Diffraction and the Wave Theory of Light — Question 1

Question 1

A finite source distance contributes its own Fresnel curvature

A point source is S before an aperture and an axial detector is L beyond it. At aperture coordinate u, the path is $r(u)=\sqrt{S^2+u^2}+\sqrt{L^2+u^2}$. Let $S=0.60\,\mathrm m$, $L=1.20\,\mathrm m$, aperture half-width $b=0.20\,\mathrm{mm}$, and $\lambda=500\,\mathrm{nm}$.

1. Expand the path through order $u^2$ and define the effective curvature distance R.

2. Calculate the edge-to-center phase and compare with the plane-wave illumination limit.

Question 1 - Solution

A finite source distance contributes its own Fresnel curvature

(a) Method and calculation

$$r(u)\simeq S+L+\frac{u^2}{2}\left(\frac1S+\frac1L\right),\quad\boxed{R=\frac{SL}{S+L}=0.400\,\mathrm m}.$$$$\Delta\phi(u)=\frac{2\pi}{\lambda}\frac{u^2}{2R}=\frac{\pi u^2}{\lambda R}.$$

(b) Interpretation and further calculation

$$\boxed{\Delta\phi(b)=0.200\pi\ \mathrm{rad}}.$$$$S\to\infty:\ R\to L,\quad\boxed{\Delta\phi_{\mathrm{plane}}=\pi b^2/(\lambda L)=\pi/15}.$$

Physical check

Both incident and outgoing curvature affect the phase budget; ignoring the finite source underestimates the phase variation by a factor of three.