José Sáez-Landete

A Fast Windowing-Based Technique Exploiting Spline Functions for Designing Modulated Filter Banks

A very fast technique to design prototype filters for modulated filter banks without using time-consuming multivariable optimization is introduced. In the proposed method, the prototype filter is optimized by using the windowing technique, with the novelty of exploiting spline functions in the transition band of the ideal filter, instead of using the conventional brick-wall filter. A study of the optimization techniques and three different objective functions existing in the literature has been carried out, and more suitable redefinitions of these objective functions are employed to achieve as optimized prototype filters as possible. The resulting filter banks closely satisfy the perfect reconstruction property, as is illustrated by means of examples.

Invariant grating pseudoimaging using polychromatic light and a finite extension source

The Talbot effect is a well studied phenomenon by which grating pseudoimages appear at certain periodic distances when monochromatic light is used. Recently, numerical simulations have shown a new phenomenon; when a polychromatic light beam is used in a double grating system, the intensity of the pseudoimages presents a transverse-profile that remains unaffected over a wide range of propagation distances. This effect can be used to increase the tolerances of gratings based optical devices, such as displacement measurement systems, interferometers, and spectrometers. The pseudoimages formation with a polychromatic and finite extension light source is analytically and experimentally demonstrated. Relatively simple analytical expressions for the intensity and the contrast allow us to predict when pseudoimages present a constant contrast and when they disappear. Furthermore, we experimentally obtain the pseudoimages using the proposed configuration, corroborating the theoretical predictions.

Design of two-dimensional zero reference codes with a genetic algorithm

In mask-alignment systems a reference signal is needed to align the mask with the silicon wafers. The optical reference signal is the autocorrelation of two two-dimensional (2D) codes with binary transmittance. For a long time, one-dimensional codes have been used in grating-measurement systems to obtain a reference signal. The design of this type of code has needed a great computational effort, which limits the size of the code to about 100 elements. Recently, we have applied genetic algorithms to design codes with arbitrary length. We propose the application of these algorithms to design 2D codes to generate 2D optical signals used in mask-alignment systems.

Design of two-dimensional zero reference codes by means of a global optimization method.

A method to obtain the absolute measure of the position is by means of the autocorrelation of two zero reference marks. In one-axis measurement systems one dimensional mark are used and the design of these marks is relatively complex. When the movement is in two-axes, two dimensional reference marks are required and they are even harder to design. We report a method of global optimization to calculate the optimal two dimensional zero reference marks which generate the reference signal with the highest central peak. This method proves to be a powerful tool for solving this problem.

Design of Two-Dimensional Optical Alignment Signals Robust to Diffractive Effects

Mask alignment is one of the most critical processes in photolithography. Prior to the shadow projection, the alignment between the lithographic mask and the silicon wafer is needed. In contact and proximity photolithography, a method to achieve the alignment with submicron or even nanometer resolution consists of superimposing two identical 2-D zero reference codes and registering the optical output signal. In order to increase the resolution of the system, the size of the code must be reduced and the diffractive effects become strong. The signal is then degraded and the precision of the alignment is reduced. In this paper, the effect of the diffraction in 2-D codes is analyzed, the degradation of the signal is characterized and its effect is modelled by means of a simple and fast computing parameter. Finally, we propose a genetic algorithm to optimize this parameter and design 2-D codes robust to diffractive effects. We propose the use of these codes to increase the resolution of alignment systems.

Optimal design of optical reference signals by use of a genetic algorithm

A new technique for the generation of optical reference signals with optimal properties is presented. In grating measurement systems a reference signal is needed to achieve an absolute measurement of the position. The optical signal is the autocorrelation of two codes with binary transmittance. For a long time, the design of this type of code has required great computational effort, which limits the size of the code to approximately 30 elements. Recently, the application of the dividing rectangles (DIRECT) algorithm has allowed the automatic design of codes up to 100 elements. Because of the binary nature of the problem and the parallel processing of the genetic algorithms, these algorithms are efficient tools for obtaining codes with particular autocorrelation properties. We design optimum zero reference codes with arbitrary length by means of a genetic algorithm enhanced with a restricted search operator.

Generation of Optical Reference Signals Robust to Diffractive Effects

In grating measurement systems, a reference signal is needed to achieve an absolute measurement of the position. The zero reference signals are normally obtained illuminating two identical superimposed zero reference codes (ZRCs) and registering the transmitted light by means of a photodiode. As one ZRC moves with respect to the other, the two codes overlap and the signal registered is the autocorrelation of the ZRC transmittance. In high resolution systems, the diffraction effects degrade the geometrical shadow of the first ZRC as it propagates to the second one. As a result, the autocorrelation is also degraded and the amplitude of the reference signal is greatly reduced. In this letter, we present a method for designing ZRCs with minimum diffractive effects. The method is based on the optimization of ZRCs by means of a genetic algorithm.

Assessment of NPR MDFT filter banks for subband coding and data transmission

Nearly perfect reconstruction (N-PR) modified discrete Fourier Transform (MDFT) filter banks (FBs) have been widely applied to signal coding and their use for data transmission in multicarrier communications has also been addressed in the literature. However, comparison and assessment of the performance of alternative designs for uniform N-PR MDFT-based systems is hampered by the lack of explicit formulas for easily calculating their expected performance. In this sense, this work introduces a unified formulation which can be used for the three different kinds of uniform N-PR MDFT filter banks which have been described in the literature. We demonstrate that the relationship between output and input signals is identical for all of them when a filter bank (FB) configuration is considered. From this expression, amplitude distortion and aliasing errors can be measured. With respect to filter bank multicarrier (FBMC) systems, two new measures that bound the maximum level of intercarrier and intersymbol interferences are defined.

Analytical model of a double grating system with partial temporal and spatial coherence

Grating pseudo images are formed in cascade grating systems and are useful in different fields and in many different applications, such as interferometry, optical encoding of position, etc. There are several processes for creating images of a grating by using only gratings as imaging elements, being the most known the Talbot effect. In other configurations one grating acts as an imaging element for another grating, such as the Lau effect where two identical gratings are illuminated with an extended monochromatic light source, and a pseudoimage of the first one is formed at infinity. In Generalized Grating Imaging, pseudoimages form, without the need for lenses, at finite distances of the gratings. One disadvantage that a grating pseudoimaging system presents for most applications is the fact that the contrast of self- and pseudo-images strongly depends on the distance between gratings. This makes the optical devices less tolerant to positioning and/or manufacturing. It has been shown that, for Talbot effect, the use of polychromatic light can eliminate the dependence of the contrast on the grating position. A similar result has been numerically demonstrated for some pseudoimages in a double grating system with spatial and temporal partially coherent light. In this work we present an analytical model for Ronchi gratings that justifies and explains the numerical results previously obtained.

Analog and digital filters with α-splines☆

Abstract Analytical expressions of analog and digital α-spline filters with continuous transition bands are derived. Previous works based on β-spline functions consider the order of the spline function, which must be a natural number, as a parameter to sharpen the transition band. However, the inclusion of an α-spline model provides greater flexibility since its order can be a real number, it depends on three parameters, and it can be used to model the transition band in the frequency domain. This paper presents the formulation of analog α-spline filters and the corresponding discretization to obtain the frequency and the impulse responses of their digital counterparts. There is also shown the design of α-spline digital filters minimizing two different least integral squared errors. As these methods do not provide direct control over the stopband attenuation, a third method which leads to principally flat digital filters is introduced. Several numerical examples demonstrate the efficiency and flexibility of the proposed techniques.