Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Backward phase-matching for nonlinear optical generation in negative-index materials


Metamaterials have enabled the realization of unconventional electromagnetic properties not found in nature, which provokes us to rethink the established rules of optics in both the linear and nonlinear regimes. One of the most intriguing phenomena in nonlinear metamaterials is ‘backward phase-matching’, which describes counter-propagating fundamental and harmonic waves in a negative-index medium. Predicted nearly a decade ago, this process is still awaiting a definitive experimental confirmation at optical frequencies. Here, we report optical measurements showing backward phase-matching by exploiting two distinct modes in a nonlinear plasmonic waveguide, where the real parts of the mode refractive indices are 3.4 and −3.4 for the fundamental and the harmonic waves respectively. The observed peak conversion efficiency at the excitation wavelength of 780 nm indicates the fulfilment of the phase-matching condition of k2ω = 2kω and n2ω = −nω, where the coherent harmonic wave emerges along a direction opposite to that of the incoming fundamental light.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Prices vary by article type



Prices may be subject to local taxes which are calculated during checkout

Figure 1: Phase-matching conditions for second-harmonic generation in nonlinear optical media.
Figure 2: Operating point for backward phase-matching in a plasmonic waveguide.
Figure 3: Experimental design for backward phase-matching in a waveguide with a dual-layered dielectric core.
Figure 4: Frequency-doubled signals emerging from the plasmonic waveguide.
Figure 5: Backward phase-matched second-harmonic generation.


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

    Google Scholar 

  2. Boyd, R. W. Nonlinear Optics 3rd edn (Academic Press, 2008).

    Google Scholar 

  3. Cai, W. & Shalaev, V. M. Optical Metamaterials: Fundamentals and Applications (Springer, 2010).

    Book  Google Scholar 

  4. Soukoulis, C. M. & Wegener, M. Past achievements and future challenges in the development of three-dimensional photonic metamaterials. Nature Photon. 5, 523–530 (2011).

    Article  CAS  Google Scholar 

  5. Liu, Y. M. & Zhang, X. Metamaterials: A new frontier of science and technology. Chem. Soc. Rev. 40, 2494–2507 (2011).

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  7. Shalaev, V. M. Optical negative-index metamaterials. Nature Photon. 1, 41–48 (2007).

    Article  CAS  Google Scholar 

  8. Hess, O. et al. Active nanoplasmonic metamaterials. Nature Mater. 11, 573–584 (2012).

    Article  CAS  Google Scholar 

  9. Zheludev, N. I. & Kivshar, Y. S. From metamaterials to metadevices. Nature Mater. 11, 917–924 (2012).

    Article  CAS  Google Scholar 

  10. Meinzer, N., Barnes, W. L. & Hooper, I. R. Plasmonic meta-atoms and metasurfaces. Nature Photon. 8, 889–898 (2014).

    Article  CAS  Google Scholar 

  11. Schuller, J. A. et al. Plasmonics for extreme light concentration and manipulation. Nature Mater. 9, 193–204 (2010).

    Article  CAS  Google Scholar 

  12. Lapine, M., Shadrivov, I. V. & Kivshar, Y. S. Colloquium: Nonlinear metamaterials. Rev. Mod. Phys. 86, 1093–1123 (2014).

    Article  CAS  Google Scholar 

  13. Kauranen, M. & Zayats, A. V. Nonlinear plasmonics. Nature Photon. 6, 737–748 (2012).

    Article  CAS  Google Scholar 

  14. Klein, M. W., Enkrich, C., Wegener, M. & Linden, S. Second-harmonic generation from magnetic metamaterials. Science 313, 502–504 (2006).

    Article  CAS  Google Scholar 

  15. Kim, E., Wang, F., Wu, W., Yu, Z. N. & Shen, Y. R. Nonlinear optical spectroscopy of photonic metamaterials. Phys. Rev. B 78, 113102 (2008).

    Article  Google Scholar 

  16. Wurtz, G. A. et al. Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality. Nature Nanotech. 6, 106–110 (2011).

    Article  Google Scholar 

  17. Linden, S. et al. Collective effects in second-harmonic generation from split-ring-resonator arrays. Phys. Rev. Lett. 109, 015502 (2012).

    Article  CAS  Google Scholar 

  18. Reinhold, J. et al. Contribution of the magnetic resonance to the third harmonic generation from a fishnet metamaterial. Phys. Rev. B 86, 115401 (2012).

    Article  Google Scholar 

  19. Suchowski, H. et al. Phase mismatch-free nonlinear propagation in optical zero-index materials. Science 342, 1223–1226 (2013).

    Article  CAS  Google Scholar 

  20. Agranovich, V. M., Shen, Y. R., Baughman, R. H. & Zakhidov, A. A. Linear and nonlinear wave propagation in negative refraction metamaterials. Phys. Rev. B 69, 165112 (2004).

    Article  Google Scholar 

  21. Popov, A. K. & Shalaev, V. M. Negative-index metamaterials: Second-harmonic generation, Manley–Rowe relations and parametric amplification. Appl. Phys. B 84, 131–137 (2006).

    Article  CAS  Google Scholar 

  22. Shadrivov, I. V., Zharov, A. A. & Kivshar, Y. S. Second-harmonic generation in nonlinear left-handed metamaterials. J. Opt. Soc. Am. B 23, 529–534 (2006).

    Article  CAS  Google Scholar 

  23. Rose, A., Huang, D. & Smith, D. R. Controlling the second harmonic in a phase-matched negative-index metamaterial. Phys. Rev. Lett. 107, 063902 (2011).

    Article  Google Scholar 

  24. Gu, X. H., Korotkov, R. Y., Ding, Y. J. J., Kang, J. U. & Khurgin, J. B. Backward second-harmonic generation in periodically poled lithium niobate. J. Opt. Soc. Am. B 15, 1561–1566 (1998).

    Article  CAS  Google Scholar 

  25. Anderson, D. B. & Boyd, J. T. Wideband CO2 laser second harmonic generation phase matched in GaAs thin-film waveguides. Appl. Phys. Lett. 19, 266–268 (1971).

    Article  CAS  Google Scholar 

  26. Yariv, A. Coupled-mode theory for guided-wave optics. IEEE J. Quantum Electron. 9, 919–933 (1973).

    Article  CAS  Google Scholar 

  27. Lezec, H. J., Dionne, J. A. & Atwater, H. A. Negative refraction at visible frequencies. Science 316, 430–432 (2007).

    Article  CAS  Google Scholar 

  28. Shin, H. & Fan, S. H. All-angle negative refraction for surface plasmon waves using a metal–dielectric–metal structure. Phys. Rev. Lett. 96, 073907 (2006).

    Article  Google Scholar 

  29. Alu, A. & Engheta, N. Optical nanotransmission lines: Synthesis of planar left-handed metamaterials in the infrared and visible regimes. J. Opt. Soc. Am. B 23, 571–583 (2006).

    Article  CAS  Google Scholar 

  30. Dionne, J. A., Verhagen, E., Polman, A. & Atwater, H. A. Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries. Opt. Express 16, 19001–19017 (2008).

    Article  CAS  Google Scholar 

  31. Feigenbaum, E., Kaminski, N. & Orenstein, M. Negative dispersion: A backward wave or fast light? Nanoplasmonic examples. Opt. Express 17, 18934–18939 (2009).

    Article  CAS  Google Scholar 

  32. Yang, T. & Crozier, K. B. Analysis of surface plasmon waves in metal–dielectric–metal structures and the criterion for negative refractive index. Opt. Express 17, 1136–1143 (2009).

    Article  CAS  Google Scholar 

  33. Verhagen, E., de Waele, R., Kuipers, L. & Polman, A. Three-dimensional negative index of refraction at optical frequencies by coupling plasmonic waveguides. Phys. Rev. Lett. 105, 223901 (2010).

    Article  Google Scholar 

  34. Davoyan, A. R., Shadrivov, I. V. & Kivshar, Y. S. Quadratic phase matching in nonlinear plasmonic nanoscale waveguides. Opt. Express 17, 20063–20068 (2009).

    Article  CAS  Google Scholar 

  35. Stegeman, G. I. & Seaton, C. T. Nonlinear integrated-optics. J. Appl. Phys. 58, R57–R78 (1985).

    Article  CAS  Google Scholar 

  36. Terhune, R. W., Maker, P. D. & Savage, C. M. Optical harmonic generation in calcite. Phys. Rev. Lett. 8, 404–406 (1962).

    Article  CAS  Google Scholar 

  37. Cai, W., Vasudev, A. P. & Brongersma, M. L. Electrically controlled nonlinear generation of light with plasmonics. Science 333, 1720–1723 (2011).

    Article  CAS  Google Scholar 

  38. Kang, L. et al. Electrifying photonic metamaterials for tunable nonlinear optics. Nature Commun. 5, 4680 (2014).

    Article  CAS  Google Scholar 

  39. Ito, H. & Inaba, H. Efficient phase-matched 2nd-harmonic generation method in 4-layered optical-waveguide structure. Opt. Lett. 2, 139–141 (1978).

    Article  CAS  Google Scholar 

Download references


This work was performed in part at the Georgia Tech Institute for Electronics and Nanotechnology, a member of the National Nanotechnology Infrastructure Network, which is supported by the National Science Foundation. W.C. acknowledges the start-up fund from the Georgia Institute of Technology and the generous gift by OPE LLC in support of the scientific research in the Cai Lab. S.P.R. acknowledges the support of the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1148903. M.L.B. acknowledges support from the AFOSR MURI on Integrated Hybrid Nanophotonic Circuits, Grant FA9550-12-1-0024.

Author information

Authors and Affiliations



W.C. and M.L.B. conceived the idea and designed the experiment. S.L., D.T.S. and Y.C. fabricated the sample. S.L., L.K. and S.P.R. carried out measurements. All authors contributed to the interpretation of results and participated in the preparation of manuscript.

Corresponding author

Correspondence to Wenshan Cai.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Information

Supplementary Information (PDF 827 kb)

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lan, S., Kang, L., Schoen, D. et al. Backward phase-matching for nonlinear optical generation in negative-index materials. Nature Mater 14, 807–811 (2015).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing