Diffraction and the Wave Theory of Light — Question 2

Question 2

Successive Fresnel half-period zones have nearly equal areas

A plane wave illuminates a circular aperture, and an axial point is distance $L=1.00\,\mathrm m$ away. Define zone boundary $r_m$ by $\sqrt{L^2+r_m^2}-L=m\lambda/2$, with $\lambda=600\,\mathrm{nm}$. Use the paraxial regime $r_m\ll L$.

1. Derive the approximate radius and area of the mth annular zone.

2. Find the first and fourth boundary radii and estimate the relative correction to $r_4^2$ from the exact geometry.

Question 2 - Solution

Successive Fresnel half-period zones have nearly equal areas

(a) Method and calculation

$$\frac{r_m^2}{2L}\simeq\frac{m\lambda}{2}\Longrightarrow\boxed{r_m^2\simeq m\lambda L}.$$$$\boxed{\mathcal A_m=\pi(r_m^2-r_{m-1}^2)\simeq\pi\lambda L}.$$

(b) Interpretation and further calculation

$$\boxed{r_1=0.774597\,\mathrm{mm},\quad r_4=1.54919\,\mathrm{mm}}.$$$$r_m^2=m\lambda L+m^2\lambda^2/4\Longrightarrow\boxed{\epsilon_4=4\lambda/(4L)=6.0\times10^{-7}}.$$

Physical check

Equal annular areas help explain comparable contributions from successive zones, while their phases alternate by approximately pi.