Computing Fresnel integrals via modified trapezium rules

In this paper we propose methods for computing Fresnel integrals based on truncated trapezium rule approximations to integrals on the real line, these trapezium rules modified to take into account poles of the integrand near the real axis. Our starting point is a method for computation of the error function of complex argument due to Matta and Reichel (J Math Phys 34:298–307, 1956) and Hunter and Regan (Math Comp 26:539–541, 1972). We construct approximations which we prove are exponentially convergent as a function of N , the number of quadrature points, obtaining explicit error bounds which show that accuracies of 10−15 uniformly on the real line are achieved with N=12 , this confirmed by computations. The approximations we obtain are attractive, additionally, in that they maintain small relative errors for small and large argument, are analytic on the real axis (echoing the analyticity of the Fresnel integrals), and are straightforward to implement

Computation of the complex error function using modified trapezoidal rules

In this paper we propose a method for computing the Faddeeva function w(z) := \re^{-z^2}\erfc(-\ri\,z) via truncated modified trapezoidal rule approximations to integrals on the real line. Our starting point is the method due to Matta and Reichel ({\em Math.\ Comp.} {\bf 25} (1971), pp.~339--344) and Hunter and Regan ({\em Math.\ Comp.} {\bf 26} (1972), pp.~339--541). Addressing shortcomings flagged by Weideman ({\em SIAM.\ J.\ Numer.\ Anal. } {\bf 31} (1994), pp.~1497--1518), we construct approximations which we prove are exponentially convergent as a function of $N+1$, the number of quadrature points, obtaining error bounds which show that accuracies of $2\times 10^{-15}$ in the computation of $w(z)$ throughout the complex plane are achieved with $N = 11$, this confirmed by computations. These approximations, moreover, provably achieve small relative errors throughout the upper complex half-plane where $w(z)$ is non-zero. Numerical tests suggest that this new method is competitive, in accuracy and computation times, with existing methods for computing $w(z)$ for complex $z$

The exponentially convergent trapezoidal rule

It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators

Modelling of light scattering by cirrus ice crystals using geometric optics combined with diffraction of facets

A new 3D model of light scattering applicable to dielectric faceted objects is presented. The model combines Geometric Optics with diffraction on individual facets yet maintains the low computational expense of standard Geometric Optics. The current implementation of the model is explained and then applied to the problem of light scattering by ice crystals in cirrus clouds. Accurate modelling of the scattering properties of such crystals is crucial to better understanding of cirrus radiative properties and hence to climate modelling and weather forecasting. Calculations using the new model are compared to a separation of variables method and the Improved Geometric Optics method with encouraging results. The model shows significant improvements over standard Geometric Optics. The size applicability of the new model is discussed. The model is applied to a range of crystal geometries that have been observed in cirrus including the hexagonal column, the hollow column, the droxtal and the bullet rosette. For each geometry the phase function and degree of linear polarization are presented and discussed. Ice analogue crystals grown at the University of Hertfordshire have optical properties very close to ice but are stable at room temperature. The geometries of three ice analogue crystals are reconstructed and the single scattering properties of the reconstructions are presented. 2D scattering patterns calculated using the model are compared to laboratory photographs of scattering patterns on a screen created by an ice analogue hexagonal column. The agreement is shown to be very good. By applying the model to a range of geometries, it is shown that the results in the form of 2D scattering patterns can potentially be used to aid particle characterization. By combining the model with a Monte Carlo radiative transfer code, comparisons are made with aircraft radiance measurements of cirrus provided by the Met Office. The improvements over standard Geometric Optics are found to persist following a radiative transfer treatment

Gratings: Theory and Numeric Applications, Second Revisited Edition

International audienceThe second Edition of the Book contains 13 chapters, written by an international team of specialist in electromagnetic theory, numerical methods for modelling of light diffraction by periodic structures having one-, two-, or three-dimensional periodicity, and aiming numerous applications in many classical domains like optical engineering, spectroscopy, and optical telecommunications, together with newly born fields such as photonics, plasmonics, photovoltaics, metamaterials studies, cloaking, negative refraction, and super-lensing. Each chapter presents in detail a specific theoretical method aiming to a direct numerical application by university and industrial researchers and engineers.In comparison with the First Edition, we have added two more chapters (ch.12 and ch.13), and revised four other chapters (ch.6, ch.7, ch.10, and ch.11

Modelling and Characterization of Guiding Micro-structured Devices for Integrated Optics

In this thesis we show several modelling tools which are used to study nonlinear photonic band-gap structures and microcavities. First of all a nonlinear CMT and BPM were implemented to test the propagation of spatial solitons in a periodic device, composed by an array of parallel straight waveguides. In addition to noteworthy theoretical considerations, active functionalities are possible by exploiting these nonlinear regimes. Another algorithm was developed for the three-dimensional modelling of photonic cavities with cylindrical symmetry, such as microdisks. This method is validated by comparison with FDTD. We also show the opportunity to confine a field in a region of low refractive index lying in the centre of a silicon microdisk. High Q-factor and small mode volumes are achieved. Finally the characterization of microdisks in SOI with Q-factor larger than 50000 is presente

13th Annual Review of Progress in Applied Computational Electromagnetics at the Naval Postgraduate School, Monterey, CA, March 17-21, 1997, Conference Proceedings Volumes I & II

Includes Volumes 1 &