Furthermore, we will see that the condition of simultaneous negative ε and negative μ is neither su¥cient nor necessary (in the usual mathematical sense) for obtaining a negative phase velocity of light, c c n= 0 / , in general, e˜ective media ( /c0 0 01= ε µ is the vacuum speed of light and n is the refractive index of the medium). In chiral metamaterials, a negative phase velocity of light can, in principle, even result if electric permittivity and magnetic permeability are positive at the same time (“not necessary”). Intuitively, the eigen-polarizations for chiral media are le¹-handed circular polarization (LCP) and right-handed circular polarization (RCP) of light. Œe stronger the e˜ects of chirality, the larger the di˜erence between the refractive indices for LCP and RCP, n n n= −

− + . Œeir mean value, n, is žxed by the square root of the product of the electric permittivity and the magnetic permeability. Œus, with increasing di˜erence n n

− +− , one of the two refractive indices eventually becomes negative. In natural systems such as a solution of chiral sugar molecules, the di˜erence | |n n

− +− is on the order of 10−4 and, hence, much smaller than their mean value of order one. In contrast, in metamaterials, the index difference can approach or even exceed unity within a certain frequency range.