Explicit finite-difference vector beam propagation method based on the iterated Crank-Nicolson scheme

Abstract

We introduce and develop a new explicit vector beam propagation method, based on the iterated Crank-Nicolson scheme, which is an established numerical method in the area of computational relativity. The proposed approach results in a fast and robust method, characterized by simplicity, efficiency, and versatility. It is free of limitations inherent in implicit beam propagation methods, which are associated with poor convergence or uneconomical use of memory in the solution of large sparse linear systems, and thus it can tackle problems of considerable size and complexity. The advantages offered by this approach are demonstrated by analyzing a multimode interference coupler and a twin-core photonic crystal fiber. A possible wide-angle generalization is also provided.

Core-Confined Beam Propagation Method for Guiding and Leaky Structures

“…One possibility is to have a computational grid to sample the core of the waveguiding structure, and a singlesite boundary layer of "ghost" points that will be slaved to the inner solution in such a way that a discrete approximation to (9) will be satisfied. Such an approach meshes well with Crank-Nicolson-based BPM methods utilizing either sparse or iterative linear solvers, and also with other methods, e.g., the explicit iterated Crank-Nicolson method [27]. We will illustrate an implementation with the method of lines.…”

“…For the reference calculations, we solve the beam propagation equations with the iterated Crank-Nicolson method [27], and calculate quasi-TM modes across a range of wavelengths by propagation in imaginary distance. Then, we compare results obtained in the semivectorial approach, in which the grid is restricted to the silicon nitride core (including the silica slot), and boundary conditions in the form of (16) and (17) are applied.…”

“…1. Lines represent results obtained with a standard beam propagation method (iterated Crank-Nicolson[27]), in which the computational domain encompassed large regions of substrate and air around the waveguide core. Symbols represent results obtained with present method that only discretizes the inner waveguide core.…”

On iterated Crank–Nicolson methods for hyperbolic and parabolic equations

“…Also another version of the iterated CN method was proposed in [5] and it was found that the last algorithm has better properties even providing only the first order of accuracy. Different versions of the iterated CN method were used in numerical relativity by a number of authors (e.g., [3,4,[6][7][8] and references therein).…”

Ανάπτυξη Υπολογιστικών Τεχνικών Για Την Ανάλυση Και Σχεδίαση Μικροκυματικών Και Φωτονικών Διατάξεων Με Την Ενσωμάτωση Στοιχείων Συντονισμού Και Δομών Με Τεχνητές Ηλεκτρομαγνητικές Ιδιότητες

“…Η λύση και στα δύο ζητήματα δόθηκε από τη μέθοδο των πεπερασμένων διαφορών (Finite Difference Method, FDM). Η πρώτη προσέγγιση της διανυσματικής FD-BPM πραγματοποιήθηκε μέσω ενός σχήματος πεπερασμένων διαφορών Crank-Nicolson [116], ενώ έκτοτε προτάθηκαν διάφορες προσεγγίσεις διατύπωσης της μεθόδου διάδοσης δέσμης με τη χρησιμοποίηση διαφόρων σχημάτων πεπερασμένων διαφορών [117][118][119][120]. Εξαιτίας της δυνατότητας για γρήγορη και εύκολη (από πλευράς υπολογιστικών πόρων) επίλυση που παρέχει, η BPM είναι αναμφίβολα μια χρήσιμη εναλλακτική αριθμητικής ανάλυσης, ενώ οι όποιες προσεγγίσεις που εισάγονται κατά την υλοποίηση της μεθόδου (π.χ.…”