Abstract

We present an advanced formulation of the Fourier modal method for analyzing the second-harmonic generation in multilayers of periodic arrays of nanostructures. In our method, we solve Maxwell’s equations in a curvilinear coordinate system, in which the interfaces are defined by surfaces of constant coordinates. Thus, we can apply the correct Fourier factorization rules as well as adaptive spatial resolution to nanostructures with complex cross sections. We extend here the factorization rules to the second-harmonic susceptibility tensor expressed in the curvilinear coordinates. The combination of adaptive curvilinear coordinates and factorization rules allows for efficient calculation of the second-harmonic intensity, as demonstrated for one- and two-dimensional periodic nanostructures.

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

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2017 (2)

M. Hentschel, M. Schäferling, X. Duan, H. Giessen, and N. Liu, “Chiral plasmonics,” Sci. Adv. 3, e1602735 (2017).
[Crossref] [PubMed]

K. Weber, M. L. Nesterov, T. Weiss, M. Scherer, M. Hentschel, J. Vogt, C. Huck, W. Li, M. Dressel, H. Giessen, and F. Neubrech, “Wavelength dcaling in antenna-enhanced infrared spectroscopy: Towards the far-IR and THz region,” ACS Photonics 4, 45–51 (2017).
[Crossref]

2016 (4)

B. Metzger, M. Hentschel, and H. Giessen, “Ultrafast nonlinear plasmonic spectroscopy: From dipole nanoantennas to complex hybrid plasmonic structures,” ACS Photonics 3, 1336–1350 (2016).
[Crossref]

M. L. Nesterov, X. Yin, M. Schäferling, H. Giessen, and T. Weiss, “The role of plasmon-generated near fields for enhanced circular dichroism spectroscopy,” ACS Photonics 4, 578–583 (2016).
[Crossref]

T. Schumacher, M. Brandstetter, D. Wolf, K. Kratzer, M. Hentschel, H. Giessen, and M. Lippitz, “The optimal antenna for nonlinear spectroscopy of weakly and strongly scattering nanoobjects,” Appl. Phys. B 122, 91 (2016).
[Crossref]

T. Weiss, M. Mesch, M. Schäferling, H. Giessen, W. Langbein, and E. A. Muljarov, “From dark to bright: First-order perturbation theory with analytical mode normalization for plasmonic nanoantenna arrays applied to refractive index sensing,” Phys. Rev. Lett. 116, 237401 (2016).
[Crossref] [PubMed]

2014 (2)

D. Dregely, K. Lindfors, M. Lippitz, N. Engheta, M. Totzeck, and H. Giessen, “Imaging and steering an optical wireless nanoantenna link,” Nat. Commun. 2, 4354 (2014).

H. Aouani, M. Rahmani, M. Navarro-Cía, and S. A. Maier, “Third-harmonic-upconversion enhancement from a single semiconductor nanoparticle coupled to a plasmonic antenna,” Nat. Nanotechnol. 9, 290–294 (2014).
[Crossref] [PubMed]

2013 (1)

2012 (5)

J. Küchenmeister, T. Zebrowski, and K. Busch, “A construction guide to analytically generated meshes for the Fourier modal method,” Opt. Express 20, 17319–17347 (2012).
[Crossref] [PubMed]

N. P. de Leon, B. J. Shields, C. L. Yu, D. E. Englund, A. V. Akimov, M. D. Lukin, and H. Park, “Tailoring light-matter interaction with a nanoscale plasmon resonator,” Phys. Rev. Lett. 108, 226803 (2012).
[Crossref] [PubMed]

S. V. Lobanov, T. Weiss, D. Dregely, H. Giessen, N. A. Gippius, and S. G. Tikhodeev, “Emission properties of an oscillating point dipole from a gold Yagi-Uda nanoantenna array,” Phys. Rev. B 85, 155137 (2012).
[Crossref]

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012).
[Crossref]

K. Thyagarajan, S. Rivier, A. Lovera, and O. J. F. Martin, “Enhanced second-harmonic generation from double resonant plasmonic antennae,” Opt. Express 20, 12860–12865 (2012).
[Crossref] [PubMed]

2011 (1)

2010 (3)

2009 (3)

2008 (1)

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. García-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett. 101, 157403 (2008).
[Crossref] [PubMed]

2007 (3)

2006 (2)

W. L. Barnes, “Surface plasmon-polariton length scales: a route to sub-wavelength optics,” J. Opt. A-Pure Appl. Op. 8, S87–S93 (2006).
[Crossref]

J. A. H. van Nieuwstadt, M. Sandtke, R. H. Harmsen, F. B. Segerink, J. C. Prangsma, S. Enoch, and L. Kuipers, “Strong modification of the nonlinear optical response of metallic subwavelength hole arrays,” Phys. Rev. Lett. 97, 146102 (2006).
[Crossref] [PubMed]

2004 (1)

2003 (2)

A. Moreau, G. Granet, F. I. Baida, and D. V. Labeke, “Light transmission by subwavelength square coaxial aperture arrays in metallic films,” Opt. Express 11, 1131–1136 (2003).
[Crossref] [PubMed]

L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A-Pure Appl. Op. 5(4), 345 (2003).
[Crossref]

2002 (3)

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

G. Granet and J. P. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A-Pure Appl. Op. 4, S145–S149 (2002).
[Crossref]

P. Gay-Balmaz and O. J. F. Martin, “Electromagnetic resonances in individual and coupled split-ring resonators,” Appl. Phys. 92, 2929–2936 (2002).
[Crossref]

1999 (2)

G. Granet, “Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution,” J. Opt. Soc. Am. A 16, 2510–2516 (1999).
[Crossref]

D. M. Whittaker and I. S. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
[Crossref]

1996 (1)

1995 (1)

D. M. Sullivan, “Nonlinear FDTD formulations using Z transforms,” IEEE T. Microw. Theory 43, 676–682 (1995).
[Crossref]

1988 (1)

K. Hayata, M. Nagai, and M. Koshiba, “Finite-element formalism for nonlinear slab-guided waves,” IEEE T. Microw. Theory 36, 1207–1215 (1988).
[Crossref]

Aizpurua, J.

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. García-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett. 101, 157403 (2008).
[Crossref] [PubMed]

Akimov, A. V.

N. P. de Leon, B. J. Shields, C. L. Yu, D. E. Englund, A. V. Akimov, M. D. Lukin, and H. Park, “Tailoring light-matter interaction with a nanoscale plasmon resonator,” Phys. Rev. Lett. 108, 226803 (2012).
[Crossref] [PubMed]

Antos, R.

Aouani, H.

H. Aouani, M. Rahmani, M. Navarro-Cía, and S. A. Maier, “Third-harmonic-upconversion enhancement from a single semiconductor nanoparticle coupled to a plasmonic antenna,” Nat. Nanotechnol. 9, 290–294 (2014).
[Crossref] [PubMed]

Bai, B.

Baida, F. I.

Barnes, W. L.

W. A. Murray and W. L. Barnes, “Plasmonic Materials,” Adv. Mater. 19, 3771–3782 (2007).
[Crossref]

W. L. Barnes, “Surface plasmon-polariton length scales: a route to sub-wavelength optics,” J. Opt. A-Pure Appl. Op. 8, S87–S93 (2006).
[Crossref]

Boyd, R. W.

R. W. Boyd, Nonlinear optics, 3rd ed. (Academic Press, 2008).

Brandstetter, M.

T. Schumacher, M. Brandstetter, D. Wolf, K. Kratzer, M. Hentschel, H. Giessen, and M. Lippitz, “The optimal antenna for nonlinear spectroscopy of weakly and strongly scattering nanoobjects,” Appl. Phys. B 122, 91 (2016).
[Crossref]

Busch, K.

Bykov, D. A.

Cornelius, T. W.

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. García-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett. 101, 157403 (2008).
[Crossref] [PubMed]

Culshaw, I. S.

D. M. Whittaker and I. S. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
[Crossref]

Curto, A. G.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329, 930–933 (2010).
[Crossref] [PubMed]

de Leon, N. P.

N. P. de Leon, B. J. Shields, C. L. Yu, D. E. Englund, A. V. Akimov, M. D. Lukin, and H. Park, “Tailoring light-matter interaction with a nanoscale plasmon resonator,” Phys. Rev. Lett. 108, 226803 (2012).
[Crossref] [PubMed]

Doskolovich, L. L.

Dregely, D.

D. Dregely, K. Lindfors, M. Lippitz, N. Engheta, M. Totzeck, and H. Giessen, “Imaging and steering an optical wireless nanoantenna link,” Nat. Commun. 2, 4354 (2014).

S. V. Lobanov, T. Weiss, D. Dregely, H. Giessen, N. A. Gippius, and S. G. Tikhodeev, “Emission properties of an oscillating point dipole from a gold Yagi-Uda nanoantenna array,” Phys. Rev. B 85, 155137 (2012).
[Crossref]

Dressel, M.

K. Weber, M. L. Nesterov, T. Weiss, M. Scherer, M. Hentschel, J. Vogt, C. Huck, W. Li, M. Dressel, H. Giessen, and F. Neubrech, “Wavelength dcaling in antenna-enhanced infrared spectroscopy: Towards the far-IR and THz region,” ACS Photonics 4, 45–51 (2017).
[Crossref]

Duan, X.

M. Hentschel, M. Schäferling, X. Duan, H. Giessen, and N. Liu, “Chiral plasmonics,” Sci. Adv. 3, e1602735 (2017).
[Crossref] [PubMed]

Engheta, N.

D. Dregely, K. Lindfors, M. Lippitz, N. Engheta, M. Totzeck, and H. Giessen, “Imaging and steering an optical wireless nanoantenna link,” Nat. Commun. 2, 4354 (2014).

Englund, D. E.

N. P. de Leon, B. J. Shields, C. L. Yu, D. E. Englund, A. V. Akimov, M. D. Lukin, and H. Park, “Tailoring light-matter interaction with a nanoscale plasmon resonator,” Phys. Rev. Lett. 108, 226803 (2012).
[Crossref] [PubMed]

Enoch, S.

J. A. H. van Nieuwstadt, M. Sandtke, R. H. Harmsen, F. B. Segerink, J. C. Prangsma, S. Enoch, and L. Kuipers, “Strong modification of the nonlinear optical response of metallic subwavelength hole arrays,” Phys. Rev. Lett. 97, 146102 (2006).
[Crossref] [PubMed]

Essig, S.

García-Etxarri, A.

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. García-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett. 101, 157403 (2008).
[Crossref] [PubMed]

Gay-Balmaz, P.

P. Gay-Balmaz and O. J. F. Martin, “Electromagnetic resonances in individual and coupled split-ring resonators,” Appl. Phys. 92, 2929–2936 (2002).
[Crossref]

Giessen, H.

M. Hentschel, M. Schäferling, X. Duan, H. Giessen, and N. Liu, “Chiral plasmonics,” Sci. Adv. 3, e1602735 (2017).
[Crossref] [PubMed]

K. Weber, M. L. Nesterov, T. Weiss, M. Scherer, M. Hentschel, J. Vogt, C. Huck, W. Li, M. Dressel, H. Giessen, and F. Neubrech, “Wavelength dcaling in antenna-enhanced infrared spectroscopy: Towards the far-IR and THz region,” ACS Photonics 4, 45–51 (2017).
[Crossref]

T. Schumacher, M. Brandstetter, D. Wolf, K. Kratzer, M. Hentschel, H. Giessen, and M. Lippitz, “The optimal antenna for nonlinear spectroscopy of weakly and strongly scattering nanoobjects,” Appl. Phys. B 122, 91 (2016).
[Crossref]

B. Metzger, M. Hentschel, and H. Giessen, “Ultrafast nonlinear plasmonic spectroscopy: From dipole nanoantennas to complex hybrid plasmonic structures,” ACS Photonics 3, 1336–1350 (2016).
[Crossref]

M. L. Nesterov, X. Yin, M. Schäferling, H. Giessen, and T. Weiss, “The role of plasmon-generated near fields for enhanced circular dichroism spectroscopy,” ACS Photonics 4, 578–583 (2016).
[Crossref]

T. Weiss, M. Mesch, M. Schäferling, H. Giessen, W. Langbein, and E. A. Muljarov, “From dark to bright: First-order perturbation theory with analytical mode normalization for plasmonic nanoantenna arrays applied to refractive index sensing,” Phys. Rev. Lett. 116, 237401 (2016).
[Crossref] [PubMed]

D. Dregely, K. Lindfors, M. Lippitz, N. Engheta, M. Totzeck, and H. Giessen, “Imaging and steering an optical wireless nanoantenna link,” Nat. Commun. 2, 4354 (2014).

S. V. Lobanov, T. Weiss, D. Dregely, H. Giessen, N. A. Gippius, and S. G. Tikhodeev, “Emission properties of an oscillating point dipole from a gold Yagi-Uda nanoantenna array,” Phys. Rev. B 85, 155137 (2012).
[Crossref]

T. Weiss, N. A. Gippius, S. G. Tikhodeev, G. Granet, and H. Giessen, “Derivation of plasmonic resonances in the Fourier modal method with adaptive spatial resolution and matched coordinates,” J. Opt. Soc. Am. A 28, 238–244 (2011).
[Crossref]

T. Weiss, N. A. Gippius, S. G. Tikhodeev, G. Granet, and H. Giessen, “Efficient calculation of the optical properties of stacked metamaterials with a Fourier modal method,” J. Opt. A-Pure Appl. Op. 11, 114019 (2009).
[Crossref]

T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, and H. Giessen, “Matched coordinates and adaptive spatial resolution in the Fourier modal method,” Opt. Express 17, 8051–8061 (2009).
[Crossref] [PubMed]

Gippius, N. A.

S. V. Lobanov, T. Weiss, D. Dregely, H. Giessen, N. A. Gippius, and S. G. Tikhodeev, “Emission properties of an oscillating point dipole from a gold Yagi-Uda nanoantenna array,” Phys. Rev. B 85, 155137 (2012).
[Crossref]

T. Weiss, N. A. Gippius, S. G. Tikhodeev, G. Granet, and H. Giessen, “Derivation of plasmonic resonances in the Fourier modal method with adaptive spatial resolution and matched coordinates,” J. Opt. Soc. Am. A 28, 238–244 (2011).
[Crossref]

T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, and H. Giessen, “Matched coordinates and adaptive spatial resolution in the Fourier modal method,” Opt. Express 17, 8051–8061 (2009).
[Crossref] [PubMed]

T. Weiss, N. A. Gippius, S. G. Tikhodeev, G. Granet, and H. Giessen, “Efficient calculation of the optical properties of stacked metamaterials with a Fourier modal method,” J. Opt. A-Pure Appl. Op. 11, 114019 (2009).
[Crossref]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Granet, G.

Harmsen, R. H.

J. A. H. van Nieuwstadt, M. Sandtke, R. H. Harmsen, F. B. Segerink, J. C. Prangsma, S. Enoch, and L. Kuipers, “Strong modification of the nonlinear optical response of metallic subwavelength hole arrays,” Phys. Rev. Lett. 97, 146102 (2006).
[Crossref] [PubMed]

Hayata, K.

K. Hayata, M. Nagai, and M. Koshiba, “Finite-element formalism for nonlinear slab-guided waves,” IEEE T. Microw. Theory 36, 1207–1215 (1988).
[Crossref]

Hentschel, M.

M. Hentschel, M. Schäferling, X. Duan, H. Giessen, and N. Liu, “Chiral plasmonics,” Sci. Adv. 3, e1602735 (2017).
[Crossref] [PubMed]

K. Weber, M. L. Nesterov, T. Weiss, M. Scherer, M. Hentschel, J. Vogt, C. Huck, W. Li, M. Dressel, H. Giessen, and F. Neubrech, “Wavelength dcaling in antenna-enhanced infrared spectroscopy: Towards the far-IR and THz region,” ACS Photonics 4, 45–51 (2017).
[Crossref]

T. Schumacher, M. Brandstetter, D. Wolf, K. Kratzer, M. Hentschel, H. Giessen, and M. Lippitz, “The optimal antenna for nonlinear spectroscopy of weakly and strongly scattering nanoobjects,” Appl. Phys. B 122, 91 (2016).
[Crossref]

B. Metzger, M. Hentschel, and H. Giessen, “Ultrafast nonlinear plasmonic spectroscopy: From dipole nanoantennas to complex hybrid plasmonic structures,” ACS Photonics 3, 1336–1350 (2016).
[Crossref]

Huck, C.

K. Weber, M. L. Nesterov, T. Weiss, M. Scherer, M. Hentschel, J. Vogt, C. Huck, W. Li, M. Dressel, H. Giessen, and F. Neubrech, “Wavelength dcaling in antenna-enhanced infrared spectroscopy: Towards the far-IR and THz region,” ACS Photonics 4, 45–51 (2017).
[Crossref]

Ishihara, T.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
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F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. García-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett. 101, 157403 (2008).
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M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012).
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Koshiba, M.

K. Hayata, M. Nagai, and M. Koshiba, “Finite-element formalism for nonlinear slab-guided waves,” IEEE T. Microw. Theory 36, 1207–1215 (1988).
[Crossref]

Kratzer, K.

T. Schumacher, M. Brandstetter, D. Wolf, K. Kratzer, M. Hentschel, H. Giessen, and M. Lippitz, “The optimal antenna for nonlinear spectroscopy of weakly and strongly scattering nanoobjects,” Appl. Phys. B 122, 91 (2016).
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A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329, 930–933 (2010).
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Kufner, A.

A. Kufner and J. Kadlec, Fourier Series, 1st ed. (Iliffe Books, 1971).

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J. A. H. van Nieuwstadt, M. Sandtke, R. H. Harmsen, F. B. Segerink, J. C. Prangsma, S. Enoch, and L. Kuipers, “Strong modification of the nonlinear optical response of metallic subwavelength hole arrays,” Phys. Rev. Lett. 97, 146102 (2006).
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Labeke, D. V.

Langbein, W.

T. Weiss, M. Mesch, M. Schäferling, H. Giessen, W. Langbein, and E. A. Muljarov, “From dark to bright: First-order perturbation theory with analytical mode normalization for plasmonic nanoantenna arrays applied to refractive index sensing,” Phys. Rev. Lett. 116, 237401 (2016).
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K. Weber, M. L. Nesterov, T. Weiss, M. Scherer, M. Hentschel, J. Vogt, C. Huck, W. Li, M. Dressel, H. Giessen, and F. Neubrech, “Wavelength dcaling in antenna-enhanced infrared spectroscopy: Towards the far-IR and THz region,” ACS Photonics 4, 45–51 (2017).
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Lindfors, K.

D. Dregely, K. Lindfors, M. Lippitz, N. Engheta, M. Totzeck, and H. Giessen, “Imaging and steering an optical wireless nanoantenna link,” Nat. Commun. 2, 4354 (2014).

Lippitz, M.

T. Schumacher, M. Brandstetter, D. Wolf, K. Kratzer, M. Hentschel, H. Giessen, and M. Lippitz, “The optimal antenna for nonlinear spectroscopy of weakly and strongly scattering nanoobjects,” Appl. Phys. B 122, 91 (2016).
[Crossref]

D. Dregely, K. Lindfors, M. Lippitz, N. Engheta, M. Totzeck, and H. Giessen, “Imaging and steering an optical wireless nanoantenna link,” Nat. Commun. 2, 4354 (2014).

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M. Hentschel, M. Schäferling, X. Duan, H. Giessen, and N. Liu, “Chiral plasmonics,” Sci. Adv. 3, e1602735 (2017).
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S. V. Lobanov, T. Weiss, D. Dregely, H. Giessen, N. A. Gippius, and S. G. Tikhodeev, “Emission properties of an oscillating point dipole from a gold Yagi-Uda nanoantenna array,” Phys. Rev. B 85, 155137 (2012).
[Crossref]

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Lukin, M. D.

N. P. de Leon, B. J. Shields, C. L. Yu, D. E. Englund, A. V. Akimov, M. D. Lukin, and H. Park, “Tailoring light-matter interaction with a nanoscale plasmon resonator,” Phys. Rev. Lett. 108, 226803 (2012).
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Magnusson, R.

Maier, S. A.

H. Aouani, M. Rahmani, M. Navarro-Cía, and S. A. Maier, “Third-harmonic-upconversion enhancement from a single semiconductor nanoparticle coupled to a plasmonic antenna,” Nat. Nanotechnol. 9, 290–294 (2014).
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Martin, O. J. F.

K. Thyagarajan, S. Rivier, A. Lovera, and O. J. F. Martin, “Enhanced second-harmonic generation from double resonant plasmonic antennae,” Opt. Express 20, 12860–12865 (2012).
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P. Gay-Balmaz and O. J. F. Martin, “Electromagnetic resonances in individual and coupled split-ring resonators,” Appl. Phys. 92, 2929–2936 (2002).
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T. Weiss, M. Mesch, M. Schäferling, H. Giessen, W. Langbein, and E. A. Muljarov, “From dark to bright: First-order perturbation theory with analytical mode normalization for plasmonic nanoantenna arrays applied to refractive index sensing,” Phys. Rev. Lett. 116, 237401 (2016).
[Crossref] [PubMed]

Metzger, B.

B. Metzger, M. Hentschel, and H. Giessen, “Ultrafast nonlinear plasmonic spectroscopy: From dipole nanoantennas to complex hybrid plasmonic structures,” ACS Photonics 3, 1336–1350 (2016).
[Crossref]

Moreau, A.

Muljarov, E. A.

T. Weiss, M. Mesch, M. Schäferling, H. Giessen, W. Langbein, and E. A. Muljarov, “From dark to bright: First-order perturbation theory with analytical mode normalization for plasmonic nanoantenna arrays applied to refractive index sensing,” Phys. Rev. Lett. 116, 237401 (2016).
[Crossref] [PubMed]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Murray, W. A.

W. A. Murray and W. L. Barnes, “Plasmonic Materials,” Adv. Mater. 19, 3771–3782 (2007).
[Crossref]

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K. Hayata, M. Nagai, and M. Koshiba, “Finite-element formalism for nonlinear slab-guided waves,” IEEE T. Microw. Theory 36, 1207–1215 (1988).
[Crossref]

Navarro-Cía, M.

H. Aouani, M. Rahmani, M. Navarro-Cía, and S. A. Maier, “Third-harmonic-upconversion enhancement from a single semiconductor nanoparticle coupled to a plasmonic antenna,” Nat. Nanotechnol. 9, 290–294 (2014).
[Crossref] [PubMed]

Nesterov, M. L.

K. Weber, M. L. Nesterov, T. Weiss, M. Scherer, M. Hentschel, J. Vogt, C. Huck, W. Li, M. Dressel, H. Giessen, and F. Neubrech, “Wavelength dcaling in antenna-enhanced infrared spectroscopy: Towards the far-IR and THz region,” ACS Photonics 4, 45–51 (2017).
[Crossref]

M. L. Nesterov, X. Yin, M. Schäferling, H. Giessen, and T. Weiss, “The role of plasmon-generated near fields for enhanced circular dichroism spectroscopy,” ACS Photonics 4, 578–583 (2016).
[Crossref]

Neubrech, F.

K. Weber, M. L. Nesterov, T. Weiss, M. Scherer, M. Hentschel, J. Vogt, C. Huck, W. Li, M. Dressel, H. Giessen, and F. Neubrech, “Wavelength dcaling in antenna-enhanced infrared spectroscopy: Towards the far-IR and THz region,” ACS Photonics 4, 45–51 (2017).
[Crossref]

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. García-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett. 101, 157403 (2008).
[Crossref] [PubMed]

Osten, W.

Park, H.

N. P. de Leon, B. J. Shields, C. L. Yu, D. E. Englund, A. V. Akimov, M. D. Lukin, and H. Park, “Tailoring light-matter interaction with a nanoscale plasmon resonator,” Phys. Rev. Lett. 108, 226803 (2012).
[Crossref] [PubMed]

Paul, T.

Plumey, J. P.

G. Granet and J. P. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A-Pure Appl. Op. 4, S145–S149 (2002).
[Crossref]

Pomerantz, M.

Prangsma, J. C.

J. A. H. van Nieuwstadt, M. Sandtke, R. H. Harmsen, F. B. Segerink, J. C. Prangsma, S. Enoch, and L. Kuipers, “Strong modification of the nonlinear optical response of metallic subwavelength hole arrays,” Phys. Rev. Lett. 97, 146102 (2006).
[Crossref] [PubMed]

Priambodo, P. S.

Pucci, A.

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. García-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett. 101, 157403 (2008).
[Crossref] [PubMed]

Purvinis, G.

Quidant, R.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329, 930–933 (2010).
[Crossref] [PubMed]

Rafler, S.

Rahmani, M.

H. Aouani, M. Rahmani, M. Navarro-Cía, and S. A. Maier, “Third-harmonic-upconversion enhancement from a single semiconductor nanoparticle coupled to a plasmonic antenna,” Nat. Nanotechnol. 9, 290–294 (2014).
[Crossref] [PubMed]

Rivier, S.

Rockstuhl, C.

Ruoff, J.

Sandtke, M.

J. A. H. van Nieuwstadt, M. Sandtke, R. H. Harmsen, F. B. Segerink, J. C. Prangsma, S. Enoch, and L. Kuipers, “Strong modification of the nonlinear optical response of metallic subwavelength hole arrays,” Phys. Rev. Lett. 97, 146102 (2006).
[Crossref] [PubMed]

Schäferling, M.

M. Hentschel, M. Schäferling, X. Duan, H. Giessen, and N. Liu, “Chiral plasmonics,” Sci. Adv. 3, e1602735 (2017).
[Crossref] [PubMed]

M. L. Nesterov, X. Yin, M. Schäferling, H. Giessen, and T. Weiss, “The role of plasmon-generated near fields for enhanced circular dichroism spectroscopy,” ACS Photonics 4, 578–583 (2016).
[Crossref]

T. Weiss, M. Mesch, M. Schäferling, H. Giessen, W. Langbein, and E. A. Muljarov, “From dark to bright: First-order perturbation theory with analytical mode normalization for plasmonic nanoantenna arrays applied to refractive index sensing,” Phys. Rev. Lett. 116, 237401 (2016).
[Crossref] [PubMed]

Scherer, M.

K. Weber, M. L. Nesterov, T. Weiss, M. Scherer, M. Hentschel, J. Vogt, C. Huck, W. Li, M. Dressel, H. Giessen, and F. Neubrech, “Wavelength dcaling in antenna-enhanced infrared spectroscopy: Towards the far-IR and THz region,” ACS Photonics 4, 45–51 (2017).
[Crossref]

Schumacher, T.

T. Schumacher, M. Brandstetter, D. Wolf, K. Kratzer, M. Hentschel, H. Giessen, and M. Lippitz, “The optimal antenna for nonlinear spectroscopy of weakly and strongly scattering nanoobjects,” Appl. Phys. B 122, 91 (2016).
[Crossref]

Schuster, T.

Segerink, F. B.

J. A. H. van Nieuwstadt, M. Sandtke, R. H. Harmsen, F. B. Segerink, J. C. Prangsma, S. Enoch, and L. Kuipers, “Strong modification of the nonlinear optical response of metallic subwavelength hole arrays,” Phys. Rev. Lett. 97, 146102 (2006).
[Crossref] [PubMed]

Shields, B. J.

N. P. de Leon, B. J. Shields, C. L. Yu, D. E. Englund, A. V. Akimov, M. D. Lukin, and H. Park, “Tailoring light-matter interaction with a nanoscale plasmon resonator,” Phys. Rev. Lett. 108, 226803 (2012).
[Crossref] [PubMed]

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D. M. Sullivan, “Nonlinear FDTD formulations using Z transforms,” IEEE T. Microw. Theory 43, 676–682 (1995).
[Crossref]

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A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329, 930–933 (2010).
[Crossref] [PubMed]

Thyagarajan, K.

Tikhodeev, S. G.

S. V. Lobanov, T. Weiss, D. Dregely, H. Giessen, N. A. Gippius, and S. G. Tikhodeev, “Emission properties of an oscillating point dipole from a gold Yagi-Uda nanoantenna array,” Phys. Rev. B 85, 155137 (2012).
[Crossref]

T. Weiss, N. A. Gippius, S. G. Tikhodeev, G. Granet, and H. Giessen, “Derivation of plasmonic resonances in the Fourier modal method with adaptive spatial resolution and matched coordinates,” J. Opt. Soc. Am. A 28, 238–244 (2011).
[Crossref]

T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, and H. Giessen, “Matched coordinates and adaptive spatial resolution in the Fourier modal method,” Opt. Express 17, 8051–8061 (2009).
[Crossref] [PubMed]

T. Weiss, N. A. Gippius, S. G. Tikhodeev, G. Granet, and H. Giessen, “Efficient calculation of the optical properties of stacked metamaterials with a Fourier modal method,” J. Opt. A-Pure Appl. Op. 11, 114019 (2009).
[Crossref]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Totzeck, M.

D. Dregely, K. Lindfors, M. Lippitz, N. Engheta, M. Totzeck, and H. Giessen, “Imaging and steering an optical wireless nanoantenna link,” Nat. Commun. 2, 4354 (2014).

Turunen, J.

van Hulst, N. F.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329, 930–933 (2010).
[Crossref] [PubMed]

van Nieuwstadt, J. A. H.

J. A. H. van Nieuwstadt, M. Sandtke, R. H. Harmsen, F. B. Segerink, J. C. Prangsma, S. Enoch, and L. Kuipers, “Strong modification of the nonlinear optical response of metallic subwavelength hole arrays,” Phys. Rev. Lett. 97, 146102 (2006).
[Crossref] [PubMed]

Vogt, J.

K. Weber, M. L. Nesterov, T. Weiss, M. Scherer, M. Hentschel, J. Vogt, C. Huck, W. Li, M. Dressel, H. Giessen, and F. Neubrech, “Wavelength dcaling in antenna-enhanced infrared spectroscopy: Towards the far-IR and THz region,” ACS Photonics 4, 45–51 (2017).
[Crossref]

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A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329, 930–933 (2010).
[Crossref] [PubMed]

Weber, K.

K. Weber, M. L. Nesterov, T. Weiss, M. Scherer, M. Hentschel, J. Vogt, C. Huck, W. Li, M. Dressel, H. Giessen, and F. Neubrech, “Wavelength dcaling in antenna-enhanced infrared spectroscopy: Towards the far-IR and THz region,” ACS Photonics 4, 45–51 (2017).
[Crossref]

Weiss, T.

K. Weber, M. L. Nesterov, T. Weiss, M. Scherer, M. Hentschel, J. Vogt, C. Huck, W. Li, M. Dressel, H. Giessen, and F. Neubrech, “Wavelength dcaling in antenna-enhanced infrared spectroscopy: Towards the far-IR and THz region,” ACS Photonics 4, 45–51 (2017).
[Crossref]

M. L. Nesterov, X. Yin, M. Schäferling, H. Giessen, and T. Weiss, “The role of plasmon-generated near fields for enhanced circular dichroism spectroscopy,” ACS Photonics 4, 578–583 (2016).
[Crossref]

T. Weiss, M. Mesch, M. Schäferling, H. Giessen, W. Langbein, and E. A. Muljarov, “From dark to bright: First-order perturbation theory with analytical mode normalization for plasmonic nanoantenna arrays applied to refractive index sensing,” Phys. Rev. Lett. 116, 237401 (2016).
[Crossref] [PubMed]

S. V. Lobanov, T. Weiss, D. Dregely, H. Giessen, N. A. Gippius, and S. G. Tikhodeev, “Emission properties of an oscillating point dipole from a gold Yagi-Uda nanoantenna array,” Phys. Rev. B 85, 155137 (2012).
[Crossref]

T. Weiss, N. A. Gippius, S. G. Tikhodeev, G. Granet, and H. Giessen, “Derivation of plasmonic resonances in the Fourier modal method with adaptive spatial resolution and matched coordinates,” J. Opt. Soc. Am. A 28, 238–244 (2011).
[Crossref]

T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, and H. Giessen, “Matched coordinates and adaptive spatial resolution in the Fourier modal method,” Opt. Express 17, 8051–8061 (2009).
[Crossref] [PubMed]

T. Weiss, N. A. Gippius, S. G. Tikhodeev, G. Granet, and H. Giessen, “Efficient calculation of the optical properties of stacked metamaterials with a Fourier modal method,” J. Opt. A-Pure Appl. Op. 11, 114019 (2009).
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D. M. Whittaker and I. S. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
[Crossref]

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T. Schumacher, M. Brandstetter, D. Wolf, K. Kratzer, M. Hentschel, H. Giessen, and M. Lippitz, “The optimal antenna for nonlinear spectroscopy of weakly and strongly scattering nanoobjects,” Appl. Phys. B 122, 91 (2016).
[Crossref]

Yablonskii, A. L.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Yin, X.

M. L. Nesterov, X. Yin, M. Schäferling, H. Giessen, and T. Weiss, “The role of plasmon-generated near fields for enhanced circular dichroism spectroscopy,” ACS Photonics 4, 578–583 (2016).
[Crossref]

Yu, C. L.

N. P. de Leon, B. J. Shields, C. L. Yu, D. E. Englund, A. V. Akimov, M. D. Lukin, and H. Park, “Tailoring light-matter interaction with a nanoscale plasmon resonator,” Phys. Rev. Lett. 108, 226803 (2012).
[Crossref] [PubMed]

Zayats, A. V.

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012).
[Crossref]

Zebrowski, T.

Zhou, M.

ACS Photonics (3)

M. L. Nesterov, X. Yin, M. Schäferling, H. Giessen, and T. Weiss, “The role of plasmon-generated near fields for enhanced circular dichroism spectroscopy,” ACS Photonics 4, 578–583 (2016).
[Crossref]

K. Weber, M. L. Nesterov, T. Weiss, M. Scherer, M. Hentschel, J. Vogt, C. Huck, W. Li, M. Dressel, H. Giessen, and F. Neubrech, “Wavelength dcaling in antenna-enhanced infrared spectroscopy: Towards the far-IR and THz region,” ACS Photonics 4, 45–51 (2017).
[Crossref]

B. Metzger, M. Hentschel, and H. Giessen, “Ultrafast nonlinear plasmonic spectroscopy: From dipole nanoantennas to complex hybrid plasmonic structures,” ACS Photonics 3, 1336–1350 (2016).
[Crossref]

Adv. Mater. (1)

W. A. Murray and W. L. Barnes, “Plasmonic Materials,” Adv. Mater. 19, 3771–3782 (2007).
[Crossref]

Appl. Phys. (1)

P. Gay-Balmaz and O. J. F. Martin, “Electromagnetic resonances in individual and coupled split-ring resonators,” Appl. Phys. 92, 2929–2936 (2002).
[Crossref]

Appl. Phys. B (1)

T. Schumacher, M. Brandstetter, D. Wolf, K. Kratzer, M. Hentschel, H. Giessen, and M. Lippitz, “The optimal antenna for nonlinear spectroscopy of weakly and strongly scattering nanoobjects,” Appl. Phys. B 122, 91 (2016).
[Crossref]

IEEE T. Microw. Theory (2)

D. M. Sullivan, “Nonlinear FDTD formulations using Z transforms,” IEEE T. Microw. Theory 43, 676–682 (1995).
[Crossref]

K. Hayata, M. Nagai, and M. Koshiba, “Finite-element formalism for nonlinear slab-guided waves,” IEEE T. Microw. Theory 36, 1207–1215 (1988).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. A-Pure Appl. Op. (4)

L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A-Pure Appl. Op. 5(4), 345 (2003).
[Crossref]

W. L. Barnes, “Surface plasmon-polariton length scales: a route to sub-wavelength optics,” J. Opt. A-Pure Appl. Op. 8, S87–S93 (2006).
[Crossref]

G. Granet and J. P. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A-Pure Appl. Op. 4, S145–S149 (2002).
[Crossref]

T. Weiss, N. A. Gippius, S. G. Tikhodeev, G. Granet, and H. Giessen, “Efficient calculation of the optical properties of stacked metamaterials with a Fourier modal method,” J. Opt. A-Pure Appl. Op. 11, 114019 (2009).
[Crossref]

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

J. Opt. Soc. Am. B (2)

Nat. Commun. (1)

D. Dregely, K. Lindfors, M. Lippitz, N. Engheta, M. Totzeck, and H. Giessen, “Imaging and steering an optical wireless nanoantenna link,” Nat. Commun. 2, 4354 (2014).

Nat. Nanotechnol. (1)

H. Aouani, M. Rahmani, M. Navarro-Cía, and S. A. Maier, “Third-harmonic-upconversion enhancement from a single semiconductor nanoparticle coupled to a plasmonic antenna,” Nat. Nanotechnol. 9, 290–294 (2014).
[Crossref] [PubMed]

Nat. Photonics (1)

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012).
[Crossref]

Opt. Express (6)

Opt. Lett. (1)

Phys. Rev. B (3)

D. M. Whittaker and I. S. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
[Crossref]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

S. V. Lobanov, T. Weiss, D. Dregely, H. Giessen, N. A. Gippius, and S. G. Tikhodeev, “Emission properties of an oscillating point dipole from a gold Yagi-Uda nanoantenna array,” Phys. Rev. B 85, 155137 (2012).
[Crossref]

Phys. Rev. Lett. (4)

J. A. H. van Nieuwstadt, M. Sandtke, R. H. Harmsen, F. B. Segerink, J. C. Prangsma, S. Enoch, and L. Kuipers, “Strong modification of the nonlinear optical response of metallic subwavelength hole arrays,” Phys. Rev. Lett. 97, 146102 (2006).
[Crossref] [PubMed]

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. García-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett. 101, 157403 (2008).
[Crossref] [PubMed]

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Sci. Adv. (1)

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[Crossref] [PubMed]

Science (1)

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[Crossref] [PubMed]

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Figures (4)

Fig. 1
Fig. 1 Lateral view of a layer of nonlinear material with circular air holes. This structure is one of the examples that will be treated in this paper. As illustrated, the linear field induces a nonlinear polarization that results in a volumetric source at the second harmonic, which we discretize in a set of planar emitting layers. For each discrete source at a position x 0 3, we calculate the propagation of the field through the structure using the scattering matrix formalism. Then, we integrate coherently over all contributions to obtain the total far field. The inset in the right part depicts a schematic of the scattering matrix formalism, with the amplitude vectors A+ and A of downward and upward propagating or decaying eigenmodes in a certain layer [cf. equation (5)], respectively. The boxes with Sb and St indicate the scattering matrices of the sub-structures above and below a layer.
Fig. 2
Fig. 2 Example of matched coordinates for the structure depicted in Fig. 1. Note that the matched coordinates include the material interfaces as surfaces of constant coordinates with an increased spatial resolution in their vicinity. The coordinate transformation is included in the numerical calculations in the form of redefined permittivity, permeability, and nonlinear susceptibility tensors.
Fig. 3
Fig. 3 (a) One-dimensional periodic asymmetric grating made of GaAs and surrounded by air. This structure consists of two layers, where each layer has a of thickness of 100 nm and a period of 1 μm. The upper part of the grating has a width of 0.66 μm; the lower part has a width of 0.75 μm. The second-harmonic emission originates in the GaAs region. We consider normal incidence with the incident electric field aligned along the x1 direction and an incoming power of 1W per unit cell area. (b) Linear transmission (red solid line), second-harmonic unpolarized intensity emitted above (black dashed line) and below (gray solid line) the structure as well as resonance energies (black arrows). (c) Relative error of the second-harmonic unpolarized intensity below the system at a pump energy of 670 meV for three different implementations: Simple factorization rules (dark blue line), full factorization rules with (light blue line) and without (blue line) adaptive spatial resolution.
Fig. 4
Fig. 4 (a) Second-harmonic unpolarized intensity above (black dashed line) and below (gray solid line) the structure, and linear transmittance spectra (red line) of the structure presented in Fig. 1. The thickness of the structure is 80 nm, the period is 1.400 μm along the x and y directions and the radius of the holes is 600 nm. (b–c) Convergence behavior of the calculated second-harmonic unpolarized intensity with the pump field at (a) 580 meV (dark blue line) and (b) 842 meV (blue line). The triangles in (a) mark the energies at which the convergence curves have been calculated.

Equations (36)

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α β γ β E γ = i k 0 μ α β H β ,
α β γ β H γ = i k 0 ( ε α β E β + 4 π P α ) + 4 π c j α .
P α ( 2 ω ) = χ ( 2 ) , α β γ ( 2 ω ; ω , ω ) E β ( ω ) E γ ( ω ) ,
i 3 F ( x 1 , x 2 , x 3 ) = M ( x 1 , x 2 ) F ( x 1 , x 2 , x 3 ) .
γ m F m = M F m .
F ( x 1 , x 2 , x 3 ) = m A m + F m + ( x 1 , x 2 ) e i γ m + ( x 3 x + 3 ) + A m F m ( x 1 , x 2 ) e i γ m ( x 3 x 3 ) ,
( A l + A l ) A l , l out = ( S l , l + + S l , l + S l , l + S l , l ) S l , l ( A l + A l ) A l , l in .
A l , l out = Σ l , l ( x 0 3 ) J 0 .
A l , l out = x l 3 x l 3 d x 3 Σ l , l ( x 3 ) J ( x 3 ) .
D α = ε α α E α + ε α β E β + 4 π χ ( 2 ) , α σ ρ E ^ σ E ^ ρ P α , β α .
E α = ( ε α α ) 1 D α ( ε α α ) 1 ε α β E β ( ε α α ) 1 4 π χ ( 2 ) , α σ ρ E ^ σ E ^ ρ , β α .
[ D α ] α = [ ( ε α α ) 1 ] α 1 ( [ E α ] α + [ ( ε α α ) 1 ε α β ] α [ E β ] α + 4 π [ ( ε α α ) 1 χ ( 2 ) , α σ ρ E ^ σ E ^ ρ ] α ) , β α .
[ D β ] α = [ ε β α ( ε α α ) 1 ] α [ D α ] α + [ ε β γ ε β α ( ε α α ) 1 ε α γ ] α [ E γ ] α β α & γ α + 4 π [ χ ( 2 ) , β σ ρ ε β α ( ε α α ) 1 χ ( 2 ) , α σ ρ E ^ σ E ^ ρ ] α .
B l τ ± A , B ρ σ = { ( A τ τ ) 1 , ρ = τ , σ = τ ; ( A τ τ ) 1 A τ σ , ρ = τ , σ τ ; A ρ τ ( A τ τ ) 1 , ρ τ , σ = τ ; A ρ σ ± A ρ τ ( A τ τ ) 1 A τ σ , ρ τ , σ τ ,
ε ˜ = l 2 + F 2 l 2 l 1 + F l l 1 ε ,
G m τ ± ( B ) C , G α ρ σ = { B τ τ C τ ρ σ , α = τ ; C α ρ σ ± B α τ C τ ρ σ , α τ .
E ^ α = ( ε ^ α α ) 1 ( D ^ α ε ^ α β E ^ β ) , β α .
Γ j τ ± ( B ) G , Γ α ρ σ = { G α τ τ B τ τ B τ τ , ρ = τ , σ = τ ; G α τ σ B τ τ ± G α τ τ B τ τ B τ σ , ρ = τ , σ τ ; G α ρ τ B τ τ ± G α τ τ B τ ρ B τ τ , ρ τ , σ = τ ; G α ρ σ ± G α τ σ B τ ρ ± G α ρ τ B τ σ + G α τ τ B τ ρ B τ σ , ρ τ , σ τ .
T τ ( ε , ε ^ ) χ ( 2 ) = j τ + ( F τ l τ ε ^ ) m τ + ( F τ l τ ε ) F τ m τ ( l τ ε ) j τ ( l τ ε ^ ) χ ( 2 ) ,
χ ˜ ( 2 ) = T 2 ( l 1 + F 1 l 1 ε , l 1 + F 1 l 1 ε ^ ) T 1 ( ε , ε ^ ) χ ( 2 ) .
M ρ σ = ( x 1 , x 2 , x 3 ) ( x ¯ 1 , x ¯ 2 , x ¯ 3 ) M ¯ α β x ρ x ¯ α x σ x ¯ β , N ρ σ τ = ( x 1 , x 2 , x 3 ) ( x ¯ 1 , x ¯ 2 , x ¯ 3 ) N ¯ α β γ x ρ x ¯ α x σ x ¯ β x τ x ¯ γ .
χ ( 2 ) = 10 7 ( 0 0 0 5.73 0 0 0 0 0 0 5.73 0 0 0 0 0 0 5.73 ) cm statV .
χ ( 2 ) = 10 9 ( 0 0 0 0 13 6.6 6.6 6.6 0 13 0 0 13 13 75.9 0 0 0 ) cm statV .
F α ( x 1 , x 2 , x 3 ) = m , n F α , m n ( x 3 ) e i k 1 , m x 1 + k 2 , n x 2 .
k α , m = k α , 0 + G α , m , G α , m = m 2 π P α , k 1 , 0 = N sin θ cos ϕ , k 2 , 0 = N sin θ sin ϕ ,
D m n ρ = p , q ε ˜ m n , p q ρ σ E σ , p q + 4 π P m n ρ ,
P m n ρ = p , q , p , q χ ˜ m n , p , q , p , q ( 2 ) , ρ α β E ^ α , p , q E ^ β , p , q .
M = [ μ ˇ 23 K 2 K 1 ε ˇ 31 μ ˇ 23 K 1 K 1 ε ˇ 32 μ ˇ 13 K 2 K 2 ε ˇ 31 μ ˇ 13 K 1 K 2 ε ˇ 32 k 0 ε ˇ 21 1 k 0 K 1 μ ˇ 33 K 2 k 0 ε ˇ 22 + 1 k 0 K 1 μ ˇ 33 K 1 k 0 ε ˇ 11 1 k 0 K 2 μ ˇ 33 K 2 k 0 ε ˇ 12 + 1 k 0 K 2 μ ˇ 33 K 1 k 0 μ ˇ 21 + 1 k 0 K 1 ε ˇ 33 K 2 k 0 μ ˇ 22 1 k 0 K 1 ε ˇ 33 K 1 k 0 μ ˇ 11 + 1 k 0 K 2 ε ˇ 33 K 2 k 0 μ ˇ 12 1 k 0 K 2 ε ˇ 33 K 1 ε ˇ 23 K 2 K 1 μ ˇ 31 ε ˇ 23 K 1 K 1 μ ˇ 32 ε ˇ 13 K 2 K 2 μ ˇ 31 ε ˇ 13 K 1 K 2 μ ˇ 32 ] ,
( A l + ( x 0 3 + L ) A l ( x 0 3 + L ) ) = ( e i Γ l + L 0 0 e i Γ l L ) P l ( L ) [ A l + ( x 0 3 ) A l ( x 0 3 ) ] .
( A l + A l ) = F l 1 F l 1 I l , l 1 ( A l 1 + A l 1 ) .
X ( X + + X + X + X ) .
X * S = [ ( W + + ) 1 S + + ( W + + ) 1 W + S + S X + ( W + + ) 1 S + + S X S X + ( W + + ) 1 W + ] .
W + ± = S + X ± X + ± .
J ( x 3 ) = 4 π i F 0 1 { K 1 ( ε ˜ 0 33 ) 1 P 3 K 2 ( ε ˜ 0 33 ) 1 P 3 k 0 [ ε ˜ 0 23 ( ε ˜ 0 33 ) 1 P 3 P 2 ] k 0 [ P 1 ε ˜ 0 13 ( ε ˜ 0 33 ) 1 P 3 ] } ,
Σ l , l ( x 0 3 ) = [ S b + + V e i Γ 0 + ( L x 0 3 ) S b + + V e i Γ 0 + L S t + e i Γ 0 x 0 3 S t U e i Γ 0 L S b + e i Γ 0 + ( L x 0 3 ) S t U e i Γ 0 ( x 0 3 ) ] .
U = ( 𝟙 e i Γ 0 L S b + e i Γ 0 + L S t + ) 1 , V = ( 𝟙 e i Γ 0 + L S t + e i Γ 0 L S b + ) 1 .

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