The thickness \(d\) needed to introduce a phase difference \(\Delta \phi = \pi/2\) in a quarter-wave plate is given by:
\[d = \frac{\lambda}{4 \times |n_e - n_o|}\]
where \(n_e\) and \(n_o\) are the refractive indices for the extraordinary and ordinary rays, respectively. Substituting the given values:
\[d = \frac{5893 \times 10^{-10} \, \text{m}}{4 \times |1.65836 - 1.48641|} \approx 8.57 \times 10^{-6} \, \text{m} = 8.57 \times 10^{-4} \, \text{mm}\]