Laguerre-Gaussian Basis Research Articles

A RAPID ACCURATE TECHNIQUE TO CALCULATE THE GROUP DELAY, DISPERSION AND DISPERSION SLOPE OF ARBITRARY RADIAL REFRACTIVE INDEX PROFILE WEAKLY-GUIDING OPTICAL FIBERS

This paper introduces a new numerical method to calculate the group delay, chromatic dispersion and dispersion slope of weakly-guiding optical fibers with arbitrary radial refractive index profiles. It is based on the analytic differentiation of the propagation coefficient up to the third order. The simulation results are compared to experimental data, with those calculated by other approaches and exact data where possible. Due to the analytical differentiation of the matrix equation, the method is more accurate compared to other approaches, it is also much faster than numerical differentiation as allows avoiding repeated solution of the eigenvalue problem to calculate the derivatives of the propagation coefficient. The precision of the method is limited only by the approximation errors of the mode solver. The Galerkin method with Laguerre-Gauss basis functions is used to determine the propagation coefficients of weakly-guiding structures. The new method enables fiber manufacturers to rapidly design dispersion characteristics of graded index, step index, single- and multiple-clad fibers, as well as few-mode and bend insensitive fibers.

Task-based performance analysis of FBP, SART and ML for digital breast tomosynthesis using signal CNR and Channelised Hotelling Observers

We assess the performance of filtered backprojection (FBP), the simultaneous algebraic reconstruction technique (SART) and the maximum likelihood (ML) algorithm for digital breast tomosynthesis (DBT) under variations in key imaging parameters, including the number of iterations, number of projections, angular range, initial guess, and radiation dose. This is the first study to compare these algorithms for the application of DBT. We present a methodology for the evaluation of DBT reconstructions, and use it to conduct preliminary experiments investigating trade-offs between the selected imaging parameters. This investigation includes trade-offs not previously considered in the DBT literature, such as the use of a stationary detector versus a C-arm imaging geometry. A real breast CT volume serves as a ground truth digital phantom from which to simulate X-ray projections under the various acquisition parameters. The reconstructed image quality is measured using task-based metrics, namely signal CNR and the AUC of a Channelised Hotelling Observer with Laguerre-Gauss basis functions. The task at hand is the detection of a simulated mass inserted into the breast CT volume. We find that the image quality in limited view tomography is highly dependent on the particular acquisition and reconstruction parameters used. In particular, we draw the following conclusions. First, we find that optimising the FBP filter design and SART relaxation parameter yields significant improvements in reconstruction quality from the same projection data. Second, we show that the convergence rate of the maximum likelihood algorithm, optimised with paraboloidal surrogates and conjugate gradient ascent (ML-PSCG), can be greatly accelerated using view-by-view updates. Third, we find that the optimal initial guess is algorithm dependent. In particular, we obtained best results with a zero initial guess for SART, and an FBP initial guess for ML-PSCG. Fourth, when the exposure per view is constant, increasing the total number of views within a given angular range improves the reconstruction quality, albeit with diminishing returns. When the total dose of all views combined is constant, there is a trade-off between increased sampling using a larger number of views and increased levels of quantum noise in each view. Fifth, we do not observe significant differences when testing various access ordering schemes, presumably due to the limited angular range of DBT. Sixth, we find that adjusting the z-resolution of the reconstruction can improve image quality, but that this resolution is best adjusted by using post-reconstruction binning, rather than by declaring lower-resolution voxels. Seventh, we find that the C-arm configuration yields higher image quality than a stationary detector geometry, the difference being most outspoken for the FBP algorithm. Lastly, we find that not all prototype systems found in the literature are currently being run under the best possible system or algorithm configurations. In other words, the present study demonstrates the critical importance (and reward) of using optimisation methodologies such as the one presented here to maximise the DBT reconstruction quality from a single scan of the patient.

Optimization of a tomosynthesis system for the detection of lung nodules

Mathematical observers that track human performance can be used to reduce the number of human observer studies needed to optimize imaging systems. The performance of human observers for the detection of a 3.6 mm lung nodule in anatomical backgrounds was measured as a function of varying tomosynthetic angle and compared with mathematical observers. The human observer results showed a dramatic increase in the percent of correct responses, from 80% in the projection images to 96% in the projection images with a tomosynthetic angle of just 3 degrees. This result suggests the potential usefulness of the scanned beam digital x-ray system for this application. Given the small number of images (40) used per tomosynthetic angle and the highly nonstationary statistical nature of the backgrounds, the nonprewhitening eye observer achieved a higher performance than the channelized Hotelling observer using a Laguerre-Gauss basis. The channelized Hotelling observer with internal noise and the eye filter matched to the projection data were shown to track human performance as the tomosynthetic angle changed. The validation of these mathematical observers extends their applicability to the optimization of tomosynthesis systems.

An efficient method for computing optical waveguides with discontinuous refractive index profiles using spectral collocation method with domain decomposition

An accurate and efficient solution method using spectral collocation method with domain decomposition is proposed for computing optical waveguides with discontinuous refractive index profiles. The use of domain decomposition divides the usual single domain into a few subdomains at the interfaces of discontinuous refractive index profiles. Each subdomain can be expanded by a suitable set of orthogonal basis functions and patched at these interfaces by matching the physical boundary conditions. In addition, a new technique incorporating the effective index method and the Wentzel-Kramers-Brillouin method for the a-priori determination of the scaling factor in Hermite-Gauss or Laguerre-Gauss basis functions is introduced to considerably save computational time without experimenting with it. This method shares the same desirable property of the spectral collocation method of providing a fast and accurate solution but avoids the oscillatory solutions and improves the poor convergence problem of the simple spectral collocation method with single domain where regions of discontinuous refractive index profiles exist. Applications to several two- and three-dimensional waveguide structures having exact or accurate approximate solutions are given to test the accuracy and efficiency; all the results are found to be in excellent agreement.

Modal properties of circular and noncircular optical waveguides

Abstract We give a review and a comparison of recent methods of analyzing circular and noncircular optical waveguides. Comparison among competing methodologies is made as follows: Galerkin's method is used with Laguerre-Gauss basis functions in circular geometry to examine the modal solution in a step index fiber, and comparison with the exact solution is made. A W-fiber, which has no exact solution, is then examined. Rectangular geometry is considered, and discussion centers on the use of Galerkin's method using trigonometric basis functions and Hermite-Gauss basis functions. Re difficulty arising from the use of basis functions that do not decay exponentially for large argument (trigonometric functions) is illustrated. Finally, a square step index waveguide is used to illustrate a comparison between a variational method that uses the Gaussian approximation as the starting point, and Galerkin's method using Hermite-Gauss basis functions. We conclude that the variational method does well in predicting the...

An accurate splice loss analysis for single-mode graded-index fibers with mismatched parameters

Expressing the fundamental scalar mode by a series of Laguerre-Gaussian basis functions, a simple formula is derived for the splice loss between two perfectly aligned single-mode graded-index nonidentical fibers. As an example, the splice loss tolerances for dispersion-shifted fibers due to mismatch of V-parameters as well as mismatch of profile shape have been compared. For zero-dispersion fibers at lambda =1.55 mu m, an optimum range for the relative index difference has been obtained in the case of profile mismatch.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>