Relation Cross-Scans

diffraction-aperture-relation-cross-scan

A diffraction-aperture cross-scan varies slit width or grating spacing across rows, showing how the order angle moves through a * sin(theta) = m * lambda or d * sin(theta) = m * lambda row by row. The arithmetic is exact only for the stated wavelengths, spacings, orders, and rows in the ideal single-aperture or grating model. Real observations add alignment, aperture geometry, coherence/bandwidth, detector calibration, material/reflection phase assumptions, and measurement uncertainty.

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.

asin⁡θ=5 m⋅25=2 ma\sin\theta=5\ \text{m}\cdot\frac{2}{5}=2\ \text{m}
Accepted aperture rowThe aperture width and sine marker are checked together.sinea=5 sine=2/5

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.

aλsin⁡θmaccepted5 m2 m15105 m2 m25115 m2 m3510\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.

aλsin⁡θmaccepted2 m2 m1115 m2 m251110 m2 m1511\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 m⇒sin⁡θ=1;a=10 m⇒sin⁡θ=15a=2\ \text{m}\Rightarrow \sin\theta=1;\qquad a=10\ \text{m}\Rightarrow \sin\theta=\frac{1}{5}
Aperture inverse contrastBoth checked rows land on the same wavelength minimum.sinesinea=2 sine=1a=10 sine=1/5