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A stable complex Jacobi iterative solution of 3D semivectorial wide-angle beam propagation using the iterated Crank-Nicholson method

Khai Le Quang (UGent) and Peter Bienstman (UGent)
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Center for nano- and biophotonics (NB-Photonics)
Abstract
An extension of the recently proposed three-dimensional (3D) wide-angle (WA) beam propagation method (BPM) using complex Jacobi iteration (CJI) taken into account polarization effects is presented. The resulting iterative BPM is faster than BPMs based on the traditional direct matrix inversion for waveguides with unchanging refractive index profiles during propagation direction. However, for varying refractive index waveguides the iterative method suffered from the fact that the iteration count between two successive cross-sections increases dramatically during the propagation direction. To overcome this problem, we propose the utility of the iterated Crank-Nicholson method. At each propagation step, the propagation equation is divided in multiple stages by the iterated Crank-Nicholson method and then each stage is recast in terms of a Helmholtz equation with source term, which is solved effectively by the complex Jacobi iterative method. The resulting approach is stable and well-suited for large structures with long propagation paths.
Keywords
Iterated Crank-Nicholson, Complex Jacobi iteration, 3D semivectorial BPM, PADE APPROXIMANT OPERATORS

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Citation

Please use this url to cite or link to this publication:

MLA
Le Quang, Khai, and Peter Bienstman. “A Stable Complex Jacobi Iterative Solution of 3D Semivectorial Wide-angle Beam Propagation Using the Iterated Crank-Nicholson Method.” OPTICAL AND QUANTUM ELECTRONICS 41.4 (2009): 215–221. Print.
APA
Le Quang, K., & Bienstman, P. (2009). A stable complex Jacobi iterative solution of 3D semivectorial wide-angle beam propagation using the iterated Crank-Nicholson method. OPTICAL AND QUANTUM ELECTRONICS, 41(4), 215–221.
Chicago author-date
Le Quang, Khai, and Peter Bienstman. 2009. “A Stable Complex Jacobi Iterative Solution of 3D Semivectorial Wide-angle Beam Propagation Using the Iterated Crank-Nicholson Method.” Optical and Quantum Electronics 41 (4): 215–221.
Chicago author-date (all authors)
Le Quang, Khai, and Peter Bienstman. 2009. “A Stable Complex Jacobi Iterative Solution of 3D Semivectorial Wide-angle Beam Propagation Using the Iterated Crank-Nicholson Method.” Optical and Quantum Electronics 41 (4): 215–221.
Vancouver
1.
Le Quang K, Bienstman P. A stable complex Jacobi iterative solution of 3D semivectorial wide-angle beam propagation using the iterated Crank-Nicholson method. OPTICAL AND QUANTUM ELECTRONICS. 2009;41(4):215–21.
IEEE
[1]
K. Le Quang and P. Bienstman, “A stable complex Jacobi iterative solution of 3D semivectorial wide-angle beam propagation using the iterated Crank-Nicholson method,” OPTICAL AND QUANTUM ELECTRONICS, vol. 41, no. 4, pp. 215–221, 2009.
@article{4339962,
  abstract     = {An extension of the recently proposed three-dimensional (3D) wide-angle (WA) beam propagation method (BPM) using complex Jacobi iteration (CJI) taken into account polarization effects is presented. The resulting iterative BPM is faster than BPMs based on the traditional direct matrix inversion for waveguides with unchanging refractive index profiles during propagation direction. However, for varying refractive index waveguides the iterative method suffered from the fact that the iteration count between two successive cross-sections increases dramatically during the propagation direction. To overcome this problem, we propose the utility of the iterated Crank-Nicholson method. At each propagation step, the propagation equation is divided in multiple stages by the iterated Crank-Nicholson method and then each stage is recast in terms of a Helmholtz equation with source term, which is solved effectively by the complex Jacobi iterative method. The resulting approach is stable and well-suited for large structures with long propagation paths.},
  author       = {Le Quang, Khai and Bienstman, Peter},
  issn         = {0306-8919},
  journal      = {OPTICAL AND QUANTUM ELECTRONICS},
  keywords     = {Iterated Crank-Nicholson,Complex Jacobi iteration,3D semivectorial BPM,PADE APPROXIMANT OPERATORS},
  language     = {eng},
  number       = {4},
  pages        = {215--221},
  title        = {A stable complex Jacobi iterative solution of 3D semivectorial wide-angle beam propagation using the iterated Crank-Nicholson method},
  url          = {http://dx.doi.org/10.1007/s11082-009-9346-6},
  volume       = {41},
  year         = {2009},
}

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