We present a study of the second-order nonlinear optical properties of metal-based metamaterials. A hydrodynamic model for electronic response is used, in which nonlinear surface contributions are expressed in terms of the bulk polarization. The model is in good agreement with published experimental results, and clarifies the mechanisms contributing to the nonlinear response. In particular, we show that the reported enhancement of second-harmonic in split-ring resonator based media is driven by the electric rather than the magnetic properties of the structure.PACS numbers: 42.65. Ky, 81.05.Xj, 78.67.Pt, 73.20.Mf Metamaterials (MMs) are artificially structured media whose collective electromagnetic properties derive from the geometry of sub-wavelength inclusions. To date, the most common MM designs have made use of inclusions formed by conducting materials that function as subwavelength electrical circuits. These conductor based MMs have proven adept at mimicking a wide variety of linear electromagnetic responses, providing a new venue to explore otherwise inaccessible concepts 1 . In the context of nonlinear response, however, artificial materials may offer even greater opportunities, due to the inherently inhomogeneous local field distribution that exists within and around MM inclusions. By carefully structuring the inclusion geometry, extremely large field enhancement regions can be produced that can dominate and enhance the effective nonlinear response of the composite.The enhancement of nonlinear processes by MMs has been demonstrated at radio and microwave frequencies, using packaged components, such as varactor diodes, to introduce nonlinearity into the gaps of metal MM inclusions 2 . However, to achieve nonlinear optical materials at higher wavelengths, a simple scaling of these prototype structures to higher frequencies (e.g., beyond a few terahertz) will not suffice. First, the response of most metals changes from conductor-like to dielectric-like at frequencies above a few terahertz, with absorption increasing significantly as the fields are able to penetrate further into the metal. Second, packaged semiconductor components are not readily available at frequencies above 100 GHz.While metals and conductors may possess undesirable properties at optical wavelengths, such as increased absorption, they also possess unique and potentially advantageous properties. In addition to large field enhancements, metal nanostructures also support intrinsic nonlinearities that relate to the dynamics of free and bound charge carriers. As a result, metals possess some of the largest nonlinear susceptibilities known. Examples include the large χ (3) values of gold or silver, for example, suggesting that metals can serve both to form the linear MM response by tailoring the structure, while serving as the source of nonlinearity for nonlinear optical MMs.The second-order nonlinearity in metals arises from both volume and surface contributions. Nonlinear surface contributions are strictly related to the response of the elec...

We present a numerical investigation of the second-order nonlinear optical properties of metal-based metamaterial nanoresonators. The nonlinear optical response of the metal is described by a hydrodynamic model, with the effects of electron pressure in the electron gas also taken into account. We show that as the pressure term tends to zero the amount of converted second-harmonic field tends to an asymptotic value. In this limit it becomes possible to rewrite the nonlinear surface contributions as functions of the value of the polarization vector inside the bulk region. Nonlocality thus can be incorporated into numerical simulations without actually utilizing the nonlocal equation of motion or solving for the rapidly varying fields that occur near the metal surface. We use our model to investigate the second-harmonic generation process with three-dimensional gold nanoparticle arrays and show that nanocrescents can easily attain conversion efficiencies of ∼6.0 × 10 −8 for pumping peak intensities of a few tens of MW/cm 2 .