Abstract

We present a numerical method to accurately model the electro-optic interaction in anisotropic materials. Specifically, we combine a full-vectorial finite-difference optical mode solver with a radio-frequency solver to analyze the overlap between optical modes and applied electric field. This technique enables a comprehensive understanding on how electro-optic effects modify individual elements in the permittivity tensor of a material. We demonstrate the interest of this approach by designing a modulator that leverages the Pockels effect in a hybrid silicon-BaTiO3 slot waveguide. Optimized optical confinement in the active BaTiO3 layer as well as design of travelling-wave index-matched electrodes is presented. Most importantly, we show that the overall electro-optic modulation is largely governed by off-diagonal elements in the permittivity tensor. As most of active electro-optic materials are anisotropic, this method paves the way to better understand the physics of electro-optic effects and to improve optical modulators.

© 2015 Optical Society of America

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    [Crossref]
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2014 (1)

C. Xiong, W. H. Pernice, J. H. Ngai, J. W. Reiner, D. Kumah, F. J. Walker, C. H. Ahn, and H. X. Tang, “Active silicon integrated nanophotonics: ferroelectric BaTiO3 devices,” Nano Lett. 14(3), 1419–1425 (2014).
[Crossref] [PubMed]

2013 (2)

S. Abel, M. Sousa, C. Rossel, D. Caimi, M. D. Rossell, R. Erni, J. Fompeyrine, and C. Marchiori, “Controlling tetragonality and crystalline orientation in BaTiO3 nano-layers grown on Si,” Nanotechnology 24(28), 285701 (2013).
[Crossref] [PubMed]

S. Abel, T. Stöferle, C. Marchiori, C. Rossel, M. D. Rossell, R. Erni, D. Caimi, M. Sousa, A. Chelnokov, B. J. Offrein, and J. Fompeyrine, “A strong electro-optically active lead-free ferroelectric integrated on silicon,” Nat. Commun. 4, 1671 (2013).
[Crossref] [PubMed]

2012 (2)

D. J. Thomson, F. Y. Gardes, J.-M. Fedeli, S. Zlatanovic, Y. Hu, B. P. P. Kuo, E. Myslivets, N. Alic, S. Radic, G. Z. Mashanovich, and G. T. Reed, “50-Gb/s silicon optical modulator,” IEEE Photon. Technol. Lett. 24(4), 234–236 (2012).
[Crossref]

G. Fan, Y. Li, B. Han, Q. Wang, X. Liu, and Z. Zhen, “Study of an Electro-Optic Polymer Modulator,” J. Lightwave Technol. 30(15), 2482–2487 (2012).
[Crossref]

2011 (2)

B. Chmielak, M. Waldow, C. Matheisen, C. Ripperda, J. Bolten, T. Wahlbrink, M. Nagel, F. Merget, and H. Kurz, “Pockels effect based fully integrated, strained silicon electro-optic modulator,” Opt. Express 19(18), 17212–17219 (2011).
[Crossref] [PubMed]

G. Niu, S. Yin, G. Saint-Girons, B. Gautier, P. Lecoeur, V. Pillard, G. Hollinger, and B. Vilquin, “Epitaxy of BaTiO3 thin film on Si (001) using a SrTiO3 buffer layer for non-volatile memory application,” Microelectron. Eng. 88(7), 1232–1235 (2011).
[Crossref]

2010 (1)

2008 (2)

B. Fallahkhair, K. S. Li, and T. E. Murphy, “Vector Finite Difference Mode solver for Anisotropic Dielectric Waveguides,” J. Lightwave Technol. 26(11), 1423–1431 (2008).
[Crossref]

C. Rhodes, M. Cerruti, A. Efremenko, M. Losego, D. E. Aspnes, J. P. Maria, and S. Franzen, “Dependence of plasmon polaritons on the thickness of indium tin oxide thin films,” J. Appl. Phys. 103(9), 093108 (2008).
[Crossref]

2007 (3)

B. W. Wessels, “Ferroelectric epitaxial thin films for integrated optics,” Annu. Rev. Mater. Res. 37(1), 659–679 (2007).
[Crossref]

P. Bienstman, S. Selleri, I. Rosa, H. P. Uranus, W. C. Hopman, R. Costa, A. Melloni, C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: A mode solver comparison,” Opt. Quantum Electron. 38(9-11), 731–759 (2007).
[Crossref]

J. D. Bull and N. A. F. Jaeger, “Parasitic Mode Conversion in Z-Propagating Lithium-Niobate Waveguides,” J. Lightwave Technol. 25(1), 387–393 (2007).
[Crossref]

2005 (1)

D. G. Sun, Z. Liu, Y. Huang, S. T. Ho, D. J. Towner, and B. W. Wessels, “Performance simulation for ferroelectric thin-film based waveguide electro-optic modulators,” Opt. Commun. 255(4-6), 319–330 (2005).
[Crossref]

2004 (1)

P. Tang, D. J. Towner, A. L. Meier, and B. W. Wessels, “Low-voltage, polarization-insensitive, electro-optic modulator based on a polydomain barium titanate thin film,” Appl. Phys. Lett. 85(20), 4615–4617 (2004).
[Crossref]

2003 (1)

T. Hamano, D. J. Towner, and B. W. Wessels, “Relative dielectric constant of epitaxial BaTiO3 thin films in the GHz frequency range,” Appl. Phys. Lett. 83(25), 5274–5276 (2003).
[Crossref]

2002 (1)

1998 (1)

1994 (1)

M. Zgonik, P. Bernasconi, M. Duelli, R. Schlesser, P. Günter, M. Garrett, D. Rytz, Y. Zhu, and X. Wu, “Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals,” Phys. Rev. B 50(9), 5941–5949 (1994).

1992 (1)

G. R. Hadley, “Transparent boundary condition for the beam propagation method,” IEEE J. Quantum Electron. 28(1), 363–370 (1992).
[Crossref]

1990 (1)

H. M. Duiker, P. D. Beale, J. F. Scott, C. A. Paz de Araujo, B. M. Melnick, J. D. Cuchiaro, and L. D. McMillan, “Fatigue and switching in ferroelectric memories - theory and experiment,” J. Appl. Phys. 68(11), 5783–5791 (1990).
[Crossref]

Abel, S.

S. Abel, T. Stöferle, C. Marchiori, C. Rossel, M. D. Rossell, R. Erni, D. Caimi, M. Sousa, A. Chelnokov, B. J. Offrein, and J. Fompeyrine, “A strong electro-optically active lead-free ferroelectric integrated on silicon,” Nat. Commun. 4, 1671 (2013).
[Crossref] [PubMed]

S. Abel, M. Sousa, C. Rossel, D. Caimi, M. D. Rossell, R. Erni, J. Fompeyrine, and C. Marchiori, “Controlling tetragonality and crystalline orientation in BaTiO3 nano-layers grown on Si,” Nanotechnology 24(28), 285701 (2013).
[Crossref] [PubMed]

Ahn, C. H.

C. Xiong, W. H. Pernice, J. H. Ngai, J. W. Reiner, D. Kumah, F. J. Walker, C. H. Ahn, and H. X. Tang, “Active silicon integrated nanophotonics: ferroelectric BaTiO3 devices,” Nano Lett. 14(3), 1419–1425 (2014).
[Crossref] [PubMed]

Alic, N.

D. J. Thomson, F. Y. Gardes, J.-M. Fedeli, S. Zlatanovic, Y. Hu, B. P. P. Kuo, E. Myslivets, N. Alic, S. Radic, G. Z. Mashanovich, and G. T. Reed, “50-Gb/s silicon optical modulator,” IEEE Photon. Technol. Lett. 24(4), 234–236 (2012).
[Crossref]

Andreani, C.

P. Bienstman, S. Selleri, I. Rosa, H. P. Uranus, W. C. Hopman, R. Costa, A. Melloni, C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: A mode solver comparison,” Opt. Quantum Electron. 38(9-11), 731–759 (2007).
[Crossref]

Aspnes, D. E.

C. Rhodes, M. Cerruti, A. Efremenko, M. Losego, D. E. Aspnes, J. P. Maria, and S. Franzen, “Dependence of plasmon polaritons on the thickness of indium tin oxide thin films,” J. Appl. Phys. 103(9), 093108 (2008).
[Crossref]

Beale, P. D.

H. M. Duiker, P. D. Beale, J. F. Scott, C. A. Paz de Araujo, B. M. Melnick, J. D. Cuchiaro, and L. D. McMillan, “Fatigue and switching in ferroelectric memories - theory and experiment,” J. Appl. Phys. 68(11), 5783–5791 (1990).
[Crossref]

Bernasconi, P.

M. Zgonik, P. Bernasconi, M. Duelli, R. Schlesser, P. Günter, M. Garrett, D. Rytz, Y. Zhu, and X. Wu, “Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals,” Phys. Rev. B 50(9), 5941–5949 (1994).

Bienstman, P.

P. Bienstman, S. Selleri, I. Rosa, H. P. Uranus, W. C. Hopman, R. Costa, A. Melloni, C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: A mode solver comparison,” Opt. Quantum Electron. 38(9-11), 731–759 (2007).
[Crossref]

Bolten, J.

Bull, J. D.

Caimi, D.

S. Abel, M. Sousa, C. Rossel, D. Caimi, M. D. Rossell, R. Erni, J. Fompeyrine, and C. Marchiori, “Controlling tetragonality and crystalline orientation in BaTiO3 nano-layers grown on Si,” Nanotechnology 24(28), 285701 (2013).
[Crossref] [PubMed]

S. Abel, T. Stöferle, C. Marchiori, C. Rossel, M. D. Rossell, R. Erni, D. Caimi, M. Sousa, A. Chelnokov, B. J. Offrein, and J. Fompeyrine, “A strong electro-optically active lead-free ferroelectric integrated on silicon,” Nat. Commun. 4, 1671 (2013).
[Crossref] [PubMed]

Cerruti, M.

C. Rhodes, M. Cerruti, A. Efremenko, M. Losego, D. E. Aspnes, J. P. Maria, and S. Franzen, “Dependence of plasmon polaritons on the thickness of indium tin oxide thin films,” J. Appl. Phys. 103(9), 093108 (2008).
[Crossref]

Chelnokov, A.

S. Abel, T. Stöferle, C. Marchiori, C. Rossel, M. D. Rossell, R. Erni, D. Caimi, M. Sousa, A. Chelnokov, B. J. Offrein, and J. Fompeyrine, “A strong electro-optically active lead-free ferroelectric integrated on silicon,” Nat. Commun. 4, 1671 (2013).
[Crossref] [PubMed]

Chmielak, B.

Costa, R.

P. Bienstman, S. Selleri, I. Rosa, H. P. Uranus, W. C. Hopman, R. Costa, A. Melloni, C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: A mode solver comparison,” Opt. Quantum Electron. 38(9-11), 731–759 (2007).
[Crossref]

Cuchiaro, J. D.

H. M. Duiker, P. D. Beale, J. F. Scott, C. A. Paz de Araujo, B. M. Melnick, J. D. Cuchiaro, and L. D. McMillan, “Fatigue and switching in ferroelectric memories - theory and experiment,” J. Appl. Phys. 68(11), 5783–5791 (1990).
[Crossref]

Dems, M.

P. Bienstman, S. Selleri, I. Rosa, H. P. Uranus, W. C. Hopman, R. Costa, A. Melloni, C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: A mode solver comparison,” Opt. Quantum Electron. 38(9-11), 731–759 (2007).
[Crossref]

Duelli, M.

M. Zgonik, P. Bernasconi, M. Duelli, R. Schlesser, P. Günter, M. Garrett, D. Rytz, Y. Zhu, and X. Wu, “Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals,” Phys. Rev. B 50(9), 5941–5949 (1994).

Duiker, H. M.

H. M. Duiker, P. D. Beale, J. F. Scott, C. A. Paz de Araujo, B. M. Melnick, J. D. Cuchiaro, and L. D. McMillan, “Fatigue and switching in ferroelectric memories - theory and experiment,” J. Appl. Phys. 68(11), 5783–5791 (1990).
[Crossref]

Efremenko, A.

C. Rhodes, M. Cerruti, A. Efremenko, M. Losego, D. E. Aspnes, J. P. Maria, and S. Franzen, “Dependence of plasmon polaritons on the thickness of indium tin oxide thin films,” J. Appl. Phys. 103(9), 093108 (2008).
[Crossref]

Erni, R.

S. Abel, M. Sousa, C. Rossel, D. Caimi, M. D. Rossell, R. Erni, J. Fompeyrine, and C. Marchiori, “Controlling tetragonality and crystalline orientation in BaTiO3 nano-layers grown on Si,” Nanotechnology 24(28), 285701 (2013).
[Crossref] [PubMed]

S. Abel, T. Stöferle, C. Marchiori, C. Rossel, M. D. Rossell, R. Erni, D. Caimi, M. Sousa, A. Chelnokov, B. J. Offrein, and J. Fompeyrine, “A strong electro-optically active lead-free ferroelectric integrated on silicon,” Nat. Commun. 4, 1671 (2013).
[Crossref] [PubMed]

Fallahkhair, B.

Fan, G.

Fan, G. F.

Fedeli, J.-M.

D. J. Thomson, F. Y. Gardes, J.-M. Fedeli, S. Zlatanovic, Y. Hu, B. P. P. Kuo, E. Myslivets, N. Alic, S. Radic, G. Z. Mashanovich, and G. T. Reed, “50-Gb/s silicon optical modulator,” IEEE Photon. Technol. Lett. 24(4), 234–236 (2012).
[Crossref]

Fompeyrine, J.

S. Abel, T. Stöferle, C. Marchiori, C. Rossel, M. D. Rossell, R. Erni, D. Caimi, M. Sousa, A. Chelnokov, B. J. Offrein, and J. Fompeyrine, “A strong electro-optically active lead-free ferroelectric integrated on silicon,” Nat. Commun. 4, 1671 (2013).
[Crossref] [PubMed]

S. Abel, M. Sousa, C. Rossel, D. Caimi, M. D. Rossell, R. Erni, J. Fompeyrine, and C. Marchiori, “Controlling tetragonality and crystalline orientation in BaTiO3 nano-layers grown on Si,” Nanotechnology 24(28), 285701 (2013).
[Crossref] [PubMed]

Franzen, S.

C. Rhodes, M. Cerruti, A. Efremenko, M. Losego, D. E. Aspnes, J. P. Maria, and S. Franzen, “Dependence of plasmon polaritons on the thickness of indium tin oxide thin films,” J. Appl. Phys. 103(9), 093108 (2008).
[Crossref]

Gardes, F. Y.

D. J. Thomson, F. Y. Gardes, J.-M. Fedeli, S. Zlatanovic, Y. Hu, B. P. P. Kuo, E. Myslivets, N. Alic, S. Radic, G. Z. Mashanovich, and G. T. Reed, “50-Gb/s silicon optical modulator,” IEEE Photon. Technol. Lett. 24(4), 234–236 (2012).
[Crossref]

Garrett, M.

M. Zgonik, P. Bernasconi, M. Duelli, R. Schlesser, P. Günter, M. Garrett, D. Rytz, Y. Zhu, and X. Wu, “Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals,” Phys. Rev. B 50(9), 5941–5949 (1994).

Gautier, B.

G. Niu, S. Yin, G. Saint-Girons, B. Gautier, P. Lecoeur, V. Pillard, G. Hollinger, and B. Vilquin, “Epitaxy of BaTiO3 thin film on Si (001) using a SrTiO3 buffer layer for non-volatile memory application,” Microelectron. Eng. 88(7), 1232–1235 (2011).
[Crossref]

Günter, P.

M. Zgonik, P. Bernasconi, M. Duelli, R. Schlesser, P. Günter, M. Garrett, D. Rytz, Y. Zhu, and X. Wu, “Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals,” Phys. Rev. B 50(9), 5941–5949 (1994).

Hadley, G. R.

G. R. Hadley, “Transparent boundary condition for the beam propagation method,” IEEE J. Quantum Electron. 28(1), 363–370 (1992).
[Crossref]

Hamano, T.

T. Hamano, D. J. Towner, and B. W. Wessels, “Relative dielectric constant of epitaxial BaTiO3 thin films in the GHz frequency range,” Appl. Phys. Lett. 83(25), 5274–5276 (2003).
[Crossref]

Han, B.

Ho, S. T.

D. G. Sun, Z. Liu, Y. Huang, S. T. Ho, D. J. Towner, and B. W. Wessels, “Performance simulation for ferroelectric thin-film based waveguide electro-optic modulators,” Opt. Commun. 255(4-6), 319–330 (2005).
[Crossref]

Hollinger, G.

G. Niu, S. Yin, G. Saint-Girons, B. Gautier, P. Lecoeur, V. Pillard, G. Hollinger, and B. Vilquin, “Epitaxy of BaTiO3 thin film on Si (001) using a SrTiO3 buffer layer for non-volatile memory application,” Microelectron. Eng. 88(7), 1232–1235 (2011).
[Crossref]

Hopman, W. C.

P. Bienstman, S. Selleri, I. Rosa, H. P. Uranus, W. C. Hopman, R. Costa, A. Melloni, C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: A mode solver comparison,” Opt. Quantum Electron. 38(9-11), 731–759 (2007).
[Crossref]

Hu, Y.

D. J. Thomson, F. Y. Gardes, J.-M. Fedeli, S. Zlatanovic, Y. Hu, B. P. P. Kuo, E. Myslivets, N. Alic, S. Radic, G. Z. Mashanovich, and G. T. Reed, “50-Gb/s silicon optical modulator,” IEEE Photon. Technol. Lett. 24(4), 234–236 (2012).
[Crossref]

Huang, Y.

D. G. Sun, Z. Liu, Y. Huang, S. T. Ho, D. J. Towner, and B. W. Wessels, “Performance simulation for ferroelectric thin-film based waveguide electro-optic modulators,” Opt. Commun. 255(4-6), 319–330 (2005).
[Crossref]

Hugonin, J. P.

P. Bienstman, S. Selleri, I. Rosa, H. P. Uranus, W. C. Hopman, R. Costa, A. Melloni, C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: A mode solver comparison,” Opt. Quantum Electron. 38(9-11), 731–759 (2007).
[Crossref]

Jaeger, N. A. F.

Kumah, D.

C. Xiong, W. H. Pernice, J. H. Ngai, J. W. Reiner, D. Kumah, F. J. Walker, C. H. Ahn, and H. X. Tang, “Active silicon integrated nanophotonics: ferroelectric BaTiO3 devices,” Nano Lett. 14(3), 1419–1425 (2014).
[Crossref] [PubMed]

Kuo, B. P. P.

D. J. Thomson, F. Y. Gardes, J.-M. Fedeli, S. Zlatanovic, Y. Hu, B. P. P. Kuo, E. Myslivets, N. Alic, S. Radic, G. Z. Mashanovich, and G. T. Reed, “50-Gb/s silicon optical modulator,” IEEE Photon. Technol. Lett. 24(4), 234–236 (2012).
[Crossref]

Kurz, H.

Lalanne, P.

P. Bienstman, S. Selleri, I. Rosa, H. P. Uranus, W. C. Hopman, R. Costa, A. Melloni, C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: A mode solver comparison,” Opt. Quantum Electron. 38(9-11), 731–759 (2007).
[Crossref]

Lecoeur, P.

G. Niu, S. Yin, G. Saint-Girons, B. Gautier, P. Lecoeur, V. Pillard, G. Hollinger, and B. Vilquin, “Epitaxy of BaTiO3 thin film on Si (001) using a SrTiO3 buffer layer for non-volatile memory application,” Microelectron. Eng. 88(7), 1232–1235 (2011).
[Crossref]

Li, K. S.

Li, Y.

Liu, X.

Liu, Z.

D. G. Sun, Z. Liu, Y. Huang, S. T. Ho, D. J. Towner, and B. W. Wessels, “Performance simulation for ferroelectric thin-film based waveguide electro-optic modulators,” Opt. Commun. 255(4-6), 319–330 (2005).
[Crossref]

Losego, M.

C. Rhodes, M. Cerruti, A. Efremenko, M. Losego, D. E. Aspnes, J. P. Maria, and S. Franzen, “Dependence of plasmon polaritons on the thickness of indium tin oxide thin films,” J. Appl. Phys. 103(9), 093108 (2008).
[Crossref]

Marchiori, C.

S. Abel, T. Stöferle, C. Marchiori, C. Rossel, M. D. Rossell, R. Erni, D. Caimi, M. Sousa, A. Chelnokov, B. J. Offrein, and J. Fompeyrine, “A strong electro-optically active lead-free ferroelectric integrated on silicon,” Nat. Commun. 4, 1671 (2013).
[Crossref] [PubMed]

S. Abel, M. Sousa, C. Rossel, D. Caimi, M. D. Rossell, R. Erni, J. Fompeyrine, and C. Marchiori, “Controlling tetragonality and crystalline orientation in BaTiO3 nano-layers grown on Si,” Nanotechnology 24(28), 285701 (2013).
[Crossref] [PubMed]

Maria, J. P.

C. Rhodes, M. Cerruti, A. Efremenko, M. Losego, D. E. Aspnes, J. P. Maria, and S. Franzen, “Dependence of plasmon polaritons on the thickness of indium tin oxide thin films,” J. Appl. Phys. 103(9), 093108 (2008).
[Crossref]

Mashanovich, G. Z.

D. J. Thomson, F. Y. Gardes, J.-M. Fedeli, S. Zlatanovic, Y. Hu, B. P. P. Kuo, E. Myslivets, N. Alic, S. Radic, G. Z. Mashanovich, and G. T. Reed, “50-Gb/s silicon optical modulator,” IEEE Photon. Technol. Lett. 24(4), 234–236 (2012).
[Crossref]

Masi, M.

Matheisen, C.

McMillan, L. D.

H. M. Duiker, P. D. Beale, J. F. Scott, C. A. Paz de Araujo, B. M. Melnick, J. D. Cuchiaro, and L. D. McMillan, “Fatigue and switching in ferroelectric memories - theory and experiment,” J. Appl. Phys. 68(11), 5783–5791 (1990).
[Crossref]

Meier, A. L.

P. Tang, D. J. Towner, A. L. Meier, and B. W. Wessels, “Low-voltage, polarization-insensitive, electro-optic modulator based on a polydomain barium titanate thin film,” Appl. Phys. Lett. 85(20), 4615–4617 (2004).
[Crossref]

Melloni, A.

P. Bienstman, S. Selleri, I. Rosa, H. P. Uranus, W. C. Hopman, R. Costa, A. Melloni, C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: A mode solver comparison,” Opt. Quantum Electron. 38(9-11), 731–759 (2007).
[Crossref]

Melnick, B. M.

H. M. Duiker, P. D. Beale, J. F. Scott, C. A. Paz de Araujo, B. M. Melnick, J. D. Cuchiaro, and L. D. McMillan, “Fatigue and switching in ferroelectric memories - theory and experiment,” J. Appl. Phys. 68(11), 5783–5791 (1990).
[Crossref]

Merget, F.

Murphy, T. E.

Myslivets, E.

D. J. Thomson, F. Y. Gardes, J.-M. Fedeli, S. Zlatanovic, Y. Hu, B. P. P. Kuo, E. Myslivets, N. Alic, S. Radic, G. Z. Mashanovich, and G. T. Reed, “50-Gb/s silicon optical modulator,” IEEE Photon. Technol. Lett. 24(4), 234–236 (2012).
[Crossref]

Nagel, M.

Ngai, J. H.

C. Xiong, W. H. Pernice, J. H. Ngai, J. W. Reiner, D. Kumah, F. J. Walker, C. H. Ahn, and H. X. Tang, “Active silicon integrated nanophotonics: ferroelectric BaTiO3 devices,” Nano Lett. 14(3), 1419–1425 (2014).
[Crossref] [PubMed]

Niu, G.

G. Niu, S. Yin, G. Saint-Girons, B. Gautier, P. Lecoeur, V. Pillard, G. Hollinger, and B. Vilquin, “Epitaxy of BaTiO3 thin film on Si (001) using a SrTiO3 buffer layer for non-volatile memory application,” Microelectron. Eng. 88(7), 1232–1235 (2011).
[Crossref]

Obayya, S. S. A.

P. Bienstman, S. Selleri, I. Rosa, H. P. Uranus, W. C. Hopman, R. Costa, A. Melloni, C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: A mode solver comparison,” Opt. Quantum Electron. 38(9-11), 731–759 (2007).
[Crossref]

Offrein, B. J.

S. Abel, T. Stöferle, C. Marchiori, C. Rossel, M. D. Rossell, R. Erni, D. Caimi, M. Sousa, A. Chelnokov, B. J. Offrein, and J. Fompeyrine, “A strong electro-optically active lead-free ferroelectric integrated on silicon,” Nat. Commun. 4, 1671 (2013).
[Crossref] [PubMed]

Orobtchouk, R.

Panajotov, K.

P. Bienstman, S. Selleri, I. Rosa, H. P. Uranus, W. C. Hopman, R. Costa, A. Melloni, C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: A mode solver comparison,” Opt. Quantum Electron. 38(9-11), 731–759 (2007).
[Crossref]

Pavesi, L.

Paz de Araujo, C. A.

H. M. Duiker, P. D. Beale, J. F. Scott, C. A. Paz de Araujo, B. M. Melnick, J. D. Cuchiaro, and L. D. McMillan, “Fatigue and switching in ferroelectric memories - theory and experiment,” J. Appl. Phys. 68(11), 5783–5791 (1990).
[Crossref]

Pernice, W. H.

C. Xiong, W. H. Pernice, J. H. Ngai, J. W. Reiner, D. Kumah, F. J. Walker, C. H. Ahn, and H. X. Tang, “Active silicon integrated nanophotonics: ferroelectric BaTiO3 devices,” Nano Lett. 14(3), 1419–1425 (2014).
[Crossref] [PubMed]

Pillard, V.

G. Niu, S. Yin, G. Saint-Girons, B. Gautier, P. Lecoeur, V. Pillard, G. Hollinger, and B. Vilquin, “Epitaxy of BaTiO3 thin film on Si (001) using a SrTiO3 buffer layer for non-volatile memory application,” Microelectron. Eng. 88(7), 1232–1235 (2011).
[Crossref]

Pinto, D.

P. Bienstman, S. Selleri, I. Rosa, H. P. Uranus, W. C. Hopman, R. Costa, A. Melloni, C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: A mode solver comparison,” Opt. Quantum Electron. 38(9-11), 731–759 (2007).
[Crossref]

Pogossian, S. P.

Powell, O.

Radic, S.

D. J. Thomson, F. Y. Gardes, J.-M. Fedeli, S. Zlatanovic, Y. Hu, B. P. P. Kuo, E. Myslivets, N. Alic, S. Radic, G. Z. Mashanovich, and G. T. Reed, “50-Gb/s silicon optical modulator,” IEEE Photon. Technol. Lett. 24(4), 234–236 (2012).
[Crossref]

Reed, G. T.

D. J. Thomson, F. Y. Gardes, J.-M. Fedeli, S. Zlatanovic, Y. Hu, B. P. P. Kuo, E. Myslivets, N. Alic, S. Radic, G. Z. Mashanovich, and G. T. Reed, “50-Gb/s silicon optical modulator,” IEEE Photon. Technol. Lett. 24(4), 234–236 (2012).
[Crossref]

Reiner, J. W.

C. Xiong, W. H. Pernice, J. H. Ngai, J. W. Reiner, D. Kumah, F. J. Walker, C. H. Ahn, and H. X. Tang, “Active silicon integrated nanophotonics: ferroelectric BaTiO3 devices,” Nano Lett. 14(3), 1419–1425 (2014).
[Crossref] [PubMed]

Rhodes, C.

C. Rhodes, M. Cerruti, A. Efremenko, M. Losego, D. E. Aspnes, J. P. Maria, and S. Franzen, “Dependence of plasmon polaritons on the thickness of indium tin oxide thin films,” J. Appl. Phys. 103(9), 093108 (2008).
[Crossref]

Ripperda, C.

Rosa, I.

P. Bienstman, S. Selleri, I. Rosa, H. P. Uranus, W. C. Hopman, R. Costa, A. Melloni, C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: A mode solver comparison,” Opt. Quantum Electron. 38(9-11), 731–759 (2007).
[Crossref]

Rossel, C.

S. Abel, M. Sousa, C. Rossel, D. Caimi, M. D. Rossell, R. Erni, J. Fompeyrine, and C. Marchiori, “Controlling tetragonality and crystalline orientation in BaTiO3 nano-layers grown on Si,” Nanotechnology 24(28), 285701 (2013).
[Crossref] [PubMed]

S. Abel, T. Stöferle, C. Marchiori, C. Rossel, M. D. Rossell, R. Erni, D. Caimi, M. Sousa, A. Chelnokov, B. J. Offrein, and J. Fompeyrine, “A strong electro-optically active lead-free ferroelectric integrated on silicon,” Nat. Commun. 4, 1671 (2013).
[Crossref] [PubMed]

Rossell, M. D.

S. Abel, T. Stöferle, C. Marchiori, C. Rossel, M. D. Rossell, R. Erni, D. Caimi, M. Sousa, A. Chelnokov, B. J. Offrein, and J. Fompeyrine, “A strong electro-optically active lead-free ferroelectric integrated on silicon,” Nat. Commun. 4, 1671 (2013).
[Crossref] [PubMed]

S. Abel, M. Sousa, C. Rossel, D. Caimi, M. D. Rossell, R. Erni, J. Fompeyrine, and C. Marchiori, “Controlling tetragonality and crystalline orientation in BaTiO3 nano-layers grown on Si,” Nanotechnology 24(28), 285701 (2013).
[Crossref] [PubMed]

Rytz, D.

M. Zgonik, P. Bernasconi, M. Duelli, R. Schlesser, P. Günter, M. Garrett, D. Rytz, Y. Zhu, and X. Wu, “Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals,” Phys. Rev. B 50(9), 5941–5949 (1994).

Saint-Girons, G.

G. Niu, S. Yin, G. Saint-Girons, B. Gautier, P. Lecoeur, V. Pillard, G. Hollinger, and B. Vilquin, “Epitaxy of BaTiO3 thin film on Si (001) using a SrTiO3 buffer layer for non-volatile memory application,” Microelectron. Eng. 88(7), 1232–1235 (2011).
[Crossref]

Schlesser, R.

M. Zgonik, P. Bernasconi, M. Duelli, R. Schlesser, P. Günter, M. Garrett, D. Rytz, Y. Zhu, and X. Wu, “Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals,” Phys. Rev. B 50(9), 5941–5949 (1994).

Scott, J. F.

H. M. Duiker, P. D. Beale, J. F. Scott, C. A. Paz de Araujo, B. M. Melnick, J. D. Cuchiaro, and L. D. McMillan, “Fatigue and switching in ferroelectric memories - theory and experiment,” J. Appl. Phys. 68(11), 5783–5791 (1990).
[Crossref]

Selleri, S.

P. Bienstman, S. Selleri, I. Rosa, H. P. Uranus, W. C. Hopman, R. Costa, A. Melloni, C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: A mode solver comparison,” Opt. Quantum Electron. 38(9-11), 731–759 (2007).
[Crossref]

Sousa, M.

S. Abel, M. Sousa, C. Rossel, D. Caimi, M. D. Rossell, R. Erni, J. Fompeyrine, and C. Marchiori, “Controlling tetragonality and crystalline orientation in BaTiO3 nano-layers grown on Si,” Nanotechnology 24(28), 285701 (2013).
[Crossref] [PubMed]

S. Abel, T. Stöferle, C. Marchiori, C. Rossel, M. D. Rossell, R. Erni, D. Caimi, M. Sousa, A. Chelnokov, B. J. Offrein, and J. Fompeyrine, “A strong electro-optically active lead-free ferroelectric integrated on silicon,” Nat. Commun. 4, 1671 (2013).
[Crossref] [PubMed]

Stöferle, T.

S. Abel, T. Stöferle, C. Marchiori, C. Rossel, M. D. Rossell, R. Erni, D. Caimi, M. Sousa, A. Chelnokov, B. J. Offrein, and J. Fompeyrine, “A strong electro-optically active lead-free ferroelectric integrated on silicon,” Nat. Commun. 4, 1671 (2013).
[Crossref] [PubMed]

Sun, D. G.

D. G. Sun, Z. Liu, Y. Huang, S. T. Ho, D. J. Towner, and B. W. Wessels, “Performance simulation for ferroelectric thin-film based waveguide electro-optic modulators,” Opt. Commun. 255(4-6), 319–330 (2005).
[Crossref]

Tang, H. X.

C. Xiong, W. H. Pernice, J. H. Ngai, J. W. Reiner, D. Kumah, F. J. Walker, C. H. Ahn, and H. X. Tang, “Active silicon integrated nanophotonics: ferroelectric BaTiO3 devices,” Nano Lett. 14(3), 1419–1425 (2014).
[Crossref] [PubMed]

Tang, P.

P. Tang, D. J. Towner, A. L. Meier, and B. W. Wessels, “Low-voltage, polarization-insensitive, electro-optic modulator based on a polydomain barium titanate thin film,” Appl. Phys. Lett. 85(20), 4615–4617 (2004).
[Crossref]

Thomson, D. J.

D. J. Thomson, F. Y. Gardes, J.-M. Fedeli, S. Zlatanovic, Y. Hu, B. P. P. Kuo, E. Myslivets, N. Alic, S. Radic, G. Z. Mashanovich, and G. T. Reed, “50-Gb/s silicon optical modulator,” IEEE Photon. Technol. Lett. 24(4), 234–236 (2012).
[Crossref]

Towner, D. J.

D. G. Sun, Z. Liu, Y. Huang, S. T. Ho, D. J. Towner, and B. W. Wessels, “Performance simulation for ferroelectric thin-film based waveguide electro-optic modulators,” Opt. Commun. 255(4-6), 319–330 (2005).
[Crossref]

P. Tang, D. J. Towner, A. L. Meier, and B. W. Wessels, “Low-voltage, polarization-insensitive, electro-optic modulator based on a polydomain barium titanate thin film,” Appl. Phys. Lett. 85(20), 4615–4617 (2004).
[Crossref]

T. Hamano, D. J. Towner, and B. W. Wessels, “Relative dielectric constant of epitaxial BaTiO3 thin films in the GHz frequency range,” Appl. Phys. Lett. 83(25), 5274–5276 (2003).
[Crossref]

Uranus, H. P.

P. Bienstman, S. Selleri, I. Rosa, H. P. Uranus, W. C. Hopman, R. Costa, A. Melloni, C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: A mode solver comparison,” Opt. Quantum Electron. 38(9-11), 731–759 (2007).
[Crossref]

Vescan, L.

Vilquin, B.

G. Niu, S. Yin, G. Saint-Girons, B. Gautier, P. Lecoeur, V. Pillard, G. Hollinger, and B. Vilquin, “Epitaxy of BaTiO3 thin film on Si (001) using a SrTiO3 buffer layer for non-volatile memory application,” Microelectron. Eng. 88(7), 1232–1235 (2011).
[Crossref]

Vonsovici, A.

Wahlbrink, T.

Waldow, M.

Walker, F. J.

C. Xiong, W. H. Pernice, J. H. Ngai, J. W. Reiner, D. Kumah, F. J. Walker, C. H. Ahn, and H. X. Tang, “Active silicon integrated nanophotonics: ferroelectric BaTiO3 devices,” Nano Lett. 14(3), 1419–1425 (2014).
[Crossref] [PubMed]

Wang, Q.

Wessels, B. W.

B. W. Wessels, “Ferroelectric epitaxial thin films for integrated optics,” Annu. Rev. Mater. Res. 37(1), 659–679 (2007).
[Crossref]

D. G. Sun, Z. Liu, Y. Huang, S. T. Ho, D. J. Towner, and B. W. Wessels, “Performance simulation for ferroelectric thin-film based waveguide electro-optic modulators,” Opt. Commun. 255(4-6), 319–330 (2005).
[Crossref]

P. Tang, D. J. Towner, A. L. Meier, and B. W. Wessels, “Low-voltage, polarization-insensitive, electro-optic modulator based on a polydomain barium titanate thin film,” Appl. Phys. Lett. 85(20), 4615–4617 (2004).
[Crossref]

T. Hamano, D. J. Towner, and B. W. Wessels, “Relative dielectric constant of epitaxial BaTiO3 thin films in the GHz frequency range,” Appl. Phys. Lett. 83(25), 5274–5276 (2003).
[Crossref]

Wu, X.

M. Zgonik, P. Bernasconi, M. Duelli, R. Schlesser, P. Günter, M. Garrett, D. Rytz, Y. Zhu, and X. Wu, “Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals,” Phys. Rev. B 50(9), 5941–5949 (1994).

Xiong, C.

C. Xiong, W. H. Pernice, J. H. Ngai, J. W. Reiner, D. Kumah, F. J. Walker, C. H. Ahn, and H. X. Tang, “Active silicon integrated nanophotonics: ferroelectric BaTiO3 devices,” Nano Lett. 14(3), 1419–1425 (2014).
[Crossref] [PubMed]

Yin, S.

G. Niu, S. Yin, G. Saint-Girons, B. Gautier, P. Lecoeur, V. Pillard, G. Hollinger, and B. Vilquin, “Epitaxy of BaTiO3 thin film on Si (001) using a SrTiO3 buffer layer for non-volatile memory application,” Microelectron. Eng. 88(7), 1232–1235 (2011).
[Crossref]

Zgonik, M.

M. Zgonik, P. Bernasconi, M. Duelli, R. Schlesser, P. Günter, M. Garrett, D. Rytz, Y. Zhu, and X. Wu, “Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals,” Phys. Rev. B 50(9), 5941–5949 (1994).

Zhen, Z.

Zhu, Y.

M. Zgonik, P. Bernasconi, M. Duelli, R. Schlesser, P. Günter, M. Garrett, D. Rytz, Y. Zhu, and X. Wu, “Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals,” Phys. Rev. B 50(9), 5941–5949 (1994).

Zlatanovic, S.

D. J. Thomson, F. Y. Gardes, J.-M. Fedeli, S. Zlatanovic, Y. Hu, B. P. P. Kuo, E. Myslivets, N. Alic, S. Radic, G. Z. Mashanovich, and G. T. Reed, “50-Gb/s silicon optical modulator,” IEEE Photon. Technol. Lett. 24(4), 234–236 (2012).
[Crossref]

Annu. Rev. Mater. Res. (1)

B. W. Wessels, “Ferroelectric epitaxial thin films for integrated optics,” Annu. Rev. Mater. Res. 37(1), 659–679 (2007).
[Crossref]

Appl. Phys. Lett. (2)

P. Tang, D. J. Towner, A. L. Meier, and B. W. Wessels, “Low-voltage, polarization-insensitive, electro-optic modulator based on a polydomain barium titanate thin film,” Appl. Phys. Lett. 85(20), 4615–4617 (2004).
[Crossref]

T. Hamano, D. J. Towner, and B. W. Wessels, “Relative dielectric constant of epitaxial BaTiO3 thin films in the GHz frequency range,” Appl. Phys. Lett. 83(25), 5274–5276 (2003).
[Crossref]

IEEE J. Quantum Electron. (1)

G. R. Hadley, “Transparent boundary condition for the beam propagation method,” IEEE J. Quantum Electron. 28(1), 363–370 (1992).
[Crossref]

IEEE Photon. Technol. Lett. (1)

D. J. Thomson, F. Y. Gardes, J.-M. Fedeli, S. Zlatanovic, Y. Hu, B. P. P. Kuo, E. Myslivets, N. Alic, S. Radic, G. Z. Mashanovich, and G. T. Reed, “50-Gb/s silicon optical modulator,” IEEE Photon. Technol. Lett. 24(4), 234–236 (2012).
[Crossref]

J. Appl. Phys. (2)

C. Rhodes, M. Cerruti, A. Efremenko, M. Losego, D. E. Aspnes, J. P. Maria, and S. Franzen, “Dependence of plasmon polaritons on the thickness of indium tin oxide thin films,” J. Appl. Phys. 103(9), 093108 (2008).
[Crossref]

H. M. Duiker, P. D. Beale, J. F. Scott, C. A. Paz de Araujo, B. M. Melnick, J. D. Cuchiaro, and L. D. McMillan, “Fatigue and switching in ferroelectric memories - theory and experiment,” J. Appl. Phys. 68(11), 5783–5791 (1990).
[Crossref]

J. Lightwave Technol. (6)

Microelectron. Eng. (1)

G. Niu, S. Yin, G. Saint-Girons, B. Gautier, P. Lecoeur, V. Pillard, G. Hollinger, and B. Vilquin, “Epitaxy of BaTiO3 thin film on Si (001) using a SrTiO3 buffer layer for non-volatile memory application,” Microelectron. Eng. 88(7), 1232–1235 (2011).
[Crossref]

Nano Lett. (1)

C. Xiong, W. H. Pernice, J. H. Ngai, J. W. Reiner, D. Kumah, F. J. Walker, C. H. Ahn, and H. X. Tang, “Active silicon integrated nanophotonics: ferroelectric BaTiO3 devices,” Nano Lett. 14(3), 1419–1425 (2014).
[Crossref] [PubMed]

Nanotechnology (1)

S. Abel, M. Sousa, C. Rossel, D. Caimi, M. D. Rossell, R. Erni, J. Fompeyrine, and C. Marchiori, “Controlling tetragonality and crystalline orientation in BaTiO3 nano-layers grown on Si,” Nanotechnology 24(28), 285701 (2013).
[Crossref] [PubMed]

Nat. Commun. (1)

S. Abel, T. Stöferle, C. Marchiori, C. Rossel, M. D. Rossell, R. Erni, D. Caimi, M. Sousa, A. Chelnokov, B. J. Offrein, and J. Fompeyrine, “A strong electro-optically active lead-free ferroelectric integrated on silicon,” Nat. Commun. 4, 1671 (2013).
[Crossref] [PubMed]

Opt. Commun. (1)

D. G. Sun, Z. Liu, Y. Huang, S. T. Ho, D. J. Towner, and B. W. Wessels, “Performance simulation for ferroelectric thin-film based waveguide electro-optic modulators,” Opt. Commun. 255(4-6), 319–330 (2005).
[Crossref]

Opt. Express (1)

Opt. Quantum Electron. (1)

P. Bienstman, S. Selleri, I. Rosa, H. P. Uranus, W. C. Hopman, R. Costa, A. Melloni, C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: A mode solver comparison,” Opt. Quantum Electron. 38(9-11), 731–759 (2007).
[Crossref]

Phys. Rev. B (1)

M. Zgonik, P. Bernasconi, M. Duelli, R. Schlesser, P. Günter, M. Garrett, D. Rytz, Y. Zhu, and X. Wu, “Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals,” Phys. Rev. B 50(9), 5941–5949 (1994).

Other (12)

J. M. Liu, Photonic Devices (Cambridge University, 2005.

R. Orobtchouk, P. Labeye, X. Hu, S. Malhouitre, Ph. Grosse, and J.-M. Fedeli, “Design, realization, and characterization of a silicon photonics coherent mixer for PDM-QPSK optical communications,” presented at the Photonics Europeen Optical Society Annual Meeting TOM 2-Silicon, Scotland, 25–28 September 2012.

R. Syms and J. Cozens Syms, Optical Guided Waves and Devices (McGraw-Hill, 1992).

M. D. Janezic, D. F. Williams, V. Blaschke, A. Karamcheti, and C. S. Chang, “Permittivity characterization of low-k thin films from transmission-line measurements,” in Proceedings of IEEE Conference on Transactions on Microwave Theory and Techniques 51, 132–136 (2003).
[Crossref]

H. Kogelnik, “Theory of optical waveguides,” in Guided Wave Optoelectronics, 2nd Ed., T. Tamir ed. (Springer, 1990,), pp. 7–88.

M. Minakata, “Recent progress of 40-GHz high-speed LiNbO3 optical modulator,” ITCom 2001: Intern. Symp. on the Convergence of IT and Communications. International Society for Optics and Photonics (2001).

J. H. Mathews, Numerical Methods for Mathematics, Science and Engineering (Prentice-Hall, 1992).

T. E. Shoup, A Practical Guide to Computer Methods for Engineers (Prentice-Hall, 1979).

S. Rosloniec, Fundamental Numerical Methods for Electrical Engineering (Springer, 2008).

E. D. Palik, Handbook of Optical Constants of Solids, (Academic press, 1998, Vol. III).

M. N. Afsar and K. J. Button, “Precise Millimeter-Wave Measurements of Complex Refractive Index, Complex Dielectric Permittivity and Loss Tangent of GaAs, Si, SiO2, A12O3, BeO, Macor, and Glass,” in Proceedings of IEEE Conference on Microwave Theory and Techniques 31, 217–223 (1983).

J. F. Scott, Ferroelectric Memories (Springer, 2000).

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Figures (9)

Fig. 1
Fig. 1

Cross-section of the active zone of a BTO-based slot-waveguide electro-optical Mach-Zehnder modulator on a standard SOI substrate.

Fig. 2
Fig. 2

Optimization of the slot waveguide geometry: (a) Optical confinement factor Γ versus W and Ta-Si; (b) Optical effective index versus W and Ta-Si; (c) Optical loss versus W and Ta-Si; (d) Optimized TM-like mode power profile

Fig. 3
Fig. 3

Evolution of propagation losses of slot mode versus electrode spacing D

Fig. 4
Fig. 4

Evolution of (a) Γeo and (b) microwave effective index n e f f m versus the thickness Te and width We of the electrode for a gap D of about 0.3 µm. The dashed line indicates geometries where value of n e f f m matches that of n e f f o .

Fig. 5
Fig. 5

Electric voltage plot with 20 V applied across the electrodes and the gap spacing between the two electrodes G is 1.36 µm with D = 0.3µm.

Fig. 6
Fig. 6

Variation of the five non-zero permittivity tensor elements Δεxx, Δεyy, Δεzz, Δεyz, Δεzy within BTO layer as a result of a 20 V bias voltage and D = 0.3µm.

Fig. 7
Fig. 7

(a) The optical effective index variation Δ n e f f o increases gradually with bias voltage from 0 to 20V, and TM-like mode shows a higher variation compared to TE-like mode; (b) Key optical modulator parameter Vπ·L decreases gradually with bias voltage from 0 to 20V, and TM-like mode shows a smaller Vπ·L value.

Fig. 8
Fig. 8

Nomenclature convention of the grid point and meshing for the finite difference scheme

Fig. 9
Fig. 9

Mesh grid of a metallic contact (i1 = 7, i2 = 12 and j1 = 5, j2 = 8).

Tables (1)

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Table 1 Optical Constants of Thin Films of Materials

Equations (46)

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Δ ( 1 ε x x 1 ε y y 1 ε z z 1 ε y z 1 ε x z 1 ε x y ) = 1 ε 0 ( 0 0 r 13 0 0 r 23 0 0 r 33 0 r 42 0 r 51 0 0 0 0 0 ) ( E x e E y e E z e ) ,
( ε x x ' 0 0 0 ε y y ' ε y z ' 0 ε z y ' ε z z ' ) = ( ε x x 1 + ε x x r 13 E z e 0 0 0 ε y y ( 1 + ε z z r 33 E z e ) Δ ε y y ε z z r 42 E y e Δ 0 ε y y ε z z r 42 E y e Δ ε z z ( 1 + ε y y r 13 E z e ) Δ ) ,
ε i j ' ε i j Δ ε i j ,
{ k 0 2 ε z z H y + 2 y 2 ( H y ) + ε z z z ( 1 ε x x z ( H y ) ) + ε z y y ( 1 ε x x z ( H y ) ) k 0 2 ε z y H z ε z y y ( 1 ε x x y ( H z ) ) + 2 y z ( H z ) ε z z z ( 1 ε x x y ( H z ) ) = β 2 H y , k 0 2 ε y z H y ε y z z ( 1 ε x x z ( H y ) ) + 2 y z ( H y ) ε y y y ( 1 ε x x z ( H y ) ) + k 0 2 ε y y H z + ε y y y ( 1 ε x x y ( H z ) ) + 2 z 2 ( H z ) + ε y z z ( 1 ε x x y ( H z ) ) = β 2 H z .
{ k 0 2 ε y y E y + y ( 1 ε x x y ( ε y y ( E y ) ) ) + 2 z 2 ( E y ) + y ( 1 ε x x z ( ε z y ( E y ) ) ) + k 0 2 ε y z E z + y ( 1 ε x x y ( ε y z ( E z ) ) ) + y ( 1 ε x x z ( ε z z ( E z ) ) ) 2 y z ( E z ) = β 2 E y , k 0 2 ε z y E y + z ( 1 ε x x z ( ε z y ( E y ) ) ) + z ( 1 ε x x y ( ε y y ( E y ) ) ) 2 y z ( E y ) + k 0 2 ε z z E z + 2 y 2 ( E z ) + z ( 1 ε x x z ( ε z z ( E z ) ) ) + z ( 1 ε x x y ( ε y z ( E z ) ) ) = β 2 E z .
β 2 ( A y A z ) = [ b y y b y z b z y b z z ] ( A y A z ) ,
Γ   =   B T O Re ( E   ×   H * ) x d S total Re ( E   ×   H * ) x d S ,
Δ β = ω ε 0 P + ( ( [ Δ ε ] E ) . E * ) d y d z ,
P = + ( E H * + H E * ) . e x d y d z .
Δ β r 42 ε O ε e ω ε 0 P + ( E y e ( E z . E y * + E y . E z * ) ) d y d z .
Γ E O   =   G V B T O E y e ( E z . E y * + E y . E z * ) d y d z P ,
r e f f   =   λ G n e f f o 3 Γ E O V π L ,
( [ ε ] Φ ) = 0 ,
y ( ε y y y Φ + ε y z z Φ ) + z ( ε z y y Φ + ε z z z Φ ) = 0 ,
L = λ 2 Δ n e f f o
a i j = X = P , N , S , W , E , N W , N E , S W , S E a i j X H j X ,
b i j = X = P , N , S , W , E , N W , N E , S W , S E b i j X E j X
a y y P = k 0 2 ε z z P ( 2 e w ) ( 4 s ε z z P n 2 ( n + s ) ( ε x x N + ε x x P ) + 4 n ε z z P s 2 ( n + s ) ( ε x x S + ε x x P ) 2 ε z z P ( n s ) 2 n 2 s 2 ε x x P ) + ( e w ) . ( n s ) e . w . n . s ( ε z y P ε x x P ) .
a y y W = ( 2 w ( e + w ) ) + e . ( n s ) w . n . s . ( e + w ) ( ε z y P ε x x W ) . a y y E = ( 2 e ( e + w ) ) w . ( n s ) e . n . s . ( e + w ) ( ε z y P ε x x E ) .
a y y S = ( 4 n ε z z P s 2 ( n + s ) ( ε x x S + ε x x P ) 2 ( n s ) ε z z P s 2 ( n + s ) ε x x P ) + n . ( e w ) e . w . s . ( n + s ) ( ε z y P ε x x P ) .
a y y N = ( 4 s ε z z P n 2 ( n + s ) ( ε x x N + ε x x P ) + 2 ( n s ) ε z z P n 2 ( n + s ) ε x x P ) s . ( e w ) e . w . n . ( n + s ) ( ε z y P ε x x P ) .
a y y N E = w . s e . n . ( e + w ) ( n + s ) ( ε z y P ε x x E ) . a y y S E = w . n e . s . ( e + w ) ( n + s ) ( ε z y P ε x x E ) .
a y y N W = e . s w . n . ( e + w ) ( n + s ) ( ε z y P ε x x W ) . a y y S W = e . n w . s . ( e + w ) ( n + s ) ( ε z y P ε x x W ) .
a y z P = k 0 2 ε z y P + ( 4 w ε z y P e 2 ( e + w ) ( ε x x E + ε x x P ) + 4 e ε z y P w 2 ( e + w ) ( ε x x W + ε x x P ) 2 ε z y P ( e w ) 2 e 2 w 2 ε x x P ) + ( e w ) . ( n s ) e . w . n . s ( 1 ε z z P ε x x P ) .
a y z W = ( 4 e ε z y P w 2 ( e + w ) ( ε x x W + ε x x P ) 2 ( e w ) ε z y P w 2 ( e + w ) ε x x P ) + e . ( n s ) w . n . s . ( e + w ) ( 1 ε z z P ε x x P ) .
a y z E = ( 4 w ε z y P e 2 ( e + w ) ( ε x x E + ε x x P ) + 2 ( e w ) ε z y P e 2 ( e + w ) ε x x P ) w . ( n s ) e . n . s . ( e + w ) ( 1 ε z z P ε x x P ) .
a y z S = n . ( e w ) e . w . s . ( n + s ) ( 1 ε z z P ε x x S ) . a y z N = s . ( e w ) e . w . n . ( n + s ) ( 1 ε z z P ε x x N ) .
a y z N E = w . s e . n . ( e + w ) ( n + s ) ( 1 ε z z P ε x x N ) . a y z S E = w . n e . s . ( e + w ) ( n + s ) ( 1 ε z z P ε x x S ) .
a y z N W = e . s w . n . ( e + w ) ( n + s ) ( 1 ε z z P ε x x N ) . a y z S W = e . n w . s . ( e + w ) ( n + s ) ( 1 ε z z P ε x x S ) .
b y y P = k 0 2 ε y y P ( 4 w ε y y P e 2 ( e + w ) ( ε x x E + ε x x P ) + 4 e ε y y P w 2 ( e + w ) ( ε x x W + ε x x P ) 2 ε y y P ( e w ) 2 e 2 w 2 ε x x P ) ( 2 n s ) + ( e w ) . ( n s ) e . w . n . s ( ε z y P ε x x P ) .
b y y W = ( 4 e ε y y W w 2 ( e + w ) ( ε x x W + ε x x P ) 2 ( e w ) ε y y W w 2 ( e + w ) ε x x P ) + e . ( n s ) w . n . s . ( e + w ) ( ε z y W ε x x W ) .
b y y E = ( 4 w ε y y E e 2 ( e + w ) ( ε x x E + ε x x P ) + 2 ( e w ) ε y y E e 2 ( e + w ) ε x x P ) w . ( n s ) e . n . s . ( e + w ) ( ε z y E ε x x E ) .
b y y S = ( 2 s ( n + s ) ) + n . ( e w ) e . w . s . ( n + s ) ( ε z y S ε x x P ) . b y y N = ( 2 n ( n + s ) ) s . ( e w ) e . w . n . ( n + s ) ( ε z y N ε x x P ) .
b y y N E = 1 ( e + w ) ( n + s ) ( w . s . ε z y N E e . n . ε x x E ) . b y y S E = w . n e . s . ( e + w ) ( n + s ) ( ε z y S E ε x x E ) .
b y y N W = e . s w . n . ( e + w ) ( n + s ) ( ε z y N W ε x x W ) . b y y S W = e . n w . s . ( e + w ) ( n + s ) ( ε z y S W ε x x W ) .
b y z P = k 0 2 ε y z P ( 4 w ε y z P e 2 ( e + w ) ( ε x x E + ε x x P ) + 4 e ε y z P w 2 ( e + w ) ( ε x x W + ε x x P ) 2 ε y z P ( e w ) 2 e 2 w 2 ε x x P ) + ( e w ) . ( n s ) e . w . n . s ( ε z z P ε x x P 1 ) .
b y z W = ( 4 e ε y z W w 2 ( e + w ) ( ε x x W + ε x x P ) 2 ( e w ) ε y z W w 2 ( e + w ) ε x x P ) + e . ( n s ) w . n . s . ( e + w ) ( ε z z W ε x x W 1 ) .
b y z E = ( 4 w ε y z E e 2 ( e + w ) ( ε x x E + ε x x P ) + 2 ( e w ) ε y z E e 2 ( e + w ) ε x x P ) w . ( n s ) e . n . s . ( e + w ) ( ε z z E ε x x E 1 ) .
b y z S = n . ( e w ) e . w . s . ( n + s ) ( ε z z S ε x x P 1 ) . b y z N = s . ( e w ) e . w . n . ( n + s ) ( ε z z N ε x x P 1 ) .
b y z N E = w . s e . n . ( e + w ) ( n + s ) ( ε z z N E ε x x E 1 ) . b y z S E = w . n e . s ( e + w ) ( n + s ) ( ε z z S E ε x x E 1 ) .
b y z N W = e . s w . n . ( e + w ) ( n + s ) ( ε z z N W ε x x W 1 ) . b y z S W = e . n w . s . ( e + w ) ( n + s ) ( ε z z S W ε x x W 1 ) .
X = P , N , S , W , E , N W , N E , S W , S E a x Φ X = 0
a P   =   ( w ( ε y y E + ε y y P ) e 2 ( e + w ) + e ( ε y y W + ε y y P ) w 2 ( e + w ) ( e w ) 2 ε y y P e 2 w 2 ) ( s ( ε z z N + ε z z P ) n 2 ( n + s ) + n ( ε z z S + ε z z P ) s 2 ( n + s ) ( n s ) 2 ε z z P n 2 s 2 ) + ( e 2 w 2 ) ( n 2 s 2 ) ( ε y z P + ε z y P ) ( e + w ) ( n + s ) e w n s .
a E   =   ( w ε y y E + e ε y y P e 2 ( e + w ) ) + w ( n 2 s 2 ) ( ε y z E + ε z y P ) e n s . a W   =   e ε y y W + w ε y y P w 2 ( e + w ) e ( n 2 s 2 ) ( ε y z W + ε z y P ) w n s .
a N   =   ( s ε z z N + n ε z z P n 2 ( n + s ) ) + s ( e 2 w 2 ) ( ε y z P + ε z y N ) e w n . a s   =   ( n ε z z S + s ε z z P s 2 ( n + s ) ) n ( e 2 w 2 ) ( ε y z P + ε z y S ) e w s .
a N E   =   w s e n ε y z E + ε z y N ( e + w ) ( n + s ) . a S E   =   w n e s ε y z E + ε z y S ( e + w ) ( n + s ) . a N W   =   e s w n ε y z W + ε z y N ( e + w ) ( n + s ) . a S W   =   e n w s ε y z W + ε z y N ( e + w ) ( n + s ) .

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