= Comb self-referencing
<Comb self-referencing> compares two parts of one <optical frequency comb>. Coherent broadening supplies an octave, a factor of two in <frequency>. <Second-harmonic generation> doubles a red tooth; a <photodiode> detects its <beat> with an original blue tooth.<Introduction to Self-reference Comb Laser by Comblaser Laboratory, Department of Physics, National Central University>[Introduction to Self-reference Comb Laser, Fig.1 explanation]{ref} Applying the tooth formula gives
$$
2f_n-f_{2n}=2(n f_r+f_0)-(2n f_r+f_0)=f_0.
$$
The doubled branch carries $2f_0$, while the original high-frequency branch carries $f_0$: their difference retains exactly one offset.
\Image[/optical-frequency-comb-self-reference.png]
{title=Read the tooth spacing and offset through separate electronic signals}
{description=Original functional <comb self-referencing> scheme. Solid optical paths bring doubled red light and original blue light to one detector. Dashed electronic paths connect both readouts to a common reference and feedback controls. Actuators can affect both parameters.}
The displayed difference uses the order doubled tooth minus original tooth. An ordinary electronic <frequency counter> returns a nonnegative <beat> magnitude; spectral selection and known electronic shifts must identify whether it represents $f_0$, $f_r-f_0$, or a shifted version. Coherent broadening, overlapping <photon polarization>, spatial alignment and temporal overlap are required. Merely producing a broad patch of colored <light> does not establish a usable offset signal.
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