# In a frequency comb, how is $n$ determined?

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 $$\Delta \lambda / \lambda < f_r / f_n?$$

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.