## Reformulation of the plane wave method to model photonic crystals

Optics Express, Vol. 11, Issue 22, pp. 2905-2910 (2003)

http://dx.doi.org/10.1364/OE.11.002905

Acrobat PDF (380 KB)

### Abstract

A new formulation of the plane-wave method to study the characteristics (dispersion curves and field patterns) of photonic crystal structures is proposed. The expression of the dielectric constant is written using the superposition of two regular lattices, the former for the perfect structure and the latter for the defects. This turns out to be simpler to implement than the classical one, based on the supercell concept. Results on mode coupling effects in two-dimensional photonic crystal waveguides are studied and successfully compared with those provided by a Finite Difference Time Domain method. In particular the approach is shown able to determine the existence of “mini-stop bands” and the field patterns of the various interfering modes.

© 2003 Optical Society of America

## 1. Introduction

5. S. Guo and S. Albin, “A simple plane wave implementation method for photonic crystal calculations,” Opt. Express **11**, 167 (2003). [CrossRef] [PubMed]

8. M. Kafesaki, M. Agio, and C. M. Soukoulis, “Waveguides in finite-height two-dimensional photonic crystals,” J. Opt. Soc. Am. B **19**, 2232 (2002). [CrossRef]

4. P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavity in photonic crystals: Mode symmetry, tunability and coupling efficiency,” Phy. Rev. B **54**, 7837–7842 (1996). [CrossRef]

5. S. Guo and S. Albin, “A simple plane wave implementation method for photonic crystal calculations,” Opt. Express **11**, 167 (2003). [CrossRef] [PubMed]

9. M. Agio and C. M. Soukoulis, “Ministop bands in single defect photonic crystal waveguides,” Phys. Rev. E **64**, 055603 (2001). [CrossRef]

13. M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phy. Rev. B **64**, 155113 (2001). [CrossRef]

## 2. Reformulation of the plane wave method

*z*direction, lattice constant

*a*, radius

*R*and dielectric constant

*ε*. The dielectric constant of the background material is

_{a}*ε*. The PC is characterized by the lattice vectors {

_{b}*,*

**a**_{1}*} and the reciprocal lattice vectors {*

**a**_{2}*,*

**b**_{1}*} [1,2].*

**b**_{2}*g*), expanded in Fourier series in the reciprocal lattice, is

**is the spatial vector position and**

*r**=*

**G***h*

_{1}*+*

**b**_{1}*h*

_{2}*, with*

**b**_{2}*h*,

_{1}*h*arbitrary integer numbers. The coefficient

_{2}*ε*is defined as

_{g}(**G**)*=*

**G***0*, while, for

*≠*

**G***0*, it holds:

*J*

_{1}is the first order Bessel function.

*ε*=

_{d}*ε*-

_{b}*ε*while the background material has a dielectric constant equal to 0. The defect perfect lattice can be characterized by the vectors

_{a}*=*

**c**_{1}*n*

_{1}*and*

**a**_{1}*=*

**c**_{2}*n*

_{2}*, with*

**a**_{2}*n*,

_{1}*n*arbitrary integer numbers. The reciprocal lattice vectors are

_{2}*=*

**d**_{1}*/*

**b**_{1}*n*,

_{1}*=*

**d**_{2}*/*

**b**_{2}*n*. The generic vector in the reciprocal lattice is now defined as

_{2}*=*

**S***l*+

_{1}**d**_{1}*l*, where

_{2}**d**_{2}*l*are arbitrary integer numbers. The relative dielectric constant of the defect perfect lattice, denoted by subscript

_{1},l_{2}*s*, can then be expressed as

*ε*is then

_{s}(**S**)*=*

**S***0*, while, for

*≠*

**S***0*, it holds

*n*=

_{1}*1*and

*n*=

_{2}*3*. Figure 1(b) reports the superposition of the two reciprocal lattices, the former (larger gray circles) referring to the perfect lattice, the latter (smaller green dots) referring to the defect perfect lattice. Note that the nodes of the perfect reciprocal lattice are a subset of the defect perfect reciprocal lattice.

*ε*is the coefficient that describes the superposition of two reciprocal lattice. For nodes belonging to both reciprocal lattices,

_{c}(**S**)*ε*can be written

_{c}(**S**)*=*

**S***0*, while, for

*≠*

**S***0*

*l*=

_{1}*m n*and

_{1}*l*=

_{2}*n n*, with

_{2}*m*,

*n*arbitrary integers different from 0.

*n*and

_{1}*n*, without the need to repeat the supercell construction procedure.

_{2}*is used instead of*

**S***:*

**G***ω*is the eigen-angular frequency and

*the wave vector while the eigenfunctions of the electric and magnetic fields can be expanded in Fourier series [1,2]:*

**k**12. C. J. M. Smith, R. M. De La Rue, M. Rattier, S. Olivier, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdrè, and U. Osterle “Coupled guide and cavity in a two-dimensional photonic crystal,” Appl. Phys. Lett. **78**, 1487–1489 (2001). [CrossRef]

## 3. Results and comparison between different methods

*E*parallel to the rod axes) propagating along the

*k*direction (in the

*x*-

*y*plane). The method is applied considering waveguides repeated every 6 lattice periods

*a*and 1089 plane waves. In this case, few minutes of CPU time are enough to run the code on a 1800 MHz Pentium IV Computer.

*y*direction of Fig. 1, varying

*between 0 and*

**k***π*/

*a*. The gray areas are the projected band structure of the perfect crystal that limit the band gap between 0.46

*a*/

*λ*and 0.54

*a*/

*λ*. The solid lines are the dispersion curves of the waveguide modes. The red line, marked by index 1, refers to the fundamental mode.

*a*/

*λ*and 0.49

*a*/

*λ*, the so called “anticrossing” phenomenon occurs, involving higher order modes. This generates two “mini-stop bands” [9

9. M. Agio and C. M. Soukoulis, “Ministop bands in single defect photonic crystal waveguides,” Phys. Rev. E **64**, 055603 (2001). [CrossRef]

13. M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phy. Rev. B **64**, 155113 (2001). [CrossRef]

*a*/

*λ*, Fig. 2(c) shows a close-up of the mini gap caused by the coupling between the fundamental mode and two higher order modes which are represented by blue and green lines, with index 2 and 3, respectively. Mode 3 is odd, while modes 1 and 2 are even. The field patterns of the waveguide modes corresponding to the five different points, marked in Fig. 2(c), are shown in the Figs. 2(d), 2(e), 2(f), 2(g), 2(h). In particular, Figs. 2(d), 2(e), 2(f) refer to the electric displacement of the fundamental mode. In the intermediate region (field of Fig. 2(e)), coupling of modes 1, 2 and 3 changes the field pattern, which results in a kind of zig-zag path caused by the interference of the odd and even modes. The even pattern of mode 2 is shown in Fig. 2(g), while the odd pattern of mode 3 is shown in Fig. 2(h).

8. M. Kafesaki, M. Agio, and C. M. Soukoulis, “Waveguides in finite-height two-dimensional photonic crystals,” J. Opt. Soc. Am. B **19**, 2232 (2002). [CrossRef]

*Δx*=

*Δy*=

*0.005µm*and 8 GT-PML layers. Agreement between the two set of results is very good. Finally Fig. 3(a) and 3(b) show how the 2D-FDTD accounts for propagation of the electric field in the structure at 0.47

*a*/

*λ*and 0.49

*a*/

*λ*, which are respectively in a allowed and in a “mini-stop band”: the attenuation of the field in the latter case is evident.

## 4. Conclusions

## Acknowledgments

## References and links

1. | K. Sakoda, |

2. | S. G. Johnson and J. D. Joannopoulos, ( |

3. | H. Benisty, “Modal analysis of optical guides with two-dimensional photonic band gap boundaries,” J. Appl. Phys. |

4. | P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavity in photonic crystals: Mode symmetry, tunability and coupling efficiency,” Phy. Rev. B |

5. | S. Guo and S. Albin, “A simple plane wave implementation method for photonic crystal calculations,” Opt. Express |

6. | A. Taflove, |

7. | F. Fogli, J. Pagazaurtundua Alberte, G. Bellanca, and P. Bassi, “Analysis of Finite 2-D Photonic Bandgap Lightwave Devices using the FD-TD Method,” Proc. of IEEE-WFOPC 2000, Pavia, June 8–9, 236–241 (2000). |

8. | M. Kafesaki, M. Agio, and C. M. Soukoulis, “Waveguides in finite-height two-dimensional photonic crystals,” J. Opt. Soc. Am. B |

9. | M. Agio and C. M. Soukoulis, “Ministop bands in single defect photonic crystal waveguides,” Phys. Rev. E |

10. | S. Olivier, H. Benisty, C. Weisbuch, C. J. M. Smith, T. F. Krauss, and R. Houdrè, “Coupled-mode theory and propagation losses in photonic crystal waveguides,” Opt. Express |

11. | S. Olivier, M. Rattier, H. Benisty, C. Weisbuch, C. J. M. Smith, R. M. De La Rue, T. F. Krauss, U. Osterle, and R. Houdrè, “Mini-stopbands of one-dimensional system: The channel waveguide in a two-dimensional photonic crystal,” Phys. Rev. B. |

12. | C. J. M. Smith, R. M. De La Rue, M. Rattier, S. Olivier, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdrè, and U. Osterle “Coupled guide and cavity in a two-dimensional photonic crystal,” Appl. Phys. Lett. |

13. | M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phy. Rev. B |

**OCIS Codes**

(130.0130) Integrated optics : Integrated optics

(230.7380) Optical devices : Waveguides, channeled

**ToC Category:**

Research Papers

**History**

Original Manuscript: September 15, 2003

Revised Manuscript: October 24, 2003

Published: November 3, 2003

**Citation**

Rossella Zoli, Marco Gnan, Davide Castaldini, Gaetano Bellanca, and Paolo Bassi, "Reformulation of the plane wave method to model photonic crystals," Opt. Express **11**, 2905-2910 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-22-2905

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### References

- K. Sakoda, Optical properties of Photonic Crystals, Springer (2001).
- S. G. Johnson, J. D. Joannopoulos, Photonic crystals The Road from Theory to Practice, Kluwer Academic Publishers, (2001).
- H. Benisty, �??Modal analysis of optical guides with two-dimensional photonic band gap boundaries,�?? J. Appl. Phys. 79, 7483-7492 (1996). [CrossRef]
- P. R. Villeneuve, S. Fan, J. D. Joannopoulos, �??Microcavity in photonic crystals: Mode symmetry, tenability and coupling efficiency,�?? Phy. Rev. B 54, 7837-7842 (1996). [CrossRef]
- S. Guo, S. Albin, �??A simple plane wave implementation method for photonic crystal calculations,�?? Opt. Express 11, 167 (2003). [CrossRef] [PubMed]
- A.Taflove, Computational electrodynamics �?? The Finite Difference Time-Domain Method, (Artech House, Norwood, MA, 1995).
- F. Fogli, J. Pagazaurtundua Alberte, G. Bellanca, P. Bassi, �??Analysis of Finite 2-D Photonic Bandgap Lightwave Devices using the FD-TD Method,�?? Proc. of IEEE-WFOPC 2000, Pavia, June 8-9, 236-241 (2000).
- M. Kafesaki, M. Agio, C. M. Soukoulis, �??Waveguides in finite-height two-dimensional photonic crystals,�?? J. Opt. Soc. Am. B 19, 2232 (2002). [CrossRef]
- M. Agio, C. M. Soukoulis, �??Ministop bands in single defect photonic crystal waveguides,�?? Phys. Rev. E 64, 055603 (2001). [CrossRef]
- S. Olivier, H. Benisty, C. Weisbuch, C. J. M. Smith, T. F . Krauss, R. Houdrè, �??Coupled-mode theory and propagation losses in photonic crystal waveguides,�?? Opt. Express 11, 1490-1496 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-13-1490.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-13-1490.</a> [CrossRef] [PubMed]
- S. Olivier, M. Rattier, H. Benisty, C. Weisbuch, C. J. M. Smith, R. M. De La Rue, T. F. Krauss, U. Osterle, R. Houdrè, �??Mini-stopbands of one-dimensional system: The channel waveguide in a two-dimensional photonic crystal,�?? Phys. Rev. B. 63, 11311 (2001). [CrossRef]
- C. J. M. Smith, R. M. De La Rue, M. Rattier ,S. Olivier, H. Benisty, C. Weisbuch, T. F . Krauss, R. Houdrè, U. Osterle �??Coupled guide and cavity in a two-dimensional photonic crystal,�?? Appl. Phys. Lett. 78, 1487-1489 (2001). [CrossRef]
- M. Qiu, K. Azizi, A. Karlsson, M. Swillo, B. Jaskorzynska, �??Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,�?? Phy. Rev. B 64, 155113 (2001). [CrossRef]

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