Numerical instability of the C method when applied to coated gratings and methods to avoid it

Area of Science:

• Optics and Photonics
• Computational Electromagnetics
• Materials Science

Background:

• The coordinate transformation (C) method is widely used for analyzing optical gratings.
• Numerical instability has been observed when applying the C method to thinly coated gratings.

Purpose of the Study:

• To identify the origin of numerical instability in the C method for thinly coated gratings.
• To propose solutions for circumventing this instability.
• To compare the performance of recursive and nonrecursive algorithms.

Main Methods:

• Analysis of the eigenvector matrix conditioning in the C method.
• Development and testing of novel methods to improve numerical stability.
• Comparative performance evaluation of recursive and nonrecursive matrix algorithms.

Main Results:

• Numerical instability arises from the ill-conditioned eigenvector matrix at high truncation numbers, not field dependence in coated layers.
• Two effective methods were identified to circumvent the observed numerical instability.
• All popular recursive matrix algorithms exhibited poor immunity to the instability, performing worse than the full matrix (nonrecursive) algorithm.

Conclusions:

• The C method's numerical instability in thinly coated gratings is linked to matrix conditioning.
• Recommended methods and the nonrecursive algorithm offer robust solutions for stable grating analysis.