Abstract
The discovery of optical second harmonic generation in 1961 started modern nonlinear optics1,2,3. Soon after, R. C. Miller found empirically that the nonlinear susceptibility could be predicted from the linear susceptibilities. This important relation, known as Miller’s Rule4,5, allows a rapid determination of nonlinear susceptibilities from linear properties. In recent years, metamaterials, artificial materials that exhibit intriguing linear optical properties not found in natural materials6, have shown novel nonlinear properties such as phase-mismatch-free nonlinear generation7, new quasi-phase matching capabilities8,9 and large nonlinear susceptibilities8,9,10. However, the understanding of nonlinear metamaterials is still in its infancy, with no general conclusion on the relationship between linear and nonlinear properties. The key question is then whether one can determine the nonlinear behaviour of these artificial materials from their exotic linear behaviour. Here, we show that the nonlinear oscillator model does not apply in general to nonlinear metamaterials. We show, instead, that it is possible to predict the relative nonlinear susceptibility of large classes of metamaterials using a more comprehensive nonlinear scattering theory, which allows efficient design of metamaterials with strong nonlinearity for important applications such as coherent Raman sensing, entangled photon generation and frequency conversion.
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References
Franken, P. A., Hill, A. E., Peters, C. W. & Weinreich, G. Generation of optical harmonics. Phys. Rev. Lett. 7, 118–119 (1961).
Armstrong, J. A., Bloembergen, N., Ducuing, J. & Pershan, P. S. Interactions between light waves in a nonlinear dielectric. Phys. Rev. 127, 1918–1939 (1962).
Bloembergen, N. & Pershan, P. S. Light waves at the boundary of nonlinear media. Phys. Rev. 128, 606–622 (1962).
Garrett, C. & Robinson, F. Miller’s phenomenological rule for computing nonlinear susceptibilities. IEEE J. Quantum Electron. 2, 328–329 (1966).
Miller, R. C. Optical second harmonic generation in piezoelectric crystals. Appl. Phys. Lett. 5, 17–19 (1964).
Shalaev, V. M. Optical negative-index metamaterials. Nature Photon. 1, 41–48 (2007).
Suchowski, H. et al. Phase mismatch-free nonlinear propagation in optical zero-index materials. Science 342, 1223–1226 (2013).
Rose, A., Huang, D. & Smith, D. R. Controlling the second harmonic in a phase-matched negative-index metamaterial. Phys. Rev. Lett. 107, 063902 (2011).
Rose, A., Larouche, S., Poutrina, E. & Smith, D. R. Nonlinear magnetoelectric metamaterials: Analysis and homogenization via a microscopic coupled-mode theory. Phys. Rev. A 86, 033816 (2012).
Sukhorukov, A. A., Solntsev, A. S., Kruk, S. S., Neshev, D. N. & Kivshar, Y. S. Nonlinear coupled-mode theory for periodic plasmonic waveguides and metamaterials with loss and gain. Opt. Lett. 39, 462–465 (2014).
Byer, R. L. Nonlinear optical phenomena and materials. Annu. Rev. Mater. Sci. 4, 147–190 (1974).
Scandolo, S. & Bassani, F. Miller’s rule and the static limit for second-harmonic generation. Phys. Rev. B 51, 6928–6931 (1995).
Bell, M. I. Frequency dependence of Miller’s rule for nonlinear susceptibilities. Phys. Rev. B 6, 516–521 (1972).
Cataliotti, F. S., Fort, C., Hänsch, T. W., Inguscio, M. & Prevedelli, M. Electromagnetically induced transparency in cold free atoms: Test of a sum rule for nonlinear optics. Phys. Rev. A 56, 2221–2224 (1997).
Miles, R. & Harris, S. Optical third-harmonic generation in alkali metal vapors. IEEE J. Quantum Electron. 9, 470–484 (1973).
Matranga, C. & Guyot-Sionnest, P. Absolute intensity measurements of the optical second-harmonic response of metals from 0.9 to 2.5 eV. J. Chem. Phys. 115, 9503–9512 (2001).
Rapapa, N. P. & Scandolo, S. Universal constraints for the third-harmonic generation susceptibility. J. Phys. Condens. Matter 8, 6997–7004 (1996).
Hentschel, M., Utikal, T., Giessen, H. & Lippitz, M. Quantitative modeling of the third harmonic emission spectrum of plasmonic nanoantennas. Nano Lett. 12, 3778–3782 (2012).
Metzger, B., Hentschel, M., Lippitz, M. & Giessen, H. Third-harmonic spectroscopy and modeling of the nonlinear response of plasmonic nanoantennas. Opt. Lett. 37, 4741–4743 (2012).
Niesler, F. B. P., Feth, N., Linden, S. & Wegener, M. Second-harmonic optical spectroscopy on split-ring-resonator arrays. Opt. Lett. 36, 1533–1535 (2011).
Canfield, B. K. et al. Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers. Nano Lett. 7, 1251–1255 (2007).
Husu, H. et al. Metamaterials with tailored nonlinear optical response. Nano Lett. 12, 673–677 (2012).
Dadap, J. I., Shan, J. & Heinz, T. F. Theory of optical second-harmonic generation from a sphere of centrosymmetric material: Small-particle limit. J. Opt. Soc. Am. B 21, 1328–1347 (2004).
Poutrina, E., Huang, D., Urzhumov, Y. & Smith, D. R. Nonlinear oscillator metamaterial model: Numerical and experimental verification. Opt. Express 19, 8312–8319 (2011).
Bassani, F. & Lucarini, V. General properties of optical harmonic generation from a simple oscillator model. Nuovo Cimento D 20, 1117–1125 (1998).
Lippitz, M., van Dijk, M. A. & Orrit, M. Third-harmonic generation from single gold nanoparticles. Nano Lett. 5, 799–802 (2005).
Roke, S., Bonn, M. & Petukhov, A. V. Nonlinear optical scattering: The concept of effective susceptibility. Phys. Rev. B 70, 115106 (2004).
Gentile, M. et al. Investigation of the nonlinear optical properties of metamaterials by second harmonic generation. Appl. Phys. B 105, 149–162 (2011).
Husnik, M. et al. Quantitative experimental determination of scattering and absorption cross-section spectra of individual optical metallic nanoantennas. Phys. Rev. Lett. 109, 233902 (2012).
Castro-Lopez, M., Brinks, D., Sapienza, R. & van Hulst, N. F. Aluminum for nonlinear plasmonics: Resonance-driven polarized luminescence of Al, Ag, and Au nanoantennas. Nano Lett. 11, 4674–4678 (2011).
Acknowledgements
This work was supported by the US Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division under contract no. DE-AC02-05CH11231. J.R. acknowledges a fellowship from the Samsung Scholarship Foundation, Republic of Korea.
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K.O. and H.S. conducted the experiments. K.O. performed the theoretical calculations. J.R. fabricated the samples. K.O., H.S., X.Y. and X.Z. prepared the manuscript. X.Z. guided the research. All authors contributed to discussions.
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O’Brien, K., Suchowski, H., Rho, J. et al. Predicting nonlinear properties of metamaterials from the linear response. Nature Mater 14, 379–383 (2015). https://doi.org/10.1038/nmat4214
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DOI: https://doi.org/10.1038/nmat4214
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