## Perfectly matched layer absorption boundary condition in planewave based transfer-scattering matrix method for photonic crystal device simulation

Optics Express, Vol. 16, Issue 15, pp. 11548-11554 (2008)

http://dx.doi.org/10.1364/OE.16.011548

Acrobat PDF (201 KB)

### Abstract

The performance of the perfectly matched layer absorption boundary condition is fully exploited when it is applied to the planewave based transfer-scattering matrix method in photonic crystal device simulation. The mode profile of one dimensional dielectric waveguide and the optical properties of sub-wavelength aluminum grating with semi-infinite substrate are studied to illustrate the accuracy and power of this approach.

© 2008 Optical Society of America

## 1. Introduction

1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. **58**, 2059 (1987). [CrossRef] [PubMed]

2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. **58**, 2486 (1987). [CrossRef] [PubMed]

9. K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys Rev. Lett. **65**, 3125 (1990). [CrossRef] [PubMed]

10. J. B. Pendry, “Photonic band structures,” J. Mod. Opt. **41**, 209 (1994). [CrossRef]

13. M. Li, Z. Y. Li, K. M. Ho, J. R. Cao, and M. Miyawaki, “High-efficiency calculations for three-dimensional photonic crystal cavities,” Opt. Lett. **31**, 262 (2006). [PubMed]

13. M. Li, Z. Y. Li, K. M. Ho, J. R. Cao, and M. Miyawaki, “High-efficiency calculations for three-dimensional photonic crystal cavities,” Opt. Lett. **31**, 262 (2006). [PubMed]

14. M. Li, X. Hu, Z. Ye, K. M. Ho, J. R. Cao, and M. Miyawaki, “Higher-order incidence transfer matrix method used in three-dimensional photonic crystal coupled-resonator array simulation,” Opt. Lett. **31**, 3498 (2006). [CrossRef] [PubMed]

15. Z. Ye, X. Hu, M. Li, and K. M. Ho, “Propagation of guided modes in curved nanoribbon waveguides,” Appl. Phys. Lett. **89**, 241108 (2006). [CrossRef]

13. M. Li, Z. Y. Li, K. M. Ho, J. R. Cao, and M. Miyawaki, “High-efficiency calculations for three-dimensional photonic crystal cavities,” Opt. Lett. **31**, 262 (2006). [PubMed]

## 2. A review of the concept of PML and benchmark for the Z-axis PML

16. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic-waves,” J. Computational Physics, **114**, 185 (1994). [CrossRef]

17. J.P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Computational Physics, **127**, 363, (1996) [CrossRef]

18. Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas and Propagation **43**, 1460 (1995). [CrossRef]

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

**p**wave incidence are given in Fig. 1(a) and Fig. 1(b). The propagation direction is from left to right with incident angle θ, and the interface of the Z-axis PMLs (XY plane at Z=0) is perpendicular to Z axis. The isotropic region in the left side of the interface has dielectric constant

*ε*

_{1}and magnetic permeability

*μ*

_{1}.

*ε*

_{2}and magnetic permeability tensor

*μ*

_{2}are given by Eq. (1) with

*s*=

_{z}*a*+

*bi*, any complex number [7]. For ideal Z-axis PML, there will be no reflection for planewaves of any incidence angle at all frequencies, and the transmitted wave in the PML will be exponential attenuated by a factor exp(-

*α*) with α ∼

*bz*cos

*θ*/

*λ*(

*b*imaginary part of

*s*, z thickness of PML,

_{z}*λ*wavelength, and

*θ*incident angle).

*0.5a*(

_{0}*a*the lattice constant) after air. The parameter

_{0}*s*is chosen to be

_{z}*4*+

*4i*. A wide range of normalized frequency

*a*/

_{0}*λ*is calculated from 0.01 to 10 for normal incidence. The reflectance and transmittance amplitudes of

**s**wave and

**p**wave are similar and only s wave spectra are shown at Fig. 2(a). The reflectance amplitudes for both s and

**p**waves are lower than 10

^{-10}; and the perfectly matched condition (no reflection at all frequency) is achieved at normal incidence. The transmittance amplitudes are determined by the attenuation factor

*exp(-α)*with

*α*∼

*bz*cos

*θ*/

*λ*; and the exponential decrease of transmittance is expected with respect to normalized frequency

*a*/

_{0}*λ*.

*0.3–0.5.*We will focus on the performance of PML at normalized frequency

*ϖa*/(2

_{0}*πc*) = 0.4 with thickness

*0.5a*and

_{0}*s*=4 + 4

_{z}*i*;. The transmittance and reflectance amplitudes as functions of the incident angle are shown in Fig. 2(b) the reflectance amplitudes are always below 10

^{-10}; and the perfectly matched condition is achieved for all incidence angles. The transmittance amplitudes increases exponentially for large incidence angle and approaches 100% due to the fact that the attenuation factor

*α*∼

*bz*cos

*θ*/

*λ*approach zero as

*θ*approaches 90 degree; but for a moderate incidence angle (for example 30 degrees), the transmittance amplitudes are still below 1%.

*s*and the other is to increase the thickness of the PML. The transmittance and reflectance amplitudes of both methods are illustrated in Fig. 3: double the thickness of PML and double the imaginary part of

_{z}*s*. As shown in Fig. 3(a), the performance of PML is improved for all incident angles, but when the incident angle approaches 90 degree, the transmittance amplitudes always approaches 100%; while the perfectly matched condition (no reflection) still remains valid for all angles [Fig. 3 (b)]. Further studies such as grading of PML are required to further improvement of the transmittance attenuation performance. Even without grading, the reflectance amplitudes are below 10

_{z}^{-10}which is already better than many of the complicated grading PML approaches in FDTD [7].

## 3. XY-side PML and its application on one-dimensional dielectric waveguide structure

## 4. Application of PML to sub-wavelength metal grating at visible wavelengths

**p**wave shows strong interference at the finite substrate case [Fig. 5(a)]. This strong interference disappears for the infinite substrate and a smooth reflection of

**p**wave is observed [Fig. 5(b)] with a minimum reflection rate (approximately zero) at wavelength 0.45μm. The corresponding results of s-polarized waves for both finite and infinite substrates are approximately constant (90%) over the whole visible frequency range. At a wavelength of 0.45μm the s and

**p**polarization beams can be separated. To get a better understanding of the EM propagation at this wavelength, the electric field mode profiles for the infinite thick substrate grating are plotted for both s and

**p**waves at Fig. 6. The s wave is Ey dominated, and strong reflection occurs at the front of the grating which corresponds to high reflectance. On the other hand, the

**p**wave is Ex dominated and the electromagnetic energy can propagate through the grating and substrate.

## 5. Conclusion

*s*is a real function of coordinate [15

_{z}15. Z. Ye, X. Hu, M. Li, and K. M. Ho, “Propagation of guided modes in curved nanoribbon waveguides,” Appl. Phys. Lett. **89**, 241108 (2006). [CrossRef]

21. X. Hu, C. T. Chan, J. Zi, M. Li, and K. M. Ho, “Diamagnetic response of metallic photonic crystals at infrared and visible Frequencies,” Phys. Rev. Lett. **96**, 223901 (2006). [CrossRef] [PubMed]

## References and links

1. | E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. |

2. | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. |

3. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, |

4. | S. P. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, “Control of light emission by 3D photonic crystals,” Science |

5. | O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science |

6. | E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Nature |

7. | A. Taflove and S. C. Hagness, |

8. | J. Jin, |

9. | K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys Rev. Lett. |

10. | J. B. Pendry, “Photonic band structures,” J. Mod. Opt. |

11. | Z. Y. Li and K. M. Ho, “Application of structural symmetries in the plane-wave-based transfer-matrix method for three-dimensional photonic crystal waveguides,” Phys. Rev. |

12. | Z. Y. Li and K. M. Ho, “Bloch mode reflection and lasing threshold in semiconductor nanowire laser arrays,” Phys. Rev. |

13. | M. Li, Z. Y. Li, K. M. Ho, J. R. Cao, and M. Miyawaki, “High-efficiency calculations for three-dimensional photonic crystal cavities,” Opt. Lett. |

14. | M. Li, X. Hu, Z. Ye, K. M. Ho, J. R. Cao, and M. Miyawaki, “Higher-order incidence transfer matrix method used in three-dimensional photonic crystal coupled-resonator array simulation,” Opt. Lett. |

15. | Z. Ye, X. Hu, M. Li, and K. M. Ho, “Propagation of guided modes in curved nanoribbon waveguides,” Appl. Phys. Lett. |

16. | J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic-waves,” J. Computational Physics, |

17. | J.P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Computational Physics, |

18. | Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas and Propagation |

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

20. | A. Yariv and P. Yeh, |

21. | X. Hu, C. T. Chan, J. Zi, M. Li, and K. M. Ho, “Diamagnetic response of metallic photonic crystals at infrared and visible Frequencies,” Phys. Rev. Lett. |

22. | S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(050.2770) Diffraction and gratings : Gratings

(220.4830) Optical design and fabrication : Systems design

(050.5298) Diffraction and gratings : Photonic crystals

(230.5298) Optical devices : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: June 3, 2008

Revised Manuscript: July 10, 2008

Manuscript Accepted: July 12, 2008

Published: July 18, 2008

**Citation**

Ming Li, Xinhua Hu, Zhuo Ye, Kai-Ming Ho, Jiangrong Cao, and Mamoru Miyawaki, "Perfectly matched layer absorption boundary condition in planewave based transfer-scattering matrix method for photonic crystal device simulation," Opt. Express **16**, 11548-11554 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-15-11548

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### References

- E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059 (1987). [CrossRef] [PubMed]
- S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486 (1987). [CrossRef] [PubMed]
- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University Press, 1995).
- S. P. Ogawa, M. Imada, S. Yoshimoto, M. Okano and S. Noda, "Control of light emission by 3D photonic crystals," Science 305, 227 (2004).
- O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional photonic band-gap defect mode laser," Science 284, 1819 (1999). [CrossRef] [PubMed]
- E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, "Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal," Nature 407, 6807 (2000) [PubMed]
- A. Taflove and S. C. Hagness, Computational Electrodynamics (Artech Houses, 2000).
- J. Jin, The Finite Element Method in Electromagnetics (Wiley and Sons, 2002).
- K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys Rev. Lett. 65, 3125 (1990). [CrossRef] [PubMed]
- J. B. Pendry, "Photonic band structures," J. Mod. Opt. 41, 209 (1994). [CrossRef]
- Z. Y. Li and K. M. Ho, "Application of structural symmetries in the plane-wave-based transfer-matrix method for three-dimensional photonic crystal waveguides," Phys. Rev. B 68, 245117 (2003).
- Z. Y. Li and K. M. Ho, "Bloch mode reflection and lasing threshold in semiconductor nanowire laser arrays," Phys. Rev. B 71, 045315 (2005).
- M. Li, Z. Y. Li, K. M. Ho, J. R. Cao, and M. Miyawaki, "High-efficiency calculations for three-dimensional photonic crystal cavities," Opt. Lett. 31, 262 (2006). [PubMed]
- M. Li, X. Hu, Z. Ye, K. M. Ho, J. R. Cao and M. Miyawaki, "Higher-order incidence transfer matrix method used in three-dimensional photonic crystal coupled-resonator array simulation," Opt. Lett. 31, 3498 (2006). [CrossRef] [PubMed]
- Z. Ye, X. Hu, M. Li and K. M. Ho, "Propagation of guided modes in curved nanoribbon waveguides," Appl. Phys. Lett. 89, 241108 (2006). [CrossRef]
- J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic-waves," J. Computational Physics, 114, 185 (1994). [CrossRef]
- J.P. Berenger, "Three-dimensional perfectly matched layer for the absorption of electromagnetic waves," J. Computational Physics, 127, 363, (1996) [CrossRef]
- Z. S. Sacks, D. M. Kingsland, R. Lee and J. F. Lee, "A perfectly matched anisotropic absorber for use as an absorbing boundary condition," IEEE Trans. Antennas and Propagation 43,1460 (1995). [CrossRef]
- S. D. Gedney, "An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices," IEEE Trans. Antennas and Propagation 44, 1630 (1996). [CrossRef]
- A. Yariv and P. Yeh, Optical Waves in Crystal, (Wiley, 1984).
- X. Hu, C. T. Chan, J. Zi, M. Li and K. M. Ho, "Diamagnetic response of metallic photonic crystals at infrared and visible Frequencies," Phys. Rev. Lett. 96, 223901 (2006). [CrossRef] [PubMed]
- S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith and J. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006).

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