The sampling theorem can be described as a economic way of representing a limited bandwidth function. A sample of
points is chosen and an interpolation function of these points is used to represent the function. The great importance of
this fact is that paves the way to discrete computation. The sample points act as a sort of “highlight” points of the
original function, and computation involving the entire function may be restricted to calculation using only the sample
points. Moreover we were able to find some refining of the classical outline of the sampling theorem that improves its
precision bringing also some physical insight into the core of the theorem. Of course also these consideration are not
restricted to optics, they can be construed as general properties of the signal theory. But we were optically minded at all
times and most of the applications are in optics.
The discovery of the Fast Fourier transform (FFT) algorithm by Cooley and Tukey meant for diffraction computation
what the invention of computers meant for computation in general. The computation time reduction is more significant
for large input data, but generally FFT reduces the computation time with several orders of magnitude. This was the
beginning of an entire revolution in optical signal processing and resulted in an abundance of fast algorithms for
diffraction computation in a variety of situations. The property that allowed the creation of these fast algorithms is that,
as it turns out, most diffraction formulae contain at their core one or more Fourier transforms which may be rapidly
calculated using the FFT. The key in discovering a new fast algorithm is to reformulate the diffraction formulae so that
to identify and isolate the Fourier transforms it contains. In this way, the fast scaled transformation, the fast Fresnel
transformation and the fast Rayleigh-Sommerfeld transform were designed. Remarkable improvements were the
generalization of the DFT to scaled DFT which allowed freedom to choose the dimensions of the output window for the
Fraunhofer-Fourier and Fresnel diffraction, the mathematical concept of linearized convolution which thwarts the
circular character of the discrete Fourier transform and allows the use of the FFT, and last but not least the linearized
discrete scaled convolution, a new concept of which we claim priority.
In this paper a Fourier transform digital holography experimental arrangement is presented. It is actually a hybrid
arrangement, half digital half analog. The Fourier hologram was constructed using the analogous means of the so called
lensless configuration. The hologram was recorded digitally by a camera with a large CCD array in stead of the
recording medium. The recording of the image was analyzed with a computer and the original image was reconstructed
by means of the discrete Fourier transform.
In this paper a simple method for determining the wavelength of an unknown source, (a problem of great theoretical and
practical importance), based on the Moire fringes phenomenon and Fourier analysis is presented and put into practice.
The accuracy and the simplicity of the problem makes it attractive and competitive.
We discuss an experiment for detecting small deformations by speckle interferometry. Vibration modes of an
aluminium plate are observed by digital speckle pattern interferometry (DSPI). A Mach-Zender interferometer
arrangement is used and the speckle interferograms are recorded with a CCD camera and processed on a computer.
These fringes depend on the path differences due to the vibration of the aluminium plate from its original state.
Vibration amplitudes between 0.3-0.6 &mgr; were measured for seven vibration modes.
In this paper we investigate the vibrations of a square aluminium plate by speckle interferometry means. Modes of
vibration of this plate are shown as speckle interferograms. As usually is the case with such interferograms, enhancement
and filtering of these images is needed after recording. The speckle index and the signal-to-noise ratio (SNR) of the preprocessed
interferograms before and after filtering are calculated. An improvement of the SNR between 1.37 and 1.81 is
obtained for the vibration modes presented here.
This paper is concerned with the determination of in-plane displacements and deformations, by using digital speckle
correlation. A special algorithm for determining the position of the maximum of the correlation function is presented. An
example on how to apply this method is presented.
We present a practical numerical method for processing the fringes obtained when two waves, with a quadratic phase difference function, interfere. This kind of fringe includes straight equispaced fringes and Newton's rings as particular cases. The numerical method we present is based on the discrete Fresnel (Fourier) transform of the data and has the same precision as least square fitting (LSF). Compared to the LSF method, this new method is better, as it is more efficient and does not require initial approximations for the fringe parameters to be determined.
This paper presents a numerical method for processing the fringes obtained when two waves, with a quadratic phase
difference function, interfere. As a particular case of this kind of fringes are the Newton's rings. The numerical method
we present is based on the discrete Fresnel (Fourier) transform of the data and it has the same precision as the least
square fitting (LSF).
Laser interferometer displacement measuring transducers have a well-defined traceability route to the definition of the
meter. The laser interferometer is de-facto length scale for applications in micro and nano technologies. However their
physical unit -half lambda is too large for nanometric resolution.
Fringe interpolation-usual technique to improve the resolution-lack of reproducibility could be avoided using the
principles of absolute distance measurement. Absolute distance refers to the use of interferometric techniques for
determining the position of an object without the necessity of measuring continuous displacements between points.
The interference pattern as produced by the interference of two point-like coherent sources is fitted to a geometric model
so as to determine the longitudinal location of the target by minimizing least square errors. The longitudinal coordinate
of the target was measured with accuracy better than 1 nm, for a target position range of 0.4μm.
The design and fabrication of two binary phase diffractive optical elements is presented. We use iterative
Fourier transform algorithms and a geometrical optics approach to design the diffractive elements. The
outgoing beam shapes of the designed elements are compared by computer simulations. The advantages
and disadvantages of each design are presented.
Presented in this paper are numerical algorithms necessary to determine the surface error by means of optical
interferometry. These algorithms are based on digital processing of phase-modulated fringe patterns, and are using the
discrete Fourier transform method.
Our paper provides analytical expressions for the statistical errors related to statistical processing of digitally recorded Newton's rings interference patterns by least squares fitting. These results completes some of our previous papers concerned with Newton's rings fringe patterns processing, which well describe an iterative numerical algorithm that we commonly use for fringe processing.
A grating type interferometer uses a grazing reflection on the testing surface. The grazing reflectance could be very large and the projected roughness height on the incident light directions is small. We built such an interferometer using a diffractive optical element (DOE) realized in National Institute of Microtechnology. We obtain resolvable quality fringes for different materials from industrial quality metal pieces to wood. Contour map of a testing flatness metallic proof is obtained followng a 4 step algorithm: smoothing for both direction of interferogram, skeletonizing interference fringes, computing heights that correspond to the deviation from a plan surface for each fringe point; representing 3D contour lines of equal (400 nm) height.
Our paper concerns with statistical processing of digitally recorded straight equispaced fringe patterns. We determine the highest degree of accuracy that can be achieved in estimating fringe parameters by statistical processing in given statistical fluctuation conditions affecting the recorded image.
Our paper concerns with statistical processing of digitally recorded straight equispaced fringe patterns by a numerical method based on discrete Fourier transforming (DFT) of the input data, which has the advantage of faster computation than the usual least square fitting method, that we have presented in a previous paper. This new method leads to the same accuracy as the least square fitting method and it is more convenient to use for processing fringe pattern with high harmonic order features.
A numeric algorithm for processing Newton's rings fringe patterns is presented. The interference images of this type have a characteristic appearance which can be described mathematically by a function depending on a set of parameters. The algorithm consists in finding the parameters of this mathematical expression by means of fitting the pattern using the least squares method, specially implemented with an iterative procedure. Unlike other processing methods which also use statistical calculus, this algorithm efficiently utilizes the whole information contained in the image and ensures the highest degree of accuracy, in given statistical fluctuation conditions affecting the image.
The phenomenon of phase retardation in the waist region of a gaussian beam (known as the Guoy effect) is described and demonstrated in the visible wavelength range. Two gaussian beams, originating from the same laser, are made to interfere in a region around the waist of one of them and far from the waist of the other. The relative phase is measured by processing the interference patterns recorded at different locations on the axis. A comparison with the theoretical results is carried out.
Speckle interferometry for non destructive testing of out of plane or in-plane stresses or deformations of rough mechanical parts is a powerful and modern technique. Basics of speckle phenomena and interferometry in specked light are reviewed. Electronic speckle pattern interferometry for vibration analysis and a Duffy-Young digital camera for in- plane measurement are presented.
Wavelength measurement is a critical topic in many applications. Stationary interferometers, such as Fizeau and Murty, can be successfully used, considering the proportionality between the wavelength and the fringe spacing in the interference pattern.In this work we present a 1D algorithm for the calculation of the fringe spacing and error sources. The final accuracy that can be achieved is also assessed. The experimental data are taken from fringe pattern recorded with a Murty interferometer.
A new method of processing Newton's rings fringe patterns is presented. After the center of the circular rings is found, a special type of pixel intensity spectrum is calculated, in which the 2D pattern is reduced to a 1D profile showing a periodic structure of fringes. By further processing the parameters of the initial interferogram can be easily extracted. The statistical nature of the method leads to a higher accuracy and a better immunity to noise.
A new self-calibration principle for phase shift interferometry is introduced, involving a whole-field consideration of the information contained by the interferograms. The principle is illustrated for the three-sample case, which was previously known as not having a self-calibration capability. Three related self- calibrating algorithms for phase shift interferometry are proposed, all of them based on this principle. The input of the algorithms is the set of three interferograms, and the output consists in both the correct phase shifts and the phase map of the wavefront being analyzed. No information on the actual phase shifts has to be supplied.
The method and experimental results of circularity measurements by optical triangulation are presented. The measuring system is a laser based one, using a bi-cell detector with rectangular active surface shape and diagonal gap. The measured data were fitted with a parametric function depending on the position of the tested object center, relative to the center of the rotating table. A resolution of 2.5 micrometers /mW is reported. This sensitivity is satisfactory for the majority of industrial applications-- circularity measurement and centering.