Abstract

This tutorial aims to provide an extensive overview of methods of generating and shaping light at new frequencies by using nonlinear metasurfaces. We first review methods of manipulating light by using linear metasurfaces, on the basis of local control of the amplitude and phase of transmitted and reflected light. To extend these principles to nonlinear metasurfaces, we first introduce the mechanisms and principles underlying nonlinear interactions in metasurfaces. We then show how to use these principles to control the phase, amplitude, and polarization of emitted nonlinear radiation and how, through careful spatial arrangement of single nonlinear elements on a metasurface, it is possible to tailor the shape of the light emitted through nonlinear interaction.

© 2018 Optical Society of America

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I. Staude and J. Schilling, “Metamaterial-inspired silicon nanophotonics,” Nat. Photonics 11, 274–284 (2017).
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O. Avayu, E. Almeida, Y. Prior, and T. Ellenbogen, “Composite functional metasurfaces for multispectral achromatic optics,” Nat. Commun. 8, 14992 (2017).
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F. Walter, G. Li, C. Meier, S. Zhang, and T. Zentgraf, “Ultrathin nonlinear metasurface for optical image encoding,” Nano Lett. 17, 3171–3175 (2017).
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G. Li, S. Zhang, and T. Zentgraf, “Nonlinear photonic metasurfaces,” Nat. Rev. Mater. 2, 17010 (2017).
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M. Khorasaninejad, A. Ambrosio, P. Kanhaiya, and F. Capasso, “Broadband and chiral binary dielectric meta-holograms,” Sci. Adv. 2, e1501258 (2016).
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J. Wang and J. Du, “Plasmonic and dielectric metasurfaces: design, fabrication and applications,” Appl. Sci. 6, 239 (2016).
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S. Keren-Zur, O. Avayu, L. Michaeli, and T. Ellenbogen, “Nonlinear beam shaping with plasmonic metasurfaces,” ACS Photonics 3, 117–123 (2016).
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E. Almeida, G. Shalem, and Y. Prior, “Subwavelength nonlinear phase control and anomalous phase matching in plasmonic metasurfaces,” Nat. Commun. 7, 10367 (2016).
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E. Almeida, O. Bitton, and Y. Prior, “Nonlinear metamaterials for holography,” Nat. Commun. 7, 12533 (2016).
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N. Nookala, J. Lee, M. Tymchenko, J. Sebastian Gomez-Diaz, F. Demmerle, G. Boehm, K. Lai, G. Shvets, M.-C. Amann, A. Alu, and M. Belkin, “Ultrathin gradient nonlinear metasurface with a giant nonlinear response,” Optica 3, 283–288 (2016).
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N. Segal, S. Keren-Zur, N. Hendler, and T. Ellenbogen, “Controlling light with metamaterial-based nonlinear photonic crystals,” Nat. Photonics 9, 180–184 (2015).
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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).
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P. Genevet and F. Capasso, “Holographic optical metasurfaces: a review of current progress,” Rep. Prog. Phys. 78, 024401 (2015).
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E. Almeida and Y. Prior, “Rational design of metallic nanocavities for resonantly enhanced four-wave mixing,” Sci. Rep. 5, 10033 (2015).
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R. Kolkowski, L. Petti, M. Rippa, C. Lafargue, and J. Zyss, “Octupolar plasmonic meta-molecules for nonlinear chiral watermarking at subwavelength scale,” ACS Photonics 2, 899–906 (2015).
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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, 379–383 (2015).
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P. Ginzburg, A. V. Krasavin, G. A. Wurtz, and A. V. Zayats, “Nonperturbative hydrodynamic model for multiple harmonics generation in metallic nanostructures,” ACS Photonics 2, 8–13 (2015).
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M. Celebrano, X. Wu, M. Baselli, S. Großmann, P. Biagioni, A. Locatelli, C. De Angelis, G. Cerullo, R. Osellame, B. Hecht, L. Duò, F. Ciccacci, and M. Finazzi, “Mode matching in multiresonant plasmonic nanoantennas for enhanced second harmonic generation,” Nat. Nanotechnol. 10, 412–417 (2015).
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G. Li, S. Chen, N. Pholchai, B. Reineke, P. W. H. Wong, E. Y. B. Y. B. Pun, K. W. Cheah, T. Zentgraf, and S. Zhang, “Continuous control of the nonlinearity phase for harmonic generations,” Nat. Mater. 14, 607–612 (2015).
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R. Kolkowski, J. Szeszko, B. Dwir, E. Kapon, and J. Zyss, “Effects of surface plasmon polariton-mediated interactions on second harmonic generation from assemblies of pyramidal metallic nano-cavities,” Opt. Express 22, 30592–30606 (2014).
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S. P. Rodrigues, S. Lan, L. Kang, Y. Cui, and W. Cai, “Nonlinear imaging and spectroscopy of chiral metamaterials,” Adv. Mater. 26, 6157–6162 (2014).
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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, 033901 (2014).
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P. R. West, J. L. Stewart, A. V. Kildishev, V. M. Shalaev, V. V. Shkunov, F. Strohkendl, Y. A. Zakharenkov, R. K. Dodds, and R. Byren, “All-dielectric subwavelength metasurface focusing lens,” Opt. Express 22, 26212–26221 (2014).
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M. W. Knight, N. S. King, L. Liu, H. O. Everitt, P. Nordlander, and N. J. Halas, “Aluminum for plasmonics,” ACS Nano 8, 834–840 (2014).
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N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13, 139–150 (2014).
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2013 (5)

N. Shitrit, I. Yulevich, E. Maguid, D. Ozeri, D. Veksler, V. Kleiner, and E. Hasman, “Spin-optical metamaterial route to spin-controlled photonics,” Science 340, 724–726 (2013).
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Z. Zhao, J. Wang, S. Li, and A. E. Willner, “Metamaterials-based broadband generation of orbital angular momentum carrying vector beams,” Opt. Lett. 38, 932–934 (2013).
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D. Bar-Lev and J. Scheuer, “Efficient second harmonic generation using nonlinear substrates patterned by nano-antenna arrays,” Opt. Express 21, 29165–29178 (2013).
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A. Salomon, M. Zielinski, R. Kolkowski, J. Zyss, and Y. Prior, “Size and shape resonances in second harmonic generation from silver nanocavities,” J. Phys. Chem. C 117, 22377–22382 (2013).
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2012 (19)

H. Husu, R. Siikanen, J. Mäkitalo, J. Lehtolahti, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Metamaterials with tailored nonlinear optical response,” Nano Lett. 12, 673–677 (2012).
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F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12, 4932–4936 (2012).
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N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12, 6328–6333 (2012).
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N. V. Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108, 233902 (2012).
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2011 (9)

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K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics 5, 335–342 (2011).
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M. Castro-Lopez, D. Brinks, R. Sapienza, and N. F. Van Hulst, “Aluminum for nonlinear plasmonics: resonance-driven polarized luminescence of Al, Ag, and Au nanoantennas,” Nano Lett. 11, 4674–4678 (2011).
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H. Husu, J. Mäkitalo, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Particle plasmon resonances in L-shaped gold nanoparticles,” Opt. Express 18, 16601–16606 (2010).
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2009 (6)

T. Hanke, G. Krauss, D. Träutlein, B. Wild, R. Bratschitsch, and A. Leitenstorfer, “Efficient nonlinear light emission of single gold optical antennas driven by few-cycle near-infrared pulses,” Phys. Rev. Lett. 103, 257404 (2009).
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2008 (4)

A. Georgiou, J. Christmas, N. Collings, J. Moore, and W. A. Crossland, “Aspects of hologram calculation for video frames,” J. Opt. A 10, 035302 (2008).
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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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H. Guo, N. Liu, L. Fu, T. P. Meyrath, T. Zentgraf, H. Schweizer, and H. Giessen, “Resonance hybridization in double split-ring resonator metamaterials,” Opt. Express 15, 12095–12101 (2007).
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2006 (6)

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W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett. 6, 1027–1030 (2006).
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W. Fan, S. Zhang, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, “Second harmonic generation from patterned GaAs inside a subwavelength metallic hole array,” Opt. Express 14, 9570–9575 (2006).
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2005 (3)

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 7, S110–S117 (2005).
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E. M. Kim, S. S. Elovikov, T. V. Murzina, A. A. Nikulin, O. A. Aktsipetrov, M. A. Bader, and G. Marowsky, “Surface-enhanced optical third-harmonic generation in Ag island films,” Phys. Rev. Lett. 95, 227402 (2005).
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P. Torok and P. R. T. Munro, “The use of Gauss-Laguerre vector beams in STED microscopy,” Opt. Express 12, 3605–3617 (2004).
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B. K. Canfield, S. Kujala, K. Jefimovs, J. Turunen, and M. Kauranen, “Linear and nonlinear optical responses influenced by broken symmetry in an array of gold nanoparticles,” Opt. Express 12, 5418–5423 (2004).
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S. Roke, M. Bonn, and A. V. Petukhov, “Nonlinear optical scattering: the concept of effective susceptibility,” Phys. Rev. B 70, 115106 (2004).
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L. Zhao, K. L. Kelly, and G. C. Schatz, “The extinction spectra of silver nanoparticle arrays: influence of array structure on plasmon resonance wavelength and width,” J. Phys. Chem. B 107, 7343–7350 (2003).
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Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Space-variant Pancharatnam-Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett. 27, 1141–1143 (2002).
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Figures (15)

Figure 1
Figure 1 The amplitude and phase of the free electron cloud motion versus the wavelength. Inset: a sketch of a plasmonic nanorod whose electron cloud has been displaced. The displacement behavior of the free electrons at a nanostructure caused by an oscillating EM field can be treated as a driven damped harmonic oscillator. In the vicinity of the resonance wavelength, the phase of the electron motion relative to the driven field changes significantly.
Figure 2
Figure 2 Plasmon hybridization of metallic nanoparticles. Two identical spherical metallic nanoparticles hybridize, thus producing four spectrally separated modes of the interacting system. The charge distributions over the two particles are shown for the different possible modes of the coupled and uncoupled states. The depicted mode splitting can be derived by using either the plasmon hybridization model [92,95,96] or the coupled dipole approximation [77,97,98]. Reprinted with permission from Myroshnychenko et al., Chem. Soc. Rev. 37, 1792–1805 (2008) [77].
Figure 3
Figure 3 Detour phase. The wavefront can be controlled by means of the transverse spatial arrangement of the transparent and opaque areas in accordance with the acquired detour phase. The illustrated device is called a Fresnel zone plate (FZP). An FZP focuses light to a distance f by permitting and blocking the passage of light through regions that result in constructive and destructive interference, respectively, at the desired focal point.
Figure 4
Figure 4 Geometrical phase. Two paths with different polarization changes, resulting in the acquisition of a relative geometrical phase, are shown on the Poincaré sphere. Light in the RCP state is shone on a surface with two nanorod orientations, and the projection of the interference between the two beams in the LCP state is measured. The acquired relative phase between the two different paths is ϕg=ΔΩ/2, where ΔΩ is the solid angle encompassed by the paths.
Figure 5
Figure 5 Examples of the linear shaping of light by using metasurfaces. (a) Artist’s view of a multilayered chromatically corrected metasurface lens. Through the vertical stacking of three different Fresnel zone plates based on plasmonic nanoparticles designed to operate in the red, green, and blue regimes of the visible spectrum, a spot of white light at the focal point can be generated under white light illumination. Images of the focal region of the lens are shown in (b) for three laser wavelengths of 450 nm, 550 nm, and 650 nm to show the chromatic aberration correction. (c) Finite-difference time-domain simulation of a wavefront created by light scattered from eight “V”-shaped nanoantennas with steady amplitude and phase variations over the full 2π range. Phase control is achieved via coupled resonances at the same nanoantenna. The resulting device can conceptually guide light in any desired direction through appropriate spacing of the scattering nanoantennas. (d) Scanning electron microscope image of a dielectric metasurface that operates as a blazed grating on the basis of the GP concept. (e) The measured diffraction patterns of the metasurface shown in (d) under illumination with RCP light (top), linearly polarized light (middle), and LCP light (bottom) at a wavelength of 550 nm. (f) Illustration of a generalized Brewster effect achieved with a silicon metasurface. By means of the interplay between electrical and magnetic dipoles, interference can be exploited to eliminate the reflection of s-polarized light while enhancing the reflection of p-polarized light, as shown in the simulated reflection curves. (g) Illustration of a dielectric metasurface designed for independent polarization and phase control at each unit cell. The metasurface is composed of elliptical amorphous silicon posts of the same height but different diameters and orientations. The orientation of each ellipse relative to the incident light polarization affords amplitude control, and the diameter of each ellipse determines the phase acquired during propagation through the surface. (a), (b) Reproduced from [108] under the terms of the Creative Commons Attribution 4.0 International License. (c) From Yu et al., Science 334, 333–337 (2011) [35]. Reprinted with permission from AAAS. (d),(e) From Lin et al., Science 345, 298–302 (2014) [70]. Reprinted with permission from AAAS. (f) From Kuznetsov et al., Science 354, aag2472 (2016) [91]. Reprinted with permission from AAAS. (g) Reprinted by permission from Macmillan Publishers Ltd.: Arbabi et al., Nat. Nanotechnol. 10, 937–943 (2015) [99]. Copyright 2015.
Figure 6
Figure 6 Different types of nonlinear metasurfaces. (a) Multiquantum well structure coupled to a plasmonic structure, resulting in a giant enhancement of the nonlinear optical interaction. (b) L-shaped metallic resonators. The arrangement of the orientations defines the nonlinear polarization interaction. (c) G-shaped resonators exhibiting optical chiral dichroism. (d) Triangular prism nanocavities. (e) Structure consisting of coupled resonators. The nonlinear field that is generated at the rod penetrates the discs and is emitted as light. The asymmetry of the structure controls the emission efficiency. (a) Reprinted with permission from [53]. Copyright 2016 Optical Society of America. (b) Reprinted with permission from Husu et al., Nano Lett. 12, 673–677 (2012) [124]. Copyright 2012 American Chemical Society. (c) Reprinted with permission from Valev et al., Nano Lett. 9, 3945–3948 (2009) [125]. Copyright 2009 American Chemical Society. (d) Reprinted with permission from Salomon et al., J. Phys. Chem. C 117, 22377–22382 (2013) [126]. Copyright 2013 American Chemical Society. (e) Reprinted with permission from Gennaro et al., Nano Lett. 16, 5278–5285 (2016) [127]. Copyright 2016 American Chemical Society.
Figure 7
Figure 7 Models for the estimation of SHG efficiency. (a) Numerical simulation of the SHG from a nanorod in accordance with the hydrodynamic model of an electron gas. The SH mode acts as two opposite dipoles, which interfere destructively with each other and consequently do not radiate to the far field. (b) The SHG from an SRR simulated in the same way as in (a). Owing to the symmetry breaking, two SH dipoles along the arms are excited, radiate constructively, and emit to the far field. (c) Nonlinear scattering model for predicting the efficiency of SHG. The overlap integral of the participating radiating modes, i.e., E(ω) and E(2ω), is proportional to the effective nonlinear coefficient. (c) Reprinted by permission from Macmillan Publishers Ltd.: O’Brien et al., Nat. Mater. 14, 379–383 (2015) [141]. Copyright 2015.
Figure 8
Figure 8 Controlling the nonlinear tensor at the meta-atom level. (a) SHG from SRR arrays as a function of the lattice constant and wavelength, showing the effect of long-range coupling between the SRRs on the overall intensity. (b) Amplitude control through the variation of the ratio between the total length of an SRR and the length of its arms. (c) Field distribution of the SH modes from inverted SRRs showing opposite SH field phases. (d) The aspect ratio of a rectangular hole in gold defines the phase of the emitted FWM signal. As a result, a gradual change in the aspect ratio along the metasurface causes the emission angle to tilt. (e) The phase of the four-wave mixing signal is defined by summing the accumulated linear phase for each of the fundamental modes. (f) SEM images of chiral metamolecules used to achieve nonlinear chiral dichroism. (g) Under excitation with circularly polarized light, the rotation of an inclusion by an angle ϕ induces a geometrical phase in the nonlinear emission that is different for each emitted nonlinear circular polarization. (a) Reprinted with permission from Fig. 2(c) of Linden et al., Phys. Rev. Lett. 109, 015502 (2012) [170]. Copyright 2012 by the American Physical Society. (b) Reprinted by permission from Macmillan Publishers Ltd.: O’Brien et al., Nat. Mater. 14, 379–383 (2015) [141]. Copyright 2015. (c) Reprinted with permission from Keren-Zur et al., ACS Photonics 3, 117–123 (2016) [50]. Copyright 2016 American Chemical Society. (d), (e) Reproduced from [51] under the terms of the Creative Commons Attribution 4.0 International License. With copyright permission. (f) Reproduced from [54] under the terms of the CCreative Commons Attribution 4.0 International License. With copyright permission. (g) Reprinted with permission from Kolkowski et al., ACS Photonics 2, 899–906 (2015) [147]. Copyright 2015 American Chemical Society.
Figure 9
Figure 9 Nonlinear diffraction. (a) Periodic modulation of the effective nonlinear coefficient can be achieved by a periodic variation in the orientation of the SRRs. As a result, the reciprocal space is spanned by the lattice momentum vectors. (b), (c) Two photons with momentum k1 at (b) normal incidence and (c) oblique incidence result in SHG emission that complies with momentum conservation of the sum of the momenta of the incident photons and the lattice momentum, in agreement with Raman–Nath diffraction. (d) SEM image of a nonlinear metasurface photonic crystal consisting of periodically inverted SRRs. (e) Fourier space imaging of SH emission from the sample depicted in (d), showing diffraction corresponding to the periodicity and wavelength. (f) All-optical scanning up to θSH=30° obtained in accordance with Eq. (25) by adjusting the wavelength and period (i.e., KΛ). Reprinted by permission from Macmillan Publishers Ltd.: Segal et al., Nat. Photonics 9, 180–184 (2015) [48]. Copyright 2015.
Figure 10
Figure 10 Two-dimensional metasurface-based nonlinear photonic crystals. (a) SEM image of a square lattice nonlinear metasurface, with the corresponding square SH diffraction pattern shown in (b). (c) SEM image of a nonlinear hexagonal lattice and (d) its corresponding hexagonal SH diffraction pattern. Reprinted by permission from Macmillan Publishers Ltd.: Segal et al., Nat. Photonics 9, 180–184 (2015) [48]. Copyright 2015.
Figure 11
Figure 11 Nonlinear metasurfaces based lenses. (a) Nonlinear metasurface Fresnel zone plate composed of gold SRRs. (b) SH imaging of the metasurface depicted in (a) and the plane 1 mm away from the surface, showing the focusing of the SH light to a focal point. (c) SEM image of an FWM lens metasurface composed of rectangular nanoholes, imitating the phase front of a lens. (d) Intensity profile of FWM light emitted from the structure depicted in (c), showing the focusing of the light. (a), (b) Reprinted by permission from Macmillan Publishers Ltd.: Segal et al., Nat. Photonics 9, 180–184 (2015) [48]. Copyright 2015. (c), (d) Reproduced from [51] under the terms of theCreative Commons Attribution 4.0 International License. With copyright permission.
Figure 12
Figure 12 Nonlinear image encoding. (a) Nonlinear watermarking in an arbitrary shape using chiral metamolecules as pixels. Under excitation with right circularly polarized light, the desired arbitrary image is visible in the SH emission, whereas under excitation with left circularly polarized light, its complementary image is shown. (b) Nonlinear metamolecules consisted of two identical threefold rotational symmetry nonlinear meta-atoms. The relative rotation angle between the meta-atoms defines a different GP at the SH, and due to interference between the emitted waves, an effective SHG amplitude. (c) Illustration of a nonlinear metasurface encoded with metamolecules as described in (b) to give an arbitrary image on the metasurface with SH light. (a) Reprinted with permission from Kolkowski et al., ACS Photonics 2, 899–906 (2015) [147]. Copyright 2015 American Chemical Society. (b), (c) Reprinted with permission from Walter et al., Nano Lett. 17, 3171–3175 (2017) [174]. Copyright 2017 American Chemical Society.
Figure 13
Figure 13 Nonlinear binary CGH beam shaping. (a) Binary quadratic susceptibility map for the generation of an Airy beam hologram following Eq. (27). (b), (c) SEM image of a nonlinear metasurface for the generation of an Airy beam as in (a), consisting of SRRs. (d) Simulated and (e) measured SH Airy beams generated by the metasurface presented in (b), (c). Reprinted with permission from Keren-Zur et al., ACS Photonics 3, 117–123 (2016) [50]. Copyright 2016 American Chemical Society.
Figure 14
Figure 14 Perfect nonlinear beam shaping. (a) Slow variation of the geometrical structure of the meta-atoms along the metasurface, consistent with Fig. 8(b), in combination with inversion of their orientation, results in a linearly varying quadratic susceptibility. (b) SH Hermite–Gauss beam as imaged from the metasurface illustrated in (a) when excited with a Gaussian FF beam, and (c) the far-field image of the propagated beam, which maintains the same form. Reprinted with permission from Keren-Zur et al., ACS Photonics 3, 117–123 (2016) [50]. Copyright 2016 American Chemical Society.
Figure 15
Figure 15 Nonlinear holography of arbitrary shapes. (a) Two THG holograms generated from the same double-layered metasurface. Each hologram corresponds to a different orthogonal polarization of the FF excitation light. (b) SEM image of the metasurface used to generate the holograms shown in (a). The front layer consists of V-shaped antennas, each with a different opening angle and size, which are used to control the phase while maintaining a constant amplitude of the cubic susceptibility. The back layer is also visible, with a different set of V-shaped antennas, rotated by 90° and thus corresponding to the orthogonal polarization. (c) Three THG holograms generated by a three-layered nonlinear metasurface. Each hologram interacts with a different FF polarization (0°, 45°, 90°) and is focused on a different holographic plane. (d) Illustration of a linear hologram generated by a nonlinear metasurface and (e) two different nonlinear holograms with opposite circular polarizations. All three are generated by the same nonlinear metasurface. (a)–(c) Reproduced from [52] under the terms of the Creative Commons Attribution 4.0 International License. (d), (e) Reproduced from [54] under the terms of the Creative Commons Attribution 4.0 International License.

Equations (29)

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x2U+y2U+2ikzU=0,
αi=Vε(ω)εmedεmed+Li(ε(ω)εmed),
mλeff=2L+2ϕr,
λeff=n1+n2(λλp),
Δϕ=Δx2πλsin(θ),
Δx·sin(θ)>λ.
ϕg=12ΔΩ,
P(E)=ε0(χ(1)E+χ(2)E2+χ(3)E3+),
2En2c22Et2=1ε0c22P(2)t2.
Pi(2)(ω1+ω2,x,y)=ε0χijk(2)(ω1+ω2;ω1,ω2,x,y)Ej(ω1,x,y)Ek(ω2,x,y),
me*n[vt+(v·)v+γv]=en(E+v×H)p,
P¨+1n0e[(·P˙)P˙+(P˙·)P˙]+γP˙=n0e2me*Eeme*E(·P)+eme*P˙×H+kBT(·P)me*.
KNL=iωn0e[t^(PP)+n^123ω+iγ2ω+iγ(P)2],
χeff(2)Sχ(2)E,ω2(r)E,2ω(r)dS,
ϕ(n)=jϕj,
χRRR(2)(Δϕ)=χRRR(2)eiΔϕ,χLRR(2)(Δϕ)=χLRR(2)e3iΔϕ,χRLL(2)(Δϕ)=χRLL(2)e3iΔϕ,χLLL(2)(Δϕ)=χLLL(2)eiΔϕ,
βeff(ω3;ω1,ω2)=βs(ω3;ω1,ω2)(1S(ω1)αs(ω1))(1S(ω2)αs(ω2))(1S(ω3)αs(ω3)),
kG=iki+G,
Pω3(x,y,z0)=ε0χ(2)(x,y,z0)Eω1Eω2ei(k1+k2)·r,
χ(2)(x,y)=χ(2)mamei2πmΛx,
Pω1+ω2(x,y,z0)=ε0χ(2)mamEω1Eω2ei(k1+k2+mKx)·r,
k3,m=k1+k2+mKx
k3,m=(n(ω3)ω3c)2|k3,m|2,
sinθSFG,m=k3,m|k3|=cn(ω3)k1+k2+mKΛω1+ω2.
sinθ2,m=sinθ1+mλ12Λ,
χ(2)(x,y)=χ(2)sign{cos[2πxΛϕ(x,y)]cos[πq(x,y)]},
χ(2)(r)=χSRR(2)sign{cos[2πxΛϕ(r)]}.
P(2)(x,y)=2ε0χmax(2)aEi2xex2+y2w2,
P(2)(r,ϕ)=ε0χ(2)Ei2ex2+y2w2+ilϕ.

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