LU decomposition of matrices with augmented dense constraints

Sparse matrices composed of a central band and augmented dense rows and columns are becoming prevalent in the numerical solution of a large class of boundary and initial-value problems. A Fortran Subroutine ARROW is presented for the LU decomposition and solution of linear equation systems with such a structure. The computational speed of the program is compared in MFLOPS (millions of floating point operations per second) to the LINPACK benchmark for the solution of a dense linear system and is found to be of comparable speed on both supercomputers and minicomputers. Use of the Basic Linear Algebra Subroutines (BLAS) available on most machines significantly enhances the speed of ARROW .     

Fast optical ray tracing using multiple DSPs

Optical ray tracing is a computationally intensive operation that is central both to the design of optical systems and to analyzing their performance once built. The authors have previously reported on the use of parallel digital signal processors (DSPs) to reduce the time required to perform ray tracing in analyzing the performance of the moderate resolution imaging spectroradiometer (MODIS), which is presently in orbit on multiple spacecraft. The earlier work was incomplete, providing only a conservative estimate of the performance improvement that could be achieved with one to four DSPs. This paper reports on the completed project and extends the earlier work to eight DSPs. As predicted in the earlier paper, not all rays make it through the entire optical system. Many are lost along the way. This is one factor that led to reduced processing time. Another is the use of an optimizing compiler. In this paper, the authors present results showing the separate effect of each of these two independent factors on the overall processing time. The most significant finding is the extraordinarily linear relationship between the number of DSPs available and the speed of the ray tracing. By using eight DSPs, the processing time is reduced from two weeks to less than one and a half days, an improvement of nearly a whole order of magnitude. Low-cost high-speed ray tracing is now feasible using off-the-shelf plug-in processor boards.     

Interpretation of nondepolarizing Mueller matrices based on singular-value decomposition

A product decomposition of a nondepolarizing Mueller matrix consisting of the sequence of three factors--a first linear retarder, a horizontal or vertical "retarding diattenuator," and a second linear retarder--is proposed. Each matrix factor can be readily identified with one or two basic polarization devices such as partial polarizers and retardation waveplates. The decomposition allows for a straightforward interpretation and parameterization of an experimentally determined Mueller matrix in terms of an arrangement of polarization devices and their characteristic parameters: diattenuations, retardances, and axis azimuths. Its application is illustrated on an experimentally determined Mueller matrix.     

Fast decomposition of matrices generated by the boundary element method

A new method to reduce the solution time of matrices generated by the Boundary Element Method is presented here. The method involves converting the fully populated system into a banded system by lumping certain coefficients of the matrix into fictitious nodes and then constraining these nodes to accurately represent each coefficient. The major advantages of lumping over the substructuring method are that lumping can be applied to arbitrarily shaped geometries and infinite-domain problems and that it preserves the diagonal-dominance of the matrix. It is shown here that the proposed algorithm reduces the rate of increase of solution time t of an n-degree-of-freedom problem from t ∝ n³ to t ∝ n². Although the algorithm is for thermal problems, its extension to mechanical problems is straightforward. The procedure can easily be incorporated into existing boundary-element-based packages.     

Second-order derivatives of optical path length of ray with respect to variable vector of source ray

A method is proposed for determining the second-order derivatives (i.e., the Hessian matrix) of the optical path length of a ray with respect to the variable vector of the source ray in an optical system comprising both flat and spherical boundary surfaces. Several wavefront aberration problems are investigated using the Hessian matrix proposed in this study and the Jacobian (first-order derivatives) matrix presented in the literature. It is found that when using the Hessian matrix the precision of wavefront aberration is significantly improved when evaluated up to the quadratic term of the Taylor series expansion. The methodology proposed in this study not only provides the means to investigate the principal curvatures of the wavefront along a ray, but also yields the information required to determine the irradiance and caustics of both axisymmetric and nonaxisymmetric optical systems.     

Atmospheric ray path modeling for radiative transfer algorithms

A new method for the determination of ray paths as well as resulting path segments and partial gas columns within a layered atmosphere is presented. Any singularity at the tangent point is avoided. No use is made of the gross spherical symmetry of the Earth's atmosphere. Using this approach we examine the impact of the Earth's oblate shape and horizontal atmospheric inhomogeneities on infrared limb spectra.     

Locality and decomposition of regular optical interconnection patterns

Methods that a designer can use to optimize the placement of nodes in a large switching network to decrease the requirements on holographic interconnections are investigated. Localized interconnections between subdivided switches are combined with simpler global interconnections. The interconnections between subdivided switches can be implemented by use of metallic traces on smart-pixel arrays. The global interconnections would be provided by optical free-space techniques. Several advantages arise from this procedure: (1) The regular interconnection pattern is decomposed into several pipes (collection of light beams that form a complete pattern) without loss of functionality. (2) The interconnection pattern may be optimized by variation of the placement of the switches in a switching network (e.g., to obtain a minimum deflection angle). (3) The interconnection pattern may be adjusted to the need of an algorithm by an additional parameter (the dimension). The application to photonic switching networks and signal processing is discussed.     

Modulation transfer functions of optical fibers

Translated from Izmeritel'naya Tekhnika, No. 6, pp. 24–25, June, 1992.     

Unique decomposition of element stiffness numeric matrices for three-noded plate bending triangles

This work is concerned with the numeric stiffness matrices of three-noded triangular plate bending finite elements; in particular with those numeric stiffness matrices, which are freedom-deficient and comply with the conditions of the patch test Subsequent to initial transformation of the rotation connectors for such matrices it is evident that there must exist an unique decomposition to the stiffness matrix K̆ ₆∈ℝ^6×6 of the corresponding Kirchhoff six-noded constant bending moment triangle. In K̆ ₆ all six trial functions are themselves synonymous with those which describe the patch test. The transformation matrices of decomposition, and subsequent restoration upon modification or design, are derived explicitly and are succinct in application. Decomposition of the numeric stiffness matrix leads to exceptional versatility in objective modification, e.g. design of the matrix by adaptive process. Attention is confined here to the stiffness matrices K̆ ₉∈ℝ^9×9 of nine-degree-of-freedom three-noded Kirchhoff plate bending triangles with their single-degree-of-freedom deficiency. The decomposition of the element stiffness matrix immediately reveals those six coefficients which are available for design. They control the effect of transverse shear and are the constituents of a symmetric positive-definite matrix M ₃∈ℝ^3×3 which is designated the ‘mechanism restraint’ matrix. It is necessary only that the designed coefficients are such that the matrix M ₃ remains symmetric and positive definite so as to ensure retention of patch test satisfaction on restoration to the newly designed K ₉. The illustrative examples provide a first perception of the leap in expectation which is enabled by design of the numeric K ₉ when uninhibited by formal method. Thus, the feasibility is illustrated of simple adaptive design of K ₉ with objective to recover cubically varying w displacements over an equilateral patch of equal triangles. This recovery is readily achieved by ad hoc inverse method but raises the issue of uniqueness in design. In highlighting the characteristics of M ₃ it is evident that there remains a wealth of opportunity for further research before adaptive design of the element stiffness matrix, within an arbitrary prevailing w displacement field, can become a practical reality. An appendix lists the Fortran computer codings which are used in the examples to calculate the stiffness matrix K̆ ₆ of the six-noded constant bending moment Kirchhoff triangle as well as the explicit transformation matrices for decomposition and restoration of the numeric K ₉ stiffness matrix. © 1998 John Wiley & Sons, Ltd.     

Stress transfer between fibrillated polyalkene films and cement matrices

Experimental results indicate that the ‘average bond’ strength between polyalkenes and cement increases with increasing film volume fraction. It is suggested that this dynamic transfer of stress from fibril to matrix is achieved because of the misfit between fibril and matrix resulting from the non-uniform cross-section of the fibrils. This mechanism is equally applicable to both fibrillated films and monofilaments.     

Determination of second-order derivatives of a skew ray with respect to the variables of its source ray in optical prism systems

The second-order derivative of a scalar function with respect to a variable vector is known as the Hessian matrix. We present a computational scheme based on the principles of differential geometry for determining the Hessian matrix of a skew ray as it travels through a prism system. A comparison of the proposed method and the conventional finite difference (FD) method is made at last. It is shown that the proposed method has a greater inherent accuracy than FD methods based on ray-tracing data. The proposed method not only provides a convenient means of investigating the wavefront shape within complex prism systems, but it also provides a potential basis for determining the higher order derivatives of a ray by further taking higher order differentiations.     

Phase-only asymmetric optical cryptosystem based on random modulus decomposition

We propose a phase-only asymmetric optical cryptosystem based on random modulus decomposition (RMD). The cryptosystem is presented for effectively improving the capacity to resist various attacks, including the attack of iterative algorithms. On the one hand, RMD and phase encoding are combined to remove the constraints that can be used in the attacking process. On the other hand, the security keys (geometrical parameters) introduced by Fresnel transform can increase the key variety and enlarge the key space simultaneously. Numerical simulation results demonstrate the strong feasibility, security and robustness of the proposed cryptosystem. This cryptosystem will open up many new opportunities in the application fields of optical encryption and authentication.     

Adaptive Hermite-Gauss decomposition method to analyze optical dielectric waveguides

A novel spectral method with variable transformation, the adaptive Hermite-Gauss decomposition method (A-HGDM), has been developed and applied to the analysis of three-dimensional (3D) dielectric structures. The proposed method includes an optimization strategy to automatically find the quasi-optimum numerical parameters of the variable transformation with low computational effort. The technique has been tested by analyzing two typical 3D dielectric structures: the rectangular step-index waveguide and the rib-waveguide directional coupler. In both cases, the A-HGDM increases the accuracy of the Hermite-Gauss decomposition method (HGDM), especially when the mode is near cutoff, and improves the computational efficiency of previously published optimization strategies (optimized HGDM).     

Second-order derivatives of a ray with respect to the variables of its source ray in optical systems containing spherical boundary surfaces

The second-order derivative matrix of a scalar function with respect to a variable vector is called a Hessian matrix, which is a square matrix. Our research group previously presented a method for determination of the first-order derivatives (i.e., the Jacobian matrix) of a skew ray with respect to the variable vector of an optical system. This paper extends our previous methodology to determine the second-order derivatives (i.e., the Hessian matrix) of a skew ray with respect to the variable vector of its source ray when this ray is reflected/refracted by spherical boundary surfaces. The traditional finite-difference methods using ray-tracing data to compute the Hessian matrix suffer from various cumulative rounding and truncation errors. The proposed method uses differential geometry, giving it an inherently greater accuracy. The proposed Hessian matrix methodology has potential use in optimization methods where the merit function is defined as ray aberrations. It also can be used to investigate the shape of the wavefront for a ray traveling through an optical system.     

Reconstruction of optical characteristics of waveguide lenses by the use of ray tracing

A method that uses the data of ray tracing for optical waveguide lens diagnostics is described. This method permits a direct reconstruction of the optical characteristics of a waveguide without the optical or the physical thickness being measured. Conditions are determined for the mathematical problem of diagnostics by ray tracing to have a unique solution, and a technique to obtain a numerical solution from noisy experimental data is described.     

Intensity-based modal decomposition of optical beams in terms of Hermite-Gaussian functions

We show that when an arbitrary optical beam is decomposed into a superposition of Hermite-Gaussian functions, it is sufficient to record a number of intensity profiles sampled at various transverse planes to uniquely determine the relative modal weights. This result follows from the parity relation and the nature of the Gouy phase, in addition to the orthogonality of the Fourier-transformed intensity profiles associated with the Hermite-Gaussian modes.