V. Agranovich and V. Kravtsov, “Notes on crystal optics of superlattices,” Solid State Commun. 55, 85–90 (1985).

[Crossref]

L. Carletti, D. Rocco, A. Locatelli, C. D. Angelis, V. F. Gili, M. Ravaro, I. Favero, G. Leo, M. Finazzi, L. Ghirardini, M. Celebrano, G. Marino, and A. V. Zayats, “Controlling second-harmonic generation at the nanoscale with monolithic AlGaAs-on-AlOx antennas,” Nanotechnology 28, 114005 (2017).

[Crossref]

R. Pollard, A. Murphy, W. Hendren, P. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett. 102, 127405 (2009).

[Crossref]

T. Stefaniuk, N. Olivier, A. Belardini, C. P. T. McPolin, C. Sibilia, A. A. Wronkowska, A. Wronkowski, T. Szoplik, and A. V. Zayats, “Self-assembled silver-germanium nanolayer metamaterial with the enhanced nonlinear response,” Adv. Opt. Mater. 5, 1700753 (2017).

[Crossref]

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7, 948–957 (2013).

[Crossref]

P. Segovia, G. Marino, A. V. Krasavin, N. Olivier, G. A. Wurtz, P. A. Belov, P. Ginzburg, and A. V. Zayats, “Hyperbolic metamaterial antenna for second-harmonic generation tomography,” Opt. Express 23, 30730–30738 (2015).

[Crossref]

N. N. Lepeshkin, A. Schweinsberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced nonlinear optical response of one-dimensional metal-dielectric photonic crystals,” Phys. Rev. Lett. 93, 123902 (2004).

[Crossref]

D. J. Bergman, “Nonlinear behavior and 1/f noise near a conductivity threshold: effects of local microgeometry,” Phys. Rev. B 39, 4598–4609 (1989).

[Crossref]

N. N. Lepeshkin, A. Schweinsberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced nonlinear optical response of one-dimensional metal-dielectric photonic crystals,” Phys. Rev. Lett. 93, 123902 (2004).

[Crossref]

P. Campagnola and C.-Y. Dong, “Second harmonic generation microscopy: principles and applications to disease diagnosis,” Laser Photon. Rev. 5, 13–26 (2011).

[Crossref]

D. de Ceglia, S. Campione, M. A. Vincenti, F. Capolino, and M. Scalora, “Low-damping epsilon-near-zero slabs: nonlinear and nonlocal optical properties,” Phys. Rev. B 87, 155140 (2013).

[Crossref]

D. de Ceglia, S. Campione, M. A. Vincenti, F. Capolino, and M. Scalora, “Low-damping epsilon-near-zero slabs: nonlinear and nonlocal optical properties,” Phys. Rev. B 87, 155140 (2013).

[Crossref]

L. Carletti, D. Rocco, A. Locatelli, C. D. Angelis, V. F. Gili, M. Ravaro, I. Favero, G. Leo, M. Finazzi, L. Ghirardini, M. Celebrano, G. Marino, and A. V. Zayats, “Controlling second-harmonic generation at the nanoscale with monolithic AlGaAs-on-AlOx antennas,” Nanotechnology 28, 114005 (2017).

[Crossref]

L. Carletti, D. Rocco, A. Locatelli, C. D. Angelis, V. F. Gili, M. Ravaro, I. Favero, G. Leo, M. Finazzi, L. Ghirardini, M. Celebrano, G. Marino, and A. V. Zayats, “Controlling second-harmonic generation at the nanoscale with monolithic AlGaAs-on-AlOx antennas,” Nanotechnology 28, 114005 (2017).

[Crossref]

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

[Crossref]

L. Nicholls, F. J. Rodríguez-Fortuño, M. E. Nasir, R. M. Cordova-Castro, N. Olivier, G. A. Wurtz, and A. V. Zayats, “Ultrafast synthesis and switching of light polarization in nonlinear anisotropic metamaterials,” Nat. Photonics 11, 628–633 (2017).

[Crossref]

M. Scalora, M. A. Vincenti, D. de Ceglia, and J. W. Haus, “Nonlocal and quantum-tunneling contributions to harmonic generation in nanostructures: electron-cloud-screening effects,” Phys. Rev. A 90, 013831 (2014).

[Crossref]

D. de Ceglia, S. Campione, M. A. Vincenti, F. Capolino, and M. Scalora, “Low-damping epsilon-near-zero slabs: nonlinear and nonlocal optical properties,” Phys. Rev. B 87, 155140 (2013).

[Crossref]

M. A. Vincenti, D. de Ceglia, and M. Scalora, “Nonlinear dynamics in low permittivity media: the impact of losses,” Opt. Express 21, 29949–29954 (2013).

[Crossref]

A. V. Krasavin, M. E. Nasir, W. Dickson, and A. V. Zayats, “Reactive tunnel junctions in electrically-driven plasmonic nanorod metamaterials,” Nat. Nanotechnol. 13, 159–164 (2018).

[Crossref]

M. E. Nasir, S. Peruch, N. Vasilantonakis, W. P. Wardley, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Tuning the effective plasma frequency of nanorod metamaterials from visible to telecom wavelengths,” Appl. Phys. Lett. 107, 121110 (2015).

[Crossref]

N. Vasilantonakis, M. E. Nasir, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Bulk plasmon-polaritons in hyperbolic nanorod metamaterial waveguides,” Laser Photon. Rev. 9, 345–353 (2015).

[Crossref]

A. Neira, N. Olivier, M. E. Nasir, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Eliminating material constraints for nonlinearity with plasmonic metamaterials,” Nat. Commun. 6, 7757 (2015).

[Crossref]

P. Campagnola and C.-Y. Dong, “Second harmonic generation microscopy: principles and applications to disease diagnosis,” Laser Photon. Rev. 5, 13–26 (2011).

[Crossref]

N. Engheta, “Circuits with light at nanoscales: optical nanocircuits inspired by metamaterials,” Science 317, 1698–1702 (2007).

[Crossref]

R. Pollard, A. Murphy, W. Hendren, P. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett. 102, 127405 (2009).

[Crossref]

L. Carletti, D. Rocco, A. Locatelli, C. D. Angelis, V. F. Gili, M. Ravaro, I. Favero, G. Leo, M. Finazzi, L. Ghirardini, M. Celebrano, G. Marino, and A. V. Zayats, “Controlling second-harmonic generation at the nanoscale with monolithic AlGaAs-on-AlOx antennas,” Nanotechnology 28, 114005 (2017).

[Crossref]

L. Carletti, D. Rocco, A. Locatelli, C. D. Angelis, V. F. Gili, M. Ravaro, I. Favero, G. Leo, M. Finazzi, L. Ghirardini, M. Celebrano, G. Marino, and A. V. Zayats, “Controlling second-harmonic generation at the nanoscale with monolithic AlGaAs-on-AlOx antennas,” Nanotechnology 28, 114005 (2017).

[Crossref]

L. Carletti, D. Rocco, A. Locatelli, C. D. Angelis, V. F. Gili, M. Ravaro, I. Favero, G. Leo, M. Finazzi, L. Ghirardini, M. Celebrano, G. Marino, and A. V. Zayats, “Controlling second-harmonic generation at the nanoscale with monolithic AlGaAs-on-AlOx antennas,” Nanotechnology 28, 114005 (2017).

[Crossref]

L. Carletti, D. Rocco, A. Locatelli, C. D. Angelis, V. F. Gili, M. Ravaro, I. Favero, G. Leo, M. Finazzi, L. Ghirardini, M. Celebrano, G. Marino, and A. V. Zayats, “Controlling second-harmonic generation at the nanoscale with monolithic AlGaAs-on-AlOx antennas,” Nanotechnology 28, 114005 (2017).

[Crossref]

G. Marino, P. Segovia, A. V. Krasavin, P. Ginzburg, N. Olivier, G. A. Wurtz, and A. V. Zayats, “Second-harmonic generation from hyperbolic plasmonic nanorod metamaterial slab,” Laser Photon. Rev. 12, 1700189 (2018).

[Crossref]

P. Ginzburg, D. Roth, M. E. Nasir, P. Segovia, A. V. Krasavin, J. Levitt, L. M. Hirvonen, B. Wells, K. Suhling, D. Richards, V. A. Podolskiy, and A. V. Zayats, “Spontaneous emission in non-local materials,” Light Sci. Appl. 6, e16273 (2017).

[Crossref]

A. V. Krasavin, P. Ginzburg, and A. V. Zayats, “Free-electron optical nonlinearities in plasmonic nanostructures: a review of the hydrodynamic description,” Laser Photon. Rev. 12, 1700082 (2017).

[Crossref]

P. Segovia, G. Marino, A. V. Krasavin, N. Olivier, G. A. Wurtz, P. A. Belov, P. Ginzburg, and A. V. Zayats, “Hyperbolic metamaterial antenna for second-harmonic generation tomography,” Opt. Express 23, 30730–30738 (2015).

[Crossref]

B. M. Wells, W. Guo, and V. A. Podolskiy, “Homogenization of nanowire-based composites with anisotropic unit-cell and layered substructure,” MRS Commun. 6, 23–29 (2016).

[Crossref]

M. Scalora, M. A. Vincenti, D. de Ceglia, and J. W. Haus, “Nonlocal and quantum-tunneling contributions to harmonic generation in nanostructures: electron-cloud-screening effects,” Phys. Rev. A 90, 013831 (2014).

[Crossref]

J. W. Haus, N. Kalyaniwalla, R. Inguva, M. Bloemer, and C. M. Bowden, “Nonlinear-optical properties of conductive spheroidal particle composites,” J. Opt. Soc. Am. B 6, 797–807 (1989).

[Crossref]

R. Pollard, A. Murphy, W. Hendren, P. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett. 102, 127405 (2009).

[Crossref]

P. Ginzburg, D. Roth, M. E. Nasir, P. Segovia, A. V. Krasavin, J. Levitt, L. M. Hirvonen, B. Wells, K. Suhling, D. Richards, V. A. Podolskiy, and A. V. Zayats, “Spontaneous emission in non-local materials,” Light Sci. Appl. 6, e16273 (2017).

[Crossref]

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

[Crossref]

D. Stroud and P. M. Hui, “Nonlinear susceptibilities of granular matter,” Phys. Rev. B 37, 8719–8724 (1988).

[Crossref]

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7, 948–957 (2013).

[Crossref]

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

[Crossref]

C. Kern, M. Kadic, and M. Wegener, “Experimental evidence for sign reversal of the hall coefficient in three-dimensional metamaterials,” Phys. Rev. Lett. 118, 016601 (2017).

[Crossref]

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

[Crossref]

C. Kern, M. Kadic, and M. Wegener, “Experimental evidence for sign reversal of the hall coefficient in three-dimensional metamaterials,” Phys. Rev. Lett. 118, 016601 (2017).

[Crossref]

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7, 948–957 (2013).

[Crossref]

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

[Crossref]

G. Marino, P. Segovia, A. V. Krasavin, P. Ginzburg, N. Olivier, G. A. Wurtz, and A. V. Zayats, “Second-harmonic generation from hyperbolic plasmonic nanorod metamaterial slab,” Laser Photon. Rev. 12, 1700189 (2018).

[Crossref]

A. V. Krasavin, M. E. Nasir, W. Dickson, and A. V. Zayats, “Reactive tunnel junctions in electrically-driven plasmonic nanorod metamaterials,” Nat. Nanotechnol. 13, 159–164 (2018).

[Crossref]

A. V. Krasavin, P. Ginzburg, and A. V. Zayats, “Free-electron optical nonlinearities in plasmonic nanostructures: a review of the hydrodynamic description,” Laser Photon. Rev. 12, 1700082 (2017).

[Crossref]

P. Ginzburg, D. Roth, M. E. Nasir, P. Segovia, A. V. Krasavin, J. Levitt, L. M. Hirvonen, B. Wells, K. Suhling, D. Richards, V. A. Podolskiy, and A. V. Zayats, “Spontaneous emission in non-local materials,” Light Sci. Appl. 6, e16273 (2017).

[Crossref]

P. Segovia, G. Marino, A. V. Krasavin, N. Olivier, G. A. Wurtz, P. A. Belov, P. Ginzburg, and A. V. Zayats, “Hyperbolic metamaterial antenna for second-harmonic generation tomography,” Opt. Express 23, 30730–30738 (2015).

[Crossref]

V. Agranovich and V. Kravtsov, “Notes on crystal optics of superlattices,” Solid State Commun. 55, 85–90 (1985).

[Crossref]

L. Carletti, D. Rocco, A. Locatelli, C. D. Angelis, V. F. Gili, M. Ravaro, I. Favero, G. Leo, M. Finazzi, L. Ghirardini, M. Celebrano, G. Marino, and A. V. Zayats, “Controlling second-harmonic generation at the nanoscale with monolithic AlGaAs-on-AlOx antennas,” Nanotechnology 28, 114005 (2017).

[Crossref]

N. N. Lepeshkin, A. Schweinsberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced nonlinear optical response of one-dimensional metal-dielectric photonic crystals,” Phys. Rev. Lett. 93, 123902 (2004).

[Crossref]

P. Ginzburg, D. Roth, M. E. Nasir, P. Segovia, A. V. Krasavin, J. Levitt, L. M. Hirvonen, B. Wells, K. Suhling, D. Richards, V. A. Podolskiy, and A. V. Zayats, “Spontaneous emission in non-local materials,” Light Sci. Appl. 6, e16273 (2017).

[Crossref]

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

[Crossref]

L. Carletti, D. Rocco, A. Locatelli, C. D. Angelis, V. F. Gili, M. Ravaro, I. Favero, G. Leo, M. Finazzi, L. Ghirardini, M. Celebrano, G. Marino, and A. V. Zayats, “Controlling second-harmonic generation at the nanoscale with monolithic AlGaAs-on-AlOx antennas,” Nanotechnology 28, 114005 (2017).

[Crossref]

G. Marino, P. Segovia, A. V. Krasavin, P. Ginzburg, N. Olivier, G. A. Wurtz, and A. V. Zayats, “Second-harmonic generation from hyperbolic plasmonic nanorod metamaterial slab,” Laser Photon. Rev. 12, 1700189 (2018).

[Crossref]

L. Carletti, D. Rocco, A. Locatelli, C. D. Angelis, V. F. Gili, M. Ravaro, I. Favero, G. Leo, M. Finazzi, L. Ghirardini, M. Celebrano, G. Marino, and A. V. Zayats, “Controlling second-harmonic generation at the nanoscale with monolithic AlGaAs-on-AlOx antennas,” Nanotechnology 28, 114005 (2017).

[Crossref]

P. Segovia, G. Marino, A. V. Krasavin, N. Olivier, G. A. Wurtz, P. A. Belov, P. Ginzburg, and A. V. Zayats, “Hyperbolic metamaterial antenna for second-harmonic generation tomography,” Opt. Express 23, 30730–30738 (2015).

[Crossref]

T. Stefaniuk, N. Olivier, A. Belardini, C. P. T. McPolin, C. Sibilia, A. A. Wronkowska, A. Wronkowski, T. Szoplik, and A. V. Zayats, “Self-assembled silver-germanium nanolayer metamaterial with the enhanced nonlinear response,” Adv. Opt. Mater. 5, 1700753 (2017).

[Crossref]

G. W. Milton, The Theory of Composites, 1st ed. (Cambridge University, 2002).

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

[Crossref]

R. Pollard, A. Murphy, W. Hendren, P. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett. 102, 127405 (2009).

[Crossref]

A. V. Krasavin, M. E. Nasir, W. Dickson, and A. V. Zayats, “Reactive tunnel junctions in electrically-driven plasmonic nanorod metamaterials,” Nat. Nanotechnol. 13, 159–164 (2018).

[Crossref]

P. Ginzburg, D. Roth, M. E. Nasir, P. Segovia, A. V. Krasavin, J. Levitt, L. M. Hirvonen, B. Wells, K. Suhling, D. Richards, V. A. Podolskiy, and A. V. Zayats, “Spontaneous emission in non-local materials,” Light Sci. Appl. 6, e16273 (2017).

[Crossref]

L. Nicholls, F. J. Rodríguez-Fortuño, M. E. Nasir, R. M. Cordova-Castro, N. Olivier, G. A. Wurtz, and A. V. Zayats, “Ultrafast synthesis and switching of light polarization in nonlinear anisotropic metamaterials,” Nat. Photonics 11, 628–633 (2017).

[Crossref]

A. Neira, N. Olivier, M. E. Nasir, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Eliminating material constraints for nonlinearity with plasmonic metamaterials,” Nat. Commun. 6, 7757 (2015).

[Crossref]

M. E. Nasir, S. Peruch, N. Vasilantonakis, W. P. Wardley, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Tuning the effective plasma frequency of nanorod metamaterials from visible to telecom wavelengths,” Appl. Phys. Lett. 107, 121110 (2015).

[Crossref]

N. Vasilantonakis, M. E. Nasir, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Bulk plasmon-polaritons in hyperbolic nanorod metamaterial waveguides,” Laser Photon. Rev. 9, 345–353 (2015).

[Crossref]

A. Neira, N. Olivier, M. E. Nasir, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Eliminating material constraints for nonlinearity with plasmonic metamaterials,” Nat. Commun. 6, 7757 (2015).

[Crossref]

L. Nicholls, F. J. Rodríguez-Fortuño, M. E. Nasir, R. M. Cordova-Castro, N. Olivier, G. A. Wurtz, and A. V. Zayats, “Ultrafast synthesis and switching of light polarization in nonlinear anisotropic metamaterials,” Nat. Photonics 11, 628–633 (2017).

[Crossref]

G. Marino, P. Segovia, A. V. Krasavin, P. Ginzburg, N. Olivier, G. A. Wurtz, and A. V. Zayats, “Second-harmonic generation from hyperbolic plasmonic nanorod metamaterial slab,” Laser Photon. Rev. 12, 1700189 (2018).

[Crossref]

T. Stefaniuk, N. Olivier, A. Belardini, C. P. T. McPolin, C. Sibilia, A. A. Wronkowska, A. Wronkowski, T. Szoplik, and A. V. Zayats, “Self-assembled silver-germanium nanolayer metamaterial with the enhanced nonlinear response,” Adv. Opt. Mater. 5, 1700753 (2017).

[Crossref]

L. Nicholls, F. J. Rodríguez-Fortuño, M. E. Nasir, R. M. Cordova-Castro, N. Olivier, G. A. Wurtz, and A. V. Zayats, “Ultrafast synthesis and switching of light polarization in nonlinear anisotropic metamaterials,” Nat. Photonics 11, 628–633 (2017).

[Crossref]

A. Neira, N. Olivier, M. E. Nasir, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Eliminating material constraints for nonlinearity with plasmonic metamaterials,” Nat. Commun. 6, 7757 (2015).

[Crossref]

P. Segovia, G. Marino, A. V. Krasavin, N. Olivier, G. A. Wurtz, P. A. Belov, P. Ginzburg, and A. V. Zayats, “Hyperbolic metamaterial antenna for second-harmonic generation tomography,” Opt. Express 23, 30730–30738 (2015).

[Crossref]

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

[Crossref]

M. E. Nasir, S. Peruch, N. Vasilantonakis, W. P. Wardley, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Tuning the effective plasma frequency of nanorod metamaterials from visible to telecom wavelengths,” Appl. Phys. Lett. 107, 121110 (2015).

[Crossref]

N. N. Lepeshkin, A. Schweinsberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced nonlinear optical response of one-dimensional metal-dielectric photonic crystals,” Phys. Rev. Lett. 93, 123902 (2004).

[Crossref]

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7, 948–957 (2013).

[Crossref]

P. Ginzburg, D. Roth, M. E. Nasir, P. Segovia, A. V. Krasavin, J. Levitt, L. M. Hirvonen, B. Wells, K. Suhling, D. Richards, V. A. Podolskiy, and A. V. Zayats, “Spontaneous emission in non-local materials,” Light Sci. Appl. 6, e16273 (2017).

[Crossref]

B. M. Wells, W. Guo, and V. A. Podolskiy, “Homogenization of nanowire-based composites with anisotropic unit-cell and layered substructure,” MRS Commun. 6, 23–29 (2016).

[Crossref]

B. M. Wells, A. V. Zayats, and V. A. Podolskiy, “Nonlocal optics of plasmonic nanowire metamaterials,” Phys. Rev. B 89, 035111 (2014).

[Crossref]

R. Pollard, A. Murphy, W. Hendren, P. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett. 102, 127405 (2009).

[Crossref]

R. Wangberg, J. Elser, E. E. Narimanov, and V. A. Podolskiy, “Nonmagnetic nanocomposites for optical and infrared negative-refractive-index media,” J. Opt. Soc. Am. B 23, 498–505 (2006).

[Crossref]

R. Pollard, A. Murphy, W. Hendren, P. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett. 102, 127405 (2009).

[Crossref]

L. Carletti, D. Rocco, A. Locatelli, C. D. Angelis, V. F. Gili, M. Ravaro, I. Favero, G. Leo, M. Finazzi, L. Ghirardini, M. Celebrano, G. Marino, and A. V. Zayats, “Controlling second-harmonic generation at the nanoscale with monolithic AlGaAs-on-AlOx antennas,” Nanotechnology 28, 114005 (2017).

[Crossref]

P. Ginzburg, D. Roth, M. E. Nasir, P. Segovia, A. V. Krasavin, J. Levitt, L. M. Hirvonen, B. Wells, K. Suhling, D. Richards, V. A. Podolskiy, and A. V. Zayats, “Spontaneous emission in non-local materials,” Light Sci. Appl. 6, e16273 (2017).

[Crossref]

L. Carletti, D. Rocco, A. Locatelli, C. D. Angelis, V. F. Gili, M. Ravaro, I. Favero, G. Leo, M. Finazzi, L. Ghirardini, M. Celebrano, G. Marino, and A. V. Zayats, “Controlling second-harmonic generation at the nanoscale with monolithic AlGaAs-on-AlOx antennas,” Nanotechnology 28, 114005 (2017).

[Crossref]

L. Nicholls, F. J. Rodríguez-Fortuño, M. E. Nasir, R. M. Cordova-Castro, N. Olivier, G. A. Wurtz, and A. V. Zayats, “Ultrafast synthesis and switching of light polarization in nonlinear anisotropic metamaterials,” Nat. Photonics 11, 628–633 (2017).

[Crossref]

P. Ginzburg, D. Roth, M. E. Nasir, P. Segovia, A. V. Krasavin, J. Levitt, L. M. Hirvonen, B. Wells, K. Suhling, D. Richards, V. A. Podolskiy, and A. V. Zayats, “Spontaneous emission in non-local materials,” Light Sci. Appl. 6, e16273 (2017).

[Crossref]

M. Scalora, M. A. Vincenti, D. de Ceglia, and J. W. Haus, “Nonlocal and quantum-tunneling contributions to harmonic generation in nanostructures: electron-cloud-screening effects,” Phys. Rev. A 90, 013831 (2014).

[Crossref]

D. de Ceglia, S. Campione, M. A. Vincenti, F. Capolino, and M. Scalora, “Low-damping epsilon-near-zero slabs: nonlinear and nonlocal optical properties,” Phys. Rev. B 87, 155140 (2013).

[Crossref]

M. A. Vincenti, D. de Ceglia, and M. Scalora, “Nonlinear dynamics in low permittivity media: the impact of losses,” Opt. Express 21, 29949–29954 (2013).

[Crossref]

N. N. Lepeshkin, A. Schweinsberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced nonlinear optical response of one-dimensional metal-dielectric photonic crystals,” Phys. Rev. Lett. 93, 123902 (2004).

[Crossref]

G. Marino, P. Segovia, A. V. Krasavin, P. Ginzburg, N. Olivier, G. A. Wurtz, and A. V. Zayats, “Second-harmonic generation from hyperbolic plasmonic nanorod metamaterial slab,” Laser Photon. Rev. 12, 1700189 (2018).

[Crossref]

P. Ginzburg, D. Roth, M. E. Nasir, P. Segovia, A. V. Krasavin, J. Levitt, L. M. Hirvonen, B. Wells, K. Suhling, D. Richards, V. A. Podolskiy, and A. V. Zayats, “Spontaneous emission in non-local materials,” Light Sci. Appl. 6, e16273 (2017).

[Crossref]

P. Segovia, G. Marino, A. V. Krasavin, N. Olivier, G. A. Wurtz, P. A. Belov, P. Ginzburg, and A. V. Zayats, “Hyperbolic metamaterial antenna for second-harmonic generation tomography,” Opt. Express 23, 30730–30738 (2015).

[Crossref]

Y. Shen, The Principles of Nonlinear Optics (Wiley, 1984).

T. Stefaniuk, N. Olivier, A. Belardini, C. P. T. McPolin, C. Sibilia, A. A. Wronkowska, A. Wronkowski, T. Szoplik, and A. V. Zayats, “Self-assembled silver-germanium nanolayer metamaterial with the enhanced nonlinear response,” Adv. Opt. Mater. 5, 1700753 (2017).

[Crossref]

T. Stefaniuk, N. Olivier, A. Belardini, C. P. T. McPolin, C. Sibilia, A. A. Wronkowska, A. Wronkowski, T. Szoplik, and A. V. Zayats, “Self-assembled silver-germanium nanolayer metamaterial with the enhanced nonlinear response,” Adv. Opt. Mater. 5, 1700753 (2017).

[Crossref]

D. Stroud and P. M. Hui, “Nonlinear susceptibilities of granular matter,” Phys. Rev. B 37, 8719–8724 (1988).

[Crossref]

P. Ginzburg, D. Roth, M. E. Nasir, P. Segovia, A. V. Krasavin, J. Levitt, L. M. Hirvonen, B. Wells, K. Suhling, D. Richards, V. A. Podolskiy, and A. V. Zayats, “Spontaneous emission in non-local materials,” Light Sci. Appl. 6, e16273 (2017).

[Crossref]

T. Stefaniuk, N. Olivier, A. Belardini, C. P. T. McPolin, C. Sibilia, A. A. Wronkowska, A. Wronkowski, T. Szoplik, and A. V. Zayats, “Self-assembled silver-germanium nanolayer metamaterial with the enhanced nonlinear response,” Adv. Opt. Mater. 5, 1700753 (2017).

[Crossref]

M. E. Nasir, S. Peruch, N. Vasilantonakis, W. P. Wardley, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Tuning the effective plasma frequency of nanorod metamaterials from visible to telecom wavelengths,” Appl. Phys. Lett. 107, 121110 (2015).

[Crossref]

N. Vasilantonakis, M. E. Nasir, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Bulk plasmon-polaritons in hyperbolic nanorod metamaterial waveguides,” Laser Photon. Rev. 9, 345–353 (2015).

[Crossref]

M. Scalora, M. A. Vincenti, D. de Ceglia, and J. W. Haus, “Nonlocal and quantum-tunneling contributions to harmonic generation in nanostructures: electron-cloud-screening effects,” Phys. Rev. A 90, 013831 (2014).

[Crossref]

D. de Ceglia, S. Campione, M. A. Vincenti, F. Capolino, and M. Scalora, “Low-damping epsilon-near-zero slabs: nonlinear and nonlocal optical properties,” Phys. Rev. B 87, 155140 (2013).

[Crossref]

M. A. Vincenti, D. de Ceglia, and M. Scalora, “Nonlinear dynamics in low permittivity media: the impact of losses,” Opt. Express 21, 29949–29954 (2013).

[Crossref]

M. E. Nasir, S. Peruch, N. Vasilantonakis, W. P. Wardley, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Tuning the effective plasma frequency of nanorod metamaterials from visible to telecom wavelengths,” Appl. Phys. Lett. 107, 121110 (2015).

[Crossref]

C. Kern, M. Kadic, and M. Wegener, “Experimental evidence for sign reversal of the hall coefficient in three-dimensional metamaterials,” Phys. Rev. Lett. 118, 016601 (2017).

[Crossref]

P. Ginzburg, D. Roth, M. E. Nasir, P. Segovia, A. V. Krasavin, J. Levitt, L. M. Hirvonen, B. Wells, K. Suhling, D. Richards, V. A. Podolskiy, and A. V. Zayats, “Spontaneous emission in non-local materials,” Light Sci. Appl. 6, e16273 (2017).

[Crossref]

B. M. Wells, W. Guo, and V. A. Podolskiy, “Homogenization of nanowire-based composites with anisotropic unit-cell and layered substructure,” MRS Commun. 6, 23–29 (2016).

[Crossref]

B. M. Wells, A. V. Zayats, and V. A. Podolskiy, “Nonlocal optics of plasmonic nanowire metamaterials,” Phys. Rev. B 89, 035111 (2014).

[Crossref]

T. Stefaniuk, N. Olivier, A. Belardini, C. P. T. McPolin, C. Sibilia, A. A. Wronkowska, A. Wronkowski, T. Szoplik, and A. V. Zayats, “Self-assembled silver-germanium nanolayer metamaterial with the enhanced nonlinear response,” Adv. Opt. Mater. 5, 1700753 (2017).

[Crossref]

T. Stefaniuk, N. Olivier, A. Belardini, C. P. T. McPolin, C. Sibilia, A. A. Wronkowska, A. Wronkowski, T. Szoplik, and A. V. Zayats, “Self-assembled silver-germanium nanolayer metamaterial with the enhanced nonlinear response,” Adv. Opt. Mater. 5, 1700753 (2017).

[Crossref]

G. Marino, P. Segovia, A. V. Krasavin, P. Ginzburg, N. Olivier, G. A. Wurtz, and A. V. Zayats, “Second-harmonic generation from hyperbolic plasmonic nanorod metamaterial slab,” Laser Photon. Rev. 12, 1700189 (2018).

[Crossref]

L. Nicholls, F. J. Rodríguez-Fortuño, M. E. Nasir, R. M. Cordova-Castro, N. Olivier, G. A. Wurtz, and A. V. Zayats, “Ultrafast synthesis and switching of light polarization in nonlinear anisotropic metamaterials,” Nat. Photonics 11, 628–633 (2017).

[Crossref]

A. Neira, N. Olivier, M. E. Nasir, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Eliminating material constraints for nonlinearity with plasmonic metamaterials,” Nat. Commun. 6, 7757 (2015).

[Crossref]

N. Vasilantonakis, M. E. Nasir, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Bulk plasmon-polaritons in hyperbolic nanorod metamaterial waveguides,” Laser Photon. Rev. 9, 345–353 (2015).

[Crossref]

M. E. Nasir, S. Peruch, N. Vasilantonakis, W. P. Wardley, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Tuning the effective plasma frequency of nanorod metamaterials from visible to telecom wavelengths,” Appl. Phys. Lett. 107, 121110 (2015).

[Crossref]

P. Segovia, G. Marino, A. V. Krasavin, N. Olivier, G. A. Wurtz, P. A. Belov, P. Ginzburg, and A. V. Zayats, “Hyperbolic metamaterial antenna for second-harmonic generation tomography,” Opt. Express 23, 30730–30738 (2015).

[Crossref]

R. Pollard, A. Murphy, W. Hendren, P. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett. 102, 127405 (2009).

[Crossref]

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

[Crossref]

A. V. Krasavin, M. E. Nasir, W. Dickson, and A. V. Zayats, “Reactive tunnel junctions in electrically-driven plasmonic nanorod metamaterials,” Nat. Nanotechnol. 13, 159–164 (2018).

[Crossref]

G. Marino, P. Segovia, A. V. Krasavin, P. Ginzburg, N. Olivier, G. A. Wurtz, and A. V. Zayats, “Second-harmonic generation from hyperbolic plasmonic nanorod metamaterial slab,” Laser Photon. Rev. 12, 1700189 (2018).

[Crossref]

T. Stefaniuk, N. Olivier, A. Belardini, C. P. T. McPolin, C. Sibilia, A. A. Wronkowska, A. Wronkowski, T. Szoplik, and A. V. Zayats, “Self-assembled silver-germanium nanolayer metamaterial with the enhanced nonlinear response,” Adv. Opt. Mater. 5, 1700753 (2017).

[Crossref]

A. V. Krasavin, P. Ginzburg, and A. V. Zayats, “Free-electron optical nonlinearities in plasmonic nanostructures: a review of the hydrodynamic description,” Laser Photon. Rev. 12, 1700082 (2017).

[Crossref]

P. Ginzburg, D. Roth, M. E. Nasir, P. Segovia, A. V. Krasavin, J. Levitt, L. M. Hirvonen, B. Wells, K. Suhling, D. Richards, V. A. Podolskiy, and A. V. Zayats, “Spontaneous emission in non-local materials,” Light Sci. Appl. 6, e16273 (2017).

[Crossref]

L. Nicholls, F. J. Rodríguez-Fortuño, M. E. Nasir, R. M. Cordova-Castro, N. Olivier, G. A. Wurtz, and A. V. Zayats, “Ultrafast synthesis and switching of light polarization in nonlinear anisotropic metamaterials,” Nat. Photonics 11, 628–633 (2017).

[Crossref]

L. Carletti, D. Rocco, A. Locatelli, C. D. Angelis, V. F. Gili, M. Ravaro, I. Favero, G. Leo, M. Finazzi, L. Ghirardini, M. Celebrano, G. Marino, and A. V. Zayats, “Controlling second-harmonic generation at the nanoscale with monolithic AlGaAs-on-AlOx antennas,” Nanotechnology 28, 114005 (2017).

[Crossref]

A. Neira, N. Olivier, M. E. Nasir, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Eliminating material constraints for nonlinearity with plasmonic metamaterials,” Nat. Commun. 6, 7757 (2015).

[Crossref]

N. Vasilantonakis, M. E. Nasir, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Bulk plasmon-polaritons in hyperbolic nanorod metamaterial waveguides,” Laser Photon. Rev. 9, 345–353 (2015).

[Crossref]

M. E. Nasir, S. Peruch, N. Vasilantonakis, W. P. Wardley, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Tuning the effective plasma frequency of nanorod metamaterials from visible to telecom wavelengths,” Appl. Phys. Lett. 107, 121110 (2015).

[Crossref]

P. Segovia, G. Marino, A. V. Krasavin, N. Olivier, G. A. Wurtz, P. A. Belov, P. Ginzburg, and A. V. Zayats, “Hyperbolic metamaterial antenna for second-harmonic generation tomography,” Opt. Express 23, 30730–30738 (2015).

[Crossref]

B. M. Wells, A. V. Zayats, and V. A. Podolskiy, “Nonlocal optics of plasmonic nanowire metamaterials,” Phys. Rev. B 89, 035111 (2014).

[Crossref]

R. Pollard, A. Murphy, W. Hendren, P. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett. 102, 127405 (2009).

[Crossref]

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

[Crossref]

T. Stefaniuk, N. Olivier, A. Belardini, C. P. T. McPolin, C. Sibilia, A. A. Wronkowska, A. Wronkowski, T. Szoplik, and A. V. Zayats, “Self-assembled silver-germanium nanolayer metamaterial with the enhanced nonlinear response,” Adv. Opt. Mater. 5, 1700753 (2017).

[Crossref]

M. E. Nasir, S. Peruch, N. Vasilantonakis, W. P. Wardley, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Tuning the effective plasma frequency of nanorod metamaterials from visible to telecom wavelengths,” Appl. Phys. Lett. 107, 121110 (2015).

[Crossref]

J. W. Haus, N. Kalyaniwalla, R. Inguva, M. Bloemer, and C. M. Bowden, “Nonlinear-optical properties of conductive spheroidal particle composites,” J. Opt. Soc. Am. B 6, 797–807 (1989).

[Crossref]

M. Fiebig, V. V. Pavlov, and R. V. Pisarev, “Second-harmonic generation as a tool for studying electronic and magnetic structures of crystals: review,” J. Opt. Soc. Am. B 22, 96–118 (2005).

[Crossref]

R. Wangberg, J. Elser, E. E. Narimanov, and V. A. Podolskiy, “Nonmagnetic nanocomposites for optical and infrared negative-refractive-index media,” J. Opt. Soc. Am. B 23, 498–505 (2006).

[Crossref]

N. Vasilantonakis, M. E. Nasir, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Bulk plasmon-polaritons in hyperbolic nanorod metamaterial waveguides,” Laser Photon. Rev. 9, 345–353 (2015).

[Crossref]

P. Campagnola and C.-Y. Dong, “Second harmonic generation microscopy: principles and applications to disease diagnosis,” Laser Photon. Rev. 5, 13–26 (2011).

[Crossref]

G. Marino, P. Segovia, A. V. Krasavin, P. Ginzburg, N. Olivier, G. A. Wurtz, and A. V. Zayats, “Second-harmonic generation from hyperbolic plasmonic nanorod metamaterial slab,” Laser Photon. Rev. 12, 1700189 (2018).

[Crossref]

A. V. Krasavin, P. Ginzburg, and A. V. Zayats, “Free-electron optical nonlinearities in plasmonic nanostructures: a review of the hydrodynamic description,” Laser Photon. Rev. 12, 1700082 (2017).

[Crossref]

P. Ginzburg, D. Roth, M. E. Nasir, P. Segovia, A. V. Krasavin, J. Levitt, L. M. Hirvonen, B. Wells, K. Suhling, D. Richards, V. A. Podolskiy, and A. V. Zayats, “Spontaneous emission in non-local materials,” Light Sci. Appl. 6, e16273 (2017).

[Crossref]

B. M. Wells, W. Guo, and V. A. Podolskiy, “Homogenization of nanowire-based composites with anisotropic unit-cell and layered substructure,” MRS Commun. 6, 23–29 (2016).

[Crossref]

L. Carletti, D. Rocco, A. Locatelli, C. D. Angelis, V. F. Gili, M. Ravaro, I. Favero, G. Leo, M. Finazzi, L. Ghirardini, M. Celebrano, G. Marino, and A. V. Zayats, “Controlling second-harmonic generation at the nanoscale with monolithic AlGaAs-on-AlOx antennas,” Nanotechnology 28, 114005 (2017).

[Crossref]

A. Neira, N. Olivier, M. E. Nasir, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Eliminating material constraints for nonlinearity with plasmonic metamaterials,” Nat. Commun. 6, 7757 (2015).

[Crossref]

A. V. Krasavin, M. E. Nasir, W. Dickson, and A. V. Zayats, “Reactive tunnel junctions in electrically-driven plasmonic nanorod metamaterials,” Nat. Nanotechnol. 13, 159–164 (2018).

[Crossref]

L. Nicholls, F. J. Rodríguez-Fortuño, M. E. Nasir, R. M. Cordova-Castro, N. Olivier, G. A. Wurtz, and A. V. Zayats, “Ultrafast synthesis and switching of light polarization in nonlinear anisotropic metamaterials,” Nat. Photonics 11, 628–633 (2017).

[Crossref]

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

[Crossref]

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7, 948–957 (2013).

[Crossref]

P. Segovia, G. Marino, A. V. Krasavin, N. Olivier, G. A. Wurtz, P. A. Belov, P. Ginzburg, and A. V. Zayats, “Hyperbolic metamaterial antenna for second-harmonic generation tomography,” Opt. Express 23, 30730–30738 (2015).

[Crossref]

M. A. Vincenti, D. de Ceglia, and M. Scalora, “Nonlinear dynamics in low permittivity media: the impact of losses,” Opt. Express 21, 29949–29954 (2013).

[Crossref]

M. Scalora, M. A. Vincenti, D. de Ceglia, and J. W. Haus, “Nonlocal and quantum-tunneling contributions to harmonic generation in nanostructures: electron-cloud-screening effects,” Phys. Rev. A 90, 013831 (2014).

[Crossref]

D. Stroud and P. M. Hui, “Nonlinear susceptibilities of granular matter,” Phys. Rev. B 37, 8719–8724 (1988).

[Crossref]

D. J. Bergman, “Nonlinear behavior and 1/f noise near a conductivity threshold: effects of local microgeometry,” Phys. Rev. B 39, 4598–4609 (1989).

[Crossref]

D. de Ceglia, S. Campione, M. A. Vincenti, F. Capolino, and M. Scalora, “Low-damping epsilon-near-zero slabs: nonlinear and nonlocal optical properties,” Phys. Rev. B 87, 155140 (2013).

[Crossref]

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

[Crossref]

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

[Crossref]

B. M. Wells, A. V. Zayats, and V. A. Podolskiy, “Nonlocal optics of plasmonic nanowire metamaterials,” Phys. Rev. B 89, 035111 (2014).

[Crossref]

R. Pollard, A. Murphy, W. Hendren, P. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett. 102, 127405 (2009).

[Crossref]

N. N. Lepeshkin, A. Schweinsberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced nonlinear optical response of one-dimensional metal-dielectric photonic crystals,” Phys. Rev. Lett. 93, 123902 (2004).

[Crossref]

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

[Crossref]

C. Kern, M. Kadic, and M. Wegener, “Experimental evidence for sign reversal of the hall coefficient in three-dimensional metamaterials,” Phys. Rev. Lett. 118, 016601 (2017).

[Crossref]

N. Engheta, “Circuits with light at nanoscales: optical nanocircuits inspired by metamaterials,” Science 317, 1698–1702 (2007).

[Crossref]

V. Agranovich and V. Kravtsov, “Notes on crystal optics of superlattices,” Solid State Commun. 55, 85–90 (1985).

[Crossref]

Note that these nonlocal corrections originate from the composite nature of the plasmonic metamaterial and not from the optical response of its components, which is assumed to be described by local εAu and εh.

The commercial software (COMSOL, www.comsol.com) implements a model of a periodic Au nanorod array with εAu given by Drude model, εAl2O3≈2.74, and geometrical parameters (r, a) deduced from the structures used in the experiments.

We assume that the fields propagate in xz plane; s-polarized light has non-zero components of Ey,Hx,Hz, while the p-polarized light has components of Ex,Ez,Hy fields. We limit our study to the regime when the metamaterial is excited by a single electromagnetic wave that is either p or s polarized. The detailed investigation of more complicated excitation geometries and the analysis of tensorial properties of χα;βγ(2) will be the subject of future work.

G. W. Milton, The Theory of Composites, 1st ed. (Cambridge University, 2002).

M. A. Noginov and V. A. Podolskiy, eds., Tutorials in Metamaterials (CRC Press, 2012).

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

Y. Shen, The Principles of Nonlinear Optics (Wiley, 1984).