Canonical transforms for paraxial wave optics

Peter Hawkes, Erwin K. Kasper

Reynaldo Castillo

This thesis concerns a general theory of beam propa-gation in optics.Well-known basic ideas are critically reviewed, and the analogy between quantum mechanics and paraxial optics is established through a number of n~w refinements.A parabolic equation for the electrodynamics wave relation is derived using the relativistic front-description of paraxial optics.The •packet-like beam solutions to this equation are shown to be hermite functions of complex argument.1 I The role and applications of canonical transformations in paraxial optics is investigated, attention is focused on some extensions of linear transformations mapping the position and momentum operators "q, '~ to a pair sands of canonically conjugate but not necessarily Hermitian operators.It is shown that for a linear transformation a related space 'jt. of entire functions exists with an inner product over the complex plane, such that the pair s, scan be realized in the Schrbdinger representation.A unitary mapping onto the ordinary Hilbert space of square-integrable functions over :l,t through an integral transform is provided.The well known Fresnel transform is a special case when sands are the harmonic oscillator raising and lowering operators, whose eigen-' functions are the Hermite functions of complex argument.With these functions the use of mathematical distributions in Fourier optics is extended.A generalization of the Cauchy problem is presented, which expresses the field in the plane l = o in terms of the field in any plane \ =~1 > o.The solution is obtained through an extensive use of the Hermite expansion of a distribution.While this expansion is not new, the fact that is not made in the context of the Hilbert space it is.The results obtained establish the applicability of special functions in optics, as well as a precise knowledge of the meaning and accuracy of paraxial wave optics.Applications of the theory to homogeneous and inhomogeneous medium are presented in particular in a selfcontained chapter about relativistic self-focussing in a non-linear medium.This work is primarily concerned with describing 1 1 1 an alternative approach to the study of a pair of canonical operators and it provides new insight over the role of symplectic geometry in quantum mechanics and optics.Chapter I canon i ea 1 tr ans for•ms.Const rue t ion an<1 app l i ea ti ons.< 19) .

Gerard Nienhuis

H. Bacry, Michel Cadilhac

The evolution of a light wave along an optical system, in paraxial approximation, can be described with the aid of a group of operators acting as canonical transformations in the framework of nonrelativistic quantum mechanics in two dimensions. This group is isomorphic to the ten-dimensional metaplectic group Mp(2). The homomorphism Mp(2)\ensuremath{\rightarrow}Sp(2) (symplectic group) gives a description of the geometrical-optics approximation.

Kurt Bernardo Wolf

John T. Sheridan, Bryan M. Hennelly, Damien P. Kelly

We discuss the significance of the Wigner Optics (WiO) and the Wigner Distribution Function (WDF), in understanding problems usually dealt with using Wave Optics (WaO). We first present a derivation of the WDF transport equation, equivalent to the Helmhotz equation. We then discuss the corresponding first order WDF differential equation and the 'paraxial’ equation. The Fresnel transform can be derived from the Paraxial Approximation (PA). However another integral transform is required to describe our first order equation. This equation is not a paraxial transform but does take account of the characteristics of the field. We clarify how these transforms are linked to electromagnetic theory. Finally we justify the practical significance we attach to WiO by describe how a new way of understanding motion measurement systems can be achieved using these ideas.