= Solution
{c}
In <second-harmonic generation>, the quadratic <nonlinear polarization> contains products of pairs of comb components. Their <angular frequencies> are
$$
\omega_p+\omega_q=2\omega_{\rm ce}+(p+q)\Delta\omega,
$$
including $2\omega_p$ when $p=q$. Original comb components have <angular frequencies> $\omega_r=\omega_{\rm ce}+r\Delta\omega$, hence
$$
(\omega_p+\omega_q)-\omega_r
=\omega_{\rm ce}+(p+q-r)\Delta\omega.
$$
When the doubled low-<frequency> region overlaps the original high-<frequency> region, choosing $r=p+q$ gives the <beat> at $|\omega_{\rm ce}|$. More generally the nearest positive <beat> has <angular frequency>
$$
\boxed{\omega_{\rm beat,min}
=\min_{k\in\mathbb Z}|\omega_{\rm ce}+k\Delta\omega|,}
$$
which is $|\omega_{\rm ce}|$ for the stipulated small offset. For $0\leq\omega_{\rm ce}<\Delta\omega$ it is $\min(\omega_{\rm ce},\Delta\omega-\omega_{\rm ce})$, 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 $\omega_{\rm ce}=0$ modulo $\Delta\omega$, giving $\boxed{\phi_{\rm slip}=0\pmod{2\pi}}$. A phase-sensitive lock or calibrated approach to zero resolves the sign ambiguity of an unsigned <beat> measurement.<Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis by David J. Jones and others>[2000, comb self-referencing and stabilization of the pulse-to-pulse carrier phase]{ref}
The coherent <optical spectrum> must span an octave, $\omega_{\max}\gtrsim2\omega_{\min}$, 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>.
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