Diffraction — Question 4

Question 4

Paired Huygens sources explain the first dark direction

Divide a uniformly illuminated slit of width a into two equal halves. Pair each point u in the lower half with the point $u+a/2$ in the upper half. In the far field at angle $\theta$, the pair path difference is $(a/2)\sin\theta$. Assume equal amplitudes and scalar waves.

1. Derive the smallest positive angle at which every such pair cancels.

2. For $a=0.80\,\mu\mathrm m$ and $\lambda=0.60\,\mu\mathrm m$, find that angle and decide whether the same construction has a physical first-zero angle if a is reduced to $0.40\,\mu\mathrm m$.

Question 4 - Solution

Paired Huygens sources explain the first dark direction

(a) Method and calculation

$$\Delta\phi_{\mathrm{pair}}=\frac{2\pi}{\lambda}\frac a2\sin\theta=\pi.$$$$\boxed{a\sin\theta_1=\lambda};\quad E(u)+E(u+a/2)=0.$$

(b) Interpretation and further calculation

$$\boxed{\theta_1=\sin^{-1}(0.60/0.80)=48.590^\circ}.$$$$a=0.40\,\mu\mathrm m:\quad\sin\theta_1=1.50>1\Rightarrow\boxed{\text{no propagating first-zero direction in this model}}.$$

Physical check

The absence of a first zero does not mean the absence of diffraction; the angular distribution becomes very broad.