Optimal quantization method for uneven-phase diffractive optical elements by use of a modified iterative Fourier-transform algorithm

A method based on an iterative Fourier-transform algorithm (IFTA) with phase optimization for phase-only multilevel diffractive optical elements (DOEs) is presented. Phase optimization by minimizing the mean-squared error composed of an amplitude-weighted probability-density function is performed in the IFTA iterations to ensure that the wavefront difference is minimized and thus to improve the DOEs' performance. Using the proposed method, we obtained a small standard deviation of diffraction efficiency of 20 uneven-phase DOEs, showing that DOEs with high diffraction efficiency did not vary significantly with the initial conditions. The simulation results of even- and uneven-phase four-level DOEs designed by use of direct and stepwise quantization IFTA methods are compared.     

Automatic symmetrical iterative Fourier-transform algorithm for the design of diffractive optical elements for highly precise laser beam shaping

A symmetrical iterative Fourier-transform algorithm (IFTA) using a combination of phase and amplitude freedom for the design of diffractive optical elements for highly precise laser beam shaping is presented. We compare this method with the basic IFTA and the symmetrical IFTA exclusively using phase freedom, and the basic IFTA using both phase and amplitude freedom, by employing these methods for super-Gaussian beam shaping. While the latter three methods fail to produce satisfactory solutions, the first method results in a beam non-uniformity error of 0.44% and a theoretical efficiency of 97.2%. Moreover, the new approach avoids the use of the trial-and-error method for finding the appropriate modified Fourier-domain constraints during the iteration, which can be difficult for some beam-shaping problems.     

Fourier fringe processing by use of an interpolated Fourier-transform technique

Recently a powerful Fourier transform technique was introduced that was able to extract the phase from only one image. However, because the method is based on the two-dimensional Fourier transform, it inherently suffers from leakage effects. A novel procedure is proposed that does not exhibit this distortion. The procedure uses localized information and estimates both the unknown frequencies and the phases of the fringe pattern (using an interpolated fast Fourier transform method). This allows us to demodulate the fringe pattern without any distortion. The proposed technique is validated on both computer simulations and the profile measurements of a tube.     

Practical example of the correction of Fourier-transform spectra for detector nonlinearity

HgCdTe photoconductive detectors can display a nonlinear response when illuminated. In interferometric applications, this behavior must be accounted for in the data transformation process to avoid errors in the measurement of the spectral distribution of the incident radiation. A model for the distortion of the interferogram is proposed and applied to solar observations made by the Atmospheric Trace Molecule Spectroscopy (ATMOS) Fourier-transform spectrometer during orbital sunrise and sunset from the Space Shuttle. Empirical estimation of the dc current level is necessary for this instrument, and satisfactory nonlinearity correction is obtained for several of the primary ATMOS optical filters. For ATMOS broadband optical filters that cover more than one half the alias bandwidth, the model is inadequate because of the presence of antialiasing electronic filters within the instrument, and it is necessary to resort to estimation and subtraction of the residual baseline offset. In either case the remaining base line offsets are typically smaller than 1%, which is satisfactory, although offset remains a significant systematic source of error in the estimation of the abundance of telluric and solar constituents from the spectra.     

Application of an iterative global approximation technique to structural optimizations

An iterative global approximation technique based on the Kriging method is proposed. The technique is validated through analytical test cases and then applied to solve two practical optimization problems: the optimization of aluminium-foam filled absorbers against crashworthiness requirements and the optimization of composite stiffened panels against buckling and strength constraints. The absorbers of the first application consist of two co-axial aluminium alloy tubes filled with lightweight aluminium foam. They were optimized to collapse at a controlled force level and to be the lightest possible. Explicit Finite element analyses were performed to evaluate the structural behavior of the absorbers in the sample points used to build the approximation. In the second application stiffened panels were optimized against buckling and strength constraints. The Tsai-Wu criterion was used to estimate first-ply failures as strength limit of the structure. Non-linear Riks analyses were performed with ABAQUS/Standard to evaluate the shell behavior in the sample points used to build the response surfaces. Basing on the obtained results the proposed iterative procedure seems a promising alternative to the classic a-priori building of response surface allowing better accuracy and saving of sample points.     

Atmospheric correction over case 2 waters with an iterative fitting algorithm

A modular atmospheric correction algorithm is proposed that uses atmospheric and water contents models to predict the visible and near-infrared reflectances observed by a satellite over water. These predicted values are compared with the satellite reflectances at each pixel, and the model parameters changed iteratively with an error minimization algorithm. The default atmospheric model uses single-scattering theory with a correction for multiple scattering based on lookup tables. With this model we used parameters of the proportions of three tropospheric aerosol types. For the default water content model we need the parameters of the concentrations of chlorophyll, inorganic sediment, and gelbstoff. The diffuse attenuation and backscatter coefficients attributed to these constituents are calculated and used to derive the water-leaving reflectance. Products include water-leaving reflectance, concentrations of water constituents, and aerosol optical depth and type. We demonstrate the application of the method to sea-viewing wide field-of-view sensor by using model data.     

Kalman-filter-based algorithms of spectrometric datacorrection-Part I: an iterative algorithm of deconvolution

This series of two papers aims to present the different solutions of the problem of improving the resolution of spectrometric measurements via numerical processing of spectrometric data subject both to systematic instrumental errors and to random measurement errors. It is assumed that the model of the spectrometric data has the form of a convolution-type equation of the first kind. The method for improving the resolution consists in numerically solving this equation using the acquired data. In this first paper of the series, an algorithm of correction is proposed which is based on the iterative use of the Kalman filter incorporating a non-negativity constraint. Its applicability to the problem of correction is assessed not only from a purely metrological point of view (accuracy, resolution) but also with respect to its suitability for implementation as a VLSI processor dedicated to measuring systems. For this latter reason a time-invariant model of the data and a steady-state version of the Kalman filter is used. The efficiency of this approach to correction is demonstrated using both synthetic and real-world data     

Spectrum leveling by an iterative algorithm with a dummy area for synthesizing the kinoform

This paper describes a new iterative algorithm for synthesizing the kinoform so that the Fourier spectrum of an object is leveled by adjusting the information of a dummy area introduced into the object's domain, spatially isolated from the signal area. Theoretical consideration of the effect of the dummy area derives the required size of the dummy area, and computer simulations prove it to be valid, although restricted to an object composed of binary numbers. Also, it is shown that highly efficient use of the incident light is possible to achieve. Experimental results verify the proposed scheme.     

Passive Fourier-transform infrared spectroscopy of chemical plumes: an algorithm for quantitative interpretation and real-time background removal

We present a ratioing algorithm for quantitative analysis of the passive Fourier-transform infrared spectrum of a chemical plume. We show that the transmission of a near-field plume is given by τ(plume) = (L(obsd) - L(bb-plume))/(L(bkgd) - L(bb-plume)), where τ(plume) is the frequency-dependent transmission of the plume, L(obsd) is the spectral radiance of the scene that contains the plume, L(bkgd) is the spectral radiance of the same scene without the plume, and L(bb-plume) is the spectral radiance of a blackbody at the plume temperature. The algorithm simultaneously achieves background removal, elimination of the spectrometer internal signature, and quantification of the plume spectral transmission. It has applications to both real-time processing for plume visualization and quantitative measurements of plume column densities. The plume temperature (L(bb-plume)), which is not always precisely known, can have a profound effect on the quantitative interpretation of the algorithm and is discussed in detail. Finally, we provide an illustrative example of the use of the algorithm on a trichloroethylene and acetone plume.     

Design of an efficient broadband far-infrared Fourier-transform spectrometer

As part of a feasibility study for a far-infrared Fourier-transform spaceborne spectrometer, the criteria that drive the choice of the instrument configuration have been identified as broadband operation, dual input and output ports, optics of the interferometer with full tilt compensation, and measurement of both planes of polarization of the source on a single detector. Despite the fact that some of these requirements are apparently difficult to reconcile, a new configuration of the polarizing interferometer that meets all the above requirements has been identified. The considerations that led to the design of this new configuration are discussed.     

An accelerated multiplicative iterative algorithm in image reconstruction

Based on the ML‐EM (maximum likelihood expectation maximization) algorithm and AWLS (one kind of multiplicative weighted least square) reconstruction, a new algorithm named RMITC (rapid multiplicative iteration with total‐count conservation) is proposed. The new method assumes a higher order correction factor and incorporates a total‐count conservation constraint to obtain better images reconstructed while achieving a higher speed of convergence. Computer simulated phantom data and real positron emission tomography (PET) transmission data were used to compare the new method with other reconstruction algorithms, such as ML‐EM and AWLS. Results demonstrated that the new method is faster and better quantitatively than both ML‐EM and AWLS. © 2002 Wiley Periodicals, Inc. Int J Imaging Syst Technol 12, 97–100, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ima.10016     

Artifact removal in complex frequency domain optical coherence tomography with an iterative least-squares phase-shifting algorithm

We present what we believe to be a new computational algorithm for complex frequency domain optical coherence tomography that can effectively suppress artifacts that are caused by uncertainty in phase shift due to sample motion and errors in reference phases. The algorithm treats the phase-shifting values as additional unknowns, and we can determine their exact values by analyzing interference fringes using the numerical least-squares technique. A series of simulations and experiments prove that this algorithm can effectively remove strong mirror-image artifacts because it is unaffected by random phase fluctuation.     

Characterization of strong fiber Bragg gratings using an applied thermal chirp and iterative algorithm

Coupling coefficients of various grating types and strengths are calculated from measurements of the complex reflectivity using an applied thermal chirp and optical frequency domain reflectometry (OFDR). The complex reflectivity is then utilized by a layer peeling algorithm to determine the coupling coefficient of the thermally chirped grating. A guess of the temperature profile enables the coupling coefficient of the unchirped grating to be estimated. An iterative algorithm is then used to converge on the exact coupling coefficient, employing an error minimization method applied to the reflectivity spectra. This technique removes the need for a reference grating while preserving the spatial resolution obtained with the initial OFDR measurement. Successful reconstruction of gratings with integrated |κ|L ~ 9.0 are demonstrated with a spatial resolution of less than 100 μm.     

Optimal multicriteria approach to the iterative fourier transform algorithm

We propose a unified approach to the multicriteria design of diffractive optics. A multicriteria version of the direct binary search that allows the user to adjust the compromise between the diffraction efficiency and the signal-to-noise ratio already exists. This technique has proved to be extremely powerful but also very time consuming. We extend this multicriteria approach to the iterative Fourier transform algorithm, which helps to reduce the computation time dramatically, especially for multilevel domains. Simulations as well as experimental validations are provided.     

Computer modeling of the effects of apertures in the Fourier-transform plane of Fourier-transform imaging systems

Two computer models are presented that can be used to determine the image formed by a 4-? Fourier imaging system with an aperture in the Fourier-transform plane of the system. These models were designed to aid in the selection of system parameters in a Fourier-transform hologram recording system (namely, object size, feature ratio, focal length, and storage area) required to obtain a retransformed image of a given quality. The results of the simulations are verified in the laboratory.     

Spatially adaptive iterative algorithm for the restoration of astronomical images

This article develops an iterative spatially adaptive regularized image restoration algorithm. The proposed algorithm is based on the minimization of a weighted smoothing functional. The weighting matrices are defined as functions of the partially restored image at each iteration step. As a result, no prior knowledge about the image and the noise is required, but the weighting matrices as well as the regularization parameter are updated based on the restored image at every step. Conditions for the convexity of the weighted smoothing functional and for the convergence of the iterative algorithm are established for a unique global solution which does not depend on initial conditions. Experimental results are shown with astronomical images which demonstrate the effectiveness of the proposed algorithm.     

An alternating iterative algorithm for the Cauchy problem in anisotropic elasticity

The alternating iterative algorithm proposed by Kozlov et al. [An iterative method for solving the Cauchy problem for elliptic equations. USSR Comput Math Math Phys 1991;31:45–52] for obtaining approximate solutions to the Cauchy problem in two-dimensional anisotropic elasticity is analysed and numerically implemented using the boundary element method (BEM). The ill-posedness of this inverse boundary value problem is overcome by employing an efficient regularising stopping. The numerical results confirm that the iterative BEM produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data.     

Performance analyses of an infrared single-mode all-fiber-optic Fourier-transform spectrometer

Stretching one of a pair of fiber arms can be done to realize optical phase modulation for an IR single-mode all-fiber-optic Fourier-transform spectrometer (FTS). But this operation will inevitably limit the physical performance of a FTS. We study these limits theoretically and experimentally. The optical path difference (OPD) will be dispersive. At the first-order approximation, this OPD dispersion will result in a shift in the recovered spectra. The spectral resolution and the sampling distance will also be dispersive. Linear birefringence introduced when a curved fiber is stretched will affect the final spectra. This effect can be eliminated by real-time compensation and (or) by system design. Errors encountered uniquely in the all-fiber-optic FTS in the optical phase domain, such as the fiber-parameter errors, nonlinearity of the piezoelectric cylinder, and their effects on the spectra are analyzed, from which we deduce the requirements for calibration. Finally, calibration methods for optical phase modulation are discussed.     

Rigorous electromagnetic design of finite-aperture diffractive optical elements by use of an iterative optimization algorithm

We propose a rigorous electromagnetic design of two-dimensional and finite-aperture diffractive optical elements (DOEs) that employs an effective iterative optimization algorithm in conjunction with a rigorous electromagnetic computational model: the finite-difference time-domain method. The iterative optimization process, the finite-difference time-domain method, and the angular spectrum propagation method are discussed in detail. Without any approximation based on the scalar theory, the algorithm can produce rigorous design results, both numerical and graphical, with fast convergence, reasonable computational cost, and good design quality. Using our iterative algorithm, we designed a diffractive cylindrical lens and a 1-to-2-beam fanner for normal-incidence TE-mode illumination, thus showing that the optimization algorithm is valid and competent for rigorously designing diffractive optical elements. Concerning the problem of fabrication, we also evaluated the performance of the DOE when the DOE profile is discrete.