Mode locking

Mode locking fixes relative phases between longitudinal normal modes of a laser, allowing short optical pulses. The same optical spectrum with wandering relative phases would not supply that stable pulse train.^[ref]

Here is an original finite-mode model, with the convention that a positive frequency has time dependence $e^{2\pi ift}$:These field coefficients follow a Gaussian function and have aligned phases at $t=0$. Choose $f_0=f_r/4$ and write $T=1/f_r$. Since $e^{2\pi in}=1$ for every integer $n$,The pulse light intensity repeats every $T$ although the field needs four periods to repeat. At the pulse centers, $E(0)=1$, $E(T)=i$, $E(2T)=-1$ and $E(3T)=-i$. A detector of light intensity alone therefore cannot distinguish this offset from zero. This model has far fewer teeth than a practical optical comb; its normalized time and frequency scales are chosen to make the carrier oscillations visible.

A short proof of the converse uses the Fourier series. If $E(t+T)=e^{i\Delta \varphi }E(t)$, then $E(t)e^{-i\Delta \varphi t/T}$ is periodic with period $T$. Its Fourier series has frequencies $n/T$; restoring the removed factor shifts every tooth by $\Delta \varphi /(2\pi T)$. A finite observation time broadens the spectral peaks; it does not change this ideal infinite-train relation.

The circulating optical pulse itself changes the cavity response so that a short, intense waveform is favored. A saturable absorber reduces loss at high intensity; a semiconductor saturable-absorber mirror is one implementation. No external periodic timing drive is required, although stable operation still depends on gain and loss dynamics.

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.

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.