Abstract

We introduce a computational scheme for analyzing higher harmonic generation in nonlinear optical periodic nanostructures that are composed of multiple layers. Such nanostructures, i.e., photonic crystals, plasmonic nanostructures, or metamaterials, are currently in the focus of interest to enhance the efficiency of various nonlinear processes. We exploit an adapted Fourier modal method combined with a modified scattering-matrix algorithm to numerically model these processes. We explicitly present a numerically stable formulation of the scheme. The strength and the applicability of the algorithm are outlined at some selected examples.

© 2010 Optical Society of America

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  43. The amplitude transmission coefficients are normalized to the input pump field which has an intensity of |E|2=1012 V2/m2.
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  47. Because of the limited discretization distance in the numerical calculations the amplitude transmission coefficient does not reach a value of zero.
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  49. R. S. Weis and T. K. Gaylord, “Lithium niobate: summary of physical properties and crystal structure,” Appl. Phys. Lett. 37, 191–203 (1985).

2009 (5)

2008 (4)

M. F. Saleh, L. D. Negro, and B. E. A. Saleh, “Second-order parametric interactions in 1-D photonic-crystal microcavity structures,” Opt. Express 16, 5261–5276 (2008).
[CrossRef] [PubMed]

E. Kim, F. Wang, W. Wu, Z. Yu, and Y. R. Shen, “Nonlinear optical spectroscopy of photonic metamaterials,” Phys. Rev. B 78, 113102 (2008).
[CrossRef]

R. Iliew, C. Etrich, T. Pertsch, and F. Lederer, “Slow-light enhanced collinear second-harmonic generation in two-dimensional photonic crystals,” Phys. Rev. B 77, 115124 (2008).
[CrossRef]

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008).
[CrossRef] [PubMed]

2007 (4)

M. W. Klein, N. Feth, M. Wegener, and S. Linden, “Experiments on second- and third-harmonic generation from magnetic metamaterials,” Opt. Express 15, 5238–5247 (2007).
[CrossRef] [PubMed]

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers,” Nano Lett. 7, 1251–1255 (2007).
[CrossRef] [PubMed]

K. Busch, J. Niegemann, M. Pototschnig, and L. Tkeshelashvili, “A Krylov-subspace based solver for the linear and nonlinear Maxwell equations,” Phys. Status Solidi B 244, 3479–3496 (2007).
[CrossRef]

B. Bai and J. Turunen, “Fourier modal method for the analysis of second-harmonic generation in two-dimensionally periodic structures containing anisotropic materials,” J. Opt. Soc. Am. B 24, 1105–1112 (2007).
[CrossRef]

2006 (3)

I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, “Second-harmonic generation in nonlinear left-handed metamaterials,” J. Opt. Soc. Am. B 23, 529–534 (2006).
[CrossRef]

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

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313, 502–504 (2006).
[CrossRef] [PubMed]

2005 (4)

B. K. Canfield, S. Kujala, K. Jefimovs, T. Vallius, J. Turunen, and M. Kauranen, “Polarization effects in the linear and nonlinear optical responses of gold nanoparticle arrays,” J. Opt. A, Pure Appl. Opt. 7, S110–S117 (2005).
[CrossRef]

V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Experimental verification of a negative index of refraction,” Opt. Lett. 30, 3356–3358 (2005).
[CrossRef]

B. Maes, P. Bienstman, and R. Baets, “Modeling second-harmonic generation by use of mode expansion,” J. Opt. Soc. Am. B 22, 1378–1383 (2005).
[CrossRef]

N. Bonod, E. Popov, and M. Neviere, “Fourier factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect,” Opt. Commun. 244, 389–398 (2005).
[CrossRef]

2004 (1)

M. Cherchi, “Exact analytic expressions for electromagnetic propagation and optical nonlinear generation in finite one-dimensional periodic multilayers,” Phys. Rev. E 69, 066602 (2004).
[CrossRef]

2003 (3)

L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A, Pure Appl. Opt. 5, 345–355 (2003).
[CrossRef]

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef] [PubMed]

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94, 6447–6455 (2003).
[CrossRef]

2002 (1)

2001 (2)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

E. Popov and B. Bozhkov, “Corrugated waveguides as resonance optical filters—advantages and limitations,” J. Opt. Soc. Am. A 18, 1758–1764 (2001).
[CrossRef]

2000 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef] [PubMed]

1999 (1)

S. G. Johnson, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[CrossRef]

1997 (1)

1996 (2)

1995 (2)

1994 (1)

1991 (1)

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic band structure: the face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295–2298 (1991).
[CrossRef] [PubMed]

1990 (2)

Z. Zhang and S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
[CrossRef] [PubMed]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

1988 (2)

1987 (2)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localisation of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

1986 (1)

E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

1985 (1)

R. S. Weis and T. K. Gaylord, “Lithium niobate: summary of physical properties and crystal structure,” Appl. Phys. Lett. 37, 191–203 (1985).

1983 (1)

R. Reinisch and M. Nevière , “Electromagnetic theory of diffraction in nonlinear optics and surface-enhanced nonlinear optical effects,” Phys. Rev. B 28, 1870–1885 (1983).
[CrossRef]

Abdenour, A.

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

Baets, R.

Bai, B.

B. Bai and J. Turunen, “Fourier modal method for the analysis of second-harmonic generation in two-dimensionally periodic structures containing anisotropic materials,” J. Opt. Soc. Am. B 24, 1105–1112 (2007).
[CrossRef]

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers,” Nano Lett. 7, 1251–1255 (2007).
[CrossRef] [PubMed]

Bartal, G.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008).
[CrossRef] [PubMed]

Becouarn, L.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94, 6447–6455 (2003).
[CrossRef]

Bienstman, P.

Bonod, N.

N. Bonod, E. Popov, and M. Neviere, “Fourier factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect,” Opt. Commun. 244, 389–398 (2005).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003).

Bozhkov, B.

Brueck, S. R. J.

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

Busch, K.

F. B. P. Niesler, N. Feth, S. Linden, J. Niegemann, J. Gieseler, K. Busch, and M. Wegener, “Second-harmonic generation from split-ring resonators on a GaAs substrate,” Opt. Lett. 34, 1997–1999 (2009).
[CrossRef] [PubMed]

K. Busch, J. Niegemann, M. Pototschnig, and L. Tkeshelashvili, “A Krylov-subspace based solver for the linear and nonlinear Maxwell equations,” Phys. Status Solidi B 244, 3479–3496 (2007).
[CrossRef]

Cai, W.

Canfield, B. K.

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers,” Nano Lett. 7, 1251–1255 (2007).
[CrossRef] [PubMed]

B. K. Canfield, S. Kujala, K. Jefimovs, T. Vallius, J. Turunen, and M. Kauranen, “Polarization effects in the linear and nonlinear optical responses of gold nanoparticle arrays,” J. Opt. A, Pure Appl. Opt. 7, S110–S117 (2005).
[CrossRef]

Chan, C. T.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Cherchi, M.

M. Cherchi, “Exact analytic expressions for electromagnetic propagation and optical nonlinear generation in finite one-dimensional periodic multilayers,” Phys. Rev. E 69, 066602 (2004).
[CrossRef]

Chettiar, U. K.

Chipouline, A.

Cotter, N. P. K.

Coutaz, J. L.

Drachev, V. P.

Enkrich, C.

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313, 502–504 (2006).
[CrossRef] [PubMed]

Etrich, C.

R. Iliew, C. Etrich, T. Pertsch, and F. Lederer, “Slow-light enhanced collinear second-harmonic generation in two-dimensional photonic crystals,” Phys. Rev. B 77, 115124 (2008).
[CrossRef]

Eyres, L. A.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94, 6447–6455 (2003).
[CrossRef]

Fainman, Y.

Fan, S.

S. G. Johnson, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[CrossRef]

Fan, W.

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

Fejer, M. M.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94, 6447–6455 (2003).
[CrossRef]

Feth, N.

Gaylord, T. K.

Genov, D. A.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008).
[CrossRef] [PubMed]

Gerard, B.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94, 6447–6455 (2003).
[CrossRef]

Gieseler, J.

Giessen, H.

Gippius, N. A.

Gmitter, T. J.

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic band structure: the face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295–2298 (1991).
[CrossRef] [PubMed]

Granet, G.

Grann, E. B.

Harris, J. S.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94, 6447–6455 (2003).
[CrossRef]

Helgert, C.

Ho, K. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Hoyer, W.

Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79, 235109 (2009).
[CrossRef]

Hübner, U.

Husu, H.

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers,” Nano Lett. 7, 1251–1255 (2007).
[CrossRef] [PubMed]

Iliew, R.

R. Iliew, C. Etrich, T. Pertsch, and F. Lederer, “Slow-light enhanced collinear second-harmonic generation in two-dimensional photonic crystals,” Phys. Rev. B 77, 115124 (2008).
[CrossRef]

Jefimovs, K.

B. K. Canfield, S. Kujala, K. Jefimovs, T. Vallius, J. Turunen, and M. Kauranen, “Polarization effects in the linear and nonlinear optical responses of gold nanoparticle arrays,” J. Opt. A, Pure Appl. Opt. 7, S110–S117 (2005).
[CrossRef]

Joannopoulos, J. D.

S. G. Johnson, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
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S. G. Johnson, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
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B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers,” Nano Lett. 7, 1251–1255 (2007).
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E. Kim, F. Wang, W. Wu, Z. Yu, and Y. R. Shen, “Nonlinear optical spectroscopy of photonic metamaterials,” Phys. Rev. B 78, 113102 (2008).
[CrossRef]

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I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, “Second-harmonic generation in nonlinear left-handed metamaterials,” J. Opt. Soc. Am. B 23, 529–534 (2006).
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B. K. Canfield, S. Kujala, K. Jefimovs, T. Vallius, J. Turunen, and M. Kauranen, “Polarization effects in the linear and nonlinear optical responses of gold nanoparticle arrays,” J. Opt. A, Pure Appl. Opt. 7, S110–S117 (2005).
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T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94, 6447–6455 (2003).
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B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers,” Nano Lett. 7, 1251–1255 (2007).
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W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, Jr., K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett. 6, 1027–1030 (2006).
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Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79, 235109 (2009).
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W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, Jr., K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett. 6, 1027–1030 (2006).
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J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
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K. Busch, J. Niegemann, M. Pototschnig, and L. Tkeshelashvili, “A Krylov-subspace based solver for the linear and nonlinear Maxwell equations,” Phys. Status Solidi B 244, 3479–3496 (2007).
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E. Kim, F. Wang, W. Wu, Z. Yu, and Y. R. Shen, “Nonlinear optical spectroscopy of photonic metamaterials,” Phys. Rev. B 78, 113102 (2008).
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T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94, 6447–6455 (2003).
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B. K. Canfield, S. Kujala, K. Jefimovs, T. Vallius, J. Turunen, and M. Kauranen, “Polarization effects in the linear and nonlinear optical responses of gold nanoparticle arrays,” J. Opt. A, Pure Appl. Opt. 7, S110–S117 (2005).
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S. G. Johnson, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
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T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94, 6447–6455 (2003).
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E. Kim, F. Wang, W. Wu, Z. Yu, and Y. R. Shen, “Nonlinear optical spectroscopy of photonic metamaterials,” Phys. Rev. B 78, 113102 (2008).
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J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008).
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J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008).
[CrossRef] [PubMed]

W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, Jr., K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett. 6, 1027–1030 (2006).
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J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008).
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I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, “Second-harmonic generation in nonlinear left-handed metamaterials,” J. Opt. Soc. Am. B 23, 529–534 (2006).
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Appl. Phys. Lett. (1)

R. S. Weis and T. K. Gaylord, “Lithium niobate: summary of physical properties and crystal structure,” Appl. Phys. Lett. 37, 191–203 (1985).

J. Appl. Phys. (1)

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94, 6447–6455 (2003).
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J. Opt. A, Pure Appl. Opt. (2)

B. K. Canfield, S. Kujala, K. Jefimovs, T. Vallius, J. Turunen, and M. Kauranen, “Polarization effects in the linear and nonlinear optical responses of gold nanoparticle arrays,” J. Opt. A, Pure Appl. Opt. 7, S110–S117 (2005).
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L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A, Pure Appl. Opt. 5, 345–355 (2003).
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J. Opt. Soc. Am. A (8)

J. Opt. Soc. Am. B (5)

Nano Lett. (2)

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers,” Nano Lett. 7, 1251–1255 (2007).
[CrossRef] [PubMed]

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

Nature (1)

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008).
[CrossRef] [PubMed]

Opt. Acta (1)

E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

Opt. Commun. (1)

N. Bonod, E. Popov, and M. Neviere, “Fourier factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect,” Opt. Commun. 244, 389–398 (2005).
[CrossRef]

Opt. Express (3)

Opt. Lett. (4)

Phys. Rev. B (5)

E. Kim, F. Wang, W. Wu, Z. Yu, and Y. R. Shen, “Nonlinear optical spectroscopy of photonic metamaterials,” Phys. Rev. B 78, 113102 (2008).
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Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79, 235109 (2009).
[CrossRef]

S. G. Johnson, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
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Phys. Rev. E (1)

M. Cherchi, “Exact analytic expressions for electromagnetic propagation and optical nonlinear generation in finite one-dimensional periodic multilayers,” Phys. Rev. E 69, 066602 (2004).
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Phys. Rev. Lett. (7)

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localisation of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

Z. Zhang and S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
[CrossRef] [PubMed]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic band structure: the face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295–2298 (1991).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef] [PubMed]

Phys. Status Solidi B (1)

K. Busch, J. Niegemann, M. Pototschnig, and L. Tkeshelashvili, “A Krylov-subspace based solver for the linear and nonlinear Maxwell equations,” Phys. Status Solidi B 244, 3479–3496 (2007).
[CrossRef]

Science (2)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313, 502–504 (2006).
[CrossRef] [PubMed]

Other (4)

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003).

Because of the limited discretization distance in the numerical calculations the amplitude transmission coefficient does not reach a value of zero.

In the chosen orientation of the GaAs crystal all susceptibility tensor elements of form χ1ab and χ2ab with a,b=1,2(x,y) are vanishing. Illuminating the structure with a plane wave under normal incidence does not allow for any x or y polarized SHG. Only a z-polarized SH will appear (χ312=χ321≠0). Therefore, no zeroth order SH plane wave is able to propagate.

The amplitude transmission coefficients are normalized to the input pump field which has an intensity of |E|2=1012 V2/m2.

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Figures (4)

Fig. 1
Fig. 1

Schematic picture of a bi-periodic multilayer structure. In general, all included layers possess a periodical anisotropic material distribution in lateral dimension (x- and y-directions). All layers have to share a common periodicity D x × D y —a super period. The material distribution is invariant along the third spatial dimension ( z ) throughout a single layer. The height of each layer is arbitrary and denoted by h l ; l is the layer index running from 0 to L 1 . The adjacent regions are assumed to be linear, homogeneous, and isotropic having permittivities of ϵ 1 and ϵ L . μ 1 and μ L are assumed to be unity.

Fig. 2
Fig. 2

(a) Two-dimensional (2D)-periodic photonic crystal slab made of gallium arsenide. The operational wavelength is λ 0 = 1.9 μ m . The periods are D x = D y = 0.65 λ 0 . The wire elements which form the photonic crystal slab have widths of w x = 0.9 λ 0 and w y = 0.05 λ 0 . The height of the structure is subject to variations. For the sake of simplicity the slab is embedded in air. (b) Transmission spectra of the zeroth order fundamental (blue dashed) and first order SH (green solid) fields at λ 0 = 1.9 μ m under normal incidence. (c) Transmission spectrum of the generated SH field in a non-logarithmic scale around a slab height of 1.98 λ 0 in detail.

Fig. 3
Fig. 3

Multilayer metallic grating structure deposited on a LiNbO 3 substrate. The entire structure is invariant in y-direction. The period is D x = 0.61 μ m . The metallic bars are 0.40 μ m in width and 30 nm in height. The voids are assumed to be partially filled by LiNbO 3 with a height of 15 nm. The right sketch shows the 2D projection of the structure. The three separate layers are indicated. The substrate is assumed to be made of LiNbO 3 as well, where only its linear optical properties are accounted for. It allows one to distinguish the effect of the plasmonic structure and the effect induced by the substrate on the generated SH signal.

Fig. 4
Fig. 4

(a) Transmission spectra of the zeroth order fundamental (blue dashed curve), the zeroth order SH (red dashed-dotted curve), and the first order SH fields (green solid curve) under normal incidence. (b) and (c) Transmission of the zeroth (red dashed-dotted curve) and first (green solid curve) order SH field at 1.41 μ m . The dependence on the assumed nonlinear substrate height (layer 3) is explicitly shown. (b) shows the very near field behavior from 0 to 500 nm and (c) shows the field characteristics around a layer thickness of 1000 μ m .

Equations (75)

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

F = n = 1 2 F n ( r ) exp ( i ω n t ) + c .c . ,
× E n = + i ω n B n ,     B n = 0 ,
× H n = i ω n D n ,     D n = 0.
B n = μ 0 μ n H n ,
D n = ϵ 0 ϵ n E n + P n NL ,
× E n = + i ω n c μ n H ̃ n ,
× H ̃ n = i ω n c ( ϵ n E n + ϵ 0 1 P n NL ) .
P 2 , i NL = ϵ 0 j 1 , , j N χ i j 1 j N E 1 , j 1 E 1 , j N r E 1 , j N r + 1 E 1 , j N ,
P 2 , i NL ( 2 ω ) = ϵ 0 j , k χ i j k ( 2 ω ; ω , ω ) E 1 , j E 1 , k .
f ( l ) ( x , y , z ) = m = U x U x n = U y U y f m n ( l ) ( z ) exp [ i ( α m x + β n y ) ] .
α m = α 0 + m K x ,
β n = β 0 + n K y ,
f ( l ) ( x , y , z ) = m n f m n ( l )   exp ( i γ ( l ) z ) exp [ i ( α m x + β n y ) ] .
M ( l ) X ( l ) = i z X ( l ) = γ ( l ) X ( l ) ,
f m n = s = 1 S / 2 a s f s , m n +   exp ( i γ s + z ̃ ) + b s f s , m n   exp ( i γ s ( z ̃ h ) ) ,
( E 1 m n E 2 m n H 1 m n H 2 m n ) ( z ̃ ) = W ϕ ( z ̃ ) ( a s b s ) = ( E + E H + H ) ( e s + 0 0 e s ) ( a s b s ) ,
e s + ( z ̃ ) diag [ exp ( i γ s + z ̃ ) ] ,
e s ( z ̃ ) diag [ exp ( i γ s ( z ̃ h ) ) ] .
W = ( E + E H + H )
χ ̂ i j 1 j N ( r ) = m , n χ ̂ m n i j 1 j N   exp [ i ( m K x x + n K y y ) ] .
d s = { 0 , s ϵ Σ + h , s ϵ Σ , }
P ̂ i NL = ϵ 0 { s } = s 1 , , s N S , , S m , n P ̂ i m n { s }   exp [ i t = 1 N γ ̃ s t ( z ̃ d s t ) ] exp [ i ( α ̂ m x + β ̂ n y ) ] ,
γ ̃ s t = { γ s t , t = 1 , , γ s t , t = + 1 , , N . }
α ̂ m = M α 0 + m K x α ̂ 0 + m K x ,
β ̂ n = M β 0 + n K y β ̂ 0 + n K y ,
P ̂ i m n { s } = ( t = 1 v s t t = + 1 N v s t ) p 1 , , p N q 1 , , q N χ ̂ m t p t , n t q t i j 1 j N E j 1 p 1 q 1 s 1 E j p q s E j + 1 ( p + 1 ) ( q + 1 ) s + 1 E j N ( p N ) ( q N ) s N .
E 3 = 1 k 0 ε ̱ 33 ( α H 2 β H 1 ) ε ̱ 31 E 1 ε ̱ 32 E 2 .
f ̂ i ( r ) = m , n f ̂ i m n ( z ̃ ) exp [ i ( α ̂ m x + β ̂ n y ) ] .
i z ̃ X ̂ ( z ̃ ) = M ̂ X ̂ ( z ̃ ) + Q ̂ ( z ̃ ) .
Q ̂ ( z ̃ ) = { s } ( α ̂ ϵ ̱ ̂ 33 P ̂ 3 m n { s } β ̂ ϵ ̱ ̂ 33 P ̂ 3 m n { s } k ̂ 0 ( ϵ ̱ ̂ 23 P ̂ 3 m n { s } P ̂ 2 m n { s } ) k ̂ 0 ( ϵ ̱ ̂ 13 P ̂ 3 m n { s } + P ̂ 1 m n { s } ) ) exp [ i t γ ̃ s t ( z ̃ d s t ) ] { s } P ̂ { s }   exp [ i t γ ̃ s t ( z ̃ d s t ) ] .
X ̃ H ( z ̃ ) = ( E ̂ + E ̂ H ̂ + H ̂ ) ( e ̂ q + 0 0 e ̂ q ) ( a ̂ q b ̂ q ) = W ̂ ϕ ̂ ( a ̂ q b ̂ q ) .
Υ ̂ ( z ̃ ) = W ̂ ϕ ̂ Ψ ̂ ( z ̃ ) ,
z ̃ Ψ ̂ = i ϕ ̂ 1 W ̂ 1 Q ̂ ,
z ̃ Ψ ̂ p = i { s } , q W ̂ p q 1 P ̂ q { s }   exp { i [ ( t γ ̃ s t ( z ̃ d s t ) ) γ ̂ p ( z ̃ d p ) ] } .
Ψ ̂ p = { s } , q Γ p { s } ( z ̃ ) W ̂ p q 1 P q { s }   exp { i [ γ ̂ p z ̃ + t γ ̃ s t ( z ̃ d s t ) ] } { s } , q Γ p { s } ( 0 ) W ̂ p q 1 P ̂ q { s }   exp ( i t γ ̃ s t d s t ) ,     p ϵ Σ + ,
Ψ ̂ p = { s } , q Γ p { s } ( z ̃ ) W ̂ p q 1 P ̂ q { s }   exp { i [ γ ̂ p ( z ̃ h ) + t γ ̃ s t ( z ̃ d s t ) ] } { s } , q Γ p { s } ( h ) W ̂ p q 1 P ̂ q { s }   exp [ i t γ ̃ s t ( h d s t ) ] ,     p ϵ Σ .
Γ p { s } ( z ) = { ( t γ ̃ s t γ ̂ p ) 1 , ( t γ ̃ s t γ ̂ p ) 0 i z , ( t γ ̃ s t γ ̂ p ) = 0 , }
Υ ̂ p ( z ̃ ) = { s } , q , p W ̂ p p Γ p { s } ( z ̃ ) W ̂ p q 1 P ̂ q { s }   exp [ i t γ ̃ s t ( z ̃ d s t ) ] { s } , q , p W ̂ p p W ̂ p q 1 P ̂ q { s } { Γ p { s } ( 0 ) exp ( i t γ ̃ s t d s t ) exp ( i γ ̂ p z ̃ ) , p ϵ Σ + Γ p { s } ( h ) exp [ i t γ ̃ s t ( h d s t ) ] exp [ i γ ̂ p ( z ̃ h ) ] , p ϵ Σ . }
Ω { s } , p = Γ p { s } ( z ̃ ) exp [ i t γ ̃ s t ( z ̃ d s t ) ] ,
Ω { s } , p + = Γ p { s } ( 0 ) exp ( i t γ ̃ s t d s t ) ,     p ϵ Σ + ,
Ω { s } , p = Γ p { s } ( h ) exp [ + i t γ ̃ s t ( h d s t ) ] ,     p ϵ Σ ,
Υ ̂ ( z ̃ ) = W ̂   diag { W ̂ 1 P ̂ Ω ( z ̃ ) } W ̂ ϕ ̂ ( z ̃ ) diag { W ̂ 1 P ̂ ( Ω + , Ω ) } .
E i ( 1 ) ( r ) = A i   exp [ i ( α 0 x + β 0 y + γ 00 ( 1 ) z ) ] + m n R i , m n   exp [ i ( α m x + β n y γ m n ( 1 ) z ) ] ,
E i ( L ) ( r ) = m n T i , m n   exp [ i ( α m x + β n y + γ m n ( L ) ( z H ) ) ] ,
γ m n ( 1 , L ) = ε ( 1 , L ) ω 2 / c 2 α m 2 β n 2 .
Re ( γ m n ( 1 , L ) ) + Im ( γ m n ( 1 , L ) ) > 0.
X ( 1 , L ) = W ( 1 , L ) ϕ ( 1 , L ) ( z ) ( a q ( 1 , L ) b q ( 1 , L ) ) ,
E ̂ i ( 1 ) ( r ) = A ̂ i   exp [ i ( α ̂ 0 x + β ̂ 0 y + γ ̂ 00 ( 1 ) z ) ] + m n R ̂ i , m n   exp [ i ( α ̂ m x + β ̂ n y γ ̂ m n ( 1 ) z ) ] ,
E ̂ i ( L ) ( r ) = m n T ̂ i , m n   exp [ i ( α ̂ m x + β ̂ n y + γ ̂ m n ( L ) ( z H ) ) ] ,
W ̂ ( l ) ( a ̱ ̂ q ( l ) b ̂ q ( l ) ) + Υ ̂ ( l ) ( h ( l ) ) = W ̂ ( l + 1 ) ( a ̂ q ( l + 1 ) b ̱ ̂ q ( l + 1 ) ) + Υ ̂ ( l + 1 ) ( 0 ) ,
Υ ̂ ( 1 , L ) = 0 ,
( a ̂ q ( l + 1 ) b ̱ ̂ q ( l + 1 ) ) = ( W ̂ ( l + 1 ) ) 1 W ̂ ( l ) ( a ̱ ̂ q ( l ) b ̂ q ( l ) ) + [ Υ ̂ ( l ) ( h ( l ) ) ( W ̂ ( l + 1 ) ) 1 Υ ̂ ( l + 1 ) ( 0 ) ] τ ̂ layer ( a ̱ ̂ q ( l ) b ̂ q ( l ) ) + υ ̂ layer .
( a ̱ ̂ q ( l ) b ̂ q ( l + 1 ) ) = ϕ ̂ ( l ) ( h ( l ) ) ( a ̂ q ( l ) b ̱ ̂ q ( l ) ) τ ̂ free ( a ̂ q ( l ) b ̱ ̂ q ( l ) ) + υ ̂ free .
X ( 1 , L ) ( z ) = ( E 1 m n ( 1 , L ) E 2 m n ( 1 , L ) H 1 m n ( 1 , L ) H 2 m n ( 1 , L ) ) ( z ) = W ( 1 , L ) ϕ ( 1 , L ) ( z ) ( a ( 1 , L ) b ( 1 , L ) ) .
W ( 1 , L ) = ( I I A ( 1 , L ) B ( 1 , L ) C ( 1 , L ) A ( 1 , L ) A ( 1 , L ) B ( 1 , L ) C ( 1 , L ) A ( 1 , L ) ) ,
k 0 A m n ( 1 , L ) = α m β n γ m n ( 1 , L ) ,
k 0 B m n ( 1 , L ) = β n 2 γ m n ( 1 , L ) γ m n ( 1 , L ) ,
k 0 C m n ( 1 , L ) = α m 2 γ m n ( 1 , L ) + γ m n ( 1 , L ) ,
ϕ ( 1 , L ) ( z ) = ( D ( 1 , L ) 0 0 0 0 D ( 1 , L ) 0 0 0 0 ( D ( 1 , L ) ) 1 0 0 0 0 ( D ( 1 , L ) ) 1 ) ,
D m n ( 1 ) = exp ( i γ m n ( 1 ) z ) ,
D m n ( L ) = exp ( i γ m n ( L ) ( z H ) ) .
a ( 1 ) = ( a 1 , m n ( 1 ) a 2 , m n ( 1 ) ) = ( A 1 δ m 0 δ n 0 A 2 δ m 0 δ n 0 ) ,     b ( 1 ) = ( b 1 , m n ( 1 ) b 2 , m n ( 1 ) ) = ( R 1 , m n R 2 , m n ) ,
a ( L ) = ( a 1 , m n ( L ) a 2 , m n ( L ) ) = ( T 1 , m n T 2 , m n ) ,     b ( L ) = ( b 1 , m n ( L ) b 2 , m n ( L ) ) = ( 0 0 ) .
( a ( 1 ) b ( 1 ) ) = τ ( 0 ) ( a ( 0 ) b ( 0 ) ) + υ ( 0 ) ,
( a ( 2 ) b ( 2 ) ) = τ ( 1 ) ( a ( 1 ) b ( 1 ) ) + υ ( 1 ) .
( a ( 2 ) b ( 2 ) ) = [ τ ( 1 ) τ ( 0 ) ] ( a ( 0 ) b ( 0 ) ) + [ υ ( 1 ) + τ ( 1 ) υ ( 0 ) ] = τ ( a ( 0 ) b ( 0 ) ) + υ ,
( a ( n + 1 ) b ( n + 1 ) ) = τ ( n ) ( a ( n ) b ( n ) ) + υ ( n )
( a ( N ) b ( N ) ) = [ τ ( N 1 ) τ ( 0 ) ] ( a ( 0 ) b ( 0 ) ) + { n = 0 N 2 [ τ ( N 1 ) τ ( n + 1 ) ] υ ( n ) + υ ( N 1 ) } .
( a ( 1 ) b ( 1 ) ) = τ ( a ( 0 ) b ( 0 ) ) + υ = ( τ 11 τ 12 τ 21 τ 22 ) ( a ( 0 ) b ( 0 ) ) + ( υ 1 υ 2 ) ,
( a ( 1 ) b ( 0 ) ) = σ ( a ( 0 ) b ( 1 ) ) + ξ = ( σ 11 σ 12 σ 21 σ 22 ) ( a ( 0 ) b ( 1 ) ) + ( ξ 1 ξ 2 ) .
σ = ( ( τ 11 τ 12 τ 22 1 τ 21 ) τ 12 τ 22 1 τ 22 1 τ 21 τ 22 1 ) ,     ξ = ( I τ 12 τ 22 1 0 τ 22 1 ) ( υ 1 υ 2 ) .
( a ( 1 ) b ( 0 ) ) = σ 1 ( a ( 0 ) b ( 1 ) ) + ξ 1 ,     ( a ( 2 ) b ( 1 ) ) = σ 2 ( a ( 1 ) b ( 2 ) ) + ξ 2 ,
σ = σ 2 σ 1 = ( σ 11 2 ( I σ 12 1 σ 21 2 ) 1 σ 11 1 σ 12 2 + σ 11 2 ( I σ 12 1 σ 21 2 ) 1 σ 12 1 σ 22 2 σ 21 1 + σ 22 1 ( I σ 21 2 σ 12 1 ) 1 σ 21 2 σ 11 1 σ 22 1 ( I σ 21 2 σ 12 1 ) 1 σ 22 2 ) ,
ξ = ξ 2 ξ 1 = ( σ 11 2 ( I σ 12 1 σ 21 2 ) 1 0 σ 22 1 ( I σ 21 2 σ 12 1 ) 1 σ 21 2 I ) ( ξ 1 1 ξ 2 1 ) + ( I σ 11 2 ( I σ 12 1 σ 21 2 ) 1 σ 12 1 0 σ 22 1 ( I σ 21 2 σ 12 1 ) 1 ) ( ξ 1 2 ξ 2 2 ) ,
( a ( 2 ) b ( 0 ) ) = σ ( a ( 0 ) b ( 2 ) ) + ξ .

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