## Fast convergent and unconditionally stable Galerkin’s method with adaptive Hermite–Gauss expansion for guided-mode extraction in two-dimensional photonic crystal based waveguides

JOSA B, Vol. 26, Issue 1, pp. 169-175 (2009)

http://dx.doi.org/10.1364/JOSAB.26.000169

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### Abstract

It has been recently shown that guided modes in two-dimensional photonic crystal based structures can be fast and efficiently extracted by using the Galerkin’s method with Hermite–Gauss basis functions. Although quite efficient and reliable for photonic crystal line defect waveguides, difficulties are likely to arise for more complicated geometries, e.g., for coupled resonator optical waveguides. First, unwanted numerical instability may well occur if a large number of basis functions are retained in the calculation. Second, the method could have a slow convergence rate with respect to the truncation order of the electromagnetic field expansion. Third, spurious solutions are not unlikely to appear. All these three important issues are here resolved by applying the unconditionally stable *S*-matrix propagation method, by proposing an adaptive algorithm to expedite the convergence rate of the expansion through duly scaled Hermite–Gauss basis functions, and by providing an effective algorithm for the elimination of spurious modes.

© 2008 Optical Society of America

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(130.0130) Integrated optics : Integrated optics

(130.5296) Integrated optics : Photonic crystal waveguides

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: September 16, 2008

Manuscript Accepted: October 24, 2008

Published: December 24, 2008

**Citation**

Peyman Sarrafi and Khashayar Mehrany, "Fast convergent and unconditionally stable Galerkin's method with adaptive Hermite-Gauss expansion for guided-mode extraction in two-dimensional photonic crystal based waveguides," J. Opt. Soc. Am. B **26**, 169-175 (2009)

http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-26-1-169

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