## Photonic band structure computations

Optics Express, Vol. 8, Issue 3, pp. 167-172 (2001)

http://dx.doi.org/10.1364/OE.8.000167

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### Abstract

We introduce a novel algorithm for band structure computations based on multigrid methods. In addition, we demonstrate how the results of these band structure calculations may be used to compute group velocities and effective photon masses. The results are of direct relevance to studies of pulse propagation in such materials.

© Optical Society of America

## 1 Introduction

1. C.M. Soukoulis (Ed.), *Photonic Band Gap Materials*, *NATO ASI Series E*315, Kluwer Academic Publishers1996 [CrossRef]

2. A. Birner et al., “Macroporous Silicon: A Two-Dimensional Photonic Bandgap Material Suitable for the Near-Infrared Spectral Range,” phys. stat. sol (a) **165**, 111–117 (1998) [CrossRef]

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

## 2 Two-dimensional photonic crystals

*d*. The rods are made from an isotropic dielectric material (dielectric constant

*∊*

_{a}) and are embedded in a matrix of dielectric material characterized by an isotropic dielectric constant

*∊*

_{b}. The direction of the rods defines the z-axis. For electromagnetic waves propagating in the xy-plane, the two transverse polarizations decouple leading to two separate scalar problems. For the TM-polarization (E-field parallel to the rods, i.e.,

*E*⃗ (

*r*⃗)≡(0, 0,

*E*(

*r*⃗)), we obtain from Maxwell’s equations

*H*⃗ (

*r*⃗)≡(0, 0,

*H*(

*r*⃗)), we have

*∂*

_{x}≡

*∂*/

*∂x*,

*∂*

_{y}≡

*∂*/

*∂*

_{y}, and

*r*⃗ (

*x, y*) is a two-dimensional vector. The lattice defined through the centers of cylinders is given by the set

*R*={

*R*⃗=

*n*

_{1}

*a*⃗

_{1}+

*n*

_{2}

*a*⃗2,

*n*

_{1},

*n*

_{2}∊

*N*} of two-dimenisonal lattice vectors

*R*⃗ that are generated by the primitive translations

*a*⃗

_{1}and

*a*⃗

_{2}. The corresponding reciprocal (dual) lattice is defined through

*G*={

*G*⃗:

*G*⃗·

*R*⃗=2

*πn, n*∊

*N,R*⃗

*∊R*}. The dielectric function

*∊*(

*r*⃗)=

*∊*

_{b}-(

*∊*

_{a}-

*∊*

_{b})∑

_{R}⃗

*θ*(

*d*/2-|

*r*⃗-

*R*⃗|) is a lattice periodic function which contains all the information of the photonic crystal. Here,

*θ*(

*r*) denotes the Heaviside step function. Eq. (1) and (2) represent differential equations with periodic coefficients. Therefore, their solutions obey the Bloch-Floquet theorem

*i*=1, 2. The wave vector

*k*⃗∊1.B

*Z*is a vector of the first Brillouin zone (BZ) known as the crystal momentum.

*G*, thereby transforming the differential equations into a infinite matrix eigenvalue problem, which is then truncated and solved numerically. Details of this plane wave method (PWM) for isotropic systems can, for instance, be found in [4

4. K. Busch and S. John, “Photonic band gap formation in certain self-organizing systems,” Phys. Rev. E **58**, 3896–3908 (1998) [CrossRef]

5. K. Busch and S. John. “Liquid crystal photonic band gap materials: The tunable electromagnetic vacuum,” Phys. Rev. Lett. **83**, 967–970 (1999) [CrossRef]

## 3 Efficient computation of the mode structure

*k*⃗ may be cast into the matrix form of a linear system of equations

_{i}=

*c*

^{2}and

*U*

_{i}represents the discretized fields

*E*

_{k⃗}(

*r*⃗) and

*H*

_{k⃗}(

*r*⃗) in case of TM- and TE-polarization, respectively. The precise form of the operator matrix

*L*

_{k⃗}and the vectors

*U*

_{i}depend on the type of polarization (TE or TM) and the type of discretization (finite difference or finite element method). Mathematically, the problem is then to find approximations to the first few eigenfrequencies (bands) Λ

_{1}≤Λ

_{2}≤Λ

_{3}≤… and associated Bloch functions

*U*

_{1};

*U*

_{2};

*U*

_{3}…. Due to the local nature of the underlying differential operators, the operator matrix

*L*

_{k⃗}is sparse, which suggests that Eq. (5) should be solved iteratively by Rayleigh quotient iteration or Lanczos. Such approaches of treating the discrete problem as a purely algebraic one can result in loss of valuable information, especially concerning the smoothness of the Bloch functions. In general, the Bloch functions corresponding to the desired first bands are very smooth, so that they are well approximated on coarser grids. Certain multigrid methods take full advantage of this smoothness and are therefore very effective for solving such problems [6]. In addition, multigrid methods can be applied to nonlinear problems such as Eq. (5), where both eigenfrequency and Bloch functions are unknown. To do so, Eq. (5) has to be amended by a ortho-normalization condition [7

7. A. Brandt, S. McCormick, and J. Ruge, “Multigrid methods for differential eigenproblems,” SIAM J. Sci. Stat. Comput. **4**, 244–260 (1983). [CrossRef]

*V*denotes the volume of the Wigner-Seitz cell (WSC) and

*δ*

_{nm}is the Kronecker symbol for the band indices n and m.

*L*

_{k⃗}(Jacobi- or Gauss-Seidel iteration [6]) is usually slow to converge, it is quick to reduce high-frequency error components [6]. This allows the problem to be transfered to a coarser grid where the error can be resolved with much less work. Not only is the relaxation cheaper per sweep on coarser grids, but the solution process is also much more effective. The coarse grid equation can be solved by relaxation and appeal to still coarser grids. The coarsest grid used is chosen so that solution of the problem there is inexpensive compared to the work performed on the fine grid. The number of relaxation sweeps needed to smooth the error on each grid is generally small.

_{i}

*i*=1,2,…,

*q*. The eigenfrequencies are updated after every relaxation step on the base level using the Rayleigh quotient

*L*

_{k⃗}and

*U*

_{i}on the lowest level and 〈.,.〉 stands for the discrete version of scalar products defined in Eq. (6). This V-cycle scheme is repeated until stable values for eigenfrequencies Λ

_{i}and Bloch-functions

*U*

_{i}emerge (generally two V-cycles are sufficient) [7

7. A. Brandt, S. McCormick, and J. Ruge, “Multigrid methods for differential eigenproblems,” SIAM J. Sci. Stat. Comput. **4**, 244–260 (1983). [CrossRef]

*∊*

_{a}=13) in air (

*∊*

_{b}=1). Then, the primitive translations are given by

*a*⃗

_{1}=(

*a*, 0) and

*a*⃗

_{2}=(0,

*a*), where

*a*is the lattice constant. The radius

*r*of the rods is

*r*/

*a*=0.45. The photonic band structure of this structure is shown in Fig. 2. In Tab. 1, we compare the band structure data

*N*of plane waves. We observe, that for large numbers of plane waves, the PWM aproaches the values of the MG-method to within 1% but requires about 100% more CPU-time. In addition, the result from the MG-computations suggest that a 128×128 mesh is sufficient to obtain converged results. It should be noted that in this example PWM was used to compute the band structure only. If an evaluation of the electromagnetic mode structure were

## 4 Computation of group velocities and effective photon masses

8. C. Martijn de Sterke and J.E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic media,” Phys. Rev. A **38**, 5149–5165 (1988) [CrossRef]

*u*

_{k⃗}(

*r*⃗) is a lattice periodic function. The equation of motion for

*E*

_{k⃗}(

*r*⃗), Eq. (1) may now be transformed into an equation of motion for

*u*

_{k⃗}(

*r*⃗)

*Ĥ*(

*k*⃗)=Δ+2

*i*∇·

*k*⃗+

*k*⃗

^{2}, where Δ=

*u*

_{k⃗+q}⃗

^{(r⃗)}(|

*q*⃗≪|

*π/a*) into the form

*i*(∇+

*ik*⃗). Eq. (11) together with the fact that |

*q*⃗|≪

*π/a*suggests that we treat the second and third part on its l.h.s as a perturbation on

*Ĥ*(

*k*⃗) such that the eigenfrequency

*ω*

_{k⃗+q}⃗ may be regarded as a perturbed eigenvalue and compared to a Taylor-expansion of

*ω*

_{k⃗+q}⃗ around

*k*⃗

*υ*⃗

_{k⃗}=∂

_{k}⃗

*ω*

_{k⃗}and the inverse effective photon mass tensors (group velocity dispersion tensors)

*∂*

_{ki}

*∂*

_{kj}

*ω*

_{k⃗},

*i*=1, 2 are thus reduced to second order perturbation theory [8

8. C. Martijn de Sterke and J.E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic media,” Phys. Rev. A **38**, 5149–5165 (1988) [CrossRef]

## 5 Discussion

## 6 Acknowledgments

## References and links

1. | C.M. Soukoulis (Ed.), |

2. | A. Birner et al., “Macroporous Silicon: A Two-Dimensional Photonic Bandgap Material Suitable for the Near-Infrared Spectral Range,” phys. stat. sol (a) |

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

4. | K. Busch and S. John, “Photonic band gap formation in certain self-organizing systems,” Phys. Rev. E |

5. | K. Busch and S. John. “Liquid crystal photonic band gap materials: The tunable electromagnetic vacuum,” Phys. Rev. Lett. |

6. | P. Wesseling, |

7. | A. Brandt, S. McCormick, and J. Ruge, “Multigrid methods for differential eigenproblems,” SIAM J. Sci. Stat. Comput. |

8. | C. Martijn de Sterke and J.E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic media,” Phys. Rev. A |

**OCIS Codes**

(260.2110) Physical optics : Electromagnetic optics

(350.3950) Other areas of optics : Micro-optics

**ToC Category:**

Focus Issue: Photonic bandgap calculations

**History**

Original Manuscript: November 13, 2000

Published: January 29, 2001

**Citation**

Daniel Hermann, Meikel Frank, Kurt Busch, and Peter Wolfle, "Photonic band structure computations," Opt. Express **8**, 167-172 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-3-167

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### References

- C.M. Soukoulis (Ed.), Photonic Band Gap Materials, NATO ASI Series E 315, (Kluwer Academic Publishers, 1996). [CrossRef]
- A. Birner et al., "Macroporous Silicon: A Two-Dimensional Photonic Bandgap Material Suitable for the Near-Infrared Spectral Range," phys. stat. sol (a) 165, 111-117 (1998) [CrossRef]
- K.M. Ho, C.T. Chan, and C.M. Soukoulis, "Existence of a photonic band gap in periodic structures," Phys. Rev. Lett. 65, 3152-3155 (1990). [CrossRef] [PubMed]
- K. Busch and S. John, "Photonic band gap formation in certain self-organizing systems," Phys. Rev. E 58, 3896-3908 (1998). [CrossRef]
- K. Busch and S. John. "Liquid crystal photonic band gap materials: The tunable electromagnetic vacuum," Phys. Rev. Lett. 83, 967-970 (1999). [CrossRef]
- P. Wesseling, An Introduction to Multigrid Methods, (John Wiley & Sons, 1992).
- A. Brandt, S. McCormick, and J. Ruge, "Multigrid methods for differential eigenproblems," SIAM J. Sci. Stat. Comput. 4, 244-260 (1983). [CrossRef]
- C. Martijn de Sterke and J.E. Sipe, "Envelope-function approach for the electrodynamics of non-linear periodic media," Phys. Rev. A 38, 5149-5165 (1988) [CrossRef]

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