## Simple plane wave implementation for photonic crystal calculations

Optics Express, Vol. 11, Issue 2, pp. 167-175 (2003)

http://dx.doi.org/10.1364/OE.11.000167

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### Abstract

A simple implementation of plane wave method is presented for modeling photonic crystals with arbitrary shaped ‘atoms’. The Fourier transform for a single ‘atom’ is first calculated either by analytical Fourier transform or numerical FFT, then the shift property is used to obtain the Fourier transform for any arbitrary supercell consisting of a finite number of ‘atoms’. To ensure accurate results, generally, two iterating processes including the plane wave iteration and grid resolution iteration must converge. Analysis shows that using analytical Fourier transform when available can improve accuracy and avoid the grid resolution iteration. It converges to the accurate results quickly using a small number of plane waves. Coordinate conversion is used to treat non-orthogonal unit cell with non-regular ‘atom’ and then is treated by standard numerical FFT. MATLAB source code for the implementation requires about less than 150 statements, and is freely available at

© 2002 Optical Society of America

## 1. Introduction

1. S. G. Johnson and J. D. Joannopoulos, “Block iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173–190 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173 [CrossRef] [PubMed]

## 2. Theory of PWM

5. K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in FCC dielectric media,” Phys. Rev. Lett. **65**, 2646–2649 (1990). [CrossRef] [PubMed]

_{0}) space can be rewritten as Helmholz’s equation:

*c*is the speed of light in vacuum.

_{̂}represents the two unit axis perpendicular to the propagation direction k⃗ + G⃗. (

*e*̂

_{1},

*e*̂

_{2}, k⃗ + G⃗) are perpendicular to each other.

*h*

_{G,λ}is the coefficient of the

*H*component along the axes

*e*̂

_{λ}.

## 3. Implementation

*e*̂

_{1},

*e*̂

_{2}can be calculated easily. The key part is to obtain the Fourier coefficient matrix ε(G⃗ - G⃗′) according to Eq.(3). It is obtained by calculating the Fourier coefficient of a single atom first using analytical or numerical Fourier transform, and then calculating the Fourier coefficients for the supercell using shift property, finally re-arranging to get the coefficients ε(G⃗ - G⃗′) and inversing to get ε

^{-1}(G⃗ - G⃗′).

### 3.1 ‘Atoms’ with regular shape

*R*, the dielectric constant for the cylinder is ε

_{a}, the background dielectric constant is ε

_{b}, the lattice structure can be represented by the two lattice basis vector

*a*⃗

_{1}and

*a*⃗

_{2}. The area of the unit cell is calculated as

*A*= |

*a*⃗

_{1}×

*a*⃗

_{2}|, the Fourier transform of the unit cell is:

*J*

_{1}is the 1

^{st}order Bessel function,

*G*is the modulus of G⃗, ƒ is a fraction parameter:

*G*are different.

### 3.2 ‘Atoms’ with arbitrary shape

*a*⃗

_{1},

*a*⃗

_{2},

*a*⃗

_{3}are:

*r*) and the column vector is r⃗ =

*m*

*a*⃗

_{1}+

*n*

*a*⃗

_{2}+

*l*

*a*⃗

_{3}, where

*m, n*and

*l*are coordinates along the basis vectors. In Cartesian coordinates:

*g*] is called the metric for the oblique coordinates [7, 8

8. A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. **43**, 773–793 (1996) [CrossRef]

### 3.3 Shift property of Fourier transform

_{G}, if ε(

*r*) is shifted by an amount

*r*

_{0}, then its Fourier transform must be multiplied by

_{i}is the location of ‘atom’

*i*in the supercell.

*G*grid points by doing simple additions and subtractions, requiring only the Fourier coefficients of each single kind of atom. This is especially advantageous for a large supercell with many periodic or random atoms in it.

*a*, the primitive lattice vector basis are defined as

*a*⃗

_{1}= [0,1,1]

*a*/2,

*a*⃗

_{2}= [1,0,1]

*a*/2,

*a*⃗

_{3}= [1,1,0]

*a*/2. The locations of the two atoms in the primitive cell are chosen as r⃗0 = [-1,-1,-1]

*a*/8 and r⃗1 = [1,1,1]

*a*/8 to keep inversion symmetry. The primitive reciprocal lattice vectors b⃗

_{1},b⃗

_{2},b⃗

_{3}are calculated according to (5) in Appendix B of Ref. [10].

*R*is the radius of the sphere,

*V*= |

*a*⃗

_{1}·

*a*⃗

_{2}×

*a*⃗

_{3}|.

*R*=√3/8

*a*, ε

_{a}=13, ε

_{b}=1 using our simple program with 343 plane waves, which is in excellent agreement with the result in Ref. [3

3. K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. **65**, 3152–3155 (1990). [CrossRef] [PubMed]

*n*

_{1}b⃗

_{1}+

*n*

_{2}b⃗ +

*n*

_{3}b⃗

_{3},|

*n*

_{i}|≤

*n*and in this example

*n*=3.

## 4. Convergence, accuracy and stability

*a*, where

*a*is the lattice constant, and the dielectric constant for GaAs is 13.0. All the eigen-frequencies with

*k*-point located at

**M**(see the inset of Fig. 1(c)) are calculated.

*norm*[

*X*(

*n*)-

*X*(

*n*- 1)], where

*X*(n) is the eigen-frequency vector of the first 10 bands for the

*n*th iteration, and the number of plane waves used is

*N*

_{PW}=(2n+1)

^{2}.

4. R. D. Meade and A. M. Rappe et a1., “Accurate theoretical analysis of photonic band gap materials,” Phys. Rev. B **48**, 8434–8437 (1993). [CrossRef]

9. D. Hermann et al., “Photonic band structure computations,” Opt. Express **8**, 167–172 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-167 [CrossRef] [PubMed]

*a*, where

*a*is the lattice constant. A 7×7 supercell is used to approximate the crystal structure. The defect frequency in the band gap at k=(0,0,0) was calculated using different number of plane waves. The Fourier transform of this 7×7 supercell is easily obtained using Eq. (5) and Eqs. (11–12). The calculated defect frequencies are listed in Table 2. The iteration error is calculated as

*norm*[

*X*(

*n*)-

*X*(

*n*- 1)], where

*X*(

*n*) is the eigenfrequency vector of the first 50 bands for the nth iteration. The band structure of a 7×7 supercell is folded 7

^{2}times and the defect band is band 49 in this case.

*H*or

*E*can be obtained conveniently at the same time using the calculated eigen-vector

*h*

_{G,λ}and Eq. (2). Defects with one or more cylinders of different sizes [11] or dielectric constants can be treated as different ‘atoms’ and their Fourier transform are obtained the same way as Eq. (5) using different

*R*or ε and then added to the supercell according to Eqs. (11–12). Similar mode fields as in Ref. [11] can be obtained, but are not shown here. Other defects such as waveguides can be treated in the same way.

## 5. Discussion

*O*(

*n*

^{3}) to

*O*(

*n*

^{2}) or less, a good example is Ref. [1

1. S. G. Johnson and J. D. Joannopoulos, “Block iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173–190 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173 [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | S. G. Johnson and J. D. Joannopoulos, “Block iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

2. | K. M. Leung, “Plane wave calculation of photonic band structures” in |

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

4. | R. D. Meade and A. M. Rappe et a1., “Accurate theoretical analysis of photonic band gap materials,” Phys. Rev. B |

5. | K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in FCC dielectric media,” Phys. Rev. Lett. |

6. | D. C. Champeney, |

7. | Geoge Arfken, |

8. | A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. |

9. | D. Hermann et al., “Photonic band structure computations,” Opt. Express |

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

11. | P. R. Villeneuve and S. Fan et al., “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(350.3950) Other areas of optics : Micro-optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: December 19, 2002

Revised Manuscript: January 21, 2003

Published: January 27, 2003

**Citation**

Shangping Guo and Sacharia Albin, "Simple plane wave implementation for photonic crystal calculations," Opt. Express **11**, 167-175 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-2-167

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### References

- S. G. Johnson and J. D. Joannopoulos, �??Block iterative frequency-domain methods for Maxwell's equations in a planewave basis,�?? Opt. Express 8, 173-190 (2001), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173</a> [CrossRef] [PubMed]
- K. M. Leung, "Plane wave calculation of photonic band structures" in Photonic band gaps and localizations, C. M. Soukoulis. ed. (Plenum Press NY 1993).
- K. M. Ho, C. T. Chan, and C. M. Soukoulis, �??Existence of a photonic gap in periodic dielectric structures,�?? Phys. Rev. Lett. 65, 3152-3155 (1990). [CrossRef] [PubMed]
- R. D. Meade, A. M. Rappe et al., �??Accurate theoretical analysis of photonic band gap materials,�?? Phys. Rev. B 48, 8434-8437 (1993). [CrossRef]
- K. M. Leung and Y. F. Liu, �??Full vector wave calculation of photonic band structures in FCC dielectric media,�?? Phys. Rev. Lett. 65, 2646-2649 (1990). [CrossRef] [PubMed]
- D. C. Champeney, Fourier transforms and their physical applications (Academic Press, 1973) Chap. 3.
- Geoge Arfken, Mathematical methods for physicists, 3rd edition, (Academic Press, 1985), Chap. 2.
- A. J. Ward, J. B. Pendry, �??Refraction and geometry in Maxwell�??s equations,�?? J. Mod. Opt. 43, 773-793 (1996) [CrossRef]
- D. Hermann, M. Frank, K. Busch, and P. Wolfle,�??Photonic band structure computations,�?? Opt. Express 8, 167-172 (2001), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-167">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-167</a> [CrossRef] [PubMed]
- J. D.. Joannopoulos et al., Photonic crystals - Molding the flow of light ( Princeton University Press 1995).
- P. R. Villeneuve, S. Fan et al., �??Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,�?? Phys. Rev. B 54, 7837-7842 (1996).

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