Diffraction by a Circular Aperture — Question 6

Question 6

The Bessel expansion gives the curvature of the central Airy peak

For a circular aperture, $I/I_c=[2J_1(u)/u]^2$ with $u=\pi D\sin\theta/\lambda$. Use $J_1(u)=u/2-u^3/16+u^5/384+O(u^7)$. Let $D=10.0\,\mathrm{mm}$ and $\lambda=500\,\mathrm{nm}$.

1. Derive the central intensity expansion through fourth order.

2. Use its quadratic term to estimate the angle causing a one-percent drop from the peak, and compare with the Rayleigh angle.

Question 6 - Solution

The Bessel expansion gives the curvature of the central Airy peak

(a) Method and calculation

$$2J_1(u)/u=1-u^2/8+u^4/192+O(u^6).$$$$\boxed{I/I_c=1-u^2/4+5u^4/192+O(u^6)}.$$

(b) Interpretation and further calculation

$$u^2/4=0.01\Longrightarrow u=0.20\Longrightarrow\boxed{\theta\simeq0.20\lambda/(\pi D)=3.18310\times10^{-6}\,\mathrm{rad}}.$$$$\theta_R=1.22\lambda/D=6.10\times10^{-5}\,\mathrm{rad}\Longrightarrow\boxed{\theta/\theta_R\simeq0.05218}.$$

Physical check

The radius for a one-percent loss is much smaller than the radius of the first dark ring; different image-quality measures probe different parts of the pattern.