Linear and nonlinear wave propagation in negative refraction metamaterials

We discuss linear and nonlinear optical wave propagation in a left-handed medium (LHM) or medium of negative refraction (NRM). We use the approach of characterizing the medium response totally by a generalized electric polarization [with a dielectric permittivity $\tilde{\varepsilon }(\omega ,\overrightarrow{k})]$ that can be decomposed into curl and noncurl parts. The description has a one-to-one correspondence with the usual approach characterizing the LHM response with a dielectric permittivity $\varepsilon <0\mathrm{}$ and a magnetic permeability $\mu <0.\mathrm{}$ The latter approach is less physically transparent in the optical frequency region because the usual definition of magnetization loses its physical meaning. Linear wave propagation in a LHM or NRM is characterized by negative refraction and negative group velocity that could be clearly manifested by ultrashort pulse propagation in such a medium. Nonlinear optical effects in a LHM can be predicted from the same calculations adopted for ordinary media using our general approach.