Diffraction by a Double Slit — Question 3

Question 3

Exactly missing orders require a rational separation-to-width ratio

A double slit has $d/a=\sqrt2$. Nominal bright orders satisfy $d\sin\theta=m\lambda$; envelope zeros satisfy $a\sin\theta=p\lambda$ for nonzero integer p. Assume any orders discussed are propagating.

1. Prove that no nonzero nominal bright order is exactly missing.

2. If fabrication changes the ratio to $d/a=7/5$, identify the first positive missing order and the corresponding envelope zero.

Question 3 - Solution

Exactly missing orders require a rational separation-to-width ratio

(a) Method and calculation

$$\text{Coincidence}\Longrightarrow\frac da=\frac mp.$$$$\sqrt2\notin\mathbb Q\Longrightarrow\boxed{\text{no nonzero exact missing order}}.$$

(b) Interpretation and further calculation

$$d/a=7/5:\quad5m=7p\Longrightarrow m=7j,\ p=5j.$$$$\boxed{m=7,\quad p=5\quad\text{for the first positive coincidence}}.$$

Physical check

An irrational ratio can still produce very weak orders close to an envelope zero; absence of exact cancellation does not guarantee visibility.