= Etalon round-trip delay
An \b[etalon round-trip delay] is the extra envelope travel time associated with one additional internal round trip in an <etalon>. At normal incidence, for a transparent plate of thickness $L$ and a sufficiently narrowband <optical pulse>,
$$
\tau_{\rm rt}=\frac{2n_gL}{c},\qquad
n_g=n+\omega\frac{dn}{d\omega}.
$$
The relevant quantity is <group refractive index>, since the delay concerns the pulse envelope. The extra propagation phase is $\phi(\omega)=2\omega n(\omega)L/c$; its derivative gives $d\phi/d\omega=\tau_{\rm rt}$. This assumes negligible frequency-dependent reflection phase and treats dispersion locally.
For an original chosen plate with $L=3.0\,\mathrm{mm}$ and $n_g=1.60$, the delay is about $32.0\,\mathrm{ps}$. A pulse transmitted after one extra round trip appears that much later than the directly transmitted pulse. Another extra round trip adds the same delay in this local model, producing an echo sequence. The <optical postpulse> article separately describes the reflection losses that determine echo energies.
The frequency-domain connection gives an independent consistency check. Neighboring transmission resonances differ in round-trip phase by $2\pi$, so the local <free spectral range> satisfies
$$
\Delta\nu_{\rm FSR}\simeq\frac1{\tau_{\rm rt}}.
$$
The example predicts about $31.2\,\mathrm{GHz}$. This <etalon delay inferred from free spectral range> uses the same local dispersion assumptions; the spacing can vary across a broad spectrum.
The historical third-order correlator calibrated its delay coordinate by inserting a BK7 <etalon> of known thickness into an arm and measuring the trace shift.<A Single Shot 3rd Order Cross-Correlator for Pulse Contrast and Pulse Shape Measurements by J. Collier and others>[printed p. 195, Description, group-delay calibration; Results and Figure 4, delayed replica]{ref} A plate inserted into an arm also changes the direct propagation delay relative to the displaced air path; that single-pass insertion delay is distinct from the echo's extra internal round trip. Oblique incidence, dispersive coatings and broad-band pulse reshaping require a more complete propagation model.
Back to article page
