In a double slit experiment, a plate having thickness t and...

When a light ray passes through a medium of thickness $t$ and refractive index $\mu$, the optical path length is $\mu t$.

In the absence of the plate, the optical path length for the same distance in vacuum (or air) is $t$.

The change in the optical path due to the introduction of the plate is the difference between the optical path with the plate and the optical path without the plate.

Change in optical path $= \mu t - t = (\mu - 1)t$.