Question #43583

In a Young's Double Slit Experiment (YDSE), two identical slits emit coherent light of intensity $I_0$ and optical angular frequency $\omega_{opt}$. An electro-optic modulator is placed in front of one of the slits, introducing a time-varying phase shift $\phi(t) = \alpha V(t)$, where $\alpha$ is a positive constant in $\text{rad/V}$ and $V(t)$ is the applied voltage. The voltage is supplied by an ideal $L-C$ circuit (inductor $L$, capacitor $C$). At $t=0$, the capacitor is fully charged to a voltage $V_0$ and the current is zero. Assume the $L-C$ oscillation frequency is much smaller than $\omega_{opt}$. Given the integral identity $\frac{1}{2\pi} \int_0^{2\pi} \cos(x \cos \theta) d\theta = J_0(x)$, where $J_0(x)$ is the zeroth-order Bessel function of the first kind, what is the time-averaged intensity $\langle I \rangle$ at the central point on the screen (where the geometrical path difference is zero) over one complete period of the $L-C$ oscillation?