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A frequency comb from a mode-locked laser produces a series of spectral lines with $f_n = nf_r + f_{ce}.$

$f_r$ is the frequency of pulses coming out of the laser and can be measured directly via an electronic counter. There is a trick for measuring $f_{ce}$ involving squaring the electric field and counting a beat frequency.

How is $n$ measured? For example, is the light put through a spectroscope giving a measurement of $\lambda_n$ that is accurate enough that we can infer $n$ from it, because the error $\Delta \lambda / \lambda < f_r / f_n?$

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You basically just count the lines.

You use a high-precision spectrometer (i.e. one whose instrument linewidth is smaller than that of your comb) and you get the optical spectrum of the beam. This will get you a direct measurement of $$ k_n = 2\pi/\lambda_n = 2\pi \ f_n/c $$ for a bunch of lines spanning (say) $10\,000\leq n \leq 20\,000$. Generally this spectrum won't go all the way down to the DC regime (hence the lower bound on $n$) but it will get you a sharp enough measurement on the $k_n$ and therefore on $\Delta k = k_{n+1}-k_n$ (which will be constant throughout) that you can extrapolate the set of $k_n$'s that you do observe down to $n=0$ and keep enough precision in your extrapolation that the uncertainty in $n$ from the extrapolation is smaller than $1$ and therefore zero.

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