Abstract

Benefiting from efficient local phase and amplitude control at the subwavelength scale, metasurfaces offer a new platform for computer generated holography with high spatial resolution. Three-dimensional and high efficient holograms have been realized by metasurfaces constituted by subwavelength meta-atoms with spatially varying geometries or orientations. Metasurfaces have been recently extended to the nonlinear optical regime to generate holographic images in harmonic generation waves. Thus far, there has been no vector field simulation of nonlinear metasurface holograms because of the tremendous computational challenge in numerically calculating the collective nonlinear responses of the large number of different subwavelength meta-atoms in a hologram. Here, we propose a general phenomenological method to model nonlinear metasurface holograms based on the assumption that every meta-atom could be described by a localized nonlinear polarizability tensor. Applied to geometric nonlinear metasurfaces, we numerically model the holographic images formed by the second-harmonic waves of different spins. We show that, in contrast to the metasurface holograms operating in the linear optical regime, the wavelength of incident fundamental light should be slightly detuned from the fundamental resonant wavelength to optimize the efficiency and quality of nonlinear holographic images. The proposed modeling provides a general method to simulate nonlinear optical devices based on metallic metasurfaces.

© 2016 Optical Society of America

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2016 (5)

E. Almeida, G. Shalem, and Y. Prior, “Subwavelength nonlinear phase control and anomalous phase matching in plasmonic metasurfaces,” Nat. Commun. 7, 10367 (2016).
[Crossref] [PubMed]

N. Nookala, J. Lee, M. Tymchenko, J. S. Gomez-Diaz, F. Demmerle, G. Boehm, K. Lai, G. Shvets, M.-C. Amann, A. Alù, and M. Belkin, “Ultrathin gradient nonlinear metasurface with a giant nonlinear response,” Optica 3(3), 283–288 (2016).
[Crossref]

E. Almeida, O. Bitton, and Y. Prior, “Nonlinear metamaterials for holography,” Nat. Commun. 7, 12533 (2016).
[Crossref] [PubMed]

W. Ye, F. Zeuner, X. Li, B. Reineke, S. He, C.-W. Qiu, J. Liu, Y. Wang, S. Zhang, and T. Zentgraf, “Spin and wavelength multiplexed nonlinear metasurface holography,” Nat. Commun. 7, 11930 (2016).
[Crossref] [PubMed]

W. Ye, Q. Guo, Y. Xiang, D. Fan, and S. Zhang, “Phenomenological modeling of geometric metasurfaces,” Opt. Express 24(7), 7120–7132 (2016).
[Crossref] [PubMed]

2015 (6)

G. Zheng, H. Mühlenbernd, M. Kenney, G. Li, T. Zentgraf, and S. Zhang, “Metasurface holograms reaching 80% efficiency,” Nat. Nanotechnol. 10(4), 308–312 (2015).
[Crossref] [PubMed]

G. Li, S. Chen, N. Pholchai, B. Reineke, P. W. H. Wong, E. Y. B. Pun, K. W. Cheah, T. Zentgraf, and S. Zhang, “Continuous control of the nonlinearity phase for harmonic generations,” Nat. Mater. 14(6), 607–612 (2015).
[Crossref] [PubMed]

M. Tymchenko, J. S. Gomez-Diaz, J. Lee, N. Nookala, M. A. Belkin, and A. Alù, “Gradient nonlinear Pancharatnam-Berry metasurfaces,” Phys. Rev. Lett. 115(20), 207403 (2015).
[Crossref] [PubMed]

K. O’Brien, H. Suchowski, J. Rho, A. Salandrino, B. Kante, X. Yin, and X. Zhang, “Predicting nonlinear properties of metamaterials from the linear response,” Nat. Mater. 14(4), 379–383 (2015).
[Crossref] [PubMed]

N. Segal, S. Keren-Zur, N. Hendler, and T. Ellenbogen, “Controlling light with metamaterial-based nonlinear photonic crystals,” Nat. Photonics 9(3), 180–184 (2015).
[Crossref]

O. Wolf, S. Campione, A. Benz, A. P. Ravikumar, S. Liu, T. S. Luk, E. A. Kadlec, E. A. Shaner, J. F. Klem, M. B. Sinclair, and I. Brener, “Phased-array sources based on nonlinear metamaterial nanocavities,” Nat. Commun. 6, 7667 (2015).
[Crossref] [PubMed]

2014 (5)

K. Konishi, T. Higuchi, J. Li, J. Larsson, S. Ishii, and M. Kuwata-Gonokami, “Polarization-controlled circular second-harmonic generation from metal hole arrays with threefold rotational symmetry,” Phys. Rev. Lett. 112(13), 135502 (2014).
[Crossref] [PubMed]

S. Chen, G. Li, F. Zeuner, W. H. Wong, E. Y. B. Pun, T. Zentgraf, K. W. Cheah, and S. Zhang, “Symmetry-selective third-harmonic generation from plasmonic metacrystals,” Phys. Rev. Lett. 113(3), 033901 (2014).
[Crossref] [PubMed]

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
[Crossref] [PubMed]

N. Meinzer, W. L. Barnes, and I. R. Hooper, “Plasmonic meta-atoms and metasurfaces,” Nat. Photonics 8(12), 889–898 (2014).
[Crossref]

M. Lapine, I. V. Shadrivov, and Y. S. Kivshar, “Colloquium: nonlinear metamaterials,” Rev. Mod. Phys. 86(3), 1093–1123 (2014).
[Crossref]

2013 (2)

X. Ni, A. V. Kildishev, and V. M. Shalaev, “Metasurface holograms for visible light,” Nat. Commun. 4, 2807 (2013).
[Crossref]

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K. Cheah, C. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nat. Commun. 4, 2808 (2013).
[Crossref]

2012 (4)

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

X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science 335(6067), 427 (2012).
[Crossref] [PubMed]

S. Linden, F. B. P. Niesler, J. Förstner, Y. Grynko, T. Meier, and M. Wegener, “Collective effects in second-harmonic generation from split-ring-resonator arrays,” Phys. Rev. Lett. 109(1), 015502 (2012).
[Crossref] [PubMed]

C. Ciracì, E. Poutrina, M. Scalora, and D. R. Smith, “Origin of second-harmonic generation enhancement in optical split-ring resonators,” Phys. Rev. B 85(20), 201403 (2012).
[Crossref]

2011 (1)

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

2008 (1)

A. Georgiou, J. Christmas, N. Collings, J. Moore, and W. A. Crossland, “Aspects of hologram calculation for video frames,” J. Opt. A, Pure Appl. Opt. 10(3), 035302 (2008).
[Crossref]

2006 (1)

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313(5786), 502–504 (2006).
[Crossref] [PubMed]

2002 (1)

1993 (1)

E. M. M. Manders, F. J. Verbeek, and J. A. Aten, “Measurement of co-localization of objects in dual-colocur confocal images,” J. Microsc. 169(3), 375–382 (1993).
[Crossref]

1986 (1)

1972 (2)

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

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).

Aieta, F.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Akahori, H.

Almeida, E.

E. Almeida, O. Bitton, and Y. Prior, “Nonlinear metamaterials for holography,” Nat. Commun. 7, 12533 (2016).
[Crossref] [PubMed]

E. Almeida, G. Shalem, and Y. Prior, “Subwavelength nonlinear phase control and anomalous phase matching in plasmonic metasurfaces,” Nat. Commun. 7, 10367 (2016).
[Crossref] [PubMed]

Alù, A.

Amann, M.-C.

Aten, J. A.

E. M. M. Manders, F. J. Verbeek, and J. A. Aten, “Measurement of co-localization of objects in dual-colocur confocal images,” J. Microsc. 169(3), 375–382 (1993).
[Crossref]

Bai, B.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K. Cheah, C. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nat. Commun. 4, 2808 (2013).
[Crossref]

Barnes, W. L.

N. Meinzer, W. L. Barnes, and I. R. Hooper, “Plasmonic meta-atoms and metasurfaces,” Nat. Photonics 8(12), 889–898 (2014).
[Crossref]

Belkin, M.

Belkin, M. A.

M. Tymchenko, J. S. Gomez-Diaz, J. Lee, N. Nookala, M. A. Belkin, and A. Alù, “Gradient nonlinear Pancharatnam-Berry metasurfaces,” Phys. Rev. Lett. 115(20), 207403 (2015).
[Crossref] [PubMed]

Benz, A.

O. Wolf, S. Campione, A. Benz, A. P. Ravikumar, S. Liu, T. S. Luk, E. A. Kadlec, E. A. Shaner, J. F. Klem, M. B. Sinclair, and I. Brener, “Phased-array sources based on nonlinear metamaterial nanocavities,” Nat. Commun. 6, 7667 (2015).
[Crossref] [PubMed]

Biener, G.

Bitton, O.

E. Almeida, O. Bitton, and Y. Prior, “Nonlinear metamaterials for holography,” Nat. Commun. 7, 12533 (2016).
[Crossref] [PubMed]

Boehm, G.

Boltasseva, A.

X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science 335(6067), 427 (2012).
[Crossref] [PubMed]

Bomzon, Z.

Brener, I.

O. Wolf, S. Campione, A. Benz, A. P. Ravikumar, S. Liu, T. S. Luk, E. A. Kadlec, E. A. Shaner, J. F. Klem, M. B. Sinclair, and I. Brener, “Phased-array sources based on nonlinear metamaterial nanocavities,” Nat. Commun. 6, 7667 (2015).
[Crossref] [PubMed]

Campione, S.

O. Wolf, S. Campione, A. Benz, A. P. Ravikumar, S. Liu, T. S. Luk, E. A. Kadlec, E. A. Shaner, J. F. Klem, M. B. Sinclair, and I. Brener, “Phased-array sources based on nonlinear metamaterial nanocavities,” Nat. Commun. 6, 7667 (2015).
[Crossref] [PubMed]

Capasso, F.

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
[Crossref] [PubMed]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Cheah, K.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K. Cheah, C. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nat. Commun. 4, 2808 (2013).
[Crossref]

Cheah, K. W.

G. Li, S. Chen, N. Pholchai, B. Reineke, P. W. H. Wong, E. Y. B. Pun, K. W. Cheah, T. Zentgraf, and S. Zhang, “Continuous control of the nonlinearity phase for harmonic generations,” Nat. Mater. 14(6), 607–612 (2015).
[Crossref] [PubMed]

S. Chen, G. Li, F. Zeuner, W. H. Wong, E. Y. B. Pun, T. Zentgraf, K. W. Cheah, and S. Zhang, “Symmetry-selective third-harmonic generation from plasmonic metacrystals,” Phys. Rev. Lett. 113(3), 033901 (2014).
[Crossref] [PubMed]

Chen, S.

G. Li, S. Chen, N. Pholchai, B. Reineke, P. W. H. Wong, E. Y. B. Pun, K. W. Cheah, T. Zentgraf, and S. Zhang, “Continuous control of the nonlinearity phase for harmonic generations,” Nat. Mater. 14(6), 607–612 (2015).
[Crossref] [PubMed]

S. Chen, G. Li, F. Zeuner, W. H. Wong, E. Y. B. Pun, T. Zentgraf, K. W. Cheah, and S. Zhang, “Symmetry-selective third-harmonic generation from plasmonic metacrystals,” Phys. Rev. Lett. 113(3), 033901 (2014).
[Crossref] [PubMed]

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K. Cheah, C. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nat. Commun. 4, 2808 (2013).
[Crossref]

Chen, X.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K. Cheah, C. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nat. Commun. 4, 2808 (2013).
[Crossref]

Christmas, J.

A. Georgiou, J. Christmas, N. Collings, J. Moore, and W. A. Crossland, “Aspects of hologram calculation for video frames,” J. Opt. A, Pure Appl. Opt. 10(3), 035302 (2008).
[Crossref]

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]

Ciracì, C.

C. Ciracì, E. Poutrina, M. Scalora, and D. R. Smith, “Origin of second-harmonic generation enhancement in optical split-ring resonators,” Phys. Rev. B 85(20), 201403 (2012).
[Crossref]

Collings, N.

A. Georgiou, J. Christmas, N. Collings, J. Moore, and W. A. Crossland, “Aspects of hologram calculation for video frames,” J. Opt. A, Pure Appl. Opt. 10(3), 035302 (2008).
[Crossref]

Crossland, W. A.

A. Georgiou, J. Christmas, N. Collings, J. Moore, and W. A. Crossland, “Aspects of hologram calculation for video frames,” J. Opt. A, Pure Appl. Opt. 10(3), 035302 (2008).
[Crossref]

Demmerle, F.

Ellenbogen, T.

N. Segal, S. Keren-Zur, N. Hendler, and T. Ellenbogen, “Controlling light with metamaterial-based nonlinear photonic crystals,” Nat. Photonics 9(3), 180–184 (2015).
[Crossref]

Emani, N. K.

X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science 335(6067), 427 (2012).
[Crossref] [PubMed]

Enkrich, C.

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313(5786), 502–504 (2006).
[Crossref] [PubMed]

Fan, D.

Förstner, J.

S. Linden, F. B. P. Niesler, J. Förstner, Y. Grynko, T. Meier, and M. Wegener, “Collective effects in second-harmonic generation from split-ring-resonator arrays,” Phys. Rev. Lett. 109(1), 015502 (2012).
[Crossref] [PubMed]

Gaburro, Z.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Genevet, P.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Georgiou, A.

A. Georgiou, J. Christmas, N. Collings, J. Moore, and W. A. Crossland, “Aspects of hologram calculation for video frames,” J. Opt. A, Pure Appl. Opt. 10(3), 035302 (2008).
[Crossref]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).

Gomez-Diaz, J. S.

Grynko, Y.

S. Linden, F. B. P. Niesler, J. Förstner, Y. Grynko, T. Meier, and M. Wegener, “Collective effects in second-harmonic generation from split-ring-resonator arrays,” Phys. Rev. Lett. 109(1), 015502 (2012).
[Crossref] [PubMed]

Guo, Q.

Hasman, E.

He, S.

W. Ye, F. Zeuner, X. Li, B. Reineke, S. He, C.-W. Qiu, J. Liu, Y. Wang, S. Zhang, and T. Zentgraf, “Spin and wavelength multiplexed nonlinear metasurface holography,” Nat. Commun. 7, 11930 (2016).
[Crossref] [PubMed]

Hendler, N.

N. Segal, S. Keren-Zur, N. Hendler, and T. Ellenbogen, “Controlling light with metamaterial-based nonlinear photonic crystals,” Nat. Photonics 9(3), 180–184 (2015).
[Crossref]

Higuchi, T.

K. Konishi, T. Higuchi, J. Li, J. Larsson, S. Ishii, and M. Kuwata-Gonokami, “Polarization-controlled circular second-harmonic generation from metal hole arrays with threefold rotational symmetry,” Phys. Rev. Lett. 112(13), 135502 (2014).
[Crossref] [PubMed]

Hooper, I. R.

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Figures (4)

Fig. 1
Fig. 1 Schematic illustration of the model of nonlinear metasurface.
Fig. 2
Fig. 2 Optical properties of the reference metasurface constituted by periodic array of gold U-shaped split ring resonators. (a) The unit-cell structure of metasurface. The lattice sizes for the metasurfaces are a = 360nm along the x-axis and y-axis directions. Length, width and thickness of U-shape SRR are 180nm, 120nm and 30nm, and the split size is 60nm × 60nm. (b) Linear transmission spectra of X-polarized and Y-polarized light normally incident from substrate onto the metasurface. (d-f) Transverse electric filed vector distributions of three eigenmodes of the SRR on the plane with 2nm above the surface of SRR. Where, the small arrows denote directions of transverse electric filed vectors, the different color hue denotes relative amplitudes of them. (g) SHG efficiency spectra of X-polarized and Y-polarized fundamental light with power density 13MWcm−2 normally incident from substrate onto the metasurface. (c, h, i) The effective linear and second-order polarizability components of the SSR retrieved from the COMSOL simulations, respectively.
Fig. 3
Fig. 3 Theoretical results of co-circular polarized SHG hologram based on the geometric metasurface to reconstruct Chinese character for ‘“Left” in the far field. (a) Orientation angle distribution φ(x, y) of 101 × 101 arrays of gold SRRs is designed to generate the holographic image. (b) The optical efficiency spectra of the transmitted SH-RCP (denoted by ■) and the holographic image carried by it (denoted by ●) for a fundamental RCP plane-wave normally incident on the designed metasurface with average pump intensity 13 MWcm−2. (c) The overlay coefficient between SHG holographic images and the target image of Chinese character for the incident fundamental wave with different wavelengths. (d) The spatial patterns of the transmitted SH-RCP with different wavelengths. The most right panel shows the corresponding target image.
Fig. 4
Fig. 4 Theoretical results of cross-circular polarized SHG hologram based on the geometric metasurface to reconstruct Chinese character for ‘“Right” in the far field. (a) Orientation angle distribution φ(x, y) of 101 × 101 arrays of gold SRRs is designed to generate the holographic image. (b) The optical efficiency spectra of the transmitted SH-LCP (denoted by ▲) and the holographic image carried by it (denoted by ▼) for a fundamental RCP plane-wave normally incident on the designed metasurface with average pump intensity 13 MWcm−2. (c) The overlay coefficient between SHG holographic images and the target image of Chinese character for the incident fundamental wave with different wavelengths. (d) The spatial patterns of the transmitted SH-LCP with different wavelengths. The most right panel shows the corresponding target image.

Equations (32)

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α mn L = α SS e mn S e mn S + α PP e mn P e mn P ,
α mn (2) = α PPP e mn P e mn P e mn P + α PSS e mn P e mn S e mn S + α SPS e mn S e mn P e mn S + α SSP e mn S e mn S e mn P .
α mn L = α + ( e + e + e e + )+ α ( e i2 φ mn e + e + + e +i2 φ mn e e ),
α mn (2) = i[ κ + e i3 φ mn e e e +ν e i φ mn e e + e + κ e i φ mn e e + e + ] i[ κ + e i3 φ mn e + e + e + +ν e i φ mn e + e e + + κ e i φ mn e + e e ],
α ± = α SS ± α PP 2 ; κ ± = α PSS α PPP ±2 α SPS 2 2 ; ν= α PSS + α PPP 2 .
χ L (X)= m,n δ( x x m )δ( y y n ) α mn L = p,q e i 2pπ L x x+i 2qπ L y y α ¯ L ( p,q ) ,
χ (2) (X)= m,n δ( x x m )δ( y y n ) α mn (2) = p,q e i 2pπ L x x+i 2qπ L y y α ¯ (2) ( p,q ) ,
α ¯ L ( p,q )= S 1 m,n e i 2pπ L x x m i 2qπ L y y n α mn L , α ¯ (2) ( p,q )= S 1 m,n e i 2pπ L x x m i 2qπ L y y n α mn (2) .
E (x,y) S+ + E (x,y) S = E (x,y) C+ = E (x,y) =[ E x E y ],
e z ×( H (x,y) C+ H (x,y) S+ H (x,y) S )= σ d = ε 0 t ( χ E (x,y) ).
E (x,y) ( x,y,ω )= p,q E (x,y),( p,q ) ω exp( i k p x+i k q y ) ,
k pq = k p e x + k q e y =( k x in + 2pπ / L x ) e x +( k y in + 2qπ / L y ) e y .
E (x,y),( m,n ) ω,C+ =2 [ Π ω ] ( m,n ),( 0,0 ) 1 Y S ( k x in , k y in ,ω ) E (x,y),( 0,0 ) ω,S+ ,
E (x,y),( m,n ) ω,S =2 [ Π ω ] ( m,n ),( 0,0 ) 1 Y S ( k x in , k y in ,ω ) E (x,y),( 0,0 ) ω,S+ E (x,y),( 0,0 ) ω,S+ δ m,0 δ n,0 .
Π ( p,q ),( m,n ) ω =[ Y C ( k p , k q ,ω )+ Y S ( k p , k q ,ω ) ] δ p,m δ q,n +iω ε 0 α ¯ L ( pm,qn,ω ),
Y S(C) ( k x , k y ,ω )= ε S(C) / μ S(C) ω n S(C) ω 2 n S(C) 2 ( k x 2 + k y 2 ) c 2 [ ω 2 n S(C) 2 k y 2 c 2 k x k y c 2 k x k y c 2 ω 2 n S(C) 2 k x 2 c 2 ].
E (x,y) ( x,y,2ω )= p,q E (x,y),( p,q ) 2ω exp( i k p x+i k q y ) ,
k pq = k p e x + k q e y =( 2 k x in + 2pπ / L x ) e x +( 2 k y in + 2qπ / L y ) e y .
E (x,y),( m,n ) 2ω,S = E (x,y),( m,n ) 2ω,C+ ,
E (x,y),( m,n ) 2ω,C+ = [ Π 2ω ] ( m,n ),( p,q ) 1 ( i2ω ε 0 )[ l,h α ¯ (2) ( pl,qh,2ω ) : ( E (x,y) ω,C+ E (x,y) ω,C+ ) ( l,h ) ],
( E (x,y) ω,C+ E (x,y) ω,C+ ) ( l,h ) = m,n E (x,y),( lm,hn ) ω,C+ E (x,y),( m,n ) ω,C+ .
η SHG = p,q S z,2ω + ( k p , k q ) S z,ω + ( k x in , k y in ) = p,q 0.5Re( E x,( p,q ) 2ω,C+ H y,( p,q ) 2ω,C+ * E y,( p,q ) 2ω,C+ H x,( p,q ) 2ω,C+ * ) 0.5Re( E x,( 0,0 ) ω,S+ H y,( 0,0 ) ω,S+ * E y,( 0,0 ) ω,S+ H x,( 0,0 ) ω,S+ * ) .
α SS ( ω ) a 2 = c iω ( n S + n C 2 n S t xx ), α PP ( ω ) a 2 = c iω ( n S + n C 2 n S t yy ),
t xx = E x,( 0,0 ) ω,C+ E x,( 0,0 ) ω,S+ , t yy = E y,( 0,0 ) ω,C+ E y,( 0,0 ) ω,S+ .
I d 2ω e x(y) =i2ω P x(y) 2ω =i2ω ε 0 ε 0 ε rAu 2 2e n 0 m d S (m) [ ( 3ω+i γ e ) 2( 2ω+i γ e ) E n,ω (m)2 ( e n (m) e x(y) )+ E n,ω (m) E t1,ω (m)2 ( e t1 (m) e x(y) )+ E n,ω (m) E t2,ω (m)2 ( e t2 (m) e x(y) ) ].
P x 2ω =2 ε 0 α SPS E y,( 0,0 ) ω,C+ E x,( 0,0 ) ω,C+ ,
P y 2ω = ε 0 α PSS E x,( 0,0 ) ω,C+ E x,( 0,0 ) ω,C+ + ε 0 α PPP E y,( 0,0 ) ω,C+ E y,( 0,0 ) ω,C+ .
α PSS a 2 = P y 2ω ( e x In ) a 2 ε 0 t xx 2 , α PPP a 2 = P y 2ω ( e y In ) a 2 ε 0 t yy 2 , α SPS a 2 = P x 2ω ( e 45 0 In ) a 2 ε 0 t xx t yy .
η= p,q f 0 (p,q) S z,2ω + ( k p , k q ) p,q f 0 (p,q) 2 p,q S z,2ω + ( k p , k q ) 2 .
σ d(p,q) 2ω =( i2ω ε 0 )[ l,h α ¯ (2) ( pl,qh,2ω ) : ( E (x,y) ω,C+ E (x,y) ω,C+ ) ( l,h ) ].
σ d(p,q) 2ω ( i2ω ε 0 )[ ( l,h ) α ¯ (2) ( pl,qh,2ω ): E (x,y),( 0,0 ) ω,C+ E (x,y),( l,h ) ω,C+ ].
e + σ d(p,q) 2ω 2ω ε 0 S m,n e i 2pπ L x x m i 2qπ L y y n κ + e i3 φ mn ( E (x,y),( 0,0 ) ω,C+ e )( E (x,y),( 0,0 ) ω,C+ e ) + 2ω ε 0 S m,n e i 2pπ L x x m i 2qπ L y y n ν e i φ mn [ E (x,y) C+ ( x m , y n ,ω ) e + ]( E (x,y),( 0,0 ) ω,C+ e ).

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