The Representation Of The Misaligned Optical Systems And The Diffraction Of Zone Plate By The Fractional Orders

Fractional Fourier transform was originally introduced by Namias in 1980 as a mathematical tool for solving theoretical physical problems. Since then a lot of works have been done on its properties, optical implementations, and applications by Ozaktas, Mendlovic and Lohmann et al. The fractional Fourier transform has become important branch of optics.In this thesis, firstly, some parts of optics (the fractional Fourier transformation and its application to the image encryption) have been summarized. Then, several primary research methods and basic theories used in this thesis are emphasized in detail. Respectively, a simple introduction to the basic method of matrix optics is given; we obtain the Fresnel diffraction formula by solving the Helmholtz equation, infer image transform formula for lens systems, and describe relevant content and result of the standard Fourier transform with its optical implementations. Then, we make an exhaustive discussion for type II basic units proposed by Lohmann and give its optical implementations for the fractional Fourier transforms. In the end of Chapter 1, we introduce an encryption technology on 4f system with double random phases masks. This method is important for study encryption technology of zone plate in the optical field with the fractional Fourier system and double random phases masks, and the investigate of a image hiding and encryption method in fractional Fourier domains in this thesis.Under the condition of rotational symmetry, fractional Fourier transform can be rewritten as fractional Hankel transform. The relationship between the fractional Hankel transform and the diffraction of misaligned optical system is established in Chapter 2. By settling and discussing the diffraction integral formulae of the misaligned ABCD optical system, the fractional Hankel transform of input function can be obtained if selecting a suitable reference plane under given conditions.In Chapter 3, the beam waist to waist transformation of Gaussian beams between input and output reference planes described by the scaled fractional Fourier transform is analyzed. We obtain the transfer matrix of ABCD optical system, which corresponds to the scaled fractional Fourier transform. The results show that the beam waist to waist transformation of Gaussian beams can be described by the scaled fractional Fourier transform when the ABCD optical system has a suitable transfer matrix. The relationship between the input and output waist planes, with some particular cases when a Gaussian beam through a thin lens, is also discussed.In Chapter 4, we analyze the diffraction result of optical field behind Fresnel zone plate, and theoretically deduce its focal positions and transform matrix of its diffraction part of optical field. It is shown that, on some certain conditions, its diffraction distributions are the mixture of Fourier spectrum and fractional Fourier spectra. The diffraction of Cosine zone plate is also analyzed in this chapter.In Chapter 5, we analyze the diffraction result of optical field after a Cosine zone plate, and theoretically deduce its transform matrix. On some certain conditions, its diffraction distribution is a mixture of fractional Fourier spectra. Then we use the Cosine.zone plate and its diffraction result to image encryption. Possible optical image encryption and decryption implementations are proposed, and some numerical simulation results are also provided.In Chapter 6, we propose a new technique for digital image encryption and hiding based on fractional Fourier transforms with double random phases. An original hidden image is encrypted two times and the keys are increased to strengthen information protection. Color image hiding and encryption with wavelength multiplexing is proposed by embedding and encryption in R, G, B three channels. The robustness against occlusion attacks is analyzed, and computer simulations are presented with the corresponding results.