## The complex Jacobi iterative method for three-dimensional wide-angle beam propagation

Optics Express, Vol. 16, Issue 21, pp. 17021-17030 (2008)

http://dx.doi.org/10.1364/OE.16.017021

Acrobat PDF (139 KB)

### Abstract

A new complex Jacobi iterative technique adapted for the solution of three-dimensional (3D) wide-angle (WA) beam propagation is presented. The beam propagation equation for analysis of optical propagation in waveguide structures is based on a novel modified Padé(1,1) approximant operator, which gives evanescent waves the desired damping. The resulting approach allows more accurate approximations to the true Helmholtz equation than the standard Padé approximant operators. Furthermore, a performance comparison of the traditional direct matrix inversion and this new iterative technique for WA-beam propagation method is reported. It is shown that complex Jacobi iteration is faster and better-suited for large problems or structures than direct matrix inversion.

© 2008 Optical Society of America

## 1. Introduction

1. R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. **6**, 150–161 (2000). [CrossRef]

2. C. Ma and E. V. Keuren, “A simple three dimensional wide-angle beam propagation method,” Opt. Express **14**, 4668–4674 (2006). [CrossRef] [PubMed]

4. M. D. Feit and J. A. Fleck Jr., “Analysis of rib waveguides and couplers by the propagation method,” J. Opt. Soc. Am. A **7**, 73–79 (1990). [CrossRef]

5. F. A. Milinazzo, C. A. Zala, and G. H. Brooke, “Rational square root approximations for parabolic equation algorithms,” J. Acoust. Soc. Am. **101**, 760–766 (1997). [CrossRef]

6. Y. Y. Lu and P. L. Ho, “Beam propagation method using a [(p-1)/p] Padé approximant of the propagator,” Opt. Lett. **27**, 683–685 (2002). [CrossRef]

8. G. R. Hadley, “Wide-angle beam propagation using Padé approximant operators,” Opt. Lett. **17**, 1426–1428 (1992). [CrossRef] [PubMed]

9. S. Sujecki, “Wide-angle, finite-difference beam propagation in oblique coordinate system,” J. Opt. Soc. Am. **25**, 138–145 (2007). [CrossRef]

2. C. Ma and E. V. Keuren, “A simple three dimensional wide-angle beam propagation method,” Opt. Express **14**, 4668–4674 (2006). [CrossRef] [PubMed]

10. C. Ma and E. V. Keuren, “A three-dimensional wide-angle beam propagation method for optical waveguide structures,” Opt. Express **15**, 402–407 (2007). [CrossRef] [PubMed]

11. G. R. Hadley, “A complex Jacobi iterative method for the indefinite Helmholtz equation,” J. Comp. Phys. **203**, 358–370 (2005). [CrossRef]

8. G. R. Hadley, “Wide-angle beam propagation using Padé approximant operators,” Opt. Lett. **17**, 1426–1428 (1992). [CrossRef] [PubMed]

## 2. Modified Padé approximant operators

8. G. R. Hadley, “Wide-angle beam propagation using Padé approximant operators,” Opt. Lett. **17**, 1426–1428 (1992). [CrossRef] [PubMed]

*k*=

*k*

_{0}

*n*,

_{ref}*n*the refractive index refractive index profile,

*n*the reference refractive index,

_{ref}*k*

_{0}the vacuum wavevector.

**17**, 1426–1428 (1992). [CrossRef] [PubMed]

12. Y. Y. Lu, “A complex coefficient rational approximation on of **27**, 141–156 (1998). [CrossRef]

12. Y. Y. Lu, “A complex coefficient rational approximation on of **27**, 141–156 (1998). [CrossRef]

## 3. WA-beam propagation formulation

### 3.1 Basic equation

13. G. R. Hadley, “Low-truncation-error finite difference equations for photonics simulation I: beam propagation,” J. Lightwave Technol. **16**, 134–141 (1998). [CrossRef]

### 3.2. WA-BPM using CJI

*k*

*Δz*. This is a condition that favors rapid convergence for the CJI method.

### 3.3 WA-BPM using DMI

10. C. Ma and E. V. Keuren, “A three-dimensional wide-angle beam propagation method for optical waveguide structures,” Opt. Express **15**, 402–407 (2007). [CrossRef] [PubMed]

*M*

^{2}by

*M*

^{2}matrix equation for an M by M mesh grid. However, each row of the coefficient matrix has no more than five non-zero values. As a result, this sparse matrix equation can be efficiently solved using various methods [3]. In our calculations, the sparse matrix solver-UMFPACK package has been used [14

14. T.A. Davis, *Direct Methods for Sparse Linear Systems* (SIAM, 2006). [CrossRef]

## 4. Convergence studies of CJI

17. Z. Ju, J. Fu, and E. Feng, “A simple wide-angle beam-propagation method for integrated optics,” Microwave Opt. Technol. Lett. **14**, 345–347 (1997). [CrossRef]

*d*

_{1}=2

*µm*and

*d*

_{2}=1

*µm*. In Fig. 3 we show that the CJI method converges faster with WA-BPM based on the modified Padé(1,1) approximant than those of Hadley(1,1). However, it suffers from the fact that the iteration count between two successive 2D cross sections increases throughout the propagation direction. To overcome this problem we propose the use of a perfectly matched layer (PML), which can absorb incident radiation without any additional parasitic reflections, regardless of wavelength, incident angle or polarization as boundary conditions [15

15. J. P. Bérenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. **114**, 185–200 (1994). [CrossRef]

16. P. Vandersteengen, B. Maes, P. Bienstman, and R. Baets, “Using the complex Jacobi method to simulate Kerr non-linear photonic components,” Opt. Quantum Electron. **38**, 35–44 (2006). [CrossRef]

## 5. Benchmark results

17. Z. Ju, J. Fu, and E. Feng, “A simple wide-angle beam-propagation method for integrated optics,” Microwave Opt. Technol. Lett. **14**, 345–347 (1997). [CrossRef]

*w*=0.2µm is propagated through 30µm at wavelength λ=1.55µm in a medium of refractive index n=3.4 with the propagation step size of Δz=0.1µm. With a very strict propagation error tolerance of 10

^{-9}the CJI method only took 84.8 seconds, whereas the DMI method took 296.5 seconds.

17. Z. Ju, J. Fu, and E. Feng, “A simple wide-angle beam-propagation method for integrated optics,” Microwave Opt. Technol. Lett. **14**, 345–347 (1997). [CrossRef]

*µm*(requiring frequent matrix inversion) the DMI method performed the propagation in 1414.8 seconds while the CJI method took only 584.1 seconds. It is obvious that for these 2D waveguide structures the CJI method is faster than DMI.

18. P. C. Lee and E. Voges, “Three-dimensional semi-vectorial wide-angle beam propagation method,” J. Lightwave Technol. **12**, 215–225 (1994) [CrossRef]

19. Y. Tsuji, M. Koshiba, and T. Shiraishi, “Finite element beam propagation method for three-dimensional optical waveguide structures,” J. Lightwave Technol. **15**, 1728–1734 (1997). [CrossRef]

*w*=2

*µm*and

*h*=1.1

*µm*, as seen in Fig. 4 of [19

19. Y. Tsuji, M. Koshiba, and T. Shiraishi, “Finite element beam propagation method for three-dimensional optical waveguide structures,” J. Lightwave Technol. **15**, 1728–1734 (1997). [CrossRef]

_{f}=3.44 and a thickness

*t*=0.2

*µm*while the refractive index of substrate and cover is n

_{s}=3.4 and n

_{c}=1, respectively. The Gaussian beam with a waist radius

*w*

_{0}=0.3

*µm*has been injected into the rib waveguide at wavelength λ=1.55

*µm*. Due to the large memory required for DMI, the small computational window of

*2x2µm*is discretized with a grid size of Δx= Δy=0.1

*µm*, and the short path length of 2

*µm*is discretized with a propagation step size Δz=0.1

*µm*. The resulting runtime of DMI is 177.9 seconds while runtime for CJI is only 4.7 seconds.

*h*=1

_{1}*µm*. The other structure parameters are the same as the above straight rib waveguide. The fundamental TE mode of the ridge waveguide of width

*w*=2

*µm*at 1.55

*µm*wavelength is used as the excited field at z=0. The propagation step size is Δz=0.1

*µm*. The field pattern at z=3

*µm*calculated by DMI and CJI are depicted in Fig. 5(a) and 5(b), respectively. Due to the high effective loss in the propagation medium the complex Jacobi method performed the propagation in only 5.9 seconds while DMI required 268.9 seconds.

*k*

*Δz*the CJI technique offers rapid convergence and shorter runtimes.

## 4. Conclusions

20. G. R. Hadley, “Multistep method for wide-angle beam propagation,” Opt. Lett. **17**, 1743–1745 (1992). [CrossRef] [PubMed]

## Acknowledgments

## References and Links

1. | R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. |

2. | C. Ma and E. V. Keuren, “A simple three dimensional wide-angle beam propagation method,” Opt. Express |

3. | W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vettering, |

4. | M. D. Feit and J. A. Fleck Jr., “Analysis of rib waveguides and couplers by the propagation method,” J. Opt. Soc. Am. A |

5. | F. A. Milinazzo, C. A. Zala, and G. H. Brooke, “Rational square root approximations for parabolic equation algorithms,” J. Acoust. Soc. Am. |

6. | Y. Y. Lu and P. L. Ho, “Beam propagation method using a [(p-1)/p] Padé approximant of the propagator,” Opt. Lett. |

7. | T. Anada, T. Hokazono, T. Hiraoka, J. P. Hsu, T. M. Benson, and P. Sewell, “Very-wide-angle beam propagation methods for integrated optical circuits,” IEICE Trans. Electron. |

8. | G. R. Hadley, “Wide-angle beam propagation using Padé approximant operators,” Opt. Lett. |

9. | S. Sujecki, “Wide-angle, finite-difference beam propagation in oblique coordinate system,” J. Opt. Soc. Am. |

10. | C. Ma and E. V. Keuren, “A three-dimensional wide-angle beam propagation method for optical waveguide structures,” Opt. Express |

11. | G. R. Hadley, “A complex Jacobi iterative method for the indefinite Helmholtz equation,” J. Comp. Phys. |

12. | Y. Y. Lu, “A complex coefficient rational approximation on of |

13. | G. R. Hadley, “Low-truncation-error finite difference equations for photonics simulation I: beam propagation,” J. Lightwave Technol. |

14. | T.A. Davis, |

15. | J. P. Bérenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. |

16. | P. Vandersteengen, B. Maes, P. Bienstman, and R. Baets, “Using the complex Jacobi method to simulate Kerr non-linear photonic components,” Opt. Quantum Electron. |

17. | Z. Ju, J. Fu, and E. Feng, “A simple wide-angle beam-propagation method for integrated optics,” Microwave Opt. Technol. Lett. |

18. | P. C. Lee and E. Voges, “Three-dimensional semi-vectorial wide-angle beam propagation method,” J. Lightwave Technol. |

19. | Y. Tsuji, M. Koshiba, and T. Shiraishi, “Finite element beam propagation method for three-dimensional optical waveguide structures,” J. Lightwave Technol. |

20. | G. R. Hadley, “Multistep method for wide-angle beam propagation,” Opt. Lett. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(220.2560) Optical design and fabrication : Propagating methods

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: August 1, 2008

Revised Manuscript: September 17, 2008

Manuscript Accepted: October 1, 2008

Published: October 10, 2008

**Citation**

Khai Q. Le, R. Godoy-Rubio, Peter Bienstman, and G. Ronald Hadley, "The complex Jacobi iterative method for three-dimensional wide-angle beam propagation," Opt. Express **16**, 17021-17030 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-21-17021

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### References

- R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, "Numerical techniques for modeling guided-wave photonic devices," IEEE J. Sel. Top. Quantum Electron. 6, 150-161 (2000). [CrossRef]
- C. Ma and E. V. Keuren, "A simple three dimensional wide-angle beam propagation method," Opt. Express 14, 4668-4674 (2006). [CrossRef] [PubMed]
- W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vettering, Numerical recipes: The art of scientific computing (Cambridge University Press, New York, 1986).
- M. D. Feit and J. A. Fleck Jr., "Analysis of rib waveguides and couplers by the propagation method," J. Opt. Soc. Am. A 7, 73-79 (1990). [CrossRef]
- F. A. Milinazzo, C. A. Zala, and G. H. Brooke, "Rational square root approximations for parabolic equation algorithms," J. Acoust. Soc. Am. 101, 760-766 (1997). [CrossRef]
- Y. Y. Lu and P. L. Ho, "Beam propagation method using a [(p-1)/p] Padé approximant of the propagator," Opt. Lett. 27, 683-685 (2002). [CrossRef]
- T. Anada, T. Hokazono, T. Hiraoka, J. P. Hsu, T. M. Benson, and P. Sewell, "Very-wide-angle beam propagation methods for integrated optical circuits," IEICE Trans. Electron.E 82-C, 1154-1158 (1999).
- G. R. Hadley, "Wide-angle beam propagation using Padé approximant operators," Opt. Lett. 17, 1426-1428 (1992). [CrossRef] [PubMed]
- S. Sujecki, "Wide-angle, finite-difference beam propagation in oblique coordinate system," J. Opt. Soc. Am. 25, 138-145 (2007). [CrossRef]
- C. Ma and E. V. Keuren, "A three-dimensional wide-angle beam propagation method for optical waveguide structures," Opt. Express 15, 402-407 (2007). [CrossRef] [PubMed]
- G. R. Hadley, "A complex Jacobi iterative method for the indefinite Helmholtz equation," J. Comp. Phys. 203, 358-370 (2005). [CrossRef]
- Y. Y. Lu, "A complex coefficient rational approximation of1+x ," Appl. Numer. Math. 27, 141-156 (1998). [CrossRef]
- G. R. Hadley, "Low-truncation-error finite difference equations for photonics simulation I: beam propagation," J. Lightwave Technol. 16, 134-141 (1998). [CrossRef]
- T. A. Davis, Direct Methods for Sparse Linear Systems (SIAM, 2006). [CrossRef]
- J. P. Bérenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994). [CrossRef]
- P. Vandersteengen, B. Maes, P. Bienstman, and R. Baets, "Using the complex Jacobi method to simulate Kerr non-linear photonic components," Opt. Quantum Electron. 38, 35-44 (2006). [CrossRef]
- Z. Ju, J. Fu, and E. Feng, "A simple wide-angle beam-propagation method for integrated optics," Microwave Opt. Technol. Lett. 14, 345-347 (1997). [CrossRef]
- P. C. Lee and E. Voges, "Three-dimensional semi-vectorial wide-angle beam propagation method," J. Lightwave Technol. 12, 215-225 (1994) [CrossRef]
- Y. Tsuji, M. Koshiba, and T. Shiraishi, "Finite element beam propagation method for three-dimensional optical waveguide structures," J. Lightwave Technol. 15, 1728-1734 (1997). [CrossRef]
- G. R. Hadley, "Multistep method for wide-angle beam propagation," Opt. Lett. 17, 1743-1745 (1992). [CrossRef] [PubMed]

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