Abstract

We demonstrate a Si photonic crystal waveguide Mach–Zehnder modulator that incorporates meander-line electrodes to compensate for the phase mismatch between slow light and RF signals. We first employed commonized ground electrodes in the modulator to suppress undesired fluctuations in the electro-optic (EO) response due to coupled slot-line modes of RF signals. Then, we theoretically and experimentally investigated the effect of the phase mismatch on the EO response. We confirmed that meander-line electrodes improve the EO response, particularly in the absence of internal reflection of the RF signals. The cut-off frequency of this device can reach 27 GHz, which allows high-speed modulation up to 50 Gbps.

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

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References

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  1. G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
    [Crossref]
  2. D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. 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(6), 85–94 (2013).
    [Crossref]
  3. H. Xu, X. Li, X. Xiao, Z. Li, Y. Yu, and J. Yu, “Demonstration and Characterization of High-Speed Silicon Depletion-Mode Mach–Zehnder Modulators,” IEEE J. Sel. Top. Quantum Electron. 20(4), 23–32 (2014).
    [Crossref]
  4. X. Tu, K. F. Chang, T. Y. Liow, J. Song, X. Luo, L. Jia, Q. Fang, M. Yu, G. Q. Lo, P. Dong, and Y. K. Chen, “Silicon optical modulator with shield coplanar waveguide electrodes,” Opt. Express 22(19), 23724–23731 (2014).
    [Crossref] [PubMed]
  5. H. Yu and W. Bogaerts, “An Equivalent Circuit Model of the Traveling Wave Electrode for Carrier-Depletion-Based Silicon Optical Modulators,” J. Lightwave Technol. 30(11), 1602–1609 (2012).
    [Crossref]
  6. J. Shin, S. R. Sakamoto, and N. Dagli, “Conductor Loss of Capacitively Loaded Slow Wave Electrodes for High-Speed Photonic Devices,” J. Lightwave Technol. 29(1), 48–52 (2011).
    [Crossref]
  7. R. Ding, Y. Liu, Y. Ma, Y. Yang, Q. Li, A. E. J. Lim, G. Q. Lo, K. Bergman, T. B. Jones, and M. Hochberg, “High-Speed Silicon Modulator With Slow-Wave Electrodes and Fully Independent Differential Drive,” J. Lightwave Technol. 32(12), 2240–2247 (2014).
    [Crossref]
  8. 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), 14263–14287 (2015).
    [Crossref] [PubMed]
  9. Y. Terada, T. Tatebe, Y. Hinakura, and T. Baba, “Si Photonic Crystal Slow-Light Modulators with Periodic p–n Junctions,” J. Lightwave Technol. 35(9), 1684–1692 (2017).
    [Crossref]
  10. Y. Hinakura, Y. Terada, T. Tamura, and T. Baba, “Wide spectral characteristics of Si photonic crystal Mach-Zehnder modulator fabricated by complementary metal–oxide–semiconductor process,” Photonics 3(2), 17 (2016).
  11. K. Hojo, Y. Terada, N. Yazawa, T. Watanabe, and T. Baba, “Compact QPSK and PAM modulators with Si photonic crystal slow light phase shifters,” IEEE Photonics Technol. Lett. 28(13), 1438–1441 (2016).
    [Crossref]
  12. T. Tamura, K. Kondo, Y. Terada, Y. Hinakura, N. Ishikura, and T. Baba, “Silica-clad silicon photonic crystal waveguides for wideband dispersion-free slow light,” J. Lightwave Technol. 33(7), 3034–3040 (2015).
  13. K. Kubota, J. Noda, and O. Mikami, “Traveling wave optical modulator using a directional coupler LiNbO3waveguide,” IEEE J. Quantum Electron. 16(7), 754–760 (1980).
    [Crossref]
  14. I. Kim, M. R. T. Tan, and S. Y. Wang, “Analysis of a new microwave low-loss and velocity-matched III-V transmission line for traveling-wave electrooptic modulators,” J. Lightwave Technol. 8(5), 728–738 (1990).
    [Crossref]
  15. G. E. Ponchak, J. Papapolymerou, and M. M. Tentzeris, “Excitation of coupled slotline mode in finite-ground CPW with unequal ground-plane widths,” IEEE Trans. Microw. Theory Tech. 53(2), 713–717 (2005).
    [Crossref]

2017 (1)

2016 (2)

Y. Hinakura, Y. Terada, T. Tamura, and T. Baba, “Wide spectral characteristics of Si photonic crystal Mach-Zehnder modulator fabricated by complementary metal–oxide–semiconductor process,” Photonics 3(2), 17 (2016).

K. Hojo, Y. Terada, N. Yazawa, T. Watanabe, and T. Baba, “Compact QPSK and PAM modulators with Si photonic crystal slow light phase shifters,” IEEE Photonics Technol. Lett. 28(13), 1438–1441 (2016).
[Crossref]

2015 (2)

2014 (3)

2013 (1)

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. 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(6), 85–94 (2013).
[Crossref]

2012 (1)

2011 (1)

2010 (1)

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[Crossref]

2005 (1)

G. E. Ponchak, J. Papapolymerou, and M. M. Tentzeris, “Excitation of coupled slotline mode in finite-ground CPW with unequal ground-plane widths,” IEEE Trans. Microw. Theory Tech. 53(2), 713–717 (2005).
[Crossref]

1990 (1)

I. Kim, M. R. T. Tan, and S. Y. Wang, “Analysis of a new microwave low-loss and velocity-matched III-V transmission line for traveling-wave electrooptic modulators,” J. Lightwave Technol. 8(5), 728–738 (1990).
[Crossref]

1980 (1)

K. Kubota, J. Noda, and O. Mikami, “Traveling wave optical modulator using a directional coupler LiNbO3waveguide,” IEEE J. Quantum Electron. 16(7), 754–760 (1980).
[Crossref]

Baba, T.

Y. Terada, T. Tatebe, Y. Hinakura, and T. Baba, “Si Photonic Crystal Slow-Light Modulators with Periodic p–n Junctions,” J. Lightwave Technol. 35(9), 1684–1692 (2017).
[Crossref]

Y. Hinakura, Y. Terada, T. Tamura, and T. Baba, “Wide spectral characteristics of Si photonic crystal Mach-Zehnder modulator fabricated by complementary metal–oxide–semiconductor process,” Photonics 3(2), 17 (2016).

K. Hojo, Y. Terada, N. Yazawa, T. Watanabe, and T. Baba, “Compact QPSK and PAM modulators with Si photonic crystal slow light phase shifters,” IEEE Photonics Technol. Lett. 28(13), 1438–1441 (2016).
[Crossref]

T. Tamura, K. Kondo, Y. Terada, Y. Hinakura, N. Ishikura, and T. Baba, “Silica-clad silicon photonic crystal waveguides for wideband dispersion-free slow light,” J. Lightwave Technol. 33(7), 3034–3040 (2015).

Bergman, K.

Bogaerts, W.

Chagnon, M.

Chang, K. F.

Chen, Y. K.

Dagli, N.

Ding, R.

Dong, P.

Fang, Q.

Fedeli, J. M.

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. 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(6), 85–94 (2013).
[Crossref]

Gardes, F. Y.

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. 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(6), 85–94 (2013).
[Crossref]

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[Crossref]

Ghosh, S.

Hinakura, Y.

Hochberg, M.

Hojo, K.

K. Hojo, Y. Terada, N. Yazawa, T. Watanabe, and T. Baba, “Compact QPSK and PAM modulators with Si photonic crystal slow light phase shifters,” IEEE Photonics Technol. Lett. 28(13), 1438–1441 (2016).
[Crossref]

Hu, Y.

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. 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(6), 85–94 (2013).
[Crossref]

Ishikura, N.

Jia, L.

Jones, T. B.

Kim, I.

I. Kim, M. R. T. Tan, and S. Y. Wang, “Analysis of a new microwave low-loss and velocity-matched III-V transmission line for traveling-wave electrooptic modulators,” J. Lightwave Technol. 8(5), 728–738 (1990).
[Crossref]

Kondo, K.

Kubota, K.

K. Kubota, J. Noda, and O. Mikami, “Traveling wave optical modulator using a directional coupler LiNbO3waveguide,” IEEE J. Quantum Electron. 16(7), 754–760 (1980).
[Crossref]

Li, Q.

Li, X.

H. Xu, X. Li, X. Xiao, Z. Li, Y. Yu, and J. Yu, “Demonstration and Characterization of High-Speed Silicon Depletion-Mode Mach–Zehnder Modulators,” IEEE J. Sel. Top. Quantum Electron. 20(4), 23–32 (2014).
[Crossref]

Li, Z.

H. Xu, X. Li, X. Xiao, Z. Li, Y. Yu, and J. Yu, “Demonstration and Characterization of High-Speed Silicon Depletion-Mode Mach–Zehnder Modulators,” IEEE J. Sel. Top. Quantum Electron. 20(4), 23–32 (2014).
[Crossref]

Lim, A. E. J.

Liow, T. Y.

Liu, S.

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. 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(6), 85–94 (2013).
[Crossref]

Liu, Y.

Lo, G. Q.

Luo, X.

Ma, Y.

Mashanovich, G. Z.

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. 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(6), 85–94 (2013).
[Crossref]

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[Crossref]

Mikami, O.

K. Kubota, J. Noda, and O. Mikami, “Traveling wave optical modulator using a directional coupler LiNbO3waveguide,” IEEE J. Quantum Electron. 16(7), 754–760 (1980).
[Crossref]

Nedeljkovic, M.

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. 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(6), 85–94 (2013).
[Crossref]

Noda, J.

K. Kubota, J. Noda, and O. Mikami, “Traveling wave optical modulator using a directional coupler LiNbO3waveguide,” IEEE J. Quantum Electron. 16(7), 754–760 (1980).
[Crossref]

Osman, M.

Papapolymerou, J.

G. E. Ponchak, J. Papapolymerou, and M. M. Tentzeris, “Excitation of coupled slotline mode in finite-ground CPW with unequal ground-plane widths,” IEEE Trans. Microw. Theory Tech. 53(2), 713–717 (2005).
[Crossref]

Patel, D.

Petropoulos, P.

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. 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(6), 85–94 (2013).
[Crossref]

Plant, D. V.

Ponchak, G. E.

G. E. Ponchak, J. Papapolymerou, and M. M. Tentzeris, “Excitation of coupled slotline mode in finite-ground CPW with unequal ground-plane widths,” IEEE Trans. Microw. Theory Tech. 53(2), 713–717 (2005).
[Crossref]

Porte, H.

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. 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(6), 85–94 (2013).
[Crossref]

Reed, G. T.

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[Crossref]

Sakamoto, S. R.

Samani, A.

Shin, J.

Song, J.

Tamura, T.

Y. Hinakura, Y. Terada, T. Tamura, and T. Baba, “Wide spectral characteristics of Si photonic crystal Mach-Zehnder modulator fabricated by complementary metal–oxide–semiconductor process,” Photonics 3(2), 17 (2016).

T. Tamura, K. Kondo, Y. Terada, Y. Hinakura, N. Ishikura, and T. Baba, “Silica-clad silicon photonic crystal waveguides for wideband dispersion-free slow light,” J. Lightwave Technol. 33(7), 3034–3040 (2015).

Tan, M. R. T.

I. Kim, M. R. T. Tan, and S. Y. Wang, “Analysis of a new microwave low-loss and velocity-matched III-V transmission line for traveling-wave electrooptic modulators,” J. Lightwave Technol. 8(5), 728–738 (1990).
[Crossref]

Tatebe, T.

Tentzeris, M. M.

G. E. Ponchak, J. Papapolymerou, and M. M. Tentzeris, “Excitation of coupled slotline mode in finite-ground CPW with unequal ground-plane widths,” IEEE Trans. Microw. Theory Tech. 53(2), 713–717 (2005).
[Crossref]

Terada, Y.

Y. Terada, T. Tatebe, Y. Hinakura, and T. Baba, “Si Photonic Crystal Slow-Light Modulators with Periodic p–n Junctions,” J. Lightwave Technol. 35(9), 1684–1692 (2017).
[Crossref]

Y. Hinakura, Y. Terada, T. Tamura, and T. Baba, “Wide spectral characteristics of Si photonic crystal Mach-Zehnder modulator fabricated by complementary metal–oxide–semiconductor process,” Photonics 3(2), 17 (2016).

K. Hojo, Y. Terada, N. Yazawa, T. Watanabe, and T. Baba, “Compact QPSK and PAM modulators with Si photonic crystal slow light phase shifters,” IEEE Photonics Technol. Lett. 28(13), 1438–1441 (2016).
[Crossref]

T. Tamura, K. Kondo, Y. Terada, Y. Hinakura, N. Ishikura, and T. Baba, “Silica-clad silicon photonic crystal waveguides for wideband dispersion-free slow light,” J. Lightwave Technol. 33(7), 3034–3040 (2015).

Thomson, D. J.

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. 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(6), 85–94 (2013).
[Crossref]

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[Crossref]

Tu, X.

Veerasubramanian, V.

Wang, S. Y.

I. Kim, M. R. T. Tan, and S. Y. Wang, “Analysis of a new microwave low-loss and velocity-matched III-V transmission line for traveling-wave electrooptic modulators,” J. Lightwave Technol. 8(5), 728–738 (1990).
[Crossref]

Watanabe, T.

K. Hojo, Y. Terada, N. Yazawa, T. Watanabe, and T. Baba, “Compact QPSK and PAM modulators with Si photonic crystal slow light phase shifters,” IEEE Photonics Technol. Lett. 28(13), 1438–1441 (2016).
[Crossref]

Xiao, X.

H. Xu, X. Li, X. Xiao, Z. Li, Y. Yu, and J. Yu, “Demonstration and Characterization of High-Speed Silicon Depletion-Mode Mach–Zehnder Modulators,” IEEE J. Sel. Top. Quantum Electron. 20(4), 23–32 (2014).
[Crossref]

Xu, H.

H. Xu, X. Li, X. Xiao, Z. Li, Y. Yu, and J. Yu, “Demonstration and Characterization of High-Speed Silicon Depletion-Mode Mach–Zehnder Modulators,” IEEE J. Sel. Top. Quantum Electron. 20(4), 23–32 (2014).
[Crossref]

Yang, X.

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. 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(6), 85–94 (2013).
[Crossref]

Yang, Y.

Yazawa, N.

K. Hojo, Y. Terada, N. Yazawa, T. Watanabe, and T. Baba, “Compact QPSK and PAM modulators with Si photonic crystal slow light phase shifters,” IEEE Photonics Technol. Lett. 28(13), 1438–1441 (2016).
[Crossref]

Yu, H.

Yu, J.

H. Xu, X. Li, X. Xiao, Z. Li, Y. Yu, and J. Yu, “Demonstration and Characterization of High-Speed Silicon Depletion-Mode Mach–Zehnder Modulators,” IEEE J. Sel. Top. Quantum Electron. 20(4), 23–32 (2014).
[Crossref]

Yu, M.

Yu, Y.

H. Xu, X. Li, X. Xiao, Z. Li, Y. Yu, and J. Yu, “Demonstration and Characterization of High-Speed Silicon Depletion-Mode Mach–Zehnder Modulators,” IEEE J. Sel. Top. Quantum Electron. 20(4), 23–32 (2014).
[Crossref]

Zimmermann, L.

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. 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(6), 85–94 (2013).
[Crossref]

IEEE J. Quantum Electron. (1)

K. Kubota, J. Noda, and O. Mikami, “Traveling wave optical modulator using a directional coupler LiNbO3waveguide,” IEEE J. Quantum Electron. 16(7), 754–760 (1980).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (2)

D. J. Thomson, F. Y. Gardes, S. Liu, H. Porte, L. Zimmermann, J. M. Fedeli, Y. 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(6), 85–94 (2013).
[Crossref]

H. Xu, X. Li, X. Xiao, Z. Li, Y. Yu, and J. Yu, “Demonstration and Characterization of High-Speed Silicon Depletion-Mode Mach–Zehnder Modulators,” IEEE J. Sel. Top. Quantum Electron. 20(4), 23–32 (2014).
[Crossref]

IEEE Photonics Technol. Lett. (1)

K. Hojo, Y. Terada, N. Yazawa, T. Watanabe, and T. Baba, “Compact QPSK and PAM modulators with Si photonic crystal slow light phase shifters,” IEEE Photonics Technol. Lett. 28(13), 1438–1441 (2016).
[Crossref]

IEEE Trans. Microw. Theory Tech. (1)

G. E. Ponchak, J. Papapolymerou, and M. M. Tentzeris, “Excitation of coupled slotline mode in finite-ground CPW with unequal ground-plane widths,” IEEE Trans. Microw. Theory Tech. 53(2), 713–717 (2005).
[Crossref]

J. Lightwave Technol. (6)

Nat. Photonics (1)

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[Crossref]

Opt. Express (2)

Photonics (1)

Y. Hinakura, Y. Terada, T. Tamura, and T. Baba, “Wide spectral characteristics of Si photonic crystal Mach-Zehnder modulator fabricated by complementary metal–oxide–semiconductor process,” Photonics 3(2), 17 (2016).

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Figures (6)

Fig. 1
Fig. 1 Fabricated Si PCW MZMs. (a) Normal electrode device without commonizing grounds.In the inset, the p- and n-doped regions are colored. (b) Normal electrode device with commonized grounds. (c) Meander-line electrode device with commonized grounds. Insets show the details of a bend in the meander-line electrode and the separated p-n junction.
Fig. 2
Fig. 2 Measured S11 of fabricated devices.
Fig. 3
Fig. 3 Calculation model of (a) normal electrode device, and (b) meander-line electrode device.
Fig. 4
Fig. 4 Calculated S21 of the EO frequency response for L = 200 μm, α = 0, Γg = 0, nRF = 4. (a), (b) Normal electrode device. (c), (d) Meander-line electrode device of Ld = 1186 μm, nd = 2. (a), (c) ΓL = 1, (b), (d) ΓL = 0.
Fig. 5
Fig. 5 Calculated f3dB for L = 200 μm, Γg = 0, nd = 2, nRF = 4, ng = 20, and various ΓL.
Fig. 6
Fig. 6 (a), (c) Measured EO frequency response and (b), (d) corresponding ng spectrum. (a), (b) Normal electrode device, (c), (d) Meander-line electrode device. VDC = −2 V.

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

V(z,f)= V f e γz + V b e γz ,
γ=α+j β RF , and
β RF = 2πf n RF c ,
V(z,f)= Z in V g ( e γz + Γ L e γz2γL ) ( Z g + Z in )(1+ Γ L e 2γL ) = Z 0 V g ( e γz + Γ L e γz2γL ) ( Z 0 + Z g )(1 Γ g Γ L e 2γL ) ,
Γ L = Z L Z 0 Z L + Z 0 , Γ g = Z g Z 0 Z g + Z 0 ,and
Z in = Z 0 1+ Γ L e 2γL 1 Γ L e 2γL ,
V eff (z,f)= Z 0 V g [ e ( j β o γ )z + Γ L e ( j β o +γ )z2γL ] ( Z 0 + Z g )(1 Γ g Γ L e 2γL ) and
β o = 2πf n g c
V ave (f)= 0 L V eff dz L = Z 0 V g ( e j φ + sinc φ + + Γ L e j φ e 2γL sinc φ ) ( Z 0 + Z g )(1 Γ g Γ L e 2γL ) and
φ ± = ( β o ±jγ )L 2
G(f)= 1 1+j2πf( Z g + R pn ) C pn
η(f)=| V ave (f)G(f) V ave (0)G(0) |
S 21 (f)[dB]=20 log 10 η(f)
V eff1 ( z,f )= Z 0 V g { e ( j β o γ )z + Γ L e ( j β o +γ )z2γLj2 φ d } ( Z 0 + Z g )(1 Γ g Γ L e 2γLj2 φ d ) (0zL/2)
V eff2 ( z,f )= Z 0 V g { e ( j β o γ )zj φ d + Γ L e ( j β o +γ )z2γLj φ d } ( Z 0 + Z g )(1 Γ g Γ L e 2γLj2 φ d ) (L/2zL)
φ d = 2πf n d L d c
V ave (f)= 0 L/2 V eff1 dz L/2 + L/2 L V eff2 dz L/2
= Z 0 V g { ( e j φ + 2 + e j3 φ + 2 j φ d )sinc φ + 2 + Γ L e 2γL ( e j φ 2 j2 φ d + e j3 φ + 2 j φ d )sinc φ 2 } 2( Z 0 + Z g )(1 Γ g Γ L e 2γLj2 φ d )

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