## Accurate analysis of planar optical waveguide devices using higher-order FDTD scheme

Optics Express, Vol. 14, Issue 24, pp. 11796-11803 (2006)

http://dx.doi.org/10.1364/OE.14.011796

Acrobat PDF (276 KB)

### Abstract

A higher-order finite-difference time-domain (HO-FDTD) numerical method is proposed for the time-domain analysis of planar optical waveguide devices. The anisotropic perfectly matched layer (APML) absorbing boundary condition for the HO-FDTD scheme is implemented and the numerical dispersion of this scheme is studied. The numerical simulations for the parallel-slab directional coupler are presented and the computing results using this scheme are in highly accordance with analytical solutions. Compared with conventional FDTD method, this scheme can save considerable computational resource without sacrificing solution accuracy and especially could be applied in the accurate analysis of optical devices.

© 2006 Optical Society of America

## 1. Introduction

2. S. Chu and S. K Chaudhuri, “A finite-difference time-domain method for the design and analysis of guidedwave optical structures,” IEEE, J. Lightwave Technol. **7**, 2033–2038 (1989). [CrossRef]

3. W. Huang, C. L. Xu, and A. Goss, “A scalar finite-difference time-domain approach to guided-wave optics,” IEEE Photon. Technol. Lett. **3**, 524–526 (1991). [CrossRef]

4. K. L. Shlager and J. B. Schneider, “Comparison of the dispersion properties of several low dispersion finite-difference time-domain algorithms,” IEEE Trans. Antennas and Propag. **51**, 642–653 (2003). [CrossRef]

6. M. Krumpholz and L. P. B. Katehi. “MRTD New time-domain schemes based on multiresolution analysis.” IEEE Trans. Microwave Theory Tech. 44, 555–571 (1996). [CrossRef]

9. K. L. Shlager and J. B. Schneider, “Comparison of the dispersion properties of higher order FDTD schemes and equivalent-sized MRTD schemes,” IEEE Trans. Antennas and Propag. **52**, 1095–1104 (2004). [CrossRef]

## 2. Higher-order FDTD schemes

*f*(

*x*) and doing some manipulations, we can obtain (2

*M*) th-order approximation for the first-order derivation of

*f*(

*x*)

*l*)’s are the coefficient weights for (2

*M*) th-order finite difference schemes and can be calculated by [10]

*a*(

*l*) for

*l*≥0 can be calculated by Eq. (3), and those for

*l*<0 can be obtained by

*a*(-

*l*)=-

*a*(

*l*-1). Now replacing the first-order time derivative in Eq. (1) with the 2nd-order central difference approximation and the first-order spatial derivatives in Eq. (1) with (2

*M*) th-order central difference approximation, we can obtain the general (2, 2

*M*) th-order FDTD formula of Eq. (1):

*t*is limited by the modified Courant stability condition [11

11. E. M. Tentzeris, R. L. Robertson, and J. F. Harvey, “Stability and dispersion analysis of Battle -Lemarie-Based MRTD schemes,” IEEE Trans. Microwave Theory Tech. **47**, 1004–1012 (1999). [CrossRef]

*v*

_{max}is the maximum of propagation speed in the computational region.

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

13. S. D. Gedney. “An anisotropic perfectly matched layer-absorbing medium for truncation of FDTD lattices,” IEEE Trans. Antennas and Propag. **44**, 1630–1639 (1996). [CrossRef]

*s*and

_{x}*s*are the stretched coordinate variables in the x and y directions, they can be chosen as

_{y}*σ*is the conductivity profile in the PML region along the

_{η}*η*direction.

13. S. D. Gedney. “An anisotropic perfectly matched layer-absorbing medium for truncation of FDTD lattices,” IEEE Trans. Antennas and Propag. **44**, 1630–1639 (1996). [CrossRef]

14. A. Yefet and P. G. Petropoulos, “A staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell’s equations,” J. Comput. Phys. **168**, 286–315 (2001). [CrossRef]

## 3. Numerical dispersion

*M*) can be obtained

*k*and

_{x}*k*are the x and y components of the numeric wave vector

_{y}*k*, respectively,

*ω*is the angular frequency and

*c*is the speed of light.

*x*=Δ

*y*=Δ

*s*and

*ϕ*define the direction of propagation, i.e.

*k*=

_{x}*k*cos

*ϕ*,

*k*=

_{y}*k*sin

*ϕ*, the dispersion relation can be expressed as

*v*can be calculated by Newton’s method. The dispersion error

_{p}*v*can be evaluated by the following expression

_{e}*t*=0.1Δ

*s*/

*c*and Δ

*t*=0.5Δ

*s*/

*c*, respectively. From Fig. 2, we can see that when Δ

*t*=0.5Δ

*s*/

*c*the maximum of dispersion error is about -6% for the standard FDTD, i.e. FDTD (2,2) and 1% for HO-FDTD. This means that those HO-FDTD schemes can exhibit the excellent dispersion characteristics when compared with the standard FDTD scheme. Figure 3 shows the numerical error versus the direction of propagation for Δ

*s*=

*λ*/10 when Δ

*t*=0.1Δ

*s*/

*c*and Δ

*t*=0.5Δ

*s*/

*c*, respectively. It is found that the dispersion characteristic of FDTD (2, 4) is better than FDTD(2, 6) and FDTD(2, 8) when Δ

*t*=0.5Δ

*s*/

*c*, but HO-FDTD schemes less depend on the direction of propagation.

## 4. Numerical results

*n*

_{1}=2.42 and

*n*

_{2}=2. The dimension of the coupler is

*A*=32

*µm*and

*B*=6

*µm*. The thickness of each slab is

*D*=0.4

*µm*, and the gap between the slabs is

*S*=0.3

*µm*. The spatial distribution of excitation source is chosen to be similar to that of the TE

_{0}mode and the wavelength of the excitation source in free space is 1.5

*λ*=

*µm*. The whole analysis region is surrounded by ten-layer PMLs and the time-step is Δ

*t*=0.5Δ

*s*/

*c*.

*s*=0.05

*µm*. The power exchange between the two waveguides is clearly illustrated. By calculating the distance between the maximum and the minimum of the field amplitudes in the slab along the propagation direction, the coupling length is obtained as 10.775

*L*=

_{c}*µm*. It is in good agreement with the analytical solution of 10.806

*µm*[15]. The relative error is about -0.287%.

*s*=0.1

*µm*), normal-cell (Δ

*s*=0.05

*µm*) and fine-cell (Δ

*s*=0.025

*µm*). The calculated relative error reduces as the cell size becomes smaller, and is less than 1% when fine-cell is used. Compared with the standard FDTD scheme, these HO-FDTD schemes demonstrate obvious advantage in calculating efficiency and computing accuracy. As the dispersion characteristics of the HOFDTD scheme is better than the standard FDTD scheme, the HO-FDTD scheme need less grids and can save a lot of computer resource, while the same accuracy is attained. For example, the relative error calculated by FDTD (2, 4) scheme with coarse-cells is -1.906%, while the error calculated by FDTD (2, 2) scheme with normal cells is -2.6%, but both the computational time ratio and the number of cells ratio are about 1:4. Moreover, the increase in the number of order doesn’t provide obvious improvement in the accuracy of the solution, and an additional computational time is used. Thus the FDTD (2, 4) scheme is the best one when both calculating efficiency and computing accuracy are considered.

## 5. Conclusion

## References and links

1. | K. S. Yee, “Numerical solution of initial boundary value problem involving Maxwell’s equation in isotropic media,” IEEE Trans. Antennas Propag. |

2. | S. Chu and S. K Chaudhuri, “A finite-difference time-domain method for the design and analysis of guidedwave optical structures,” IEEE, J. Lightwave Technol. |

3. | W. Huang, C. L. Xu, and A. Goss, “A scalar finite-difference time-domain approach to guided-wave optics,” IEEE Photon. Technol. Lett. |

4. | K. L. Shlager and J. B. Schneider, “Comparison of the dispersion properties of several low dispersion finite-difference time-domain algorithms,” IEEE Trans. Antennas and Propag. |

5. | Q. H. Liu, “The pseudospectral time-domain (PSTD) method: a new algorithm for solution of Maxwell’s equations,” IEEE Antennas Propag. Soc. Int. Symp. |

6. | M. Krumpholz and L. P. B. Katehi. “MRTD New time-domain schemes based on multiresolution analysis.” IEEE Trans. Microwave Theory Tech. 44, 555–571 (1996). [CrossRef] |

7. | J. Fang, “Time Domain Finite Difference Computation for Maxwell’s Equations,” PhD thesis, (Univ. California, Berkeley, 1989). |

8. | C. W. Manry, S. L. Broschat, and J. B. Schneider, “Higher-order FDTD methods for large problems,” Appl. Comput. Electromagn. Soc. J. |

9. | K. L. Shlager and J. B. Schneider, “Comparison of the dispersion properties of higher order FDTD schemes and equivalent-sized MRTD schemes,” IEEE Trans. Antennas and Propag. |

10. | M. L. Ghrist, “High order finite difference methods for wave equations,” M. S. dissertation, (University of Colorado, 1997). |

11. | E. M. Tentzeris, R. L. Robertson, and J. F. Harvey, “Stability and dispersion analysis of Battle -Lemarie-Based MRTD schemes,” IEEE Trans. Microwave Theory Tech. |

12. | J. P. Berenger, “A perfectly match layer for the absorption of electromagnetic waves,” J. Comput. Phys. |

13. | S. D. Gedney. “An anisotropic perfectly matched layer-absorbing medium for truncation of FDTD lattices,” IEEE Trans. Antennas and Propag. |

14. | A. Yefet and P. G. Petropoulos, “A staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell’s equations,” J. Comput. Phys. |

15. | K. Okamoto, |

**OCIS Codes**

(200.0200) Optics in computing : Optics in computing

(230.7390) Optical devices : Waveguides, planar

**ToC Category:**

Optical Devices

**History**

Original Manuscript: August 15, 2006

Revised Manuscript: October 9, 2006

Manuscript Accepted: November 11, 2006

Published: November 27, 2006

**Citation**

Fanmin Kong, Kang Li, and Xin Liu, "Accurate analysis of planar optical waveguide devices using higher-order FDTD scheme," Opt. Express **14**, 11796-11803 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-24-11796

Sort: Year | Journal | Reset

### References

- K. S. Yee, "Numerical solution of initial boundary value problem involving Maxwell’s equation in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1996).
- S. Chu and S. K. Chaudhuri, "A finite-difference time-domain method for the design and analysis of guided-wave optical structures," IEEE, J. Lightwave Technol. 7, 2033-2038 (1989). [CrossRef]
- W. Huang, C. L. Xu and A. Goss, "A scalar finite-difference time-domain approach to guided-wave optics," IEEE Photon. Technol. Lett. 3, 524-526 (1991). [CrossRef]
- K. L. Shlager and J. B. Schneider, "Comparison of the dispersion properties of several low dispersion finite-difference time-domain algorithms," IEEE Trans. Antennas and Propag. 51, 642-653 (2003). [CrossRef]
- Q. H. Liu, "The pseudospectral time-domain (PSTD) method: a new algorithm for solution of Maxwell’s equations," IEEE Antennas Propag. Soc. Int. Symp. 1,122-125 (1997).
- M. Krumpholz and L. P. B. Katehi. "MRTD New time-domain schemes based on multiresolution analysis." IEEE Trans. Microwave Theory Tech. 44, 555-571 (1996). [CrossRef]
- J. Fang, "Time Domain Finite Difference Computation for Maxwell’s Equations," PhD thesis, (Univ. California, Berkeley, 1989).
- C. W. Manry, S. L. Broschat, and J. B. Schneider, "Higher-order FDTD methods for large problems," Appl. Comput. Electromagn. Soc. J. 10, 17-29 (1995).
- K. L. Shlager and J. B. Schneider, "Comparison of the dispersion properties of higher order FDTD schemes and equivalent-sized MRTD schemes," IEEE Trans. Antennas and Propag. 52,1095-1104 (2004). [CrossRef]
- M. L. Ghrist, "High order finite difference methods for wave equations," M. S. dissertation, (University of Colorado, 1997).
- E. M. Tentzeris, R. L. Robertson, and J. F. Harvey, "Stability and dispersion analysis of Battle -Lemarie-Based MRTD schemes," IEEE Trans. Microwave Theory Tech. 47, 1004-1012 (1999). [CrossRef]
- J. P. Berenger, "A perfectly match layer for the absorption of electromagnetic waves," J. Comput. Phys. 114,185-200 (1994). [CrossRef]
- S. D. Gedney. "An anisotropic perfectly matched layer-absorbing medium for truncation of FDTD lattices," IEEE Trans. Antennas and Propag. 44, 1630-1639 (1996). [CrossRef]
- A. Yefet and P. G. Petropoulos, "A staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell’s equations," J. Comput. Phys. 168, 286-315 (2001). [CrossRef]
- K. Okamoto, Fundamentals of optical waveguides, (Academic Press, 2000).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.