Abstract

Plasma dispersion modulators (PDMs), such as carrier injection modulators and carrier depletion modulators, are widely used for high speed phase modulation in silicon photonic circuits, but they suffer from spurious intensity modulation. This can be a problem in coherent communication systems that make use of complex multi-level quadrature modulation formats, as well as analog applications such as microwave photonics. In this article, a method to achieve pure phase modulation using PDMs is proposed based on a configurable modulator circuit. The configurable modulator is implemented as a Mach-Zehnder interferometer with a PDM and tunable couplers (TCs). The spurious intensity modulation of the phase modulated lightwave can be compensated by tuning the coupling ratios of the TCs and the phase delay between the two arms of the MZI. Simulation results show that for a depletion modulator, the 1.26 dB spurious intensity modulation can be suppressed down to 0.023 dB within a phase range of 0.4$\pi$, and for injection modulator, the 1.27 dB spurious intensity modulation can be suppressed down to 0.07 dB within a phase range of 0.76$\pi$.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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2019 (2)

2018 (3)

H. Deng, P. Lu, S. J. Mihailov, and J. Yao, “High-Speed and High-Resolution Interrogation of a Strain and Temperature Random Grating Sensor,” J. Lightwave Technol. 36(23), 5587–5592 (2018).
[Crossref]

J. Witzens, “High-Speed Silicon Photonics Modulators,” Proc. IEEE 106(12), 2158–2182 (2018).
[Crossref]

W. Bogaerts and L. Chrostowski, “Silicon Photonics Circuit Design: Methods, Tools and Challenges,” Laser Photonics Rev. 12(4), 1700237 (2018).
[Crossref]

2017 (1)

2016 (1)

2015 (3)

K. Suzuki, G. Cong, K. Tanizawa, S.-H. Kim, K. Ikeda, S. Namiki, and H. Kawashima, “Ultra-high-extinction-ratio 2 × 2 silicon optical switch with variable splitter,” Opt. Express 23(7), 9086 (2015).
[Crossref]

D. Patel, S. Ghosh, M. Chagnon, A. Samani, V. Veerasubramanian, M. Osman, and D. V. Plant, “Design, analysis, and transmission system performance of a 41 GHz silicon photonic modulator,” Opt. Express 23(11), 14263 (2015).
[Crossref]

Y. Xing, T. Ako, J. P. George, D. Korn, H. Yu, P. Verheyen, M. Pantouvaki, G. Lepage, P. Absil, A. Ruocco, C. Koos, J. Leuthold, K. Neyts, J. Beeckman, and W. Bogaerts, “Digitally controlled phase shifter using an SOI slot waveguide with liquid crystal infiltration,” IEEE Photonics Technol. Lett. 27(12), 1269–1272 (2015).
[Crossref]

2014 (2)

2012 (2)

2011 (1)

M. Nedeljkovic, R. Soref, and G. Z. Mashanovich, “Free-carrier electrorefraction and electroabsorption modulation predictions for silicon over the 1-14-$\mu$μm infrared wavelength range,” IEEE Photonics J. 3(6), 1171–1180 (2011).
[Crossref]

2007 (1)

1987 (1)

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

Abrams, N.

Absil, P.

Y. Xing, T. Ako, J. P. George, D. Korn, H. Yu, P. Verheyen, M. Pantouvaki, G. Lepage, P. Absil, A. Ruocco, C. Koos, J. Leuthold, K. Neyts, J. Beeckman, and W. Bogaerts, “Digitally controlled phase shifter using an SOI slot waveguide with liquid crystal infiltration,” IEEE Photonics Technol. Lett. 27(12), 1269–1272 (2015).
[Crossref]

H. Yu, M. Pantouvaki, J. V. Campenhout, D. Korn, K. Komorowska, P. Dumon, Y. Li, P. Verheyen, P. Absil, L. Alloatti, D. Hillerkuss, J. Leuthold, R. Baets, and W. Bogaerts, “Performance tradeoff between lateral and interdigitated doping patterns for high speed carrier-depletion based silicon modulators,” Opt. Express 20(12), 12926 (2012).
[Crossref]

Ako, T.

Y. Xing, T. Ako, J. P. George, D. Korn, H. Yu, P. Verheyen, M. Pantouvaki, G. Lepage, P. Absil, A. Ruocco, C. Koos, J. Leuthold, K. Neyts, J. Beeckman, and W. Bogaerts, “Digitally controlled phase shifter using an SOI slot waveguide with liquid crystal infiltration,” IEEE Photonics Technol. Lett. 27(12), 1269–1272 (2015).
[Crossref]

Alloatti, L.

Baehr-Jones, T.

Baets, R.

Beeckman, J.

Y. Xing, T. Ako, J. P. George, D. Korn, H. Yu, P. Verheyen, M. Pantouvaki, G. Lepage, P. Absil, A. Ruocco, C. Koos, J. Leuthold, K. Neyts, J. Beeckman, and W. Bogaerts, “Digitally controlled phase shifter using an SOI slot waveguide with liquid crystal infiltration,” IEEE Photonics Technol. Lett. 27(12), 1269–1272 (2015).
[Crossref]

Bennett, B.

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

Bergman, K.

Bienstman, P.

W. Bogaerts, P. D. Heyn, T. V. Vaerenbergh, K. D. Vos, S. K. Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. V. Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photonics Rev. 6(1), 47–73 (2012).
[Crossref]

Bogaerts, W.

W. Bogaerts and L. Chrostowski, “Silicon Photonics Circuit Design: Methods, Tools and Challenges,” Laser Photonics Rev. 12(4), 1700237 (2018).
[Crossref]

A. Ribeiro, W. Bogaerts, M. Nedeljkovic, Y. Hu, D. J. Thomson, K. Li, P. R. Wilson, S. Chen, and S. S. Hsu, “Digitally controlled multiplexed silicon photonics phase shifter using heaters with integrated diodes,” Opt. Express 25(24), 29778 (2017).
[Crossref]

Y. Xing, T. Ako, J. P. George, D. Korn, H. Yu, P. Verheyen, M. Pantouvaki, G. Lepage, P. Absil, A. Ruocco, C. Koos, J. Leuthold, K. Neyts, J. Beeckman, and W. Bogaerts, “Digitally controlled phase shifter using an SOI slot waveguide with liquid crystal infiltration,” IEEE Photonics Technol. Lett. 27(12), 1269–1272 (2015).
[Crossref]

W. Bogaerts, P. D. Heyn, T. V. Vaerenbergh, K. D. Vos, S. K. Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. V. Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photonics Rev. 6(1), 47–73 (2012).
[Crossref]

H. Yu, M. Pantouvaki, J. V. Campenhout, D. Korn, K. Komorowska, P. Dumon, Y. Li, P. Verheyen, P. Absil, L. Alloatti, D. Hillerkuss, J. Leuthold, R. Baets, and W. Bogaerts, “Performance tradeoff between lateral and interdigitated doping patterns for high speed carrier-depletion based silicon modulators,” Opt. Express 20(12), 12926 (2012).
[Crossref]

Bolten, J.

Campenhout, J. V.

Capmany, J.

D. Marpaung, J. Yao, and J. Capmany, “Integrated microwave photonics,” Nat. Photonics 13(2), 80–90 (2019).
[Crossref]

Cartledge, J. C.

Chagnon, M.

Chen, S.

Cheng, Q.

Chetrit, Y.

Chrostowski, L.

W. Bogaerts and L. Chrostowski, “Silicon Photonics Circuit Design: Methods, Tools and Challenges,” Laser Photonics Rev. 12(4), 1700237 (2018).
[Crossref]

Ciftcioglu, B.

Claes, T.

W. Bogaerts, P. D. Heyn, T. V. Vaerenbergh, K. D. Vos, S. K. Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. V. Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photonics Rev. 6(1), 47–73 (2012).
[Crossref]

Cong, G.

Dalton, L. R.

Deng, H.

Dumon, P.

Elder, D. L.

Englund, D.

Freude, W.

Galland, C.

Gao, Y.

Gazman, A.

George, J. P.

Y. Xing, T. Ako, J. P. George, D. Korn, H. Yu, P. Verheyen, M. Pantouvaki, G. Lepage, P. Absil, A. Ruocco, C. Koos, J. Leuthold, K. Neyts, J. Beeckman, and W. Bogaerts, “Digitally controlled phase shifter using an SOI slot waveguide with liquid crystal infiltration,” IEEE Photonics Technol. Lett. 27(12), 1269–1272 (2015).
[Crossref]

Ghosh, S.

Harris, N. C.

Heyn, P. D.

W. Bogaerts, P. D. Heyn, T. V. Vaerenbergh, K. D. Vos, S. K. Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. V. Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photonics Rev. 6(1), 47–73 (2012).
[Crossref]

Hillerkuss, D.

Hochberg, M.

Hsu, S. S.

Hu, Y.

Ikeda, K.

Izhaky, N.

Kashi, A. S.

Kawashima, H.

Kim, S.-H.

Knights, A. P.

Koeber, S.

Komorowska, K.

Koos, C.

Y. Xing, T. Ako, J. P. George, D. Korn, H. Yu, P. Verheyen, M. Pantouvaki, G. Lepage, P. Absil, A. Ruocco, C. Koos, J. Leuthold, K. Neyts, J. Beeckman, and W. Bogaerts, “Digitally controlled phase shifter using an SOI slot waveguide with liquid crystal infiltration,” IEEE Photonics Technol. Lett. 27(12), 1269–1272 (2015).
[Crossref]

M. Lauermann, R. Palmer, S. Koeber, P. C. Schindler, D. Korn, T. Wahlbrink, J. Bolten, M. Waldow, D. L. Elder, L. R. Dalton, J. Leuthold, W. Freude, and C. Koos, “Low-power silicon-organic hybrid (SOH) modulators for advanced modulation formats,” Opt. Express 22(24), 29927 (2014).
[Crossref]

Korn, D.

Lauermann, M.

Lepage, G.

Y. Xing, T. Ako, J. P. George, D. Korn, H. Yu, P. Verheyen, M. Pantouvaki, G. Lepage, P. Absil, A. Ruocco, C. Koos, J. Leuthold, K. Neyts, J. Beeckman, and W. Bogaerts, “Digitally controlled phase shifter using an SOI slot waveguide with liquid crystal infiltration,” IEEE Photonics Technol. Lett. 27(12), 1269–1272 (2015).
[Crossref]

Leuthold, J.

Li, K.

Li, Y.

Liao, L.

Liu, A.

Lu, P.

Ma, Y.

Manzhosov, E.

Marpaung, D.

D. Marpaung, J. Yao, and J. Capmany, “Integrated microwave photonics,” Nat. Photonics 13(2), 80–90 (2019).
[Crossref]

Mashanovich, G. Z.

M. Nedeljkovic, R. Soref, and G. Z. Mashanovich, “Free-carrier electrorefraction and electroabsorption modulation predictions for silicon over the 1-14-$\mu$μm infrared wavelength range,” IEEE Photonics J. 3(6), 1171–1180 (2011).
[Crossref]

Meng, X.

Mihailov, S. J.

Mower, J.

Namiki, S.

Nedeljkovic, M.

A. Ribeiro, W. Bogaerts, M. Nedeljkovic, Y. Hu, D. J. Thomson, K. Li, P. R. Wilson, S. Chen, and S. S. Hsu, “Digitally controlled multiplexed silicon photonics phase shifter using heaters with integrated diodes,” Opt. Express 25(24), 29778 (2017).
[Crossref]

M. Nedeljkovic, R. Soref, and G. Z. Mashanovich, “Free-carrier electrorefraction and electroabsorption modulation predictions for silicon over the 1-14-$\mu$μm infrared wavelength range,” IEEE Photonics J. 3(6), 1171–1180 (2011).
[Crossref]

Neyts, K.

Y. Xing, T. Ako, J. P. George, D. Korn, H. Yu, P. Verheyen, M. Pantouvaki, G. Lepage, P. Absil, A. Ruocco, C. Koos, J. Leuthold, K. Neyts, J. Beeckman, and W. Bogaerts, “Digitally controlled phase shifter using an SOI slot waveguide with liquid crystal infiltration,” IEEE Photonics Technol. Lett. 27(12), 1269–1272 (2015).
[Crossref]

Nguyen, H.

Osman, M.

Palmer, R.

Paniccia, M.

Pantouvaki, M.

Y. Xing, T. Ako, J. P. George, D. Korn, H. Yu, P. Verheyen, M. Pantouvaki, G. Lepage, P. Absil, A. Ruocco, C. Koos, J. Leuthold, K. Neyts, J. Beeckman, and W. Bogaerts, “Digitally controlled phase shifter using an SOI slot waveguide with liquid crystal infiltration,” IEEE Photonics Technol. Lett. 27(12), 1269–1272 (2015).
[Crossref]

H. Yu, M. Pantouvaki, J. V. Campenhout, D. Korn, K. Komorowska, P. Dumon, Y. Li, P. Verheyen, P. Absil, L. Alloatti, D. Hillerkuss, J. Leuthold, R. Baets, and W. Bogaerts, “Performance tradeoff between lateral and interdigitated doping patterns for high speed carrier-depletion based silicon modulators,” Opt. Express 20(12), 12926 (2012).
[Crossref]

Patel, D.

Plant, D. V.

Ribeiro, A.

Rubin, D.

Rumley, S.

Ruocco, A.

Y. Xing, T. Ako, J. P. George, D. Korn, H. Yu, P. Verheyen, M. Pantouvaki, G. Lepage, P. Absil, A. Ruocco, C. Koos, J. Leuthold, K. Neyts, J. Beeckman, and W. Bogaerts, “Digitally controlled phase shifter using an SOI slot waveguide with liquid crystal infiltration,” IEEE Photonics Technol. Lett. 27(12), 1269–1272 (2015).
[Crossref]

Samani, A.

Schindler, P. C.

Selvaraja, S. K.

W. Bogaerts, P. D. Heyn, T. V. Vaerenbergh, K. D. Vos, S. K. Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. V. Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photonics Rev. 6(1), 47–73 (2012).
[Crossref]

Shen, Y.

Soref, R.

M. Nedeljkovic, R. Soref, and G. Z. Mashanovich, “Free-carrier electrorefraction and electroabsorption modulation predictions for silicon over the 1-14-$\mu$μm infrared wavelength range,” IEEE Photonics J. 3(6), 1171–1180 (2011).
[Crossref]

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

Strom Glick, M.

Suzuki, K.

Tanizawa, K.

Thomson, D. J.

Thourhout, D. V.

W. Bogaerts, P. D. Heyn, T. V. Vaerenbergh, K. D. Vos, S. K. Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. V. Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photonics Rev. 6(1), 47–73 (2012).
[Crossref]

Vaerenbergh, T. V.

W. Bogaerts, P. D. Heyn, T. V. Vaerenbergh, K. D. Vos, S. K. Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. V. Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photonics Rev. 6(1), 47–73 (2012).
[Crossref]

Veerasubramanian, V.

Verheyen, P.

Y. Xing, T. Ako, J. P. George, D. Korn, H. Yu, P. Verheyen, M. Pantouvaki, G. Lepage, P. Absil, A. Ruocco, C. Koos, J. Leuthold, K. Neyts, J. Beeckman, and W. Bogaerts, “Digitally controlled phase shifter using an SOI slot waveguide with liquid crystal infiltration,” IEEE Photonics Technol. Lett. 27(12), 1269–1272 (2015).
[Crossref]

H. Yu, M. Pantouvaki, J. V. Campenhout, D. Korn, K. Komorowska, P. Dumon, Y. Li, P. Verheyen, P. Absil, L. Alloatti, D. Hillerkuss, J. Leuthold, R. Baets, and W. Bogaerts, “Performance tradeoff between lateral and interdigitated doping patterns for high speed carrier-depletion based silicon modulators,” Opt. Express 20(12), 12926 (2012).
[Crossref]

Vos, K. D.

W. Bogaerts, P. D. Heyn, T. V. Vaerenbergh, K. D. Vos, S. K. Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. V. Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photonics Rev. 6(1), 47–73 (2012).
[Crossref]

Wahlbrink, T.

Waldow, M.

Wang, Z.

Wilson, P. R.

Witzens, J.

J. Witzens, “High-Speed Silicon Photonics Modulators,” Proc. IEEE 106(12), 2158–2182 (2018).
[Crossref]

Xing, Y.

Y. Xing, T. Ako, J. P. George, D. Korn, H. Yu, P. Verheyen, M. Pantouvaki, G. Lepage, P. Absil, A. Ruocco, C. Koos, J. Leuthold, K. Neyts, J. Beeckman, and W. Bogaerts, “Digitally controlled phase shifter using an SOI slot waveguide with liquid crystal infiltration,” IEEE Photonics Technol. Lett. 27(12), 1269–1272 (2015).
[Crossref]

Yao, J.

Yu, H.

Y. Xing, T. Ako, J. P. George, D. Korn, H. Yu, P. Verheyen, M. Pantouvaki, G. Lepage, P. Absil, A. Ruocco, C. Koos, J. Leuthold, K. Neyts, J. Beeckman, and W. Bogaerts, “Digitally controlled phase shifter using an SOI slot waveguide with liquid crystal infiltration,” IEEE Photonics Technol. Lett. 27(12), 1269–1272 (2015).
[Crossref]

H. Yu, M. Pantouvaki, J. V. Campenhout, D. Korn, K. Komorowska, P. Dumon, Y. Li, P. Verheyen, P. Absil, L. Alloatti, D. Hillerkuss, J. Leuthold, R. Baets, and W. Bogaerts, “Performance tradeoff between lateral and interdigitated doping patterns for high speed carrier-depletion based silicon modulators,” Opt. Express 20(12), 12926 (2012).
[Crossref]

IEEE J. Quantum Electron. (1)

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

IEEE Photonics J. (1)

M. Nedeljkovic, R. Soref, and G. Z. Mashanovich, “Free-carrier electrorefraction and electroabsorption modulation predictions for silicon over the 1-14-$\mu$μm infrared wavelength range,” IEEE Photonics J. 3(6), 1171–1180 (2011).
[Crossref]

IEEE Photonics Technol. Lett. (1)

Y. Xing, T. Ako, J. P. George, D. Korn, H. Yu, P. Verheyen, M. Pantouvaki, G. Lepage, P. Absil, A. Ruocco, C. Koos, J. Leuthold, K. Neyts, J. Beeckman, and W. Bogaerts, “Digitally controlled phase shifter using an SOI slot waveguide with liquid crystal infiltration,” IEEE Photonics Technol. Lett. 27(12), 1269–1272 (2015).
[Crossref]

J. Lightwave Technol. (3)

Laser Photonics Rev. (2)

W. Bogaerts and L. Chrostowski, “Silicon Photonics Circuit Design: Methods, Tools and Challenges,” Laser Photonics Rev. 12(4), 1700237 (2018).
[Crossref]

W. Bogaerts, P. D. Heyn, T. V. Vaerenbergh, K. D. Vos, S. K. Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. V. Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photonics Rev. 6(1), 47–73 (2012).
[Crossref]

Nat. Photonics (1)

D. Marpaung, J. Yao, and J. Capmany, “Integrated microwave photonics,” Nat. Photonics 13(2), 80–90 (2019).
[Crossref]

Opt. Express (7)

H. Yu, M. Pantouvaki, J. V. Campenhout, D. Korn, K. Komorowska, P. Dumon, Y. Li, P. Verheyen, P. Absil, L. Alloatti, D. Hillerkuss, J. Leuthold, R. Baets, and W. Bogaerts, “Performance tradeoff between lateral and interdigitated doping patterns for high speed carrier-depletion based silicon modulators,” Opt. Express 20(12), 12926 (2012).
[Crossref]

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[Crossref]

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Figures (11)

Fig. 1.
Fig. 1. (a) Measured intensity modulation and (b) phase modulation response of the used depletion modulator model (0 V to 6 V) [9]; (c) modulation curve of the depletion modulator in polar diagram (0 V to 6 V);(d) Measured intensity modulation and (e) phase modulation response of the used injection modulator model (0.8 V to 0.96 V); (f) modulation curve of the injection modulator in polar diagram (0.8 V to 0.9 V);
Fig. 2.
Fig. 2. Schematic of the proposed configurable modulator. PS: phase shifter; PDM: plasma dispersion modulator; TC: tunable coupler.
Fig. 3.
Fig. 3. Phasor diagrams for a carrier depletion modulator based configurable modulator. (a) All light feeds into the depletion modulator; (b) half of the light feeds into the modulator and the other half goes into the phase shifter with zero phase shift; (c) half of the light feeds into the modulator and half goes to the phase shifter with a nonzero phase shift; (d) one of the optimized conditions for pure phase modulation.
Fig. 4.
Fig. 4. Phasor diagrams for a carrier injection modulator based configurable modulator. (a) All light feeds into the injection modulator; (b) half of the light feeds into the modulator and the other half goes to the phase shifter with zero phase shift; (c) half of the light feeds into the modulator and half goes to the phase shifter with a nonzero phase shift; (d) one of the optimized conditions.
Fig. 5.
Fig. 5. (a) Spurious intensity modulation with depletion modulator: sweep $\kappa _1$ and $\kappa _2$ at $\phi _s = 0$; (b) Spurious intensity modulation with depletion modulator: sweep $\kappa _1$ and $\kappa _2$ at $\phi _s = -0.2 \pi$.
Fig. 6.
Fig. 6. Simulation results with depletion modulator for minimal spurious intensity modulation: sweep splitting ratio and offset phase shifts. Operation points: A: $\kappa = 0.092$ and $\phi _s = -0.384\pi$; B: $\kappa = 0.42$ and $\phi _s = 0.8\pi$; C: $\kappa = 0.42$ and $\phi _s = 0.05\pi$; D: $\kappa = 0.42$ and $\phi _s = -0.8\pi$.
Fig. 7.
Fig. 7. Simulation results with depletion modulator within 0 V to 6 V. (a) spurious intensity modulation; (b) nonlinearity of phase modulation; (c) minimal intensity; (d) phase modulation range. Green curve show the performance of the used depletion modulator.
Fig. 8.
Fig. 8. DC response of a depletion modulator (blue) and the configurable MZI modulator working at $\kappa _1 = \kappa _2 = 0.092$ and $\phi _s = -0.384\pi$ (green).
Fig. 9.
Fig. 9. Simulation results with injection modulator within 0.8 V to 0.9 V. (a) spurious intensity modulation; (b) nonlinearity of phase modulation; (c) minimal intensity; (d) phase modulation range. Green curve show the performance of the used injection modulator.
Fig. 10.
Fig. 10. Pure phase modulation based on an injection modulator. $\kappa _1 = \kappa _2 = 0.063$ and $\phi _s = -0.9\pi$.
Fig. 11.
Fig. 11. (a)Minimal intensity modulation and (b) phase modulation range for the injection modulator and the optimized configurable modulator at different voltage ranges (0.8 V to Max voltage).

Equations (6)

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Δ n = 5.4 × 10 22 Δ N 1.011 1.53 × 10 18 Δ P 0.838
Δ α = 8.88 × 10 21 Δ N 1.167 + 5.84 × 10 20 Δ P 1.109
[ E c 1 E c 2 ] = [ ( 1 κ ) j κ j κ ( 1 κ ) ] × [ E c 3 E c 4 ]
[ E a 1 E a 2 ] = [ α ( V ) e j ϕ m ( V ) 0 0 e ( j ϕ s ) ] × [ E a 3 E a 4 ]
[ E o u t 1 E o u t 2 ] = [ ( 1 κ 2 ) j κ 2 j κ 2 ( 1 κ 2 ) ] × [ α ( V ) e j ϕ m ( V ) 0 0 e j ϕ s ] × [ ( 1 κ 1 ) j κ 1 j κ 1 ( 1 κ 1 ) ] × [ E i n 1 E i n 2 ] = [ ( 1 κ 1 ) ( 1 κ 2 ) α ( V ) e j ϕ m ( V ) κ 1 κ 2 e j ϕ s j ( 1 κ 1 ) κ 2 α ( V ) e j ϕ m ( V ) + j κ 1 ( 1 κ 2 ) e j ϕ s j κ 1 ( 1 κ 2 ) α ( V ) e j ϕ m ( V ) + j ( 1 κ 1 ) κ 2 e j ϕ s κ 1 κ 2 α ( V ) e j ϕ m ( V ) + ( 1 κ 1 ) ( 1 κ 2 ) e j ϕ s ] × [ E i n 1 E i n 2 ]
E o u t 1 = [ ( 1 κ 1 ) ( 1 κ 2 ) α ( V ) e j ϕ m ( V ) κ 1 κ 2 e j ϕ s ] E i n 1 = [ ( 1 κ 1 ) ( 1 κ 2 ) α ( V ) e j ϕ m ( V ) + κ 1 κ 2 e j ( ϕ s π ) ] E i n 1

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