  In this paper, we first study the Lifshitz-dilaton holographic
superconductors with nonlinear Born-Infeld (BI) gauge field and obtain the
critical temperature of the system for different values of Lifshitz dynamical
exponent, $z$, and nonlinear parameter $b$. We find that for fixed value of
$b$, the critical temperature decreases with increasing $z$. This indicates
that the increase of anisotropy between space and time prevents the phase
transition. Also, for fixed value of $z$, the critical temperature decrease
with increasing $b$. Then, we investigate the optical properties of ($2+1$) and
($3+1$)-dimensional BI-Lifshitz holographic superconductors in the the presence
of dilaton field. We explore the refractive index of the system. For $z=1$ and
$(2+1)$-dimensional holographic superconductor, we observe negative real part
for permittivity $\textrm{Re}[\epsilon]$ as frequency $\omega $ decreases.
Thus, in low frequency region our superconductor exhibit metamaterial property.
This behaviour is independent of the nonlinear parameter and can be seen for
either linear ($b=0$) and nonlinear ($b\neq 0$) electrodynamics. Interestingly,
for ($3+1$)-dimensional Lifshitz-dilaton holographic superconductors, we
observe metamaterial behavior neither in the presence of linear nor nonlinear
electrodynamics.
