$(20$ points$)$ An unpolarized white light illuminates a linear polarizer $(45),$ a birefringence crystal and then a linear polarizer $(-45).$ Assume that we have a birefringent crystal that can be controlled optically. Its positioned at a $45$ angle between two crossed polarizers. The geometrical path length through the crystal is $5\mathrm{.}00$ mm $.$ What value the birefringence $(|$ne$-$no$|)$ makes the transmitted light appear:

a$.$ Yellow $(600nm)$

b$.$ Blue $)$