## A three-dimensional wide-angle BPM for optical waveguide structures

Optics Express, Vol. 15, Issue 2, pp. 402-407 (2007)

http://dx.doi.org/10.1364/OE.15.000402

Acrobat PDF (95 KB)

### Abstract

Algorithms for effective modeling of optical propagation in three- dimensional waveguide structures are critical for the design of photonic devices. We present a three-dimensional (3-D) wide-angle beam propagation method (WA-BPM) using Hoekstra’s scheme. A sparse matrix algebraic equation is formed and solved using iterative methods. The applicability, accuracy and effectiveness of our method are demonstrated by applying it to simulations of wide-angle beam propagation, along with a technique for shifting the simulation window to reduce the dimension of the numerical equation and a threshold technique to further ensure its convergence. These techniques can ensure the implementation of iterative methods for waveguide structures by relaxing the convergence problem, which will further enable us to develop higher-order 3-D WA-BPMs based on Padé approximant operators.

© 2007 Optical Society of America

## 1. Introduction

21. W. P. Huang and C. L. Xu, “A wide-angle vector beam propagation method,” IEEE Photon. Technol. Lett. **4**,1118–1120 (1992). [CrossRef]

*Ax*=

*b*, iterative methods have convergence problems if either the coefficient matrix

*A*is ill-conditioned or the elements of the vector

*b*have a wide range of values [6]. A large simulation window is needed for wide-angle simulations, which may result in an ill-conditioned large size algebraic equation with a corresponding convergence problem. In order to overcome this problem, we introduce a simulation window shifting scheme to decrease the dimension of the equation and a threshold technique to reduce the volatility of field distribution

*b*.

## 2. Formulation

*k*denotes the free-space wavenumber,

_{0}*E*(

*x,y,z*)exp(

*iωt*) is the electric field component with angular frequency

*ω*, and

*n(x, y, z)*is the refractive index profile. If we assume the slowly varying envelope approximation (SVEA), then

*E(x, y, z)*can be separated into two parts: the complex field amplitude

*ψ(x,y,z)*(the axially slowly varying envelope term) and a propagation factor

*exp(-ik*(the rapidly varying phase term). The field is then expressed as

_{0}n_{0}z)*a*= 2

*k*

_{0}

*n*

_{0}and

*b*=

*k*

^{2}

_{0}(

*n*

^{2}-

*n*

^{2}). In the usual paraxial formulation, the second-order derivative ∣∂

^{2}

*ψ*/∂

*z*

^{2}∣ in Eq. (3) is dropped, giving

*z*direction into the following

*z*positions

*l*and

*l*+1 to find expressions for the first derivative with respect to

*z*

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

*M*

^{2}by

*M*

^{2}matrix equation for an

*M*by

*M*mesh grid, which requires a large amount of computer resources, further resulting in the inapplicability of direct matrix equation solvers, if the matrix is fully stored. For example, a relatively small simulation window with 200 by 200 mesh grids will have a 40,000 by 40,000 matrix equation, which requires 25.6 GB dynamic storage space only for the coefficient matrix of Eq. (7). Additional matrices of the same size are typically needed in direct matrix equation solvers. As a result, direct methods would be prohibitively expensive and in some cases impossible even with the best available computing power. Some direct solvers can store the matrix sparsely, i.e. only the non-zero elements. However, Gaussian elimination method or LU decomposition method in these direct solvers might generate non-sparse triangular matrices, which will require a large amount of computer memory as the matrix is fully stored. Whether the generated triangular matrices are sparse or not is quite problem dependent. Fortunately, each row of the coefficient matrix of Eq. (7) has no more than five non-zero values. As a result, this sparse matrix equation can be efficiently solved using iterative methods [6].

*A*is ill-conditioned or the vector

*b*is widely distributed. In our simulations, all nonzero elements of the coefficient matrix are in a narrow range, indicating that the matrix is not ill-conditioned. However, the values of the elements of the vector

*b*are widely distributed due to the wide range of the field [

*Ψ*] in the equation. For example, a normalized Gaussian beam with a waist radius of 3.0 μm drops exponentially to the order of 10

^{-16}roughly 18 μm transversely away from the center of the pulse. In order to implement the simulations using iterative methods, we narrow the field distribution by setting the elements of the field profile to zero if they are below some threshold. This reduces the volatility of the field profile to a large extent. Simulations performed without using the threshold technique confirm that iterative solvers will not converge if the threshold technique is not used.

## 3. Simulations and discussions

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

*w*= 3.0 μm propagating in free space (unity refractive index) with a 30-degree tilt with respect to

_{0}*z*axis and along a direction mid way between the

*x*and

*y*axes. We also analyze the propagation in a single-mode channel waveguide, with the same structure and parameters as the second example in reference 20, namely, a 4 μm by 4 μm channel waveguide with the refractive indices of the core and the cladding of 1.55 and 1.52 respectively, again tilted at 30-degrees with respect to

*z*axis and 45 degrees with respect to the

*x*and

*y*axes. The wavelength is 0.85 μm for both cases. Table 1 shows the comparison of the relative L

^{2}norm errors and relative position shift with respect to the analytical profile for both cases between the output profiles at

*z*= 60 μm the simulation results obtained using the classical BPM (hereafter referred to as method 1) [1

01. M. D. Feit and J. A. Fleck Jr., “Light propagation in graded-index optical fibers,” Appl. Opt. **17**,3990–3998 (1978). [CrossRef] [PubMed]

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

^{2}norm errors and better relative position shifts. Simulations were also performed using different step sizes (

*Δz*) in the range from 0.05 μm to 0.25 μm, only resulting in slight or even indistinguishable differences in output. The dependence of relative L

^{2}norm errors and relative position shifts on the values of the threshold were tested in the range of 10

^{-4}-10

^{-16}for the Gaussian beam propagation case, and no significant change was found. This is also validated in the simulations of the channel waveguide simulation, in which the refractive index contrast between the core and the cladding may change the range of the field profile. However, no matter how the range of the field profile is changed, a different threshold can be selected to ensure the convergence. Based on this technique, zeros can be directly inserted to the edge of the new simulation windows if the threshold is satisfied when shifting the simulation window.

^{a}

*Δy*is set equal to

*Δx*, and

*Δz*= 0.2 μm for all simulations.

^{b}all errors and shifts are in percentage.

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

^{-6}.

*z*axis. This method yields good convergence and low errors. The simulation window shifting technique is useful for wide-angle simulations of large photonic structures. The threshold technique should be adopted to ensure convergence for any relatively large simulation windows which may result in widely distributed field profile. The threshold technique, however, cannot be applied if there is a considerable amount of radiation leaving the simulation window, from which no widely distributed field profile results. We are currently trying to adapt the new algorithm to a semi-vectorial formulation and investigating other large sparse matrix equation solving methods. This method also enables us to develop higher order 3-D wide-angle algorithms using Padé approximant operators and multistep method as was developed by Hadley for 2-D simulations [16

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

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

## Acknowledgments

## References and links

01. | M. D. Feit and J. A. Fleck Jr., “Light propagation in graded-index optical fibers,” Appl. Opt. |

02. | D. Yevick, “A guide to electric field propagation techniques for guided-wave optics,” Opt. Quantum. Electron. |

03. | J. Van Roey, J. van der Donk, and P. E. Lagasse, “Beam-propagation method: analysis and assessment,” J. Opt. Soc. Am. |

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

05. | Y. Chung and N. Dagli, “An assessment of finite difference beam propagation,” J. Quantum. Electron. |

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

07. | G. R. Hadley, “Transparent boundary condition for beam propagation,” Opt. Lett. |

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

09. | C. Vassallo, “Reformulation for the beam-propagation method,” J.Opt. Soc. Am. A |

10. | J. Gerdes and R. Pregla, “Beam-propagation algorithm based on the method of lines,” J. Opt. Soc. Amer. A |

11. | R. P. Ratowsky and J. A. Fleck Jr., “Accurate numerical solution of the Helmholtz equation by iterative Lanczos reduction,” Opt. Lett. |

12. | P. -C. Lee, D. Schulz, and E. Voges, “Three-dimensional finite difference beam propagation algorithms for photonic devices,” J. Lightwave Technol. |

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

14. | A. Sharma and A. Agrawal, “New method for nonparaxial beam propagation,” J. Opt. Soc. Am. A |

15. | S. L. Chui and Y. Y. Lu, “NWide-angle full-vector beam propagation method based on an alternating direction implicit preconditioner,” J. Opt. Soc. Am. A |

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

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

18. | H. J. W. M. Hoekstra, G. J. M. Krijnen, and P. V. Lambeck, “On the accuracy of the finite difference method for applications in beam propagating techniques,” Opt. Commun. |

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

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

21. | W. P. Huang and C. L. Xu, “A wide-angle vector beam propagation method,” IEEE Photon. Technol. Lett. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(130.2790) Integrated optics : Guided waves

**ToC Category:**

General

**History**

Original Manuscript: December 8, 2006

Revised Manuscript: January 11, 2007

Manuscript Accepted: January 16, 2007

Published: January 22, 2007

**Citation**

Changbao Ma and Edward Van Keuren, "A three-dimensional wide-angle BPM for optical waveguide structures," Opt. Express **15**, 402-407 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-2-402

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### References

- M. D. Feit and J. A. FleckJr., "Light propagation in graded-index optical fibers," Appl. Opt. 17, 3990-3998 (1978). [CrossRef] [PubMed]
- D. Yevick, "A guide to electric field propagation techniques for guided-wave optics," Opt. Quantum. Electron. 26, S185-S197 (1994). [CrossRef]
- J. Van Roey, J. van der Donk, and P. E. Lagasse, "Beam-propagation method: analysis and assessment," J. Opt. Soc. Am. 71, 803-810 (1981). [CrossRef]
- 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]
- Y. Chung and N. Dagli, "An assessment of finite difference beam propagation," J. Quantum. Electron. 26, 1335-1339 (1990). [CrossRef]
- 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).
- G. R. Hadley, "Transparent boundary condition for beam propagation," Opt. Lett. 16, 624-626 (1991). [CrossRef] [PubMed]
- M. D. Feit and J. A. FleckJr., "Analysis of rib waveguides and couplers by the propagation method," J. Opt. Soc. Am A 7, 73-79 (1990). [CrossRef]
- C. Vassallo, "Reformulation for the beam-propagation method," J.Opt. Soc. Am. A 10, 2208-2216 (1993). [CrossRef]
- J. Gerdes and R. Pregla, "Beam-propagation algorithm based on the method of lines," J. Opt. Soc. Amer. A 8, 389-394 (1991). [CrossRef]
- R. P. Ratowsky and J. A. FleckJr., "Accurate numerical solution of the Helmholtz equation by iterative Lanczos reduction," Opt. Lett. 16, 787-789 (1991). [CrossRef] [PubMed]
- P. -C. Lee, D. Schulz, and E. Voges, "Three-dimensional finite difference beam propagation algorithms for photonic devices," J. Lightwave Technol. 10, 1832-1838 (1992). [CrossRef]
- P. -C. Lee and E. Voges, "Three-dimensional semi-vectorial wave-angle beam propagation method," J. Lightwave Technol. 12, 215-225 (1994). [CrossRef]
- A. Sharma and A. Agrawal, "New method for nonparaxial beam propagation," J. Opt. Soc. Am. A 21, 1082-1087 (2004). [CrossRef]
- S. L. Chui and Y. Y. Lu, "Wide-angle full-vector beam propagation method based on an alternating direction implicit preconditioner," J. Opt. Soc. Am. A 21, 420-425 (2004). [CrossRef]
- G. R. Hadley, "Wide-angle beam propagation using Padé approximant operators," Opt. Lett. 17, 1426-1428 (1992). [CrossRef] [PubMed]
- G. R. Hadley, "Multistep method for wide-angle beam propagation," Opt. Lett. 17, 1743-1745 (1992). [CrossRef] [PubMed]
- H. J. W. M. Hoekstra, G. J. M. Krijnen and P. V. Lambeck, "On the accuracy of the finite difference method for applications in beam propagating techniques," Opt. Commun. 94, 506-508 (1992). [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]
- C. Ma and E. V. Keuren, "A simple three dimensional wide-angle beam propagation method," Opt. Express 14, 4668-4674 (2006). [CrossRef] [PubMed]
- W. P. Huang and C. L. Xu, "A wide-angle vector beam propagation method," IEEE Photon. Technol. Lett. 4, 1118-1120 (1992). [CrossRef]

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