General conditions of periodic phase elements self-images forming (Talbot effect) in the fractional Fourier transform (FrFT) domain is given. Analytical solution of the FrFT images intensity distribution for the different forms (binary, linear, parabolic and others) of periodic elements low-level cell profiles is presented. Intensity difference ▵Ι measuring of the FrFT periodic self-image allow to determine the phase difference ▵φ of periodic elements low-level cell profile. Theory of the FrFT images forming of periodic phase elements based on signal distribution method is given. We use ambiguity function Aff* (χ0;ω0) in difference conjugate coordinates (χ0;ω0) as base functional of the periodic phase element distribution. The FrFT distribution Aupup*(χ0;ω0) corresponds to the rotation matrix Tφ which describe rotation of the input signal distribution on an angle φ=pπ/2, p=0÷ - the FrFT parameter. The signal distribution method allow to obtain general formula of intensity distribution of the periodic phase element FrFT image. Theoretically proved that at condition F0/tanφ=1, where F0=T2/4λd - Fresnel number, T - phase element period, λ - wave-length, d - length, periodic phase elements self-images are forming in the FrFT domain. In this case interference term is written as δ - function and intensity distribution I(χ) of the FrFT self-images is forming as superposition of the cross displaced on a quarter of period self-images of neighboring phase low-cells. Analysis of the FrFT self-images forming at condition 2F0/tanφ is also given. The results of numerical calculations of the periodic phase elements self-images at the different values of the FrFT parameter p are presented. Analytical dependence of the FrFT self-images contrast from phase difference ▵φ is obtained and the questions about phase microrelief parameters restoration of the phase element low-cell are discussed.
The theory of periodic phase elements images forming is described based on the method of the coordinate-frequency distribution. The invariant conditions of periodic elements self-images forming which are determined by the ratio of the Fresnel number F0 to tan(pπ/2) (where p is the FrFT parameter) are investigated in the FrFT domain. The analytic expressions for the calculation of periodic phase elements at different values of the invariant parameter F0/ tan &Jgr; are obtained. It is shown that the FrFT self-image of elementary cell forms as a result of the finite number of the cross displaced elementary cells superposition. The results of numerical calculations of the periodic phase elements self-images in the FrFT domain are presented. The mechanism of constant intensity levels forming depending on the value of invariant parameter is explained.
The FFT images optical superposition regularities are investigated for the general case of two shifted and modulated by the plane wave optical signals. Principal possibility of the optical superposition of two images in the FFT domain at an arbitrary value of parameter p is theoretically proved. Numerical results which demonstrate forming of the height-contrast interference band pattern in the FFT images optical superposition point are given. Domains of the FFT realization in the optical systems are investigated.