= Etalon delay inferred from free spectral range
{title2=$\tau_{\rm rt}\simeq1/\Delta\nu_{\rm FSR}$}
An \b[etalon delay inferred from free spectral range] estimates an <etalon round-trip delay> from the frequency spacing of adjacent transmission resonances. When round-trip group delay varies little across that spacing,
$$
\tau_{\rm rt}\simeq\frac1{\Delta\nu_{\rm FSR}}.
$$
The measured quantity is resonance spacing, not the width of one resonance. The manufacturer tutorial explicitly distinguishes <free spectral range> from <optical finesse>, which compares that spacing with an individual peak's width.<Etalon Tutorial by Iridian Spectral Technologies>[Free Spectral Range and Finesse sections]{ref}
The relation follows from <wave phase>. Let $\phi(\nu)$ be the total <etalon round-trip phase>, including relevant reflection phases. Adjacent resonances satisfy $\phi(\nu_{m+1})-\phi(\nu_m)=2\pi$. Defining $\tau_{\rm rt}(\nu)=(2\pi)^{-1}d\phi/d\nu$ gives the exact interval relation
$$
\int_{\nu_m}^{\nu_{m+1}}\tau_{\rm rt}(\nu)\,d\nu=1.
$$
Thus the reciprocal spacing is the delay averaged over that frequency interval. Interpreting it as a local pulse-envelope delay requires sufficiently weak variation across the interval and relevant pulse spectrum.
For a uniform plate at normal incidence with negligible frequency-dependent reflection phase, $\phi=4\pi\nu n(\nu)L/c$. Differentiation gives $\tau_{\rm rt}=2n_gL/c$, where $n_g=n+\nu\,dn/d\nu$ is the <group refractive index>. Treating $n$ as frequency independent recovers the manufacturer tutorial's elementary plate formula; using a phase index in a strongly dispersive material can instead give the wrong delay.
An original measured-spacing example takes $\Delta\nu_{\rm FSR}=25.0\,\mathrm{GHz}$. The inferred delay is $40.0\,\mathrm{ps}$. A spacing standard uncertainty of $0.2\,\mathrm{GHz}$ contributes approximately $0.32\,\mathrm{ps}$ to the delay uncertainty through $u(\tau)=u(\Delta\nu)/(\Delta\nu)^2$. With stipulated $n_g=1.50$ and negligible coating delay, the corresponding plate thickness is about $4.00\,\mathrm{mm}$.
In a spectral experiment, identify adjacent peaks of the same resonance family and calibrate the frequency coordinate. If the coordinate is <wavelength>, use $\Delta\nu=c|1/\lambda_1-1/\lambda_2|$ rather than treating a wavelength interval as a frequency interval. Beam angle, dispersion and reflection phase belong in the propagation model.
A time-domain check looks for replicas separated by the predicted extra round trip. The historical single-shot correlator experiment inserted a known-thickness BK7 etalon to calibrate delay and separately placed an uncoated etalon in the input to create a double-reflection postpulse.<A Single Shot 3rd Order Cross-Correlator for Pulse Contrast and Pulse Shape Measurements by J. Collier and others>[printed p. 195, Description, Results and Figure 4]{ref} The single-pass delay caused by inserting glass into an arm is distinct from the additional round-trip separation of successive transmitted replicas.
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