Question #43230

A circular aperture of radius $a$ is illuminated by a coherent monochromatic plane wave of wavelength $\lambda$. To modify the standard Airy diffraction pattern, a thin, partially reflecting dielectric structure is placed across the aperture. This structure causes the transmitted electric field at the aperture plane to be a coherent superposition of a directly transmitted wave and an internally reflected wave. The directly transmitted wave has a uniform real amplitude $E_0$. The internally reflected wave has an amplitude $E_0/2$ and a radially varying relative phase shift $\Delta\phi(r) = \frac{\pi}{2} \left(\frac{r}{a}\right)^2$ due to the structure's varying thickness, where $r$ is the radial distance from the center. If the on-axis intensity of the Fraunhofer diffraction pattern for a completely empty aperture of radius $a$ (with uniform amplitude $E_0$) is $I_0$, what is the on-axis intensity $I$ of the modified aperture?