= Solution
<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 $\omega_p=\omega_{q_0}+m\Delta\omega$. Then
$$
\boxed{E(t)=e^{-i(\omega_{q_0}t-\phi_{q_0})}\sum_{m=-N/2}^{N/2}e^{-(m\Delta\omega/\delta)^2}e^{-im\Delta\omega t}.}
$$
There are $N+1$ terms with the printed inclusive indexing. The sum is periodic with period
$$
\boxed{T=\frac{2\pi}{\Delta\omega},}
$$
which 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 $A(\tau)\cos(\omega_{q_0}\tau+\varphi_j)$. Its cosine phase is $\varphi_j=\omega_{q_0}jT-\phi_{q_0}$. Since $q_0\Delta\omega T=2\pi q_0$,
$$
\boxed{\varphi_{j+1}-\varphi_j=\omega_{\rm ce}T\pmod{2\pi}.}
$$
Equivalently, the phase $\phi_j$ in the complex convention $E_j=A(\tau)e^{-i\omega_{q_0}\tau+i\phi_j}$ advances by $-\omega_{\rm ce}T$. 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.
\Image[/past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2016-mode-locked-gaussian-pulses.png]
{title=Identical pulse envelopes with changing carrier phase; the plotted phase labels use the complex-field convention}
If $\delta/\Delta\omega\gg1$ 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:
$$
\sum_m e^{-(m\Delta\omega/\delta)^2}e^{-im\Delta\omega t}\simeq\frac1{\Delta\omega}\int_{-\infty}^{\infty}e^{-\Omega^2/\delta^2}e^{-i\Omega t}\,d\Omega=\sqrt\pi\frac\delta{\Delta\omega}e^{-(\delta t/2)^2}.
$$
The <Fourier transform of a Gaussian> therefore gives
$$
\boxed{E(t)\simeq\sqrt\pi\frac\delta{\Delta\omega}e^{-(\delta t/2)^2}e^{-i(\omega_{q_0}t-\phi_{q_0})}.}
$$
This 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 $e^{-\delta^2t^2/2}$, while the spectral intensity envelope is proportional to $e^{-2\Omega^2/\delta^2}$. Their <full widths at half maximum> are
$$
\tau=\frac{2\sqrt{2\ln2}}\delta,\qquad\Delta\omega_L=\delta\sqrt{2\ln2},\qquad\boxed{\Delta\omega_L\tau=4\ln2.}
$$
In ordinary frequency this <time-bandwidth product> is $\Delta\nu_L\tau=2\ln2/\pi\simeq0.441$. The spectral phases are flat, so this is a <transform-limited pulse>; <chirp> would generally increase its temporal width.
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