thin-film-path-relation-cross-scan - Wave Optics: Interference and Diffraction Step by Step

The thin-film optical path formula is scanned through index and thickness before any phase verdict is read. 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 checked thin-film path

The reflected film path is a double pass through the film. The training model keeps the index exact so the path ledger can be read without decimals.

\text{path}=2nt=2\cdot2\cdot1\ \text{m}=4\ \text{m}

Index sweep: three indices, same thickness

Hold the physical thickness fixed. The optical path grows in the same ratio as the refractive index because the path is two times index times thickness.

\begin{array}{c|c|c|c}n&t&\text{path}&\text{order}\\1&1\ \text{m}&2\ \text{m}&1\\2&1\ \text{m}&4\ \text{m}&2\\3&1\ \text{m}&6\ \text{m}&3\\\end{array}

Thickness sweep: three thicknesses, same index

Now hold the index fixed and change only thickness. The optical path grows directly with thickness, so the order count grows with it.

\begin{array}{c|c|c|c}n&t&\text{path}&\text{order}\\2&1\ \text{m}&4\ \text{m}&2\\2&2\ \text{m}&8\ \text{m}&4\\2&3\ \text{m}&12\ \text{m}&6\\\end{array}

The index and thickness diagrams show the same multiplier

The first panel reaches a longer path by changing index. The second reaches a longer path by changing thickness. The formula does not claim those are the same material; it shows which factor made the optical path larger.

n=3\Rightarrow \text{path}=6\ \text{m};\qquad t=3\ \text{m}\Rightarrow \text{path}=12\ \text{m}