  There is a growing interest in investigating modified theories of gravity,
primarily, with the aim of explaining the universe's accelerated expansion,
which has been confirmed by several independent observations. Compact objects,
like neutron stars, exhibit strong gravity effects and therefore are used to
study modified gravity theories. We use the $f(R)=R+aR^2$ model, where R is the
Ricci scalar and $a$ is a free parameter. This model has been studied both
perturbatively and non-perturbatively. However, it was found that perturbative
methods results in nonphysical solutions for the neutron stars. In this paper,
we examine neutron star properties, such as mass, radius, tidal deformability
in non-perturbative $f(R)$ gravity model with density dependant relativistic
equation of state with different particle compositions. The strange particles
in the core of NS in the form of ${\bf \Lambda}$ hyperons, $K^-$ condensate,
and quarks are considered. We have observed that while the mass-radius relation
allows for a wide range of parameter $a$, when tidal deformability is
considered, the parameter $a$ is constrained down by one order.
