= Etalon coating group delay
{title2=$\tau_c$}
An \b[etalon coating group delay] is the frequency derivative of the reflection phase contributed by the two coatings during one internal round trip. At fixed incidence and polarization, write the internal <reflection coefficients> as $r_j(\omega)=|r_j(\omega)|e^{i\theta_j(\omega)}$. With the phase convention of <etalon round-trip phase>,
$$
\tau_c(\omega)=\frac{d}{d\omega}[\theta_1(\omega)+\theta_2(\omega)],\qquad
\tau_{\rm rt}=\frac{2n_gL}{c}+\tau_c.
$$
The phases must refer to consistently chosen surfaces: moving a reference plane transfers a propagation contribution between the terms. Only the consistently defined total is directly comparable with the resonance spacing.
A constant coating phase changes the positions of resonances but contributes zero <group delay>. A linear phase produces constant group delay. Curvature produces <group delay dispersion>:
$$
\frac{d\tau_c}{d\omega}=\frac{d^2(\theta_1+\theta_2)}{d\omega^2}.
$$
A highly reflecting coating therefore cannot be characterized by reflectance alone. The manufacturer's ultrafast-mirror documentation separately specifies reflection dispersion and shows multiple-bounce arrangements for compensation.<Chirped Mirrors by Thorlabs>[Dispersion-Compensating Mirrors, phase/GD/GDD characterization and Fig. 1.1]{ref}
For an original constant-slope example, suppose propagation contributes $10\,\mathrm{ps}$ and the coatings together contribute $0.20\,\mathrm{ps}$. The local <free spectral range> is approximately $1/(10.20\,\mathrm{ps})=98.04\,\mathrm{GHz}$, rather than $100\,\mathrm{GHz}$. Ignoring coating delay would overassign that $0.20\,\mathrm{ps}$ to propagation when inferring a spacer thickness or <group index>. The values are stipulated, not a coating specification.
For an independent curvature example, let the summed coating phase contain $D(\omega-\omega_0)^2/2$, with $D=100\,\mathrm{fs^2}$. Across a frequency interval $10\,\mathrm{THz}$, its group delay changes by $D\,2\pi\Delta f\approx6.28\,\mathrm{fs}$. The derivative is with respect to angular frequency; omitting $2\pi$ changes the result.
Experimentally, one may measure adjacent resonance frequencies and compare their spacing with an independently characterized spacer and <group index>. A discrepancy alone cannot uniquely identify coating delay: spacer dispersion, incidence angle and geometry also affect the result. <Etalon delay inferred from free spectral range> gives an interval-averaged phase slope; identifying it with a single-frequency derivative requires sufficiently slow variation. The simple manufacturer's spacer relation explicitly neglects spacer dispersion.<Etalon Tutorial by Iridian Spectral Technologies>[Free Spectral Range section and displayed plate relation]{ref}
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