Degree course: 
Corso di Second cycle degree in PHYSICS
Academic year when starting the degree: 
Academic year in which the course will be held: 
Course type: 
Compulsory subjects, characteristic of the class
Second semester
Standard lectures hours: 
Detail of lecture’s hours: 
Lesson (60 hours)

Basic knowledge of electromagnetism and quantum mechanics.

Final Examination: 

Oral examination

Voto Finale

The main goal of the course is to provide an introduction to the field of quantum electronics. The Maxwell-Bloch model is introduced to describe the nonlinear interaction between an optical beam and a two-level atomic system under nearly resonant conditions. The nonlinear dynamics is studied both in a traveling wave configuration and when the atomic system is enclosed in an optical resonator, considering the cases of an absorbing medium and an amplifier. The last part of the course is dedicated to more advanced topics, such as the multi-mode nonlinear dynamics arising in lasers (Risken-Nummendal instabilities) and the spontaneous formation of patterns in the transverse profile of the field in passive cavity with a nonlinear Kerr medium.

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
- 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
- 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

- Lecture notes from the teacher (e-learning platform).
- Recommended text book:
L. Lugiato, F. Prati M. Brambilla, Nonlinear Optical Systems, Cambridge University Press (2015).
F. Prati e L. A. Lugiato, Appunti di fisica dei laser, cap. 2-5, cap. 7

Others references
A. Yariv, Quantum Electronics, Cap, 15, Wiley & sons, 3rd ed. (1989)
R. W. Boyd, Nonlinear Optics, cap. 3, 6 e 7, 3rd ed., Accademic,Press (2008)

Lectures on the blackboard with occasional use of powerpoints transparencies. Exercises with the numerical simulation of some optical nonlinear systems studied during the lectures.

Receive under appointment (office V4.7 fourth floor via Valleggio 11)
e-mail: enrico.brambilla@uninsubria.it