EQMResearch group
Level 4 · Light in layered structures

Transfer-matrix method (TMM)

2×2 layer matrices

The math trick we use to predict reflection through any 1D stack: hand the wave from layer to layer with a small matrix at every interface.

Build on:Thin-film interference,Refractive index n(E)

One matrix per layer

Once you have more than two layers, doing interference by hand turns into a mess. The transfer-matrix method (TMM) is the trick that keeps the math under control. Each layer (and each interface) is represented by a 2×2 matrix that maps the (forward, backward) amplitudes of the wave on its left to those on its right.

airn₁M1n₂M2n₃M3n₄M4substrateincidentreflectedtransmittedM_total = M₁ · M₂ · M₃ · M₄ → r, t → R = |r|²
The wave is handed from layer to layer by a small matrix product. The whole stack reduces to one matrix; the reflection coefficient r and the transmission t fall straight out of it.

The whole stack in one product

Multiply all the layer matrices and you get an effective 2×2 matrix for the entire stack. The reflection and transmission coefficientsr and t are simple ratios of its entries, andR = |r|² is what we plot.

The simulator uses an existing implementation of TMM (the Pythontmm package) — but the conceptual recipe is exactly the one sketched above. Every reflectance dip in the app comes from this one calculation, repeated at every photon energy.

Key takeaways
  • Each layer = one 2×2 matrix.
  • Whole stack = the matrix product.
  • R(E) = |r(E)|² evaluated for every E in the chosen window.
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