The study of properties of the diffracted field, when two Cantor diffraction gratings are superimposed is important for establishing the relationships between the geometry of each fractal grating and the corresponding structure of the diffracted field. Here, we consider simple examples related with the Moire effect, since the Cantor gratings used are built through periodic functions.
In this paper, the case for studying is the characterization of field in the Fraunhofer region, into each envelope of diffraction, using the self-similarity function as a function of the angular direction. For this case it is shown that it is more appropriate the calculation on the modulus of the electromagnetic field, instead of using the intensity distribution. Also, we use a Cantor set obtained as a product superposition of cosine functions.
We extend the results obtained for different cases of the product superposition between periodic functions to the case of circular symmetry. The characteristics of focalization for such case is studied. We name Fresnel-Cantor zone plate to the structures obtained.
We study the geometrical phase transition for fractal transmittances, to determine the limits from which it must be considered like fractal structure, according with the density of diffractive elements contained in such planar transmittance. For such objective two considerations are made: (1) the diffracted field from the transmittance is analyzed at the Fourier plane, (2) the autocorrelation of the same is studied by means of the joint transform correlator.
We study the properties of the self-similarity function for the intensity distribution of field when different types of gratings, fractal, periodic and aleatory, are used. For this we introduce some definitions considering different points of view that allows us the construction of the diffraction gratings with each geometry. Such structures are applied for the calculation of the electromagnetic field propagation in the Fresnel and Fraunhofer regions.
In this work, we superimpose tow and more random images of irregular particles, using different statistics and concentration for the texture distributions. This is achieved using different random generators to obtain binary images in the distribution of these particles. For these case, the box counting dimension is calculated and we obtain the relation between this dimension and the corresponding structure. In this way, we obtain a fractal characterization for the superposition of these binary images.
In this work, the changes in the self-similar functions for a grating under possible structural changes are investigated using the shearing interferometry method. Such changes can be: deformations, introduction of random perturbations or changes in the fractal dimension. The results obtained have applications to relate such changes with the corresponding fractal parameters, which can be extended for the case of rough surfaces.
In this paper, several textures with different degrees of self-similarity are accomplished by using random generators. This implies different statistics in the grey level distribution into the images. Their lacunarity is numerically measured and a correspondence between the randomization degree and the obtained lacunarity for each case is established, using a new definition for the lacunarity parameter. These measurements are applied in textures obtained from a laser system, generated for rough metallic surfaces and by biological systems as well.
We are interested in the study of the diffraction from complex apertures with the use of conformal mapping transformation. We use polygonal structures and the Riemann mapping theorem to transform these structures into a circular region and so to develop the calculation of the diffracted fields along the Fresnel region. We show some cases for fractal apertures with Koch boundary.
The method for the calculation of scattered field from Cantor corrugation is developed using the impedance boundary condition. Some consideration about the coordinates transformation are included and the effects on the equations that permits the calculation of fields are discussed.
We use a simple formalism, based on the convolution between rectangular and gaussian functions, to represent the propagation of beams with high energy, and the CO2 laser, used in material processing. We show the validity of such approximation comparing with samples makes on transparent material.
The method of conformal mapping, through the Schwartz-Christoffel transformation, is applied to solve the scattering of electromagnetic beams from planar surface with a Koch corrugation. We use the integral method with the calculation of admittance of the transformed plane which is the kernel in the equations. The problem is planted for both polarization for the far field.
The phenomena for which the laws of geometrical optics are invalid in the case of beams are already very well-known. Such effects are known as non-speculars and many works about them already exist in the literature. Classically, the angular and longitudinal displacements are considered the basic non-specular effects, but there are the focal and width changes that, in any sense, are related with the previous ones. The foundation of these non-specular effects are related with the energetic interchanges that take place at the inrerface between two media, and an entropic foundation has been developed recently. From the optical point of view, the non-specular effects have been studied because they are important for guided waves and integrated optics. An experimental verification is more simple in the microwaves region because the range in angular change has a small value (the order of magnitude is 10-3 radians), and the lateral displacement is comparable with the wavelength of the incident radiation. In this work, an impedance method (through the input impedance of the interface) is used to characterize different non-conventional structures: film on a medium with complex refraction index or multilayers with fractal distribution. The characterization of the reflection and transmission from the equivalent interface can be studied; also, very important is the relation between the angular and longitudinal changes with the entropy function for these cases.
A theoretical formalism to study the scattering by reflection from a vitreous film on a planar medium with graded refraction index is developed. It is employed the impedance function, which characterizes the interface between two media and the diffuse diffraction pattern is obtained The medium is characterized using the function sech(x) in the refraction index and then, general and particular conclusions can be obtained starting from the surface impedance function.
In this paper, we obtain 2D fractal as a cyclic process which define different fine structures using a functional from that can represent circle and square as limit cases. The self similarity for these cases is introduced intensity patterns in the Franhoufer region are shown for different configurations.
In this paper, some considerations about the scalar and vectorial theory related with the interaction fractal object and electromagnetic wave are shown. Different combination is in the density function, that permits us to obtain the Cantor function, can be used for this objectives.
Some regular fractals, as Cantor bars and Sierpinski carpet, can be obtained as multiplicative superposition of periodical functions. Adding an exponent to each of this functions we can obtain a system to apply in optics for image processing, because different combinations can be achieved. A parameter to characterize the fractal structures in the Fresnel and Fraunhofer regions is introduced. It is called the in-order self-similarity function, which permit us to determine the periodical components filtered from the initial structure. The application is developed mainly for 2D fractals as the Sierpinski carpet.
The formalism to study the scattering by a planar interface with spatial variations in the refraction index is developed. The results obtained are applied to two cases of theoretical interests, which are not currently implemented with this method: thermal gratings and spatial variations in the magnetic permeability. Surface impedance is plotted as a function of the spatial coordinate and the angular component of incident field. Also, scattered intensity is calculated for each case using a Gaussian beam as incident field.
Talbot effect and interferometrical fringes phenomena combined with computer-generated moire are employed in order to enhance the detection of the displacements of one surface. The high sensitivity achieved, easily assembly and the possibility to change automatically the reference pattern tourn competitive such methods as compared to other but complex. The range of application for each case is estimated.
Some regular fractal objects can be obtained from a density function, as a product of periodical components. In the optical processing the manipulation of fundamental components of the structures involving this method is very important. We applied the filter operation to each 2D components of the Sierpinski carpet using an opto-digital device and speckle modulated by Young fringes method.
Surface roughness determination is of great interest for many applications. Several methods can be found in the literature, but most of them rely on indirect evaluation of the information, such as photographic techniques. We propose a method to measure surlace roughness that takes advantage of digital speckle pattern interferometry for obtaining the data, and of digital image processing for evaluating it. After defining the problem, a theoretical description is presented, and finally it is compared with experimental results, showing good agreement.