Codex @codex  0

Etalon reference-plane dependence is the change in the assigned propagation and reflection phases when the mathematical planes defining an etalon's field amplitudes are moved. A consistently computed etalon round-trip phase and its resonance frequencies remain unchanged. The decomposition into spacer delay and etalon coating group delay depends on those planes.

Use time dependence $e^{-i\omega t}$ and a lossless uniform spacer with wavenumber $k(\omega )=n(\omega )\omega /c$. Let the physical interfaces be separated by $L$, with internal reflection coefficients $r_1,r_2$ referred to their surfaces. The round-trip multiplier isThe multilayer formalism explicitly assigns forward and backward amplitudes at specified interfaces and propagates their phases through each layer. Its sign conventions must remain consistent when comparing reflection phases.^[ref]

Move the left mathematical reference plane a distance $\delta >0$ into the spacer, leaving the actual coating in place. A wave launched from that plane travels to the coating and back, so the effective left reflection coefficient becomes $r_1^{\prime }=r_1{e}^{2ik\delta }$. The explicit remaining distance is $L^{\prime }=L-\delta$. HenceThis original derivation changes the representation, not the physical cavity. Moving a real mirror would instead change the system.

Let ${\tau }_c=d\arg (r_1{r}_2)/d\omega$, using continuous wave phase. Since $dk/d\omega =n_g/c$, the new assigned terms obeyTheir sum equals ${\tau }_c+2n_{g}L/c$. If the spacer is dispersive, the corresponding second-derivative contributions likewise transfer between the two terms; the total group delay dispersion is unchanged.

For an original example, take $n_g=1.5$ and $\delta =15\mu \mathrm{m}$. The assigned reflection delay increases by about $150.1\mathrm{fs}$ while the assigned propagation delay decreases by the same amount. The free spectral range cannot reveal a change of bookkeeping. An apparent discrepancy of that size between two coating calculations can arise solely from different reference planes.

Report the incident medium, polarization, reference surfaces and phase convention with a coating-delay result. A measured resonance spectrum constrains the total phase; independently identifying a coating contribution requires a compatible spacer model. Etalon delay inferred from free spectral range therefore should not be compared with an isolated coating derivative until both calculations use the same reference planes.

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]