= Optical frequency measurement with a comb
{title2=$f_L=n f_r+f_0+b$}
Mix an unknown single-frequency <laser> with a nearby comb tooth on a <photodiode>. Define the signed <beat> $b=f_L-f_n$, so
$$
f_L=n f_r+f_0+b.
$$
The <beat> arises from the cross term in <light intensity>:
$$
\left|A e^{2\pi i f_Lt}+B e^{2\pi i f_nt}\right|^2
=|A|^2+|B|^2+2\operatorname{Re}\!\left(AB^*e^{2\pi i b t}\right).
$$
An electronic detector can follow this slow difference even though it cannot resolve either optical oscillation. The square-law expression also shows why a real-valued intensity trace by itself loses the sign of $b$. A small known upward shift of the chosen tooth decreases $|b|$ when the unknown lies above it and increases $|b|$ when it lies below, provided the tooth assignment remains unchanged.
For an original example, stipulate
$$
f_r=250\,\mathrm{MHz},\qquad f_0=20\,\mathrm{MHz},\qquad b=+30\,\mathrm{MHz},\qquad n=2{,}000{,}000.
$$
Then $f_n=500{,}000.020\,\mathrm{GHz}$ and $f_L=500{,}000.050\,\mathrm{GHz}$. Choosing the opposite sign would give $499{,}999.990\,\mathrm{GHz}$ instead. The stipulated optical <frequency> is countable because the large optical value is expressed using the <integer> $n$ and three electronic <frequencies>; the diagram's two controls establish the comb, and this additional <beat> places the unknown against it.
Both comb readouts use one calibrated <microwave> reference. Feedback fixes $f_r$ and $f_0$, determining each tooth once $n$ is identified. A <frequency counter> measures the slow <beat>, transferring the calibration to the optical <frequency>.<Passion for Precision by Theodor W. Hänsch>[Passion for Precision, p.158: microwave link]{ref}
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