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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.^[ref] With the positive-exponential convention derived above, $f_0=f_r\Delta \varphi /(2\pi )$ modulo $f_r$. Changing $f_0$ translates every tooth equally; differentiating $f_n=n{f}_r+f_0$ at fixed $n$ givesThus at $n=2\times 10^6$, 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.

An optical frequency comb has evenly spaced optical spectral lines, or teeth. A mode-locked laser supplies short optical pulses at repetition frequency $f_r$, the tooth spacing. The carrier-envelope offset frequency $f_0$ positions the whole ruler:Only teeth within the emitted optical spectrum are present; the extrapolated intercept need not be an emitted tooth.^[ref]

The achievement associated with John L. Hall and Theodor W. Hänsch is a practical bridge between optical frequencies and countable electronic frequencies. Instead of constructing a separate chain of lasers for each optical measurement, a broad comb supplies many linked reference teeth. Stable lasers, short optical pulses and strong nonlinear optics in optical fibers came together around 1999–2000.^[ref]