 Solution

In second-harmonic generation, the quadratic nonlinear polarization contains products of pairs of comb components. Their angular frequencies areincluding $2{\omega }_p$ when $p=q$. Original comb components have angular frequencies ${\omega }_r={\omega }_{\mathrm{ce}}+r\Delta \omega$, henceWhen the doubled low-frequency region overlaps the original high-frequency region, choosing $r=p+q$ gives the beat at $|{\omega }_{\mathrm{ce}}|$. More generally the nearest positive beat has angular frequencywhich is $|{\omega }_{\mathrm{ce}}|$ for the stipulated small offset. For $0\le {\omega }_{\mathrm{ce}}<\Delta \omega$ it is $\min ({\omega }_{\mathrm{ce}},\Delta \omega -{\omega }_{\mathrm{ce}})$, provided the corresponding teeth are in the overlap.

In comb self-referencing, overlap the doubled red part and the original blue part on a photodiode. The beat measures the carrier-envelope offset frequency. Feedback on an intracavity dispersion control or pump setting can set ${\omega }_{\mathrm{ce}}=0$ modulo $\Delta \omega$, giving $\boxed{{\varphi }_{\mathrm{slip}}=0(\mathrm{mod}2\pi )}$. A phase-sensitive lock or calibrated approach to zero resolves the sign ambiguity of an unsigned beat measurement.^[ref]

The coherent optical spectrum must span an octave, ${\omega }_{\max }\gtrsim 2{\omega }_{\min }$, with sufficient overlapping spectral strength after second-harmonic generation. This calls for very short broadband optical pulses, or coherent external spectral broadening. The two branches must preserve optical coherence, overlap in time and polarization, and have adequate phase matching; bandwidth alone does not guarantee a detectable beat.