= Etalon round-trip phase
{title2=$\phi_{\rm rt}$}
An \b[etalon round-trip phase] is the phase accumulated by a circulating optical field after one complete internal round trip, relative to its earlier passage. At normal incidence through a uniform plate of thickness $L$,
$$
\phi_{\rm rt}(\omega)=\frac{2\omega n(\omega)L}{c}+\theta_1(\omega)+\theta_2(\omega),
$$
where the last terms are the phases of the two internal amplitude-reflection coefficients. Propagation and reflection both contribute; a coating phase is not automatically zero.
The resonance condition follows from a field sum. If $r_1r_2e^{2ikL}=a e^{i\phi_{\rm rt}}$, with $0\le a<1$, successive transmitted paths contain factors $1,a e^{i\phi_{\rm rt}},a^2e^{2i\phi_{\rm rt}},\ldots$. Their <geometric series> has denominator $1-ae^{i\phi_{\rm rt}}$. Its squared magnitude contains
$$
|1-ae^{i\phi_{\rm rt}}|^2=1+a^2-2a\cos\phi_{\rm rt}.
$$
For slowly varying amplitudes, transmission peaks occur near $\phi_{\rm rt}=2\pi m$. The primary manufacturer tutorial describes the resulting <etalon> transmission peaks and distinguishes their spacing from their width.<Etalon Tutorial by Iridian Spectral Technologies>[Free Spectral Range and Finesse sections]{ref}
A constant phase offset shifts resonance positions without changing the local slope. In an original model with constant round-trip delay $20\,\mathrm{ps}$, adding reflection phase $0.4\,\mathrm{rad}$ shifts a given resonance by approximately $-0.4/(2\pi\times20\,\mathrm{ps})=-3.18\,\mathrm{GHz}$. The <free spectral range> remains $50\,\mathrm{GHz}$ in this constant-slope model.
Frequency-dependent phase has a different effect. Differentiating with respect to angular frequency gives
$$
\tau_{\rm rt}=\frac{d\phi_{\rm rt}}{d\omega}
=\frac{2n_gL}{c}+\frac{d\theta_1}{d\omega}+\frac{d\theta_2}{d\omega}.
$$
The last two terms form an <etalon coating group delay>. A stipulated $20\,\mathrm{ps}$ propagation contribution plus $2\,\mathrm{ps}$ coating contribution gives local resonance spacing about $1/(22\,\mathrm{ps})=45.45\,\mathrm{GHz}$, rather than 50 GHz. These are illustrative phase functions, not measured coating specifications.
Use a continuous branch of <wave phase> when differentiating: jumps of $2\pi$ in a wrapped display are not physical delay spikes. The <etalon delay inferred from free spectral range> article derives the interval relation between phase slope and adjacent resonances. Finite pulse bandwidth, angular spread and frequency-dependent reflection amplitudes determine how closely the simple resonance model describes an actual <optical pulse>.
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