Mode-Locking

A laser cavity does not oscillate on a single frequency but on a whole comb of longitudinal modes, evenly spaced by the free spectral range Δν = c/2L that the resonator articles fix, and spread across whatever band the gain medium amplifies. Left alone, these modes run with independent, drifting phases: their fields beat against one another into a noisy, quasi-continuous output that flickers but never sharpens. Mode-locking clamps those phases into a fixed relationship. The instant the phases are made equal — or made to ramp linearly with mode number — the modes add in step at one moment each round trip, building a single towering spike of light that circulates and leaks out through the coupler once per trip. What makes the spike so sharp is a Fourier duality between frequency and time. N equally spaced, phase-locked modes are the discrete twin of a frequency comb, and the transform of a comb is a train of pulses: the spacing in time is the reciprocal of the mode spacing, T = 2L/c, while each pulse's width is the reciprocal of the total locked bandwidth, τ_p ≈ 1/(NΔν). Lock more modes — find a broader-gain medium — and the pulse shrinks. This is why titanium-sapphire, amplifying across nearly 100 THz, mode-locks to a few-femtosecond flash while a narrowband gas laser reaches only picoseconds. Everything that follows — active modulators, saturable absorbers, Kerr lenses, Haus's master equation — is engineering to make the pulsed state win over the continuous.

⚡ Key Takeaways

Mode-locking fixes the relative phases of the many longitudinal modes within a laser's gain bandwidth so they add coherently once per round trip, radiating a periodic train of ultrashort pulses.
The repetition period equals the round-trip time $T=2L/c$ (mode spacing $\Delta\nu=c/2L$ ), while the duration scales as $\tau_p\sim 1/\Delta\nu_{\mathrm{osc}}$ , the inverse of the total locked bandwidth.
Locking $N$ equal modes gives a peak $N$ times the average and $N^2$ times a single mode — the same energy squeezed into pulses shorter than the round trip by $N$ .
Active mode-locking uses a modulator driven at the mode spacing; passive mode-locking uses an intensity-dependent loss (saturable absorber, SESAM, or Kerr-lens self-focusing) that responds on the pulse's own timescale and wins by orders of magnitude.
A transform-limited pulse obeys $\tau_p\,\Delta\nu\gtrsim K$ with $K\approx0.441$ (Gaussian) or $0.315$ ( $\operatorname{sech}^2$ ); broadband Ti:sapphire ( $\Delta\nu\sim100$ THz) reaches sub-10-fs pulses.
Haus's master equation folds gain, loss, bandwidth, dispersion, and self-phase modulation into one envelope equation whose stationary $\operatorname{sech}$ soliton sets the steady pulse.
Key idea: mode-locking is constructive interference engineered in the frequency domain — force $N$ longitudinal modes into a fixed phase relation and their Fourier sum becomes a pulse train of period $2L/c$ and width $\sim 1/(N\Delta\nu)$ ; the broader the locked bandwidth, the shorter the pulse.

💡 The Core Insight

Mode-locking is constructive interference engineered in the frequency domain: force $N$ longitudinal modes into a fixed phase relation and their Fourier sum becomes a pulse train with period $2L/c$ and width $\sim 1/(N\Delta\nu)$ — the broader the locked bandwidth, the shorter the pulse.

The physical picture

Picture each longitudinal mode as a phasor turning at its own frequency. In a free-running laser the phasors point every which way, so their sum wanders about a modest average — CW-like noise. Mode-locking pins them to line up at one instant each round trip; there they sum coherently into a tall spike, and everywhere else they interfere to almost nothing. The spike circulates and is emitted as a pulse every $2L/c$ . Nothing is added — the same modes, the same energy — but the phase discipline reshapes a smear into a sequence of intense flashes.

Figure

The Fourier duality at the heart of mode-locking: N phase-locked modes spaced by Δν = 100 MHz in frequency transform into a pulse train spaced by T = 2L/c = 10 ns in time, each pulse narrower than the period by the number of locked modes, τ_p ≈ T/N

The time–bandwidth reciprocity is the whole plot: a wide, flat, phase-coherent spectrum transforms into a narrow pulse. Double the locked modes and you double the bandwidth, halve the width, and lift the peak as $N^2$ . A laser can never make a pulse shorter than the inverse of its gain bandwidth — the hunt for ultrashort pulses is a hunt for broadband gain. And the phases must genuinely lock: a plain multimode laser has the same bandwidth yet gives only noise, because its phases drift at random.

Figure

The only thing mode-locking changes is the phases. With random phases the same N modes beat into a noisy quasi-continuous output averaging N times one mode intensity; with the phases locked, the identical modes over the identical bandwidth collapse into a clean pulse train whose peaks reach N² times one mode — here 49× for N = 7 — the same energy concentrated in time

A self-consistent pulse must reproduce itself every round trip: whatever does the locking must trim the pulse edges just enough each pass to counter the spreading from finite bandwidth and dispersion, so it settles where shortening balances broadening. Passive nonlinearities shorten on the pulse's own femtosecond timescale, which is why they leave active modulators far behind.

The core equations

The starting point is bookkeeping: write the total field as the sum over every oscillating longitudinal mode, each with its own amplitude, evenly spaced frequency, and — what matters — its own phase. Mode-locking is entirely a statement about that last symbol: free-running, the $\phi_n$ drift independently; locked, they are pinned to a common value or linear ramp.

E(t)=\sum_{n=-M}^{M} E_n\, e^{i(\omega_0+n\,\Delta\omega)t + i\phi_n} \qquad (1)

where: $E_n$ — complex amplitude of mode $n$ ; $\omega_0$ — central optical angular frequency; $\Delta\omega=2\pi\,\Delta\nu=2\pi c/2L$ — angular mode spacing (the FSR); $\phi_n$ — phase of mode $n$ , the knob mode-locking fixes; $N=2M+1$ — number of modes. The output is a beat of $N$ tones, and only their phase relationship decides whether that beat is noise or a pulse. Set every amplitude equal and every phase to zero and the sum collapses to a Dirichlet kernel — the function that governs an $N$ -slit grating, here in time rather than angle. Its square is the instantaneous intensity.

I(t)=|E_0|^2\,\frac{\sin^2(N\,\Delta\omega\, t/2)}{\sin^2(\Delta\omega\, t/2)},\qquad I_{\mathrm{peak}}=N^2|E_0|^2 \qquad (2)

where: $E_0$ — common mode amplitude; $N$ — number of locked modes; $\Delta\omega$ — angular mode spacing. The fast numerator over the slow denominator spikes wherever the denominator vanishes — a picket fence of pulses reaching $N^2$ times a single mode's intensity. The kernel spikes whenever the slow denominator vanishes, recurring at a fixed interval — the repetition period, nothing more than the round-trip time the circulating spike takes to return to the output coupler.

T=\frac{2\pi}{\Delta\omega}=\frac{1}{\Delta\nu}=\frac{2L}{c} \qquad (3)

where: $T$ — time between pulses; $L$ — one-way length of a linear cavity; $c$ — speed of light. The repetition rate $f_{\mathrm{rep}}=1/T=c/2L$ is identical to the mode spacing — the same number in the frequency domain. Between two spikes the fast numerator oscillates $N$ times, so each spike is narrower than the period by $N$ — the width is set by the bandwidth, not the mode spacing.

\tau_p \approx \frac{T}{N} = \frac{1}{N\,\Delta\nu}=\frac{1}{\Delta\nu_{\mathrm{osc}}} \qquad (4)

where: $\tau_p$ — pulse duration (peak-to-first-null, or FWHM up to a shape factor); $\Delta\nu_{\mathrm{osc}}=N\,\Delta\nu$ — total oscillating bandwidth. The lesson is blunt: to shorten the pulse, widen the spectrum. No pulse can be shorter than the reciprocal of its spectral width; the constant depends on shape, and a transform-limited pulse sits at equality — the same bound that ties any conjugate Fourier pair.

\tau_p\,\Delta\nu_{\mathrm{osc}} \ge K,\quad K=\begin{cases}0.441 & \text{Gaussian}\\ 0.315 & \operatorname{sech}^2\end{cases} \qquad (5)

where: $K$ — shape-dependent time–bandwidth constant (time FWHM times frequency FWHM); $\Delta\nu_{\mathrm{osc}}$ — spectral FWHM. A measured $\tau_p\,\Delta\nu$ well above $K$ signals residual chirp — bandwidth present but not usefully compressed. To make the pulsed state self-sustaining, a passive cavity needs an element that punishes the weak continuous background more than the intense pulse. A saturable absorber does exactly that: opaque to low intensity, it bleaches transparent once the light is strong.

q(I)=\frac{q_0}{1+I/I_{\mathrm{sat}}} \qquad (6)

where: $q(I)$ — intensity-dependent absorber loss; $q_0$ — small-signal (unbleached) loss; $I$ — instantaneous intensity; $I_{\mathrm{sat}}$ — absorber saturation intensity. The pulse peak digs itself a low-loss window the CW background cannot reach, and grows at that background's expense. In steady state the circulating pulse must reproduce itself each round trip despite gain, loss, finite bandwidth, dispersion, and nonlinearity all acting at once. Haus's master equation packs every per-round-trip effect into a single equation for the envelope.

T_R\frac{\partial A}{\partial T}=\Big[(g-l)+\big(D_g+iD_2\big)\frac{\partial^2}{\partial t^2}-i\delta_K|A|^2+\gamma|A|^2\Big]A \qquad (7)

where: $A(T,t)$ — pulse envelope, $T$ the slow round-trip time and $t$ the fast time within a pulse; $T_R$ — round-trip time; $g,l$ — saturated gain and linear loss per pass; $D_g=g/\Omega_g^2$ — gain dispersion from finite bandwidth ( $\Omega_g$ the gain HWHM); $D_2$ — group-velocity dispersion; $\delta_K$ — self-phase-modulation (Kerr) coefficient; $\gamma$ — self-amplitude-modulation (fast saturable absorber) coefficient. Each bracketed term is one effect per pass; the stationary pulse is where they cancel.

Derivation APulse train from equal phase-locked modes

The claim is sharp: force $N$ equally spaced modes to share one amplitude and one phase, and their superposition is not merely "peaked" but a precisely defined periodic train whose peak is exactly $N^2$ times a single mode, whose average is $N$ times a single mode, and whose width is the round-trip time over $N$ . We collapse the geometric series and read off the peak, period, and width in turn — every headline number from one line of algebra.

Step 1. Impose $E_n=E_0$ , $\phi_n=0$ in equation (1) and pull out the carrier, leaving $E(t)=E_0 e^{i\omega_0 t}\sum_{n=-M}^{M} e^{in\Delta\omega t}$ . Why: the carrier only rides the envelope at $\omega_0$ ; the interference that builds the pulse lives entirely in the residual sum.

Step 2. The residual sum is a finite geometric series of $N=2M+1$ terms, closing into the Dirichlet kernel $\sum_{n=-M}^{M} e^{in\theta}=\dfrac{\sin(N\theta/2)}{\sin(\theta/2)}$ , $\theta=\Delta\omega t$ . Why: the same kernel as the $N$ -slit grating, so a mode-locked pulse sharpens by the same mathematics as a grating's maxima.

Step 3. Square the envelope for the intensity, $I(t)=|E_0|^2\dfrac{\sin^2(N\Delta\omega t/2)}{\sin^2(\Delta\omega t/2)}$ . Why: the detector and absorber respond to intensity, not field, so the observable train is the squared kernel.

Step 4. As $\theta\to 0$ both sines go linear, their ratio approaches $N$ , and $I_{\mathrm{peak}}=N^2|E_0|^2$ ; these maxima recur at $t=mT$ , $T=2\pi/\Delta\omega=2L/c$ . Why: one limit fixes both the $N^2$ peak and the round-trip repetition period.

Step 5. The first zero of the numerator sits at $N\Delta\omega t/2=\pi$ , giving width $\tau_p=2\pi/(N\Delta\omega)=T/N=1/(N\Delta\nu)$ . Why: the width is the reciprocal of the full locked bandwidth — "broader spectrum, shorter pulse".

Step 6. Energy conservation makes the time-average just $N$ modes of $|E_0|^2$ , so $I_{\mathrm{avg}}=N|E_0|^2$ and $I_{\mathrm{peak}}/I_{\mathrm{avg}}=N$ . Why: locking rearranges energy without creating it — the peak rises by $N$ because the pulse fills a fraction $1/N$ of the period.

Derivation BActive amplitude-modulation mode-locking and the Kuizenga-Siegman width

How short the pulse becomes is settled by a tug of war every round trip. The finite gain bandwidth acts as a spectral filter that trims the pulse's frequency wings and broadens it in time, while the modulator opens a brief window synchronized to the round trip, trimming its temporal wings and narrowing it. The steady width is where the two balance, and the Kuizenga-Siegman formula exposes a weak bandwidth scaling that keeps active mode-locking at picoseconds while passive schemes reach femtoseconds.

Step 1. Model the finite gain as a parabolic spectral filter, $\tilde g(\omega)\approx g\left(1-\omega^2/\Omega_g^2\right)$ , $\Omega_g=\pi\Delta\nu_g$ . Why: a Gaussian through a curved-top gain profile loses its spectral edges and stretches in time — the broadening the modulator must fight.

Step 2. Expand the modulator about its transmission peak, $T_{\mathrm{mod}}(t)=1-\delta[1-\cos\Omega_m t]\approx 1-\tfrac12\delta\,\Omega_m^2 t^2$ , driven at $\Omega_m=2\pi f_m$ , $f_m=f_{\mathrm{rep}}$ . Why: near the loss minimum the gate is parabolic in time, so light arriving early or late is cut — the modulator scissors the temporal wings each pass.

Step 3. Take a Gaussian ansatz $A(t)=A_0\exp(-t^2/2\tau^2)$ , which both parabolic operations preserve to leading order. Why: the Gaussian is their fixed point, so self-consistency reduces to one equation in $\tau$ .

Step 4. Demand steady state — filter broadening equals modulator narrowing per pass — giving $\tau^4\propto g/(\delta\,\Omega_g^2\,\Omega_m^2)$ . Why: at balance the width neither grows nor shrinks trip to trip — a stationary pulse.

Step 5. Solving gives $\displaystyle \tau_p \approx \left(\frac{2\ln 2}{\pi}\right)^{1/2}\left(\frac{g}{\delta}\right)^{1/4}\frac{1}{\sqrt{f_m\,\Delta\nu_g}}$ . Why: the width falls only as the inverse square root of the bandwidth — the weak scaling that keeps active mode-locking short of the transform limit $1/\Delta\nu_g$ .

Derivation CThe soliton pulse of the Haus master equation

The shortest pulses come from a balance that lives in the optical phase, not the amplitude. In a medium with anomalous dispersion, Kerr self-phase modulation and group-velocity dispersion cancel order by order — as for a fibre soliton — so the pulse propagates without spreading, while gain, loss, and a fast saturable absorber stabilize it and set its energy. Reducing the master equation to its conservative core recovers the nonlinear Schrödinger equation, whose fundamental $\operatorname{sech}$ solution shrinks as the pulse energy grows — the soliton area theorem.

Step 1. Seek a stationary solution $A(T,t)=A(t)\,e^{i\psi T/T_R}$ so the slow derivative becomes a pure phase, turning equation (7) into $i\psi A=\big[(g-l)+(D_g+iD_2)\partial_t^2-i\delta_K|A|^2+\gamma|A|^2\big]A$ . Why: a steady pulse changes only its overall phase per round trip, never its shape.

Step 2. Separate the conservative, purely imaginary part, $iD_2 A'' - i\delta_K|A|^2 A=i\psi A$ : the nonlinear Schrödinger equation. Why: the real terms $(g-l)$ , $D_g$ , $\gamma$ only feed and shape slowly; the form is set by dispersion balancing Kerr nonlinearity.

Step 3. Its fundamental solution is $A(t)=A_0\operatorname{sech}(t/\tau)$ ; substituting fixes $|D_2|/\tau^2=\tfrac12\delta_K A_0^2$ , requiring anomalous dispersion $D_2<0$ . Why: only the right sign lets self-phase modulation and group delay cancel, so the pulse holds its shape.

Step 4. With pulse energy $E_p=2A_0^2\tau$ the balance becomes $\tau=\dfrac{4|D_2|}{\delta_K\,E_p}$ . Why: duration falls with energy and nonlinearity — pump harder and the soliton compresses itself, while $\gamma$ and gain filtering suppress the noise that would seed a second pulse.

Limiting cases

Case	Result	Meaning
Many modes, equal phase	$I_{\mathrm{peak}}/I_{\mathrm{avg}} = N,\quad I_{\mathrm{peak}}/I_{\mathrm{single}} = N^2$	Locking $N$ equal modes concentrates energy: peak is $N\times$ the average and $N^2\times$ a single mode — same energy, reshaped in time.
Transform-limited pulse	$\tau_p\,\Delta\nu \gtrsim K,\ K\approx 0.441\ (\text{Gaussian}),\ 0.315\ (\operatorname{sech}^2)$	Shortest duration a given bandwidth allows; equality holds only for an unchirped pulse of that shape.
Repetition and period	$f_{\mathrm{rep}}=\dfrac{c}{2L},\qquad T=\dfrac{2L}{c}$	Fixed by the round-trip time; a 1.5 m linear cavity gives $f_{\mathrm{rep}}=100$ MHz, $T=10$ ns.
Active vs passive shortening	$\tau_{\mathrm{active}}\propto(\Delta\nu_g\, f_m)^{-1/2},\qquad \tau_{\mathrm{passive}}\propto \Delta\nu_g^{-1}$	Active (Kuizenga-Siegman) scaling is weak in bandwidth; passive/soliton reaches the full transform limit and so goes orders of magnitude shorter.

⚠️ More oscillating modes automatically means a shorter pulse

Bandwidth alone buys you nothing — the phases must be locked. An ordinary multimode laser carries the same $N$ modes over the same $\Delta\nu_{\mathrm{osc}}$ as its mode-locked twin, yet its output is noise, because the phases $\phi_n$ drift independently; only when a mechanism pins them does the Fourier sum become a clean pulse of width $1/\Delta\nu_{\mathrm{osc}}$ . A related trap is confusing mode-locking with Q-switching. Q-switching spoils and restores the cavity $Q$ to dump the stored inversion into one giant nanosecond pulse per cycle, at low repetition rate; mode-locking phase-locks many continuously oscillating modes into a fast femto- to picosecond train at $c/2L$ . Different physics, timescales, and energetics — do not swap the words.

Worked examples

1 · Anatomy of a Ti:sapphire pulse train (confidence builder)

A Kerr-lens mode-locked Ti:sapphire oscillator has a 1.5 m linear cavity, gain bandwidth $\Delta\nu_g\approx100$ THz near 800 nm, and 400 mW average power. Find its repetition rate, number of locked modes, transform-limited duration, and peak power.

Step 1. From the round trip, $f_{\mathrm{rep}}=c/2L=(3.00\times10^8)/(2\times1.5)=\mathbf{100\ MHz}$ , so $T=1/f_{\mathrm{rep}}=\mathbf{10\ ns}$ .

Step 2. The modes under the gain bandwidth number $N=\Delta\nu_g/\Delta\nu=(1\times10^{14})/(1\times10^{8})=\mathbf{10^6\ modes}$ — a million phasors clamped into step.

Step 3. The transform-limited $\operatorname{sech}^2$ duration is $\tau_p=K/\Delta\nu_g=0.315/(1\times10^{14})=\mathbf{3.15\ fs}$ . One optical cycle at 800 nm lasts $\lambda/c=2.67$ fs, so this is barely more than a single cycle — a genuine few-cycle pulse, matching real sub-10-fs Ti:sapphire oscillators. (The coarser $1/\Delta\nu_g\approx10$ fs from equation (4) is the first-null width; the factor $K$ sharpens it.)

Step 4. Pulse energy is average power over rate, $E_p=P_{\mathrm{avg}}/f_{\mathrm{rep}}=0.400/(1\times10^{8})=\mathbf{4\ nJ}$ . For a $\operatorname{sech}^2$ pulse the peak power is $P_{\mathrm{peak}}\approx0.88\,E_p/\tau_p=0.88\times(4.0\times10^{-9})/(3.15\times10^{-15})\approx\mathbf{1.1\ MW}$ . A 0.4 W average becomes over a megawatt of peak — a factor of order $N\sim10^6$ , the peak-to-average ratio the Dirichlet kernel predicts.

2 · Why Kerr-lens beats a modulator by a factor of a thousand

Mode-lock the same Ti:sapphire medium ( $\Delta\nu_g=100$ THz) two ways: actively, with a modulator at $f_m=f_{\mathrm{rep}}=100$ MHz and $g\approx\delta$ , and passively at the transform limit. Compare durations.

Step 1. Evaluate the Kuizenga-Siegman width from Derivation 2. The prefactor is $(2\ln 2/\pi)^{1/2}=0.664$ , and with $g\approx\delta$ the factor $(g/\delta)^{1/4}=1$ .

Step 2. The bandwidth factor is $(f_m\,\Delta\nu_g)^{-1/2}=(10^{22})^{-1/2}=10^{-11}$ s, so $\tau_{\mathrm{active}}=0.664\times10^{-11}\ \text{s}\approx\mathbf{6.6\ ps}$ .

Step 3. The passive transform limit from Example 1 is $\tau_{\mathrm{passive}}=\mathbf{3.15\ fs}$ , so the ratio is $6.6\times10^{-12}/3.15\times10^{-15}\approx\mathbf{2100}$ .

Step 4. This is the whole reason Kerr-lens mode-locking exists: active modulation reaches only $\sim6.6$ ps, using a vanishing sliver of the 100 THz because its shortening scales as the weak $(\Delta\nu_g\,f_m)^{-1/2}$ . The Kerr lens, an effective saturable absorber as fast as the pulse itself, drives the width to the full transform limit $\propto1/\Delta\nu_g$ — about three orders of magnitude shorter from the identical medium.

Physics of the portrait

Mode-locking has two founding lineages — the experimentalists who first coaxed lasers into pulsing, and the theorist who explained, in one equation, why any of it works. The portrait belongs to the second.

Hermann Haus

1925–2003 · the master-equation theory of mode-locking

Hermann A. Haus, born in Ljubljana in 1925 and for decades an Institute Professor at MIT, gave ultrafast optics the theoretical spine it lacked. Through the 1970s the field had a zoo of mechanisms — modulators, saturable dyes, colliding pulses — but no unifying account of why a pulse, rather than a noisy continuum, was the natural steady state. Haus supplied it: in a pair of 1975 papers he treated a fast and then a slow saturable absorber as a per-round-trip balance of shaping effects, and over two decades distilled the subject into what is now called the Haus master equation — a single envelope equation in which gain, loss, finite bandwidth, dispersion, self-phase modulation, and self-amplitude modulation each appear as one term. Its stationary $\operatorname{sech}$ solution is the soliton laser pulse, predicting duration, chirp, and stability from material constants alone. With Erich Ippen and James Fujimoto he tied the theory to the Ti:sapphire and fibre lasers of modern ultrafast science and to the quantum noise that limits a comb's coherence. Awarded the U.S. National Medal of Science, he was still refining the formalism — his 2000 review appeared three years before his death — when the comb revolution it underpins earned its own Nobel Prize.

⏳ History

1964

— Hargrove, Fork and Pollack demonstrate active mode-locking of a helium-neon laser by synchronous intracavity acousto-optic modulation

1966

— DeMaria and co-workers achieve passive mode-locking with a saturable dye, producing the first picosecond pulses

1972

— Ippen, Shank and Dienes passively mode-lock a continuous-wave dye laser, opening steady femtosecond-scale operation

1981

— Fork, Greene and Shank build the colliding-pulse mode-locked dye laser and break the 100-femtosecond barrier

1991

— Spence, Kean and Sibbett discover Kerr-lens mode-locking in titanium-sapphire, soon reaching sub-10-femtosecond pulses

2005

— Hänsch and Hall share the Nobel Prize for the optical frequency comb, the direct offspring of the mode-locked pulse train

Where this lives today

The most consequential thing a mode-locked laser makes is not a pulse but a ruler. In the frequency domain the train is a comb of razor-sharp lines at $\nu_m=f_{\mathrm{ceo}}+m\,f_{\mathrm{rep}}$ , with $f_{\mathrm{rep}}=c/2L$ set by the cavity and $f_{\mathrm{ceo}}$ the carrier-envelope-offset frequency. Stabilize those two radio-frequency numbers — the offset via $f$ -to- $2f$ self-referencing, which needs an octave of bandwidth and hence the broadest mode-locking — and every optical line inherits microwave-clock accuracy. This is the engine of optical atomic clocks reaching uncertainties near $10^{-18}$ , of astro-combs that calibrate exoplanet spectrographs, and of dual-comb spectrometers. At the other extreme, the few-cycle pulses broadband mode-locking allows are the seed of attosecond science: focused into a gas jet, a carrier-envelope-stabilized Ti:sapphire pulse drives high-harmonic generation, and the extreme-ultraviolet burst can be tens of attoseconds long — the shortest events humans make, used to film electrons rearranging inside atoms. The same megawatt-peak trains machine materials athermally, cut corneas in LASIK surgery, and drive two-photon microscopy. Every one rests on this article's reciprocity: lock the phases, widen the band, and time collapses into a pulse.

Going further

Texts. Saleh and Teich, Fundamentals of Photonics, 3rd ed., Chs. 15–16 · Siegman, Lasers (1986), Chs. 27–28 · Demtröder, Laser Spectroscopy, Vol. 1, Ch. 6 · Hecht, Optics, 5th ed., §13 · Loudon, The Quantum Theory of Light, 3rd ed. · Haus, "Mode-locking of lasers," IEEE J. Sel. Top. Quantum Electron. 6, 1173 (2000).

Provenance. Equations follow Saleh and Teich (chs. 15–16), Demtröder (ch. 6), and Siegman (chs. 27–28); the master-equation and soliton results follow Haus (2000). A verifier should recompute: $I_{\mathrm{peak}}=N^2|E_0|^2$ and $\tau_p=1/(N\Delta\nu)$ ; $f_{\mathrm{rep}}=c/2L=100$ MHz for $L=1.5$ m; $K=0.441$ (Gaussian) and $0.315$ ( $\operatorname{sech}^2$ ); $\tau_p=0.315/(100\ \mathrm{THz})\approx3.2$ fs against the 2.67 fs cycle at 800 nm; the active $\tau_p\propto(f_m\Delta\nu_g)^{-1/2}\approx6.6$ ps; and the soliton $\tau\propto|D_2|/E_p$ . Numerical examples are standard textbook estimates.

Prerequisites. Longitudinal Modes and Free Spectral Range · Optical Resonators and Cavity Modes · Population Inversion and Optical Gain · Cavity Q Factor and Photon Lifetime. Leads to. Optical Frequency Combs · Nonlinear Optics and Self-Phase Modulation · Group-Velocity Dispersion and Pulse Broadening · Ultrafast Spectroscopy and Attosecond Physics.

Ask the tutor: Ask me to walk through how a random-phase multimode spectrum and a mode-locked one share identical bandwidth yet look completely different in time.

Why does the pulse repeat once per round trip, not once per optical cycle?

Because a single pulse circulates in the cavity and taps the output coupler once each lap. The carrier oscillates at hundreds of terahertz, but the envelope — the pulse itself — is a localized packet that travels down the cavity, reflects, and returns in $T=2L/c$ ; each time it hits the output mirror a fraction escapes, so the train is spaced by that round-trip time. In the frequency domain the same fact is the mode spacing $\Delta\nu=c/2L$ . The carrier lives inside each pulse; the round trip sets the rhythm between them.

Where does the enormous peak power come from if no energy is added?

Nowhere new — mode-locking only rearranges the energy the laser already produces. A free-running laser spreads its power evenly; locking the phases forces it into a window of width $\tau_p\approx T/N$ , a fraction $1/N$ of the period, so the peak rises by $N$ to hold the average fixed. Relative to a single mode the peak is $N^2$ higher because $N$ fields add coherently in amplitude and intensity is amplitude squared. In the worked example a steady 0.4 W becomes a 1.1 MW peak because each 10 ns interval's energy is squeezed into about 3 fs.

What actually locks the phases physically?

Some element must make the pulsed state cheaper than the continuous one, so the laser self-organizes into it. An active modulator opens a low-loss window synchronized to the round trip; only light arriving there survives, forcing a pulse and imposing sidebands that pull neighbouring modes into phase. A passive saturable absorber needs no external timing: a strong fluctuation bleaches it, sees lower loss, grows at the weaker background's expense, and sharpens each pass until a full pulse builds. Either way the mechanism is a nonlinear preference for a peaked field, and phase coherence is its signature.

Why do broadband gain media give shorter pulses?

Because pulse duration is the reciprocal of the locked bandwidth, $\tau_p\approx1/\Delta\nu_{\mathrm{osc}}$ , which cannot exceed what the medium amplifies. A narrow-linewidth gas laser supports only a thin slice of modes, so even perfectly locked it yields picosecond pulses; titanium-sapphire amplifies across nearly 100 THz, locking a million modes to reach a few femtoseconds. This is the transform limit of equation (5): to compress in time you must spread in frequency, which is why the quest for shorter pulses has always been a quest for broader gain media.

Can a mode-locked pulse be shorter than the transform limit?

No. The time-bandwidth product $\tau_p\,\Delta\nu_{\mathrm{osc}}\ge K$ is a hard Fourier bound: for a given spectral width the shortest pulse is the transform-limited one, reached only when unchirped. Real pulses come out longer because dispersion and self-phase modulation leave a chirp — the colours are all present but arrive at slightly different times. You cannot beat the limit, but you can reach it by removing the chirp with external compression (prism pairs, chirped mirrors, gratings). If $\tau_p\,\Delta\nu$ comes out well above $K$ , the fix is compression, not more bandwidth.

Why is anomalous dispersion needed for soliton mode-locking?

Because the soliton is a truce between two shape-distorting effects that cancel only with the right sign of dispersion. Kerr self-phase modulation always chirps the pulse the same way — red-shifting the leading edge, blue-shifting the trailing edge. To undo that, dispersion must delay the blue relative to the red so the colours reconverge, which is what anomalous group-velocity dispersion ( $D_2<0$ ) does; when they balance, the pulse is a fundamental $\operatorname{sech}$ soliton of width $\tau=4|D_2|/(\delta_K E_p)$ . In the normal regime the two add and the pulse smears — which is why femtosecond oscillators engineer a net anomalous cavity dispersion with chirped mirrors or prism pairs.