Question #43727

In a Young's Double Slit Experiment, a transparent cell of length $L$ is placed immediately in front of the upper slit $S_1$. The cell is filled with a polarizable liquid whose refractive index depends on the local electric field intensity $E$ according to the Kerr effect: $\mu(x) = \mu_0 + \beta E^2(x)$, where $\mu_0$ and $\beta$ are positive constants. A specialized electrode configuration maintains a non-uniform electric potential $V(x) = -V_0 \left( \frac{x}{L} \right)^2$ along the path of the light inside the cell (from $x=0$ to $x=L$). The lower slit $S_2$ is left clear. If the wavelength of the light in vacuum is $\lambda$, what is the additional phase difference $\Delta \phi$ introduced by the cell at the center of the screen, relative to a vacuum path of the same length?