= Carrier-envelope offset frequency
{title2=$f_0=f_r\Delta\phi/(2\pi)\pmod{f_r}$}
= Carrier-envelope offset
{synonym}
The <carrier-envelope offset frequency> expresses the carrier's pulse-to-pulse <phase> slip relative to its envelope. Fixing the repetition <frequency> leaves this slip undetermined.<Defining and Measuring Optical Frequencies: The Optical Clock Opportunity – and More by John L. Hall>[Defining and Measuring Optical Frequencies: The Optical Clock Opportunity – and More, p.131: offset]{ref} With the positive-exponential convention derived above, $f_0=f_r\Delta\phi/(2\pi)$ modulo $f_r$. Changing $f_0$ translates every tooth equally; differentiating $f_n=n f_r+f_0$ at fixed $n$ gives
$$
\delta f_n=n\,\delta f_r+\delta f_0.
$$
Thus at $n=2\times10^6$, a chosen $1\,\mathrm{Hz}$ change in spacing moves that tooth by $2\,\mathrm{MHz}$ if $f_0$ stays fixed. The large multiplier is why a stable-looking pulse train is insufficient for accurate optical <metrology>.
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