Answer
When they emerge, the phase difference is $~~1.70$
Work Step by Step
We can write a general expression for the wavelength $\lambda'$ in a material with an index of refraction of $n$:
$\lambda' = \frac{\lambda}{n}$
We can find the difference in the number of cycles $\Delta N$ of each wave in the material:
$\Delta N = \frac{L}{\lambda/n_2}-\frac{L}{\lambda/n_1}$
$\Delta N = \frac{n_2~L}{\lambda}-\frac{n_1~L}{\lambda}$
$\Delta N = \frac{(1.72)(8.50~\mu m)}{500~nm}-\frac{(1.62)(8.50~\mu m)}{500~nm}$
$\Delta N = 1.70$
When they emerge, the phase difference is $~~1.70~~$ as a multiple of $\lambda$
