Application of rational Chebyshev polynomials to optical problems

We present the use of the rational Chebyshev polynomials for discretising the transverse dimension(s) of beam propagation problems within the field of nonlinear optics. How a beam propagates in an optical medium, whether linear or nonlinear, is a common problem and important in both theoretical studies and optical design. The infinite domain and convergence properties of these polynomials allows one to handle the boundary conditions with greater correctness than methods that impose periodic boundary conditions such as Fourier methods. The beam is propagated forward by exponential integration for fast and accurate numerical simulations. The techniques employed to solve the beam propagation problems are easily applied to problems in other fields with mathematically similar models. References L. N. Trefethen. Spectral Methods in Matlab. Siam, Philadelphia, PA, 2000. http://www.comlab.ox.ac.uk/nick.trefethen/spectral.html M. Frigo and S. G. Johnson. The Design and Implementation of FFTW3. Proceedings of the IEEE, 93, 2005, 216--231. doi:10.1109/JPROC.2004.840301; Fastest Fourier Transform in the West. http://www.fftw.org/ J. A. C. Weideman and B. M. Herbst. Split-step methods for the solution of the nonlinear Schrodinger equation. SIAM J. Numer. Anal., 23, 1986, 485--507. http://www.jstor.org/stable/2157521 A. Taflove and S. C. Hageness. Computational Electrodynamics: The Finite-Difference Time-Domain Method. Norwood, MA: Artech, 2000. J. P. Boyd. Chebyshev and Fourier Spectral Methods. Dover Publications, 2nd edition, 2000. J. P. Boyd. Spectral methods using rational basis functions on an infinite interval. J. Comput. Phys., 69, 1987, 112--142. doi:10.1016/0021-9991(87)90158-6 J. A. C. Weideman and S. C. Reddy. A Matlab differentiation matrix suite. ACM Transactions on Mathematical Software, 26, 2000, 465--519. doi:10.1145/365723.365727 B. Minchev and W. Wright. A review of exponential integrators for first order semi-linear problems. Preprint Numerics No. 2/2005. Norwegian University of Science and Technology. http://www.math.ntnu.no/preprint/numerics/2005/N5-2005.pdf S. Krogstad. Generalized integrating factor methods for stiff pdes. J. Comput. Phys., 203, 2002, 72--88. doi:10.1016/j.jcp.2004.08.006 A. Kassam and L. N. Trefethen. Fourth-order time-stepping for stiff pdes. SIAM J. Sci. Comput., 6, 2005, 1214--1244. doi:10.1137/S1064827502410633 S. M. Cox and P. C. Matthews. Exponential time differencing for stiff systems. J. Comput. Phys., 176, 2002, 430--455. doi:10.1006/jcph.2002.6995 H. Berland, B. Skaflestad and W. M. Wright. Expint --- A Matlab package for exponential integrators. ACM Trans. Math. Softw., 33, 2007, 4. doi:10.1145/1206040.1206044 N. N. Akhmediev and A. Ankiewicz. Solitons---Nonlinear pulses and beams. Chapman and Hall, 1997. R. A. Fisher. The wave of advance of advantageous genes. Ann. Eugenics, 7, 1937, 353--369. http://digital.library.adelaide.edu.au/dspace/handle/2440/1512