Transfer Matrix Calculator

Interactive thin-film optics calculator using 2×2 matrix formalism

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Transfer Matrix Method

The transfer matrix method provides a powerful framework for analyzing light propagation through stratified media (multi-layer thin films). Each layer and interface is represented by a 2×2 matrix that relates the forward and backward propagating electric field amplitudes.

Interface Matrix

At an interface between two media with refractive indices n₁ and n₂, the Fresnel coefficients determine the reflection (r) and transmission (t) coefficients:

r = (n₁ - n₂) / (n₁ + n₂)
t = 2n₁ / (n₁ + n₂)
I = (1/t) × [[ 1, r ],
           [ r, 1 ]]

Propagation Matrix

Propagation through a layer of thickness d with refractive index n accumulates a phase β:

β = 2πnd/λ
P = [[ exp(-iβ), 0 ],
     [ 0, exp(iβ) ]]

Total System Matrix

The total transfer matrix is the product of all interface and propagation matrices:

Mtotal = I₀ × P₁ × I₁ × P₂ × I₂ × ... × Isubstrate

The reflectance coefficient is extracted from the total matrix elements:

rtotal = M₂₁ / M₁₁
R = |rtotal|²