The optical Bloch equations for a two level atomic system
- Interaction of the 2-level atom with the e.m. field. Interaction Hamiltonian in the dipole
approximation.
- Density matrix formalism - Liouville Von Neuman evolution equation
- Derivation of the Optical Bloch equations.
- Solution of the Bloch equations driven by a monochromatic plane wave – precession of
the Bloch vector on the Bloch sphere - comparison with the results of the perturbative model.
- Self-induced transparency and propagating soliton solution in the plane-wave approximation
- Superradiance and superfluorescence – simplified model

The Maxwell-Bloch model
- Derivation of the MB equations within the plane-wave approximation. Slowly varying envelope
approximation (SVEA) and rotating wave approximation (RWA).
- Phenomenological inclusion of the irreversible processes.
- Stationary solution in the linear regime. Linear susceptibility. Comparison with the classical
Lorentz model and the perturbative model.
- Stationary solution in the nonlinear regime. Non lineare susceptibility. Saturation and power broadening
effect.
- Self-induced trasparency and soliton solution.
- Superradiance e superfluorescence – simplified model.

Atomic system in an optical cavity – linear regime
- Optical resonators. The plane-mirror ring cavity. Transmission function of the empty cavity.
- Optical resonator with a 2-level atomic system. Perturbation of the empty cavity modes
in the linear regime: mode pulling, mode pushing and mode splitting effects.

Optical cavity with an active 2-level system - laser dynamics
- Stationary solution of the laser beyond threshold – mode pulling formula for the laser frequency.
- Derivation of the dynamical laser equation in the low transmission regime, single-mode and
multi-mode dynamics – mean field approximation.
- Nonlinear dissipative systems – general discussion. Linear stability analysis of stationary
solutions – example of the damped pendulum.
- Linear stability analysis of the laser solution below threshold.
- Single mode laser instability above threshold (Lorentz-Haken).
- Multi longitudinal mode instability ( Risken Nummedal). Example of numerical simulation

Optical cavities with a passive medium – optical bistability and spontaneous pattern formation
- Optical bistability – qualititive discussion: optical hysteresis cycle, tecnological applications.
- Optical bistability within the Maxwell Bloch model: a) purely absorbitve case b) dispersive limit.
- Cavity model beyond the plane-wave pump approximation. Diffraction in the paraxial approximation.
- Lugiato-Lefever model for the Kerr medium. Turing modulational instability. Example of
spontaneous pattern formation.
- Pattern formation in dissipative optical system inside a cavity - general overview. Examples of periodic structures, localized structures and cavity solitons