diffraction-aperture-relation-cross-scan - Wave Optics: Interference and Diffraction Step by Step

The aperture-minimum formula is scanned through sine marker and aperture width using exact accepted and rejected rows. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Start from one accepted aperture row

A single-slit minimum is not an angle sketch by itself. The accepted row multiplies aperture width by a rational sine marker and compares the product with the wavelength.

a\sin\theta=5\ \text{m}\cdot\frac{2}{5}=2\ \text{m}

Sine sweep: three markers, same aperture

Hold the aperture width fixed. Only the middle sine marker makes the product equal one wavelength, so the neighboring rows stay visible as rejected contrast cases.

\begin{array}{c|c|c|c|c}a&\lambda&\sin\theta&m&\text{accepted}\\5\ \text{m}&2\ \text{m}&\frac{1}{5}&1&0\\5\ \text{m}&2\ \text{m}&\frac{2}{5}&1&1\\5\ \text{m}&2\ \text{m}&\frac{3}{5}&1&0\\\end{array}

Width sweep: three apertures, matched sine markers

Now keep the wavelength and order fixed while changing aperture width. Wider apertures require smaller sine markers; the inverse pair is visible across three accepted rows.

\begin{array}{c|c|c|c|c}a&\lambda&\sin\theta&m&\text{accepted}\\2\ \text{m}&2\ \text{m}&1&1&1\\5\ \text{m}&2\ \text{m}&\frac{2}{5}&1&1\\10\ \text{m}&2\ \text{m}&\frac{1}{5}&1&1\\\end{array}

Narrow and wide apertures close the same wavelength ledger

The narrow row reaches the minimum at the endpoint marker. The wide row reaches the same wavelength with a smaller sine marker. The diagram is a second representation of the width sweep.

a=2\ \text{m}\Rightarrow \sin\theta=1;\qquad a=10\ \text{m}\Rightarrow \sin\theta=\frac{1}{5}