Answer
When they emerge, the phase difference is $~~0.833$
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 after traveling a distance of $L_1$:
$\Delta N = (\frac{L_2}{\lambda/n_2}+\frac{L_1-L_2}{\lambda/1.00})-\frac{L_1}{\lambda/n_1}$
$\Delta N = (\frac{n_2~L_2}{\lambda}+\frac{L_1-L_2}{\lambda})-\frac{n_1~L_1}{\lambda}$
$\Delta N = (\frac{(1.60)(3.50~\mu m)}{600.0~nm}+\frac{0.50~\mu m}{600.0~nm})-\frac{(1.40)(4.00~\mu m)}{600.0~nm}$
$\Delta N = 10.1666- 9.3333$
$\Delta N = 0.833$
When they emerge, the phase difference is $~~0.833~~$ as a multiple of $\lambda$
