Paraxial quantum propagation

A quantum version of the classical paraxial approximation is established. After expanding the classical free-field wave equation in powers of θ (the characteristic opening angle of the paraxial rays), the familiar time-dependent paraxial equation appears in zeroth order. The arguments employed in deriving this approximation scheme are used in turn to identify the subspace of the photon Fock space consisting of paraxial states of the field. The quantum version of the paraxial approximation is then obtained by restricting the Maxwell field operators to this domain. Exact equal-time commutation relations are recovered as expansions in θ. In zeroth order, the theory yields a quantized analog of the classical paraxial wave equation, and formally resembles a nonrelativistic many-body theory. This formalism is applied to show that Mandel’s local-photon-number operator and Glauber’s photon-counting operator reduce, in zeroth order, to the same true-number operator. In addition, it is shown that the O(${\mathrm{\theta }}^2$) difference between them vanishes for experiments described by stationary coherent states.