Question #43725

In a Young's Double Slit Experiment, a monochromatic light source of wavelength $\lambda$ is used, and the two slits produce beams of equal intensity at the screen. A transparent dielectric slab of thickness $t_s$ is placed directly in front of one of the slits. The slab is subjected to a high-frequency electromagnetic field, which modulates its refractive index via the Kerr effect such that $n(t) = n_0 + \Delta n \sin(\omega t)$, where $n_0$ and $\Delta n$ are positive constants. The interference pattern is recorded by a slow detector with an integration time $\tau \gg \frac{2\pi}{\omega}$. Which of the following expressions represents the visibility $V = \frac{I_{max} - I_{min}}{I_{max} + I_{min}}$ of the observed fringes?

[Given: $\frac{1}{2\pi} \int_{0}^{2\pi} \cos(z \sin \theta) d\theta = J_0(z)$ and $\frac{1}{2\pi} \int_{0}^{2\pi} \sin(z \sin \theta) d\theta = 0$, where $J_0(z)$ is the zeroth-order Bessel function of the first kind.]