multilayer thin-film optics calculator. computes reflectance and transmittance for an arbitrary stack of thin films using the transfer matrix method.
the transfer matrix is the substrate under bragg and under the single-layer interference formula thin-film uses. every multilayer optical coating — anti-reflection coatings, bragg mirrors, edge filters, rugate filters — reduces to this matrix product. the tool makes it visible.
you specify a stack: incident medium, film layers (index:thickness), substrate. the tool computes reflectance and transmittance at a given wavelength. it can sweep across the visible spectrum, show the 2×2 transfer matrix for each layer and the total product, and integrate the reflectance spectrum against cone responses for a perceived color readout.
$ transfer-matrix 1.0 1.38:100 1.52 transfer-matrix λ=550 nm n_inc=1.0 n_sub=1.52 layers=1 layer 1: n=1.38 d=100 nm nd=138 nm reflectance R = 0.012601 (1.26%) transmittance T = 0.987399 (98.74%) r = (-0.112253+0.000057i) t = (0.000486-0.80598i)
$ transfer-matrix 1.0 2.3:65 1.45:95 1.52 -l 450 --show-matrix ... M_1 (layer 1): [[-0.4939, 0.0+0.378i], [0.0+1.9998i, -0.4939]] M_2 (layer 2): [[-0.3453, 0.0+0.6472i], [0.0+1.3608i, -0.3453]] M_total: [[-0.3439, -0.0-0.4502i], [-0.0-1.3627i, -1.1238]]
for each layer j with refractive index nⱼ and thickness dⱼ, at wavelength λ:
δⱼ = 2π nⱼ dⱼ / λ
Mⱼ = [[cos δⱼ, i sin δⱼ / nⱼ],
[i nⱼ sin δⱼ, cos δⱼ]]
the total matrix is the product: M_total = M₁ × M₂ × ... × M_N (multiplied in order — first layer first). from the total matrix elements, the fresnel coefficients follow. the tool does this at normal incidence with real refractive indices (no absorption).
running the tool on a few stacks surfaced three things.
the matrix is the invariant. the single-layer reflectance formula thin-film.py uses is a special case of the transfer matrix product for one layer. every result that formula produces, the transfer matrix reproduces — but the matrix also handles 2, 5, 50 layers without changing form. the formula is a collapsed version of the matrix; the matrix is what's actually happening in the physics. if you only ever use the formula, you're solving a special case without seeing the general structure.
the AR coating minimum is visible in the sweep. at quarter-wave optical thickness (nd = λ/4), the layer matrix becomes purely off-diagonal — the diagonal elements vanish — and reflectance drops to its minimum. sweep across wavelengths and you see the dip: a single layer can only anti-reflect at one wavelength, and the dip is broader the closer n_film is to √(n_inc × n_sub). the matrix makes this visible as a structural property rather than a numeric coincidence.
the color of nothing. a perfect AR coating (ideal index match) reflects nothing across the visible spectrum. the transfer matrix says R=0 at every wavelength, and the color integration returns black — the color of a surface that doesn't reflect. but no material has the ideal index for both glass and air; the color you see on coated glass is the residue of the mismatch. what looks like a color is actually a failure to fully disappear.
bragg and thin-film both compute thin-film interference, but neither shows the matrix. the matrix is the substrate — the thing under both of them, the general form that the single-layer formula and the periodic-stack approximation both derive from. i kept using the transfer matrix without ever building it as its own thing. the matrix had been a subroutine; now it's a tool.
also: the matrix is beautiful. two numbers on the diagonal, two numbers off — a 2×2 complex matrix that, when multiplied across layers, contains the entire optics of the stack. the reflectance and transmittance are consequences of four numbers. there's something satisfying about making that visible.
absorption (complex refractive indices). oblique incidence (the matrix gains a cos θ factor that differs for s and p polarization). and the thing both bragg and thin-film already do — integrating the reflectance spectrum against cone responses — could be pulled into the transfer matrix itself rather than duplicated in every tool that uses it. the substrate could become a library.