Abstract

The magnitude and origin of the electro-optic measurements in strained silicon devices has been lately the object of a great controversy. Furthermore, recent works underline the importance of the masking effect of free carriers in strained waveguides and the low interaction between the mode and the highly strained areas. In the present work, the use of a p-i-n junction and an asymmetric cladding is proposed to eliminate the unwanted carrier influence and improve the electro-optical modulation response. The proposed configuration enhances the effective refractive index due to the strain-induced Pockels effect in more than two orders of magnitude with respect to the usual configuration.

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

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2019 (3)

M. He, M. Xu, Y. Ren, J. Jian, Z. Ruan, Y. Xu, S. Gao, S. Sun, X. Wen, L. Zhou, L. Liu, C. Guo, H. Chen, S. Yu, L. Liu, and X. Cai, “High-Performance Hybrid Silicon and Lithium Niobate Mach-Zehnder Modulators for 100 Gbit/s and Beyond,” Nat. Photonics 13, 359–364 (2019).
[Crossref]

S. Abel, F. Eltes, J. E. Ortmann, A. Messner, P. Castera, T. Wagner, D. Urbonas, A. Rosa, A. M. Gutierrez, D. Tulli, P. Ma, B. Baeuerle, A. Josten, W. Heni, D. Caimi, L. Czornomaz, A. A. Demkov, J. Leuthold, P. Sanchis, and J. Fompeyrine, “Large Pockels effect in micro- and nanostructured barium titanate integrated on silicon,” Nat. Mater. 18(1), 42–47 (2019).
[Crossref] [PubMed]

C. Castellan, A. Trenti, C. Vecchi, A. Marchesini, M. Mancinelli, M. Ghulinyan, G. Pucker, and L. Pavesi, “On the origin of second harmonic generation in silicon waveguides with silicon nitride cladding,” Sci. Rep. 9(1), 1088 (2019).
[Crossref] [PubMed]

2018 (3)

M. Berciano, G. Marcaud, P. Damas, X. Le Roux, P. Crozat, C. A. Ramos, D. Pérez, D. Benedikovic, D. Marris-Morini, E. Cassan, and L. Vivien, “Fast linear electro-optic effect in a centrosymmetric semiconductor,” Commun. Phys. 1(1), 64 (2018).
[Crossref]

C. Haffner, D. Chelladurai, Y. Fedoryshyn, A. Josten, B. Baeuerle, W. Heni, T. Watanabe, T. Cui, B. Cheng, S. Saha, D. L. Elder, L. R. Dalton, A. Boltasseva, V. M. Shalaev, N. Kinsey, and J. Leuthold, “Low-loss plasmon-assisted electro-optic modulator,” Nature 556(7702), 483–486 (2018).
[Crossref] [PubMed]

T. Komljenovic, D. Huang, P. Pintus, M. A. Tran, M. L. Davenport, and J. E. Bowers, “Photonic Integrated Circuits Using Heterogeneous Integration on Silicon,” Proc. IEEE 106(12), 2246–2257 (2018).
[Crossref]

2017 (4)

E. Timurdogan, C. V. Poulton, M. J. Byrd, and M. R. Watts, “Electric field-induced second-order nonlinear optical effects in silicon waveguides,” Nat. Photonics 11(3), 200–206 (2017).
[Crossref]

C. Castellan, A. Trenti, M. Mancinelli, A. Marchesini, M. Ghulinyan, G. Pucker, and L. Pavesi, “From SHG to mid-infrared SPDC generation in strained silicon waveguides,” Proc. SPIE 10358, 1035804 (2017).

P. Damas, M. Berciano, G. Marcaud, C. Alonso, D. Marris-Morini, E. Cassan, and L. Vivien, “Comprehensive description of the electro-optic effects in strained silicon waveguides,” J. Appl. Phys. 122(15), 153105 (2017).
[Crossref]

I. Olivares, T. Angelova, and P. Sanchis, “On the influence of interface charging dynamics and stressing conditions in strained silicon devices,” Sci. Rep. 7(1), 7241 (2017).
[Crossref] [PubMed]

2016 (4)

P. Damas, D. Marris-Morini, E. Cassan, and L. Vivien, “Bond orbital description of the strain-induced second-order optical susceptibility in silicon,” Phys. Rev. B 93(16), 165208 (2016).
[Crossref]

R. Sharma, M. W. Puckett, H. H. Lin, A. Isichenko, F. Vallini, and Y. Fainman, “Effect of dielectric claddings on the electro-optic behavior of silicon waveguides,” Opt. Lett. 41(6), 1185–1188 (2016).
[Crossref] [PubMed]

M. Borghi, M. Mancinelli, M. Bernard, M. Ghulinyan, G. Pucker, and L. Pavesi, “Homodyne Detection of Free Carrier Induced Electro-Optic Modulation in Strained Silicon Resonators,” J. Lit. Technol. 34(24), 5657–5668 (2016).
[Crossref]

M. Cazzanelli and J. Schilling, “Second order optical nonlinearity in silicon by symmetry breaking,” Phys. Rev. Appl. 3(1), 011104 (2016).
[Crossref]

2015 (5)

2014 (2)

2013 (1)

2012 (2)

F. Bianco, K. Fedus, F. Enrichi, R. Pierobon, M. Cazzanelli, M. Ghulinyan, G. Pucker, and L. Pavesi, “Two-dimensional micro-Raman mapping of stress and strain distributions in strained silicon waveguides,” Semicond. Sci. Technol. 27(8), 085009 (2012).
[Crossref]

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(2), 148–154 (2012).
[Crossref] [PubMed]

2011 (2)

2010 (2)

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

M. A. Hopcroft, W. D. Nix, and T. W. Kenny, “What is the Young’s Modulus of Silicon?” J. Microelectromech. Syst. 19(2), 229–238 (2010).
[Crossref]

2006 (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(7090), 199–202 (2006).
[Crossref] [PubMed]

1965 (1)

J. J. Wortman and R. A. Evans, “Young’s Modulus, Shear Modulus, and Poisson’s Ratio in Silicon and Germanium,” J. Appl. Phys. 36(1), 153–156 (1965).
[Crossref]

Abashin, M.

Abel, S.

S. Abel, F. Eltes, J. E. Ortmann, A. Messner, P. Castera, T. Wagner, D. Urbonas, A. Rosa, A. M. Gutierrez, D. Tulli, P. Ma, B. Baeuerle, A. Josten, W. Heni, D. Caimi, L. Czornomaz, A. A. Demkov, J. Leuthold, P. Sanchis, and J. Fompeyrine, “Large Pockels effect in micro- and nanostructured barium titanate integrated on silicon,” Nat. Mater. 18(1), 42–47 (2019).
[Crossref] [PubMed]

Alonso, C.

P. Damas, M. Berciano, G. Marcaud, C. Alonso, D. Marris-Morini, E. Cassan, and L. Vivien, “Comprehensive description of the electro-optic effects in strained silicon waveguides,” J. Appl. Phys. 122(15), 153105 (2017).
[Crossref]

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(7090), 199–202 (2006).
[Crossref] [PubMed]

Angelova, T.

I. Olivares, T. Angelova, and P. Sanchis, “On the influence of interface charging dynamics and stressing conditions in strained silicon devices,” Sci. Rep. 7(1), 7241 (2017).
[Crossref] [PubMed]

Avrutsky, I.

Baeuerle, B.

S. Abel, F. Eltes, J. E. Ortmann, A. Messner, P. Castera, T. Wagner, D. Urbonas, A. Rosa, A. M. Gutierrez, D. Tulli, P. Ma, B. Baeuerle, A. Josten, W. Heni, D. Caimi, L. Czornomaz, A. A. Demkov, J. Leuthold, P. Sanchis, and J. Fompeyrine, “Large Pockels effect in micro- and nanostructured barium titanate integrated on silicon,” Nat. Mater. 18(1), 42–47 (2019).
[Crossref] [PubMed]

C. Haffner, D. Chelladurai, Y. Fedoryshyn, A. Josten, B. Baeuerle, W. Heni, T. Watanabe, T. Cui, B. Cheng, S. Saha, D. L. Elder, L. R. Dalton, A. Boltasseva, V. M. Shalaev, N. Kinsey, and J. Leuthold, “Low-loss plasmon-assisted electro-optic modulator,” Nature 556(7702), 483–486 (2018).
[Crossref] [PubMed]

Benedikovic, D.

M. Berciano, G. Marcaud, P. Damas, X. Le Roux, P. Crozat, C. A. Ramos, D. Pérez, D. Benedikovic, D. Marris-Morini, E. Cassan, and L. Vivien, “Fast linear electro-optic effect in a centrosymmetric semiconductor,” Commun. Phys. 1(1), 64 (2018).
[Crossref]

Berciano, M.

M. Berciano, G. Marcaud, P. Damas, X. Le Roux, P. Crozat, C. A. Ramos, D. Pérez, D. Benedikovic, D. Marris-Morini, E. Cassan, and L. Vivien, “Fast linear electro-optic effect in a centrosymmetric semiconductor,” Commun. Phys. 1(1), 64 (2018).
[Crossref]

P. Damas, M. Berciano, G. Marcaud, C. Alonso, D. Marris-Morini, E. Cassan, and L. Vivien, “Comprehensive description of the electro-optic effects in strained silicon waveguides,” J. Appl. Phys. 122(15), 153105 (2017).
[Crossref]

Bernard, M.

M. Borghi, M. Mancinelli, M. Bernard, M. Ghulinyan, G. Pucker, and L. Pavesi, “Homodyne Detection of Free Carrier Induced Electro-Optic Modulation in Strained Silicon Resonators,” J. Lit. Technol. 34(24), 5657–5668 (2016).
[Crossref]

M. Borghi, M. Mancinelli, F. Merget, J. Witzens, M. Bernard, M. Ghulinyan, G. Pucker, and L. Pavesi, “High-frequency electro-optic measurement of strained silicon racetrack resonators,” Opt. Lett. 40(22), 5287–5290 (2015).
[Crossref] [PubMed]

Bianco, F.

C. Schriever, F. Bianco, M. Cazzanelli, M. Ghulinyan, C. Eisenschmidt, J. Boor, A. Schmid, J. Heitmann, L. Pavesi, and J. Schilling, “Second‐Order Optical Nonlinearity in Silicon Waveguides: Inhomogeneous Stress and Interfaces,” Adv. Opt. Mater. 3(1), 129–136 (2015).
[Crossref]

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(2), 148–154 (2012).
[Crossref] [PubMed]

F. Bianco, K. Fedus, F. Enrichi, R. Pierobon, M. Cazzanelli, M. Ghulinyan, G. Pucker, and L. Pavesi, “Two-dimensional micro-Raman mapping of stress and strain distributions in strained silicon waveguides,” Semicond. Sci. Technol. 27(8), 085009 (2012).
[Crossref]

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(7090), 199–202 (2006).
[Crossref] [PubMed]

Boltasseva, A.

C. Haffner, D. Chelladurai, Y. Fedoryshyn, A. Josten, B. Baeuerle, W. Heni, T. Watanabe, T. Cui, B. Cheng, S. Saha, D. L. Elder, L. R. Dalton, A. Boltasseva, V. M. Shalaev, N. Kinsey, and J. Leuthold, “Low-loss plasmon-assisted electro-optic modulator,” Nature 556(7702), 483–486 (2018).
[Crossref] [PubMed]

Bolten, J.

Bonati, C.

Boor, J.

C. Schriever, F. Bianco, M. Cazzanelli, M. Ghulinyan, C. Eisenschmidt, J. Boor, A. Schmid, J. Heitmann, L. Pavesi, and J. Schilling, “Second‐Order Optical Nonlinearity in Silicon Waveguides: Inhomogeneous Stress and Interfaces,” Adv. Opt. Mater. 3(1), 129–136 (2015).
[Crossref]

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(7090), 199–202 (2006).
[Crossref] [PubMed]

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(2), 148–154 (2012).
[Crossref] [PubMed]

Borghi, M.

M. Borghi, M. Mancinelli, M. Bernard, M. Ghulinyan, G. Pucker, and L. Pavesi, “Homodyne Detection of Free Carrier Induced Electro-Optic Modulation in Strained Silicon Resonators,” J. Lit. Technol. 34(24), 5657–5668 (2016).
[Crossref]

M. Borghi, M. Mancinelli, F. Merget, J. Witzens, M. Bernard, M. Ghulinyan, G. Pucker, and L. Pavesi, “High-frequency electro-optic measurement of strained silicon racetrack resonators,” Opt. Lett. 40(22), 5287–5290 (2015).
[Crossref] [PubMed]

Bowers, J. E.

T. Komljenovic, D. Huang, P. Pintus, M. A. Tran, M. L. Davenport, and J. E. Bowers, “Photonic Integrated Circuits Using Heterogeneous Integration on Silicon,” Proc. IEEE 106(12), 2246–2257 (2018).
[Crossref]

Byrd, M. J.

E. Timurdogan, C. V. Poulton, M. J. Byrd, and M. R. Watts, “Electric field-induced second-order nonlinear optical effects in silicon waveguides,” Nat. Photonics 11(3), 200–206 (2017).
[Crossref]

Cai, X.

M. He, M. Xu, Y. Ren, J. Jian, Z. Ruan, Y. Xu, S. Gao, S. Sun, X. Wen, L. Zhou, L. Liu, C. Guo, H. Chen, S. Yu, L. Liu, and X. Cai, “High-Performance Hybrid Silicon and Lithium Niobate Mach-Zehnder Modulators for 100 Gbit/s and Beyond,” Nat. Photonics 13, 359–364 (2019).
[Crossref]

Caimi, D.

S. Abel, F. Eltes, J. E. Ortmann, A. Messner, P. Castera, T. Wagner, D. Urbonas, A. Rosa, A. M. Gutierrez, D. Tulli, P. Ma, B. Baeuerle, A. Josten, W. Heni, D. Caimi, L. Czornomaz, A. A. Demkov, J. Leuthold, P. Sanchis, and J. Fompeyrine, “Large Pockels effect in micro- and nanostructured barium titanate integrated on silicon,” Nat. Mater. 18(1), 42–47 (2019).
[Crossref] [PubMed]

Cassan, E.

M. Berciano, G. Marcaud, P. Damas, X. Le Roux, P. Crozat, C. A. Ramos, D. Pérez, D. Benedikovic, D. Marris-Morini, E. Cassan, and L. Vivien, “Fast linear electro-optic effect in a centrosymmetric semiconductor,” Commun. Phys. 1(1), 64 (2018).
[Crossref]

P. Damas, M. Berciano, G. Marcaud, C. Alonso, D. Marris-Morini, E. Cassan, and L. Vivien, “Comprehensive description of the electro-optic effects in strained silicon waveguides,” J. Appl. Phys. 122(15), 153105 (2017).
[Crossref]

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M. Berciano, G. Marcaud, P. Damas, X. Le Roux, P. Crozat, C. A. Ramos, D. Pérez, D. Benedikovic, D. Marris-Morini, E. Cassan, and L. Vivien, “Fast linear electro-optic effect in a centrosymmetric semiconductor,” Commun. Phys. 1(1), 64 (2018).
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C. Castellan, A. Trenti, C. Vecchi, A. Marchesini, M. Mancinelli, M. Ghulinyan, G. Pucker, and L. Pavesi, “On the origin of second harmonic generation in silicon waveguides with silicon nitride cladding,” Sci. Rep. 9(1), 1088 (2019).
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M. Borghi, M. Mancinelli, M. Bernard, M. Ghulinyan, G. Pucker, and L. Pavesi, “Homodyne Detection of Free Carrier Induced Electro-Optic Modulation in Strained Silicon Resonators,” J. Lit. Technol. 34(24), 5657–5668 (2016).
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C. Schriever, F. Bianco, M. Cazzanelli, M. Ghulinyan, C. Eisenschmidt, J. Boor, A. Schmid, J. Heitmann, L. Pavesi, and J. Schilling, “Second‐Order Optical Nonlinearity in Silicon Waveguides: Inhomogeneous Stress and Interfaces,” Adv. Opt. Mater. 3(1), 129–136 (2015).
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S. Abel, F. Eltes, J. E. Ortmann, A. Messner, P. Castera, T. Wagner, D. Urbonas, A. Rosa, A. M. Gutierrez, D. Tulli, P. Ma, B. Baeuerle, A. Josten, W. Heni, D. Caimi, L. Czornomaz, A. A. Demkov, J. Leuthold, P. Sanchis, and J. Fompeyrine, “Large Pockels effect in micro- and nanostructured barium titanate integrated on silicon,” Nat. Mater. 18(1), 42–47 (2019).
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M. Berciano, G. Marcaud, P. Damas, X. Le Roux, P. Crozat, C. A. Ramos, D. Pérez, D. Benedikovic, D. Marris-Morini, E. Cassan, and L. Vivien, “Fast linear electro-optic effect in a centrosymmetric semiconductor,” Commun. Phys. 1(1), 64 (2018).
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Figures (9)

Fig. 1
Fig. 1 (a) Sketch of the strained silicon waveguide covered with a silicon nitride layer of 700 nm and a compressive stress of 2 GPa, and (b) representation of the rotated waveguide showing the reference and waveguide coordinate systems. The reference system is aligned with the crystalline directions of a cubic crystal x’ = [100], y’ = [001], z’ = [0-10]. The four bond vectors of the silicon primitive cell are displayed in the inset.
Fig. 2
Fig. 2 (a) Sketch of the electrode configuration used to study the strain induced electro-optic effect and the corresponding effective index change for (b) TE (blue) and TM (red) modes as a function of ϕ for an applied voltage of −15 V.
Fig. 3
Fig. 3 (a) Contour plot of Δ n yy V at ϕ = 0° (top) and ϕ = 45° (bottom) for an applied voltage of −15V and (b) contour plot of the Ex (top) and Ey (bottom) DC electric field components (V/µm).
Fig. 4
Fig. 4 (a) Sketch of the configuration 1 (top) and contour plot of the Ex (middle) and Ey (bottom) components of the electric field inside the waveguide core (V/µm). (b) Sketch of the configuration 2 (top) and contour plot of the Ex (middle) and Ey (bottom) electric fields inside the waveguide core for this configuration.
Fig. 5
Fig. 5 (a) Contour plot of Δ n yy V for the configuration 1 (bottom) and configuration 2 (top) at ϕ = 0° for an applied voltage of −15 V and (b) corresponding effective index change for TE and TM modes.
Fig. 6
Fig. 6 (a) Sketch of the proposed structure with the left half of the cladding with a compressive stress of 2 GPa (light blue) and the right half with a tensile stress of 1.25 GPa (violet) and (b) contour plot of the Δ n yy V refractive index element for ϕ = 0° and an applied voltage of −15 V.
Fig. 7
Fig. 7 Effective index change for TE and TM modes as a function of ϕ for an applied voltage of −15 V.
Fig. 8
Fig. 8 Rotational dependency of the susceptibility coefficients dependent on (a) the horizontal and (b) vertical strain gradients taken at p1 and p2, respectively. The insets show the η 111 and η 112 strain gradients as well as the chosen points p1 and p2, respectively.
Fig. 9
Fig. 9 Effective index change for TE and TM modes for (a) the initial and (b) the optimized structures with an applied voltage of −15V. The depicted results are obtained either by considering all refractive index elements (solid lines) or only diagonal elements (dashed lines).

Equations (19)

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ξ 1 = d 3 ( cosϕ+sinϕ,1,sinϕcosϕ ) ξ 2 = d 3 ( cosϕsinϕ,1,sinϕ+cosϕ ) ξ 3 = d 3 ( cosϕ+sinϕ,1,sinϕcosϕ ) ξ 4 = d 3 ( cosϕsinϕ,1,sinϕ+cosϕ )
χ (2) =[ χ 11 (2) (ϕ) χ 12 (2) (ϕ) χ 13 (2) (ϕ) χ 14 (2) (ϕ) χ 15 (2) (ϕ) χ 16 (2) (ϕ) χ 21 (2) (ϕ) χ 22 (2) (ϕ) χ 23 (2) (ϕ) 0 χ 25 (2) (ϕ) χ 26 (2) (ϕ) χ 31 (2) (ϕ) χ 32 (2) (ϕ) χ 33 (2) (ϕ) χ 34 (2) (ϕ) χ 35 (2) (ϕ) χ 36 (2) (ϕ) ],
( η ij +Δ η ij ) x i x j =1
[ Δ η xx Δ η yy Δ η zz Δ η yz Δ η xz Δ η xy ]=[ r 11 r 12 r 13 r 21 r 22 r 23 r 31 r 32 r 33 r 41 0 r 43 r 51 r 52 r 53 r 61 r 62 r 63 ][ E x E y E z ].
( 1 n o 2 + r 11 E x + r 12 E y ) x 2 +( 1 n o 2 + r 21 E x + r 22 E y ) y 2 +( 1 n o 2 + r 31 E x + r 32 E y ) z 2 + 2 r 41 E x yz+2( r 51 E x + r 52 E y )xz+2( r 61 E x + r 62 E y )xy=1.
( η ij +Δ η ij )= ε 1 = ( n·n ) 1 ,
Δ n xx V 1 2 n o 3 ( r 11 E x + r 12 E y ) Δ n yy V 1 2 n o 3 ( r 21 E x + r 22 E y ) Δ n zz V 1 2 n o 3 ( r 31 E x + r 32 E y ) Δ n yz V 1 2 n o 3 r 41 E x Δ n xz V 1 2 n o 2 ( r 51 E x + r 52 E y ) Δ n xy V 1 2 n o 2 ( r 61 E x + r 62 E y ).
χ 11 (2) = K d 6 2·27 ε o { ( 7β5α+( 3αβ )cos( 4ϕ ) )· η 111 + ( 9β3α+( α3β )cos( 4ϕ ) )· η 221 + ( 11βα( α+5β )cos( 4ϕ ) )· η 331 }
χ 12 (2) = K d 6 2·27 ε o { ( 5β3α+( α+β )cos( 4ϕ ) )· η 111 + 2( 3αβ )· η 221 +( 7βα( α+β )cos( 4ϕ ) )· η 331 }
χ 13 (2) = K d 6 2·27 ε o { ( 3βα )( cos( 4ϕ )+1 )( η 111 + η 221 + η 331 ) }
χ 14 (2) = K d 6 27 ε o sin( 4ϕ )( α+β )( η 112 η 332 )
χ 15 (2) = 2K d 6 27 ε o cos( 2ϕ ){ ( 2β2α+( α+β )cos ( 2ϕ ) 2 )· η 111 + ( 3βα )· η 221 +( 4β( α+β )cos ( 2ϕ ) 2 )· η 331 }
χ 16 (2) = K d 6 27 ε o { ( 5β3α+( α+β )cos( 4ϕ ) )· η 112 + 2( 3βα )· η 222 +( 7βα( α+β )cos( 4ϕ ) )· η 332 }
χ 21 (2) = K d 6 2·27 ε o { ( 5β3α+( α+β )cos( 4ϕ ) )· η 112 + 2( 3βα )· η 222 +( 7βα( α+β )cos( 4ϕ ) )· η 332 }
χ 22 (2) = K d 6 27 ε o { ( 3βα )( η 112 + η 222 + η 332 ) }
χ 23 (2) = K d 6 2·27 ε o { ( 7βα( α+β )cos( 4ϕ ) )· η 112 + 2( 3βα )· η 222 +( 5β+( α+β )cos( 4ϕ ) )· η 332 }
χ 24 (2) =0
χ 25 (2) = 2K d 6 27 ε o cos( 2ϕ )( 3βα )( η 112 + η 222 + η 332 )
χ 26 (2) = K d 6 27 ε o { ( 5β3α+( α+β )cos( 4ϕ ) )· η 111 + 2( 3βα )· η 221 +( 7βα( α+β )cos( 4ϕ ) )· η 331 },

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