Abstract

We propose a theoretical model to describe the strain-induced linear electro-optic (Pockels) effect in centro-symmetric crystals. The general formulation is presented and the specific case of the strained silicon is investigated in detail because of its attractive properties for integrated optics. The outcome of this analysis is a linear relation between the second order susceptibility tensor and the strain gradient tensor, depending generically on fifteen coefficients. The proposed model greatly simplifies the description of the electro-optic effect in strained silicon waveguides, providing a powerful and effective tool for design and optimization of optical devices.

© 2015 Optical Society of America

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References

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2015 (2)

R. Sharma, M. W. Puckett, H.-H. Lin, F. Vallini, and Y. Fainman, “Characterizing the effects of free carriers in fully etched, dielectric-clad silicon waveguides,” Appl. Phys. Lett. 106, 241104 (2015).
[Crossref]

S. SharifAzadeh, F. Merget, M. P. Nezhad, and J. Witzens, “On the measurements of pockels effect in strained silicon,” Opt. Lett. 40, 1877–1880 (2015).
[Crossref]

2014 (4)

2013 (2)

M. Virgilio, C. L. Manganelli, G. Grosso, G. Pizzi, and G. Capellini, “Radiative recombination and optical gain spectra in biaxially strained n-type germanium,” Phys. Rev. B 87, 235313 (2013).
[Crossref]

B. Chmielak, C. Matheisen, C. Ripperda, J. Bolten, T. Wahlbrink, M. Waldow, and H. Kurz, “Investigation of local strain distribution and linear electro-optic effect in strained silicon waveguides,” Opt. Express 21, 25324–25332 (2013).
[Crossref] [PubMed]

2011 (3)

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, “Second-harmonic generation in silicon waveguides strained by silicon nitride,” Nat. Mater. 11, 148 (2011).
[Crossref] [PubMed]

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

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, 17212–17219 (2011).
[Crossref] [PubMed]

2010 (4)

J. Liu, X. Sun, R. Camacho-Aguilera, L. C. Kimerling, and J. Michel, “Ge-on-Si laser operating at room temperature,” Opt. Lett. 35, 679–681 (2010).
[Crossref] [PubMed]

M. A. Hopcroft, W. D. Nix, and T. W. Kenny, “What is the Young’s modulus of silicon?” J. Microelectromech. Syst. 19, 229–238 (2010).
[Crossref]

E. Luppi, H. Hübener, and V. Véniard, “Communications: Ab initio second-order nonlinear optics in solids,” J. Chem. Phys. 132, 241104 (2010).
[Crossref] [PubMed]

E. Luppi, H. Hübener, and V. Véniard, “Ab-initio second-order nonlinear optics in solids: second-harmonic generation spectroscopy from time-dependent density-functional theory,” Phys. Rev. B 82, 235201 (2010).
[Crossref]

2009 (1)

2006 (2)

X. Chen, N. C. Panoiu, and R. M. Osgood, “Theory of raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42, 160–170 (2006).
[Crossref]

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature 441, 199–202 (2006).
[Crossref]

2001 (1)

S. Mitchell, M. Mehendale, D. Villeneuve, and R. Boukherroub, “Second harmonic generation spectroscopy of chemically modified Si(111) surfaces,” Surf. Sci. 488, 367–378 (2001).
[Crossref]

1997 (1)

N. Fleck and J. W. Hutchinson, “Strain gradient plasticity,” Adv. Appl. Mech. 33, 295–361 (1997).
[Crossref]

1994 (1)

J. Y. Huang, “Probing inhomogeneous lattice deformation at interface of Si(111)/SiO2 by optical second-harmonic reflection and Raman spectroscopy,” Jpn. J. Appl. Phys. 33, 3878–3886 (1994).
[Crossref]

1987 (1)

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

1984 (1)

B. M. A. Rahman and B. J. Davies, “Penalty function improvement of waveguide solution by finite elements,” IEEE Trans. Microwave Theory Tech. 32, 922–928 (1984).
[Crossref]

Abashin, M.

Aktsipetrov, O. A.

Andersen, K. N.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature 441, 199–202 (2006).
[Crossref]

Bennett, B.

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

Bianco, F.

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, “Second-harmonic generation in silicon waveguides strained by silicon nitride,” Nat. Mater. 11, 148 (2011).
[Crossref] [PubMed]

Bjarklev, A.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature 441, 199–202 (2006).
[Crossref]

Bolten, J.

Borel, P. I.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature 441, 199–202 (2006).
[Crossref]

Borga, E.

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, “Second-harmonic generation in silicon waveguides strained by silicon nitride,” Nat. Mater. 11, 148 (2011).
[Crossref] [PubMed]

Boukherroub, R.

S. Mitchell, M. Mehendale, D. Villeneuve, and R. Boukherroub, “Second harmonic generation spectroscopy of chemically modified Si(111) surfaces,” Surf. Sci. 488, 367–378 (2001).
[Crossref]

Boyd, R.

R. Boyd, Non Linear Optics (Academic Press, 2010).

Camacho-Aguilera, R.

Capellini, G.

M. Virgilio, C. L. Manganelli, G. Grosso, G. Pizzi, and G. Capellini, “Radiative recombination and optical gain spectra in biaxially strained n-type germanium,” Phys. Rev. B 87, 235313 (2013).
[Crossref]

Cassan, E.

Cazzanelli, M.

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, “Second-harmonic generation in silicon waveguides strained by silicon nitride,” Nat. Mater. 11, 148 (2011).
[Crossref] [PubMed]

Chen, Q.-D.

Chen, X.

X. Chen, N. C. Panoiu, and R. M. Osgood, “Theory of raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42, 160–170 (2006).
[Crossref]

Chen, Z.-G.

Chmielak, B.

Damas, P.

Davies, B. J.

B. M. A. Rahman and B. J. Davies, “Penalty function improvement of waveguide solution by finite elements,” IEEE Trans. Microwave Theory Tech. 32, 922–928 (1984).
[Crossref]

Degoli, E.

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, “Second-harmonic generation in silicon waveguides strained by silicon nitride,” Nat. Mater. 11, 148 (2011).
[Crossref] [PubMed]

Dolgova, T. V.

Esseni, D.

D. Esseni, P. Palestri, and L. Selmi, Nanoscale MOS Transistors: Semi-classical Modeling and Applications (Cambridge University Press, 2011).
[Crossref]

Fage-Pedersen, J.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature 441, 199–202 (2006).
[Crossref]

Fainman, Y.

R. Sharma, M. W. Puckett, H.-H. Lin, F. Vallini, and Y. Fainman, “Characterizing the effects of free carriers in fully etched, dielectric-clad silicon waveguides,” Appl. Phys. Lett. 106, 241104 (2015).
[Crossref]

M. W. Puckett, J. S. T. Smalley, M. Abashin, A. Grieco, and Y. Fainman, “Tensor of the second-order nonlinear susceptibility in asymmetrically strained silicon waveguides: analysis and experimental validation,” Opt. Lett. 39, 1693–1696 (2014).
[Crossref] [PubMed]

R. Sharma, M. W. Puckett, H.-H. Lin, A. Isichenko, F. Vallini, and Y. Fainman, “Capacitively-induced free-carrier effects in nanoscale silicon waveguides for electro-optic modulation,” arXiv preprint arXiv:1508.05440, (2015).

Fleck, N.

N. Fleck and J. W. Hutchinson, “Strain gradient plasticity,” Adv. Appl. Mech. 33, 295–361 (1997).
[Crossref]

Frandsen, L. H.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature 441, 199–202 (2006).
[Crossref]

Ghulinyan, M.

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, “Second-harmonic generation in silicon waveguides strained by silicon nitride,” Nat. Mater. 11, 148 (2011).
[Crossref] [PubMed]

Grieco, A.

Grosso, G.

M. Virgilio, C. L. Manganelli, G. Grosso, G. Pizzi, and G. Capellini, “Radiative recombination and optical gain spectra in biaxially strained n-type germanium,” Phys. Rev. B 87, 235313 (2013).
[Crossref]

Hansen, O.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature 441, 199–202 (2006).
[Crossref]

Hon, N. K.

N. K. Hon, K. K. Tsia, D. R. Solli, B. Jalali, and J. B. Khurgin, “Stress-induced χ(2) in silicon comparison between theoretical and experimental values,” in Proceedings of 6th IEEE International Conference on Group IV Photonics, 2009, (San Francisco, California), pp. 234September 2009.

Hopcroft, M. A.

M. A. Hopcroft, W. D. Nix, and T. W. Kenny, “What is the Young’s modulus of silicon?” J. Microelectromech. Syst. 19, 229–238 (2010).
[Crossref]

Huang, J. Y.

J. Y. Huang, “Probing inhomogeneous lattice deformation at interface of Si(111)/SiO2 by optical second-harmonic reflection and Raman spectroscopy,” Jpn. J. Appl. Phys. 33, 3878–3886 (1994).
[Crossref]

Hübener, H.

E. Luppi, H. Hübener, and V. Véniard, “Communications: Ab initio second-order nonlinear optics in solids,” J. Chem. Phys. 132, 241104 (2010).
[Crossref] [PubMed]

E. Luppi, H. Hübener, and V. Véniard, “Ab-initio second-order nonlinear optics in solids: second-harmonic generation spectroscopy from time-dependent density-functional theory,” Phys. Rev. B 82, 235201 (2010).
[Crossref]

Hutchinson, J. W.

N. Fleck and J. W. Hutchinson, “Strain gradient plasticity,” Adv. Appl. Mech. 33, 295–361 (1997).
[Crossref]

Isichenko, A.

R. Sharma, M. W. Puckett, H.-H. Lin, A. Isichenko, F. Vallini, and Y. Fainman, “Capacitively-induced free-carrier effects in nanoscale silicon waveguides for electro-optic modulation,” arXiv preprint arXiv:1508.05440, (2015).

Izard, N.

Jacobsen, R. S.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature 441, 199–202 (2006).
[Crossref]

Jalali, B.

N. K. Hon, K. K. Tsia, D. R. Solli, B. Jalali, and J. B. Khurgin, “Stress-induced χ(2) in silicon comparison between theoretical and experimental values,” in Proceedings of 6th IEEE International Conference on Group IV Photonics, 2009, (San Francisco, California), pp. 234September 2009.

Jia, G.

Jiang, Y.

Kenny, T. W.

M. A. Hopcroft, W. D. Nix, and T. W. Kenny, “What is the Young’s modulus of silicon?” J. Microelectromech. Syst. 19, 229–238 (2010).
[Crossref]

Khurgin, J. B.

N. K. Hon, K. K. Tsia, D. R. Solli, B. Jalali, and J. B. Khurgin, “Stress-induced χ(2) in silicon comparison between theoretical and experimental values,” in Proceedings of 6th IEEE International Conference on Group IV Photonics, 2009, (San Francisco, California), pp. 234September 2009.

Kimerling, L. C.

Kosevich, A. M.

L. D. Landau, L. P. Pitaevskii, A. M. Kosevich, and E. M. Lifshitz, Theory of Elasticity (Elsevier Butterworth Heinemann, 1986).

Kristensen, M.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature 441, 199–202 (2006).
[Crossref]

Kurz, H.

Landau, L. D.

L. D. Landau, L. P. Pitaevskii, A. M. Kosevich, and E. M. Lifshitz, Theory of Elasticity (Elsevier Butterworth Heinemann, 1986).

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodinamic of Continuous Media (Elsevier Butterworth Heinemann, 1984).

Lavrinenko, A. V.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature 441, 199–202 (2006).
[Crossref]

Le Bourdais, D.

Le Roux, X.

Lecoeur, P.

Leon, S.

S. Leon, Linear Algebra with Applications (Pearson, 2009).

Li, J.

J. Li, Z. Shan, and E. Ma, “Elastic strain engineering for unprecedented materials properties,” MRS Bull. 39, 108–114 (2014).
[Crossref]

Lifshitz, E. M.

L. D. Landau, L. P. Pitaevskii, A. M. Kosevich, and E. M. Lifshitz, Theory of Elasticity (Elsevier Butterworth Heinemann, 1986).

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodinamic of Continuous Media (Elsevier Butterworth Heinemann, 1984).

Lin, H.-H.

R. Sharma, M. W. Puckett, H.-H. Lin, F. Vallini, and Y. Fainman, “Characterizing the effects of free carriers in fully etched, dielectric-clad silicon waveguides,” Appl. Phys. Lett. 106, 241104 (2015).
[Crossref]

R. Sharma, M. W. Puckett, H.-H. Lin, A. Isichenko, F. Vallini, and Y. Fainman, “Capacitively-induced free-carrier effects in nanoscale silicon waveguides for electro-optic modulation,” arXiv preprint arXiv:1508.05440, (2015).

Liu, J.

Liu, J.-M.

J.-M. Liu, Photonic Devices (Cambridge University Press, 2009).

Love, J.

A. W. Snyder and J. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

Luppi, E.

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, “Second-harmonic generation in silicon waveguides strained by silicon nitride,” Nat. Mater. 11, 148 (2011).
[Crossref] [PubMed]

E. Luppi, H. Hübener, and V. Véniard, “Communications: Ab initio second-order nonlinear optics in solids,” J. Chem. Phys. 132, 241104 (2010).
[Crossref] [PubMed]

E. Luppi, H. Hübener, and V. Véniard, “Ab-initio second-order nonlinear optics in solids: second-harmonic generation spectroscopy from time-dependent density-functional theory,” Phys. Rev. B 82, 235201 (2010).
[Crossref]

Ma, E.

J. Li, Z. Shan, and E. Ma, “Elastic strain engineering for unprecedented materials properties,” MRS Bull. 39, 108–114 (2014).
[Crossref]

Manganelli, C. L.

M. Virgilio, C. L. Manganelli, G. Grosso, G. Pizzi, and G. Capellini, “Radiative recombination and optical gain spectra in biaxially strained n-type germanium,” Phys. Rev. B 87, 235313 (2013).
[Crossref]

Maroutian, T.

Marris-Morini, D.

Mashanovich, G. Z.

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

Matheisen, C.

Mehendale, M.

S. Mitchell, M. Mehendale, D. Villeneuve, and R. Boukherroub, “Second harmonic generation spectroscopy of chemically modified Si(111) surfaces,” Surf. Sci. 488, 367–378 (2001).
[Crossref]

Merget, F.

Michel, J.

Mitchell, S.

S. Mitchell, M. Mehendale, D. Villeneuve, and R. Boukherroub, “Second harmonic generation spectroscopy of chemically modified Si(111) surfaces,” Surf. Sci. 488, 367–378 (2001).
[Crossref]

Modotto, D.

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, “Second-harmonic generation in silicon waveguides strained by silicon nitride,” Nat. Mater. 11, 148 (2011).
[Crossref] [PubMed]

Moulin, G.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature 441, 199–202 (2006).
[Crossref]

Nagel, M.

Nedeljkovic, M.

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

Nezhad, M. P.

Nix, W. D.

M. A. Hopcroft, W. D. Nix, and T. W. Kenny, “What is the Young’s modulus of silicon?” J. Microelectromech. Syst. 19, 229–238 (2010).
[Crossref]

Nye, J. F.

J. F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices (Oxford University Press, 1985).

Osgood, R. M.

X. Chen, N. C. Panoiu, and R. M. Osgood, “Theory of raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42, 160–170 (2006).
[Crossref]

Ossicini, S.

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, “Second-harmonic generation in silicon waveguides strained by silicon nitride,” Nat. Mater. 11, 148 (2011).
[Crossref] [PubMed]

Ou, H.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature 441, 199–202 (2006).
[Crossref]

Palestri, P.

D. Esseni, P. Palestri, and L. Selmi, Nanoscale MOS Transistors: Semi-classical Modeling and Applications (Cambridge University Press, 2011).
[Crossref]

Panoiu, N. C.

X. Chen, N. C. Panoiu, and R. M. Osgood, “Theory of raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42, 160–170 (2006).
[Crossref]

Pavesi, L.

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, “Second-harmonic generation in silicon waveguides strained by silicon nitride,” Nat. Mater. 11, 148 (2011).
[Crossref] [PubMed]

Peucheret, C.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature 441, 199–202 (2006).
[Crossref]

Pierobon, R.

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, “Second-harmonic generation in silicon waveguides strained by silicon nitride,” Nat. Mater. 11, 148 (2011).
[Crossref] [PubMed]

Pitaevskii, L. P.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodinamic of Continuous Media (Elsevier Butterworth Heinemann, 1984).

L. D. Landau, L. P. Pitaevskii, A. M. Kosevich, and E. M. Lifshitz, Theory of Elasticity (Elsevier Butterworth Heinemann, 1986).

Pizzi, G.

M. Virgilio, C. L. Manganelli, G. Grosso, G. Pizzi, and G. Capellini, “Radiative recombination and optical gain spectra in biaxially strained n-type germanium,” Phys. Rev. B 87, 235313 (2013).
[Crossref]

Pucker, G.

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, “Second-harmonic generation in silicon waveguides strained by silicon nitride,” Nat. Mater. 11, 148 (2011).
[Crossref] [PubMed]

Puckett, M. W.

R. Sharma, M. W. Puckett, H.-H. Lin, F. Vallini, and Y. Fainman, “Characterizing the effects of free carriers in fully etched, dielectric-clad silicon waveguides,” Appl. Phys. Lett. 106, 241104 (2015).
[Crossref]

M. W. Puckett, J. S. T. Smalley, M. Abashin, A. Grieco, and Y. Fainman, “Tensor of the second-order nonlinear susceptibility in asymmetrically strained silicon waveguides: analysis and experimental validation,” Opt. Lett. 39, 1693–1696 (2014).
[Crossref] [PubMed]

R. Sharma, M. W. Puckett, H.-H. Lin, A. Isichenko, F. Vallini, and Y. Fainman, “Capacitively-induced free-carrier effects in nanoscale silicon waveguides for electro-optic modulation,” arXiv preprint arXiv:1508.05440, (2015).

Rahman, B. M. A.

B. M. A. Rahman and B. J. Davies, “Penalty function improvement of waveguide solution by finite elements,” IEEE Trans. Microwave Theory Tech. 32, 922–928 (1984).
[Crossref]

Ripperda, C.

Selmi, L.

D. Esseni, P. Palestri, and L. Selmi, Nanoscale MOS Transistors: Semi-classical Modeling and Applications (Cambridge University Press, 2011).
[Crossref]

Shan, Z.

J. Li, Z. Shan, and E. Ma, “Elastic strain engineering for unprecedented materials properties,” MRS Bull. 39, 108–114 (2014).
[Crossref]

SharifAzadeh, S.

Sharma, R.

R. Sharma, M. W. Puckett, H.-H. Lin, F. Vallini, and Y. Fainman, “Characterizing the effects of free carriers in fully etched, dielectric-clad silicon waveguides,” Appl. Phys. Lett. 106, 241104 (2015).
[Crossref]

R. Sharma, M. W. Puckett, H.-H. Lin, A. Isichenko, F. Vallini, and Y. Fainman, “Capacitively-induced free-carrier effects in nanoscale silicon waveguides for electro-optic modulation,” arXiv preprint arXiv:1508.05440, (2015).

Smalley, J. S. T.

Snyder, A. W.

A. W. Snyder and J. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

Solli, D. R.

N. K. Hon, K. K. Tsia, D. R. Solli, B. Jalali, and J. B. Khurgin, “Stress-induced χ(2) in silicon comparison between theoretical and experimental values,” in Proceedings of 6th IEEE International Conference on Group IV Photonics, 2009, (San Francisco, California), pp. 234September 2009.

Soref, R.

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

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

Su, W.

Sun, H.-B.

Sun, X.

Tsia, K. K.

N. K. Hon, K. K. Tsia, D. R. Solli, B. Jalali, and J. B. Khurgin, “Stress-induced χ(2) in silicon comparison between theoretical and experimental values,” in Proceedings of 6th IEEE International Conference on Group IV Photonics, 2009, (San Francisco, California), pp. 234September 2009.

Vallini, F.

R. Sharma, M. W. Puckett, H.-H. Lin, F. Vallini, and Y. Fainman, “Characterizing the effects of free carriers in fully etched, dielectric-clad silicon waveguides,” Appl. Phys. Lett. 106, 241104 (2015).
[Crossref]

R. Sharma, M. W. Puckett, H.-H. Lin, A. Isichenko, F. Vallini, and Y. Fainman, “Capacitively-induced free-carrier effects in nanoscale silicon waveguides for electro-optic modulation,” arXiv preprint arXiv:1508.05440, (2015).

Véniard, V.

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, “Second-harmonic generation in silicon waveguides strained by silicon nitride,” Nat. Mater. 11, 148 (2011).
[Crossref] [PubMed]

E. Luppi, H. Hübener, and V. Véniard, “Communications: Ab initio second-order nonlinear optics in solids,” J. Chem. Phys. 132, 241104 (2010).
[Crossref] [PubMed]

E. Luppi, H. Hübener, and V. Véniard, “Ab-initio second-order nonlinear optics in solids: second-harmonic generation spectroscopy from time-dependent density-functional theory,” Phys. Rev. B 82, 235201 (2010).
[Crossref]

Villeneuve, D.

S. Mitchell, M. Mehendale, D. Villeneuve, and R. Boukherroub, “Second harmonic generation spectroscopy of chemically modified Si(111) surfaces,” Surf. Sci. 488, 367–378 (2001).
[Crossref]

Virgilio, M.

M. Virgilio, C. L. Manganelli, G. Grosso, G. Pizzi, and G. Capellini, “Radiative recombination and optical gain spectra in biaxially strained n-type germanium,” Phys. Rev. B 87, 235313 (2013).
[Crossref]

Vivien, L.

Wabnitz, S.

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, “Second-harmonic generation in silicon waveguides strained by silicon nitride,” Nat. Mater. 11, 148 (2011).
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Waldow, M.

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A. Yariv and P. Yeh, Optical Waves in Crystals (A Wiley-Interscience Publication, 1984).

Yildiz, B.

B. Yildiz, “Streching the energy landscape of oxides - effects on electrocatalysis and diffusion,” MRS Bull. 39, 147–156 (2014).
[Crossref]

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Zsigri, B.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature 441, 199–202 (2006).
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N. Fleck and J. W. Hutchinson, “Strain gradient plasticity,” Adv. Appl. Mech. 33, 295–361 (1997).
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Appl. Phys. Lett. (1)

R. Sharma, M. W. Puckett, H.-H. Lin, F. Vallini, and Y. Fainman, “Characterizing the effects of free carriers in fully etched, dielectric-clad silicon waveguides,” Appl. Phys. Lett. 106, 241104 (2015).
[Crossref]

IEEE J. Quantum Electron. (2)

X. Chen, N. C. Panoiu, and R. M. Osgood, “Theory of raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42, 160–170 (2006).
[Crossref]

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

IEEE Photon. J. (1)

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

IEEE Trans. Microwave Theory Tech. (1)

B. M. A. Rahman and B. J. Davies, “Penalty function improvement of waveguide solution by finite elements,” IEEE Trans. Microwave Theory Tech. 32, 922–928 (1984).
[Crossref]

J. Chem. Phys. (1)

E. Luppi, H. Hübener, and V. Véniard, “Communications: Ab initio second-order nonlinear optics in solids,” J. Chem. Phys. 132, 241104 (2010).
[Crossref] [PubMed]

J. Microelectromech. Syst. (1)

M. A. Hopcroft, W. D. Nix, and T. W. Kenny, “What is the Young’s modulus of silicon?” J. Microelectromech. Syst. 19, 229–238 (2010).
[Crossref]

Jpn. J. Appl. Phys. (1)

J. Y. Huang, “Probing inhomogeneous lattice deformation at interface of Si(111)/SiO2 by optical second-harmonic reflection and Raman spectroscopy,” Jpn. J. Appl. Phys. 33, 3878–3886 (1994).
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MRS Bull. (2)

J. Li, Z. Shan, and E. Ma, “Elastic strain engineering for unprecedented materials properties,” MRS Bull. 39, 108–114 (2014).
[Crossref]

B. Yildiz, “Streching the energy landscape of oxides - effects on electrocatalysis and diffusion,” MRS Bull. 39, 147–156 (2014).
[Crossref]

Nat. Mater. (1)

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, “Second-harmonic generation in silicon waveguides strained by silicon nitride,” Nat. Mater. 11, 148 (2011).
[Crossref] [PubMed]

Nature (1)

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature 441, 199–202 (2006).
[Crossref]

Opt. Express (3)

Opt. Lett. (4)

Phys. Rev. B (2)

E. Luppi, H. Hübener, and V. Véniard, “Ab-initio second-order nonlinear optics in solids: second-harmonic generation spectroscopy from time-dependent density-functional theory,” Phys. Rev. B 82, 235201 (2010).
[Crossref]

M. Virgilio, C. L. Manganelli, G. Grosso, G. Pizzi, and G. Capellini, “Radiative recombination and optical gain spectra in biaxially strained n-type germanium,” Phys. Rev. B 87, 235313 (2013).
[Crossref]

Surf. Sci. (1)

S. Mitchell, M. Mehendale, D. Villeneuve, and R. Boukherroub, “Second harmonic generation spectroscopy of chemically modified Si(111) surfaces,” Surf. Sci. 488, 367–378 (2001).
[Crossref]

Other (13)

R. Sharma, M. W. Puckett, H.-H. Lin, A. Isichenko, F. Vallini, and Y. Fainman, “Capacitively-induced free-carrier effects in nanoscale silicon waveguides for electro-optic modulation,” arXiv preprint arXiv:1508.05440, (2015).

A. Yariv, Optical Electronics (Holt McDougal, 1984).

J.-M. Liu, Photonic Devices (Cambridge University Press, 2009).

A. Yariv and P. Yeh, Optical Waves in Crystals (A Wiley-Interscience Publication, 1984).

N. K. Hon, K. K. Tsia, D. R. Solli, B. Jalali, and J. B. Khurgin, “Stress-induced χ(2) in silicon comparison between theoretical and experimental values,” in Proceedings of 6th IEEE International Conference on Group IV Photonics, 2009, (San Francisco, California), pp. 234September 2009.

R. Boyd, Non Linear Optics (Academic Press, 2010).

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodinamic of Continuous Media (Elsevier Butterworth Heinemann, 1984).

D. Esseni, P. Palestri, and L. Selmi, Nanoscale MOS Transistors: Semi-classical Modeling and Applications (Cambridge University Press, 2011).
[Crossref]

L. D. Landau, L. P. Pitaevskii, A. M. Kosevich, and E. M. Lifshitz, Theory of Elasticity (Elsevier Butterworth Heinemann, 1986).

A. W. Snyder and J. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

“COMSOL Multiphysics,” www.comsol.com .

J. F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices (Oxford University Press, 1985).

S. Leon, Linear Algebra with Applications (Pearson, 2009).

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Figures (9)

Fig. 1
Fig. 1 Cross-section of the strained silicon based MZI studied in [12, 13] and [14], respectively. In (a) the slab waveguide cross-section described in [12,13]. The cases of waveguide width (wSi) equal to 300 nm, 350 nm, 400 nm, 450 nm, and 500 nm have been investigated. In (b) the channel waveguide cross-section described in [14]. The cases of waveguide width (wSi) equal to 385 nm, 435 nm, and 468 nm have been investigated. Pictures are not to scale.
Fig. 2
Fig. 2 Strain profile of silicon waveguide used in Chmielak et al. [13]: (a) εxx, (b) εyy, (c) εzz, (d) εxy.
Fig. 3
Fig. 3 Electric field components in the case of Chmielak et al. [13]
Fig. 4
Fig. 4 Strain profile of silicon waveguide used in Damas et al. [14]: (a) εxx, (b) εyy, (c) εzz, (d) εxy.
Fig. 5
Fig. 5 Electric field components in the case of Damas et al. [14] for an incident wavelength of 1550 nm.
Fig. 6
Fig. 6 Behavior of the overlap functions for the waveguides under investigation with respect to waveguide width for (a) the device used by Chmielak et al. [13] and (b) the device used by Damas et al. [14]. In both cases only the most significant overlaps are plotted at λ = 1550nm.
Fig. 7
Fig. 7 Behavior of the overlap functions for (a) the waveguide used in Chmielak et al. [13] and (b) the waveguide used in Damas et al. [14] with respect to the wavelength for a waveguide width wSi = 385nm. Only the most significant overlaps are plotted.
Fig. 8
Fig. 8 Strain components profile εxx (a), εyy(b), εzz(c) and εxy(d) in the strained silicon waveguide [13] in presence of defect fabrication in the Si3N4 slab.
Fig. 9
Fig. 9 Comparison between the overlap factors for the case of Chmielak et al. [13] with and without fabrication defect.

Equations (39)

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P i ( 2 ) ( t ) = ε 0 0 χ i j k ( 2 ) ( τ 1 , τ 2 ) E j ( t τ 1 ) E k ( t τ 2 ) d τ 1 d τ 2 ,
P i ( 2 ) ( ω 1 + ω 2 ) = ε 0 χ i j k ( 2 ) ( ω 1 + ω 2 ; ω 1 , ω 2 ) E j ( ω 1 ) E k ( ω 2 ) ,
χ i j k ( 2 ) ( ω 1 + ω 2 ; ω 1 , ω 2 ) = χ i j k ( 2 ) ( ω 1 + ω 2 ; ω 2 , ω 1 )
χ i j k ( 2 ) ( ω 1 + ω 2 ; ω 1 , ω 2 ) = χ i j k ( 2 ) ( ω 1 ω 2 ; ω 1 , ω 2 ) * .
χ i j k ( 2 ) ( ω ; ω , 0 ) = χ i j k ( 2 ) ( ω ; 0 , ω ) , χ i j k ( 2 ) ( ω ; ω , 0 ) = χ i j k ( 2 ) ( ω ; ω , 0 ) = χ i j k ( 2 ) ( ω ; ω , 0 ) ,
P i ( 2 ) ( ω ) = 2 χ i j k ( 2 ) ( ω ; ω , 0 ) E j o p t ( ω ) E k dc .
ε i j = u i x j + u j x i
χ i j k ( 2 ) = χ i j k ( 2 ) | ε = 0 + χ i j k ( 2 ) ε α β | ε = 0 ε α β + χ i j k ( 2 ) ζ α β γ | ε = 0 ζ α β γ + 2 χ i j k ( 2 ) ε α β ε γ δ | ε = 0 ε α β ε γ δ + 2 χ i j k ( 2 ) ε α β ζ γ δ μ | ε = 0 ε α β ζ γ δ μ + 2 χ i j k ( 2 ) ζ α β γ ζ δ μ ν | ε = 0 ζ α β γ ζ δ μ ν + ,
ζ α β γ = ε α β x γ
χ i j k ( 2 ) = χ i j k ( 2 ) ζ α β γ | ε = 0 ζ α β γ + 2 χ i j k ( 2 ) ε α β ζ γ δ μ | ε = 0 ε α β ζ γ δ μ + .
χ i j k ( 2 ) ( x ; ω 1 + ω 2 ; ω 1 , ω 2 ) = T i j k α β γ ( ω 1 + ω 2 ; ω 1 , ω 2 ) ζ α β γ ( x )
Δ n eff = ε 0 c N A E i * χ i j k ( 2 ) ( ω ; ω , 0 ) E j E k dc d A ,
N = 1 2 A ( E × H * × E * × H ) i z d A ,
χ k eff E k dc = n eff Δ n eff ,
χ k eff ( ω ; ω , 0 ) = T i j k α β γ ( ω ; ω , 0 ) ζ α β γ i j ¯ ( ω ) ,
ζ α β γ i j ¯ ( ω ) = ε 0 c n eff N A E i * ( x ) ζ α β γ ( x ) E j ( x ) d A .
Y x = 169 GPa Y y = 130 GPa Y z = Y x ν x y = 0.36 ν y z = 0.28 ν x z = 0.064 G x y = 79.6 GPa G y z = G x y G x y = 50.9 GPa .
χ y eff ( ω ) = c i o i ( ω ) ,
o 3 p 7 o 3 = o 7 , o 5 p 6 o 5 = o 6 , o 11 p 12 o 11 = o 12 ,
{ 11 } 1 , { 22 } 2 , { 33 } 3 , { 23 } , { 32 } 4 , { 13 } , { 31 } 5 , { 12 } , { 21 } 6 .
T ^ i 1 i 2 i 3 i 4 = T { j 1 j 2 } j 3 { j 4 j 5 } j 6 ,
i = 3 6 1 ( i 6 1 ) + + 3 ( i 2 1 ) + i 1 .
A i j = R i 1 j 1 R i 2 j 2 R i 6 j 6
( I I ) T c = ( A ( 1 ) A ( s ) ) T c ,
i = 6 3 6 ( i 4 1 ) + 6 3 ( i 3 1 ) + 6 ( i 2 1 ) + i 1 .
T c = C T ^ c ,
0 = ( C A ( 1 ) C C A ( s ) C ) T ^ c N T ^ c ,
( I M 0 0 ) ( T ^ dep c T ^ ind c ) = 0 , ( T ^ dep c T ^ ind c ) = Λ T ^ c ,
χ 111 c 1 ζ 111 + c 4 ζ 221 + c 4 ζ 331 + 2 c 13 ζ 122 + 2 c 13 ζ 133 ; χ 112 2 c 11 ζ 121 + c 5 ζ 112 + c 2 ζ 222 + c 6 ζ 332 + 2 c 12 ζ 233 ; χ 113 2 c 14 ζ 131 + 2 c 12 ζ 232 + c 5 ζ 113 + c 6 ζ 223 + c 2 ζ 333 ; χ 221 c 2 ζ 111 + c 5 ζ 221 + c 6 ζ 331 + 2 c 11 ζ 122 + 2 c 12 ζ 133 ; χ 222 2 c 13 ζ 121 + c 4 ζ 112 + c 1 ζ 222 + c 4 ζ 332 + 2 c 13 ζ 233 ; χ 223 2 c 12 ζ 131 + 2 c 11 ζ 232 + c 6 ζ 113 + c 5 ζ 223 + c 2 ζ 333 ; χ 331 c 2 ζ 111 + c 6 ζ 221 + c 5 ζ 331 + 2 c 12 ζ 122 + 2 c 11 ζ 133 ; χ 332 2 c 12 ζ 121 + c 6 ζ 112 + c 2 ζ 222 + c 5 ζ 332 + 2 c 11 ζ 233 ; χ 333 2 c 13 ζ 131 + 2 c 13 ζ 232 + c 4 ζ 113 + c 4 ζ 223 + c 1 ζ 333 ; χ 231 2 c 9 ζ 231 + 2 c 10 ζ 132 + 2 c 10 ζ 123 ; χ 232 2 c 14 ζ 131 + 2 c 15 ζ 232 + c 7 ζ 113 + c 8 ζ 223 + c 3 ζ 333 ; χ 233 2 c 14 ζ 121 + c 7 ζ 112 + c 8 ζ 222 + c 8 ζ 332 + 2 c 15 ζ 233 ; χ 131 2 c 15 ζ 131 + 2 c 14 ζ 232 + c 8 ζ 113 + c 7 ζ 223 + c 3 ζ 333 ; χ 132 2 c 10 ζ 231 + 2 c 9 ζ 132 + 2 c 10 ζ 123 ; χ 133 c 3 ζ 111 + c 7 ζ 221 + c 8 ζ 331 + 2 c 14 ζ 122 + 2 c 15 ζ 133 ; χ 121 2 c 15 ζ 121 + c 8 ζ 112 + c 8 ζ 222 + c 7 ζ 332 + 2 c 14 ζ 232 ; χ 122 c 3 ζ 111 + c 8 ζ 221 + c 7 ζ 331 + 2 c 15 ζ 122 + 2 c 14 ζ 133 ; χ 123 2 c 10 ζ 231 + 2 c 10 ζ 132 + 2 c 9 ζ 123 .
c 1 = T ^ 3333 , c 2 = T ^ 2333 , c 3 = T ^ 4233 , c 4 = T ^ 3323 , c 5 = T ^ 2323 , c 6 = T ^ 1323 , c 7 = T ^ 5123 , c 8 = T ^ 4223 , c 9 = T ^ 6363 , c 10 = T ^ 5263 , c 11 = T ^ 3153 , c 12 = T ^ 2153 , c 13 = T ^ 1153 , c 14 = T ^ 6253 , c 15 = T ^ 5353 .
o 1 = ζ y y y y y ¯ , o 2 = ζ y y y z z ¯ + ζ y y y x x ¯ , o 3 = e { ζ x x x y x ¯ + ζ z z x y x ¯ } , o 4 = ζ x x y y y ¯ + ζ z z y y y ¯ , o 5 = 1 2 ζ x x y x x ¯ + 1 2 ζ z z y z z ¯ + 1 2 ζ x x y z z ¯ + 1 2 ζ z z y x x ¯ , o 6 = o 5 , o 7 = o 3 , o 8 = 2 e { ζ y y x x y ¯ } , o 9 = ζ x x y x x ¯ + ζ z z y z z ¯ ζ x x y z z ¯ ζ z z y x x ¯ , o 10 = 2 ζ x y x x x ¯ 2 ζ x y x z z ¯ , o 11 = ζ x y x x x ¯ ζ x y x z z ¯ , o 12 = o 11 , o 13 = 2 ζ x y x y y ¯ , o 14 = 2 e { ζ x x x y x ¯ ζ z z x y x ¯ } , o 15 = 4 e { ζ x y y y x ¯ } .
z A F c i z d A = A F c d A ,
F c = E 0 * × H + E × H 0 * .
z A ( E 0 * × H + E × H 0 ) i z d A = i ω A E 0 2 P ( 2 ) d A ,
E 0 ( r , t ) = e 0 ( x , y ; ω 0 ) e i ( k 0 z ω t ) , H 0 ( r , t ) = h 0 ( x , y ; ω 0 ) e i ( k 0 z ω t ) ,
E 0 ( r , t ) = u ( z ) e 0 ( x , y ; ω 0 ) e i ( k z ω t ) , H 0 ( r , t ) = u ( z ) h 0 ( x , y ; ω 0 ) e i ( k z ω t ) ,
P ( 2 ) = 2 ε 0 χ ( 2 ) ( ω ; ω , 0 ) : E E dc ,
u ( z ) z + i ( k k 0 ) u ( z ) = i ω u ( z ) X ,
X = 2 ε 0 A e 0 * χ ( 2 ) : e 0 E dc d A A ( e 0 × h 0 * + e 0 * × h 0 ) i z d A .

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