We survey some of the applications of soliton theory in laser physics. We briefly treat the theory of optical self-focussing and optical filamentation in neutral dielectrics and plasmas where the governing equation is the non-linear Schrödinger equation or one of its generalisations. We establish the connection of this theory with 1-dimensional Langmuir turbulence in plasmas. We treat the theory of optical self-induced transparency (SIT) at greater length and develop the Maxwell-Bloch (MB), reduced Maxwell-Bloch (RMB), SIT and sine-Gordon (s-G) equations to describe it. An optical three-wave interaction is related to the s-G equation; and reference is made to recent work on solitons in stimulated Raman scattering.
The RMB equations are solved by a Zakharov-Shabat-AKNS inverse scattering scheme. The inhomogeneously broadened RMB equations have the unusual feature that ln a is not a constant of the motion. However, the sharp line RMB equations have two infinite sets of conserved densities, and the system constitutes one more example of a completely integrable infinite dimensional Hamiltonian system. In terms of scattering data the Hamiltonian of the RMB equations separates into soliton, breather and `background', that is `radiation', contributions. The sine-Gordon equation and its separable Hamiltonian are found as a special case of the RMB equations and its Hamiltonian. Averaged Lagrangian techniques are independently used to relate the c-number MB, RMB and SIT equations and to analyse the slowly varying phase and envelope approximations by which the SIT equations are derived from the MB or RMB equations. The connection of this Lagrangian theory with the Hamiltonian theory is not established.
The Hamiltonian formalism in terms of the scattering data is used to quantise the RMB and s-G equations. The RMB, like the s-G, has the discrete energy level spectrum associated with a quantised breather; but only in the s-G limit is the quantised system easy to interpret. The quantised s-G is used to model a `coarse grained' operator theory of strictly resonant sharp line optical pulse propagation. The validity of such a description is examined.
The physics of the c-number RMB equations is also discussed and particularly the relation of the c-number breather solutions to the 2π-pulse solutions of the SIT equations. The c-number RMB breather solutions provide a more general theory of SIT valid (within the 2-level atom model) at all electromagnetic field intensities and restricted only by the low density condition which permits the neglect of back scattering.
Finally we look at four problems in resonant non-linear optics from a physical point of view. These are degenerate SIT and singular perturbation theory for it; the collision of oppositely directed resonant optical pulses; the theory of super-radiance; and the theory of optical self-focussing in resonant SIT.
