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An external periodic modulation of cavity loss, gain or phase fixes the relative phases of its longitudinal cavity modes. A transmission window produced with an electro-optic modulator at the pulse repetition frequency selects a circulating optical pulse. Modulation bandwidth and dispersion constrain the attainable pulse duration.

Adjacent longitudinal cavity modes obey $2nL=m{\lambda }_0$. Neglecting optical dispersion in estimating their spacing,Hence the gain interval accommodatespossible longitudinal cavity modes. The exact integer depends on the positions of the interval endpoints and resonances. This counts available resonances; mode competition and actual laser threshold can reduce the number that oscillates. More accurately, the spacing uses the group refractive index instead of $n$.

A broad multimode spectrum suffers differential delay through optical fiber group velocity dispersion, broadening transmitted pulses. Near $800\mathrm{nm}$, ordinary silica optical fiber attenuation is also larger than in the long-distance windows near $1.3$ and $1.55\mu \mathrm{m}$. These limitations concern this short-wavelength, simple multimode device: suitably engineered single-frequency semiconductor lasers at telecommunications wavelengths are useful communication sources.

Use mode locking to phase-lock many longitudinal cavity modes; their repeated constructive interference produces short optical pulses. A saturable absorber can select this pulsed operation. A $2\mathrm{MHz}$ fundamental repetition rate corresponds to a $500\mathrm{ns}$ round trip, or an optical round-trip length $c/R\simeq 150\mathrm{m}$; a linear air-filled cavity would therefore be about $75\mathrm{m}$ long. A compact alternative is a faster mode-locked oscillator followed by pulse selection and, if required, amplification. Ordinary Q-switching alone generally does not provide $350\mathrm{fs}$ pulses.^[ref]

Mode locking fixes the relative phases of a laser's longitudinal cavity modes, so they interfere constructively in a regularly repeating short pulse rather than producing an uncontrolled temporal waveform. Active mode locking uses an externally driven modulation of loss, gain or phase at the cavity repetition frequency; an electro-optic modulator acting as a periodic transmission gate is one example. Passive mode locking uses the pulse itself to select a high-intensity waveform: a saturable absorber has lower loss at the pulse peak and can lock the phases without an external clock. A semiconductor saturable-absorber mirror is a practical example. Gain recovery, dispersion and saturation must support a stable circulating pulse.

Using the complex-field convention in the question, add the fields of all the oscillating modes. Set $m=p-q_0$, so that ${\omega }_p={\omega }_{q_0}+m\Delta \omega$. ThenThere are $N+1$ terms with the printed inclusive indexing. The sum is periodic with periodwhich is the separation of adjacent pulses when many modes contribute. The carrier factor need not repeat over this interval.

To specify the carrier-envelope phase sign, write the real field near pulse $j$, with $\tau =t-jT$, as $A(\tau )\cos ({\omega }_{q_0}\tau +{\phi }_j)$. Its cosine phase is ${\phi }_j={\omega }_{q_0}jT-{\varphi }_{q_0}$. Since $q_0\Delta \omega T=2\pi q_0$,Equivalently, the phase ${\varphi }_j$ in the complex convention $E_j=A(\tau )e^{-i{\omega }_{q_0}\tau +i{\varphi }_j}$ advances by $-{\omega }_{\mathrm{ce}}T$. This is the negative of the cosine phase, not a different physical prediction. The carrier-envelope offset frequency fixes this pulse-to-pulse slip. It matters for a few-cycle field's peak electric waveform and for locating an optical frequency comb's lines; a stable intensity envelope alone does not fix the slip.

If $\delta /\Delta \omega \gg 1$ and $N\Delta \omega /2\gg \delta$, many closely spaced modes lie under the spectral Gaussian and the cutoffs have negligible weight. Near one pulse, replace the sum by its continuum integral:The Fourier transform of a Gaussian therefore givesThis is a local single-pulse approximation. The discrete spectrum produces its periodic replicas; extending one Gaussian over all time would lose them. The temporal light intensity is proportional to $e^{-{\delta }^2{t}^2/2}$, while the spectral intensity envelope is proportional to $e^{-2{\Omega }^2/{\delta }^2}$. Their full widths at half maximum areIn ordinary frequency this time-bandwidth product is $\Delta {\nu }_L\tau =2\ln 2/\pi \simeq 0.441$. The spectral phases are flat, so this is a transform-limited pulse; chirp would generally increase its temporal width.