The equivalent medium theory within the linear response region
In this paper, we present the equivalent medium theory by using the linear response theory. It is found that, under the condition of the linear response, a series of different media with different refractive indices $n_{i}(ω)$ and lengths $d_{i}$ can be equivalent to an effective medium with the volume-averaged refractive index $\frac{1}{D}\sum_{i=1}^{N}n_{i}(ω)d_{i}$ and the total length $D=\sum_{i=1i}^{N}d_{i}$,where $N$ is the number of different media. Based on this equivalent theory, it is a simple but very useful method to design the effective medium with any desirable dispersion properties. As an example, we present a proposal to obtain the enhancement or reduction of the refractive index without absorption and the large dispersion without obvious absorption by assembling different linear dispersive gain and absorptive media.