Propagating Theory Of Nonparaxial Beams And Its Application

Standard paraxial theory, based on the well-known parabolic equation, is able to give an accurate description of beam propagation whenever the beam waist is much smaller than, diffraction length. This condition is fulfilled in the vast majority of standard propagation conditions and this explains the wide theoretical use of parabolic equation in describing most practical situation. Nevertheless, whenever beam waist and diffraction length are comparable, the optical field acquires new structural features which are not predictable by means of paraxial theory, so that a suitable approximation scheme is required for improving it. Description of nonparaxial propagation has not only a conceptual value: there are actually relevant situations in which paraxiality condition is not fulfilled. The first systematic approach to nonparaxial propagation goes back to pioneering work of Lax et al.. Following this pioneering work, other approaches developed by Davis ,Agrawal, Pattanayak, Couture and Belanger, Takenaka , and Wünsche et al. are used to study nonparaxial propagation of light beams.In this thesis, we wish to show that, in case of propagation in free space (or in a homogeneous medium), it is possible to develop a new approach to nonparaxial propagation of light beams-Wigner distribution function (WDF) approach. On the basis of the (scalar, vectorial) Rayleigh-Sommerfeld diffraction integral, a closed-form propagation expression for the Wigner distribution function (or WDF matrix) of partially coherent nonparaxial beams in free space (or in a homogeneous medium) is derived for the first time, which has general applicable advantage. The propagation of spatially fully coherent nonparaxial beams is treated as a special case of our general result. The application of the result is illustrated with the nonparaxial propagation of partially coherent scalar anisotropic Gaussian-Schell-model (AGSM) beams, TEM₁₁-mode Hermite-Gaussian (H-G) beams, Cosh-Gaussian (CHG) beams, partially coherent vectorial anisotropic Gaussian-Schell-model (AGSM) beams, and partially coherent vectorial twisted Gaussian Schell-model (TGSM) beams. Thepropagating characters of above beams are obtained. It can be also readily shown that the paraxial results corresponding can be obtained straightforwardly. Finally, we would like to mention that our formulation is based on a three-dimensional (x, y, z) treatment. The two-dimensional (x, z) treatment can be regarded as a special case of our general result.