Abstract

Nanoplasmonic waveguides based on lithium niobate (LN) are shown to provide the light-matter interaction required for next-generation developments in nonlinear frequency-conversion nanostructures. Here, we numerically investigate second harmonic generation of a 1550 nm, 100 fs pulse in metal-LN-metal (MLNM) nanoplasmonic and LN hybrid-plasmonic (LNHP) waveguides. In comparison to a photonic LN waveguide, a 2.1 µm-long LNHP waveguide exhibits a conversion efficiency improvement of 11 times, whereas a 20 µm-long MLNM nanoplasmonic waveguide is shown to have a conversion efficiency of 1.1 × 10−4. The MLNM nanoplasmonic and LNHP waveguides have the potential to operate as sources of optical radiation for on-chip photonic systems.

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

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References

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2018 (1)

2015 (2)

S. Sederberg and A. Y. Elezzabi, “Coherent visible-light-generation enhancement in silicon-based nanoplasmonic waveguides via third-harmonic conversion,” Phys. Rev. Lett. 114(22), 227401 (2015).
[Crossref] [PubMed]

D. Kong and M. Tsubokawa, “Evaluation of slot-to-slot coupling between dielectric slot waveguides and metal-insulator-metal slot waveguides,” Opt. Express 23(15), 19082–19091 (2015).
[Crossref] [PubMed]

2014 (4)

X. Wu, S. Carbajo, K. Ravi, F. Ahr, G. Cirmi, Y. Zhou, O. D. Mücke, and F. X. Kärtner, “Terahertz generation in lithium niobate driven by Ti:sapphire laser pulses and its limitations,” Opt. Lett. 39(18), 5403–5406 (2014).
[Crossref] [PubMed]

S. Sederberg and A. Y. Elezzabi, “Ponderomotive electron acceleration in a silicon-based nanoplasmonic waveguide,” Phys. Rev. Lett. 113(16), 167401 (2014).
[Crossref] [PubMed]

J. Richter, A. Steinbrück, M. Zilk, A. Sergeyev, T. Pertsch, A. Tünnermann, and R. Grange, “Core-shell potassium niobate nanowires for enhanced nonlinear optical effects,” Nanoscale 6(10), 5200–5207 (2014).
[Crossref] [PubMed]

A. Casadei, E. F. Pecora, J. Trevino, C. Forestiere, D. Rüffer, E. Russo-Averchi, F. Matteini, G. Tutuncuoglu, M. Heiss, A. Fontcuberta i Morral, and L. Dal Negro, “Photonic-plasmonic coupling of GaAs single nanowires to optical nanoantennas,” Nano Lett. 14(5), 2271–2278 (2014).
[Crossref] [PubMed]

2012 (2)

2011 (2)

2010 (3)

2009 (1)

2006 (1)

W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett. 6(5), 1027–1030 (2006).
[Crossref]

2003 (2)

J. Z. Sanborn, C. Hellings, and T. D. Donnelly, “Breakdown of the slowly-varying-amplitude approximation: generation of backward-traveling, second-harmonic light,” J. Opt. Soc. Am. B 20(1), 152–157 (2003).
[Crossref]

R. S. Klein, G. E. Kugel, A. Maillard, K. Polgár, and A. Péter, “Absolute non-linear optical coefficients of LiNbO3 for near stoichiometric crystal compositions,” Opt. Mater. 22(2), 171–174 (2003).
[Crossref]

1997 (1)

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

1970 (1)

R. C. Miller and W. A. Nordland, “Dependence of second‐harmonic‐generation coefficients of LiNbO3 on melt composition,” Appl. Phys. Lett. 16, 174–176 (1970).
[Crossref]

1968 (1)

J. E. Bjorkholm, “Relative signs of the optical nonlinear coefficients d31 and d22 in LiNbO3,” Appl. Phys. Lett. 13(1), 36–37 (1968).
[Crossref]

1967 (1)

A. S. Barker and R. Loudon, “Dielectric properties and optical phonons in LiNbO3,” Phys. Rev. 158(2), 433–445 (1967).
[Crossref]

Abdenour, A.

W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett. 6(5), 1027–1030 (2006).
[Crossref]

Ahr, F.

Barker, A. S.

A. S. Barker and R. Loudon, “Dielectric properties and optical phonons in LiNbO3,” Phys. Rev. 158(2), 433–445 (1967).
[Crossref]

Bjorkholm, J. E.

J. E. Bjorkholm, “Relative signs of the optical nonlinear coefficients d31 and d22 in LiNbO3,” Appl. Phys. Lett. 13(1), 36–37 (1968).
[Crossref]

Brueck, S. R. J.

W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett. 6(5), 1027–1030 (2006).
[Crossref]

Carbajo, S.

Casadei, A.

A. Casadei, E. F. Pecora, J. Trevino, C. Forestiere, D. Rüffer, E. Russo-Averchi, F. Matteini, G. Tutuncuoglu, M. Heiss, A. Fontcuberta i Morral, and L. Dal Negro, “Photonic-plasmonic coupling of GaAs single nanowires to optical nanoantennas,” Nano Lett. 14(5), 2271–2278 (2014).
[Crossref] [PubMed]

Cheng, Q. Q.

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Cirmi, G.

Coutts, D. W.

Dal Negro, L.

A. Casadei, E. F. Pecora, J. Trevino, C. Forestiere, D. Rüffer, E. Russo-Averchi, F. Matteini, G. Tutuncuoglu, M. Heiss, A. Fontcuberta i Morral, and L. Dal Negro, “Photonic-plasmonic coupling of GaAs single nanowires to optical nanoantennas,” Nano Lett. 14(5), 2271–2278 (2014).
[Crossref] [PubMed]

Davoyan, A. R.

Dawes, J. M.

Donnelly, T. D.

Downes, J. E.

Elezzabi, A. Y.

S. Sederberg and A. Y. Elezzabi, “Coherent visible-light-generation enhancement in silicon-based nanoplasmonic waveguides via third-harmonic conversion,” Phys. Rev. Lett. 114(22), 227401 (2015).
[Crossref] [PubMed]

S. Sederberg and A. Y. Elezzabi, “Ponderomotive electron acceleration in a silicon-based nanoplasmonic waveguide,” Phys. Rev. Lett. 113(16), 167401 (2014).
[Crossref] [PubMed]

Z. Han, A. Y. Elezzabi, and V. Van, “Experimental realization of subwavelength plasmonic slot waveguides on a silicon platform,” Opt. Lett. 35(4), 502–504 (2010).
[Crossref] [PubMed]

Fan, W.

W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett. 6(5), 1027–1030 (2006).
[Crossref]

Fontcuberta i Morral, A.

A. Casadei, E. F. Pecora, J. Trevino, C. Forestiere, D. Rüffer, E. Russo-Averchi, F. Matteini, G. Tutuncuoglu, M. Heiss, A. Fontcuberta i Morral, and L. Dal Negro, “Photonic-plasmonic coupling of GaAs single nanowires to optical nanoantennas,” Nano Lett. 14(5), 2271–2278 (2014).
[Crossref] [PubMed]

Forestiere, C.

A. Casadei, E. F. Pecora, J. Trevino, C. Forestiere, D. Rüffer, E. Russo-Averchi, F. Matteini, G. Tutuncuoglu, M. Heiss, A. Fontcuberta i Morral, and L. Dal Negro, “Photonic-plasmonic coupling of GaAs single nanowires to optical nanoantennas,” Nano Lett. 14(5), 2271–2278 (2014).
[Crossref] [PubMed]

Grange, R.

J. Richter, A. Steinbrück, M. Zilk, A. Sergeyev, T. Pertsch, A. Tünnermann, and R. Grange, “Core-shell potassium niobate nanowires for enhanced nonlinear optical effects,” Nanoscale 6(10), 5200–5207 (2014).
[Crossref] [PubMed]

Han, Z.

Hasan, S. B.

Heiss, M.

A. Casadei, E. F. Pecora, J. Trevino, C. Forestiere, D. Rüffer, E. Russo-Averchi, F. Matteini, G. Tutuncuoglu, M. Heiss, A. Fontcuberta i Morral, and L. Dal Negro, “Photonic-plasmonic coupling of GaAs single nanowires to optical nanoantennas,” Nano Lett. 14(5), 2271–2278 (2014).
[Crossref] [PubMed]

Hellings, C.

Hu, X. P.

Ito, R.

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Kärtner, F. X.

Kitamoto, A.

Kivshar, Y. S.

Klein, R. S.

R. S. Klein, G. E. Kugel, A. Maillard, K. Polgár, and A. Péter, “Absolute non-linear optical coefficients of LiNbO3 for near stoichiometric crystal compositions,” Opt. Mater. 22(2), 171–174 (2003).
[Crossref]

Kondo, T.

Kong, D.

Krishna, S.

W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett. 6(5), 1027–1030 (2006).
[Crossref]

Kugel, G. E.

R. S. Klein, G. E. Kugel, A. Maillard, K. Polgár, and A. Péter, “Absolute non-linear optical coefficients of LiNbO3 for near stoichiometric crystal compositions,” Opt. Mater. 22(2), 171–174 (2003).
[Crossref]

Lederer, F.

Li, L.

Li, Q.

Li, T.

Lin, J.

Loudon, R.

A. S. Barker and R. Loudon, “Dielectric properties and optical phonons in LiNbO3,” Phys. Rev. 158(2), 433–445 (1967).
[Crossref]

Lu, F. F.

Maillard, A.

R. S. Klein, G. E. Kugel, A. Maillard, K. Polgár, and A. Péter, “Absolute non-linear optical coefficients of LiNbO3 for near stoichiometric crystal compositions,” Opt. Mater. 22(2), 171–174 (2003).
[Crossref]

Malloy, K. J.

W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett. 6(5), 1027–1030 (2006).
[Crossref]

Matteini, F.

A. Casadei, E. F. Pecora, J. Trevino, C. Forestiere, D. Rüffer, E. Russo-Averchi, F. Matteini, G. Tutuncuoglu, M. Heiss, A. Fontcuberta i Morral, and L. Dal Negro, “Photonic-plasmonic coupling of GaAs single nanowires to optical nanoantennas,” Nano Lett. 14(5), 2271–2278 (2014).
[Crossref] [PubMed]

Miller, R. C.

R. C. Miller and W. A. Nordland, “Dependence of second‐harmonic‐generation coefficients of LiNbO3 on melt composition,” Appl. Phys. Lett. 16, 174–176 (1970).
[Crossref]

Mücke, O. D.

Ng, V.

Nordland, W. A.

R. C. Miller and W. A. Nordland, “Dependence of second‐harmonic‐generation coefficients of LiNbO3 on melt composition,” Appl. Phys. Lett. 16, 174–176 (1970).
[Crossref]

Osgood, R. M.

W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett. 6(5), 1027–1030 (2006).
[Crossref]

Panoiu, N.-C.

W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett. 6(5), 1027–1030 (2006).
[Crossref]

Pecora, E. F.

A. Casadei, E. F. Pecora, J. Trevino, C. Forestiere, D. Rüffer, E. Russo-Averchi, F. Matteini, G. Tutuncuoglu, M. Heiss, A. Fontcuberta i Morral, and L. Dal Negro, “Photonic-plasmonic coupling of GaAs single nanowires to optical nanoantennas,” Nano Lett. 14(5), 2271–2278 (2014).
[Crossref] [PubMed]

Pertsch, T.

Péter, A.

R. S. Klein, G. E. Kugel, A. Maillard, K. Polgár, and A. Péter, “Absolute non-linear optical coefficients of LiNbO3 for near stoichiometric crystal compositions,” Opt. Mater. 22(2), 171–174 (2003).
[Crossref]

Polgár, K.

R. S. Klein, G. E. Kugel, A. Maillard, K. Polgár, and A. Péter, “Absolute non-linear optical coefficients of LiNbO3 for near stoichiometric crystal compositions,” Opt. Mater. 22(2), 171–174 (2003).
[Crossref]

Qiu, M.

Ravi, K.

Richter, J.

J. Richter, A. Steinbrück, M. Zilk, A. Sergeyev, T. Pertsch, A. Tünnermann, and R. Grange, “Core-shell potassium niobate nanowires for enhanced nonlinear optical effects,” Nanoscale 6(10), 5200–5207 (2014).
[Crossref] [PubMed]

Rockstuhl, C.

Rüffer, D.

A. Casadei, E. F. Pecora, J. Trevino, C. Forestiere, D. Rüffer, E. Russo-Averchi, F. Matteini, G. Tutuncuoglu, M. Heiss, A. Fontcuberta i Morral, and L. Dal Negro, “Photonic-plasmonic coupling of GaAs single nanowires to optical nanoantennas,” Nano Lett. 14(5), 2271–2278 (2014).
[Crossref] [PubMed]

Russo-Averchi, E.

A. Casadei, E. F. Pecora, J. Trevino, C. Forestiere, D. Rüffer, E. Russo-Averchi, F. Matteini, G. Tutuncuoglu, M. Heiss, A. Fontcuberta i Morral, and L. Dal Negro, “Photonic-plasmonic coupling of GaAs single nanowires to optical nanoantennas,” Nano Lett. 14(5), 2271–2278 (2014).
[Crossref] [PubMed]

Sanborn, J. Z.

Schiek, R.

Sederberg, S.

S. Sederberg and A. Y. Elezzabi, “Coherent visible-light-generation enhancement in silicon-based nanoplasmonic waveguides via third-harmonic conversion,” Phys. Rev. Lett. 114(22), 227401 (2015).
[Crossref] [PubMed]

S. Sederberg and A. Y. Elezzabi, “Ponderomotive electron acceleration in a silicon-based nanoplasmonic waveguide,” Phys. Rev. Lett. 113(16), 167401 (2014).
[Crossref] [PubMed]

Sergeyev, A.

J. Richter, A. Steinbrück, M. Zilk, A. Sergeyev, T. Pertsch, A. Tünnermann, and R. Grange, “Core-shell potassium niobate nanowires for enhanced nonlinear optical effects,” Nanoscale 6(10), 5200–5207 (2014).
[Crossref] [PubMed]

Shadrivov, I. V.

Shirane, M.

Shoji, I.

Song, Y.

Spence, D. J.

Steinbrück, A.

J. Richter, A. Steinbrück, M. Zilk, A. Sergeyev, T. Pertsch, A. Tünnermann, and R. Grange, “Core-shell potassium niobate nanowires for enhanced nonlinear optical effects,” Nanoscale 6(10), 5200–5207 (2014).
[Crossref] [PubMed]

Trevino, J.

A. Casadei, E. F. Pecora, J. Trevino, C. Forestiere, D. Rüffer, E. Russo-Averchi, F. Matteini, G. Tutuncuoglu, M. Heiss, A. Fontcuberta i Morral, and L. Dal Negro, “Photonic-plasmonic coupling of GaAs single nanowires to optical nanoantennas,” Nano Lett. 14(5), 2271–2278 (2014).
[Crossref] [PubMed]

Tsubokawa, M.

Tünnermann, A.

J. Richter, A. Steinbrück, M. Zilk, A. Sergeyev, T. Pertsch, A. Tünnermann, and R. Grange, “Core-shell potassium niobate nanowires for enhanced nonlinear optical effects,” Nanoscale 6(10), 5200–5207 (2014).
[Crossref] [PubMed]

Tutuncuoglu, G.

A. Casadei, E. F. Pecora, J. Trevino, C. Forestiere, D. Rüffer, E. Russo-Averchi, F. Matteini, G. Tutuncuoglu, M. Heiss, A. Fontcuberta i Morral, and L. Dal Negro, “Photonic-plasmonic coupling of GaAs single nanowires to optical nanoantennas,” Nano Lett. 14(5), 2271–2278 (2014).
[Crossref] [PubMed]

Van, V.

Wang, J.

Warrier, A. M.

Wu, M.

Wu, X.

Xie, Z. D.

Xu, J.

Yan, M.

Zhang, S.

W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett. 6(5), 1027–1030 (2006).
[Crossref]

Zhou, Y.

Zhu, S. N.

Zhu, Y. Y.

Zilk, M.

J. Richter, A. Steinbrück, M. Zilk, A. Sergeyev, T. Pertsch, A. Tünnermann, and R. Grange, “Core-shell potassium niobate nanowires for enhanced nonlinear optical effects,” Nanoscale 6(10), 5200–5207 (2014).
[Crossref] [PubMed]

Appl. Phys. Lett. (2)

J. E. Bjorkholm, “Relative signs of the optical nonlinear coefficients d31 and d22 in LiNbO3,” Appl. Phys. Lett. 13(1), 36–37 (1968).
[Crossref]

R. C. Miller and W. A. Nordland, “Dependence of second‐harmonic‐generation coefficients of LiNbO3 on melt composition,” Appl. Phys. Lett. 16, 174–176 (1970).
[Crossref]

J. Opt. Soc. Am. B (4)

Nano Lett. (2)

W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett. 6(5), 1027–1030 (2006).
[Crossref]

A. Casadei, E. F. Pecora, J. Trevino, C. Forestiere, D. Rüffer, E. Russo-Averchi, F. Matteini, G. Tutuncuoglu, M. Heiss, A. Fontcuberta i Morral, and L. Dal Negro, “Photonic-plasmonic coupling of GaAs single nanowires to optical nanoantennas,” Nano Lett. 14(5), 2271–2278 (2014).
[Crossref] [PubMed]

Nanoscale (1)

J. Richter, A. Steinbrück, M. Zilk, A. Sergeyev, T. Pertsch, A. Tünnermann, and R. Grange, “Core-shell potassium niobate nanowires for enhanced nonlinear optical effects,” Nanoscale 6(10), 5200–5207 (2014).
[Crossref] [PubMed]

Opt. Express (5)

Opt. Lett. (3)

Opt. Mater. (1)

R. S. Klein, G. E. Kugel, A. Maillard, K. Polgár, and A. Péter, “Absolute non-linear optical coefficients of LiNbO3 for near stoichiometric crystal compositions,” Opt. Mater. 22(2), 171–174 (2003).
[Crossref]

Opt. Mater. Express (1)

Phys. Rev. (1)

A. S. Barker and R. Loudon, “Dielectric properties and optical phonons in LiNbO3,” Phys. Rev. 158(2), 433–445 (1967).
[Crossref]

Phys. Rev. B (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Phys. Rev. Lett. (2)

S. Sederberg and A. Y. Elezzabi, “Coherent visible-light-generation enhancement in silicon-based nanoplasmonic waveguides via third-harmonic conversion,” Phys. Rev. Lett. 114(22), 227401 (2015).
[Crossref] [PubMed]

S. Sederberg and A. Y. Elezzabi, “Ponderomotive electron acceleration in a silicon-based nanoplasmonic waveguide,” Phys. Rev. Lett. 113(16), 167401 (2014).
[Crossref] [PubMed]

Other (3)

M. J. Weber, Handbook of optical materials (CRC, 2003).

E. D. Palik, Handbook of optical constants of solids (Academic, 1998), (I).

A. Zheltikov, A. L’Huillier, and F. Krausz, “Nonlinear optics,” in Springer handbook of lasers and optics, F. Träger, ed. (Springer Science & Business Media, 2012).

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Figures (4)

Fig. 1
Fig. 1 (a) Cross-sectional schematic of the w = 400 nm and h = 780 nm MLNM nanoplasmonic waveguide. (b) λpump = 1550 nm and (c) λSHG = 775 nm modal intensity distribution supported by the MLNM nanoplasmonic waveguide. (d) Cross-section of the w = 400 nm and h = 100 nm LNHP waveguide. (e) λpump = 1550 nm and (f) λSHG = 775 nm modal intensity distribution supported by the LNHP waveguide.
Fig. 2
Fig. 2 (a) Effective refractive indices of the MLNM nanoplasmonic and LNHP waveguides illustrating phase-matching between λpump = 1550 nm and λSHG = 775 nm. (b) Coherence length for the MLNM nanoplasmonic and LNHP waveguides. (c) Propagation lengths of the pump and SHG wavelengths for the MLNM nanoplasmonic and LNHP waveguides.
Fig. 3
Fig. 3 Magnitude of the time-averaged electric field recorded at λSHG = 775 nm for the (a) MLNM nanoplasmonic and (b) LNHP waveguides. (c) SHG time-domain electric field pulses and (d) SHG spectral density recorded in the waveguides near the positions of maximum electric fields.
Fig. 4
Fig. 4 Conversion efficiency for various lengths of the MLNM nanoplasmonic and LNHP waveguides. For comparison, the conversion efficiency is determined in a 400 nm × 509 nm phase-matched MLN waveguide with a 100 nm gold layer situated on the LN, as well as in a phase-matched 400 nm × 837 nm photonic waveguide (see insets).

Equations (11)

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d ¯ ¯ =[ d 11 d 12 d 13 d 21 d 22 d 23 d 31 d 32 d 33 d 14 d 15 d 16 d 24 d 25 d 26 d 34 d 35 d 36 ],
P x (2) (ω)=2 ε 0 d 11 { E x E x }(ω)+2 ε 0 d 12 { E y E y }(ω)+2 ε 0 d 13 { E z E z }(ω)+ 4 ε 0 d 14 { E y E z }(ω)+4 ε 0 d 15 { E x E z }(ω)+4 ε 0 d 16 { E x E y }(ω)
P y (2) (ω)=2 ε 0 d 21 { E x E x }(ω)+2 ε 0 d 22 { E y E y }(ω)+2 ε 0 d 23 { E z E z }(ω)+ 4 ε 0 d 24 { E y E z }(ω)+4 ε 0 d 25 { E x E z }(ω)+4 ε 0 d 26 { E x E y }(ω)
P z (2) (ω)=2 ε 0 d 31 { E x E x }(ω)+2 ε 0 d 32 { E y E y }(ω)+2 ε 0 d 33 { E z E z }(ω)+ 4 ε 0 d 34 { E y E z }(ω)+4 ε 0 d 35 { E x E z }(ω)+4 ε 0 d 36 { E x E y }(ω)
D x (ω)= ε 0 ε r,xx (ω) E x (ω)+2 ε 0 d 11 { E x E x }(ω)+2 ε 0 d 12 { E y E y }(ω)+ 2 ε 0 d 13 { E z E z }(ω)+4 ε 0 d 14 { E y E z }(ω)+ 4 ε 0 d 15 { E x E z }(ω)+4 ε 0 d 16 { E x E y }(ω)
D y (ω)= ε 0 ε r,yy (ω) E y (ω)+2 ε 0 d 21 { E x E x }(ω)+2 ε 0 d 22 { E y E y }(ω)+ 2 ε 0 d 23 { E z E z }(ω)+4 ε 0 d 24 { E y E z }(ω)+ 4 ε 0 d 25 { E x E z }(ω)+4 ε 0 d 26 { E x E y }(ω)
D z (ω)= ε 0 ε r,zz (ω) E z (ω)+2 ε 0 d 31 { E x E x }(ω)+2 ε 0 d 32 { E y E y }(ω)+ 2 ε 0 d 33 { E z E z }(ω)+4 ε 0 d 34 { E y E z }(ω)+ 4 ε 0 d 35 { E x E z }(ω)+4 ε 0 d 36 { E x E y }(ω)
D x (t)= ε 0 { ε r,xx E x }(t)+2 ε 0 d 11 E x 2 (t)+2 ε 0 d 12 E y 2 (t)+2 ε 0 d 13 E z 2 (t)+ 4 ε 0 d 14 E y (t) E z (t)+4 ε 0 d 15 E x (t) E z (t)+4 ε 0 d 16 E x (t) E y (t)
D y (t)= ε 0 { ε r,yy E y }(t)+2 ε 0 d 21 E x 2 (t)+2 ε 0 d 22 E y 2 (t)+2 ε 0 d 23 E z 2 (t)+ 4 ε 0 d 24 E y (t) E z (t)+4 ε 0 d 25 E x (t) E z (t)+4 ε 0 d 26 E x (t) E y (t)
D z (t)= ε 0 { ε r,zz E z }(t)+2 ε 0 d 31 E x 2 (t)+2 ε 0 d 32 E y 2 (t)+2 ε 0 d 33 E z 2 (t)+ 4 ε 0 d 34 E y (t) E z (t)+4 ε 0 d 35 E x (t) E z (t)+4 ε 0 d 36 E x (t) E y (t)
d ¯ ¯ LN =[ 0 0 0 d 22 d 22 0 d 31 d 31 d 33 0 d 31 d 22 d 31 0 0 0 0 0 ],

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