 Steven J. Byrnes

Steven J. Byrnes developed a detailed treatment of multilayer optical calculations together with the tmm Python implementation. His primary preprint derives how reflected, transmitted and absorbed fields follow from interface relations and propagation through planar layers, including absorbing media and incoherent thick layers.^[ref]

The calculation retains optical phase as well as amplitude. Reflections from two surfaces can therefore cancel even though either surface alone reflects light. Treating the two reflected powers as independent contributions would miss this interference.

For an original normal-incidence antireflection coating example, take lossless refractive indices $n_0=1$, $n_1=1.5$ and $n_s=2.25$. At a vacuum wavelength of $600\mathrm{nm}$, a coating thickness $d=\lambda /(4n_1)=100\mathrm{nm}$ gives one-way phase $\delta =2\pi n_{1}d/\lambda =\pi /2$. The Fresnel equations give amplitude coefficients $r_{01}=r_{1s}=-0.2$, and the single-layer result isThus the ideal reflected power is zero at the stipulated wavelength, whereas the uncoated air–substrate interface reflects $|(1-2.25)/(1+2.25){|}^2\approx 14.8\%$. These are lossless model numbers, not a fabricated coating's performance.

Byrnes's analysis also explains why absorbing incident or final media require careful definitions of power flow: an amplitude coefficient and a transmitted-power fraction are not interchangeable. Sign conventions, complex wavevectors and interference assumptions belong in a reproducible calculation, alongside the layer thicknesses and refractive indices.^[ref]