Question #43731

An opaque screen is placed in the $x=0$ plane with two narrow parallel slits $S_1$ and $S_2$ located at $y = d$ and $y = 3d$ respectively. The slits are infinitely long along the $z$-axis. An infinitely long straight wire with a uniform linear charge density $\lambda$ is embedded in the screen along the $z$-axis. A transparent electro-optic slab of thickness $W = \sqrt{3}d$ is placed directly in front of the screen, occupying the region $-\sqrt{3}d \le x \le 0$. The refractive index of this slab varies spatially as $n(x,y) = n_0 + \alpha |\vec{E}(x,y)|^2$, where $n_0$ and $\alpha$ are positive constants, and $\vec{E}(x,y)$ is the local electric field produced by the charged wire. A monochromatic plane wave of wavelength $\lambda_0$ (in vacuum) is incident normally on the slab from $x < -\sqrt{3}d$. After passing through the slits, the light interferes on a distant screen placed at $x = D$ ($D \gg d$). What is the phase difference between the wavelets from $S_1$ and $S_2$ at the point $P(D, 2d)$ on the distant screen?