Designing Optical Fibers: Fitting the Derivatives of a Nonlinear Pde-Eigenvalue Problem

Linda Kaufman, Seok-Min Bang, Brian Heacook, William Landon, Daniel Savacool, Nick Woronekin

Computer Science Department, William Paterson University, Passaic, USA.

DOI: 10.4236/ajcm.2012.24044                

Abstract

When trying to fit data to functions of the eigensystem of a pde-eigenvalue problem, such as Maxwell’s equation, numerical differentiation is ineffective and analytic gradients must be supplied. In our motivating example of trying to determine the chemical composition of the layers of specialty optical fibers, the function involved fitting the higher order derivatives with respect to frequency of the positive eigenvalues. The computation of the gradient was the most time consuming part of the minimization problem. It was realized that if one interchanged the order of differentiation, and differentiated first with respect to the design parameters, fewer derivatives of the eigenvectors would be required and one could take full advantage that each grid point was affected by only a few variables. As the model was expanded to cover a fiber wrapped around a spool, the bandwidth of the linearized symmetric eigenvalue problem increased. At the heart of each of the iterative methods used to find the few positive eigenvalues was a symmetric, banded, indefinite matrix. Here we present an algorithm for this problem which reduces a symmetric banded matrix to a block diagonal matrix of 1 x 1 and 2 x 2 blocks. Fillin outside the band because of pivoting for stability is prevented by a sequence of planar transformations. Computationally the algorithm is compared to the block unsymmetric banded solver and the block positive definite symmetric band solver in LAPACK.

Keywords

Symmetric; Indefinite; Eigenvalue; Gradient; Optical Fiber Design

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	W. A. Reed, “Fiber Design for the 21st Century,” Optical Society of America, San Francisco, 1999
[2]	T. A. Lenahan, “Calculation of Modes in an Optical Fiber using the Finite Element Method and EISPACK,” The Bell System Technical Journal, Vol. 62, No. 9, 1983, pp. 2663-2694
[3]	S. V. Kartalopoulous, “Introduction to DWDM Technology,” IEEE Press, Piscataway, 2000.
[4]	C. Bischof, A. Carle, G. Corliss, A. Griewank and P. Hovland, “ADIFOR-Generating Derivative Codes for Fortran Programs,” Scientific Programming, Vol. 1, No. 1, 1992, pp. 11-29.
[5]	L. Kaufman, “Eigenvalue Problems in Fiber Optics Design,” SIAM Journal on Matrix Analysis and Applications, Vol. 28, No. 1, 2006, pp.105-117. doi:10.1137/S0895479803432708
[6]	X. Huang, Z. Bai and Y. Su, “Nonlinear Rank One Modification of the Symmetric Eigenvalue Problem,” Journal of Computational Mathematics, Vol. 28, No. 2, 2010, pp. 218-234
[7]	T. Betcke, N. J. Higham, V. Mehrmann, C Schr?der and F. Tisseur, “NLEVP: A Collection of Nonlinear Eigenvalue Problems,” Manchester Institute of Mathematical Sciences Preprint, Manchester, 2008
[8]	H. Voss, K. Yildiztekin and X. Huang, “Nonlinear Low Rank Modification of a symmetric Eigenvalue Problem,” SIAM Journal on Matrix Analysis and Applications, Vol. 32, No.2, 2011, pp. 515-535. doi:10.1137/070779528
[9]	H. Schwetlick and K. Schreiber, “Nonlinear Rayleigh Functionals,” Linear Algebra and its Applications, Vol. 436, No. 10, 2011, pp. 3991-4016.
[10]	B. S. Liao, Z. Bai, L.-Q. Lee and K. Ko, “Nonlinear Rayleigh-Ritz Iterative Methods for Solving Large Scale Nonlinear Eigenvalue Problems,” Taiwanese Journal of Mathematics, Vol. 14, No. 3, 2010, pp. 869-883.
[11]	J. R. Bunch and L. Kaufman, “Some Stable Methods for Calculating Inertia and Solving Symmetric Linear Systems,” Mathematics of Computation, Vol. 31, No. 137, 1977, pp. 163-179. doi:10.1090/S0025-5718-1977-0428694-0
[12]	L. Kaufman, “The Retraction Algorithm for Factoring Banded Symmetric Matrices,” Numerical Linear Algebra with Applications, Vol. 14, No. 3, 2007, pp. 237-254. doi:10.1002/nla.529
[13]	P. Lancaster, “On Eigenvalues of Matrices Dependent on a Parameter,” Numerische Mathematik, Vol. 6, No. 1, 1964, pp. 377-387. doi:10.1007/BF01386087
[14]	L. Kaufman, “Calculating Dispersion Derivatives in Fiber-Optic Design,” Journal of Lightwave Technology, Vol. 25, No. 3, 2007, pp. 811-819. doi:10.1109/JLT.2006.889648
[15]	F. Brechet, J. Marcou, D. Pagnoux and P. Roy, “Complete Analysis of the Characteristics of Propagation into Photonic Crystal Fibers by the Finite Element Method,” Optical Fiber Technology, Vol. 6, No. 2, 2000 pp. 181-191. doi:10.1006/ofte.1999.0320
[16]	K. Saitoh, M. Koshiba, T. Hasegawa and E. Sasaoka, “Chromatic Dispersion Control in Photonic Optical Fibers: Application to Ultra-Flattened Dispersion,” Optics Express, Vol. 11, No. 8, 2003, pp. 843-852. doi:10.1364/OE.11.000843
[17]	S. Johnson and J. Joannopoulos, “Block-Iterative Frequency-Domain Methods for Maxwell’s Equations in a Planewave Basis,” Optics Express, Vol. 8, No. 3, 2001, pp. 173-190. doi:10.1364/OE.8.000173
[18]	S. Guo, F. Wu, S. Albin, H. Tai and R. Rogowski, “Loss and Dispersion Analysis of Microstructured Fibers by Finite Difference Method,” Optics Express, Vol. 12, No. 15, 2004, pp. 3341-3352. doi:10.1364/OPEX.12.003341
[19]	J. W. Fleming, “Material Dispersion in Lightguide Glasses,” Electronics Letters, Vol. 14, No. 11, 1978, pp. 326-328. doi:10.1049/el:19780222
[20]	D. Marcuse, “Field Deformation and Loss Caused by Curvature of Optical Fibers,” Journal of the Optical Society America, Vol. 66, No. 4, 1972, pp. 311-320. doi:10.1364/JOSA.66.000311
[21]	E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov and D. Sorensen, “LAPACK Users’ Guide,” 2nd Edition, Society for Industrial and Applied Mathematics, Philadelphia, 1995.