A monochromatic light is incident from air on a refracting surface of a prism of angle $75^{\circ}$ and refractive index $n_0=\sqrt{3}$. The other refracting surface of a prism is coated by a thin film of material of refractive index $n$ as shown in figure. The light suffers total internal reflection at the coated prism surface for an incidence angle of $\theta \leq 60^{\circ}$. The value of $\mathrm{n}^2$ is $\_\_\_\_$ .