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Table of Contents

9.1 Basic Principle
9.2 Coupled-Cavity and Additive-Pulse Modelocking
9.3 Slow Saturable Absorber with Dynamic Gain Saturation
9.4 Fast Saturable Absorbers and Kerr-Lens Modelocking
9.5 Soliton Modelocking with Slow Absorbers
9.6 Q-Switching Instabilities
9.7 Modern Solid-State Modelocked Lasers

9 Passive Modelocking

Passive modelocking uses an intensity-dependent element inside the cavity to favour pulsed operation without an external modulation signal. The most common mechanisms are a real saturable absorber, such as a SESAM, or an artificial saturable absorber, such as Kerr-lens modelocking.

Compared with active modelocking, passive modelocking can produce much shorter pulses. The reason is that the loss gate is generated by the pulse itself. The stronger and shorter the pulse becomes, the more effectively it can reduce its own loss. This nonlinear feedback is much stronger than a sinusoidal RF modulation.

9.1 Basic Principle

In passive modelocking, the cavity is arranged so that a short pulse experiences lower roundtrip loss than weak cw light or noise. Once a fluctuation becomes slightly more intense, it saturates the absorber more strongly, experiences less loss, and grows relative to the background. This positive feedback can lead to a stable circulating pulse.

For self-starting operation, the absorber should not be strongly saturated by the cw intracavity intensity:

I_{cw} ≪ I_{sat, A} .

For a slow absorber,

I_{sat, A} = \frac{F_{sat, A}}{τ_{A}} .

The pulse, however, should saturate the absorber in terms of fluence:

F_{p} ≳ F_{sat, A} .

This is the essential discrimination: low-intensity cw operation sees high loss, while a pulse with sufficient fluence sees reduced loss.

The pulse energy and average power are related by

P_{av} = E_{p} f_{rep} = \frac{E_{p}}{T_{R}} .

This relation is useful because many stability conditions involve pulse energy, while thermal loading and pump requirements often involve average power.

For strong-field and HHG applications, the oscillator usually does not directly provide enough pulse energy. Its job is to produce a stable, coherent seed pulse train. The pulses are then stretched, amplified, compressed, and sometimes spectrally broadened. Passive modelocking is therefore the starting point of the timing and phase coherence that later matters in attosecond experiments.

9.2 Coupled-Cavity and Additive-Pulse Modelocking

Historically, passive modelocking was also achieved by coupled-cavity or additive-pulse modelocking. In these schemes, the main cavity is coupled to an auxiliary cavity. Interference between the returning fields creates an intensity-dependent transmission or loss.

The physical idea is that nonlinear phase shifts modify the interference condition. A more intense pulse can acquire a different phase shift than low-intensity light, so it couples back into the main cavity with lower effective loss. This behaves like an artificial saturable absorber.

Although SESAMs and Kerr-lens modelocking are more common in modern solid-state ultrafast lasers, coupled-cavity modelocking is conceptually useful because it shows that passive modelocking does not require literal absorption. What matters is nonlinear loss discrimination.

9.3 Slow Saturable Absorber with Dynamic Gain Saturation

A slow saturable absorber alone does not necessarily shorten the trailing edge of the pulse, because once it is bleached it can remain transparent for the rest of the pulse. Pulse shaping can still occur if the gain also changes dynamically during the pulse.

This situation is important for dye lasers and other systems where the gain recovery and saturation can occur on relevant time scales. The leading edge of the pulse bleaches the absorber, while the pulse also depletes the gain. The absorber favours the leading edge less loss, and the depleted gain suppresses the trailing edge. Together they can form a stable pulse.

In solid-state lasers, the upper-state lifetime is often much longer than the pulse duration and roundtrip time. The gain then saturates mostly with the average power, not within a single pulse. In that case, dynamic gain saturation is weak and a slow absorber needs help from other shaping mechanisms, especially soliton dynamics.

9.4 Fast Saturable Absorbers and Kerr-Lens Modelocking

A fast saturable absorber follows the instantaneous pulse intensity. It gives lower loss at the peak of the pulse and higher loss in the wings, so it directly favours pulse shortening. A simple model is

l (I) = l_{ns} + \frac{l_{0}}{1 + I / I_{sat, A}},

where $l_{0}$ is the saturable part of the loss and $l_{ns}$ is the nonsaturable loss.

Kerr-lens modelocking (KLM) is the most important artificial fast saturable absorber in solid-state lasers. The optical Kerr effect changes the refractive index according to

n = n_{0} + n_{2} I .

The intense pulse then acts as a self-induced lens in the gain medium. If the cavity contains an aperture, or if the pumped gain volume acts as a soft aperture, the better-focused high-intensity pulse experiences lower loss or better gain overlap than cw light. This is equivalent to a fast saturable absorber.

KLM can support extremely short pulses because the Kerr response is essentially instantaneous. However, it often requires careful cavity alignment and can be harder to self-start than SESAM modelocking. Many practical lasers therefore combine mechanisms, for example using a SESAM for self-starting and Kerr effects for additional pulse shortening.

The Kerr lens can be interpreted as an intensity-dependent mode-size change. In a hard-aperture KLM laser, the aperture literally clips the low-intensity mode more strongly than the high-intensity mode. In a soft-aperture KLM laser, the pumped gain region acts as the aperture: the high-intensity Kerr-lensed pulse overlaps the gain better and therefore experiences more net gain.

9.5 Soliton Modelocking with Slow Absorbers

Many femtosecond solid-state lasers use a slow SESAM together with soliton pulse shaping. The SESAM starts and stabilises modelocking, while the pulse duration is mainly determined by the balance between negative group-delay dispersion and self-phase modulation.

The approximate soliton condition is

\frac{| D_{2} |}{τ_{p}^{2}} \sim γ P_{0},

where $D_{2}$ is the roundtrip GDD, $γ$ is the nonlinear phase coefficient, $P_{0}$ is the peak power, and $τ_{p}$ is the pulse duration. The SESAM does not have to be fast on the femtosecond time scale. It only has to provide enough loss discrimination to suppress cw operation and stabilise the pulse energy.

A cavity-averaged master equation can be written schematically as

T_{R} \frac{\partial A}{\partial T} = [g - l - q (A) + D_{g} \frac{\partial^{2}}{\partial t^{2}} + i \frac{D_{2}}{2} \frac{\partial^{2}}{\partial t^{2}} + i γ | A |^{2}] A .

Here $q (A)$ is the saturable absorber loss. In a slow absorber, $q$ depends mostly on pulse energy or fluence rather than instantaneous intensity.

This picture also explains why dispersion compensation is central to femtosecond solid-state lasers. Without the correct roundtrip GDD, SPM simply chirps and broadens the spectrum without forming a stable short pulse.

The sign convention is easy to lose track of, so the physical statement is more useful than the symbol: the nonlinear phase creates a chirp across the pulse, and the cavity dispersion must delay the newly generated frequencies in the opposite way so the pulse refocuses after each roundtrip. If the balance is too weak, the pulse broadens; if it is too strong, the pulse can split, develop sidebands, or become unstable.

9.6 Q-Switching Instabilities

A passively modelocked laser can become Q-switched if the absorber and gain also support a slower energy-storage cycle. Then the laser emits bursts of modelocked pulses under a Q-switched envelope rather than a steady pulse train.

This instability is especially relevant because solid-state gain media store energy for microseconds to milliseconds, while the absorber can bleach on a much faster time scale. If the intracavity pulse energy is not sufficient to keep the absorber and gain in a stable steady state, the system can fall into a relaxation-oscillation-like cycle.

Design choices that help suppress Q-switched modelocking include:

choosing an absorber modulation depth that is large enough for self-starting but not unnecessarily large;
matching absorber saturation fluence to the intracavity mode size;
using sufficient intracavity pulse energy in steady modelocked operation;
avoiding excessive gain saturation and relaxation-oscillation drive;
using output coupling and pump power that support a stable pulse energy.

This is why SESAM parameters cannot be chosen independently of the laser cavity. A SESAM that works well in one oscillator can produce Q-switched modelocking or fail to start in another.

9.7 Modern Solid-State Modelocked Lasers

Modern passive modelocking covers several regimes:

Bulk solid-state oscillators: Ti:sapphire and Yb-doped systems can produce femtosecond pulses with dispersion compensation and Kerr or SESAM modelocking.
Thin-disk oscillators: the gain medium has excellent thermal handling, enabling high average power directly from an oscillator.
VECSELs and MIXSELs: semiconductor gain and absorber structures are integrated or closely combined, allowing compact high-repetition-rate sources.
Dual-comb lasers: two pulse trains with slightly different repetition rates are generated in one or closely related cavities for spectroscopy and metrology.

The performance frontier is not one-dimensional. Shortest pulse duration, highest average power, highest repetition rate, lowest noise, broadest tuning, and easiest self-starting are different optimisation targets. Passive modelocking is powerful precisely because its physical mechanisms can be combined: saturable absorption for self-starting, Kerr lensing for fast nonlinear loss, soliton shaping for femtosecond pulses, and engineered dispersion for stable operation.

Few-cycle passive modelocked oscillators are especially important because the electric-field waveform becomes experimentally relevant. In amplitude gating for isolated attosecond pulse generation, only the strongest half-cycle of a few-cycle driver should contribute efficiently to HHG. That requires not only short intensity envelopes, but also stable carrier-envelope phase, which connects directly to frequency combs.