Coherence — Question 5

Question 5

An extended uniform source can erase otherwise stable fringes

A uniform incoherent line source of width W lies distance S before two slits of spacing d. A source element at coordinate h contributes phase offset $2\pi dh/(\lambda S)$. Its fringe intensity adds incoherently to those of the other elements. Define $\gamma=(1/W)\int_{-W/2}^{W/2}e^{i2\pi dh/(\lambda S)}dh$ and $V=|\gamma|$.

1. Evaluate the integral and derive the first source width producing zero contrast.

2. For $\lambda=600\,\mathrm{nm}$, $S=0.50\,\mathrm m$, and $d=0.30\,\mathrm{mm}$, calculate that width and the visibility at half that width.

Question 5 - Solution

An extended uniform source can erase otherwise stable fringes

(a) Method and calculation

$$\boxed{\gamma=\frac{\sin[\pi dW/(\lambda S)]}{\pi dW/(\lambda S)}}.$$$$\pi dW_0/(\lambda S)=\pi\Longrightarrow\boxed{W_0=\lambda S/d}.$$

(b) Interpretation and further calculation

$$\boxed{W_0=1.00\,\mathrm{mm}}.$$$$W=W_0/2:\quad\boxed{V=2/\pi=0.63662}.$$

Physical check

Each source element can produce a clear pattern, while their differently shifted patterns cancel the total modulation.