Reciprocal Vector Theory Of The Talbot Effect And Its Applications

The phenomenon of self-imaging of periodic objects, also known as Talbot effect, was first observed by H. F. Talbot in 1836. When a periodic object is illuminated with a coherent light, exact Talbot images will be found at the Talbot distance from the object. Since H. F. Talbot found the self-imaging phenomenon of the periodic object, many researchers have published their works on this subject. Now Talbot effect has been applied in a wide field of physics, such as in the interferometry, in optical information memeory, in atom optics, in Bose-Einstein condensates, and so on. The array illuminator of basing on the Talbot effect also has been applied in optical communication and optical calculation. The design of the Talbot illuminator is very simple. We merely design a phase-only grating of different high compression ratios to produce the desired array of output fields. Generally speaking, because the Talbot effect is based on the Fresnel diffracton theory, so when people calculate the phase distribution of the fractional Talbot, they usually resort to the Fresnel diffraction integral. However, the process of the calculation is very complex. So people come up with many methods to save the problem, of which the iterative method is very ease to analysis the phase-only distribution at the fractional Talbot plane, but a simple equation is also very difficult to be given using this method. In this dissertation a reciprocal vector theory for analysis of the Talbot effect of periodic objects is proposed. Using this method we deduce a general condition for determining the Talbot distance. Talbot distances of some typical arrays (a rectangular array, a centered-square array, and a hexagonal array) are derived from this condition. Further, the fractional Talbot effects of a one-dimensional grating, a square array, a centered-square array and a hexagonal array are analyzed and some simple analytical expressions for calculation of the complex amplitude distribution at any fractional Talbot planes are deduced. Based on these formulas, we design some Talbot array illuminators (TAI) with a high compression ratio. The main innovative researches are as follows: 1,A reciprocal vector theory for analysis of the diffractive self-imaging (or Talbot effect) of a two dimensional (2D) periodic object is proposed. Using this method a general condition for determining the Talbot distance is derived with the reciprocal lattice vector of the input object. As an example, some Talbot distances of some typical arrays (a rectangular array, a centered-square array, and a hexagonal array) are derived from this condition. Further, the fractional Talbot effect of the hexagonal structure is analyzed quantitatively. Finally some computer-simulated results are given for demonstration of the reciprocal lattice theory.2,Using this method we deduce a general condition for determining the fractional Talbot distance. Further, the fractional Talbot effects of a one-dimensional grating, a square array, a centered-square array and a hexagonal array are analyzed and some simple analytical expressions for calculation of the complex amplitude distribution at any fractional Talbot planes are deduced.3,Based on these simple formulas, we design some Talbot illuminators with a high compression ratio. Finally, some computer-simulated results and experimental results are also given.