Coherence — Question 6

Question 6

Two independent source points can wash out a double-slit pattern

Two mutually incoherent point sources of equal brightness lie at $h=+H/2$ and $-H/2$, distance S before the slit plane. Each alone produces $I_s[1+\cos(\delta\pm\psi)]$, with $\psi=\pi dH/(\lambda S)$.

1. Add their intensities and derive the visibility as a function of H.

2. For $d=0.50\,\mathrm{mm}$, $S=1.0\,\mathrm m$, and $\lambda=500\,\mathrm{nm}$, find the smallest positive source separation for zero visibility and the visibility when that separation is doubled.

Question 6 - Solution

Two independent source points can wash out a double-slit pattern

(a) Method and calculation

$$I_{\mathrm{total}}=2I_s+I_s[\cos(\delta+\psi)+\cos(\delta-\psi)].$$$$\boxed{I_{\mathrm{total}}=2I_s[1+\cos\psi\cos\delta],\quad V=|\cos\psi|}.$$

(b) Interpretation and further calculation

$$\psi=\pi/2\Longrightarrow\boxed{H_0=\lambda S/(2d)=0.500\,\mathrm{mm}}.$$$$H=2H_0:\quad\psi=\pi\Longrightarrow\boxed{V=1\ \text{with the fringe phase reversed}}.$$

Physical check

Spatial coherence need not decrease monotonically with source separation for a discrete two-point source.