An unpolarized light beam of intensity \(I_0\) travels along the \(z\)-axis and enters a liquid crystal display (LCD) pixel assembly. The assembly consists of a linear polarizer with a vertical transmission axis, a liquid crystal layer, and a second linear polarizer (analyzer) with a horizontal transmission axis. In the absence of an electric field, the liquid crystal layer rotates the plane of polarization of the light passing through it by \(60^\circ\) relative to the vertical. When a control electric field is applied, the liquid crystal molecules reorient such that the layer rotates the plane of polarization by only \(45^\circ\) relative to the vertical. What is the ratio of the transmitted light intensity with the electric field applied (\(I_{\text{on}}\)) to the transmitted light intensity without the electric field (\(I_{\text{off}}\))?