Abstract

Nonlinear processes are at the core of many optical technologies whose further development require optimized materials suitable for nanoscale integration. Here we demonstrate the emergence of a strong bulk second-order nonlinear response in a plasmonic nanorod composite comprised of centrosymmetric materials. We develop an effective-medium description of the underlying physics, compare its predictions to the experimental results, and analyze the limits of its applicability. We demonstrate strong tunable generation of the p-polarized second-harmonic light in response to either s- or p-polarized excitation. High second-harmonic enhancement is observed for fundamental frequencies in the epsilon-near-zero spectral range. The work demonstrates emergence of structurally tunable nonlinear optical response in plasmonic composites and presents a new nonlinear optical platform suitable for integrated nonlinear photonics.

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

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    [Crossref]
  27. 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.
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    [Crossref]
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    [Crossref]
  30. 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]
  31. 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.
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    [Crossref]
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    [Crossref]
  34. 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]
  35. 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]
  36. 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]
  37. 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.
  38. 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]
  39. 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]
  40. 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]
  41. 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]

2018 (2)

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]

2017 (6)

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]

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]

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]

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]

2016 (1)

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]

2015 (4)

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]

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]

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]

2014 (2)

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]

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

2013 (3)

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]

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]

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

2012 (1)

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

2011 (1)

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

2009 (2)

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]

2007 (1)

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

2006 (1)

2005 (1)

2004 (1)

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]

2000 (1)

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

1993 (1)

1989 (2)

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]

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]

1988 (1)

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

1985 (1)

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

1984 (1)

1972 (1)

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

Agranovich, V.

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

Angelis, C. D.

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]

Atkinson, R.

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]

Belardini, A.

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]

Belov, P.

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

Belov, P. A.

Bennink, R. S.

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]

Bergman, D. J.

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]

Bloemer, M.

Bowden, C. M.

Boyd, R. W.

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]

Campagnola, P.

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

Campione, S.

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]

Capolino, F.

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]

Carletti, L.

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]

Celebrano, M.

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]

Christy, R. W.

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

Cordova-Castro, R. M.

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]

de Ceglia, D.

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]

Dickson, W.

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]

Dong, C.-Y.

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

Dumelow, T.

Elser, J.

Engheta, N.

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

Evans, P.

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]

Favero, I.

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]

Fiebig, M.

Finazzi, M.

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]

Flytzanis, C.

Ghirardini, L.

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]

Gili, V. F.

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]

Ginzburg, P.

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]

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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]

Zayats, A.

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

Zayats, A. V.

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).
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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).
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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]

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).
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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]

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]

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[Crossref]

Adv. Opt. Mater. (1)

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]

Appl. Phys. Lett. (1)

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. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (3)

Laser Photon. Rev. (4)

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]

Light Sci. Appl. (1)

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]

MRS Commun. (1)

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]

Nanotechnology (1)

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]

Nat. Commun. (1)

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]

Nat. Nanotechnol. (1)

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]

Nat. Photonics (3)

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]

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Opt. Lett. (1)

Phys. Rev. A (1)

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Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79, 235109 (2009).
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B. M. Wells, A. V. Zayats, and V. A. Podolskiy, “Nonlocal optics of plasmonic nanowire metamaterials,” Phys. Rev. B 89, 035111 (2014).
[Crossref]

Phys. Rev. Lett. (4)

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).
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Other (7)

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).

Supplementary Material (1)

NameDescription
» Supplement 1       Details of numerical calculations and analysis of validity of effective medium theory

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Figures (5)

Fig. 1.
Fig. 1. (a) Schematic geometry of a metamaterial along with orientation of the fields and wavevectors considered in modeling and experiments. (b) SEM image of the nanorods after removal of the AAO matrix. (c), (d) Real (solid lines) and imaginary (dashed lines) parts of the effective permittivity of samples A (c) and B (d); green and yellow areas highlight the spectral ranges of the ENZ behavior for sample A and hyperbolic dispersion for both samples, repectively. (e)–(h) Linear reflection spectra for metamaterials A (e), (g) and B (f), (h): experiment (e), (f) and theoretical modeling (g), (h) using the full-wave finite-element simulations (solid lines) and the effective-medium theory (dashed lines). Angle of incidence in all figures is fixed at 45°.
Fig. 2.
Fig. 2. SHG spectra for different polarization configurations from metamaterial A at an angle of incidence of 45°: (a) experimental spectra normalized with sp SHG from z-quartz at λ0=1300  nm; (b) spectra simulated using the full-wave numerical modeling with the nonlinear polarization described by Eq. (2) (solid lines) and by the simplified model Eq. (3) (dashed lines). (c) Spectral dependence of the non-vanishing components of the effective polarizability matrix. (d) SHG spectra simulated with the nonlinear EMT model [Eqs. (4) and (5)].
Fig. 3.
Fig. 3. Same as Fig. 2 but for sample B.
Fig. 4.
Fig. 4. Spectral and angular dependences of the components of the effective nonlinear polarizability for sample A (a), (b) and sample B (c), (d) for (a), (c) p- and (b), (d) s-polarized fundamental light.
Fig. 5.
Fig. 5. Full-wave numerical modeling of the SHG spectra from low-loss analogs of sample A (a) and sample B (b). The loss is decreased by two times compared to Figs. 2(b) and 3(b).

Equations (5)

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ε=εh(1+p)εAu+(1p)εh(1+p)εh+(1p)εAu,εzz=pεAu+(1p)εh.
P2ω=12ω(2ωiτ){αrα(jωjω;αen0)eme[ε0(·Eω)Eω+jω×Bω]},
P2ω;x(p)=12ω(2ωiτ)(emjω;zBω;y+1ne[jω;zjω;xzjω;xjω;zz]),P2ω;x(s)=12ω(2ωiτ)emjω;yBω;z,P2ω;z(p)=12ω(2ωiτ)(emjω;xBω;y2jω;znejω;zz),P2ω;z(s)=12ω(2ωiτ)emjω;yBω;x.
P2ω;α=β,γ[χα;βγ(2,e)Eω;βEω;γ+χα;βγ(2,m)Eω;βBω;γ],
χx;xx(2,e)=NpLεAukxεεzz,χx;zz(2,e)=Np(pεzzωp2+εzzεAuLω2kxc2εAuLkx),χz;xz(2,e)=Np/2(Lωp2εzzc2kx2pkxεAuεεzz),χz;yx(2,m)=2Lε0εAuemω2ωp2[εAu(2ω)εb],χx;yy(2,e)=χz;yx(2,m)kxω.