Abstract

We demonstrate a hybrid silicon modulator operating up to 40 Gb/s with 11.4 dB extinction ratio. The modulator has voltage-length product of 2.4 V-mm and chirp of −0.75 over the entire bias range. As a switch, it has a switching time less than 20 ps.

© 2011 OSA

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  1. H. Park, M. N. Sysak, H.-W. Chen, A. W. Fang, D. Liang, L. Liao, B. R. Koch, J. Bovington, Y. Tang, K. Wong, M. Jacob-Mitos, R. Jones, and J. E. Bowers, “Device and Integration Technology for Silicon Photonic Transmitters,” IEEE J. Sel. Top. Quantum Electron. (to be published).
  2. A. S. 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]
  3. H.-W. Chen, Y.-H. Kuo, and J. E. Bowers, “A Hybrid Silicon-AlGaInAs Phase Modulator,” IEEE Photon. Technol. Lett. 20(23), 1920–1922 (2008).
    [Crossref]
  4. J. Van Campenhout, W. M. J. Green, S. Assefa, and Y. A. Vlasov, “Low-power, 2 x 2 silicon electro-optic switch with 110-nm bandwidth for broadband reconfigurable optical networks,” Opt. Express 17(26), 24020–24029 (2009).
    [Crossref]
  5. Q. Xu, S. Manipatruni, B. Schmidt, J. Shakya, and M. Lipson, “12.5 Gbit/s carrier-injection-based silicon micro-ring silicon modulators,” Opt. Express 15(2), 430–436 (2007).
    [Crossref] [PubMed]
  6. H. W. Chen, Y. H. Kuo, and J. E. Bowers, “High speed hybrid silicon evanescent Mach-Zehnder modulator and switch,” Opt. Express 16(25), 20571–20576 (2008).
    [Crossref] [PubMed]
  7. J. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, “Waveguide-integrated, ultra-energy GeSi electro-absorption modulators,” Nat. Photonics 2(7), 433–437 (2008).
    [Crossref]
  8. H.-W. Chen, Y.-H. Kuo, and J. E. Bowers, “25Gb/s hybrid silicon switch using a capacitively loaded traveling wave electrode,” Opt. Express 18(2), 1070–1075 (2010).
    [Crossref] [PubMed]
  9. A. W. Fang, H. Park, O. Cohen, R. Jones, M. J. Paniccia, and J. E. Bowers, “Electrically pumped hybrid AlGaInAs-silicon evanescent laser,” Opt. Express 14(20), 9203–9210 (2006).
    [Crossref] [PubMed]
  10. J. Shin, C. Ozturk, S. R. Sakamoto, Y. J. Chiu, and N. Dagli, “Novel T-Rail Electrodes for Substrate Removed Low-Voltage High-Speed GaAs/AlGaAs Electrooptic Modulators,” IEEE Trans. Microw. Theory Tech. 53(2), 636–643 (2005).
    [Crossref]
  11. S. Akiyama, H. Itoh, S. Sekiguchi, S. Hirose, T. Takeuchi, A. Kuramata, and T. Yamamoto, “InP-Based Mach-Zehnder Modulator With Capacitively Loaded Traveling-Wave Electrodes,” J. Lightwave Technol. 26(5), 608–615 (2008).
    [Crossref]
  12. 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]
  13. A. Sneh, and C. Doerr, Integrated Optical Circuits and Components - Design and Applications (Marcel Dekker, Inc., 1999), Chap. 7.
  14. F. Devaux, Y. Sorel, and J. F. Kerdiles, “Simple Measurement of fiber Dispersion and of Chirp Parameter of Intensity Modulated Light Emitter,” IEEE Photon. Technol. Lett. 11, 1937–1940 (1993).
  15. S. R. Jain, M. N. Sysak, G. Kurczveil, and J. E. Bowers, “Integration of Hybrid Silicon DFB Laser and Electro-Absorption Modulator using Quantum Well Intermixing,” in Proceedings of IEEE European Conference on Optical Communication (Turin, Italy, 2010), Tu.5C5.

2010 (1)

2009 (1)

2008 (4)

H.-W. Chen, Y.-H. Kuo, and J. E. Bowers, “A Hybrid Silicon-AlGaInAs Phase Modulator,” IEEE Photon. Technol. Lett. 20(23), 1920–1922 (2008).
[Crossref]

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

S. Akiyama, H. Itoh, S. Sekiguchi, S. Hirose, T. Takeuchi, A. Kuramata, and T. Yamamoto, “InP-Based Mach-Zehnder Modulator With Capacitively Loaded Traveling-Wave Electrodes,” J. Lightwave Technol. 26(5), 608–615 (2008).
[Crossref]

H. W. Chen, Y. H. Kuo, and J. E. Bowers, “High speed hybrid silicon evanescent Mach-Zehnder modulator and switch,” Opt. Express 16(25), 20571–20576 (2008).
[Crossref] [PubMed]

2007 (2)

2006 (1)

2005 (1)

J. Shin, C. Ozturk, S. R. Sakamoto, Y. J. Chiu, and N. Dagli, “Novel T-Rail Electrodes for Substrate Removed Low-Voltage High-Speed GaAs/AlGaAs Electrooptic Modulators,” IEEE Trans. Microw. Theory Tech. 53(2), 636–643 (2005).
[Crossref]

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]

1993 (1)

F. Devaux, Y. Sorel, and J. F. Kerdiles, “Simple Measurement of fiber Dispersion and of Chirp Parameter of Intensity Modulated Light Emitter,” IEEE Photon. Technol. Lett. 11, 1937–1940 (1993).

Akiyama, S.

Assefa, S.

Beals, M.

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

Bernardis, S.

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

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]

Bovington, J.

H. Park, M. N. Sysak, H.-W. Chen, A. W. Fang, D. Liang, L. Liao, B. R. Koch, J. Bovington, Y. Tang, K. Wong, M. Jacob-Mitos, R. Jones, and J. E. Bowers, “Device and Integration Technology for Silicon Photonic Transmitters,” IEEE J. Sel. Top. Quantum Electron. (to be published).

Bowers, J. E.

H.-W. Chen, Y.-H. Kuo, and J. E. Bowers, “25Gb/s hybrid silicon switch using a capacitively loaded traveling wave electrode,” Opt. Express 18(2), 1070–1075 (2010).
[Crossref] [PubMed]

H.-W. Chen, Y.-H. Kuo, and J. E. Bowers, “A Hybrid Silicon-AlGaInAs Phase Modulator,” IEEE Photon. Technol. Lett. 20(23), 1920–1922 (2008).
[Crossref]

H. W. Chen, Y. H. Kuo, and J. E. Bowers, “High speed hybrid silicon evanescent Mach-Zehnder modulator and switch,” Opt. Express 16(25), 20571–20576 (2008).
[Crossref] [PubMed]

A. W. Fang, H. Park, O. Cohen, R. Jones, M. J. Paniccia, and J. E. Bowers, “Electrically pumped hybrid AlGaInAs-silicon evanescent laser,” Opt. Express 14(20), 9203–9210 (2006).
[Crossref] [PubMed]

H. Park, M. N. Sysak, H.-W. Chen, A. W. Fang, D. Liang, L. Liao, B. R. Koch, J. Bovington, Y. Tang, K. Wong, M. Jacob-Mitos, R. Jones, and J. E. Bowers, “Device and Integration Technology for Silicon Photonic Transmitters,” IEEE J. Sel. Top. Quantum Electron. (to be published).

Chen, H. W.

Chen, H.-W.

H.-W. Chen, Y.-H. Kuo, and J. E. Bowers, “25Gb/s hybrid silicon switch using a capacitively loaded traveling wave electrode,” Opt. Express 18(2), 1070–1075 (2010).
[Crossref] [PubMed]

H.-W. Chen, Y.-H. Kuo, and J. E. Bowers, “A Hybrid Silicon-AlGaInAs Phase Modulator,” IEEE Photon. Technol. Lett. 20(23), 1920–1922 (2008).
[Crossref]

H. Park, M. N. Sysak, H.-W. Chen, A. W. Fang, D. Liang, L. Liao, B. R. Koch, J. Bovington, Y. Tang, K. Wong, M. Jacob-Mitos, R. Jones, and J. E. Bowers, “Device and Integration Technology for Silicon Photonic Transmitters,” IEEE J. Sel. Top. Quantum Electron. (to be published).

Cheng, J.

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

Chetrit, Y.

Chiu, Y. J.

J. Shin, C. Ozturk, S. R. Sakamoto, Y. J. Chiu, and N. Dagli, “Novel T-Rail Electrodes for Substrate Removed Low-Voltage High-Speed GaAs/AlGaAs Electrooptic Modulators,” IEEE Trans. Microw. Theory Tech. 53(2), 636–643 (2005).
[Crossref]

Ciftcioglu, B.

Cohen, O.

Dagli, N.

J. Shin, C. Ozturk, S. R. Sakamoto, Y. J. Chiu, and N. Dagli, “Novel T-Rail Electrodes for Substrate Removed Low-Voltage High-Speed GaAs/AlGaAs Electrooptic Modulators,” IEEE Trans. Microw. Theory Tech. 53(2), 636–643 (2005).
[Crossref]

Devaux, F.

F. Devaux, Y. Sorel, and J. F. Kerdiles, “Simple Measurement of fiber Dispersion and of Chirp Parameter of Intensity Modulated Light Emitter,” IEEE Photon. Technol. Lett. 11, 1937–1940 (1993).

Fang, A. W.

A. W. Fang, H. Park, O. Cohen, R. Jones, M. J. Paniccia, and J. E. Bowers, “Electrically pumped hybrid AlGaInAs-silicon evanescent laser,” Opt. Express 14(20), 9203–9210 (2006).
[Crossref] [PubMed]

H. Park, M. N. Sysak, H.-W. Chen, A. W. Fang, D. Liang, L. Liao, B. R. Koch, J. Bovington, Y. Tang, K. Wong, M. Jacob-Mitos, R. Jones, and J. E. Bowers, “Device and Integration Technology for Silicon Photonic Transmitters,” IEEE J. Sel. Top. Quantum Electron. (to be published).

Green, W. M. J.

Hirose, S.

Itoh, H.

Izhaky, N.

Jacob-Mitos, M.

H. Park, M. N. Sysak, H.-W. Chen, A. W. Fang, D. Liang, L. Liao, B. R. Koch, J. Bovington, Y. Tang, K. Wong, M. Jacob-Mitos, R. Jones, and J. E. Bowers, “Device and Integration Technology for Silicon Photonic Transmitters,” IEEE J. Sel. Top. Quantum Electron. (to be published).

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]

Jones, R.

A. W. Fang, H. Park, O. Cohen, R. Jones, M. J. Paniccia, and J. E. Bowers, “Electrically pumped hybrid AlGaInAs-silicon evanescent laser,” Opt. Express 14(20), 9203–9210 (2006).
[Crossref] [PubMed]

H. Park, M. N. Sysak, H.-W. Chen, A. W. Fang, D. Liang, L. Liao, B. R. Koch, J. Bovington, Y. Tang, K. Wong, M. Jacob-Mitos, R. Jones, and J. E. Bowers, “Device and Integration Technology for Silicon Photonic Transmitters,” IEEE J. Sel. Top. Quantum Electron. (to be published).

Kerdiles, J. F.

F. Devaux, Y. Sorel, and J. F. Kerdiles, “Simple Measurement of fiber Dispersion and of Chirp Parameter of Intensity Modulated Light Emitter,” IEEE Photon. Technol. Lett. 11, 1937–1940 (1993).

Kimerling, L. C.

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

Koch, B. R.

H. Park, M. N. Sysak, H.-W. Chen, A. W. Fang, D. Liang, L. Liao, B. R. Koch, J. Bovington, Y. Tang, K. Wong, M. Jacob-Mitos, R. Jones, and J. E. Bowers, “Device and Integration Technology for Silicon Photonic Transmitters,” IEEE J. Sel. Top. Quantum Electron. (to be published).

Kuo, Y. H.

Kuo, Y.-H.

H.-W. Chen, Y.-H. Kuo, and J. E. Bowers, “25Gb/s hybrid silicon switch using a capacitively loaded traveling wave electrode,” Opt. Express 18(2), 1070–1075 (2010).
[Crossref] [PubMed]

H.-W. Chen, Y.-H. Kuo, and J. E. Bowers, “A Hybrid Silicon-AlGaInAs Phase Modulator,” IEEE Photon. Technol. Lett. 20(23), 1920–1922 (2008).
[Crossref]

Kuramata, A.

Liang, D.

H. Park, M. N. Sysak, H.-W. Chen, A. W. Fang, D. Liang, L. Liao, B. R. Koch, J. Bovington, Y. Tang, K. Wong, M. Jacob-Mitos, R. Jones, and J. E. Bowers, “Device and Integration Technology for Silicon Photonic Transmitters,” IEEE J. Sel. Top. Quantum Electron. (to be published).

Liao, L.

A. S. 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]

H. Park, M. N. Sysak, H.-W. Chen, A. W. Fang, D. Liang, L. Liao, B. R. Koch, J. Bovington, Y. Tang, K. Wong, M. Jacob-Mitos, R. Jones, and J. E. Bowers, “Device and Integration Technology for Silicon Photonic Transmitters,” IEEE J. Sel. Top. Quantum Electron. (to be published).

Lipson, M.

Liu, A. S.

Liu, J.

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

Manipatruni, S.

Michel, J.

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

Nguyen, H.

Ozturk, C.

J. Shin, C. Ozturk, S. R. Sakamoto, Y. J. Chiu, and N. Dagli, “Novel T-Rail Electrodes for Substrate Removed Low-Voltage High-Speed GaAs/AlGaAs Electrooptic Modulators,” IEEE Trans. Microw. Theory Tech. 53(2), 636–643 (2005).
[Crossref]

Paniccia, M.

Paniccia, M. J.

Park, H.

A. W. Fang, H. Park, O. Cohen, R. Jones, M. J. Paniccia, and J. E. Bowers, “Electrically pumped hybrid AlGaInAs-silicon evanescent laser,” Opt. Express 14(20), 9203–9210 (2006).
[Crossref] [PubMed]

H. Park, M. N. Sysak, H.-W. Chen, A. W. Fang, D. Liang, L. Liao, B. R. Koch, J. Bovington, Y. Tang, K. Wong, M. Jacob-Mitos, R. Jones, and J. E. Bowers, “Device and Integration Technology for Silicon Photonic Transmitters,” IEEE J. Sel. Top. Quantum Electron. (to be published).

Pomerene, A.

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

Rubin, D.

Sakamoto, S. R.

J. Shin, C. Ozturk, S. R. Sakamoto, Y. J. Chiu, and N. Dagli, “Novel T-Rail Electrodes for Substrate Removed Low-Voltage High-Speed GaAs/AlGaAs Electrooptic Modulators,” IEEE Trans. Microw. Theory Tech. 53(2), 636–643 (2005).
[Crossref]

Schmidt, B.

Sekiguchi, S.

Shakya, J.

Shin, J.

J. Shin, C. Ozturk, S. R. Sakamoto, Y. J. Chiu, and N. Dagli, “Novel T-Rail Electrodes for Substrate Removed Low-Voltage High-Speed GaAs/AlGaAs Electrooptic Modulators,” IEEE Trans. Microw. Theory Tech. 53(2), 636–643 (2005).
[Crossref]

Sorel, Y.

F. Devaux, Y. Sorel, and J. F. Kerdiles, “Simple Measurement of fiber Dispersion and of Chirp Parameter of Intensity Modulated Light Emitter,” IEEE Photon. Technol. Lett. 11, 1937–1940 (1993).

Sun, R.

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

Sysak, M. N.

H. Park, M. N. Sysak, H.-W. Chen, A. W. Fang, D. Liang, L. Liao, B. R. Koch, J. Bovington, Y. Tang, K. Wong, M. Jacob-Mitos, R. Jones, and J. E. Bowers, “Device and Integration Technology for Silicon Photonic Transmitters,” IEEE J. Sel. Top. Quantum Electron. (to be published).

Takeuchi, T.

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]

Tang, Y.

H. Park, M. N. Sysak, H.-W. Chen, A. W. Fang, D. Liang, L. Liao, B. R. Koch, J. Bovington, Y. Tang, K. Wong, M. Jacob-Mitos, R. Jones, and J. E. Bowers, “Device and Integration Technology for Silicon Photonic Transmitters,” IEEE J. Sel. Top. Quantum Electron. (to be published).

Van Campenhout, J.

Vlasov, Y. A.

Wong, K.

H. Park, M. N. Sysak, H.-W. Chen, A. W. Fang, D. Liang, L. Liao, B. R. Koch, J. Bovington, Y. Tang, K. Wong, M. Jacob-Mitos, R. Jones, and J. E. Bowers, “Device and Integration Technology for Silicon Photonic Transmitters,” IEEE J. Sel. Top. Quantum Electron. (to be published).

Xu, Q.

Yamamoto, T.

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

H. Park, M. N. Sysak, H.-W. Chen, A. W. Fang, D. Liang, L. Liao, B. R. Koch, J. Bovington, Y. Tang, K. Wong, M. Jacob-Mitos, R. Jones, and J. E. Bowers, “Device and Integration Technology for Silicon Photonic Transmitters,” IEEE J. Sel. Top. Quantum Electron. (to be published).

IEEE Photon. Technol. Lett. (2)

H.-W. Chen, Y.-H. Kuo, and J. E. Bowers, “A Hybrid Silicon-AlGaInAs Phase Modulator,” IEEE Photon. Technol. Lett. 20(23), 1920–1922 (2008).
[Crossref]

F. Devaux, Y. Sorel, and J. F. Kerdiles, “Simple Measurement of fiber Dispersion and of Chirp Parameter of Intensity Modulated Light Emitter,” IEEE Photon. Technol. Lett. 11, 1937–1940 (1993).

IEEE Trans. Microw. Theory Tech. (1)

J. Shin, C. Ozturk, S. R. Sakamoto, Y. J. Chiu, and N. Dagli, “Novel T-Rail Electrodes for Substrate Removed Low-Voltage High-Speed GaAs/AlGaAs Electrooptic Modulators,” IEEE Trans. Microw. Theory Tech. 53(2), 636–643 (2005).
[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. (1)

Nat. Photonics (1)

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

Opt. Express (6)

Other (2)

A. Sneh, and C. Doerr, Integrated Optical Circuits and Components - Design and Applications (Marcel Dekker, Inc., 1999), Chap. 7.

S. R. Jain, M. N. Sysak, G. Kurczveil, and J. E. Bowers, “Integration of Hybrid Silicon DFB Laser and Electro-Absorption Modulator using Quantum Well Intermixing,” in Proceedings of IEEE European Conference on Optical Communication (Turin, Italy, 2010), Tu.5C5.

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Figures (6)

Fig. 1
Fig. 1

(a) Top schematic of a 500 μm MZM with slotline design. (c) SEM cross section of one arm of the MZM.

Fig. 2
Fig. 2

(a) Schematic circuit of the device with DC bias and RF driven signal. (c) Normalized transmission as a function of reverse bias at 1550 nm.

Fig. 3
Fig. 3

(a) Propagation loss improvement of a MZM with La = 500 μm. (b) Modulation bandwidth measured at −3 V with 25 Ω termination.

Fig. 4
Fig. 4

(a) Measured resonant frequency where u is uth resonant position. Inset: Measured chirp spectrum with 36 km SMF. (b) Chirp parameters at different bias condition.

Fig. 5
Fig. 5

(a) Left: 25 Gb/s electrical driven signal with 231-1 NRZ PRBS. Right: signal after modulator with 15.5 dB ER. (b) Left: 40 Gb/s electrical driven signal with 231-1 NRZ PRBS. Right: signal after modulator with 11.4 dB ER.

Fig. 6
Fig. 6

BER versus optical received power for all ports configurations at 40 Gb/s with 231-1 NRZ PRBS.

Equations (3)

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Z d = 2 B D A ± ( A + D ) 2 4       γ = α + j β = 1 L [ ln [ A + D 2 ± ( A + D 2 ) 2 1 ] ]
α = Δ ϕ 1 + Δ ϕ 2 Δ ϕ 1 Δ ϕ 2
f u 2 L = c 0 2 D λ 2 ( 1 + 2 u 2 π arctan ( α ) ) ,

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