## Acceleration of FDTD mode solver by high-performance computing techniques

Optics Express, Vol. 18, Issue 13, pp. 13679-13692 (2010)

http://dx.doi.org/10.1364/OE.18.013679

Acrobat PDF (1136 KB)

### Abstract

A two-dimensional (2D) compact finite-difference time-domain (FDTD) mode solver is developed based on wave equation formalism in combination with the matrix pencil method (MPM). The method is validated for calculation of both real guided and complex leaky modes of typical optical waveguides against the bench-mark finite-difference (FD) eigen mode solver. By taking advantage of the inherent parallel nature of the FDTD algorithm, the mode solver is implemented on graphics processing units (GPUs) using the compute unified device architecture (CUDA). It is demonstrated that the high-performance computing technique leads to significant acceleration of the FDTD mode solver with more than 30 times improvement in computational efficiency in comparison with the conventional FDTD mode solver running on CPU of a standard desktop computer. The computational efficiency of the accelerated FDTD method is in the same order of magnitude of the standard finite-difference eigen mode solver and yet require much less memory (e.g., less than 10%). Therefore, the new method may serve as an efficient, accurate and robust tool for mode calculation of optical waveguides even when the conventional eigen value mode solvers are no longer applicable due to memory limitation.

© 2010 OSA

## 1. Introduction

1. E. Schweig and W. Bridges, “Computer analysis of dielectric waveguides: A finite-difference method,” IEEE Trans. Microw. Theory Tech. **32**(5), 531–541 (1984). [CrossRef]

5. C. Xu, W. Huang, M. Stern, and S. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” IEE Proc., Optoelectron. **141**(5), 281–286 (1994). [CrossRef]

6. B. Rahman and J. Davies, “Finite-element analysis of optical and microwave waveguide problems,” IEEE Trans. Microw. Theory Tech. **32**(1), 20–28 (1984). [CrossRef]

7. J. Lee, D. Sun, and Z. Cendes, “Full-wave analysis of dielectric waveguides using tangential vector finite elements,” IEEE Trans. Microw. Theory Tech. **39**(8), 1262–1271 (1991). [CrossRef]

8. M. D. Feit and J. A. Fleck Jr., “Computation of mode properties in optical fiber waveguides by a propagating beam method,” Appl. Opt. **19**(7), 1154–1164 (1980). [CrossRef] [PubMed]

10. Y. Tsuji and M. Koshiba, “Guided-mode and leaky-mode analysis by imaginary distance beam propagation method based on finite element scheme,” J. Lightwave Technol. **18**(4), 618–623 (2000). [CrossRef]

11. A. Zhao, J. Juntunen, and A. Räisänen, “Analysis of hybrid modes in channel multilayer optical waveguides with the compact 2-D FDTD method,” Microw. Opt. Technol. Lett. **15**(6), 398–403 (1997). [CrossRef]

12. S. Xiao, R. Vahldieck, and H. Jin, “Full-wave analysis of guided wave structures using a novel 2-D FDTD,” IEEE Microw. Guid. Wave Lett. **2**(5), 165–167 (1992). [CrossRef]

8. M. D. Feit and J. A. Fleck Jr., “Computation of mode properties in optical fiber waveguides by a propagating beam method,” Appl. Opt. **19**(7), 1154–1164 (1980). [CrossRef] [PubMed]

13. G. Zhou and X. Li, “Wave equation-based semivectorial compact 2-D-FDTD method for optical waveguide modal analysis,” J. Lightwave Technol. **22**(2), 677–683 (2004). [CrossRef]

12. S. Xiao, R. Vahldieck, and H. Jin, “Full-wave analysis of guided wave structures using a novel 2-D FDTD,” IEEE Microw. Guid. Wave Lett. **2**(5), 165–167 (1992). [CrossRef]

14. T. Sarkar and O. Pereira, “Using the matrix pencil method to estimate the parameters of a sum of complex exponentials,” IEEE Antennas Propag. Mag. **37**(1), 48–55 (1995). [CrossRef]

15. Z. Sacks, D. Kingsland, R. Lee, and J. Lee, “A perfectly matched anisotropic absorber for use as an absorbingboundary condition,” IEEE Trans. Antenn. Propag. **43**(12), 1460–1463 (1995). [CrossRef]

*z*) through the FDTD method for a certain time (

*t*). When the steady-state is reached, a series of instantaneous response field values at the end of the propagation are adopted by MPM algorithm for mode parameter extractions. Since the MPM algorithm is much more efficient and accurate for the mode parameter extraction compared with the traditional FT method, only limited propagation distance is required in FDTD simulation. Since the FDTD method is an inherently data-parallel algorithm, we implement it in Graphics Processing Units (GPUs) by using Nvidia’s Compute Unified Device Architecture (CUDA) [18]. With the help of the high computing power brought by GPUs [19,20], the simulation time of the FDTD method is further reduced to less than 3% of that used for the implementation on a standard 3.0GHz CPU.

## 2. Wave Eq. (-)based compact 2D-FDTD method with UPML ABCs

12. S. Xiao, R. Vahldieck, and H. Jin, “Full-wave analysis of guided wave structures using a novel 2-D FDTD,” IEEE Microw. Guid. Wave Lett. **2**(5), 165–167 (1992). [CrossRef]

21. M. Okoniewski, “Vector wave equation 2-D-FDTD method for guided wave problems,” IEEE Microw. Guid. Wave Lett. **3**(9), 307–309 (1993). [CrossRef]

22. J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. **114**(2), 185–200 (1994). [CrossRef]

## 3. Extraction of mode parameters by matrix pencil method

*β*is an output in the mode calculation. In our algorithm, the wave equations are solved in time domain and hence

*β*is selected as an input parameter so as to the mode frequencies

14. T. Sarkar and O. Pereira, “Using the matrix pencil method to estimate the parameters of a sum of complex exponentials,” IEEE Antennas Propag. Mag. **37**(1), 48–55 (1995). [CrossRef]

## 4. Simulation results

### 4.1. Assessment and validation

#### 4.1.1. Guided mode analysis

*β*and the Gaussian distribution is launched at the position

#### 4.1.2. Leaky mode analysis

24. J. Heaton, M. Bourke, S. Jones, B. Smith, K. Hilton, G. Smith, J. Birbeck, G. Berry, S. Dewar, and D. Wight, “Optimization of deep-etched, single-mode GaAs/AlGaAs optical waveguides using controlled leakage into the substrate,” J. Lightwave Technol. **17**(2), 267–281 (1999). [CrossRef]

24. J. Heaton, M. Bourke, S. Jones, B. Smith, K. Hilton, G. Smith, J. Birbeck, G. Berry, S. Dewar, and D. Wight, “Optimization of deep-etched, single-mode GaAs/AlGaAs optical waveguides using controlled leakage into the substrate,” J. Lightwave Technol. **17**(2), 267–281 (1999). [CrossRef]

4. A. Fallahkhair, K. Li, and T. Murphy, “Vector finite difference modesolver for anisotropic dielectric waveguides,” J. Lightwave Technol. **26**(11), 1423–1431 (2008). [CrossRef]

### 4.2. Error analysis and convergence test

#### 4.2.1. Convergence with FDTD time steps

#### 4.2.2. Convergence with MPM sample number

## 4.3. Performance of the mode solver on GPUs

*z*), the compact FDTD method is employed so that significant computation time can be saved. However, in the modal analysis of an optical waveguide with relatively large dimensions, a large number of meshes are required to ensure adequate accuracy. This leads to drastic increase in computation time and render this approach not applicable as a practical tool for mode calculation on CPUs of today’s desktop computers. In order to improve the computation efficiency, this method is implemented on Nvidia GTX 295 GPUs with parallel calculation algorithms. Comparisons of the memory and time consumptions are made in implementing this method on GPUs, CPU as well as the FD mode solver, respectively.

#### 4.3.1. Memory consumption

#### 4.3.2. Time consumption

## 5. Summary

## Appendix

## Part A. Full-vectorial scheme

25. W. Huang and C. Xu, “Simulation of three-dimensional optical waveguides by a full-vectorbeam propagation method,” IEEE J. Quantum Electron. **29**(10), 2639–2649 (1993). [CrossRef]

*z*), derivatives with respect to z have been replaced with

26. W. Chew and W. Weedon, “A 3-D perfectly matched medium from modified Maxwell's equations with stretched coordinates,” Microw. Opt. Technol. Lett. **7**(13), 599–604 (1994). [CrossRef]

## Part B. Semivectorial scheme

## Part C. Scalar scheme

## Acknowledgements

## References and links

1. | E. Schweig and W. Bridges, “Computer analysis of dielectric waveguides: A finite-difference method,” IEEE Trans. Microw. Theory Tech. |

2. | M. Stern, “Semivectorial polarised finite difference method for opticalwaveguides with arbitrary index profiles,” IEE Proc., Optoelectron. |

3. | W. P. Huang, S. T. Chu, A. Goss, and S. K. Chaudhuri, “A scalar finite-difference time-domain approach to guided-wave optics,” (1991), pp. 524–526. |

4. | A. Fallahkhair, K. Li, and T. Murphy, “Vector finite difference modesolver for anisotropic dielectric waveguides,” J. Lightwave Technol. |

5. | C. Xu, W. Huang, M. Stern, and S. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” IEE Proc., Optoelectron. |

6. | B. Rahman and J. Davies, “Finite-element analysis of optical and microwave waveguide problems,” IEEE Trans. Microw. Theory Tech. |

7. | J. Lee, D. Sun, and Z. Cendes, “Full-wave analysis of dielectric waveguides using tangential vector finite elements,” IEEE Trans. Microw. Theory Tech. |

8. | M. D. Feit and J. A. Fleck Jr., “Computation of mode properties in optical fiber waveguides by a propagating beam method,” Appl. Opt. |

9. | C. Xu, W. Huang, and S. Chaudhuri, ““Efficient and accurate vector mode calculations by beam propagationmethod,” Lightwave Technology,” Journalism |

10. | Y. Tsuji and M. Koshiba, “Guided-mode and leaky-mode analysis by imaginary distance beam propagation method based on finite element scheme,” J. Lightwave Technol. |

11. | A. Zhao, J. Juntunen, and A. Räisänen, “Analysis of hybrid modes in channel multilayer optical waveguides with the compact 2-D FDTD method,” Microw. Opt. Technol. Lett. |

12. | S. Xiao, R. Vahldieck, and H. Jin, “Full-wave analysis of guided wave structures using a novel 2-D FDTD,” IEEE Microw. Guid. Wave Lett. |

13. | G. Zhou and X. Li, “Wave equation-based semivectorial compact 2-D-FDTD method for optical waveguide modal analysis,” J. Lightwave Technol. |

14. | T. Sarkar and O. Pereira, “Using the matrix pencil method to estimate the parameters of a sum of complex exponentials,” IEEE Antennas Propag. Mag. |

15. | Z. Sacks, D. Kingsland, R. Lee, and J. Lee, “A perfectly matched anisotropic absorber for use as an absorbingboundary condition,” IEEE Trans. Antenn. Propag. |

16. | S. Gedney, “An anisotropic perfectly matched layer-absorbing medium for thetruncation of FDTD lattices,” IEEE Trans. Antenn. Propag. |

17. | A. Taflove, and S. Hagness, |

18. | S. Ryoo, C. Rodrigues, S. Baghsorkhi, S. Stone, D. Kirk, and W. Hwu, “Optimization principles and application performance evaluation of a multithreaded GPU using CUDA,” in (ACM, 2008), 73–82. |

19. | S. Krakiwsky, L. Turner, and M. Okoniewski, “Acceleration of finite-difference time-domain (FDTD) using graphics processor units (GPU),” in |

20. | A. Balevic, L. Rockstroh, A. Tausendfreund, S. Patzelt, G. Goch, and S. Simon, “Accelerating simulations of light scattering based on finite-difference time-domain method with general purpose gpus,” in 2008), 327–334. |

21. | M. Okoniewski, “Vector wave equation 2-D-FDTD method for guided wave problems,” IEEE Microw. Guid. Wave Lett. |

22. | J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. |

23. | J. Goell, “A circular-harmonic computer analysis of rectangular dielectric waveguides,” Bell Syst. Tech. J. |

24. | J. Heaton, M. Bourke, S. Jones, B. Smith, K. Hilton, G. Smith, J. Birbeck, G. Berry, S. Dewar, and D. Wight, “Optimization of deep-etched, single-mode GaAs/AlGaAs optical waveguides using controlled leakage into the substrate,” J. Lightwave Technol. |

25. | W. Huang and C. Xu, “Simulation of three-dimensional optical waveguides by a full-vectorbeam propagation method,” IEEE J. Quantum Electron. |

26. | W. Chew and W. Weedon, “A 3-D perfectly matched medium from modified Maxwell's equations with stretched coordinates,” Microw. Opt. Technol. Lett. |

**OCIS Codes**

(030.4070) Coherence and statistical optics : Modes

(200.4960) Optics in computing : Parallel processing

(230.0230) Optical devices : Optical devices

(230.7370) Optical devices : Waveguides

**ToC Category:**

Physical Optics

**History**

Original Manuscript: April 26, 2010

Revised Manuscript: June 4, 2010

Manuscript Accepted: June 5, 2010

Published: June 10, 2010

**Citation**

Lin Han, Yanping Xi, and Wei-Ping Huang, "Acceleration of FDTD mode solver by high-performance computing techniques," Opt. Express **18**, 13679-13692 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-13-13679

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### References

- E. Schweig and W. Bridges, “Computer analysis of dielectric waveguides: A finite-difference method,” IEEE Trans. Microw. Theory Tech. 32(5), 531–541 (1984). [CrossRef]
- M. Stern, “Semivectorial polarised finite difference method for opticalwaveguides with arbitrary index profiles,” IEE Proc. Optoelectron. 135, 56–63 (1988). [CrossRef]
- W. P. Huang, S. T. Chu, A. Goss, and S. K. Chaudhuri, “A scalar finite-difference time-domain approach to guided-wave optics,” IEEE Photon. Technol Lett. 3, 524–526 (1991).
- A. Fallahkhair, K. Li, and T. Murphy, “Vector finite difference modesolver for anisotropic dielectric waveguides,” J. Lightwave Technol. 26(11), 1423–1431 (2008). [CrossRef]
- C. Xu, W. Huang, M. Stern, and S. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” IEE Proc. Optoelectron. 141(5), 281–286 (1994). [CrossRef]
- B. Rahman and J. Davies, “Finite-element analysis of optical and microwave waveguide problems,” IEEE Trans. Microw. Theory Tech. 32(1), 20–28 (1984). [CrossRef]
- J. Lee, D. Sun, and Z. Cendes, “Full-wave analysis of dielectric waveguides using tangential vector finite elements,” IEEE Trans. Microw. Theory Tech. 39(8), 1262–1271 (1991). [CrossRef]
- M. D. Feit and J. A. Fleck., “Computation of mode properties in optical fiber waveguides by a propagating beam method,” Appl. Opt. 19(7), 1154–1164 (1980). [CrossRef] [PubMed]
- C. Xu, W. Huang, and S. Chaudhuri, ““Efficient and accurate vector mode calculations by beam propagationmethod,” Lightwave Technology,” Journalism 11, 1209–1215 (1993).
- Y. Tsuji and M. Koshiba, “Guided-mode and leaky-mode analysis by imaginary distance beam propagation method based on finite element scheme,” J. Lightwave Technol. 18(4), 618–623 (2000). [CrossRef]
- A. Zhao, J. Juntunen, and A. Räisänen, “Analysis of hybrid modes in channel multilayer optical waveguides with the compact 2-D FDTD method,” Microw. Opt. Technol. Lett. 15(6), 398–403 (1997). [CrossRef]
- S. Xiao, R. Vahldieck, and H. Jin, “Full-wave analysis of guided wave structures using a novel 2-D FDTD,” IEEE Microw. Guid. Wave Lett. 2(5), 165–167 (1992). [CrossRef]
- G. Zhou and X. Li, “Wave equation-based semivectorial compact 2-D-FDTD method for optical waveguide modal analysis,” J. Lightwave Technol. 22(2), 677–683 (2004). [CrossRef]
- T. Sarkar and O. Pereira, “Using the matrix pencil method to estimate the parameters of a sum of complex exponentials,” IEEE Antennas Propag. Mag. 37(1), 48–55 (1995). [CrossRef]
- Z. Sacks, D. Kingsland, R. Lee, and J. Lee, “A perfectly matched anisotropic absorber for use as an absorbingboundary condition,” IEEE Trans. Antenn. Propag. 43(12), 1460–1463 (1995). [CrossRef]
- S. Gedney, “An anisotropic perfectly matched layer-absorbing medium for thetruncation of FDTD lattices,” IEEE Trans. Antenn. Propag. 44(12), 1630–1639 (1996). [CrossRef]
- A. Taflove and S. Hagness, Computational electrodynamics: the finite-difference time-domain method (Artech House Norwood, MA, 1995).
- S. Ryoo, C. Rodrigues, S. Baghsorkhi, S. Stone, D. Kirk, and W. Hwu, “Optimization principles and application performance evaluation of a multithreaded GPU using CUDA,” in (ACM, 2008), 73–82.
- S. Krakiwsky, L. Turner, and M. Okoniewski, “Acceleration of finite-difference time-domain (FDTD) using graphics processor units (GPU),” in Microwave Symposium Digest, IEEE MTT-S International, 2004), 1033- 1036.
- A. Balevic, L. Rockstroh, A. Tausendfreund, S. Patzelt, G. Goch, and S. Simon, “Accelerating simulations of light scattering based on finite-difference time-domain method with general purpose gpus,” in 2008), 327–334.
- M. Okoniewski, “Vector wave equation 2-D-FDTD method for guided wave problems,” IEEE Microw. Guid. Wave Lett. 3(9), 307–309 (1993). [CrossRef]
- J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994). [CrossRef]
- J. Goell, “A circular-harmonic computer analysis of rectangular dielectric waveguides,” Bell Syst. Tech. J. 48, 2133–2160 (1969).
- J. Heaton, M. Bourke, S. Jones, B. Smith, K. Hilton, G. Smith, J. Birbeck, G. Berry, S. Dewar, and D. Wight, “Optimization of deep-etched, single-mode GaAs/AlGaAs optical waveguides using controlled leakage into the substrate,” J. Lightwave Technol. 17(2), 267–281 (1999). [CrossRef]
- W. Huang and C. Xu, “Simulation of three-dimensional optical waveguides by a full-vectorbeam propagation method,” IEEE J. Quantum Electron. 29(10), 2639–2649 (1993). [CrossRef]
- W. Chew and W. Weedon, “A 3-D perfectly matched medium from modified Maxwell's equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7(13), 599–604 (1994). [CrossRef]

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