Steven J. Byrnes
= Steven J. Byrnes
{c}
\b[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.<Multilayer optical calculations by Steven J. Byrnes>[Abstract; §4.1, layer and interface amplitudes]{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 $\delta=2\pi n_1d/\lambda=\pi/2$. The <Fresnel equations> give amplitude coefficients $r_{01}=r_{1s}=-0.2$, and the single-layer result is
$$
r=\frac{r_{01}+r_{1s}e^{2i\delta}}{1+r_{01}r_{1s}e^{2i\delta}}=0.
$$
Thus the ideal reflected power is zero at the stipulated wavelength, whereas the uncoated air–substrate interface reflects $|(1-2.25)/(1+2.25)|^2\approx14.8\%$. 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>.<Multilayer optical calculations by Steven J. Byrnes>[Abstract and Appendix A, absorbing-media and sign-convention issues]{ref}