Study On The Fractional Fourier Transforms For Beams

The propagation and transformation of beams are topics of current interest, which play an important role in development of modern optics theory and laser technologies. At present, FRFT (Fractional Fourier Transforms) has been introduced in optics for over ten years. Although the basic theory frame of the FRFT has been established, its practical applications are more less and immature relatively.The present thesis is devoted to develop the applications of the FRFT. We applied the FRFT to study the propagation and transformation of beams. The main results achieved in this thesis can be summarized as follows:1. Combined with the present studies on optical FRFT in home and abroad, the foundational theories and some applications of optical FRFT are summarized. A comprehensive review of the object, the basic analytical methods and the present studies for beams propaga on and transformation are introduced briefly. The definition and physical meanings of WDF (Wigner Distribution Function) are also introduced.2. Based on the A. W. Lohmann's WDF definition of FRFT, The WDF method of the FRFT for laser beams is given. By using this method, we analyses the characteristics of the FRFT for Hermite-Gaussian beams, Cosh-Gaussian beams et al. Study on the transformative laws of the parameters, such as the intensity distribution, the on-axis intensity distribution, the beam width defined by the second-order moments versus with the order of the FRFT, the analytical expressions for these parameters are obtained. Detailed numerical calculations are performed to illustrate the theory analysis results.3. Starting from the matrix expressions of FRFT, combined with the generalized Huygens-Fresnel diffraction integral (Collins) formula, we analyses the characteristics of the FRFT for Laguerre-Gaussian beams, Bessel-Gaussian beams. We study on the transformative laws of the parameters of these beams versus with the order of the FRFT. The influence of the decentered parameter on the FRFT for Bessel-Gaussian beams is discussed.