Diffraction by a Circular Aperture — Question 4

Question 4

Two Rayleigh-separated Airy patterns have a calculable central dip

For one point source, $P(u)=[2J_1(u)/u]^2$ with $P(0)=1$. The first zero is $u_1=3.831705970$. Two equal incoherent sources are separated by one first-zero radius. Use $J_1(u_1/2)=0.580724582$. Each isolated peak is $I_0$.

1. Calculate the combined intensity at either source image center.

2. Calculate the midpoint intensity and its fractional drop relative to the intensity at a source image center.

Question 4 - Solution

Two Rayleigh-separated Airy patterns have a calculable central dip

(a) Method and calculation

$$I_{\mathrm{center}}=I_0[P(0)+P(u_1)]=\boxed{I_0}.$$$$P(u_1)=0\quad\text{because}\quad J_1(u_1)=0.$$

(b) Interpretation and further calculation

$$\boxed{I_{\mathrm{mid}}=2I_0\left[\frac{2J_1(u_1/2)}{u_1/2}\right]^2=0.735032I_0}.$$$$\boxed{1-I_{\mathrm{mid}}/I_{\mathrm{center}}=0.264968\simeq26.5\%}.$$

Physical check

The intensities add because the sources are incoherent; adding their field amplitudes would give a phase-dependent result.