50 Gb/s hybrid silicon traveling-wave electroabsorption modulator

Yongbo Tang; Hui-Wen Chen; Siddharth Jain; Jonathan D. Peters; Urban Westergren; John E. Bowers

doi:10.1364/OE.19.005811

50 Gb/s hybrid silicon traveling-wave electroabsorption modulator

Yongbo Tang, Hui-Wen Chen, Siddharth Jain, Jonathan D. Peters, Urban Westergren, and John E. Bowers

Optics Express
Vol. 19,
Issue 7,
pp. 5811-5816
(2011)
•https://doi.org/10.1364/OE.19.005811

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Yongbo Tang, Hui-Wen Chen, Siddharth Jain, Jonathan D. Peters, Urban Westergren, and John E. Bowers, "50 Gb/s hybrid silicon traveling-wave electroabsorption modulator," Opt. Express 19, 5811-5816 (2011)

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Related Topics
Table of Contents Category
- Optoelectronics
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Photonic integrated circuits (250.5300)
- Waveguide modulators (250.7360)

About this Article
History
- Original Manuscript: January 13, 2011
- Revised Manuscript: February 25, 2011
- Manuscript Accepted: March 4, 2011
- Published: March 14, 2011

Accessible

Open Access

Abstract

We have demonstrated a traveling-wave electroabsorption modulator based on the hybrid silicon platform. For a device with a 100 μm active segment, the small-signal electro/optical response renders a 3 dB bandwidth of around 42 GHz and its modulation efficiency reaches 23 GHz/V. A dynamic extinction ratio of 9.8 dB with a driving voltage swing of only 2 V was demonstrated at a transmission rate of 50 Gb/s. This represents a significant improvement for modulators compatible with integration of silicon-based photonic integrated circuits.

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References

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Author
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Publication

D. Liang, G. Roelkens, R. Baets, and J. Bowers, “Hybrid integrated platforms for silicon photonics,” Materials 3(3), 1782–1802 (2010).
[Crossref]
J. Michel, J. Liu, and L. Kimerling, “High-performance Ge-on-Si photodetectors,” Nat. Photonics 4(8), 527–534 (2010).
[Crossref]
http://www.intel.com/pressroom/archive/releases/20100727comp_sm.htm
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(6975), 615–618 (2004).
[Crossref] [PubMed]
A. Liu, L. Liao, D. Rubin, H. Nguyen, B. Ciftcioglu, Y. Chetrit, N. Izhaky, and M. Paniccia, “High-speed optical modulation based on carrier depletion in a silicon waveguide,” Opt. Express 15(2), 660–668 (2007).
[Crossref] [PubMed]
Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[Crossref] [PubMed]
J. Basak, L. Liao, A. Liu, D. Rubin, Y. Chetrit, H. Nguyen, D. Samara-Rubio, R. Cohen, N. Izhaky, and M. Paniccia, “Developments in gigascale silicon optical modulators using free carrier dispersion mechanisms,” Adv. Opt. Technol. 2008, 1 (2008).
[Crossref]
J. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. Kimerling, and J. Michel, “Waveguide-integrated, ultralow-energy GeSi electroabsorption modulators,” Nat. Photonics 2(7), 433–437 (2008).
[Crossref]
Y. H. Kuo, Y. K. Lee, Y. Ge, S. Ren, J. E. Roth, T. I. Kamins, D. A. Miller, and J. S. Harris, “Strong quantum-confined Stark effect in germanium quantum-well structures on silicon,” Nature 437(7063), 1334–1336 (2005).
[Crossref] [PubMed]
D. Liang and J. Bowers, “Recent progress in lasers on silicon,” Nat. Photonics 4(8), 511–517 (2010).
[Crossref]
R. Walker, “High-speed III-V semiconductor intensity modulators,” IEEE J. Quantum Electron. 27(3), 654–667 (1991).
[Crossref]
R. Lewén, S. Irmscher, U. Westergren, L. Thylén, and U. Eriksson, “Segmented transmission-line electroabsorption modulators,” J. Lightwave Technol. 22(1), 172–179 (2004).
[Crossref]
C. Doerr, L. Zhang, P. Winzer, J. Sinsky, A. Adamiecki, N. Sauer, and G. Raybon, “Compact High-Speed InP DQPSK Modulator,” IEEE Photon. Technol. Lett. 19(15), 1184–1186 (2007).
[Crossref]
Y. Chiu, T. Wu, W. Cheng, F. Lin, and J. Bowers, “Enhanced performance in traveling-wave electroabsorption modulators based on undercut-etching the active-region,” IEEE Photon. Technol. Lett. 17(10), 2065–2067 (2005).
[Crossref]
Y. H. Kuo, H. W. Chen, and J. E. Bowers, “High speed hybrid silicon evanescent electroabsorption modulator,” Opt. Express 16(13), 9936–9941 (2008).
[Crossref] [PubMed]
F. Devaux, S. Chelles, A. Ougazzaden, A. Mircea, and J. Harmand, “Electroabsorption modulators for high-bit-rate optical communications: a comparison of strained InGaAs/InAIAs and InGaAsP/InGaAsP MQW,” Semicond. Sci. Technol. 10(7), 887–901 (1995).
[Crossref]
S. Jain, M. Sysak, G. Kurczveil, and J. Bowers, “Integrated Hybrid Silicon DFB Laser-EAM array using Quantum Well Intermixing, IEEE J. Sel. Top. Quantum Electron. Submitted.
H. W. Chen, J. D. Peters, and J. E. Bowers, “Forty Gb/s hybrid silicon Mach-Zehnder modulator with low chirp,” Opt. Express 19(2), 1455–1460 (2011).
[Crossref] [PubMed]
H. Boudinov, H. H. Tan, and C. Jagadish, “Electrical isolation of n-type and p-type InP layers by proton bombardment,” J. Appl. Phys. 89(10), 5343–5347 (2001).
[Crossref]
F. Devaux, Y. Sorel, and J. F. Kerdiles, “Simple measurement of fiber dispersion and of chirp parameter of intensity modulated light emitter,” J. Lightwave Technol. 11(13), 1937–1940 (1993).
[Crossref]

2011 (1)

H. W. Chen, J. D. Peters, and J. E. Bowers, “Forty Gb/s hybrid silicon Mach-Zehnder modulator with low chirp,” Opt. Express 19(2), 1455–1460 (2011).
[Crossref] [PubMed]

2010 (3)

D. Liang, G. Roelkens, R. Baets, and J. Bowers, “Hybrid integrated platforms for silicon photonics,” Materials 3(3), 1782–1802 (2010).
[Crossref]

J. Michel, J. Liu, and L. Kimerling, “High-performance Ge-on-Si photodetectors,” Nat. Photonics 4(8), 527–534 (2010).
[Crossref]

D. Liang and J. Bowers, “Recent progress in lasers on silicon,” Nat. Photonics 4(8), 511–517 (2010).
[Crossref]

2008 (3)

J. Basak, L. Liao, A. Liu, D. Rubin, Y. Chetrit, H. Nguyen, D. Samara-Rubio, R. Cohen, N. Izhaky, and M. Paniccia, “Developments in gigascale silicon optical modulators using free carrier dispersion mechanisms,” Adv. Opt. Technol. 2008, 1 (2008).
[Crossref]

J. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. Kimerling, and J. Michel, “Waveguide-integrated, ultralow-energy GeSi electroabsorption modulators,” Nat. Photonics 2(7), 433–437 (2008).
[Crossref]

Y. H. Kuo, H. W. Chen, and J. E. Bowers, “High speed hybrid silicon evanescent electroabsorption modulator,” Opt. Express 16(13), 9936–9941 (2008).
[Crossref] [PubMed]

2007 (2)

A. Liu, L. Liao, D. Rubin, H. Nguyen, B. Ciftcioglu, Y. Chetrit, N. Izhaky, and M. Paniccia, “High-speed optical modulation based on carrier depletion in a silicon waveguide,” Opt. Express 15(2), 660–668 (2007).
[Crossref] [PubMed]

C. Doerr, L. Zhang, P. Winzer, J. Sinsky, A. Adamiecki, N. Sauer, and G. Raybon, “Compact High-Speed InP DQPSK Modulator,” IEEE Photon. Technol. Lett. 19(15), 1184–1186 (2007).
[Crossref]

2005 (3)

Y. Chiu, T. Wu, W. Cheng, F. Lin, and J. Bowers, “Enhanced performance in traveling-wave electroabsorption modulators based on undercut-etching the active-region,” IEEE Photon. Technol. Lett. 17(10), 2065–2067 (2005).
[Crossref]

Y. H. Kuo, Y. K. Lee, Y. Ge, S. Ren, J. E. Roth, T. I. Kamins, D. A. Miller, and J. S. Harris, “Strong quantum-confined Stark effect in germanium quantum-well structures on silicon,” Nature 437(7063), 1334–1336 (2005).
[Crossref] [PubMed]

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[Crossref] [PubMed]

2004 (2)

R. Lewén, S. Irmscher, U. Westergren, L. Thylén, and U. Eriksson, “Segmented transmission-line electroabsorption modulators,” J. Lightwave Technol. 22(1), 172–179 (2004).
[Crossref]

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(6975), 615–618 (2004).
[Crossref] [PubMed]

2001 (1)

H. Boudinov, H. H. Tan, and C. Jagadish, “Electrical isolation of n-type and p-type InP layers by proton bombardment,” J. Appl. Phys. 89(10), 5343–5347 (2001).
[Crossref]

1995 (1)

F. Devaux, S. Chelles, A. Ougazzaden, A. Mircea, and J. Harmand, “Electroabsorption modulators for high-bit-rate optical communications: a comparison of strained InGaAs/InAIAs and InGaAsP/InGaAsP MQW,” Semicond. Sci. Technol. 10(7), 887–901 (1995).
[Crossref]

1993 (1)

F. Devaux, Y. Sorel, and J. F. Kerdiles, “Simple measurement of fiber dispersion and of chirp parameter of intensity modulated light emitter,” J. Lightwave Technol. 11(13), 1937–1940 (1993).
[Crossref]

1991 (1)

R. Walker, “High-speed III-V semiconductor intensity modulators,” IEEE J. Quantum Electron. 27(3), 654–667 (1991).
[Crossref]

Adamiecki, A.

C. Doerr, L. Zhang, P. Winzer, J. Sinsky, A. Adamiecki, N. Sauer, and G. Raybon, “Compact High-Speed InP DQPSK Modulator,” IEEE Photon. Technol. Lett. 19(15), 1184–1186 (2007).
[Crossref]

Baets, R.

D. Liang, G. Roelkens, R. Baets, and J. Bowers, “Hybrid integrated platforms for silicon photonics,” Materials 3(3), 1782–1802 (2010).
[Crossref]

Basak, J.

Beals, M.

Bernardis, S.

Boudinov, H.

H. Boudinov, H. H. Tan, and C. Jagadish, “Electrical isolation of n-type and p-type InP layers by proton bombardment,” J. Appl. Phys. 89(10), 5343–5347 (2001).
[Crossref]

Bowers, J.

D. Liang, G. Roelkens, R. Baets, and J. Bowers, “Hybrid integrated platforms for silicon photonics,” Materials 3(3), 1782–1802 (2010).
[Crossref]

D. Liang and J. Bowers, “Recent progress in lasers on silicon,” Nat. Photonics 4(8), 511–517 (2010).
[Crossref]

S. Jain, M. Sysak, G. Kurczveil, and J. Bowers, “Integrated Hybrid Silicon DFB Laser-EAM array using Quantum Well Intermixing, IEEE J. Sel. Top. Quantum Electron. Submitted.

Bowers, J. E.

H. W. Chen, J. D. Peters, and J. E. Bowers, “Forty Gb/s hybrid silicon Mach-Zehnder modulator with low chirp,” Opt. Express 19(2), 1455–1460 (2011).
[Crossref] [PubMed]

Y. H. Kuo, H. W. Chen, and J. E. Bowers, “High speed hybrid silicon evanescent electroabsorption modulator,” Opt. Express 16(13), 9936–9941 (2008).
[Crossref] [PubMed]

Chelles, S.

Chen, H. W.

H. W. Chen, J. D. Peters, and J. E. Bowers, “Forty Gb/s hybrid silicon Mach-Zehnder modulator with low chirp,” Opt. Express 19(2), 1455–1460 (2011).
[Crossref] [PubMed]

Y. H. Kuo, H. W. Chen, and J. E. Bowers, “High speed hybrid silicon evanescent electroabsorption modulator,” Opt. Express 16(13), 9936–9941 (2008).
[Crossref] [PubMed]

Cheng, J.

Cheng, W.

Chetrit, Y.

Chiu, Y.

Ciftcioglu, B.

Cohen, O.

Cohen, R.

Devaux, F.

Doerr, C.

C. Doerr, L. Zhang, P. Winzer, J. Sinsky, A. Adamiecki, N. Sauer, and G. Raybon, “Compact High-Speed InP DQPSK Modulator,” IEEE Photon. Technol. Lett. 19(15), 1184–1186 (2007).
[Crossref]

Eriksson, U.

R. Lewén, S. Irmscher, U. Westergren, L. Thylén, and U. Eriksson, “Segmented transmission-line electroabsorption modulators,” J. Lightwave Technol. 22(1), 172–179 (2004).
[Crossref]

Ge, Y.

Harmand, J.

Harris, J. S.

Irmscher, S.

R. Lewén, S. Irmscher, U. Westergren, L. Thylén, and U. Eriksson, “Segmented transmission-line electroabsorption modulators,” J. Lightwave Technol. 22(1), 172–179 (2004).
[Crossref]

Izhaky, N.

Jagadish, C.

H. Boudinov, H. H. Tan, and C. Jagadish, “Electrical isolation of n-type and p-type InP layers by proton bombardment,” J. Appl. Phys. 89(10), 5343–5347 (2001).
[Crossref]

Jain, S.

S. Jain, M. Sysak, G. Kurczveil, and J. Bowers, “Integrated Hybrid Silicon DFB Laser-EAM array using Quantum Well Intermixing, IEEE J. Sel. Top. Quantum Electron. Submitted.

Jones, R.

Kamins, T. I.

Kerdiles, J. F.

Kimerling, L.

J. Michel, J. Liu, and L. Kimerling, “High-performance Ge-on-Si photodetectors,” Nat. Photonics 4(8), 527–534 (2010).
[Crossref]

Kuo, Y. H.

Y. H. Kuo, H. W. Chen, and J. E. Bowers, “High speed hybrid silicon evanescent electroabsorption modulator,” Opt. Express 16(13), 9936–9941 (2008).
[Crossref] [PubMed]

Kurczveil, G.

S. Jain, M. Sysak, G. Kurczveil, and J. Bowers, “Integrated Hybrid Silicon DFB Laser-EAM array using Quantum Well Intermixing, IEEE J. Sel. Top. Quantum Electron. Submitted.

Lee, Y. K.

Lewén, R.

R. Lewén, S. Irmscher, U. Westergren, L. Thylén, and U. Eriksson, “Segmented transmission-line electroabsorption modulators,” J. Lightwave Technol. 22(1), 172–179 (2004).
[Crossref]

Liang, D.

D. Liang and J. Bowers, “Recent progress in lasers on silicon,” Nat. Photonics 4(8), 511–517 (2010).
[Crossref]

D. Liang, G. Roelkens, R. Baets, and J. Bowers, “Hybrid integrated platforms for silicon photonics,” Materials 3(3), 1782–1802 (2010).
[Crossref]

Liao, L.

Lin, F.

Lipson, M.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[Crossref] [PubMed]

Liu, A.

Liu, J.

J. Michel, J. Liu, and L. Kimerling, “High-performance Ge-on-Si photodetectors,” Nat. Photonics 4(8), 527–534 (2010).
[Crossref]

Michel, J.

J. Michel, J. Liu, and L. Kimerling, “High-performance Ge-on-Si photodetectors,” Nat. Photonics 4(8), 527–534 (2010).
[Crossref]

Miller, D. A.

Mircea, A.

Nguyen, H.

Nicolaescu, R.

Ougazzaden, A.

Paniccia, M.

Peters, J. D.

H. W. Chen, J. D. Peters, and J. E. Bowers, “Forty Gb/s hybrid silicon Mach-Zehnder modulator with low chirp,” Opt. Express 19(2), 1455–1460 (2011).
[Crossref] [PubMed]

Pomerene, A.

Pradhan, S.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[Crossref] [PubMed]

Raybon, G.

C. Doerr, L. Zhang, P. Winzer, J. Sinsky, A. Adamiecki, N. Sauer, and G. Raybon, “Compact High-Speed InP DQPSK Modulator,” IEEE Photon. Technol. Lett. 19(15), 1184–1186 (2007).
[Crossref]

Ren, S.

Roelkens, G.

D. Liang, G. Roelkens, R. Baets, and J. Bowers, “Hybrid integrated platforms for silicon photonics,” Materials 3(3), 1782–1802 (2010).
[Crossref]

Roth, J. E.

Rubin, D.

Samara-Rubio, D.

Sauer, N.

C. Doerr, L. Zhang, P. Winzer, J. Sinsky, A. Adamiecki, N. Sauer, and G. Raybon, “Compact High-Speed InP DQPSK Modulator,” IEEE Photon. Technol. Lett. 19(15), 1184–1186 (2007).
[Crossref]

Schmidt, B.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[Crossref] [PubMed]

Sinsky, J.

C. Doerr, L. Zhang, P. Winzer, J. Sinsky, A. Adamiecki, N. Sauer, and G. Raybon, “Compact High-Speed InP DQPSK Modulator,” IEEE Photon. Technol. Lett. 19(15), 1184–1186 (2007).
[Crossref]

Sorel, Y.

Sun, R.

Sysak, M.

S. Jain, M. Sysak, G. Kurczveil, and J. Bowers, “Integrated Hybrid Silicon DFB Laser-EAM array using Quantum Well Intermixing, IEEE J. Sel. Top. Quantum Electron. Submitted.

Tan, H. H.

H. Boudinov, H. H. Tan, and C. Jagadish, “Electrical isolation of n-type and p-type InP layers by proton bombardment,” J. Appl. Phys. 89(10), 5343–5347 (2001).
[Crossref]

Thylén, L.

R. Lewén, S. Irmscher, U. Westergren, L. Thylén, and U. Eriksson, “Segmented transmission-line electroabsorption modulators,” J. Lightwave Technol. 22(1), 172–179 (2004).
[Crossref]

Walker, R.

R. Walker, “High-speed III-V semiconductor intensity modulators,” IEEE J. Quantum Electron. 27(3), 654–667 (1991).
[Crossref]

Westergren, U.

R. Lewén, S. Irmscher, U. Westergren, L. Thylén, and U. Eriksson, “Segmented transmission-line electroabsorption modulators,” J. Lightwave Technol. 22(1), 172–179 (2004).
[Crossref]

Winzer, P.

C. Doerr, L. Zhang, P. Winzer, J. Sinsky, A. Adamiecki, N. Sauer, and G. Raybon, “Compact High-Speed InP DQPSK Modulator,” IEEE Photon. Technol. Lett. 19(15), 1184–1186 (2007).
[Crossref]

Wu, T.

Xu, Q.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[Crossref] [PubMed]

Zhang, L.

C. Doerr, L. Zhang, P. Winzer, J. Sinsky, A. Adamiecki, N. Sauer, and G. Raybon, “Compact High-Speed InP DQPSK Modulator,” IEEE Photon. Technol. Lett. 19(15), 1184–1186 (2007).
[Crossref]

Adv. Opt. Technol. (1)

IEEE J. Quantum Electron. (1)

R. Walker, “High-speed III-V semiconductor intensity modulators,” IEEE J. Quantum Electron. 27(3), 654–667 (1991).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

S. Jain, M. Sysak, G. Kurczveil, and J. Bowers, “Integrated Hybrid Silicon DFB Laser-EAM array using Quantum Well Intermixing, IEEE J. Sel. Top. Quantum Electron. Submitted.

IEEE Photon. Technol. Lett. (2)

C. Doerr, L. Zhang, P. Winzer, J. Sinsky, A. Adamiecki, N. Sauer, and G. Raybon, “Compact High-Speed InP DQPSK Modulator,” IEEE Photon. Technol. Lett. 19(15), 1184–1186 (2007).
[Crossref]

J. Appl. Phys. (1)

H. Boudinov, H. H. Tan, and C. Jagadish, “Electrical isolation of n-type and p-type InP layers by proton bombardment,” J. Appl. Phys. 89(10), 5343–5347 (2001).
[Crossref]

J. Lightwave Technol. (2)

R. Lewén, S. Irmscher, U. Westergren, L. Thylén, and U. Eriksson, “Segmented transmission-line electroabsorption modulators,” J. Lightwave Technol. 22(1), 172–179 (2004).
[Crossref]

Materials (1)

D. Liang, G. Roelkens, R. Baets, and J. Bowers, “Hybrid integrated platforms for silicon photonics,” Materials 3(3), 1782–1802 (2010).
[Crossref]

Nat. Photonics (3)

J. Michel, J. Liu, and L. Kimerling, “High-performance Ge-on-Si photodetectors,” Nat. Photonics 4(8), 527–534 (2010).
[Crossref]

D. Liang and J. Bowers, “Recent progress in lasers on silicon,” Nat. Photonics 4(8), 511–517 (2010).
[Crossref]

Nature (3)

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[Crossref] [PubMed]

Opt. Express (3)

Y. H. Kuo, H. W. Chen, and J. E. Bowers, “High speed hybrid silicon evanescent electroabsorption modulator,” Opt. Express 16(13), 9936–9941 (2008).
[Crossref] [PubMed]

H. W. Chen, J. D. Peters, and J. E. Bowers, “Forty Gb/s hybrid silicon Mach-Zehnder modulator with low chirp,” Opt. Express 19(2), 1455–1460 (2011).
[Crossref] [PubMed]

Semicond. Sci. Technol. (1)

Other (1)

http://www.intel.com/pressroom/archive/releases/20100727comp_sm.htm

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Fig. 1 (a) Schematic structure of a hybrid silicon TW-EAM. (b) Cross section of the modulation segment with a superimposed fundamental optical mode. (c) Top-view photograph of a fabricated TW-EAM.

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Fig. 2 Device loss of a hybrid silicon TW-EAM under different biases. The inset shows the schematic of the testing structure for the loss measurement.

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Fig. 3 (a) Small signal E/O response and (b) the microwave reflection to the RF source (S₁₁) for a hybrid silicon TW-EAM with a 100 µm active segment.

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Fig. 4 (a) E/O response with −3V bias for chirp measurement; (b) Chirp parameters at different biases. A 27.92 km single-mode fiber was used.

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Fig. 5 50 Gb/s NRZ eye diagram at λ = 1550 nm with 2³¹-1 PRBS pattern for a 100 μm long hybrid silicon TW-EAM. The modulator was biased at −3 V with a driving voltage swing of only 2 V.

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Equations (1)

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L o s s (d B) = 10 \log_{10} (\frac{P_{2}}{P_{1}}) = 10 \log_{10} (\frac{I_{2}}{I_{1}}),