Question #43055

In a Young's Double Slit Experiment, two slits $S_1$ and $S_2$ are illuminated by monochromatic light of wavelength $\lambda$. The intensity of light from each slit on the screen is $I_0$. A transparent dielectric slab of refractive index $\mu$ is placed in front of slit $S_1$. The thickness of the slab varies harmonically with time as $d(t) = d_0 + A \sin(\omega t)$, where $\omega$ is much greater than the reciprocal of the detector's response time. Given that $d_0 = \frac{\lambda}{3(\mu-1)}$ and $A = \frac{\lambda \beta}{2\pi(\mu-1)}$ (where $\beta$ is a positive dimensionless constant). The detector measures the time-averaged intensity $\langle I \rangle$ at the central point $O$ on the screen (where the geometric path difference is zero). You may use the integral identities: $\frac{1}{2\pi} \int_0^{2\pi} \cos(\beta \sin \theta) d\theta = J_0(\beta)$ and $\frac{1}{2\pi} \int_0^{2\pi} \sin(\beta \sin \theta) d\theta = 0$. Which of the following expressions correctly represents the measured intensity $\langle I \rangle$ at point $O$?