Course program
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

Type of didactic activities
Lectures on the blackboard. Exercises with the numerical simulation of the laser instabilities predicted by the theory.