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The carrier-envelope phase locates the carrier oscillation relative to the envelope of an optical pulse. In $E_{\mathrm{real}}(t)=F(t)\cos ({\omega }_{0}t+{\varphi }_{\mathrm{CE}})$ with envelope maximum at zero, it is the displayed phase modulo $2\pi$. Few-cycle dynamics can depend on it even when cycle-averaged light intensity is unchanged. A pulse train's phase increment is related to its carrier-envelope offset frequency; signs depend on the complex-field convention.

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.

Use the complex electric field convention $e^{-i\omega t}$ and take the locked wave amplitudes $a({\omega }_p)$ to be real and nonnegative after extracting their common phase. Linear superposition givesThe leading factor has unit modulus, so it does not change the light intensity. The triangle inequality bounds the remaining modulus by ${\sum }_{p}a({\omega }_p)$. Consecutive occupied normal modes attain this bound together atThere is also a peak at $t=0$. The electric field obeys $E(t+T)=e^{-i{\omega }_{\mathrm{ce}}T}E(t)$, although the light intensity is periodic.

The carrier-envelope phase specifies where the carrier oscillation lies relative to the maximum of an optical pulse's envelope. With $s=t-t_m$ and $E=F(s)e^{-i{\omega }_{0}s+i{\psi }_m}$, the signed complex-field phase at the envelope maximum is ${\psi }_m$. Consequently,If instead the real electric field is written $F(s)\cos ({\omega }_{0}s+{\varphi }_{\mathrm{CE},m})$, then ${\varphi }_{\mathrm{CE},m}=-{\psi }_m$ and the conventionally quoted slip is ${\varphi }_{\mathrm{slip}}=+2\pi {\omega }_{\mathrm{ce}}/\Delta \omega$. These are the same physical shift with opposite phase definitions.

The carrier-envelope phase matters for few-cycle optical pulses, when the largest instantaneous electric field affects strong-field dynamics, for high harmonic generation, and for stabilizing an optical frequency comb. An ordinary cycle-averaged light intensity measurement does not determine it.

Put $b=\Delta {\omega }_b>0$ and $s=t-t_m$. At the peak time, $e^{-ip\Delta \omega t_m}=1$, so the exact comb sum becomesFor many occupied normal modes, $b\gg \Delta \omega$, replace the sum by an integral with density $1/\Delta \omega$. Setting $x=\omega -{\omega }_0$ and evaluating the Fourier transform of a Gaussian givesThus the isolated-optical pulse approximation hasThe optical pulse has a Gaussian function envelope and no chirp. Its carrier angular frequency is ${\omega }_0$, and its light intensity full width at half maximum is $2\sqrt{2\ln 2}/b$. Successive envelope maxima have different carrier-envelope phases, consistent with b.

Approximation in the printed assertion. A discrete optical frequency comb gives a train, not one exact isolated Gaussian pulse. Applying the Poisson summation formula to the ideal sum over all integers instead givesNear $t_m$, the $j=m$ term dominates when $bT\gg 1$. The other terms are exponentially small but not identically zero. A single Gaussian function used globally would decay between $0$ and $T$, whereas the exact light intensity has equal maxima there; this supplies a concrete reason the isolated form cannot be an exact global identity. The extension to negative frequencies is the usual narrow-positive-band analytic-field approximation when ${\omega }_0\gg b$.