Question #43243

A plane monochromatic wave of wavelength $\lambda$ is incident normally on a rectangular slit of width $a$. A transparent glass plate of uniform thickness $t$ is placed immediately behind the slit, covering it completely. The refractive index of the glass varies with temperature $T$ as $\mu(T) = \mu_0 + \gamma T$, where $\mu_0$ and $\gamma$ are positive constants. A steady temperature gradient is established along the width of the slit such that the temperature varies linearly from $T_0$ at the lower edge to $T_0 + \Delta T$ at the upper edge. Assuming thermal expansion of the glass and the slit is negligible, what is the minimum positive temperature difference $\Delta T$ required so that the intensity of light at the geometric center of the diffraction pattern (i.e., at an angle $\theta = 0^\circ$) becomes zero?