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A. Dubietis coauthored the 1992 paper introducing chirped and stretched pulse parametric amplification in beta barium borate with G. Jonušauskas and A. Piskarskas. Its publisher Abstract reports compressed pulses of $70\mathrm{fs}$ and approximately $0.9\mathrm{GW}$ output power.^[ref]

The approach connects optical parametric amplifier with the central logic of chirped pulse amplification: amplify a temporally stretched signal, then compensate its spectral phase to recover a short optical pulse. The amplified optical bandwidth matters because a narrow spectrum cannot support an arbitrarily short pulse, even with ideal compression.

For an original Gaussian pulse example, an intensity full width at half maximum $\Delta t$ and ordinary-frequency intensity width $\Delta \nu$ satisfy $\Delta t\Delta \nu =2\ln 2/\pi \approx 0.441$ for a transform-limited pulse. Thus a Gaussian spectrum of width $10\mathrm{THz}$ supports a transform-limited intensity duration of about $44.1\mathrm{fs}$. A measured duration of $100\mathrm{fs}$ with that same spectrum would have time-bandwidth product one, exceeding the transform-limited value. Residual spectral phase can account for such broadening, though the product alone does not uniquely reconstruct that phase. The stipulated bandwidth and durations are illustrative, not additional experimental data from the paper.

A transform-limited pulse needs coherent phases across its contributing frequency range. A laser's usable gain bandwidth therefore sets a duration scale through the time-bandwidth product, as displayed. The wavelength interval must first be converted to frequency; the simple conversion $c\Delta \lambda /{\lambda }^2$ is only a narrow-band approximation. Numerical constants depend on pulse shape and width convention: a Gaussian intensity envelope has $\Delta t_{\mathrm{FWHM}}\Delta {\nu }_{\mathrm{FWHM}}=2\ln 2/\pi \simeq 0.441$. A hard gain interval is not automatically a spectral FWHM. Mode locking can arrange the longitudinal modes' phases, while group velocity dispersion must be controlled to approach the transform limit. The cavity round-trip time sets pulse spacing, rather than this bandwidth-limited width.

Choose an initially transform-limited pulse with Gaussian power and complex envelopeHere ${\sigma }_0$ is the standard deviation of the power-weighted time distribution; ${\tau }_0$ is its full width at half maximum. Neither is a field-amplitude width. The spectral amplitude is proportional to $e^{-{\sigma }_0^2{\Omega }^2}$, so the spectral power has standard deviation ${\sigma }_{\Omega }=1/(2{\sigma }_0)$. The Gaussian time-bandwidth product in ordinary frequency is ${\tau }_0\Delta \nu =2\ln 2/\pi \simeq 0.4413$, where both widths are power FWHMs.

Multiplication by $e^{i{\varphi }_2{\Omega }^2/2}$ preserves spectral power and energy. Substituting ${\varphi }^{\prime }={\varphi }_2\Omega$ into the moment identity givesThis is an exact result for a Gaussian spectrum with quadratic spectral phase. It also follows by completing the square in its inverse Fourier transform.

For these chosen inputs, ${\sigma }_0=12.7398\mathrm{fs}$. The stretched pulse has $\tau =278.88\mathrm{fs}$ and root mean square width $118.4282\mathrm{fs}$. Adding $-3000\mathrm{f}{\mathrm{s}}^2$ recovers $30\mathrm{fs}$; leaving $300\mathrm{f}{\mathrm{s}}^2$ gives $40.85\mathrm{fs}$. All phase-only cases retain $1\mathrm{nJ}$.

If GDD is cancelled but ${\varphi }_3=20000\mathrm{f}{\mathrm{s}}^3$ remains, the pulse is asymmetric. Gaussian spectral moments giveThus the centroid is $15.4033\mathrm{fs}$ and the root mean square duration is $25.2354\mathrm{fs}$. Converting this last value to a FWHM with the Gaussian factor would be wrong: the power envelope is no longer Gaussian. The calculation's integrated energy agrees in time and frequency through Parseval's identity, while its centroid and variance agree with these independent moment formulas.

A real amplifier also changes spectral amplitude. For a simple small-signal example, stipulate an intensity gain profileThis specifies finite gain bandwidth, with no saturation or extra amplifier phase. Multiplying two Gaussian spectra givesFor chosen ${\sigma }_g={\sigma }_{\Omega }$, the amplified spectrum is narrower by $\sqrt{2}$, the shortest Gaussian power FWHM becomes $30\sqrt{2}=42.43\mathrm{fs}$, and the total energy is $G_{0}U/\sqrt{2}$. Perfect spectral phase correction cannot recreate the suppressed spectral amplitude. Real gain narrowing and saturation require a material-dependent amplifier model; a constant multiplier is only a limited idealization. Gérard Mourou's 2018 lecture discusses gain narrowing as a limit on the shortest amplified pulses.^[ref]

A mode-locked laser maintains definite relative wave phases among many longitudinal resonator modes. Their coherent sum forms a repeating train of short optical pulses, rather than merely producing a broad optical spectrum.

For an ordinary linear resonator, the repetition interval is approximately the round-trip travel time. More phase-aligned spectral bandwidth permits shorter pulses, subject to the time-bandwidth product and spectral shape. A photodetector can measure pulse-train timing, while an intensity autocorrelation estimates much shorter pulse durations under a stated model. Neither a broad spectrum nor a fast periodic intensity signal alone uniquely reconstructs the pulse's complex electric field.

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.

In mode locking, fixed relative phases between many longitudinal normal modes produce constructive interference during a short part of each optical resonator round trip. The round-trip time sets the repetition period. The available coherent gain bandwidth, the number of locked normal modes, the spectral phase, and residual group velocity dispersion determine the optical pulse duration.

For $M$ consecutive equally weighted normal modes of angular frequency spacing $\Delta \omega$, their wave amplitude sum is proportional toIts central peak has width of order $1/(M\Delta \omega )$, while its repetition period is $2\pi /\Delta \omega$. A transform-limited pulse therefore becomes shorter as more coherent gain bandwidth is used. This mechanism can give picosecond or femtosecond optical pulses: it does not require depletion of the whole stored population inversion during each peak, as Q-switching does. Mode locking generally accesses a much shorter time-bandwidth product scale than the gain-discharge time of Q-switching.