Fast numerical solution of nonlinear diffusion equation for the simulation of ion-exchanged micro-optics components in glass

Abstract Modulation of the refractive index of glass using ion exchange is a common fabrication method for various micro-optics components, such as waveguide devices, microlenses, and diffractive elements. The design of elements with complicated optical functions has increased the demand for fast numerical simulation of the nonlinear ion-exchange process of two monovalent ions. In this paper, a linear implicit finite-difference Crank-Nicholson-type method is implemented for thermal two- and three-dimensional ion-exchange processes. The resulting system of linear equations is solved by using various iterative methods. Grid spacing, length of the time step, and the number of iterations are chosen so that desired accuracy is obtained within minimum computing time. The implemented methods are compared with the Peaceman-Rachford Alternating Direction Implicit (PR-ADI) and the explicit Du Fort-Frankel method in modeling a computer-synthesized diffractive optical element and a waveguide channel system.

[1]  Norbert Neuroth,et al.  The Properties of Optical Glass , 1995 .

[2]  S. Iraj Najafi,et al.  Introduction to glass integrated optics , 1992 .

[3]  S. Sinzinger,et al.  Index-distributed planar microlenses for three-dimensional micro-optics fabricated by silver-sodium ion exchange in BGG35 substrates. , 1994, Applied optics.

[4]  R. Adar,et al.  Adiabatic 3-dB couplers, filters, and multiplexers made with silica waveguides on silicon , 1992 .

[5]  Roland W. Freund,et al.  A Transpose-Free Quasi-Minimal Residual Algorithm for Non-Hermitian Linear Systems , 1993, SIAM J. Sci. Comput..

[6]  R. H. Hobbs,et al.  Theory of 2-D ion exchange in glass: optimization of microlens arrays. , 1991, Applied optics.

[7]  H.-P. Nolting,et al.  Results of benchmark tests for different numerical BPM algorithms , 1995 .

[8]  P. Sonneveld.  CGS, A Fast Lanczos-Type Solver for Nonsymmetric Linear systems , 1989 .

[9]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[10]  H. V. D. Vorst,et al.  High Performance Preconditioning , 1989 .

[11]  G. Pinder,et al.  Numerical solution of partial differential equations in science and engineering , 1982 .

[12]  J. Turunen,et al.  Three-dimensionally modulated graded-index diffractive elements by thermal ion exchange in glass. , 1995, Optics letters.

[13]  R. Wood.  Introduction to glass integrated optics , 1993 .

[14]  B. A. Carré,et al.  The Determination of the Optimum Accelerating Factor for Successive Over-relaxation , 1961, Comput. J..

[15]  J. Ferziger.  Numerical methods for engineering application , 1981 .

[16]  Ramu V. Ramaswamy,et al.  Ion-exchanged glass waveguides: a review , 1988 .

[17]  M Izutsu,et al.  Optical-waveguide hybrid coupler. , 1982, Optics letters.

[18]  F. Krogh,et al.  Solving Ordinary Differential Equations , 2019, Programming for Computations - Python.