Study About Fractal Grating And Its Self-image Effect

As an optical basic problem, different of aperture has been attracted wide attentions infundamental research of optical phenomena and application studies. The required spatialdistribution of diffraction field can be obtained by setting a certain diffraction aperture.And thus, the aperture and its diffraction is used in wave-front transformation and beamshaping. Moreover, grating is an importantdiffraction opticalcomponent, and it has been widelyused in many fields including optical measurement, optical calculation, and optical spectrumanalysis. When the parallel light illuminate the grating, the self-imaging of grating appears inFresnel diffraction region is Talbot effect. With the further study on this effect in the classicaloptical, it has been applied in Talbot array illumination, phase encoding and pulse shaping. Talboteffect has also extended to nonlinear optical, atomic physics, Bose-Einstein condensation and otherfields, and it becomes important techniques in electron microscopy imaging and X-ray diffractionimaging. Therefore, the research of Talbot effect of grating has the important significance forexpanding the cross of different subjects and promoting the widely applications of Talbot effect ofgrating.As we know, Fractal is a kind of complex structure with short-range disorder andlong-range order, and it has self-similar and scale-invariant properties. Because of thestructure superiority of fractal grating, it appears in optical filter, resonant transmission,wireless communication and so on. However, the complexity of fractal, the theoretical andexperimental researches are limited greatly. Owing to how to simplify the mathematicaldescription of fractal grating becomes the key problem for analyzing diffractioncharacteristics and the distribution rule. This is significant for promoting the application offractal grating. For practically manufactured aperture and grating, the edges are rough.These rough edges must affect the distribution of diffraction. Therefore the studies aboutinfluence of the random edges and the defect of grating on the diffraction play a foundationfor evaluating Talbot imaging of rough grating. The content of this paper includes thediffraction of aperture with rough edges, Fresnel diffraction of quasi-periodic grating andits self-imaging effect, the productions of one-dimensional and two-dimensional fractalgratings and its self-imaging effect. The detailed arrangements of this paper are as follow:The first chapter introduces the scalar diffraction and the vector diffraction theories indetail. We chose a circular and the amplitude grating as examples to discuss the near-field and far-field diffraction of aperture and grating by applying Fraunhofer diffraction, Freneldiffraction and the finite-difference time-domain method. The periodic grating and thequasi-periodic grating including fractal grating and defect grating are introduced. Finally,the main research of this paper is summarized.Chapter two researches Fresnel diffraction of a square aperture with rough edges.Firstly, we analyze Fresnel diffraction of a square aperture with rough edge theoreticallyand obtain the diffraction intensity formula in Fresnel diffraction region. The simulationcalculations of the diffraction by a square aperture modulated by Gaussian random edge areperformed. The influence of the lateral correlation length and the roughness of the randomedge on the transverse and the longitudinal diffraction of the square aperture is discussed indetail. The results show that the larger the lateral correlation length and the roughness ofmodulated edge are, the larger the scattering modulation among the diffraction distribution.Moreover, for the same random parameters, the random modulation degree on Fresneldiffraction at different propagation distance is also different. The modulated squareapertures are obtained through the laser direct writing technology, and the experimentalmeasurement for the diffraction of modulated square aperture is accomplished. Theexperimental results prove the theoretic analysis and the numerical simulation ones. Theseconclusions will be instructively meaningful for the evaluation of Fresnel diffraction of thepractical aperture.Chapter three studies Talbot effect of grating with different periodicity error. Theperiodicity errors of grating include the loss and the shift of the diffraction unit and themodulation of the slit separation. The exact diffraction distributions of three kinds of thequasi-periodic gratings are obtained according to the finite-difference time-domain method.The diffraction distortions with the diffraction units lost or shifting appear always along theradial direction, and the self-image of the first and the second kinds of quasi-periodicgratings has partially self-repairing function. The ideal Talbot image of double-periodicgrating can be obtained, but the random modulation of the slit separation destroys theperiodicity of the transverse and longitude diffraction distribution. These distortionphenomena among the diffractions of the quasi-periodic gratings are explained by use ofthe light ray theory.Chapter four studies the one-dimensional fractal grating and its self-imaging effect.We put forward the method to obtain the fractal grating through product multiple periodicgratings on basis of the scale-invariant property of fractal. Fresnel diffractions of two kinds of fractal gratings are analyzed theoretically and the mathematic expressions of thediffraction intensity distributions are deduced. The gray-scale patterns of thetwo-dimensional diffraction intensity distributions of fractal gratings are provided throughthe numerical calculations. The diffraction patterns take on the periodicity along thelongitudinal and transverse directions and this indicates that the self-image of fractalgrating is really formed in Fresnel region. The experiment measurement of the diffractionintensity distribution of fractal gratings with different fractal dimension and differentfractal level is performed and the experimental results are consistent with the theoretic ones.The self-imaging effect of fractal grating may be applied in the optical code and themanufacture of the integrated circuit board.Chapter five studies Talbot effect of two-dimensional fractal grating. We put forward themethod to produce the two-dimensional fractal grating by square aperture arrays. Theamplitude fractal gratings are produced by use of the spatial light modulator, and thediffraction intensity distributions of fractal gratings with different fractal level in Fresneldiffraction feld are measured. Talbot images of fractal gratings with1-level and2-levelfractal are obtained in practical experiment. The analytic expression of Fresnel diffractionintensity of the fractal gratings is derived through decomposing fractal gratings into the sumof many periodic gratings. Theoretic results predict the self-image of fractal grating appears.The numerical calculations also show the Talbot image and the fractional Talbot image offractal grating. These results may extend the applications of fractal grating in the opticalinformation processing and laser measurement.Chapter six has the conclusions of this paper and shows the following work plan.