Math 2.0

Optics, electromagnetic theory Definition

Thin-film interference

Builds on InterferenceRefractive indexOptical path lengthSnell's law

Definition

Let a film of thickness dd and refractive index nfn_f lie between two media. The medium on the incident side has index n1n_1, and the medium on the other side has index n2n_2. Light of vacuum wavelength λ\lambda falls on the film, and θt\theta_t is the angle of the refracted ray in the film, given by Snell's law. The waves reflected at the top and at the bottom of the film have the optical path length difference Δ=2nfdcos⁡θt.\Delta = 2 n_f d \cos \theta_t. At normal incidence, a reflection off a medium of higher index adds a phase of π\pi. A reflection off a medium of lower index adds no phase. Thin-film interference is the interference of these two reflected waves. Their phase difference is 2πΔ/λ2\pi\Delta/\lambda plus the difference of the reflection phases.

Remarks

If nfn_f is larger than both n1n_1 and n2n_2, exactly one reflection adds π\pi. Then the reflection is strongest when Δ=(m+12)λ\Delta = (m + \tfrac12)\lambda and weakest when Δ=mλ\Delta = m\lambda, for integers m≥0m \ge 0. If n1<nf<n2n_1 < n_f < n_2, both reflections add π\pi, and the two conditions are exchanged. The two-beam picture neglects the waves reflected several times inside the film, which is accurate when the surfaces reflect weakly.

Example: a quarter-wave antireflection coating has n1<nf<n2n_1 < n_f < n_2 and nfd=λ/4n_f d = \lambda / 4. At normal incidence Δ=λ/2\Delta = \lambda/2, so the two reflected waves are in opposite phase and the reflection is reduced. It vanishes exactly when nf=n1n2n_f = \sqrt{n_1 n_2}, because then the two reflected amplitudes are equal.

Non-example: a glass plate several millimetres thick, lit by sunlight, shows no colours. Its path difference is much longer than the coherence length of sunlight, so the two reflected waves do not interfere.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

Show the path as a graph
Calculus of variations, concept, no evidence levelCalculus of variationsconceptEuler–Lagrange equation, concept, no evidence levelEuler–Lagrange equati…conceptGeodesic, concept, no evidence levelGeodesicconceptInterference, concept, no evidence levelInterferenceconceptNormal vector, concept, no evidence levelNormal vectorconceptOptical path length, concept, no evidence levelOptical path lengthconceptRefractive index, concept, no evidence levelRefractive indexconceptFermat's principle, statement, no evidence levelFermat's principlestatementSnell's law, statement, no evidence levelSnell's lawstatementThin-film interference, definition, no evidence levelThin-film interferencedefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Calculus of variations concept
  2. Euler–Lagrange equation concept
  3. Geodesic concept
  4. Interference concept
  5. Normal vector concept
  6. Optical path length concept
  7. Refractive index concept
  8. Fermat's principle statement
  9. Snell's law statement
  10. Thin-film interference definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Thin-film_interference
About this page
TagQD7FNARS9P
URIm2://QD7FNARS9P
Kinddefinition
WritingW0 stub
MSC78A10, 78A40
DefinesThin-film interference
UsesInterference, Refractive index, Optical path length, Snell's law
Changed3 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.