
[Crossref]

Gan, F. W.

Gardes, F.

Gardes, F. Y.

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, M. Nedeljkovic, Y. Hu, D. J. Thomson, K. Li, P. R. Wilson, S.-W. Chen, and S. S. Hsu, “Recent breakthroughs in carrier depletion based silicon optical modulators,” Nanophotonics 3, 229–245 (2014).
[Crossref]

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. F. Hu, M. Nedeljkovic, X. Yang, P. Petropoulos, and G. Z. Mashanovich, “High performance Mach–Zehnder-based silicon optical modulators,” IEEE J. Sel. Top. Quantum Electron. 19, 85–94 (2013).
[Crossref]

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4, 518–526 (2010).
[Crossref]

Goi, K.

K. Goi, A. Oka, H. Kusaka, Y. Terada, K. Ogawa, T.-Y. Liow, X. Tu, G.-Q. Lo, and D.-L. Kwong, “Low-loss high-speed silicon IQ modulator for QPSK/DQPSK in C and L bands,” Opt. Express 22, 10703–10709 (2014).
[Crossref]

K. Goi, N. Ishikura, H. Ishihara, S. Sakamoto, K. Ogawa, T.-Y. Liow, X. Tu, G.-Q. Lo, and D. L. Kwong, “Low-voltage silicon Mach-Zehnder modulator operating at high temperatures without thermo-electric cooling,” in Optical Fiber Communication Conference (Optical Society of America, 2016), paper W2A–23.

Grosse, P.

Harris, N. C.

Heinrich, W.

W. Heinrich, “Quasi-TEM description of MMIC coplanar lines including conductor-loss effects,” IEEE Trans. Microwave Theory Tech. 41, 45–52 (1993).
[Crossref]

Hietala, V. M.

Y. R. Kwon, V. M. Hietala, and K. S. Champlin, “Quasi-TEM analysis of ‘slow-wave’ mode propagation on coplanar microstructure MIS transmission lines,” IEEE Trans. Microwave Theory Tech. 35, 545–551 (1987).
[Crossref]

Hochberg, M.

Hsu, S. S.

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, M. Nedeljkovic, Y. Hu, D. J. Thomson, K. Li, P. R. Wilson, S.-W. Chen, and S. S. Hsu, “Recent breakthroughs in carrier depletion based silicon optical modulators,” Nanophotonics 3, 229–245 (2014).
[Crossref]

Hu, Y.

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, M. Nedeljkovic, Y. Hu, D. J. Thomson, K. Li, P. R. Wilson, S.-W. Chen, and S. S. Hsu, “Recent breakthroughs in carrier depletion based silicon optical modulators,” Nanophotonics 3, 229–245 (2014).
[Crossref]

D. Thomson, F. Gardes, Y. Hu, G. Mashanovich, M. Fournier, P. Grosse, J. Fedeli, and G. Reed, “High contrast 40  Gbit/s optical modulation in silicon,” Opt. Express 19, 11507–11516 (2011).
[Crossref]

Hu, Y. F.

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. F. Hu, M. Nedeljkovic, X. Yang, P. Petropoulos, and G. Z. Mashanovich, “High performance Mach–Zehnder-based silicon optical modulators,” IEEE J. Sel. Top. Quantum Electron. 19, 85–94 (2013).
[Crossref]

Ishihara, H.

K. Goi, N. Ishikura, H. Ishihara, S. Sakamoto, K. Ogawa, T.-Y. Liow, X. Tu, G.-Q. Lo, and D. L. Kwong, “Low-voltage silicon Mach-Zehnder modulator operating at high temperatures without thermo-electric cooling,” in Optical Fiber Communication Conference (Optical Society of America, 2016), paper W2A–23.

Ishikura, N.

K. Goi, N. Ishikura, H. Ishihara, S. Sakamoto, K. Ogawa, T.-Y. Liow, X. Tu, G.-Q. Lo, and D. L. Kwong, “Low-voltage silicon Mach-Zehnder modulator operating at high temperatures without thermo-electric cooling,” in Optical Fiber Communication Conference (Optical Society of America, 2016), paper W2A–23.

Jargon, J. A.

V. Milanovic, M. Ozgur, D. C. DeGroot, J. A. Jargon, M. Gaitan, and M. E. Zaghloul, “Characterization of broad-band transmission for coplanar waveguides on CMOS silicon substrates,” IEEE Trans. Microwave Theory Tech. 46, 632–640 (1998).
[Crossref]

Jayatilleka, H.

H. Jayatilleka, W. D. Sacher, and J. K. S. Poon, “Analytical model and fringing-field parasitics of carrier-depletion silicon-on-insulator optical modulation diodes,” IEEE Photon. J. 5, 2200211 (2013).
[Crossref]

Ji, R. Q.

Kroh, M.

D. Petousi, L. Zimmermann, A. Gajda, M. Kroh, K. Voigt, G. Winzer, B. Tillack, and K. Petermann, “Analysis of optical and electrical tradeoffs of traveling-wave depletion-type Si Mach–Zehnder modulators for high-speed operation,” IEEE J. Sel. Top. Quantum Electron. 21, 199–206 (2015).
[Crossref]

Kusaka, H.

Kwon, Y. R.

Y. R. Kwon, V. M. Hietala, and K. S. Champlin, “Quasi-TEM analysis of ‘slow-wave’ mode propagation on coplanar microstructure MIS transmission lines,” IEEE Trans. Microwave Theory Tech. 35, 545–551 (1987).
[Crossref]

Kwong, D. L.

T. Y. Liow, K. W. Ang, Q. Fang, J. F. Song, Y. Z. Xiong, M. B. Yu, G. Q. Lo, and D. L. Kwong, “Silicon modulators and germanium photodetectors on SOI: monolithic integration, compatibility, and performance optimization,” IEEE J. Sel. Top. Quantum Electron. 16, 307–315 (2010).
[Crossref]

K. Goi, N. Ishikura, H. Ishihara, S. Sakamoto, K. Ogawa, T.-Y. Liow, X. Tu, G.-Q. Lo, and D. L. Kwong, “Low-voltage silicon Mach-Zehnder modulator operating at high temperatures without thermo-electric cooling,” in Optical Fiber Communication Conference (Optical Society of America, 2016), paper W2A–23.

Kwong, D.-L.

Lee, P.

Lessard, S.

Li, H.

Li, K.

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, M. Nedeljkovic, Y. Hu, D. J. Thomson, K. Li, P. R. Wilson, S.-W. Chen, and S. S. Hsu, “Recent breakthroughs in carrier depletion based silicon optical modulators,” Nanophotonics 3, 229–245 (2014).
[Crossref]

Li, L.

Li, M.

X. Xiao, M. Li, Z. Li, L. Wang, Q. Yang, and S. Yu, “Substrate removed silicon Mach-Zehnder modulator for high baud rate optical intensity modulations,” in Optical Fiber Communication Conference (Optical Society of America, 2016), paper Th4H-5.

Li, Q.

Li, X.

Y. Zhou, L. Zhou, F. Su, J. Xie, H. Zhu, X. Li, and J. Chen, “Linearity measurement of a silicon single-drive push-pull Mach-Zehnder modulator,” in Conference on Lasers and Electro-Optics (CLEO): Science and Innovations (Optical Society of America, 2015), paper SW3N-6.

Li, X. W.

H. K. Zhu, L. J. Zhou, T. Wang, L. Liu, C. Y. Wong, Y. Y. Zhou, R. Yang, X. W. Li, and J. P. Chen, “Optimized silicon QPSK modulator with 64-Gb/s modulation speed,” IEEE Photon. J. 7, 1–6 (2015).
[Crossref]

Li, X. Y.

H. Xu, X. Y. Li, X. Xiao, Z. Y. Li, Y. D. Yu, and J. Z. Yu, “Demonstration and characterization of high-speed silicon depletion-mode Mach–Zehnder modulators,” IEEE J. Sel. Top. Quantum Electron. 20, 23–32 (2014).
[Crossref]

Li, Z.

X. Xiao, M. Li, Z. Li, L. Wang, Q. Yang, and S. Yu, “Substrate removed silicon Mach-Zehnder modulator for high baud rate optical intensity modulations,” in Optical Fiber Communication Conference (Optical Society of America, 2016), paper Th4H-5.

Li, Z. Y.

H. Xu, X. Y. Li, X. Xiao, Z. Y. Li, Y. D. Yu, and J. Z. Yu, “Demonstration and characterization of high-speed silicon depletion-mode Mach–Zehnder modulators,” IEEE J. Sel. Top. Quantum Electron. 20, 23–32 (2014).
[Crossref]

Lim, A. E. J.

Lim, A. E.-J.

Ling, W.

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T. Y. Liow, K. W. Ang, Q. Fang, J. F. Song, Y. Z. Xiong, M. B. Yu, G. Q. Lo, and D. L. Kwong, “Silicon modulators and germanium photodetectors on SOI: monolithic integration, compatibility, and performance optimization,” IEEE J. Sel. Top. Quantum Electron. 16, 307–315 (2010).
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Liu, L.

J. T. Wang, L. J. Zhou, H. K. Zhu, R. Yang, Y. Y. Zhou, L. Liu, T. Wang, and J. P. Chen, “Silicon high-speed binary phase-shift keying modulator with a single-drive push-pull high-speed traveling wave electrode,” Photon. Res. 3, 58–62 (2015).
[Crossref]

H. K. Zhu, L. J. Zhou, T. Wang, L. Liu, C. Y. Wong, Y. Y. Zhou, R. Yang, X. W. Li, and J. P. Chen, “Optimized silicon QPSK modulator with 64-Gb/s modulation speed,” IEEE Photon. J. 7, 1–6 (2015).
[Crossref]

Liu, S.

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. F. Hu, M. Nedeljkovic, X. Yang, P. Petropoulos, and G. Z. Mashanovich, “High performance Mach–Zehnder-based silicon optical modulators,” IEEE J. Sel. Top. Quantum Electron. 19, 85–94 (2013).
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Liu, Y.

Lo, G. Q.

R. Ding, Y. Liu, Y. J. Ma, Y. S. Yang, Q. Li, A. E. J. Lim, G. Q. Lo, K. Bergman, T. Baehr-Jones, and M. Hochberg, “High-speed silicon modulator with slow-wave electrodes and fully independent differential drive,” J. Lightwave Technol. 32, 2240–2247 (2014).
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T. Y. Liow, K. W. Ang, Q. Fang, J. F. Song, Y. Z. Xiong, M. B. Yu, G. Q. Lo, and D. L. Kwong, “Silicon modulators and germanium photodetectors on SOI: monolithic integration, compatibility, and performance optimization,” IEEE J. Sel. Top. Quantum Electron. 16, 307–315 (2010).
[Crossref]

Lo, G.-Q.

K. Goi, A. Oka, H. Kusaka, Y. Terada, K. Ogawa, T.-Y. Liow, X. Tu, G.-Q. Lo, and D.-L. Kwong, “Low-loss high-speed silicon IQ modulator for QPSK/DQPSK in C and L bands,” Opt. Express 22, 10703–10709 (2014).
[Crossref]

X. Tu, T.-Y. Liow, J. Song, X. Luo, Q. Fang, M. Yu, and G.-Q. Lo, “50-Gb/s silicon optical modulator with traveling-wave electrodes,” Opt. Express 21, 12776–12782 (2013).
[Crossref]

K. Goi, N. Ishikura, H. Ishihara, S. Sakamoto, K. Ogawa, T.-Y. Liow, X. Tu, G.-Q. Lo, and D. L. Kwong, “Low-voltage silicon Mach-Zehnder modulator operating at high temperatures without thermo-electric cooling,” in Optical Fiber Communication Conference (Optical Society of America, 2016), paper W2A–23.

Lo, P. G.-Q.

Luo, X.

Ma, Y.

Ma, Y. J.

Mantl, S.

Mashanovich, G.

Mashanovich, G. Z.

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, M. Nedeljkovic, Y. Hu, D. J. Thomson, K. Li, P. R. Wilson, S.-W. Chen, and S. S. Hsu, “Recent breakthroughs in carrier depletion based silicon optical modulators,” Nanophotonics 3, 229–245 (2014).
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D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. F. Hu, M. Nedeljkovic, X. Yang, P. Petropoulos, and G. Z. Mashanovich, “High performance Mach–Zehnder-based silicon optical modulators,” IEEE J. Sel. Top. Quantum Electron. 19, 85–94 (2013).
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G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4, 518–526 (2010).
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Milanovic, V.

V. Milanovic, M. Ozgur, D. C. DeGroot, J. A. Jargon, M. Gaitan, and M. E. Zaghloul, “Characterization of broad-band transmission for coplanar waveguides on CMOS silicon substrates,” IEEE Trans. Microwave Theory Tech. 46, 632–640 (1998).
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Moscoso-Mártir, A.

Müller, J.

Nedeljkovic, M.

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, M. Nedeljkovic, Y. Hu, D. J. Thomson, K. Li, P. R. Wilson, S.-W. Chen, and S. S. Hsu, “Recent breakthroughs in carrier depletion based silicon optical modulators,” Nanophotonics 3, 229–245 (2014).
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D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. F. Hu, M. Nedeljkovic, X. Yang, P. Petropoulos, and G. Z. Mashanovich, “High performance Mach–Zehnder-based silicon optical modulators,” IEEE J. Sel. Top. Quantum Electron. 19, 85–94 (2013).
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K. Goi, A. Oka, H. Kusaka, Y. Terada, K. Ogawa, T.-Y. Liow, X. Tu, G.-Q. Lo, and D.-L. Kwong, “Low-loss high-speed silicon IQ modulator for QPSK/DQPSK in C and L bands,” Opt. Express 22, 10703–10709 (2014).
[Crossref]

K. Goi, N. Ishikura, H. Ishihara, S. Sakamoto, K. Ogawa, T.-Y. Liow, X. Tu, G.-Q. Lo, and D. L. Kwong, “Low-voltage silicon Mach-Zehnder modulator operating at high temperatures without thermo-electric cooling,” in Optical Fiber Communication Conference (Optical Society of America, 2016), paper W2A–23.

Oka, A.

Ozgur, M.

V. Milanovic, M. Ozgur, D. C. DeGroot, J. A. Jargon, M. Gaitan, and M. E. Zaghloul, “Characterization of broad-band transmission for coplanar waveguides on CMOS silicon substrates,” IEEE Trans. Microwave Theory Tech. 46, 632–640 (1998).
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Paquet, C.

Petermann, K.

D. Petousi, L. Zimmermann, A. Gajda, M. Kroh, K. Voigt, G. Winzer, B. Tillack, and K. Petermann, “Analysis of optical and electrical tradeoffs of traveling-wave depletion-type Si Mach–Zehnder modulators for high-speed operation,” IEEE J. Sel. Top. Quantum Electron. 21, 199–206 (2015).
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D. Petousi, L. Zimmermann, A. Gajda, M. Kroh, K. Voigt, G. Winzer, B. Tillack, and K. Petermann, “Analysis of optical and electrical tradeoffs of traveling-wave depletion-type Si Mach–Zehnder modulators for high-speed operation,” IEEE J. Sel. Top. Quantum Electron. 21, 199–206 (2015).
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D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. F. Hu, M. Nedeljkovic, X. Yang, P. Petropoulos, and G. Z. Mashanovich, “High performance Mach–Zehnder-based silicon optical modulators,” IEEE J. Sel. Top. Quantum Electron. 19, 85–94 (2013).
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Plant, D. V.

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H. Jayatilleka, W. D. Sacher, and J. K. S. Poon, “Analytical model and fringing-field parasitics of carrier-depletion silicon-on-insulator optical modulation diodes,” IEEE Photon. J. 5, 2200211 (2013).
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D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. F. Hu, M. Nedeljkovic, X. Yang, P. Petropoulos, and G. Z. Mashanovich, “High performance Mach–Zehnder-based silicon optical modulators,” IEEE J. Sel. Top. Quantum Electron. 19, 85–94 (2013).
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Qiu, C.

Reed, G.

Reed, G. T.

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, M. Nedeljkovic, Y. Hu, D. J. Thomson, K. Li, P. R. Wilson, S.-W. Chen, and S. S. Hsu, “Recent breakthroughs in carrier depletion based silicon optical modulators,” Nanophotonics 3, 229–245 (2014).
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G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4, 518–526 (2010).
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Sacher, W. D.

H. Jayatilleka, W. D. Sacher, and J. K. S. Poon, “Analytical model and fringing-field parasitics of carrier-depletion silicon-on-insulator optical modulation diodes,” IEEE Photon. J. 5, 2200211 (2013).
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K. Goi, N. Ishikura, H. Ishihara, S. Sakamoto, K. Ogawa, T.-Y. Liow, X. Tu, G.-Q. Lo, and D. L. Kwong, “Low-voltage silicon Mach-Zehnder modulator operating at high temperatures without thermo-electric cooling,” in Optical Fiber Communication Conference (Optical Society of America, 2016), paper W2A–23.

Sheng, Z.

Song, J.

Song, J. F.

T. Y. Liow, K. W. Ang, Q. Fang, J. F. Song, Y. Z. Xiong, M. B. Yu, G. Q. Lo, and D. L. Kwong, “Silicon modulators and germanium photodetectors on SOI: monolithic integration, compatibility, and performance optimization,” IEEE J. Sel. Top. Quantum Electron. 16, 307–315 (2010).
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Su, F.

Y. Zhou, L. Zhou, F. Su, J. Xie, H. Zhu, X. Li, and J. Chen, “Linearity measurement of a silicon single-drive push-pull Mach-Zehnder modulator,” in Conference on Lasers and Electro-Optics (CLEO): Science and Innovations (Optical Society of America, 2015), paper SW3N-6.

Terada, Y.

Thomson, D.

Thomson, D. J.

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, M. Nedeljkovic, Y. Hu, D. J. Thomson, K. Li, P. R. Wilson, S.-W. Chen, and S. S. Hsu, “Recent breakthroughs in carrier depletion based silicon optical modulators,” Nanophotonics 3, 229–245 (2014).
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D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. F. Hu, M. Nedeljkovic, X. Yang, P. Petropoulos, and G. Z. Mashanovich, “High performance Mach–Zehnder-based silicon optical modulators,” IEEE J. Sel. Top. Quantum Electron. 19, 85–94 (2013).
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G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4, 518–526 (2010).
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D. Petousi, L. Zimmermann, A. Gajda, M. Kroh, K. Voigt, G. Winzer, B. Tillack, and K. Petermann, “Analysis of optical and electrical tradeoffs of traveling-wave depletion-type Si Mach–Zehnder modulators for high-speed operation,” IEEE J. Sel. Top. Quantum Electron. 21, 199–206 (2015).
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Voigt, K.

D. Petousi, L. Zimmermann, A. Gajda, M. Kroh, K. Voigt, G. Winzer, B. Tillack, and K. Petermann, “Analysis of optical and electrical tradeoffs of traveling-wave depletion-type Si Mach–Zehnder modulators for high-speed operation,” IEEE J. Sel. Top. Quantum Electron. 21, 199–206 (2015).
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Wang, J.

Wang, J. T.

Wang, L.

X. Xiao, M. Li, Z. Li, L. Wang, Q. Yang, and S. Yu, “Substrate removed silicon Mach-Zehnder modulator for high baud rate optical intensity modulations,” in Optical Fiber Communication Conference (Optical Society of America, 2016), paper Th4H-5.

Wang, T.

J. T. Wang, L. J. Zhou, H. K. Zhu, R. Yang, Y. Y. Zhou, L. Liu, T. Wang, and J. P. Chen, “Silicon high-speed binary phase-shift keying modulator with a single-drive push-pull high-speed traveling wave electrode,” Photon. Res. 3, 58–62 (2015).
[Crossref]

H. K. Zhu, L. J. Zhou, T. Wang, L. Liu, C. Y. Wong, Y. Y. Zhou, R. Yang, X. W. Li, and J. P. Chen, “Optimized silicon QPSK modulator with 64-Gb/s modulation speed,” IEEE Photon. J. 7, 1–6 (2015).
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Wang, X.

Wilson, P. R.

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, M. Nedeljkovic, Y. Hu, D. J. Thomson, K. Li, P. R. Wilson, S.-W. Chen, and S. S. Hsu, “Recent breakthroughs in carrier depletion based silicon optical modulators,” Nanophotonics 3, 229–245 (2014).
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D. Petousi, L. Zimmermann, A. Gajda, M. Kroh, K. Voigt, G. Winzer, B. Tillack, and K. Petermann, “Analysis of optical and electrical tradeoffs of traveling-wave depletion-type Si Mach–Zehnder modulators for high-speed operation,” IEEE J. Sel. Top. Quantum Electron. 21, 199–206 (2015).
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Wong, C. Y.

H. K. Zhu, L. J. Zhou, T. Wang, L. Liu, C. Y. Wong, Y. Y. Zhou, R. Yang, X. W. Li, and J. P. Chen, “Optimized silicon QPSK modulator with 64-Gb/s modulation speed,” IEEE Photon. J. 7, 1–6 (2015).
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Xiao, X.

H. Xu, X. Y. Li, X. Xiao, Z. Y. Li, Y. D. Yu, and J. Z. Yu, “Demonstration and characterization of high-speed silicon depletion-mode Mach–Zehnder modulators,” IEEE J. Sel. Top. Quantum Electron. 20, 23–32 (2014).
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X. Xiao, M. Li, Z. Li, L. Wang, Q. Yang, and S. Yu, “Substrate removed silicon Mach-Zehnder modulator for high baud rate optical intensity modulations,” in Optical Fiber Communication Conference (Optical Society of America, 2016), paper Th4H-5.

Xie, J.

Y. Zhou, L. Zhou, F. Su, J. Xie, H. Zhu, X. Li, and J. Chen, “Linearity measurement of a silicon single-drive push-pull Mach-Zehnder modulator,” in Conference on Lasers and Electro-Optics (CLEO): Science and Innovations (Optical Society of America, 2015), paper SW3N-6.

Xiong, Y. Z.

T. Y. Liow, K. W. Ang, Q. Fang, J. F. Song, Y. Z. Xiong, M. B. Yu, G. Q. Lo, and D. L. Kwong, “Silicon modulators and germanium photodetectors on SOI: monolithic integration, compatibility, and performance optimization,” IEEE J. Sel. Top. Quantum Electron. 16, 307–315 (2010).
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H. Xu, X. Y. Li, X. Xiao, Z. Y. Li, Y. D. Yu, and J. Z. Yu, “Demonstration and characterization of high-speed silicon depletion-mode Mach–Zehnder modulators,” IEEE J. Sel. Top. Quantum Electron. 20, 23–32 (2014).
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X. Xiao, M. Li, Z. Li, L. Wang, Q. Yang, and S. Yu, “Substrate removed silicon Mach-Zehnder modulator for high baud rate optical intensity modulations,” in Optical Fiber Communication Conference (Optical Society of America, 2016), paper Th4H-5.

Yang, R.

J. T. Wang, L. J. Zhou, H. K. Zhu, R. Yang, Y. Y. Zhou, L. Liu, T. Wang, and J. P. Chen, “Silicon high-speed binary phase-shift keying modulator with a single-drive push-pull high-speed traveling wave electrode,” Photon. Res. 3, 58–62 (2015).
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H. K. Zhu, L. J. Zhou, T. Wang, L. Liu, C. Y. Wong, Y. Y. Zhou, R. Yang, X. W. Li, and J. P. Chen, “Optimized silicon QPSK modulator with 64-Gb/s modulation speed,” IEEE Photon. J. 7, 1–6 (2015).
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D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. F. Hu, M. Nedeljkovic, X. Yang, P. Petropoulos, and G. Z. Mashanovich, “High performance Mach–Zehnder-based silicon optical modulators,” IEEE J. Sel. Top. Quantum Electron. 19, 85–94 (2013).
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Yang, Y. S.

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H. Xu, X. Y. Li, X. Xiao, Z. Y. Li, Y. D. Yu, and J. Z. Yu, “Demonstration and characterization of high-speed silicon depletion-mode Mach–Zehnder modulators,” IEEE J. Sel. Top. Quantum Electron. 20, 23–32 (2014).
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Yu, M. B.

T. Y. Liow, K. W. Ang, Q. Fang, J. F. Song, Y. Z. Xiong, M. B. Yu, G. Q. Lo, and D. L. Kwong, “Silicon modulators and germanium photodetectors on SOI: monolithic integration, compatibility, and performance optimization,” IEEE J. Sel. Top. Quantum Electron. 16, 307–315 (2010).
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X. Xiao, M. Li, Z. Li, L. Wang, Q. Yang, and S. Yu, “Substrate removed silicon Mach-Zehnder modulator for high baud rate optical intensity modulations,” in Optical Fiber Communication Conference (Optical Society of America, 2016), paper Th4H-5.

Yu, Y. D.

H. Xu, X. Y. Li, X. Xiao, Z. Y. Li, Y. D. Yu, and J. Z. Yu, “Demonstration and characterization of high-speed silicon depletion-mode Mach–Zehnder modulators,” IEEE J. Sel. Top. Quantum Electron. 20, 23–32 (2014).
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V. Milanovic, M. Ozgur, D. C. DeGroot, J. A. Jargon, M. Gaitan, and M. E. Zaghloul, “Characterization of broad-band transmission for coplanar waveguides on CMOS silicon substrates,” IEEE Trans. Microwave Theory Tech. 46, 632–640 (1998).
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Zhang, Y.

Zhou, L.

Y. Zhou, L. Zhou, F. Su, J. Xie, H. Zhu, X. Li, and J. Chen, “Linearity measurement of a silicon single-drive push-pull Mach-Zehnder modulator,” in Conference on Lasers and Electro-Optics (CLEO): Science and Innovations (Optical Society of America, 2015), paper SW3N-6.

Zhou, L. J.

J. T. Wang, L. J. Zhou, H. K. Zhu, R. Yang, Y. Y. Zhou, L. Liu, T. Wang, and J. P. Chen, “Silicon high-speed binary phase-shift keying modulator with a single-drive push-pull high-speed traveling wave electrode,” Photon. Res. 3, 58–62 (2015).
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H. K. Zhu, L. J. Zhou, T. Wang, L. Liu, C. Y. Wong, Y. Y. Zhou, R. Yang, X. W. Li, and J. P. Chen, “Optimized silicon QPSK modulator with 64-Gb/s modulation speed,” IEEE Photon. J. 7, 1–6 (2015).
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Zhou, Y.

Y. Zhou, L. Zhou, F. Su, J. Xie, H. Zhu, X. Li, and J. Chen, “Linearity measurement of a silicon single-drive push-pull Mach-Zehnder modulator,” in Conference on Lasers and Electro-Optics (CLEO): Science and Innovations (Optical Society of America, 2015), paper SW3N-6.

Zhou, Y. Y.

J. T. Wang, L. J. Zhou, H. K. Zhu, R. Yang, Y. Y. Zhou, L. Liu, T. Wang, and J. P. Chen, “Silicon high-speed binary phase-shift keying modulator with a single-drive push-pull high-speed traveling wave electrode,” Photon. Res. 3, 58–62 (2015).
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H. K. Zhu, L. J. Zhou, T. Wang, L. Liu, C. Y. Wong, Y. Y. Zhou, R. Yang, X. W. Li, and J. P. Chen, “Optimized silicon QPSK modulator with 64-Gb/s modulation speed,” IEEE Photon. J. 7, 1–6 (2015).
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Zhu, H.

Y. Zhou, L. Zhou, F. Su, J. Xie, H. Zhu, X. Li, and J. Chen, “Linearity measurement of a silicon single-drive push-pull Mach-Zehnder modulator,” in Conference on Lasers and Electro-Optics (CLEO): Science and Innovations (Optical Society of America, 2015), paper SW3N-6.

Zhu, H. K.

J. T. Wang, L. J. Zhou, H. K. Zhu, R. Yang, Y. Y. Zhou, L. Liu, T. Wang, and J. P. Chen, “Silicon high-speed binary phase-shift keying modulator with a single-drive push-pull high-speed traveling wave electrode,” Photon. Res. 3, 58–62 (2015).
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H. K. Zhu, L. J. Zhou, T. Wang, L. Liu, C. Y. Wong, Y. Y. Zhou, R. Yang, X. W. Li, and J. P. Chen, “Optimized silicon QPSK modulator with 64-Gb/s modulation speed,” IEEE Photon. J. 7, 1–6 (2015).
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D. Petousi, L. Zimmermann, A. Gajda, M. Kroh, K. Voigt, G. Winzer, B. Tillack, and K. Petermann, “Analysis of optical and electrical tradeoffs of traveling-wave depletion-type Si Mach–Zehnder modulators for high-speed operation,” IEEE J. Sel. Top. Quantum Electron. 21, 199–206 (2015).
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X. Tu, T.-Y. Liow, J. Song, X. Luo, Q. Fang, M. Yu, and G.-Q. Lo, “50-Gb/s silicon optical modulator with traveling-wave electrodes,” Opt. Express 21, 12776–12782 (2013).
[Crossref]

M. Streshinsky, R. Ding, Y. Liu, A. Novack, Y. Yang, Y. Ma, X. Tu, E. K. S. Chee, A. E.-J. Lim, and P. G.-Q. Lo, “Low power 50  Gb/s silicon traveling wave Mach–Zehnder modulator near 1300  nm,” Opt. Express 21, 30350–30357 (2013).
[Crossref]

K. Goi, A. Oka, H. Kusaka, Y. Terada, K. Ogawa, T.-Y. Liow, X. Tu, G.-Q. Lo, and D.-L. Kwong, “Low-loss high-speed silicon IQ modulator for QPSK/DQPSK in C and L bands,” Opt. Express 22, 10703–10709 (2014).
[Crossref]

D. Thomson, F. Gardes, Y. Hu, G. Mashanovich, M. Fournier, P. Grosse, J. Fedeli, and G. Reed, “High contrast 40  Gbit/s optical modulation in silicon,” Opt. Express 19, 11507–11516 (2011).
[Crossref]

P. Dong, L. Chen, and Y.-k. Chen, “High-speed low-voltage single-drive push-pull silicon Mach-Zehnder modulators,” Opt. Express 20, 6163–6169 (2012).
[Crossref]

T. Baehr-Jones, R. Ding, Y. Liu, A. Ayazi, T. Pinguet, N. C. Harris, M. Streshinsky, P. Lee, Y. Zhang, and A. E.-J. Lim, “Ultralow drive voltage silicon traveling-wave modulator,” Opt. Express 20, 12014–12020 (2012).
[Crossref]

Photon. Res. (1)

Other (3)

K. Goi, N. Ishikura, H. Ishihara, S. Sakamoto, K. Ogawa, T.-Y. Liow, X. Tu, G.-Q. Lo, and D. L. Kwong, “Low-voltage silicon Mach-Zehnder modulator operating at high temperatures without thermo-electric cooling,” in Optical Fiber Communication Conference (Optical Society of America, 2016), paper W2A–23.

X. Xiao, M. Li, Z. Li, L. Wang, Q. Yang, and S. Yu, “Substrate removed silicon Mach-Zehnder modulator for high baud rate optical intensity modulations,” in Optical Fiber Communication Conference (Optical Society of America, 2016), paper Th4H-5.

Y. Zhou, L. Zhou, F. Su, J. Xie, H. Zhu, X. Li, and J. Chen, “Linearity measurement of a silicon single-drive push-pull Mach-Zehnder modulator,” in Conference on Lasers and Electro-Optics (CLEO): Science and Innovations (Optical Society of America, 2015), paper SW3N-6.

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Figures (10)

Fig. 1.
Fig. 1. (a) Cross section of a single-drive push–pull carrier-depletion-based optical modulator. Inset shows the zoom-in of the PN junction. (b) Microwave electrical field distribution in the modulation arms.
Fig. 2.
Fig. 2. (a) Electrical components contributed by each layer. (b) Two-port network and the equivalent circuit model of the modulator.
Fig. 3.
Fig. 3. (a) Calculated L and R as a function of frequency. Comparison between calculation and simulation results for (b) characteristic impedance ( Z ), (c) propagation constant ( β ) and microwave attenuation ( α ), and (d) EE S -parameters ( S 21 and S 11 ).
Fig. 4.
Fig. 4. (a) Calculated EO 3 dB bandwidth as a function of metal line width W m t and gap separation G m t . (b) Effect of BOX layer thickness H BOX on the TWE characteristics. (c) Effect of doping separation S dop on the TWE characteristics. (d) Microwave attenuation of the MZMs designed with a single or a segmented PN junction.
Fig. 5.
Fig. 5. (a) Modulation efficiency V π L and (b) optical loss versus PN junction doping concentrations at V b = 0    V .
Fig. 6.
Fig. 6. (a) Measurement setup to characterize the modulators. (b) Optical microscope image of the MZMs.
Fig. 7.
Fig. 7. (a) Optical transmission spectra of MZM-2 at 0 and 4 V reverse biases. (b) Modulation efficiency V π L of MZM-2. (c) Real part of the characteristic impedance. (d) Imaginary part of the characteristic impedance. (e) Microwave transmission response EE S 21 . (f) Microwave reflection response EE S 11 . (g) Modulator frequency response EO S 21 .
Fig. 8.
Fig. 8. Measured 56 Gb/s OOK modulation eye diagrams for (a) MZM-1, (b) MZM-2, and (c) MZM-3.
Fig. 9.
Fig. 9. 40 Gb/s BPSK modulation eye diagrams for (a) MZM-1, (b) MZM-2, and (c) MZM-3.
Fig. 10.
Fig. 10. Measured 40 Gb/s BPSK modulation BER curves for (a) MZM-2 and (b) MZM-3. Inset shows the constellation diagram.

Tables (1)

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Table 1. Design Parameters for Three MZMs

Equations (41)

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F ( k i ) = K ( k i ) K ( k i ) ,
k 0 = 1 ( x a x b ) 2 ,
k i = 1 sinh 2 ( π x a 2 h i ) sinh 2 ( π x b 2 h i ) ,
G sub = 2 σ sub [ F 2 ( k 0 ) F ( k 1 ) F 2 ( k 0 ) F ( k 2 ) ] ,
C sub = 1 2 ϵ 0 ϵ si [ F 2 ( k 0 ) F ( k 1 ) F 2 ( k 0 ) F ( k 2 ) ] ,
C sub _ s = ϵ 0 ϵ o x W m t / h 2 ,
C sub = C sub _ s C sub C sub + C sub _ s + C sub 1 .
C BOX = 1 2 ϵ 0 ϵ o x [ F 2 ( k 0 ) F ( k 2 ) F 2 ( k 0 ) F ( k 3 ) ] .
C dep = C + C f + C c p s t + C c p s b ,
C = ϵ 0 ϵ si H rib W D ,
C f = ϵ 0 ϵ o x π ln ( 2 π H rib W D ) ,
C c p s t , b = ϵ 0 ϵ o x K ( k ) K ( k ) ,
k = W D ( W D + t n + t p ) ( W D + t n ) ( W D + t p ) ,
t p , n = 2 W D p , n ( ϵ o x ) π ,
W D = W D p + W D n = 2 ϵ 0 ϵ si q ( N A + N D N A N D ) ( V j V b 2 k B T q ) ,
R p , n = ρ p , n 1 + j ω m ρ p , n ϵ 0 ϵ si ( S dop H slab + S p , n H rib ) ,
S p = W rib 2 + Δ jun W D p ,
S n = W rib 2 Δ jun W D n ,
ρ p , n = 1 q μ p , n N A , D ,
C via = ϵ 0 ϵ o x h 4 / G via .
C m t = ϵ 0 T m t / G m t .
C air = ϵ 0 F 2 ( k 0 ) 1 2 ϵ 0 F 2 ( k 0 ) F ( k 1 ) .
R slab = ρ p + + 1 + j ω m ρ p ++ ϵ 0 ϵ s i 2 H slab W p ,
C slab = ϵ 0 ϵ si W p H slab d 1 Λ ,
L = 2 L CPW ,
R = R G + R S = 2 R CPW ,
Z = R eff + j w L eff G eff + j w C eff ,
γ = α + j β = ( R eff + j w L eff ) ( G eff + j w C eff ) .
V avg ( ω m ) = V g ( 1 + ρ 1 ) exp ( i β 0 L ) ( V + + ρ 2 V ) 2 [ exp ( γ L ) + ρ 1 ρ 2 exp ( γ L ) ] ,
V ± = exp [ ± i ( i γ β 0 ) L 2 ] sin [ ( i γ β 0 ) L / 2 ] ( i γ β 0 ) L / 2 ,
ρ 1 = Z Z s Z + Z s ,
ρ 2 = Z t Z Z t + Z ,
β 0 = ω m c n g o ,
S 21 ( ω m ) = ( 1 + Γ s ) ( 1 + Γ t ) e γ L ,
S 11 ( ω m ) = Γ s ,
Γ s = Z i n Z s Z i n + Z s ,
Γ t = Z i n Z t Z i n + Z t ,
Z i n = Z Z t Z tanh ( γ L ) Z Z t tanh ( γ L ) .
V dep ( ω m ) = V avg ( ω m ) 1 + j ω ( R p + R n ) C dep .
m ( ω m ) = | V dep ( ω m ) V dep ( ω 0 ) | .
V π 2 π λ d n eff o d V | V = V b = π ,

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