## Numerical techniques for excitation and analysis of defect modes in photonic crystals

Optics Express, Vol. 11, Issue 9, pp. 1080-1089 (2003)

http://dx.doi.org/10.1364/OE.11.001080

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### Abstract

Two numerical techniques for analysis of defect modes in photonic crystals are presented. Based on the finite-difference time-domain method (FDTD), we use plane wave incidences and point sources for excitation and analysis. Using a total-field/scattered-field scheme, an ideal plane wave incident at different angles is implemented; defect modes are selectively excited and mode symmetries are probed. All modes can be excited by an incident plane wave along a non-symmetric direction of the crystal. Degenerate modes can also be differentiated using this method. A proper arrangement of point sources with positive and negative amplitudes in the cavity flexibly excites any chosen modes. Numerical simulations have verified these claims. Evolution of each defect mode is studied using spectral filtering. The quality factor of the defect mode is estimated based on the field decay. The far-field patterns are calculated and the Q values are shown to affect strongly the sharpness of these patterns. Animations of the near-fields of the defect modes are presented to give an intuitive image of their oscillating features.

© 2003 Optical Society of America

## 1. Introduction

1. P R Villeneuve et al, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phy. Rev. B **54**, 7837 (1996) [CrossRef]

2. Shanhui Fan et al, “Channel drop filters in photonic crystals,” Opt. Express, **3**, 4 (1998), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-3-1-4 [CrossRef]

3. K. M. Ho et al, “Existence of a photonic gap in periodic dielectric structures,” Phy. Rev. Lett. **65**, 3152 (1990) [CrossRef]

1. P R Villeneuve et al, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phy. Rev. B **54**, 7837 (1996) [CrossRef]

1. P R Villeneuve et al, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phy. Rev. B **54**, 7837 (1996) [CrossRef]

4. Min Qiu et al, “Numerical method for computing defect modes in two-dimensional photonic crystals with dielectric or metallic inclusions,” Phy. Rev. B **61**, 12871 (2000) [CrossRef]

5. Kazuaki Sakoda et al, “Optical response of three-dimensional photonic lattices: solution of inhomgeneous Maxwell’s equations and their applications,” Phy. Rev. B **54**, 5732 (1996) [CrossRef]

7. Kazuaki Sakoda et al, “Numerical method for localized defect modes in photonic lattices,” Phy. Rev. B **56**, 4830 (1997) [CrossRef]

## 2. Methods of computation

_{z}, H

_{x}and H

_{y}components) based on Yee’s algorithm [9

9. K. S Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenna and Propagation **14**, 302 (1966) [CrossRef]

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

^{nd}-order FDTD scheme. A 4

^{th}order FDTD scheme was also used to reduce accumulated numerical dispersions, but we found the 2

^{nd}-order algorithm was accurate enough in this case.

*a*and 0.60

*a*, respectively, where

*a*is the lattice constant and

*a*=1.0 µm. Each unit cell is divided into a 40×40 mesh and the whole FDTD region is 200×200 with Δx=Δy=0.025 µm. 10 PML layers are used outside the structure to absorb all energy flowing into the boundary.

*A*is the amplitude, t

_{0}=1000Δt, t

_{w}=250Δt, Δt=5.89e-11µs, f

_{0}=0.35c/a and

*c*is the speed of light in vacuum. The total number of time steps is 50,000. All pulsed sources will be removed after 5000 time steps.

**54**, 7837 (1996) [CrossRef]

5. Kazuaki Sakoda et al, “Optical response of three-dimensional photonic lattices: solution of inhomgeneous Maxwell’s equations and their applications,” Phy. Rev. B **54**, 5732 (1996) [CrossRef]

3. K. M. Ho et al, “Existence of a photonic gap in periodic dielectric structures,” Phy. Rev. Lett. **65**, 3152 (1990) [CrossRef]

11. S. Guo and S. Albin, “Simple plane wave implementation for photonic crystal calculations,” Opt. Express **11**, 167 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-2-167 [CrossRef] [PubMed]

**54**, 7837 (1996) [CrossRef]

## 3. Plane wave excitation

^{st}Brillouin zone, plane wave incidence along these directions will not excite specific modes. Plane wave along other directions, such as incidence at 60° in the example excites all defect modes. Since the structure is symmetric for 90° rotation, 0° and 90° plane wave incidence will yield identical spectral information. However, 0° plane wave incidence will not excite Mode 4(2) and 90° plane wave incidence will not excite Mode 4(1), hence, degenerate modes can be differentiated.

## 4. Point sources

## 5. Far-field patterns

^{st}order monopole mode as in [1

**54**, 7837 (1996) [CrossRef]

^{st}order and 2

^{nd}order monopole are similar since the far-field pattern only measures its outgoing diffraction strength.

## 6. Mode dynamics and their Q values

**54**, 7837 (1996) [CrossRef]

*E*

_{1}and

*E*

_{0}at time step

*N*

_{1}and

*N*

_{0}. Taking Mode 1 as an example, it reaches an amplitude

*E*

_{0}=1.0 at

*N*

_{0}=14000 time steps, and its amplitude decays to

*E*

_{1}=0.40 at time step

*N*

_{1}=50,000. This gives a Q value of 648. The Q values for Mode 2, 3 and 4 are estimated to be 276, 466 and 2936, respectively. The two degenerate modes and the mixed Mode 4 have similar Q values.

## 7. Conclusions

## Acknowledgement

## References and links

1. | P R Villeneuve et al, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phy. Rev. B |

2. | Shanhui Fan et al, “Channel drop filters in photonic crystals,” Opt. Express, |

3. | K. M. Ho et al, “Existence of a photonic gap in periodic dielectric structures,” Phy. Rev. Lett. |

4. | Min Qiu et al, “Numerical method for computing defect modes in two-dimensional photonic crystals with dielectric or metallic inclusions,” Phy. Rev. B |

5. | Kazuaki Sakoda et al, “Optical response of three-dimensional photonic lattices: solution of inhomgeneous Maxwell’s equations and their applications,” Phy. Rev. B |

6. | Vladimir Kuzmiak et al, “Localized defect modes in a two-dimensional triangular photonic crystal,” Phy. Rev. B |

7. | Kazuaki Sakoda et al, “Numerical method for localized defect modes in photonic lattices,” Phy. Rev. B |

8. | Allen Taflove, |

9. | K. S Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenna and Propagation |

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

11. | S. Guo and S. Albin, “Simple plane wave implementation for photonic crystal calculations,” Opt. Express |

12. | J. D. Joannopoulos et al, |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(350.3950) Other areas of optics : Micro-optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 26, 2003

Revised Manuscript: April 27, 2003

Published: May 5, 2003

**Citation**

Shangping Guo and Sacharia Albin, "Numerical techniques for excitation and analysis of defect modes in photonic crystals," Opt. Express **11**, 1080-1089 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-9-1080

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### References

- P R Villeneuve et al, �??Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,�?? Phy. Rev. B 54, 7837 (1996) [CrossRef]
- Shanhui Fan et al, �??Channel drop filters in photonic crystals,�?? Opt. Express 3, 4 (1998), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-3-1-4">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-3-1-4</a> [CrossRef]
- K. M. Ho et al, �??Existence of a photonic gap in periodic dielectric structures,�?? Phy. Rev. Lett. 65, 3152 (1990) [CrossRef]
- Min Qiu et al, �??Numerical method for computing defect modes in two-dimensional photonic crystals with dielectric or metallic inclusions,�?? Phy. Rev. B 61, 12871 (2000) [CrossRef]
- Kazuaki Sakoda et al, �??Optical response of three-dimensional photonic lattices: solution of inhomgeneous Maxwell�??s equations and their applications,�?? Phy. Rev. B 54, 5732 (1996) [CrossRef]
- Vladimir Kuzmiak et al, �??Localized defect modes in a two-dimensional triangular photonic crystal,�?? Phy. Rev. B 57, 15242 (1998) [CrossRef]
- Kazuaki Sakoda et al, �??Numerical method for localized defect modes in photonic lattices,�?? Phy. Rev. B 56, 4830 (1997) [CrossRef]
- Allen Taflove, Computational electrodynamics, the finite difference time domain method (Artech House, 1995)
- Yee, K. S, �??Numerical solution of initial boundary value problems involving Maxwell�??s equations in isotropic media,�?? IEEE Trans. Antenna and Propagation 14, 302 (1966) [CrossRef]
- J. P. Berenger, �??A perfectly matched layer for the absorption of electromagnetic waves,�?? J. Computational Phys. 185, (1994) [CrossRef]
- S. Guo and S. Albin, �??Simple plane wave implementation for photonic crystal calculations,�?? Opt. Express 11, 167 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-2-167">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-2-167<a/> [CrossRef] [PubMed]

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