What is the total optical path length of two media in contact with...

Optical path length is given by the product of the refractive index and the geometric thickness of the medium.

For the first medium, the optical path length is $x_1 = n_1 d$.

For the second medium, the optical path length is $x_2 = n_2 \left(\dfrac{d}{4}\right).<br><br>The total optical path length is the sum of the individual optical path lengths:<br><br>$x = x_1 + x_2$<br><br>$x = n_1 d + $\dfrac{n_2 d}{4}<br><br>$ x = $\dfrac{d(4n_1 + n_2)}{4}<br><br>Answer: \dfrac{d(4n_1 + n_2)}{4}$